LMI Properties and Applications in Systems, Stability, and Control Theory

LMI Pr operties and A pplications in Systems, Stability , and Contr ol Theory Ryan James Cav erly 1 and James Richard Forbes 2 1 Assistan t Pr ofesso r , Depa rtment of Aer ospa ce Engineeri ng and Mechanics, Univer sity of Minnesota, 110 Union St. SE, Minneapolis , MN 55455, USA , r caver ly@umn .edu . 2 Associat e Pr ofess or , Department of Mecha nical Engine ering, McGill Universi ty , 817 Sherbr oo ke St. W e st, Montr eal, QC, C anada H3A 0C3 , james .rich ard.forbes@mcgill.ca . No vember 27, 2024 T able of Co ntents 1 Pr elimi naries 7 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Deﬁnitions and Fundamental LMI Properties . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Deﬁniteness of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Matrix Inequali t ies and LMIs . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.3 Relativ e Deﬁniteness of a Matrix . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.4 Strict and Nonstrict Matrix Inequaliti es . . . . . . . . . . . . . . . . . . . 12 1.3.5 Concatenation of LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.6 Con ve xity o f LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Semideﬁnite Programs (SDPs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Numerical T ools to Solve SDPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5.1 SDP Sol vers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.2 LMI P arsers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Pr operties and T ricks Aimed at Ref ormulating BMIs as LMIs 16 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Change of V ariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Congruence Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Schur Complem ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 2.4.1 Strict Schur Complem ent . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.2 Nonstrict Schur Complement . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.3 Schur Complement Lemma-Based Properties . . . . . . . . . . . . . . . . 1 8 2.5 Projection Lemma (Matrix El imination Lemma) . . . . . . . . . . . . . . . . . . . 22 2.5.1 Strict Projectio n Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1 2.5.2 Nonstrict Projectio n Lemm a . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.3 Reciprocal Projection Lemma . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.4 Projection Lemma-Based Properties . . . . . . . . . . . . . . . . . . . . . 23 2.6 Finsler’ s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6.1 Finsler’ s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6.2 Alternative Form of Finsler’ s L em ma . . . . . . . . . . . . . . . . . . . . 24 2.6.3 Modiﬁed Finsler’ s L emma . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6.4 Strict Petersen’ s Lem ma . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5 2.6.5 Nonstrict Petersen’ s Lem ma . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7.1 Examples of Dilated M atrix Inequaliti es . . . . . . . . . . . . . . . . . . . 28 2.8 Y ou n g’ s Relation (Completion of the Squares) . . . . . . . . . . . . . . . . . . . . 29 2.8.1 Y ou n g’ s Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.8.2 Reformulation of Y oung’ s Relation . . . . . . . . . . . . . . . . . . . . . 29 2.8.3 Special Cases of Y oung’ s Relation . . . . . . . . . . . . . . . . . . . . . . 29 2.8.4 Y ou n g’ s Relation-Based Properties . . . . . . . . . . . . . . . . . . . . . 32 2.8.5 Con ve x-Concav e Decomposition . . . . . . . . . . . . . . . . . . . . . . . 33 2.8.6 Iterativ e Con vex Overbounding . . . . . . . . . . . . . . . . . . . . . . . 34 2.9 Penalized Con vex Relaxatio n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.9.1 Sequential Penalized Con vex Relaxation Optimizatio n . . . . . . . . . . . 36 2.10 Coordinate Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.11 Discussi on on Reformulating BMIs as LMIs . . . . . . . . . . . . . . . . . . . . . 37 2.11.1 Reformulati ng a BMI as an Equivalent LMI . . . . . . . . . . . . . . . . . 38 2.11.2 Reformulati ng a BMI as an LMI that Im plies the Original BMI . . . . . . . 38 3 Additional LMI Prop erties and T ricks 40 3.1 The S-Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Dualization Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Singular V alues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.1 Maximum Singu l ar V alue . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.2 Maximum Singu l ar V alue of a Complex Matrix . . . . . . . . . . . . . . . 41 3.3.3 Minimum Singular V alu e . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 3.3.4 Minimum Singular V alu e of a Compl ex Mat ri x . . . . . . . . . . . . . . . 41 3.3.5 Frobenius Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.6 Nuclear Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 3.4 Eigen values of Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.1 Maximum Eig en v alue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.2 Minimum Eigen v alue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.3 Sum o f Largest Eigen values . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.4 Sum o f Absol ute V alue of Largest Eigen values . . . . . . . . . . . . . . . 43 3.4.5 W eighted Sum of Lar gest Eigen values . . . . . . . . . . . . . . . . . . . . 43 3.4.6 W eighted Sum of Absolute V alue of Lar gest Eigen values . . . . . . . . . . 43 3.5 Matrix Condit ion Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5.1 Condition Number of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5.2 Condition Number of a Positive Deﬁnite Matrix . . . . . . . . . . . . . . . 44 2 3.6 Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.7 T race of a Sym metric Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.7.1 T race of a Matrix wit h a Slack V ariable . . . . . . . . . . . . . . . . . . . 44 3.7.2 Relativ e Trac e of M at ri ces . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.8 Range of a Symm etric Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.9 Logarithm of a Positive Deﬁnite Matrix . . . . . . . . . . . . . . . . . . . . . . . 45 3.10 Douglas-Fill more-W illiams Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.11 Submatrix Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.12 Imaginary and Real P arts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.13 Quadratic Inequali ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.13.1 W eighted Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.13.2 Qu adrati c Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.14 Miscellaneou s Properties and Results . . . . . . . . . . . . . . . . . . . . . . . . 46 4 LMIs in Systems a nd Stability Theory 49 4.1 L yapu n ov Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 9 4.1.1 L yapu n ov Stabilit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.2 Asymptoti c Stabili ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.1.3 Discrete-T ime L yapunov Stability . . . . . . . . . . . . . . . . . . . . . . 5 1 4.1.4 Discrete-T ime As ymptotic Stabilit y . . . . . . . . . . . . . . . . . . . . . 52 4.1.5 Descriptor System Admiss i bility . . . . . . . . . . . . . . . . . . . . . . . 53 4.1.6 Discrete-T ime Descrip t or System Admissibi lity . . . . . . . . . . . . . . . 54 4.2 Bounded Real Lemma and the H ∞ Norm . . . . . . . . . . . . . . . . . . . . . . 55 4.2.1 Continuous-Time Bounded Real Lemma . . . . . . . . . . . . . . . . . . 55 4.2.2 Discrete-T ime Boun d ed Real Lem ma . . . . . . . . . . . . . . . . . . . . 58 4.2.3 Descriptor System Bounded Real Lemma . . . . . . . . . . . . . . . . . . 60 4.2.4 Discrete-T ime Descrip t or System Bounded Real Lemma . . . . . . . . . . 62 4.3 H 2 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.1 Continuous-Time H 2 Norm . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.2 Discrete-T ime H 2 Norm W ithou t Feedthrough . . . . . . . . . . . . . . . 66 4.3.3 Discrete-T ime H 2 Norm W ith Feedthrough . . . . . . . . . . . . . . . . . 71 4.3.4 Descriptor System H 2 Norm . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3.5 Discrete-T ime Descrip t or System H 2 Norm . . . . . . . . . . . . . . . . . 75 4.4 Generalized H 2 Norm (Induced L 2 - L ∞ Norm) . . . . . . . . . . . . . . . . . . . 7 7 4.5 Peak-to-Peak Norm (Induced L ∞ - L ∞ Norm) . . . . . . . . . . . . . . . . . . . . 77 4.6 Kalman-Y akubovich-Popov (KYP) Lemma . . . . . . . . . . . . . . . . . . . . . 78 4.6.1 KYP Lemma for QSR Diss ipativ e Systems . . . . . . . . . . . . . . . . . 78 4.6.2 Discrete-T ime KY P Lemma for QSR Dissipative Systems . . . . . . . . . 78 4.6.3 KYP (Positiv e Real) Lemm a W ithou t Feedthroug h . . . . . . . . . . . . . 79 4.6.4 KYP (Positiv e Real) Lemm a W ith Feedthrough . . . . . . . . . . . . . . . 80 4.6.5 Discrete-T ime KY P (Positive Real) L em ma W ith Feedthrough . . . . . . . 80 4.6.6 KYP Lemma for Descriptor Systems . . . . . . . . . . . . . . . . . . . . 82 4.6.7 Discrete-T ime KY P Lemma for Descriptor Systems . . . . . . . . . . . . 82 4.6.8 QSR Di ssipativity-Related Properties . . . . . . . . . . . . . . . . . . . . 82 4.7 Conic Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3 4.7.1 Conic Sector Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.7.2 Exterior Conic Sector Lemma . . . . . . . . . . . . . . . . . . . . . . . . 83 4.7.3 Modiﬁed Exterior Conic Sector Lemma . . . . . . . . . . . . . . . . . . . 84 4.7.4 Generalized KYP (GKYP) Lemma for Conic Sectors . . . . . . . . . . . . 85 4.8 Minimum Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.8.1 Minimum Gain Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.8.2 Modiﬁed Minimum Gain Lemma . . . . . . . . . . . . . . . . . . . . . . 88 4.8.3 Discrete-T ime M inimum Gain Lemma . . . . . . . . . . . . . . . . . . . . 89 4.8.4 Discrete-T ime M odiﬁed Mini m um Gain Lemma . . . . . . . . . . . . . . 90 4.9 Negati ve Imaginary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.9.1 Negati ve Im aginary Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.9.2 Discrete-T ime Negative Imaginary Lemm a . . . . . . . . . . . . . . . . . 92 4.9.3 Generalized Negati ve Imaginary Lemma . . . . . . . . . . . . . . . . . . 92 4.9.4 Negati ve Im aginary System DC Constraint . . . . . . . . . . . . . . . . . 93 4.10 Algebraic Riccati Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.10.1 Al gebraic Riccati Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.10.2 Di screte-T ime Algebraic Riccati Inequali ty . . . . . . . . . . . . . . . . . 93 4.11 Stabilizabili ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.11.1 Cont i nuous-T im e Stabili zability . . . . . . . . . . . . . . . . . . . . . . . 94 4.11.2 Di screte-T ime Stabili zabil ity . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.12 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.12.1 Cont i nuous-T im e Detectabili ty . . . . . . . . . . . . . . . . . . . . . . . . 94 4.12.2 Di screte-T ime Detectabili ty . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.13 Static Out put Feedback Stabilizability . . . . . . . . . . . . . . . . . . . . . . . . 95 4.13.1 Cont i nuous-T im e Static Output Feedback Stabilizability . . . . . . . . . . 95 4.13.2 Di screte-T ime Static Output Feedback Stabilizability . . . . . . . . . . . . 96 4.14 Strong Stabi l izability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.14.1 Cont i nuous-T im e Strong Stabilizability . . . . . . . . . . . . . . . . . . . 97 4.14.2 Di screte-T ime Strong Stabilizability . . . . . . . . . . . . . . . . . . . . . 97 4.15 System Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.15.1 Syst em Zeros without Feedthroug h . . . . . . . . . . . . . . . . . . . . . 98 4.15.2 Syst em Zeros with Feedthrough . . . . . . . . . . . . . . . . . . . . . . . 98 4.15.3 Di screte-T ime System Zeros wit h Feedthrough . . . . . . . . . . . . . . . 99 4.16 D -Stabili ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.16.1 General LMI Re gion D -Stability . . . . . . . . . . . . . . . . . . . . . . . 100 4.16.2 α -Stability Re gion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.16.3 V ertical Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.16.4 Coni c Sector Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.16.5 Circular Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 4.16.6 Ho ri zon tal Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.16.7 El liptic Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.16.8 Hy p erbolic Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.17 D -Adm i ssibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.17.1 General LMI Re gion D -Admissi bility . . . . . . . . . . . . . . . . . . . . 103 4.17.2 Circular Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 4 4.18 DC Gain of a T ransfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.19 Tr ansient Boun d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.19.1 Transient State Bound for Autonomous L TI Systems . . . . . . . . . . . . 1 0 5 4.19.2 Transient State Bound for Discrete-T ime Auto n omous L T I Systems . . . . 105 4.19.3 Transient State Bound for Non-Autonomous L TI Systems . . . . . . . . . 106 4.19.4 Transient State Bound for Discrete-T ime Non-Aut onomous L TI Syst ems . 107 4.19.5 Transient Output Bound for Au tonomous L TI Systems . . . . . . . . . . . 108 4.19.6 Transient Output Bound for Di screte-T ime Auto n omous L TI Systems . . . 108 4.19.7 Transient Output Bound for No n-Autonomous L TI Systems . . . . . . . . 109 4.19.8 Transient Output Bound for Di screte-T ime Non-Aut onomous L TI Syst ems 109 4.19.9 Transient Impulse Response Bound . . . . . . . . . . . . . . . . . . . . . 110 4.19.10 Discrete-T im e T ransient Impulse Response Bound . . . . . . . . . . . . . 11 0 4.20 Output En er gy Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.20.1 Ou t put Energy Bound for Au t onomous L TI Systems . . . . . . . . . . . . 111 4.20.2 Ou t put Energy Bound for Di screte-T ime Autonomo us L TI System s . . . . 112 4.20.3 Ou t put Energy Bound for No n -Autonomous L TI Systems . . . . . . . . . 112 4.20.4 Ou t put Energy Bound for Di screte-T ime Non-Autonom ous L TI Systems . . 113 4.21 Kharitonov-Bernstein-Haddad (KBH) Th eorem . . . . . . . . . . . . . . . . . . . 114 4.22 Stability o f Discrete-T ime System with Polytopic Uncertainty . . . . . . . . . . . 115 4.22.1 Op en-Lo o p Robust Stabilit y . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.22.2 Clos ed-Lo o p Robust Stabilit y . . . . . . . . . . . . . . . . . . . . . . . . 115 4.23 Quadratic Stabil ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.23.1 Cont i nuous-T im e Quadratic Stability . . . . . . . . . . . . . . . . . . . . 1 15 4.23.2 Di screte-T ime Quadratic Stability . . . . . . . . . . . . . . . . . . . . . . 116 4.24 Stability o f T im e-Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.24.1 Delay-Independent Conditio n . . . . . . . . . . . . . . . . . . . . . . . . 117 4.24.2 Delay-Dependent Conditio n . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.25 µ -Analysi s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7 4.26 Static Out put Feedback Algebraic Loop . . . . . . . . . . . . . . . . . . . . . . . 117 5 LMIs in Optimal Contr ol 119 5.1 The Generalized Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.1.1 The Conti nuous-T im e Generalized Plant . . . . . . . . . . . . . . . . . . . 119 5.1.2 The Di screte-T ime Generalized Plant . . . . . . . . . . . . . . . . . . . . 121 5.2 H 2 -Optimal Cont rol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 1 5.2.1 H 2 -Optimal Ful l -State Feedback Control . . . . . . . . . . . . . . . . . . 122 5.2.2 Discrete-T ime H 2 -Optimal Full-State Feedback Control . . . . . . . . . . 122 5.2.3 H 2 -Optimal Dy namic Output Feedback Control . . . . . . . . . . . . . . . 123 5.2.4 Discrete-T ime H 2 -Optimal Dynamic Output Feedback Control . . . . . . . 124 5.3 H ∞ -Optimal Cont rol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.3.1 H ∞ -Optimal Ful l -State Feedback Control . . . . . . . . . . . . . . . . . . 127 5.3.2 Discrete-T ime H ∞ -Optimal Ful l -State Feedback Control . . . . . . . . . . 128 5.3.3 H ∞ -Optimal Dy namic Output Feedback Control . . . . . . . . . . . . . . 128 5.3.4 Discrete-T ime H ∞ -Optimal Dy namic Output Feedback Control . . . . . . 131 5.4 Mixed H 2 - H ∞ -Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5 5.4.1 Mixed H 2 - H ∞ -Optimal Full-State Feedback Control . . . . . . . . . . . . 133 5.4.2 Discrete-T ime M ixed H 2 - H ∞ -Optimal Ful l -State Feedback Control . . . . 134 5.4.3 Mixed H 2 - H ∞ -Optimal Dynamic Output Feedback Control . . . . . . . . 135 5.4.4 Discrete-T ime M ixed H 2 - H ∞ -Optimal Dy namic Output Feedback Cont rol 1 3 7 6 LMIs in Optimal Estimation and Filtering 140 6.1 H 2 -Optimal State Estimati on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.1.1 H 2 -Optimal Ob server . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.1.2 Discrete-T ime H 2 -Optimal Observer . . . . . . . . . . . . . . . . . . . . . 141 6.2 H ∞ -Optimal State Estimati on . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.2.1 H ∞ –Optimal Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.2.2 Discrete-T ime H ∞ –Optimal Observer . . . . . . . . . . . . . . . . . . . . 142 6.3 Mixed H 2 - H ∞ -Optimal State Estimati on . . . . . . . . . . . . . . . . . . . . . . 143 6.3.1 Mixed H 2 - H ∞ -Optimal Observer . . . . . . . . . . . . . . . . . . . . . . 143 6.3.2 Discrete-T ime M ixed H 2 - H ∞ -Optimal Ob server . . . . . . . . . . . . . . 144 6.4 Continuous-Time and Di screte-T ime Opti mal Filtering . . . . . . . . . . . . . . . 14 5 6.4.1 H 2 -Optimal Fil ter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.4.2 Discrete-T ime H 2 -Optimal Filter . . . . . . . . . . . . . . . . . . . . . . 147 6.4.3 H ∞ -Optimal Fil ter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.4.4 Discrete-T ime H ∞ -Optimal Fil ter . . . . . . . . . . . . . . . . . . . . . . 149 A V ersion H istory 150 A.1 Updat es in V ersion 4 (Nov ember 27 , 2024) . . . . . . . . . . . . . . . . . . . . . 150 A.2 Updat es in V ersion 3 (April 2, 2021) . . . . . . . . . . . . . . . . . . . . . . . . . 152 Refer ences 154 Index 172 6 1 Pr elim inaries 1.1 Introduction Linear matri x inequali ties (LMIs) com m only appear in system s, stability , and cont rol appl i - cations. Many analysis and synthesis probl em s in these areas can be solved as feasibility o r op- timization problem s subject to LMI constraint s. Although most well-known LM I properties and manipulation tricks, such as the Schur complement and the congruence transformati on, can be found in stand ard references [ 1 – 5 ], many useful LMI p rop erties are scattered throughout t he lit er - ature. Th e purpose of this document is to collect and organize properties, tricks, and applications related to LM Is from a number of references to g et h er in a si ngle document. In this sens e, the document can be t h ought of as an “LMI encyclopedia” or “LMI cookbook. ” Proofs of the prop- erties presented in this document are n o t included when they can be found in the cited references in the interest of bre vity . Illustrative examples are included whenev er necessary to ful ly explain a certain prop erty . Mult i ple equiv alent forms of LMIs are often presented to give the reader a choice of which form may be best suited for a particular prob lem at h and. Th e equiv alency of some o f the LMIs in this document may be straightforward to more experienced readers, but the authors believ e th at some readers may beneﬁt from the presentation of mul tiple equiv alent LMIs. The docum ent is organized as follows. In t he remaining portions of Section 1 , t he notation used throughout the document is presented and some fundamental LMI properties are discussed. Sections 2 and 3 feature a coll ection of LM I properties and tricks that are i n teresting and potentially useful. Properties t hat are primarily aimed at reformulating bil inear matrix inequali t ies (BMIs) as LMIs are presented in Section 2 , while more general properties and deﬁnitions are found in Section 3 . Ap plications in volving LMIs in systems and st ability theory are i ncluded in Section 4 . Section 5 presents a number of LM I-based optimal contro l ler synt hesis methods, while Section 6 includes LM I-based optimal estimation synt h esis methods. The autho rs would like to thank t h e following individuals for alerting us o f errors, and pro- viding useful comment s and su g gestions for improvement: Logan Anderson, Leila Bridgeman, Jyot Buch, Manash Chakraborty , Stev en Dahdah, W illiam Elke, Robyn Fortune, Bruce Lee, Pe- ter Seiler . Please note that this d o cument is a work in prog ress . If you notice any errors o r inaccuracies, or have any sug g estions of content t h at should be included in this do cum ent, pl ease email either of the authors at rcaverly@u mn.edu or james.rich ard.forbes@ mcgill.ca so that changes t o future versions can be made. 1.2 Notation In th is document, m atrices are denoted by bol dface l et t ers (e.g., A ∈ R n × n ), colum n matrices are denoted by lowercase boldface letters (e.g., x ∈ R n ), s calars are denot ed by sim p le letters (e.g., γ ∈ R ), and operators are denoted by script letters (e.g., G : L 2 e → L 2 e ). The set of n by m real m atrices is denoted as R n × m , the set of n by m complex matrices is denoted as C n × m , and the set of n by n symmetric matrices is denoted as S n . The identity matrix is writ ten as 1 and a matrix ﬁlled with zeros is written as 0 . The dimensions of 1 and 0 are speciﬁed when necessary . Repeated b locks withi n symm et ri c matrices are replaced by ∗ for brevity and clarity . The conj u gate transpose or Hermitian transpose o f the matrix V ∈ C n × m is denoted by V H . T h e notation He {·} 7 is u s ed as a sh o rthand in situations with limited space, where He {· } = ( · ) + ( · ) H . The real and imaginary parts of the complex nu m ber z ∈ C are denoted as Re ( z ) and Im ( z ) , respecti vely . The Kroenecker product of two mat ri ces is deno t ed by ⊗ . Consider the square matri x A ∈ R n × n . The ei g en v alues of A are denoted by λ i ( A ) , i = 1 , 2 , . . . , n . The m atrix A is Hu rwi tz if all of i t s eigen va lues are in the open l eft-half complex plane (i.e., Re ( λ i ( A )) < 0 , i = 1 , . . . , n ). A m atrix i s Schur if all of its eigen v alues are s trictly wit hin a unit di s k centered at the origin of the complex plane (i.e., | λ i ( A ) | < 1 , i = 1 , . . . , n ). If A ∈ S n , then the min i mum eigen value of A is denot ed by λ ( A ) and its maximum eigen value is denoted by ¯ λ ( A ) . Consider t he mat ri x B ∈ R n × m . The mi nimum singu lar value of B is denoted b y σ ( B ) and the maximum singular value of B i s denoted by ¯ σ ( B ) . The range and nullspace of B are denoted by R ( B ) and N ( B ) , respectively . A st ate-space realization of the continuous-ti m e li near time-in variant (L TI) system ˙ x ( t ) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) + D u ( t ) , is often written com pactly as ( A , B , C , D ) in t his document. The argument of tim e is often omit t ed in contin uous-time state-space realizations, unless needed to pre vent ambig uity . A st ate-space realization of the discrete-time L TI system x k +1 = A d x k + B d u k , y k = C d x k + D d u k , is often written com pactly as ( A d , B d , C d , D d ) . The i n ner product spaces L 2 and L 2 e for conti nuous-time signals are deﬁned as L 2 =  x : R ≥ 0 → R n    k x k 2 2 = Z ∞ 0 x T ( t ) x ( t )d t < ∞  , L 2 e =  x : R ≥ 0 → R n    k x k 2 2 T = Z T 0 x T ( t ) x ( t )d t < ∞ , ∀ T ∈ R ≥ 0  . The i n ner product sequence spaces ℓ 2 and ℓ 2 e for di s crete-time signals are deﬁned as ℓ 2 = ( x : Z ≥ 0 → R n    k x k 2 2 = ∞ X k =0 x T k x k < ∞ ) , ℓ 2 e = ( x : Z ≥ 0 → R n    k x k 2 2 N = N X k =0 x T k x k < ∞ , ∀ N ∈ Z ≥ 0 ) . 1.3 Deﬁnitions and Fundamental LMI Properties 1.3.1 Deﬁniteness of a Matrix Deﬁnition 1 .1. [ 6 , p p. 42 9 –430] Consider t he sym metric matrix A ∈ S n . The matrix A is 8 a) pos i tive deﬁnite if x T Ax > 0 , ∀ x 6 = 0 ∈ R n , b) p o sitive semi-deﬁnit e if x T Ax ≥ 0 , ∀ x ∈ R n , c) ne gati ve deﬁnite if x T Ax < 0 , ∀ x 6 = 0 ∈ R n , d) n e gat i ve semi-deﬁnite i f x T Ax ≤ 0 , ∀ x ∈ R n , e) and indeﬁnite if x T Ax is neither posi tiv e nor negative. Theor em 1.2. [ 6 , p p . 430–431], [ 7 , p. 703] Consi der the symmetric m atrix A ∈ S n . The matrix A i s a) pos i tive deﬁnite if and o n ly if λ ( A ) > 0 , b) p o sitive semi-deﬁnit e if and only if λ ( A ) ≥ 0 , c) ne gati ve deﬁnite if and onl y if ¯ λ ( A ) < 0 , d) n e gat i ve semi-deﬁnite i f and only if ¯ λ ( A ) ≤ 0 , e) and indeﬁnite if and onl y if λ ( A ) < 0 and ¯ λ ( A ) > 0 . Pr oof. T o see why th e sign of x T Ax i s di ctated by the eigen va lues of A , let A = V Λ V − 1 , where V − 1 = V T because A is sym metric. Notice t hat x T Ax = x T V Λ V − 1 x =  V T x  T Λ V T x = z T Λ z = n X i =1 λ i ( A ) z 2 i , where z = V T x =  z 1 z 2 · · · z n  T . When ev aluatin g the sign of the quadratic form x T Ax , there i s no los s of g eneralit y in restri ct i ng A t o be sym metric. This is seen through the next two examples. Example 1.1. Consider the ske w-symmetri c matrix A = − A T ∈ R n × n . Evaluating t he quadratic form x T Ax yields x T Ax = 1 2 x T Ax + 1 2 x T Ax = 1 2 x T Ax + 1 2  x T Ax  T = 1 2 x T Ax + 1 2 x T A T x = 1 2 x T ( A − A ) x = 0 . Therefore, x T Ax = 0 for all skew-symmetic m atrices. 9 Example 1.2. Consider the matrix A ∈ R n × n , wh i ch can b e decompos ed as A = 1 2 A + 1 2 A = 1 2 A + 1 2 A + 1 2  A T − A T  = 1 2  A + A T  | {z } A sym + 1 2  A − A T  | {z } A skew , where A sym = A T sym = 1 2  A + A T  is the sym metric part of A and A ske w = − A T ske w = 1 2  A − A T  is the ske w-symm etric part of A . Evaluating the quadratic form x T Ax y ields x T Ax = x T ( A sym + A ske w ) x = x T A sym x + ✘ ✘ ✘ ✘ ✘ ✿ 0 x T A ske w x = x T A sym x . This conﬁrms that when determ i ning the d eﬁniteness of a matrix there i s no loss of generality in restricting th e matrix to be symmetric. The posit iv e deﬁniteness and positive semideﬁniteness of a m atrix are denoted by > 0 and ≥ 0 , respectiv ely . That is, A = A T > 0 is positive deﬁnit e and B = B T ≥ 0 is positive sem i deﬁnite. Similarly , the ne gative deﬁniteness and negati ve s emideﬁniteness of a matrix are denoted by < 0 and ≤ 0 , respectiv ely . That is, C = C T < 0 is negative deﬁnite and D = D T ≤ 0 is negati ve semideﬁnite. For brevity , the transpose component o f a deﬁniteness st atement is om itted in this document, for example, A = A T > 0 is sim ply written as A > 0 . 1.3.2 Matrix Inequalities and LMIs Deﬁnition 1 .3. A matrix inequality, G : R m → S n , i n the va riable x ∈ R m is an expression of the form G ( x ) = G 0 + p X i =1 f i ( x ) G i ≤ 0 , where x T =  x 1 · · · x m  , G 0 ∈ S n , and G i ∈ R n × n , i = 1 , . . . , p . Deﬁnition 1.4 . [ 8 ], [ 9 , p. 34], [ 10 ] A bilinear matrix in equ ality (BMI), H : R m → S n , in the var iable x ∈ R m is an expression of the form H ( x ) = H 0 + m X i =1 x i H i + m X i =1 m X j = 1 x i x j H i,j ≤ 0 , where x T =  x 1 · · · x m  , and H i , H i,j ∈ S n , i = 0 , . . . , m , j = 0 , . . . , m . Deﬁnition 1.5. [ 1 , p. 7], [ 3 , p. 17] An LM I, F : R m → S n , in the variable x ∈ R m is an expression of th e form F ( x ) = F 0 + m X i =1 x i F i ≤ 0 , (1.1) where x T =  x 1 · · · x m  and F i ∈ S n , i = 0 , . . . , m . 10 LMIs can alternatively be deﬁned in terms of matrix variables as foll ows. Deﬁnition 1.6 . [ 11 , p. 125] An LMI, F : R p 1 × q 1 × · · · × R p r × q r → S n , in the matrix variables X i ∈ R p i × q i , i = 1 , . . . , r , where m = P r i =1 p i q i , i s an expression of the form F ( X 1 , . . . , X r ) = F 0 + r X i =1  G i X i H i + H T i X T i G T i  ≤ 0 , (1.2) where F 0 ∈ S n , G i ∈ R n × p i , and H i ∈ R q i × n , i = 1 , . . . , r . Example 1.3. [ 1 , pp. 8–9] Consider the matrices A ∈ R n × n and Q ∈ S n , where Q > 0 . It i s desired t o ﬁnd a symm etric matrix P ∈ S n satisfying the matrix i nequality P A + A T P + Q < 0 , (1.3) where P > 0 . The matrix P is the design v ariable in this problem, and this LMI can be directly related to the deﬁnition in ( 1.2 ) by sett ing F 0 = Q , G 1 = 1 , H 1 = A , X 1 = P , and enforcing the constraint X 1 = X T 1 . This LMI can be reformul ated in the form of ( 1.1 ) by deﬁning the scalar entries of the matrix variable P as the design var iables. T o illustrate this, cons i der the case o f n = 2 so that each matrix is of dim ension 2 × 2 , and x =  p 1 p 2 p 3  T . Writ ing the matrix P in terms of a basis E i ∈ S 2 , i = 1 , 2 , 3 , yields P =  p 1 p 2 p 2 p 3  = p 1  1 0 0 0  | {z } E 1 + p 2  0 1 1 0  | {z } E 2 + p 3  0 0 0 1  | {z } E 3 . Note that the matrices E i are l i nearly independent and sym metric, thus formin g a basis for the symmetric matrix P . The matrix in equality in ( 1.3 ) can be written as p 1  E 1 A + A T E 1  + p 2  E 2 A + A T E 2  + p 3  E 3 A + A T E 3  + Q < 0 . Deﬁning F 0 = Q and F i = F T i = E i A + A T E i , i = 1 , 2 , 3 , yields F 0 + 3 X i =1 p i F i < 0 , which now resembles the deﬁnition of an LMI in ( 1.1 ). Through out thi s document , LMIs are typically w ri t ten in the matrix form of ( 1.2 ), rather than the scalar form of ( 1.1 ). 1.3.3 Relative Deﬁniteness of a Matrix The deﬁniteness of a matrix can be found relative to another matrix . For example, consider the m at ri ces A ∈ S n and B ∈ S n . T h e matrix inequality A 0 . Knowing the relativ e deﬁniteness of matrices can be useful. For example, if in the previous example we ha ve A 0 , then w e know that B > 0 . This follows from 0 < A < B . For more foun d ational facts in volving the relativ e deﬁniteness of matrices, see [ 7 , p p. 703–704]. 11 1.3.4 Strict and Nonstrict Matrix Inequalities A strict matrix inequality can be con verted to a nonstrict matrix inequality . For example, A > 0 is im p lied by A ≥ ǫ 1 , where ǫ ∈ R > 0 . Simi l arly , B < 0 is implied by B ≤ − ǫ 1 , where ǫ ∈ R > 0 . Con verting a strict m atrix inequality into a nonstrict matri x inequalit y is useful when working with LM I solvers that cannot handle strict cons traints. 1.3.5 Concatenation of LMIs A us eful property of LMIs i s that m ultiple LMIs can be concatenated together to form a single LMI. F or example, satisfyin g the LM Is A < 0 and B < 0 is equ ivalent to satisfying the concate- nated L M I  A 0 0 B  < 0 . More g enerally , satisfying the LMIs A i < 0 , i = 1 , . . . , n is equi valent to satisfying the concate- nated L M I diag { A 1 , . . . , A n } < 0 . 1.3.6 Con vexity of LMIs Deﬁnition 1.7. [ 12 , p. 138] A set, S , i n a real inner product space is con vex if for all x , y ∈ S and α ∈ R , where 0 ≤ α ≤ 1 , i t holds t hat α x + (1 − α ) y ∈ S . Lemma 1.1. [ 10 ] The s et of soluti o n s to an LM I is con vex. That is, the set S = { x ∈ R m | F ( x ) ≤ 0 } is a conv ex set, wh ere F : R m → S n is an LMI. Pr oof. Consider x , y ∈ R m and α ∈ [0 , 1] , and suppo se t h at x and y satis fy ( 1.1 ). The LMI F : R m → S n is con vex, since F ( α x + (1 − α ) y ) = F 0 + m X i =1 ( αx i + (1 − α ) y i ) F i = F 0 − α F 0 + α F 0 + α m X i =1 x i F i + (1 − α ) m X i =1 y i F i = α F 0 + α m X i =1 x i F i + (1 − α ) F 0 + (1 − α ) m X i =1 y i F i = α F ( x ) + (1 − α ) F ( y ) . 1.4 Semideﬁnite Programs (SDPs) A sem ideﬁnite program (SDP) i s a con vex optim ization probl em of the form [ 13 , p. 168] min x ∈ R m c T x (1.4) subject to F 0 + m X i =1 x i F i ≤ 0 , (1.5) 12 where x T =  x 1 · · · x m  , c ∈ R m , F i ∈ S n , i = 0 , . . . , m , and ( 1.5 ) is an LMI in the v ariable x . As shown in Example 1.3 , the LMI constraint in ( 1.5 ) can be written in matrix form, rather than the standard form. The du al probl em of the SDP described by ( 1.4 ) and ( 1.5 ) is given by [ 13 , pp. 168–169] max Z ∈ S n tr ( F 0 Z ) subject to tr ( F i Z ) + c i = 0 , i = 1 , . . . , n, Z ≥ 0 , where c T =  c 1 · · · c m  . W ithin the context of d u ality , the SDP outl ined in ( 1.4 ) and ( 1. 5 ) i s denoted as the primal problem. Further detail s o n t h e use of SDP duality withi n the con t ext of L TI systems can be found in [ 14 , 15 ]. When using matrix variables to describe an SDP’ s LM I constraints, it may be incon venient to rewrite the objective function in the form of ( 1.4 ). SDP parsers, which will be discus s ed in Section 1.5 , are capable of con verting LMIs and linear objective functi o ns in matrix form to the standard form required b y m ost SDP solvers. An example of a li near objective functi on in matrix form is J ( X ) = tr  Q T X + X T R  , where X , Q , R ∈ R n × m . More generally , a number of con vex objective functions in volving mat ri x variables that are not explicitly written in th e standard SDP form can be reformulated as SDPs. Some SDP parsers are capable of performing this conv ersion for the user . T wo examples of such o bjectiv e functions are giv en, with a brief explanation of ho w they can be reformul ated in the standard SDP form. Example 1.4. [ 13 , p. 71] Consider J ( x ) = 1 2 x T Px + q T x + r , where x , q ∈ R n , P ∈ S n , P > 0 , and r ∈ R . T wo special cases of this obj ectiv e function are list ed next. • Special case when q = 0 and r = 0 : J ( x ) = 1 2 x T Px , where x ∈ R n , P ∈ S n , and P > 0 . • Special case when P = 2 · 1 , q = 0 , and r = 0 : J ( x ) = x T x = k x k 2 2 , wh ere x ∈ R n . The op timization problem min x ∈ R m 1 2 x T Px + q T x + r subject to F ( x ) ≤ 0 , is equiva lent to the o p t imization problem min x ∈ R m ,γ ∈ R γ subject to F ( x ) ≤ 0 ,  q T x + r − γ x T ∗ − 2 P − 1  ≤ 0 , where the Schur complement , presented in Section 2.4 , is used to reformulate the quadratic objec- tiv e function into an LMI constraint. 13 Example 1.5. Consider J ( X ) = tr  X T PX + Q T X + X T R + S  , where X , Q , R ∈ R n × m , P ∈ S n , S ∈ R n × n , and P ≥ 0 . Four special cases of this o b jectiv e function are listed next. • Special case when Q = R = 0 and S = 0 : J ( X ) = tr  X T PX  , where X ∈ R n × m , P ∈ S n , and P > 0 . • Special case when P = 1 , Q = R = 0 , and S = 0 : J ( X ) = tr  X T X  = k X k 2 F , where X ∈ R n × m . • [ 1 , p. 88] Special case when P = 0 , R = 0 and S = 0 : J ( X ) = t r( Q T X ) , where X , Q ∈ R n × m . • [ 7 , p. 718] Special case when P = 1 , Q = R = 0 , S = 0 , and X ∈ S n : J ( X ) = tr( X 2 ) , where X ∈ S n . The op timization problem min X ∈ R n × m tr  X T PX + Q T X + X T R + S  subject to F ( X ) ≤ 0 . is equiva lent to the o p t imization problem min X ∈ R n × m , Z ∈ S m ,γ ∈ R γ subject to F ( X ) ≤ 0 ,  Q T X + X T R + S − Z X T ∗ − P − 1  ≤ 0 , tr( Z ) ≤ γ . where a property in volving the trace of a symmetric m atrix, as discussed in Section 3.7 , and the Schur compl ement in Section 2.4 are used to reformulate the q uadratic objective function into an LMI constraint. Another useful con vex objectiv e function is give n by J ( X ) = log (det( X − 1 )) = − log (det ( X )) , where X ∈ S n and X > 0 [ 1 , p . 14], [ 10 ]. T his objective fun ct i on cannot be readily con verted into the st and ard SDP form, but can be implemented with most SDP solvers and parsers. In particu- lar , SDPT3 [ 16 , 17 ] i s capable of directly mini mizing SDPs with objectiv e functions of the form − log (det( X )) . 1.5 Numerical T ools to Solve SDPs There are many semideﬁnite program solvers that accept LMI constraints. Most solvers require that LMI constraints be written in the standard form sho wn in ( 1.1 ). This i s often not con venient, as it is typical t o derive LMI constraints in matrix form , such as the LMI in ( 1.3 ). LMI parsers con vert LMIs in matrix form to the standard form in ( 1.1 ), allowing for a smoother transition from mathematical deriv ation to numerical im plementation. A non-exhausti ve list of SDP solvers and LMI parsers are i n cluded for reference . 14 1.5.1 SDP Solver s There are a n u mber of SDP solvers ava ilabl e. T h e authors have experience wit h Se DuMi [ 18 , 19 ], SDPT3 [ 16 , 17 ], and Mosek [ 20 ], though other solvers are av ailable, such as CSDP [ 21 , 22 ], CVXOPT [ 23 , 24 ], DD S [ 25 , 26 ], DSDP [ 27 , 28 ], LMILab [ 29 ], PENLAB [ 30 , 31 ], SCS [ 32 , 33 ], SDPA [ 34 – 36 ], SMCP [ 37 , 38 ], SDPNAL [ 39 , 40 ], and STRIDE [ 41 , 42 ]. There are advantages and disadvantages to each of these sol vers, and sometimes one solver may give a soluti o n to a giv en problem when others do not. For this reason, it is useful t o ha ve mult i ple solvers a vailable. Comparisons o f various LMI solvers and benchm ark problem s are fou n d in [ 43 – 45 ]. Many solvers, includi n g SeDuMi , SD PT3 , are av ailable for free, w h ile Mosek is a commercial software package. A free academic license of Mosek can be requested for research in academic institut i ons or educational purpo ses. 1.5.2 LMI Parsers LMI parsers all ow t h e user to deﬁne the SDP to be solved within st andard software en v iron- ments, and often in a more con venient m atrix form. A num b er of openly-dis t ributed LMI parsers are av ailable for use wi thin different software en vironments. The following is a non-exhaustive list of LMI parsers and the solvers they are known to be compatible with, sorted by software en viron- ment. • Matlab – Yalmi p [ 46 , 47 ]. Solvers: CSDP , DSDP , LMILab , Mo sek , P ENLAB , SCS , SDPA , SDPT3 , S DPNAL , and SeDuMi . – CVX [ 48 , 49 ]. Solvers: Mosek , SDPT 3 , and SeDuMi . – LMILa b [ 29 ]. Features an internal solver . – ROLMI P [ 50 , 51 ]. P arser designed for optim ization problems wit h uncertain polyno- mial matrices. Requires Yal mip . Solvers: CSDP , D SDP , LMILab , Mosek , P ENLAB , SCS , SDPA , SDPT 3 , S DPNAL , and SeDuMi . • Python – CVXPY [ 52 – 54 ]. Solvers: SCS . Other sol vers can be i nstalled separately . – PICOS [ 55 ]. Solvers: CVXO PT , Mosek , and SMCP . – Irene [ 56 ]. Solvers: CSDP , CVXOPT , DS DP , and SDPA . – PyLMI -SDP [ 57 ]. Solvers: CV XOPT and SDPA . • Julia – Conve x.jl [ 58 , 59 ]. Solvers: Mosek and SCS . – JuMP [ 60 , 61 ]. Solvers: Mosek and SCS . • NSP – NSPYa lmip [ 62 , 63 ]. Solvers: CSDP and SeDuMi . 15 2 Pr ope r ties and T ricks Aimed at Reformulating BMIs as LMIs 2.1 Introduction This section presents a compil ation of properties and methods from the literature that are pri- marily used to reformulate BMI const rain t s as LMI cons traints. Many of these properties are used in subs equ ent sections t o reformulate LM Is or transform matrix inequalities i n to LMIs. The p rop erties discussed in Sections 2.2 to 2.7 are t ypically able to reformulate a BMI as an equi valent LMI or LMIs. Use of these properties is desi rable, as they wil l not in t roduce any conservatism when reformulating a BMI. On the other hand, the properties in Sections 2.8 to 2.10 are typicall y used to obtain an LMI that implies a BMI, generally with cons erv atism . This makes the u se of these properties a less desirable, yet sometim es un avoidable, opt ion. A di scussion on when and ho w t o use a selection of the properti es presented in this section is provided in Section 2.11 . 2.2 Change of V ariables [ 1 , pp. 100–101], [ 4 , Sec. 12.3.1] A BMI can so metimes be con verted into an LMI us i ng a change of va riables. Example 2.1. [ 4 , Example 12.5, Sec. 1 2 .3.1] Consi der A ∈ R n × n , B ∈ R n × m , K ∈ R m × n , and Q ∈ S n , where Q > 0 . The matrix inequali t y g iven by QA T + A Q − QK T B T − BKQ < 0 , is bil i near in the variables Q and K . Deﬁne a change of v ariable as F = KQ t o obtain QA T + A Q − F T B T − BF < 0 , which is an LMI in the variables Q and F . Once thi s LMI is solved, the original v ariable can be recov ered by K = FQ − 1 . It is important that a change of variables is chosen t o be a one-to-one m apping in order for t h e new m atrix inequalit y t o be equiv alent to th e origi nal matrix i nequality . In Example 2.1 the change of variable F = KQ is a one-to-one mapping since Q − 1 is in vertible due to t h e constraint Q > 0 , which gives a unique s olution for the reverse change of variable K = FQ − 1 . 2.3 Congruence T ransformation [ 1 , p. 15], [ 4 , Sec. 12. 3 .2] Consider Q ∈ S n and W ∈ R n × n , where rank ( W ) = n . The matrix inequali ty Q < 0 is satisﬁed if and only if WQ W T < 0 or equiv alently W T QW < 0 , and is referred to as a congruence transformation. Example 2.2. [ 4 , Example 12.6, Sec. 12.3.2] Consider A ∈ R n × n , B ∈ R n × m , K ∈ R m × p , C T ∈ R n × p , P ∈ S n , and V ∈ S p , wh ere P > 0 and V > 0 . The matrix i n equality given by Q =  A T P + P A − PBK + C T V ∗ − 2 V  < 0 , 16 is li near in the variable V and bilinear in the variable pair ( P , K ) . Choose the matrix W = diag { P − 1 , V − 1 } to obtain an equiv alent BMI gi ven by WQW T =  P − 1 A T + AP − 1 − BKV − 1 + P − 1 C T ∗ − 2 V − 1  < 0 . (2.1) Using a change of variable X = P − 1 , U = V − 1 , and F = KV − 1 , ( 2.1 ) b ecomes WQW T =  XA T + AX − B F + XC T ∗ − 2 U  < 0 , (2.2) which is an LMI in th e v ariables X , U , and F . Once ( 2.2 ) is solved in terms of X , U , and F , the original variable K is recovered by the rev erse change of variable K = FU − 1 . A congruence transformation p reserves the deﬁnit eness of a matrix by ensurin g that Q < 0 and WQW T < 0 are equiva lent. A congruence transformation is related, but not equiv alent to a si m- ilarity transform ation TQT − 1 , whi ch preserves not only the deﬁniteness , but also the eigen values of a matri x . A congruence transformation is equiv alent to a si milarity transformati on in the special case when W T = W − 1 . 2.4 Schur Complement 2.4.1 Strict Schur Complement [ 1 , pp. 7–8], [ 4 , Sec. 12.3.3 ] Consider A ∈ S n , B ∈ R n × m , and C ∈ S m . The following st atements are equi valent. a)  A B B T C  < 0 . b) A − BC − 1 B T < 0 , C < 0 . c) C − B T A − 1 B < 0 , A < 0 . 2.4.2 Nonstrict Schur Complement [ 1 , p. 28] Consider A ∈ S n , B ∈ R n × m , and C ∈ S m . The following st atements are equi valent. a)  A B B T C  ≤ 0 . b) A − BC + B T < 0 , C ≤ 0 , B ( 1 − CC + ) = 0 , where C + is the Moore-Penrose i nv erse of C . c) C − B T A + B < 0 , A ≤ 0 , B T ( 1 − AA + ) = 0 , where A + is the Moore-Penrose i nv erse of A . 17 2.4.3 Schur Complement Lemma-Based P roperties 1. [ 3 , p. 109], [ 64 , p. 100] Consider P 11 ∈ S n , P 12 ∈ R n × m , P 22 , X ∈ S m , P 13 ∈ R n × p , P 23 ∈ R m × p , and P 33 ∈ S p . There exists X such that   P 11 P 12 P 13 ∗ P 22 + X P 23 ∗ ∗ P 33   < 0 , (2.3) if and only if  P 11 P 13 ∗ P 33  < 0 . Any matrix X ∈ S m satisfying X < − P 22 +  P T 12 P 23   P 11 P 13 ∗ P 33  − 1  P 12 P T 23  (2.4) is a solution to ( 2.3 ). That is, ( 2.4 ) = ⇒ ( 2.3 ). 2. [ 3 , pp. 109–110], [ 64 , p. 101] Consider P 11 ∈ S n , P 12 , X ∈ R n × m , P 22 ∈ S m , P 13 ∈ R n × p , P 23 ∈ R m × p , and P 33 ∈ S p . There exists X such that   P 11 P 12 + X T P 13 ∗ P 22 P 23 ∗ ∗ P 33   < 0 (2.5) if and only if  P 11 P 13 ∗ P 33  < 0 , and  P 22 P 23 ∗ P 33  < 0 . (2.6) If the two matrix inequalities in ( 2.6 ) h old, then a solut i on to ( 2.5 ) is gi ven by X = P 23 P − 1 33 P T 13 − P T 12 . Pr oof. Necessity (( 2.5 ) = ⇒ ( 2.6 )) comes from the requirement that the submatrices corre- sponding to t he principle minors of ( 2.5 ) are negativ e deﬁnite. Suf ﬁciency (( 2.6 ) = ⇒ ( 2.5 )) is shown by rewriting th e matrix inequalities of ( 2.6 ) in the equiv alent form P 11 − P 13 P − 1 33 P T 13 < 0 , and P 22 − P 23 P − 1 33 P T 23 < 0 . (2.7) Concatenating the two m atrix i nequalities in ( 2.7 ) and choosing X = P 23 P − 1 33 P T 13 − P T 12 giv es the equ ivalent matrix inequality  P 11 − P 13 P − 1 33 P T 13 P 12 − P 13 P − 1 33 P T 23 + X T ∗ P 22 − P 23 P − 1 33 P T 23  < 0 , or  P 11 P 12 + X T ∗ P 22  −  P 13 P 23  P − 1 33  P T 13 P T 23  < 0 , which is equiv alent to ( 2.5 ) u sing the Schur com p lement lemma. 18 Permutation of the colum ns and ro ws of ( 2.5 ) yields the following equiva lent resul t . [ 5 , pp. 41–42] Cons i der P 11 ∈ S n , P 12 , X ∈ R n × m , P 22 ∈ S m , P 13 ∈ R n × p , P 23 ∈ R m × p , and P 33 ∈ S p . There exists X such that   P 11 P 12 P 13 ∗ P 22 P 23 + X T ∗ ∗ P 33   < 0 (2.8) if and only if  P 11 P 12 ∗ P 22  < 0 , and  P 11 P 13 ∗ P 33  < 0 . (2.9) If the matrix i nequalities in ( 2.9 ) h old, then a s olution to ( 2.8 ) is giv en by X = P T 13 P − 1 11 P 12 − P T 23 . 3. [ 5 , p. 4 1] Consider P 11 , X ∈ S n , P 12 ∈ R n × m , and P 22 ∈ S m , where X > 0 . T h ere exists X such that   P 11 − X P 12 X ∗ P 22 0 ∗ ∗ − X   < 0 , (2.10) if and only if  P 11 P 12 ∗ P 22  < 0 . (2.11) Pr oof. The matrix inequality in ( 2.10 ) can be re writt en usi ng the Schur complement lemm a as  P 11 − X P 12 ∗ P 22  −  X 0   − X − 1   X 0  < 0  P 11 − X P 12 ∗ P 22  +  X 0 ∗ 0  < 0  P 11 P 12 ∗ P 22  < 0 . 4. [ 65 ], [ 66 , p. 3 1 9–320] Consider P 11 ∈ S n , P 12 ∈ R n × m , P 22 ∈ S m , P 23 ∈ R m × p , P 33 ∈ S p , and X ∈ R n × p . There exists X such th at   P 11 P 12 X ∗ P 22 P 23 ∗ ∗ P 33   > 0 , if and only if  P 11 P 12 ∗ P 22  > 0 , and  P 22 P 23 ∗ P 33  > 0 . 19 Pr oof. The proof is found in [ 66 ]. 5. [ 66 , p. 320] Consid er P 11 ∈ S n , P 12 ∈ R n × m , P 22 ∈ S m , P 23 ∈ R m × p , P 33 ∈ S p , E ∈ R p × n , F ∈ R p × m , and X ∈ R n × p . There exists X such that   P 11 + XE + E T X P 12 + XF X ∗ P 22 P 23 ∗ ∗ P 33   > 0 , if and only if  P 11 + E T P 33 E P 12 − E T P T 23 + E T P 33 F ∗ P 22 − P 23 F − F T P T 23 + F T P 33 F  > 0 , and  P 22 P 23 ∗ P 33  > 0 . Pr oof. The proof is found in [ 66 ]. 6. [ 67 ] Consider X ∈ S n , H ∈ R m × n , G ∈ R m × m , and P ∈ S m , where P > 0 . The matrix inequality given by  X H T ∗ G + G T − P  > 0 , (2.12) implies X > H T G − 1 PG − T H . (2.13) For G = P , this relationship becomes the Schur complement l em ma. Pr oof. Using the Schur complement lemma on ( 2.12 ) gi ves X > H T  G + G T − P  − 1 H . Using the property G + G T − P ≤ G T P − 1 G (see the special case of Y oung’ s relation i n Section 2.8.3 ), or equiv alently  G + G T − P  − 1 ≥ G − 1 PG − T giv es X > H T  G + G T − P  − 1 H ≥ H T G − 1 PG − T H , thus im plying ( 2.13 ). V ariatio ns of this property are li sted as fol l ows. (a) [ 67 ] Consider X ∈ S n , H ∈ R n × n , G ∈ R m × n , and P ∈ S m , where P > 0 . The matrix inequality gi ven by  H + H T − X G T ∗ P  > 0 , (2.14) implies X < H T G − 1 PG − T H . 20 (b) [ 68 ] Consid er A ∈ S n , B ∈ R n × m , G ∈ R m × m , P ∈ S m , and β ∈ R . The matri x inequality gi ven by  A BG ∗ − β  G + G T  + β 2 P  < 0 , implies the matrix i nequality A + BPB T < 0 . 7. [ 65 , 69 ], [ 66 , p . 321 ] Consider P 1 ∈ S n , P 2 , X ∈ S q , Q 1 ∈ R n × m , Q 2 ∈ R q × p , R 1 ∈ S m , and R 2 ∈ S p . The matrix i nequalities given by  P 1 − LXL T Q 1 ∗ R 1  > 0 ,  P 2 + X Q 2 ∗ R 2  > 0 , (2.15) are satisﬁed if and only if   P 1 + LP 2 L T Q 1 LQ 2 ∗ R 1 0 ∗ ∗ R 2   > 0 . (2.16) Pr oof. The proof is found in [ 69 ] and is very sim i lar to th e proof of Property 2 . 8. [ 65 , 69 ] Cons i der P ∈ S n , R ∈ S m , S ∈ S p , Q ∈ R n × m , X ∈ R n × p , V ∈ R m × p , and E ∈ R p × m . The matrix i nequalities given by  P Q ∗ R − VE − E T V T + E T SE  > 0 ,  R V ∗ S  > 0 , (2.17) are satisﬁed if and only if   P Q + XE X ∗ R V ∗ ∗ S   > 0 . (2.18) Pr oof. The proof is found in [ 69 ] and is ver y similar to the proof of Property 2 . 9. [ 70 ], [ 2 , p. 229] Consider P 1 , Q ∈ S n , P 2 , Q 2 ∈ R n × m , and P 3 , Q 3 ∈ S m , where P 1 > 0 , P 3 > 0 , Q 1 > 0 , and Q 3 > 0 . There exist P 2 , P 3 , Q 2 , and Q 3 such that  P 1 P 2 ∗ P 3  > 0 ,  P 1 P 2 ∗ P 3  − 1 =  Q 1 Q 2 ∗ Q 3  , (2.19) if and only if  P 1 1 ∗ Q 1  ≥ 0 , rank  P 1 1 ∗ Q 1  ≤ n + m. (2.20) Provided P 1 and Q 1 satisfy ( 2.20 ), a solution to ( 2.19 ) i s give n by P 3 = 1 , Q 2 = − Q 1 P 2 , Q 3 = P T 2 Q 1 P 2 + 1 , and P 2 satisﬁes P 2 P T 2 = P 1 − Q − 1 1 . 21 10. [ 71 , pp. 1 3 – 14] Consi der X , Y ∈ R n × m , P , Q ∈ S n , and ǫ ∈ R > 0 , where P > 0 , Q > 0 , and ǫ ≥ 1 . The m atrix i nequality given by ǫ X T P − 1 X + ǫ Y T Q − 1 Y ≥ ( X + Y ) T ( P + Q ) − 1 ( X + Y ) (2.21) holds. Pr oof. Since P > 0 , Q > 0 , and ǫ ≥ 1 , it is known that ( ǫ − 1) X T P − 1 X ≥ 0 and ( ǫ − 1) Y T Q − 1 Y ≥ 0 . These inequalities are rewritten as ǫ X T P − 1 X − X T P − 1 X ≥ 0 , ǫ Y T P − 1 Y − X T Q − 1 Y ≥ 0 . (2.22) Applying the Schur com p lement lemm a to th e expressions in ( 2.22 ) results in  P X ∗ ǫ X T P − 1 X  ≥ 0 ,  Q Y ∗ ǫ Y T Q − 1 Y  ≥ 0 . (2.23) The m atrix inequalit ies i n ( 2.23 ) imply  P + Q X + Y ∗ ǫ X T P − 1 X + ǫ Y T Q − 1 Y  ≥ 0 . (2.24) Applying the Schur com p lement lemm a to ( 2.24 ) yields ǫ X T P − 1 X + ǫ Y T Q − 1 Y − ( X + Y ) T ( P + Q ) − 1 ( X + Y ) ≥ 0 . (2.25) Rearranging ( 2.25 ) give s ( 2.21 ). 11. ( Linearizati o n Lemma [ 3 , pp. 91–92]) Consider X ∈ R n × p , S ∈ R n × m , T ∈ R m × q , Y ( v ) ∈ R m × p , Q ( v ) ∈ S n , R ( v ) ∈ S m , and U ( v ) ∈ S q , wh ere Y ( v ) , Q ( v ) , and R ( v ) depend af ﬁnely on the parameter v , and R ( v ) can be decomposed as R ( v ) = TU − 1 ( v ) T − 1 . The matrix inequalities U ( v ) > 0 and  X Y ( v )  T  Q ( v ) S S T R ( s )   X Y ( v )  < 0 are equiv alent to  X T Q ( v ) X + X T SY ( v ) + Y T ( v ) S T X X T ( v ) T ∗ − U ( v )  < 0 . 2.5 Projection Lemma (Matrix Elimination Lemma) 2.5.1 Strict Pr ojection Lemma [ 70 ], [ 1 , pp. 2 2 –23], [ 3 , pp. 110–111 ], [ 4 , Sec. 12.3.5] Consider Ψ ∈ S n , G ∈ R n × m , Λ ∈ R m × p , and H ∈ R n × p . There exists Λ such that Ψ + G Λ H T + H Λ T G T < 0 , (2.26) if and only if N T G Ψ N G < 0 , N T H Ψ N H < 0 , where R ( N G ) = N ( G T ) and R ( N H ) = N ( H T ) . 22 2.5.2 Nonstrict Projection Lemma [ 72 , p. 93] Consider Ψ ∈ S n , G ∈ R n × m , Λ ∈ R m × p , and H ∈ R n × p , wh ere R ( G ) and R ( H ) are linearly independent. There exists Λ such that Ψ + G Λ H T + H Λ T G T ≤ 0 , if and only if N T G Ψ N G ≤ 0 , N T H Ψ N H ≤ 0 , where R ( N G ) = N ( G T ) and R ( N H ) = N ( H T ) . 2.5.3 Recipr ocal Project ion Lemma [ 73 ] Consider P , Ψ ∈ S n and W , S ∈ R n × n . There exists W such t hat  Ψ + P −  W + W T  S T + W T ∗ − P  < 0 , if and only if Ψ + S + S T < 0 . 2.5.4 Project ion Lemma-Based Pr operties 1. [ 74 ] Cons i der A ∈ S n , B , J ∈ R n × m , G ∈ R m × m , and P ∈ S m . The matrix inequalit y given by  A + BJ T + JB T − J + BG ∗ −  G + G T  + P  < 0 , (2.27) implies the matrix i nequality A + BPB T < 0 . (2.28) If the matrices J and G are free (i.e., t hey are design variables), then the m atrix inequali- ties ( 2.27 ) and ( 2.28 ) are equiv alent [ 75 ]. 2. [ 76 ] Consider T ∈ S n and A , J , G , P ∈ R n × n . The matrix i nequality given by  T + A T J T + JA P − J + A T G ∗ −  G + G T   < 0 (2.29) implies the matrix i nequality T + A T P T + P A < 0 . (2.30) If the matrices J and G are free (i.e., t hey are design variables), then the m atrix inequali- ties ( 2.29 ) and ( 2.30 ) are equiv alent [ 75 ]. 23 3. [ 75 ] Consider T 1 , P ∈ S n , A , J 1 , G ∈ R n × n , T 2 ∈ R n × m , J 2 ∈ R m × n , and T 3 ∈ S m , where P > 0 and T 3 < 0 . The matri x inequality g iven by   T 1 + A T J T 1 + J 1 A T 2 + A T J T 2 P − J 1 + A T G ∗ T 3 − J 2 ∗ ∗ −  G + G T    < 0 (2.31) implies the matrix i nequality  T 1 + A T P + P A T 2 ∗ T 3  < 0 . (2.32) If the m atrices J 1 , J 2 , and G are free (i.e., they are design variables), then the matrix inequal- ities ( 2.31 ) and ( 2.32 ) are equivalent. 4. [ 77 , p. 9] Consider T ∈ S n , A , G , P ∈ R n × n , and β ∈ R , where T < 0 . The m atrix inequalit y giv en by  T β P + A T G ∗ − β  G + G T   < 0 , implies the matrix i nequality T + A T P T + P A < 0 . 2.6 Finsler’ s Lemma 2.6.1 Finsler’ s Lemma [ 1 , pp. 22–23], [ 4 , Sec. 12.3.5], [ 78 ] Consider Ψ ∈ S n , G ∈ R n × m , Λ ∈ R m × p , H ∈ R n × p , and σ ∈ R . There exists Λ su ch that Ψ + G Λ H T + H Λ T G T < 0 , if and only if there exists σ s u ch that Ψ − σ GG T < 0 , Ψ − σ HH T < 0 . 2.6.2 Alternativ e F orm o f Finsler’ s Lemma [ 78 – 80 ], [ 81 , pp. 90–97], [ 82 , pp. 41–48 ] Consider Ψ ∈ S n , Z ∈ R p × n , and x ∈ R n , where rank ( Z ) < n . The following statements are equiv alent. 1. The inequality x T Ψ x < 0 is satis ﬁed for all x s atisfying Zx = 0 , where x 6 = 0 . 2. The matrix inequality N T Z Ψ N Z < 0 is satis ﬁed, where R ( N Z ) = N ( Z ) . 24 3. There exists σ ∈ R s u ch that Ψ − σ Z T Z < 0 . 4. There exists X ∈ R p × m such that Ψ + XZ + Z T X T < 0 . 2.6.3 Modiﬁed Finsler’ s Lemma [ 83 , p. 37], [ 84 , 85 ] Consider Ψ ∈ S n , G ∈ R n × m , Λ ∈ R m × p , H ∈ R n × p , and R ∈ S p , where Λ T Λ ≤ R and R > 0 . There exists Λ such t hat Ψ + G Λ H T + H Λ T G T < 0 , (2.33) if and only if there exists ǫ ∈ R > 0 such that Ψ + ǫ − 1 GG T + ǫ HRH T < 0 . (2.34) Pr oof. The proof of ( 2.34 ) = ⇒ ( 2.33 ) follows from a compl etion of th e squares argument. The autho rs are not aw are of a complete proof of ( 2.33 ) = ⇒ ( 2 . 3 4 ), so us e this identity with caution. 2.6.4 Strict P etersen’ s Lemma [ 79 , 86 , 87 ] Consider Ψ ∈ S n , G ∈ R n × m , H ∈ R n × p , and R ∈ S p , where R ≥ 0 . Also consider the set F := { F ∈ R m × p | F T F ≤ R } . The matrix inequalit y Ψ + GFH T + HF T G T < 0 , holds for all F ∈ F if and onl y if there exists ǫ ∈ R > 0 such that Ψ + ǫ − 1 GG T + ǫ HRH T < 0 . (2.35) The m atrix inequalit y i n ( 2.35 ) can be equiva lentl y rewritten using the Schur complement as [ 88 ]  Ψ + ǫ HR H T G ∗ − ǫ 1  < 0 . A modiﬁcation to the Strict Petersen’ s l emma found i n [ 88 ] is st ated as foll ows. Consider Ψ ∈ S n , G ∈ R n × m , y ∈ R n , and R ∈ S m , where R > 0 . Also consider the set X := { x ∈ R m | x T Rx ≤ 1 } . The matrix inequali ty Ψ + G xy T + yx T G T < 0 , holds for all x ∈ X if and only if t h ere exists ǫ ∈ R > 0 such that   Ψ G y ∗ − ǫ R 0 ∗ ∗ − ǫ − 1 1   < 0 . 25 2.6.5 Nonstrict Petersen’ s Lemma [ 87 , 89 , 90 ] Consider Ψ ∈ S n , G ∈ R n × m , H ∈ R n × p , and R ∈ S p , wh ere R > 0 , G 6 = 0 , and H 6 = 0 . Also consider the set F := { F ∈ R m × p | F T F ≤ R } . The matrix inequalit y Ψ + GFH T + HF T G T ≤ 0 , holds for all F ∈ F if and onl y if there exists ǫ ∈ R > 0 such that Ψ + ǫ − 1 GG T + ǫ HRH T ≤ 0 . (2.36) The m atrix inequalit y i n ( 2.36 ) can be equiva lentl y rewritten using the Schur complement as [ 88 ]  Ψ + ǫ HR H T G ∗ − ǫ 1  ≤ 0 . The fol l owing are slight modi ﬁcations t o t h e original l emma. 1. [ 91 ] Cons i der Ψ ∈ S n , G ∈ R n × m , and H ∈ R n × p , where G 6 = 0 and H 6 = 0 . Als o consider the s et F := { F ∈ R m × p | k F k F ≤ 1 } . The matrix inequality Ψ + GFH T + HF T G T ≤ 0 , holds for all F ∈ F if and onl y if there exists ǫ ∈ R > 0 such that Ψ + ǫ − 1 GG T + ǫ HH T ≤ 0 . 2. [ 91 ] Cons i der Ψ ∈ S n , G ∈ R n × m , and H ∈ R n × m , wh ere G 6 = 0 and H 6 = 0 . A l so consider the s et F := { F ∈ S m × m | − 1 ≤ F ≤ 1 } . The matrix inequality Ψ + GFH T + HF T G T ≤ 0 , holds for all F ∈ F if and onl y if there exists ǫ ∈ R > 0 such that Ψ + ǫ − 1 GG T + ǫ HH T ≤ 0 . 2.7 Dilation Matrix inequalit ies can be dil at ed t o obt ain a lar ger matrix inequality , often with additional design variables. This can be a useful technique to separate d esi gn variables in a BMI. A common technique to dilate an LMI inv olves the use the projection lemma in rever se or the reciprocal projection lemma. For instance, consider the following example taken from [ 73 ] and inspired by the di lated bounded real lemma matrix i nequality in [ 5 , pp . 153–1 5 5] in v olvi ng the matrices P ∈ S n and A ∈ R n × n , wh ere P > 0 . The matrix in equ al i ty  P A + A T P − P P ∗ − P  < 0 , (2.37) 26 can be re writt en as  A T 1 0 1 0 1    0 P 0 ∗ − P 0 ∗ ∗ − P     A 1 1 0 0 1   < 0 . (2.38) Since P > 0 , it is also kn own that  − P 0 ∗ − P  < 0 , which can be re written as  0 1 0 0 0 1    0 P 0 ∗ − P 0 ∗ ∗ − P     0 0 1 0 0 1   < 0 . (2.39) The matrix inequalities i n ( 2.38 ) and ( 2.39 ) are in the form of the strict projection lem m a. Speciﬁ- cally , ( 2.3 8 ) is in the form of N T G ( A ) Φ ( P ) N G ( A ) < 0 , where Φ ( P ) =   0 P 0 ∗ − P 0 ∗ ∗ − P   , N G ( A ) =   A 1 1 0 0 1   . The m atrix inequalit y o f ( 2.39 ) is in the form of N T H Φ ( P ) N H < 0 , where N H =   0 0 1 0 0 1   . The proj ection lemm a states that ( 2.38 ) and ( 2.39 ) are equiv alent to Φ ( P ) + G ( A ) VH T + HV T G T ( A ) , (2.40) where N ( G T ( A )) = R ( N G ( A )) , N ( H T ) = R ( N H ) , and V ∈ R n × n . Choosi n g G ( A ) =   − 1 A T 1   , H =   1 0 0   , the m atrix inequali t y o f ( 2.40 ) can be re written as   0 P 0 ∗ − P 0 ∗ ∗ − P   +   − 1 A T 1   V  1 0 0  +   1 0 0   V T  − 1 A 1  < 0 , or equivalently   −  V + V T  V T A + P V T ∗ − P 0 ∗ ∗ − P   < 0 . (2.41) Therefore, the matrix inequality o f ( 2.38 ) with P > 0 is equiv alent to the di lated matrix inequality of ( 2.41 ). 27 2.7.1 Examples of Dilated Matrix Inequalities Examples of some useful dilat ed matrix inequalities are presented here, while dil ated forms of a number of important matrix inequalities are includ ed as equivalent matrix inequalities in their respectiv e sectio n s. 1. [ 92 ] Consider the matrices A , G ∈ R n × n , ∆ ∈ R m × n , P ∈ S n , δ 1 , δ 2 , a , b ∈ R > 0 , where P > 0 and b = a − 1 . The matrix inequalit y AP + P A T + δ 1 P + δ 2 AP A T + P ∆ T ∆ P < 0 (2.42) is equiva lent to the m atrix inequalit y       0 − P P 0 P ∆ T ∗ 0 0 − P 0 ∗ ∗ − δ − 1 1 P 0 0 ∗ ∗ ∗ − δ − 1 2 P 0 ∗ ∗ ∗ ∗ − 1       + He                  A 1 0 0 0       G  1 − b 1 b 1 1 b ∆ T             < 0 . (2.43) Moreover , for ever y solution P > 0 of ( 2.42 ), P and G = − a ( A − a 1 ) − 1 P wi ll be soluti ons of ( 2.43 ). 2. [ 77 , pp. 7–8] Consider th e m atrices A , V ∈ R n × n , P , X ∈ S n , B ∈ R n × m , C ∈ R p × n , D ∈ R p × m , R ∈ S m , and S ∈ S p , where P > 0 , R > 0 , S > 0 , and X > 0 . The matrix inequality given by       − V − V T V A + P VB 0 V ∗ − 2 P + X 0 C T 0 ∗ ∗ − R D T 0 ∗ ∗ ∗ − S 0 ∗ ∗ ∗ ∗ − X       < 0 , implies the matrix i nequality   P A + A T P PB C T ∗ − R D T ∗ ∗ − S   < 0 . 3. [ 77 , p. 9] Consider the matrices A , V ∈ R n × n , Q , X ∈ S n , B ∈ R n × m , C ∈ R p × n , D ∈ R p × m , R ∈ S m , and S ∈ S p , where Q > 0 , R > 0 , S > 0 , and X > 0 . The matrix inequalit y given by       − V − V T V T A T + Q 0 V T C V T ∗ − 2 Q + X B 0 0 ∗ ∗ − R D T 0 ∗ ∗ ∗ − S 0 ∗ ∗ ∗ ∗ − X       < 0 implies the matrix i nequality   A Q + QA T B QC T ∗ − R D T ∗ ∗ − S   < 0 . 28 2.8 Y oung’ s Relation (Completion of the Squar es) 2.8.1 Y oung’ s Relation [ 93 , 94 ] Consider X , Y ∈ R n × m and S ∈ S n , wh ere S > 0 . The matrix inequali t y g iven by X T Y + Y T X ≤ X T S − 1 X + Y T SY , is known as Y oung ’ s relation or Y oung’ s i n equality . Y ou n g’ s relation can be derive d from a completion of the squares as fol lows. 0 ≤ ( X − SY ) T S − 1 ( X − SY ) 0 ≤ X T S − 1 X + Y T SY − X T Y − Y T X X T Y + Y T X ≤ X T S − 1 X + Y T SY , which is Y oung’ s relation. 2.8.2 Reformu latio n of Y oung’ s Relation [ 94 ] Consider X , Y ∈ R n × m and S ∈ S n , wh ere S > 0 . The matrix inequali t y g iven by X T Y + Y T X ≤ 1 2 ( X + SY ) T S − 1 ( X + SY ) , is a reformulation of Y oung’ s relatio n . 2.8.3 Special Ca s es of Y o ung’ s Relation 1. Consider X , Y ∈ R n × m . A special case of Y oung’ s relation with S = 1 is give n by X T Y + Y T X ≤ X T X + Y T Y . 2. Consider ¯ X , Y ∈ R n × m and S ∈ S n , where S > 0 . A special case o f Y o u ng’ s relation with ¯ X = − X is giv en by − ¯ X T Y − Y T ¯ X ≤ ¯ X T S − 1 ¯ X + Y T SY . 3. [ 67 ] Consider G ∈ R n × n and S ∈ S n , where S > 0 . A special case of Y oung’ s relation with X = G and Y = 1 is g iv en by G T S − 1 G ≥ G + G T − S . 4. [ 7 , p. 737] Consider P , S ∈ S n , where S > 0 . A special case of Y oung’ s relation with X = X T = P and Y = 1 is given by 2 P ≤ PS − 1 P + S . 5. [ 7 , p. 732] Consider G ∈ R n × n and α ∈ R > 0 . A special case o f Y oung’ s relation wit h X = G , Y = 1 , and S = α 1 is given by α − 1 G T G ≥ G + G T − α 1 . 29 6. [ 7 , p. 73 2] Consi der G ∈ R n × n and α ∈ R > 0 . A special case o f Y oung’ s relation wit h X = G , Y = G T , and S = α 1 is given by G 2 +  G T  2 ≤ α − 1 G T G + α GG T . 7. [ 7 , p . 732] Consi d er S ∈ S n , where S > 0 . A special case of Y oung’ s relation with X = 1 , Y = 1 is giv en by 2 1 ≤ S + S − 1 . 8. [ 95 ] Consider S ∈ S n and α ∈ R , where S > 0 . A special case of Y oung’ s relatio n with X = 1 , Y = α 1 is g iven by 2 α 1 ≤ α S + S − 1 . 9. [ 83 , p. 38 ] Consider the column matrices x , y ∈ R n , and S ∈ S n , where S > 0 . A s pecial case of Y ou n g’ s relation with X = x and Y = y is gi ven by − 2 x T y ≤ x T S − 1 x + y T Sy . (2.44) 10. [ 96 ] Consider the col umn matrices x , y ∈ R n , and S ∈ S n , where S > 0 . A special case of Y ou n g’ s relation with X = x and Y = − y is given by − 2 x T y ≤ x T S − 1 x + y T Sy . 11. Consid er X ∈ R n × m , F ∈ R n × q , ¯ Y ∈ R q × m , and S ∈ S n , where S > 0 . A special case of Y ou n g’ s relation with Y = F ¯ Y i s giv en by X T F ¯ Y + ¯ Y T F T X ≤ X T S − 1 X + ¯ Y T F T SF ¯ Y . (2.45) 12. [ 5 , pp. 29–30] Consi der X ∈ R n × m , ¯ Y ∈ R n × m , F ∈ S n , and δ ∈ R > 0 , where F > 0 . A special case of Y oung’ s relation with Y = F ¯ Y and S = ( δ F ) − 1 is given by X T F ¯ Y + ¯ Y T FX ≤ δ X T FX + δ − 1 ¯ Y T F ¯ Y . 13. [ 79 ] Consi der X ∈ R n × m , F ∈ R n × q , ¯ Y ∈ R q × m , and ǫ ∈ R > 0 , where F T F ≤ 1 . A special case of the m atrix inequali t y ( 2. 4 5 ) wit h S = ǫ 1 is giv en by X T F ¯ Y + ¯ Y T F T X ≤ ǫ − 1 X T X + ǫ ¯ Y T ¯ Y . (2.46) Pr oof . Substituti n g S = ǫ 1 into ( 2.45 ) yields X T F ¯ Y + ¯ Y T F T X ≤ ǫ X T X + ǫ − 1 ¯ Y T F T F ¯ Y . (2.47) Premultiplying F T F ≤ 1 by ¯ Y T , postmu ltiplying by ¯ Y , and multiplying bot h sides by ǫ − 1 leads t o ǫ − 1 ¯ Y T F T F ¯ Y ≤ ǫ − 1 ¯ Y T ¯ Y . (2.48) Substitutin g ( 2.48 ) into ( 2.47 ) yields ( 2.46 ).  30 14. Consid er X ∈ R n × m , F ∈ R n × q , Y ∈ R q × m , and S ∈ S n , where S > 0 . Applyin g Y oung’ s relation gives the matrix in equ al i ty 1 2 ( X + FY ) T S − 1 ( X + FY ) ≤ X T S − 1 X + Y T F T S − 1 FY . (2.49) Pr oof . Expanding the left-hand side of ( 2.49 ) yields 1 2 ( X + FY ) T S − 1 ( X + FY ) = 1 2  X T S − 1 X + X T S − 1 FY + Y T F − 1 S − 1 X + Y T F T S − 1 FY  (2.50) From Y oung’ s relation it can be sho wn that X T S − 1 FY + Y T F − 1 S − 1 X ≤ X T S − 1 X + Y T F T S − 1 FY . (2.51) Substitutin g ( 2.51 ) into ( 2.50 ) gives ( 2.49 ).  15. Consid er X , Y ∈ R n × m , and S ∈ S n , where S > 0 . A sp ecial case of ( 2.49 ) with F = S is giv en by 1 2 ( X + SY ) T S − 1 ( X + SY ) ≤ X T S − 1 X + Y T SY . 16. [ 83 , p. 38], [ 96 ] Consider X ∈ R n × m , D ∈ R n × r , F ∈ R r × q , E ∈ R q × m , P ∈ S n , and ǫ ∈ R > 0 , wh ere P > 0 , F T F ≤ 1 , and P − ǫ DD T > 0 . Then th e matrix inequality give n by ( X + DFE ) T P − 1 ( X + DFE ) ≤ ǫ − 1 E T E + X T ( P − ǫ DD T ) − 1 X , (2.52) holds. Pr oof . Deﬁne W =  ǫ − 1 1 − D T P − 1 D  − 1 / 2 D T P − 1 X −  ǫ − 1 1 − D T P − 1 D  1 / 2 FE , where  ǫ − 1 1 − D T P − 1 D  − 1 / 2 exists due to the m atrix in version lemma [ 7 , p. 304] s i nce P − ǫ DD T > 0 . Expanding the terms in W T W ≥ 0 yields X T P − 1 D  ǫ − 1 1 − D T P − 1 D  − 1 D T P − 1 X − X T P − 1 DFE − E T F T D T P − 1 X + E T F T  ǫ − 1 1 − D T P − 1 D  FE ≥ 0 . Adding X T P − 1 X t o both si d es of the inequali ty and rearranging give s X T P − 1 X + X T P − 1 DFE + E T F T D T P − 1 X + E T F T D T P − 1 DFE ≤ ǫ − 1 E T F T FE + X T  P − 1 D ( ǫ − 1 1 − D T P − 1 D ) − 1 D T P − 1 + P − 1  X . (2.53) Using th e matrix i n version lemma [ 7 , p. 304], it i s known that ( P − ǫ DD T ) − 1 = P − 1 D ( ǫ − 1 1 − D T P − 1 D ) − 1 D T P − 1 + P − 1 . (2.54) Substitutin g ( 2.54 ) into ( 2.53 ), factoring th e left sid e of the inequality , and knowing F T F ≤ 1 giv es ( 2.52 ).  31 17. [ 96 , 97 ] Consider X ∈ R n × m , D ∈ R n × r , F ∈ R r × q , E ∈ R q × m , P ∈ S n , and ǫ ∈ R > 0 , wh ere P > 0 , F T F ≤ 1 , and ǫ 1 − D T PD > 0 . Then the matrix inequality given by ( X + DFE ) T P ( X + DFE ) ≤ ǫ E T E + X T PD ( ǫ 1 − D T PD ) − 1 D T PX + X T PX , (2.55) holds. Pr oof . Deﬁne W =  ǫ 1 − D T PD  − 1 / 2 D T PX −  ǫ 1 − D T PD  1 / 2 FE , where  ǫ 1 − D T PD  − 1 / 2 exists since ǫ 1 − D T PD > 0 . Expanding th e terms in W T W ≥ 0 yields X T PD  ǫ 1 − D T PD  − 1 D T PX − X T PDFE − E T F T D T PX + E T F T  ǫ 1 − D T PD  FE ≥ 0 . Adding X T PX t o both sides of the inequ al i ty and rearranging gives X T PX + X T PDFE + E T F T D T PX + E T F T D T PDFE ≤ ǫ E T F T FE + X T PD ( ǫ 1 − D T PD ) − 1 D T PX + X T PX . Factoring t he left side of the inequ al i ty and knowing F T F ≥ 1 gives ( 2.55 ).  18. [ 77 , p. 11 ] Consider N ∈ R n × n , E ∈ R n × m , H ∈ R m × p , F ∈ R p × n , J ∈ S n , and ǫ ∈ R > 0 , where J > 0 and F T F ≤ 1 . With some manipulation, a special case o f ( 2.45 ) with X = H T E T N T and ¯ Y = 1 is given by − N ( 1 − EHF ) J − 1 ( 1 − EHF ) T N T ≤ J − N − N T + ǫ − 1 NEHH T E T N T + ǫ 1 . 19. [ 77 , p. 11 ] Consider N ∈ R n × n , F ∈ R n × m , E ∈ R m × p , H ∈ R p × n , J ∈ S n , and ǫ ∈ R > 0 , where J > 0 and F T F ≤ 1 . W i th some manipul ation, a special case of ( 2.45 ) with X = NHE and ¯ Y = 1 is giv en by − N T ( 1 − FEH ) T J − 1 ( 1 − FEH ) N ≤ J − N − N T + ǫ − 1 N T H T E T EHN + ǫ 1 . 2.8.4 Y oung’ s Relation-Based Properties 1. [ 98 ] Consider X , Y ∈ R n × m and Z ∈ S m . The matrix inequalit y given by Z + X T Y + Y T X > 0 , is satisﬁed if and only if there exist Q ∈ S m , P ∈ S n , G 1 ∈ R n × n , G 2 ∈ R n × m , F ∈ R m × n , and H ∈ R m × m , wh ere Q > 0 and P > 0 , such that  P Y ∗ Q  > 0 and   Z + Q + X T PX F − X T G 1 H − X T G 2 ∗ G 1 + G T 1 − P F T + G 2 − Y ∗ ∗ H T + H − Q   > 0 . 32 2. [ 98 ] Consider X ∈ R n × n and W ∈ S n , where X is full rank and W > 0 . The matrix inequality given by X T X − W > 0 , is satis ﬁed if there exists λ ∈ R > 0 such that    λ 1 λ 1 0 ∗ X + X T W 1 2 ∗ ∗ λ 1    > 0 . 3. [ 7 , p. 737] Consider P , Q ∈ S n , wh ere P > 0 and Q > 0 . The matrix inequali ty given by P + Q ≤ PQ − 1 P + QP − 1 Q holds. 2.8.5 Con vex-Conca ve Decomposition [ 99 , 100 ] Consider X , Y ∈ R n × m and Q ∈ S m . The matrix inequalit y Q + X T Y + Y T X < 0 (2.56) is equiva lent to Q + 1 2 ( X + Y ) T ( X + Y ) | {z } G ( X , Y ) − 1 2 ( X − Y ) T ( X − Y ) | {z } H ( X , Y ) < 0 , (2.5 7 ) where matrix function G ( X , Y ) = 1 2 ( X + Y ) T ( X + Y ) is con vex and the matrix functio n − H ( X , Y ) = − 1 2 ( X − Y ) T ( X − Y ) is concav e. Suppose t h at an init ial feasible values o f X = X 0 and Y = Y 0 are known, the matrix inequaliti es in ( 2.56 ) and ( 2. 5 7 ) are satisﬁed with X = X 0 + δ X and Y = Y 0 + δ Y if " Q − H ( X 0 , Y 0 ) − 1 2  ( X 0 − Y 0 ) T ( δ X − δ Y ) + ( δ X − δ Y ) T ( X 0 − Y 0 )  ( X + Y ) T ∗ − 2 1 # < 0 . (2.58) Moreover , the conservatism o f ( 2.58 ) with respect to the m atrix inequalit i es in ( 2. 5 6 ) and ( 2.57 ) in the neighborhood of th e X 0 and Y 0 (i.e., ( 2.58 ) becomes equiv alent to ( 2.56 ) and ( 2.57 ) as δ X → 0 and δ Y → 0 ). Pr oof. The function H ( X , Y ) is re written in t erm s of perturbations from a prior so l ution X 0 , Y 0 (i.e., X = X 0 + δ X and Y = Y 0 + δ Y ), which yields H ( X , Y ) = 1 2 ( X 0 + δ X − Y 0 − δ Y ) T ( X 0 + δ X − Y 0 − δ Y ) = 1 2 ( X 0 − Y 0 ) T ( X 0 − Y 0 ) | {z } H ( X 0 , Y 0 ) + 1 2  ( X 0 − Y 0 ) T ( δ X − δ Y ) + ( δ X − δ Y ) T ( X 0 − Y 0 )  + 1 2 ( δ X − δ Y ) T ( δ X − δ Y ) | {z } H ( δ X , δ Y ) = H ( X 0 , Y 0 ) + 1 2  ( X 0 − Y 0 ) T ( δ X − δ Y ) + ( δ X − δ Y ) T ( X 0 − Y 0 )  + H ( δ X , δ Y ) . 33 Knowing that H ( δ X , δ Y ) ≥ 0 results in H ( X , Y ) ≥ H ( X 0 , Y 0 ) + 1 2  ( X 0 − Y 0 ) T ( δ X − δ Y ) + ( δ X − δ Y ) T ( X 0 − Y 0 )  . (2.59) T aking the Schur complement of G ( X , Y ) = ( X + Y ) T  1 2 1  ( X + Y ) al l ows for ( 2.57 ) to be equiv- alently writt en as  Q − H ( X , Y ) ( X + Y ) T ∗ − 2 1  < 0 . (2.60) Making use of ( 2.59 ), resul ts in ( 2.58 ) imp lying ( 2.60 ), w h ich is equiv alent to ( 2.56 ) and ( 2.57 ). 2.8.6 Iterative Con vex Overbounding [ 101 , 102 ] Iterativ e con vex overbounding is a technique based on Y oung’ s relati o n that is useful wh en solving an optimizatio n p rob lem with a BMI constraint. Consider the matrices Q = Q T ∈ R n × n , B ∈ R n × m , R ∈ R m × p , D ∈ R p × q , S ∈ R q × r , and C ∈ R r × n , wh ere S and R are design va riables in the BMI given by Q + BRDSC + C T S T D T R T B T < 0 . (2.61) Suppose that S 0 and R 0 are known to s atisfy ( 2.61 ). Th e BMI of ( 2.61 ) is i mplied by t h e LMI   Q + φ ( R , S ) + φ T ( R , S ) B ( R − R 0 ) U C T ( S − S 0 ) T V T ∗ − W − 1 0 ∗ ∗ − W   < 0 , (2.62) where φ ( R , S ) = B ( RDS 0 + R 0 DS − R 0 DS 0 ) C , W > 0 is an arbitrary matrix, D = UV , and the matrices U and V T hav e full column rank. The LMI of ( 2.62 ) is equiv alent to t he BMI of ( 2.61 ) when R = R 0 and S = S 0 , and is therefore non-conservativ e for values of R and S and are clos e to the p re viousl y k nown solution s R 0 and S 0 . Alternative ly , the BMI of ( 2.61 ) is implied by the LMI  Q + φ ( R , S ) + φ T ( R , S ) Z T U T ( R − R 0 ) T B T + V ( S − S 0 ) C ∗ − Z  < 0 , (2.63) where Z > 0 is an arbitrary matrix, D = UV , and the matrices U and V T hav e full colum n rank. Again, the LMI of ( 2.63 ) is equ ivalent to the BMI of ( 2.61 ) when R = R 0 and S = S 0 , and is therefore non-conserv ative for v alues of R and S and are close to the pre viously known solut ions R 0 and S 0 . A beneﬁt of con ve x overbounding compared to a linearization approach, i s that i n addit ion to ensuring conserv atism or error is reduced in the neighborho o d of R = R 0 and S = S 0 , the LMIs of ( 2.62 ) and ( 2.63 ) imply ( 2.61 ). Iterativ e con vex ov erboundin g is particularly useful when used to solve an optimization prob- lem wi t h BMI constraints. For example, choos e R 0 and S 0 that are ini tial feasible solution s to ( 2.61 ). Th en solve for R and S that mi nimize a speciﬁed objective function and satisfy ( 2.62 ) or ( 2.63 ), which imply ( 2.61 ) wi thout conserv atism when R = R 0 and S = S 0 . Set R 0 = R and 34 S 0 = S , and repeat un t il t h e objective function meets a speciﬁed stopping criteria. The beneﬁts of this procedure are that its individual steps are con vex optimization problems with very little conservatism in the neigh b orhood of the solution from the previous iteration, and that it tends to con ver ge qui ckly to a solution. Howe ver , there is no guarantee that the method wi l l con ver ge to e ven a local s o lution. Example 2.3. Consider a special case of ( 2.61 ) given by Q + RS + S T R T < 0 , (2.64) where Q ∈ S n , R ∈ R n × m , and S ∈ R m × n . The BMI of ( 2.64 ) is implied by the LMI   Q + RS 0 + S T 0 R T + R 0 S + S T R T 0 − R 0 S 0 − S T 0 R T 0 R − R 0 S T − S T 0 ∗ − W − 1 0 ∗ ∗ − W   < 0 , where W > 0 is an arbitrary m atrix. Alternatively , the BMI of ( 2.64 ) is impli ed by the LM I  Q + RS 0 + S T 0 R T + R 0 S + S T R T 0 − R 0 S 0 − S T 0 R T 0 Z ( R − R 0 ) T + S − S 0 ∗ − Z  < 0 , where Z > 0 is an arbitrary m atrix. 2.9 Penaliz ed Con vex Relaxation [ 103 ] Consider a BMI con straint in the variable x ∈ R m giv en b y H ( x ) = H 0 + m X i =1 x i H i + m X i =1 m X j = 1 x i x j H i,j ≤ 0 , (2.65) where x T =  x 1 · · · x m  , and H i , H i,j ∈ S n , i = 0 , . . . , m , j = 0 , . . . , m . The BMI in ( 2.65 ) can be rewritten in t erms o f a ne w li ft ed v ariable X ∈ R m × m as ¯ H ( x , X ) = H 0 + m X i =1 x i H i + m X i =1 m X j = 1 X ij H i,j ≤ 0 , (2.66) where X ij represents the entry of X in the i th row and the j th column and in order to maintain consistency with ( 2.65 ) the equality con s traint X = xx T must b e satisﬁed. Rather than working with the non-conv ex const raint X = xx T , con vex relaxation s o f thi s con- straint are provided in [ 103 ], which include t he following o p tions: 1. [ 8 ] An LMI relaxati o n of the form  X x ∗ 1  ≥ 0 . This relaxation can be used to formulate an SDP . 35 2. A cone relaxation of the form X ii − x 2 i ≥ 0 , i = 1 , . . . , m ( X ii − x 2 i )( X j j − x 2 j ) ≥ ( X ij − x i x j ) 2 , i = 1 , . . . , m, j = 1 , . . . , m. This relaxation can be used to formulate a second-order cone program (SOCP). 3. A parabolic relaxatio n of the form X ii + X j j − 2 X ij ≥ ( x i − x j ) 2 , i = 1 , . . . , m, j = 1 , . . . , m, X ii + X j j + 2 X ij ≥ ( x i + x j ) 2 , i = 1 , . . . , m, j = 1 , . . . , m. This relaxati o n can be used to formulate an opt imization probl em with con vex quadratic constraints (e.g., a quadratically -constrained quadratic program). Giv en that this do cument focuses on SDPs and LMI constraint s , the LMI relaxation i s chosen as th e focus for the remainder of this section. 2.9.1 Sequential Penalized Con vex Relaxation Optimization [ 103 , 104 ] The no n-con ve x optim ization problem min x ∈ R m c T x subject to H ( x ) ≤ 0 , where H ( x ) is a BMI constraint deﬁned in ( 2.65 ), can be solved with near -global optimalit y by formulating a sequential penali zed con vex relaxation optimization probl em of th e form min x ∈ R m , X ∈ R m × m c T x + η  tr( X ) − 2 x T 0 x + x T 0 x 0  subject to ¯ H ( x , X ) ≤ 0 ,  X x ∗ 1  ≥ 0 , where x 0 is a prior guess o f x , ¯ H ( x , X ) is deﬁned in ( 2.66 ), and η > 0 i s a scalar regularization parameter that allows for a tradeoff b etween the original objective functio n and the penalty term ensuring tight satisfaction of the constraint X = xx T . As o u tlined in [ 103 ], provided that x 0 is a feasible sol ution to the o ri g inal optim i zation problem and a su f ﬁciently large v alue of η is used, the solution to this relaxed optimization p roblem, denoted as x ∗ and X ∗ satisﬁes X ∗ = x ∗ x ∗ T and c T x ∗ ≤ c T x 0 . Thus, a sequential implem ent ation of thi s optimization can be form ulated, as described in [ 104 ] to obtain a near-global solution to the original non-con vex optim ization problem. 2.10 Coordinate Descent Coordinate descent, also known as block coordinate descent [ 105 ], is an iterative technique that can be empl oyed when faced with a BMI that is in fact an LM I when one or more of the design 36 var iables are ﬁxed. For example, cons i der the design v ariables P ∈ S n and A ∈ R n × n that deﬁne the BMI P A + A T P < 0 . (2.67) This BMI in the design variables P and A is an LM I when either P o r A is ﬁxed. When so lving an optimizatio n problem with such a BMI constraint, coordinate descent can be used, which in volves alternating between ﬁxing one variable and optimi zi n g over the oth er variable. T o highl ight the implem ent ation of a coordinate descent approach, consider the objective func- tion φ ( P , A ) : ( S n × R n × n ) 7→ R , where φ ( P , A ) is conv ex and a v alid SDP objective fun ct i on for either a ﬁx ed v alue of P or A . The opt i mization problem of minimizing φ ( P , A ) subject to ( 2.67 ) can be approached us i ng the following iterative process. 1. Choose an initial value for A . 2. Solve for P that minimizes φ ( P , A 0 ) s u bject to P A 0 + A T 0 P < 0 , where A 0 is the ﬁxed value o f A from the pre vious step. 3. Solve for A that minimizes φ ( P 0 , A ) subject to P 0 A + A T P 0 < 0 , where P 0 is the ﬁxed value o f P from the pre vious step. 4. Repea t Steps 2 and 3 until the desired con ver gence or stopping crit erion is met. This type of it erative algo ri t hm is kn own as coordinate descent. Coordinate descent is int ro - duced well in [ 106 ] and has been used in many applications, including D K -iteration [ 107 , 108 ] and ot h er cont rol design approaches (e.g., [ 109 – 115 ]). Although coordinate descent provides a practical approach to iteratively solve a BMI p rob lem, it typically is not capable of guaranteeing con ver gence to the globall y o ptimal s olution. In general, it wil l con ver ge t o a locally optimal solu- tion th at is dependent on the initial guess. This motiv ates the need for a good initial guess, i d eally in the neighbourhood of the globally optimal solution. 2.11 Discussion on Reformulatin g B MIs as LMIs Properties and tri cks were presented Sections 2.2 to 2.10 that can be used to reformulate BMIs into LMIs. Speciﬁcally , the properties in Sections 2.2 to 2.7 are t ypically able to reformulate a BMI as an equiv alent LMI or LMIs. The properties in Sections 2.8 to 2.1 0 are typically used t o obtain an LMI that impli es a BMI, g enerally with conserv atism . This section presents examples in wh i ch these properties are applied to obtain an LM I that is either equivalent to t he orig i nal BMI or impl ies the o ri ginal BMI. 37 2.11.1 Reformulating a BMI as an Equi valent LMI Example 2.4. Consider the case of a BMI in the variable Y ∈ R m × n of th e form P + Y T SY < 0 , (2.68) where P ∈ S n , S ∈ S m , and S > 0 . The Schur complement is used to obtain an equivalent LMI giv en by  P Y T ∗ − S − 1  < 0 . This LM I can also be written as  P 0 ∗ − S − 1  +  0 1  Y  1 0  +  1 0  Y T  0 1  < 0 . ( 2.69 ) Applying the Projection Lemma, it is known that there exists Y sat i sfying ( 2.69 ) if and only if P < 0 and S − 1 > 0 , since N  1 0  = R  0 1  , N  0 1  = R  1 0  , and P =  1 0   P 0 ∗ − S − 1   1 0  , − S − 1 =  0 1   P 0 ∗ − S − 1   0 1  . Notice that the Projection Lemma gives two m atrix inequali t ies t hat do n ot depend on the var iable Y . Thi s is why the Projection Lemma is also known as the Matrix Elimination Lemm a. 2.11.2 Reformulating a BMI as an LMI that Implies the Original BMI Example 2.5. As a second example, consi d er the BMI P − Y T SY < 0 , (2.70) where Y ∈ R m × n , P ∈ S n , S ∈ S m , and S > 0 . Y oung’ s relation is used to obtain an LMI in Y giv en by P − X T Y − X T Y + X T S − 1 X < 0 , (2.71) which imp lies the BMI of ( 2.70 ). Notice that ( 2.71 ) inv olves a n ew variable X ∈ R m × n . Usi n g t he Schur compl ement on ( 2.71 ) yields  P − X T Y − Y T X X T ∗ − S  < 0 , which is an LMI in Y for a ﬁxed X . It is desirable to use th e Schur comp lement of the Projection Lemma over Y oung’ s relation whenev er poss i ble, as they provides an LMI or LMIs that are equiv alent to the origi nal BMI. When using Y oung’ s relati o n, the resulting L M I imp l ies the original BMI, but is not equi valent. This in t roduces conservatism i nto an optimization p rob lem. 38 If a previously-known solution Y 0 to ( 2.70 ) is a vailable, then th e concepts of con vex-conca ve decompositio n s and con vex over boun ding can be used to reduce conservatism in th e neighbor- hood of Y 0 . In this particular example, the BMI of ( 2.7 0 ) d oes not have a con ve x p ortion t o its decompositio n and it can be sho w that it i s equivalent to t he BMI P − ( Y − Y 0 ) T S ( Y − Y 0 ) − Y T SY 0 − Y T 0 SY + Y T 0 SY 0 < 0 . (2.72) Since the term ( Y − Y 0 ) T S ( Y − Y 0 ) i s po s itiv e deﬁnite, ( 2.72 ) is impli ed by the LM I P − Y T SY 0 − Y T 0 SY + Y T 0 SY 0 < 0 . (2.73) The LMI of ( 2.73 ) is in general conservati ve, but this conserv atis m disapp ears when Y = Y 0 and is reduced when Y is clos e to Y 0 . 39 3 Additiona l LMI Pr ope rties and T r icks This sectio n presents a compilatio n of additional LMI p rop erties and tricks from t he literature. 3.1 The S-Procedu re [ 1 , pp. 23–24] , [ 4 , Sec. 12.3.4], [ 116 , 117 ] Consider x ∈ R n and the quadratic functi ons F 0 ( x ) : R n → R , F i ( x ) : R n → R , where i = 1 , . . . , m . The in equality F 0 ( x ) ≤ 0 is satisﬁed when F i ( x ) ≥ 0 , i = 1 , . . . , m , i f there exist τ i ∈ R ≥ 0 , i = 1 , . . . , m such that F 0 ( x ) + m X i =1 τ i F i ( x ) ≤ 0 . If m = 1 , then this becomes a necessary and sufﬁcient condition, t hat is, F 0 ( x ) ≤ 0 is sati s ﬁed when F 1 ( x ) ≥ 0 if and o n ly if there exists τ 1 ∈ R ≥ 0 such that F 0 ( x ) + τ 1 F 1 ( x ) ≤ 0 . Example 3.1. [ 1 , p. 24 ], [ 4 , Example 12.8, Sec. 12.3 . 4 ] Consider P ∈ S n , A ∈ R n × n , B ∈ R n × m , x ∈ R n , u ∈ R m , γ ∈ R > 0 , and τ ∈ R ≥ 0 . There exists P > 0 such that  x T u T   A T P + P A PB ∗ 0   x u  < 0 when x 6 = 0 and u s atisfy the constraint u T u ≤ γ x T C T Cx if and on ly if there exist P > 0 and τ ∈ R ≥ 0 such that  A T P + P A + τ C T C PB ∗ − τ γ − 1 1  < 0 . 3.2 Dualization Lemma [ 3 , pp. 106–107] Consider P ∈ S n and the subs p aces U , V , where P is in vertible and U + V = R n . The following are equi valent. • x T Px < 0 for all x ∈ U \ { 0 } and x T Px ≥ 0 for all x ∈ V . • x T P − 1 x > 0 for all x ∈ U ⊥ \ { 0 } and x T P − 1 x ≤ 0 for all x ∈ V ⊥ . Example 3.2. [ 3 , pp. 106–1 0 7 ] Consider the matrices Q ∈ S n , S ∈ R n × m , R ∈ S m , M ∈ R m × n , where R ≥ 0 , which deﬁne the qu adrati c matrix inequality  1 M  T  Q S S T R   1 M  < 0 . (3.1) Deﬁne P =  Q S S T R  , U = R  1 M  , and V = R  0 1  , where U + V = R n + m . Notice that ( 3.1 ) i s equiv alent to x T Px < 0 for all x ∈ U \ { 0 } . Additionally , x T Px ≥ 0 for all x ∈ V is equiv alent to  0 1  T  Q S S T R   0 1  = R ≥ 0 , 40 which i s satisﬁed based on the deﬁnition of R . By the dualization lemm a, ( 3.1 ) is satisﬁed wit h R ≥ 0 if and onl y if  − M T 1  T " ˜ Q ˜ S ˜ S T ˜ R #  − M T 1  > 0 , ˜ Q ≤ 0 , where " ˜ Q ˜ S ˜ S T ˜ R # =  Q S S T R  − 1 , U ⊥ = N  1 M T  = R  − M T 1  , and V ⊥ = N  0 1  = R  1 0  3.3 Singular V a lues 3.3.1 Maximum Singular V alue [ 1 , p. 8], [ 10 , 118 ] Consider A ∈ R n × m and γ ∈ R > 0 . The maximum si ngular v alue o f A is strictly less than γ (i.e., ¯ σ ( A ) < γ ) if and only if AA T < γ 2 1 . Using the Schur complement, AA T < γ 2 1 is equivalent to  γ 1 A ∗ γ 1  > 0 . Equiv alently , ¯ σ ( A ) < γ i f and onl y if A T A < γ 2 1 or  γ 1 A T ∗ γ 1  > 0 . 3.3.2 Maximum Singular V alue of a Complex Matrix [ 119 ] Consider A ∈ C n × m and γ ∈ R > 0 . The maximum si ngular v alue o f A is strictly less than γ (i.e., ¯ σ ( A ) < γ ) if and only i f AA H < γ 2 1 . Using the Schur compl ement, AA H < γ 2 1 is equivalent to  γ 1 A A H γ 1  > 0 . Equiv alently , ¯ σ ( A ) < γ i f and onl y if A H A < γ 2 1 o r  γ 1 A H A γ 1  > 0 . 3.3.3 Minimum Singular V alue Consider A ∈ R n × m and ν ∈ R ≥ 0 . If n ≤ m , the minimum singul ar value of A is strictly greater than ν (i.e., σ ( A ) > ν ) i f and onl y if A A T > ν 2 1 . If m ≤ n , σ ( A ) > ν if and only if A T A > ν 2 1 . 3.3.4 Minimum Singular V alue o f a Complex Matrix Consider A ∈ C n × m and ν ∈ R ≥ 0 . If n ≤ m , the minimum singul ar value of A is strictly greater than ν (i.e., σ ( A ) > ν ) i f and onl y if A A H > ν 2 1 . If m ≤ n , σ ( A ) > ν if and only if A H A > ν 2 1 . 41 3.3.5 Froben ius Norm Consider A ∈ R n × m and γ ∈ R > 0 . The Frobeniu s n orm of A is k A k F = p tr( A T A ) = p tr( AA T ) [ 6 , pp. 341–3 4 2]. The Frobenius norm is less than or equal to γ if and onl y if any of the fol lowing equiv alent conditions are satisﬁed. 1. There exists Z ∈ S n such that  Z A T ∗ 1  ≥ 0 , tr( Z ) ≤ γ 2 . 2. There exists Z ∈ S m such that  Z A ∗ 1  ≥ 0 , tr( Z ) ≤ γ 2 . 3.3.6 Nuclear Norm [ 120 , 121 ] Consider A ∈ R n × m and µ ∈ R > 0 . The nuclear norm of A is giv en by k A k ∗ = P p i =1 σ i ( A ) , where p = min( n, m ) and σ i ( A ) , i = 1 , . . . , p are th e singular values of A [ 6 , p. 466]. The nu cl ear norm of A is less than or equal to µ (i.e., k A k ∗ ≤ µ ) if and only if there exist X ∈ S n and Y ∈ S m such that  X A ∗ Y  ≥ 0 , 1 2 tr( X + Y ) ≤ µ. 3.4 Eigen valu es of Symmetric Matrices 3.4.1 Maximum Eigen value [ 1 , p. 10] Consider A ∈ S n × n and γ ∈ R . The maximum eigen value of A i s strictly less than γ (i.e., ¯ λ ( A ) < γ ) if and only if A < γ 1 . 3.4.2 Minimum Eigen value Consider A ∈ S n × n and γ ∈ R . The minimum eigen value of A is strictly greater than γ (i.e., λ ( A ) > γ ) if and only if A > γ 1 . 3.4.3 Sum of Largest Eigen values [ 122 ] Consider A ∈ S n × n , γ ∈ R , and k ∈ Z > 0 . The sum of the k largest eigen values of A , where k ≤ n , is less than γ (i.e., P k i =1 λ i ( A ) ≤ γ ) if and only if there exist X ∈ S n and z ∈ R , where X ≥ 0 , such that z 1 + X − A ≥ 0 , z k + tr( X ) ≤ γ . 42 3.4.4 Sum of Absolute V alue Largest Eigen values [ 122 ] Consider A ∈ S n × n , γ ∈ R , and k ∈ Z > 0 . The sum of t he absolu t e value of the k largest eigen va lues of A , where k ≤ n , i s less than γ (i.e., P k i =1 | λ i ( A ) | ≤ γ ) i f and on ly if there exist X , Y ∈ S n and z ∈ R , where X ≥ 0 and Y ≥ 0 , such that z 1 + X − A ≥ 0 , z 1 + Y + A ≥ 0 , z k + tr( X + Y ) ≤ γ . 3.4.5 W eighted Sum of Lar gest Eigen values [ 122 ] Consider A ∈ S n × n , γ ∈ R , k ∈ Z > 0 , and w i ∈ R > 0 , i = 1 , . . . , k , where 0 < w k ≤ w k − 1 ≤ · · · ≤ w 1 . The weighted sum of the k lar gest eigen v alues of A , where k ≤ n , is less than γ (i.e., P k i =1 w i λ i ( A ) ≤ γ ) if and only if there exist X i ∈ S n and z i ∈ R , i = 1 , . . . , k , where X i ≥ 0 , such that z i 1 + X i − ( w i − w i +1 ) A ≥ 0 , for i = 1 , . . . , k − 1 , z k 1 + X k − w k A ≥ 0 , k X i =1 ( iz i + tr( X i )) ≤ γ . 3.4.6 W eighted Sum of Absolute V alue of Largest Eig en values [ 122 ] Consider A ∈ S n × n , γ ∈ R , k ∈ Z > 0 , and w i ∈ R > 0 , i = 1 , . . . , k , where 0 < w k ≤ w k − 1 ≤ · · · ≤ w 1 . The weighted sum of the absolute value o f the k largest eigen values of A , where k ≤ n , is less t han γ (i. e., P k i =1 w i | λ i ( A ) | ≤ γ ) if and only if there exist X i , Y i ∈ S n and z i ∈ R , i = 1 , . . . , k , where X i ≥ 0 and Y i ≥ 0 , such that z i 1 + X i − ( w i − w i +1 ) A ≥ 0 , for i = 1 , . . . , k − 1 , z i 1 + Y i + ( w i − w i +1 ) A ≥ 0 , for i = 1 , . . . , k − 1 , z k 1 + X k − w k A ≥ 0 , z k 1 + Y k + w k A ≥ 0 , k X i =1 ( iz i + tr( X i + Y i )) ≤ γ . 3.5 Matrix Condition Number 3.5.1 Condition N umber of a Matrix [ 1 , pp. 37–38] Consider A ∈ R n × m and γ , µ ∈ R > 0 , wh ere the condition num ber of A is κ ( A ) . If m ≤ n , the inequality κ ( A ) ≤ γ h olds if there exists µ such t h at µ 1 ≤ A T A ≤ γ 2 µ 1 . 43 If n ≤ m , the inequal i ty κ ( A ) ≤ γ holds if th ere exists µ such that µ 1 ≤ AA T ≤ γ 2 µ 1 . 3.5.2 Condition N umber of a Positive Deﬁnite Matrix [ 1 , p. 38] Consider A ∈ S n and γ , µ ∈ R > 0 , where the condition number of A is κ ( A ) . The inequality κ ( A ) ≤ γ holds if there e xis ts µ such that µ 1 ≤ A ≤ γ µ 1 . 3.6 Spectral Radius [ 9 , p. 17] Consider A ∈ R n × n and δ ∈ R > 0 . The spectral radius of A is strictl y less than δ (i.e., ρ ( A ) < δ ) under eit her of the following necessary and sufﬁcient cond itions. 1. There exists X ∈ S n , wh ere X > 0 , such that A T XA − δ 2 X < 0 . 2. There exists X ∈ S n , wh ere X > 0 , such that AXA T − δ 2 X < 0 . Also see Section 4.25 for a similar condition related to the st ructured singular v alue. 3.7 T race of a Symm etric Matrix 3.7.1 T race of a Matrix with a Slack V ariable 1. [ 5 , pp. 46–47] Consider P ∈ S n and γ ∈ R > 0 , wh ere P > 0 and Z > 0 . The inequality given by tr( P ) < γ is satis ﬁed if and only if there e xis ts Z ∈ S n such that P < Z , tr ( Z ) < γ . 2. [ 1 , p . 8] Consider P ∈ S n , X ∈ R n × m , and γ ∈ R > 0 , where P > 0 and Z > 0 . The matrix inequality given by tr  X T P − 1 X  < γ is satis ﬁed if and only if there e xis ts Z ∈ S m such that  Z X T ∗ P  > 0 , tr( Z ) < γ . 44 3.7.2 Relative T race of Ma trices 1. [ 5 , pp. 46–47] Consider P , Q ∈ S n . The property tr( P ) < tr( Q ) holds if t he matri x inequal- ity P < Q is satis ﬁed. 2. [ 7 , p. 768], [ 123 , p. 215] Consider P , Q ∈ S n , wh ere P ≥ 0 , Q > 0 , and P ≤ Q . Then, det( P ) det( Q ) ≤ tr( P ) tr( Q ) . 3. [ 7 , p. 771], [ 124 ] Consider P , Q ∈ S n and i , j ∈ R ≥ 0 , where P ≥ 0 , Q ≥ 0 , and P ≤ Q . Then, tr  A i B j  ≤ tr  B i + j  . 4. [ 7 , p. 771], [ 124 ] Consider P , Q ∈ S n and i , j ∈ R , wh ere P > 0 , Q > 0 , P ≤ Q , j ≥ − 1 , and i + j ≥ 0 . Then, tr  A i B j  ≤ tr  B i + j  . 5. [ 7 , p. 771], [ 125 ] Consi der P , Q ∈ S n and i , j ∈ R > 0 , where P ≥ 0 , Q ≥ 0 , Q ≤ 1 , and i ≤ j . Then, tr  QP i Q  1 /i ≤ tr  QP j Q  1 /j and tr  QP i Q  1 /i ≤ tr  Q i/j P i Q i/j  1 /j . 6. [ 7 , p. 773], [ 123 , p. 213] Consider P , Q , V ∈ S n , where P ≥ 0 , Q ≥ 0 , V ≥ 0 , and P ≤ Q . Then, tr  ( V + P ) − 1 P  ≤ tr  ( V + Q ) − 1 Q  . 3.8 Range of a Symmetric Ma t rix [ 7 , p. 714] Consider P ∈ S n , wh ere P > 0 . If P ≤ Q , then R ( P ) ⊆ R ( Q ) . 3.9 Logarithm of a Positive Deﬁnite Matrix [ 7 , p. 715] Consider P , Q ∈ S n and α ∈ R > 0 , where P ≥ 0 . Th e matrix logarithm of P satisﬁes the following matri x inequality 1 − P − 1 ≤ log ( P ) ≤ α − 1 ( P α − 1 ) . 3.10 Douglas-Fillmore-W illiams Lemma [ 7 , p. 714] [ 126 , 1 2 7 ] Consider A ∈ R n × m and B ∈ R n × p . The following st atements are equi valent. 1. There exists C ∈ R p × m such that A = BC . 2. There exists α ∈ R > 0 such that AA T − α BB T ≤ 0 . 3. R ( A ) ⊆ R ( B ) . 45 3.11 Submatrix Determinants [ 119 ] Consider A ∈ S n . Let A k ∈ S k be a submatrix of A consist i ng of its ﬁrst k rows and columns, where k ≤ n . The m atrix inequalit y A > 0 is satisﬁed if and only if det ( A k ) > 0 , k = 1 , . . . , n. 3.12 Ima g inary and Real Parts [ 4 , Sec. 12.1.1] Consider Q R ∈ S n , Q I ∈ R n × n , and Q = Q H = Q R + j Q I ∈ C n × n . The m atrix i nequality Q > 0 is equiv alent to th e matrix inequality give n by  Q R Q I − Q I Q R  > 0 . 3.13 Quadratic Inequalities 3.13.1 W eighted Norm [ 10 ] Consider W ∈ S n , x , y ∈ R n , and γ ∈ R ≥ 0 , w h ere W > 0 . The inequal i ty ( x − y ) T W ( x − y ) ≤ γ is equi valent to the m atrix inequalit y given by  γ ( x − y ) T ∗ W − 1  ≥ 0 . 3.13.2 Quadratic Inequalities 1. Consider W ∈ S n , A ∈ R n × m , x , c ∈ R m , b ∈ R n , and d ∈ R , where W > 0 . The quadratic inequality ( Ax + b ) T W ( Ax + b ) − c T x − d ≤ 0 wit h W > 0 is equiv alent to the matrix inequality given by  W − 1 Ax + b ∗ c T x + d  ≥ 0 . 2. [ 7 , p. 731] Consider x ∈ R n . The matrix inequalit y given by xx T − x T x1 ≤ 0 holds. 3.14 Miscellaneous Properties and Results 1. [ 7 , p. 738] Consider P , Q ∈ S n , where P ≥ 0 and Q > 0 . Then, P ≤ Q if and on l y if PQ − 1 P ≤ P . 2. [ 128 , p. 269], [ 7 , p. 738] Consider P , Q ∈ S n , where P ≥ 0 , Q ≥ 0 , and P ≤ Q . Then, th ere exists S ∈ R n × n such that P = S T QS and S T S ≤ 1 . 3. [ 129 , 130 ], [ 7 , p. 738] Consid er P , Q , R , S ∈ S n , w h ere P ≥ 0 , Q ≥ 0 , R ≥ 0 , S > 0 , S ≤ R , and Q RQ ≤ PSP . Then, Q ≤ P . 46 4. [ 123 , pp. 289–2 9 0], [ 7 , p . 738] Consider P , Q ∈ R n × m . Then, t here e xist uni tary mat ri ces S 1 , S 2 ∈ R m × m such that q ( P + Q ) T ( P + Q ) ≤ S 1 √ P T PS T 1 + S 2 p Q T QS T 2 . This is a matri x version of the triangle inequality . 5. [ 131 , 132 ], [ 7 , p. 73 9 ] Consider P , Q ∈ S n . Then, there exists a unitary matri x S ∈ R n × n such that p QPPQ ≤ 1 2 S ( PP + Q Q ) S T . 6. [ 133 ], [ 7 , p. 739] Consider P , Q ∈ S n , where P ≥ 0 , Q > 0 , P ≤ 1 , α = λ min ( Q ) , and β = λ max ( Q ) . Then, PQP ≤ ( α + β ) 2 4 αβ Q . 7. [ 134 ], [ 7 , p . 740] Consider P , Q ∈ S n , where P ≥ 0 , Q ≥ 0 , P ≤ Q and PQ = QP . Then, PP ≤ BB . 8. [ 123 , p. 214], [ 7 , p. 740] Consider P , Q ∈ S n , where P ≥ 0 , Q ≥ 0 , and QP P Q ≤ 1 . Then, √ QP √ Q ≤ 1 . 9. [ 123 , p. 292], [ 7 , p. 741] Consider P , Q ∈ S n . Then  1 2 ( P + Q )  2 ≤ 1 2  P 2 + Q 2  . 10. [ 135 ], [ 7 , p. 741] Consider P , Q ∈ S n and α ∈ R , where P ≥ 0 and Q ≥ 0 . If ei t her • α ∈ [1 , 2] , o r • P > 0 , Q > 0 , and α ∈ [ − 1 , 0 ] ∪ [1 , 2] , then,  1 2 ( P + Q )  α ≤ 1 2 ( P α + Q α ) . 11. [ 136 , 137 ], [ 7 , p. 741] Consider P , Q ∈ S n and α , β ∈ R , where P ≥ 0 , Q ≥ 0 , and 1 ≤ α ≤ β . Then,  1 2 ( P α + Q α )  1 /α ≤  1 2  P β + Q β  1 /β . Furthermore, µ ( P , Q ) ∆ = lim γ →∞  1 2 ( P γ + Q γ )  1 /γ exists and satisﬁes P ≤ µ ( P , Q ) and Q ≤ µ ( P , Q ) . Additionall y , lim γ → 0  1 2 ( P γ + Q γ )  1 /γ = e 1 2 (log( A )+log ( B )) . 47 12. [ 79 , 86 ] Consider P , Q , Z ∈ S n and x ∈ R n , where P ≥ 0 , Q ≥ 0 , and Z > 0 . If the inequality  x T Zx  2 − 4  x T Pxx T Qx  > 0 holds for all x 6 = 0 , then there exists λ ∈ R > 0 such that λ 2 P + λ Z + Q < 0 . 13. [ 138 ] Consi der A ∈ R n × n and W , Q ∈ S n , where W > 0 . If th ere exists S ∈ S n , where S > 0 , such that SWS = SA + A T S + Q , then for any 0 < W 1 ≤ W and Q 1 ≥ Q there exists S 1 ∈ S n , wh ere S 1 ≥ S such th at S 1 W 1 S 1 = S 1 A + A T S 1 + Q 1 . 14. [ 139 ] Consider X , Y ∈ S n and r ∈ Z > 0 . There exist X 2 , Y 2 ∈ R n × r and X 3 , Y 3 ∈ S r , wh ere X 3 > 0 such t hat  X X 2 ∗ X 3  − 1 =  Y Y 2 ∗ Y 3  if and only if X − Y − 1 ≥ 0 and rank ( X − Y − 1 ) ≤ r . 15. [ 140 , p. 19] Consider M 11 , A ∈ S n , M 12 ∈ R n × m , M 22 ∈ S m , E , F 1 ∈ R n × n , and F 2 ∈ R m × n , wh ere M 11 ≥ 0 and E is inv ertible. The matrix inequal i ty  E − 1 A 1  T  M 11 M 12 ∗ M 22   E − 1 A 1  < 0 (3.2) holds i f and onl y if t h ere exist F 1 and F 2 such that  M 11 + F 1 E + E T F T 1 M 12 − F 1 A + E T F T 2 ∗ M 22 − F 2 A − A T F T 2  < 0 , (3.3) Moreover , th e following statement s hold. (a) If ( 3.2 ) ho l ds, t hen ( 3.3 ) holds with F 1 = − ( M 11 + ǫ W ) E − 1 and F 2 = − M T 12 E − 1 , where ǫ ∈ R > 0 is sufﬁciently small, W ∈ S n , and W > 0 . (b) If ( 3.2 ) h olds and M 11 > 0 , then ( 3.3 ) holds wi th F 1 = M 11 E − 1 and F 2 = − M T 12 E − 1 . 48 4 LMIs in Sy stems and Stabi l i t y The ory This section presents a compilati on of LMIs resul t s that are related to systems and stabi lity theory . 4.1 L yapunov Inequalities 4.1.1 L ya punov Stabili ty [ 7 , pp. 1201–1203], [ 1 , pp. 20–21] Consider the matri ces A ∈ R n × n and Q ∈ S n , where Q ≥ 0 . There exists P ∈ S n , where P > 0 , sati s fying the L yapunov equation A T P + P A + Q = 0 , if and only if there exists P ∈ S n , wh ere P > 0 , such that A T P + P A ≤ 0 . (4.1) If ( 4.1 ) holds, then Re { λ i ( A ) } ≤ 0 , i = 1 , . . . , n , and the equili brium poi n t ¯ x = 0 of the s y stem ˙ x = Ax is L yapunov stable. The matrix inequali ty of ( 4.1 ) is satis ﬁed under an y of t he following equiv alent necessary and sufﬁ cient conditions. 1. There exists X ∈ S n , wh ere X > 0 , such that XA T + AX ≤ 0 . 2. There exist X ∈ S n and V ∈ R n , wh ere X > 0 , such that   −  V + V T  V T A + X V T ∗ − X 0 ∗ ∗ − X   ≤ 0 . Pr oof. Identical to the proof of ( 4.5 ) i n [ 73 ], except with the u s e of the N o nstrict Projection Lemma, where G T =  − 1 A 1  and H T =  1 0 0  , and therefore R ( G ) and R ( H ) are linearly independ ent . 3. There exist X ∈ S n and V ∈ R n , wh ere X > 0 , such that   −  V + V T  V T A T + X V T ∗ − X 0 ∗ ∗ − X   ≤ 0 . Pr oof. Identical to the proof of ( 4.6 ) in [ 73 ], except with the use of the Nonstrict Projection Lemma, where G T =  − 1 A T 1  and H T =  1 0 0  , and therefore R ( G ) and R ( H ) are linearly independent. 4. [ 15 ] There does not exist Z ∈ S n , where Z > 0 , such that ZA T + AZ > 0 . 49 4.1.2 Asymptotic Stability [ 7 , p. 1201 –1203], [ 1 , p. 2] Consider the matri ces A ∈ R n × n and Q ∈ S n , where Q > 0 . There exists P ∈ S n , where P > 0 , sati s fying the L yapunov equation A T P + P A + Q = 0 , if and only if there exists P ∈ S n , wh ere P > 0 , such that A T P + P A < 0 . (4.2) If ( 4.2 ) hol ds, then Re { λ i ( A ) } < 0 , i = 1 , . . . , n , the matrix A is Hurwitz, and the equili brium point ¯ x = 0 of the syst em ˙ x = Ax is asymptotically stable. The matrix inequal i ty of ( 4.2 ) is s at i sﬁed and the matrix A is Hurwitz under any of the following equiv alent necessary and sufﬁcient cond itions. 1. There exists X ∈ S n , wh ere X > 0 , such that XA T + AX < 0 . 2. ( The S -V ariable Appr oach [ 140 , p p. 2–3], [ 141 ]) There exist P ∈ S n and F 1 , F 2 ∈ R n × n , where P > 0 , such that  F 1 A + A T F T 1 P − F 1 + A T F T 2 ∗ − ( F 2 + F T 2 )  < 0 . (4.3) 3. [ 142 ] There exist P ∈ S n and F 1 , F 2 ∈ R n × n , wh ere P > 0 , such that  F 1 P + PF T 1 A T − F 1 + PF T 2 ∗ − ( F 2 + F T 2 )  < 0 . (4.4) 4. [ 73 ] There exist Y ∈ S n and W ∈ R n × n , wh ere Y > 0 , such that  Y −  W + W T  A Y + W T ∗ − Y  < 0 . 5. [ 73 ] There exist X ∈ S n and V ∈ R n × n , wh ere X > 0 , such that   −  V + V T  V T A + X V T ∗ − X 0 ∗ ∗ − X   < 0 . (4.5) 6. [ 73 ] There exist X ∈ S n and V ∈ R n × n , wh ere X > 0 , such that   −  V + V T  V T A T + X V T ∗ − X 0 ∗ ∗ − X   < 0 . ( 4.6) 50 7. [ 142 ] There exist P ∈ S n and F 1 , F 2 , X 1 , X 2 , X 3 ∈ R n × n , wh ere P > 0 , such that   X 1 F T 1 + F 1 X T 1 P + X 1 F T 2 + F 1 X T 2 A T − X 1 + F 1 X T 3 ∗ X 2 F T 2 + F 2 X T 2 − 1 − X 2 + F 2 X T 3 ∗ ∗ − ( X 3 + X T 3 )   < 0 . (4.7) 8. There exist P ∈ S n and F 1 , F 2 , X 1 , X 2 , X 3 ∈ R n × n , wh ere P > 0 , such that   X 1 F T 1 + F 1 X T 1 A T + X 1 F T 2 + F 1 X T 2 P − X 1 + F 1 X T 3 ∗ X 2 F T 2 + F 2 X T 2 − 1 − X 2 + F 2 X T 3 ∗ ∗ − ( X 3 + X T 3 )   < 0 . Pr oof. The proof follows the same steps as the proof of ( 4.7 ) in [ 142 ], beginning with ( 4.4 ) instead of ( 4.3 ). 9. [ 143 ] There exist P ∈ S n and Y 1 , Y 2 , Y 3 , X 1 , X 2 , X 3 ∈ R n × n , wh ere P > 0 , such that   X 1 Y 1 + Y T 1 X T 1 P + X 1 Y 2 + Y T 1 X T 2 A T + X 1 Y 3 + Y T 1 X T 3 ∗ X 2 Y 2 + Y T 2 X T 2 − 1 + X 2 Y 3 + Y T 2 X T 3 ∗ ∗ X 3 Y 3 + Y T 3 X T 3   < 0 . 10. [ 14 ] There do n o t exist Z 1 , Z 2 ∈ S n , wh ere Z 1 ≥ 0 , Z 2 ≥ 0 , Z 1 6 = 0 , and Z 2 6 = 0 , such that Z 1 A T + AZ 1 − Z 2 = 0 . 4.1.3 Discrete- Time L yapunov Stability [ 7 , pp. 1203–1 2 04] Consider the matrices A d ∈ R n × n and Q ∈ S n , where Q ≥ 0 . There exists P ∈ S n , where P > 0 , sati s fying the discrete-time L yapunov equatio n A T d P A d − P + Q = 0 . if and only if there exists P ∈ S n , wh ere P > 0 , such that A T d P A d − P ≤ 0 . (4.8) If ( 4.8 ) holds, then | λ i ( A d ) | ≤ 1 , i = 1 , . . . , n , and the equilibrium point ¯ x = 0 of the system x k +1 = A d x k is L yapunov s t able. The m atrix inequality of ( 4.8 ) is satisﬁed and the eigen values of A d satisfy | λ i ( A d ) | ≤ 1 , i = 1 , . . . , n under any of the following equiv alent necessary and sufﬁcient con ditions. 1. There exists X ∈ S n , wh ere X > 0 , such that A d P A T d − P ≤ 0 . 2. There exists P ∈ S n , wh ere P > 0 , such that  P A d P ∗ P  ≥ 0 . 51 3. There exists P ∈ S n , wh ere P > 0 , such that  P A T d P ∗ P  ≥ 0 . 4. There exists P ∈ S n , wh ere P > 0 , such that  P P A T d ∗ P  ≥ 0 . 5. There exists P ∈ S n , wh ere P > 0 , such that  P P A d ∗ P  ≥ 0 . 6. [ 144 ] There exist P ∈ S n and G ∈ R n × n , where P > 0 , such that  P A T d G T ∗ G + G T − P  ≥ 0 . 4.1.4 Discrete- Time As ymptotic Stability [ 7 , pp. 1203–1204], [ 5 , pp. 97–98] Consider the matrices A d ∈ R n × n and Q ∈ S n , where Q > 0 . There exists P ∈ S n , where P > 0 , sati s fying the discrete-time L yapunov equatio n A T d P A d − P + Q = 0 . if and only if there exists P ∈ S n , wh ere P > 0 , such that A T d P A d − P < 0 . (4.9) If ( 4.9 ) h o lds, then | λ i ( A d ) | < 1 , i = 1 , . . . , n , the matrix A d is Schur, and the equi librium point ¯ x = 0 of the syst em x k +1 = A d x k is asymptotically stable. The m atrix inequality of ( 4.9 ) is satisﬁed and the eigen values o f A d satisfy | λ i ( A d ) | < 1 , i = 1 , . . . , n under any of the following equiv alent necessary and sufﬁcient con ditions. 1. There exists X ∈ S n , wh ere X > 0 , such that A d P A T d − P < 0 . 2. [ 5 , p. 97] T h ere exists P ∈ S n , wh ere P > 0 , such that  P A d P ∗ P  > 0 . 3. There exists P ∈ S n , wh ere P > 0 , such that  P A T d P ∗ P  > 0 . 52 4. [ 5 , p. 97] There exists P ∈ S n , wh ere P > 0 , such that  P P A T d ∗ P  > 0 . 5. There exists P ∈ S n , wh ere P > 0 , such that  P P A d ∗ P  > 0 . 6. [ 144 ] There exist P ∈ S n and G ∈ R n × n , where P > 0 , such that  P A T d G T ∗ G + G T − P  > 0 . 7. ( The S -V ariable Approac h [ 140 , p. 3], [ 145 ]) T h ere exist P ∈ S n and F 1 , F 2 ∈ R n × n , where P > 0 , such that  F 1 A d + A T d F T 1 − P − F 1 + A T d F T 2 ∗ P − ( F 2 + F T 2 )  < 0 . 8. [ 146 , pp. 46–47], [ 147 ] There exist P ∈ S n and F 1 , F 2 , X 1 , X 2 , X 3 ∈ R n × n , where P > 0 , such that   X 1 F 1 + F T 1 X T 1 − P X 1 F 2 + F T 1 X T 2 A T d − X 1 + F T 1 X T 3 ∗ P + X 2 F 2 + F T 2 X T 2 − 1 − X 2 + F T 2 X T 3 ∗ ∗ − ( X 3 + X T 3 )   < 0 . 9. [ 146 , pp. 46–47], [ 143 , 147 ] There exist P ∈ S n and Y 1 , Y 2 , Y 3 , X 1 , X 2 , X 3 ∈ R n × n , wh ere P > 0 , such that   X 1 Y 1 + Y T 1 X T 1 − P X 1 Y 2 + Y T 1 X T 2 A T d + X 1 Y 3 + Y T 1 X T 3 ∗ P + X 2 Y 2 + Y T 2 X T 2 − 1 + X 2 Y 3 + Y T 2 X T 3 ∗ ∗ X 3 Y 3 + Y T 3 X T 3   < 0 . 4.1.5 Descriptor System Admissibili ty Consider the descriptor system give n by E ˙ x = Ax , where E , A ∈ R n × n . The descriptor system is admis sible under any of the fol lowing equiv alent necessary and su f ﬁcient cond itions. 1. [ 148 , 149 ] There exists X ∈ R n × n , s ati sfying E T X = X T E ≥ 0 and A T X + X T A < 0 . 2. [ 150 ] There exists X ∈ R n × n , s ati sfying EX = X T E T ≥ 0 and AX + X T A T < 0 . 53 3. [ 150 ] There exist P ∈ S n , X ∈ R ( n − n e ) × n , and Z ∈ R n × ( n − n e ) , where n e = rank ( E ) and P > 0 , sati s fying E T Z = 0 and A T ( PE + ZX ) + ( PE + ZX ) T A < 0 . 4. [ 150 ] There exist P ∈ S n , X ∈ R ( n − n e ) × n , and Z ∈ R n × ( n − n e ) , where n e = rank ( E ) and P > 0 , sati s fying EZ = 0 and A  PE T + ZX  +  PE T + ZX  T A T < 0 . 5. [ 151 ] There exist P ∈ S n , S ∈ R ( n − n e ) × ( n − n e , and U , V ∈ R n × ( n − n e ) , where n e = rank ( E ) , R ( U ) = N ( E T ) , R ( V ) = N ( E ) , and P > 0 , satisfying A  PE T + VSU T  +  PE T + VSU T  T A T < 0 . 6. [ 150 ] There exist P ∈ S n , F , G ∈ R n × n , X ∈ R ( n − n e ) × n , and Z ∈ R n × ( n − n e ) , where n e = rank ( E ) and P > 0 , satisfying E T Z = 0 and  A T G T + GA ( P E + ZX ) T + A T F T − G ∗ −  F + F T   < 0 . 7. [ 150 ] There exist P ∈ S n , F , G ∈ R n × n , X ∈ R ( n − n e ) × n , and Z ∈ R n × ( n − n e ) , where n e = rank ( E ) and P > 0 , satisfying EZ = 0 and  A G + G T A T  PE T + ZX  T + AF − G T ∗ −  F + F T   < 0 . 4.1.6 Discrete- Time Descriptor System Admissibility Consider t he discrete-time descriptor system gi ven by E d x k +1 = A d x k , wh ere E d , A d ∈ R n × n . The di screte-time descriptor system is admissib le un d er any o f the following equiv alent necessary and s u f ﬁcient cond itions. 1. [ 152 , 153 ] There exists P ∈ S n , s ati sfying E T d PE d ≥ 0 and A T d P A d − E T d PE d < 0 . 2. [ 154 , 155 ] There e xist P ∈ S n , X ∈ S ( n − n e ) , and Z ∈ R n × ( n − n e ) , where n e = rank ( E d ) and P > 0 , sati s fying E T d Z = 0 and A T d  P − ZXZ T  A d − E T d PE d < 0 . 3. [ 155 ] There exist P ∈ S n , X ∈ S ( n − n e ) , and Z ∈ R n × ( n − n e ) , where n e = rank ( E d ) and P > 0 , satisfying E d Z = 0 and A d  P − ZXZ T  A T d − E T d PE d < 0 . 54 4. [ 151 ] There exist P ∈ S n , S ∈ R ( n − n e ) × ( n − n e , and U , V ∈ R n × ( n − n e ) , wh ere n e = rank ( E d ) , R ( U ) = N ( E T d ) , R ( V ) = N ( E d ) , and P > 0 , satis fying  − E d PE T d + A d VSU T + US T V T A T d A d PE T d + A d VSU T + US T V T A T d ∗ − E d PE T d + A d VSU T + US T V T A T d  < 0 . 5. [ 156 ] There exist P ∈ S n , X ∈ S ( n − n e ) , and Z ∈ R n × ( n − n e ) , where n e = rank ( E d ) and P > 0 , satisfying E d Z = 0 and A T d P A d − E T d PE d + XZA d + A T d Z T X T < 0 . 6. [ 157 ] There exist X ∈ S n and α ∈ R , satisfying E T d X = X T E d ≥ 0 and " X T ( E d − A d ) + ( E d − A d ) T X ( E d − A d ) T X ∗ E T d X + α  1 − E † d E d  # > 0 , where E † d is the pseudoin verse o f E d . 7. [ 155 ] There exist P ∈ S n , X ∈ S ( n − n e ) , F , G ∈ R n × n , and Z ∈ R n × ( n − n e ) , where n e = rank ( E d ) and P > 0 , satis fyi ng E T d Z = 0 and  − E T d PE d + A T d G T + GA d − G + A T d F T ∗ P − ZXZ T −  F + F T   < 0 . 8. [ 155 ] There exist P ∈ S n , X ∈ S ( n − n e ) , F , G ∈ R n × n , and Z ∈ R n × ( n − n e ) , where n e = rank ( E d ) and P > 0 , satis fyi ng E d Z = 0 and  − E d PE T d + A d G T + GA T d − G + A d F T ∗ P − ZXZ T −  F + F T   < 0 . 4.2 Bounded Real Lem ma and the H ∞ Norm 4.2.1 Continuous-T ime Bounded Real Lemma [ 70 ], [ 15 8 , pp. 85–86] Consider a conti nuous-time L TI system, G : L 2 e → L 2 e , wi th state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and D ∈ R p × m . The H ∞ norm of G is k G k ∞ = sup u ∈L 2 , u 6 = 0 k G u k 2 k u k 2 . The inequality k G k ∞ < γ h o lds under any of the following equiva lent necessary and sufﬁ cient conditions. 1. There exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that   P A + A T P PB C T ∗ − γ 1 D T ∗ ∗ − γ 1   < 0 . (4.10) 55 2. There exist Q ∈ S n and γ ∈ R > 0 , wh ere Q > 0 , such that   A Q + QA T B QC T ∗ − γ 1 D T ∗ ∗ − γ 1   < 0 . (4.11) 3. There exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that  P A + A T P + C T C PB + C T D ∗ − γ 2 1 + D T D  < 0 . 4. There exist Q ∈ S n and γ ∈ R > 0 , wh ere Q > 0 , such that  A Q + QA T + BB T QC T + BD T ∗ − γ 2 1 + DD T  < 0 . 5. [ 159 ] There exist P ∈ S n , V ∈ R n × n , and r , γ ∈ R > 0 , wh ere P > 0 , such that     A V + V T A T P − V T + r A V V T C T B ∗ − r  V + V T  r V T C T 0 ∗ ∗ − 1 D ∗ ∗ ∗ − γ 2 1     < 0 . 6. [ 160 ], [ 161 , pp. 46– 47] There exist P ∈ S n , F 1 , F 2 ∈ R n × n , and γ ∈ R > 0 , where P > 0 , such that     F 1 A + A T F T 1 P − F 1 + A T F T 2 F 1 B C T ∗ − ( F 2 + F T 2 ) F 2 B 0 ∗ ∗ − γ 1 D T ∗ ∗ ∗ − γ 1     < 0 . 7. [ 160 ] There exist P ∈ S n , F 1 , F 2 ∈ R n × n , and γ ∈ R > 0 , wh ere P > 0 , such that     P A + A T P P + F 1 + A T F 2 PB C T ∗ F 2 + F T 2 F T 2 B 0 ∗ ∗ − γ 2 1 D T ∗ ∗ ∗ − 1     < 0 . 8. [ 161 , pp. 46–47] There exist P ∈ S n , F 1 , F 2 , X 1 , X 2 , X 3 ∈ R n × n , and γ ∈ R > 0 , where P > 0 , such that       X 1 F 1 + F T 1 X T 1 P + X 1 F 2 + F T 1 X T 2 A T − X 1 + F T 1 X T 3 0 C T ∗ X 2 F 2 + F T 2 X T 2 − 1 − X 2 + F T 2 X T 3 0 0 ∗ ∗ −  X 3 + X T 3  B 0 ∗ ∗ ∗ − γ 1 D T ∗ ∗ ∗ ∗ − γ 1       < 0 . 56 9. [ 161 , pp. 46–47] There exist P ∈ S n , Y 1 , Y 2 , Y 3 , X 1 , X 2 , X 3 ∈ R n × n , and γ ∈ R > 0 , where P > 0 , such that       X 1 Y 1 + Y T 1 X T 1 P + X 1 Y 2 + Y T 1 X T 2 A T + X 1 Y 3 + Y T 1 X T 3 0 C T ∗ X 2 Y 2 + Y T 2 X T 2 − 1 + X 2 Y 3 + Y T 2 X T 3 0 0 ∗ ∗ X 3 Y 3 + Y T 3 X T 3 B 0 ∗ ∗ ∗ − γ 1 D T ∗ ∗ ∗ ∗ − γ 1       < 0 . 10. There exist Q ∈ S n , V 11 ∈ R n × n , V 12 ∈ R n × m , V 21 ∈ R m × n , V 22 ∈ R m × m , and γ ∈ R > 0 , where Q > 0 , such that       − ( V 11 + V T 11 ) V T 11 A T + V T 21 B T + Q V T 11 C T + V T 21 D T V T 11 − V 12 − V T 21 ∗ − Q 0 0 A V 12 + BV 22 ∗ ∗ − γ 2 1 0 CV 12 + D V 22 ∗ ∗ ∗ − Q V 12 ∗ ∗ ∗ ∗ − 1 − ( V 22 + V T 22 )       < 0 . Pr oof. Identical to the proof of ( 4.12 ) in [ 5 , p. 156], except with Ω =  V 11 V 12 V 21 V 22  . 11. There exist P ∈ S n , W 11 ∈ R n × n , W 12 ∈ R n × p , V 21 ∈ R p × n , V 22 ∈ R p × p , and γ ∈ R > 0 , where P > 0 , such that       − ( W 11 + W T 11 ) W T 11 A + W T 21 C + P W T 11 B + W T 21 D W T 11 − ( W 12 + W T 21 ) ∗ − P 0 0 A T W 12 + C T W 22 ∗ ∗ − γ 2 1 0 B T W 12 + D T W 22 ∗ ∗ ∗ − P W 12 ∗ ∗ ∗ ∗ − ( 1 + W 22 + W T 22 )       < 0 . Pr oof. Identical to the proof of ( 4.13 ), exce pt with Ω =  W 11 W 12 W 21 W 22  . The H ∞ norm of G is the minimum value of γ ∈ R > 0 that sati s ﬁes any of the above conditi ons. If ( A , B , C , D ) is a mi nimal realization, then the m atrix inequalities can be nonstrict [ 1 , pp. 26 – 27], [ 16 2 , pp. 3 08–311], [ 163 ]. The inequ ali ty k G k ∞ < γ also holds un der any of the foll o win g equivalent sufﬁcient condi- tions. 1. [ 5 , p. 156] There exist Q ∈ S n , V ∈ R n × n , and γ ∈ R > 0 , wh ere Q > 0 , such that       − ( V + V T ) V T A T + Q V T C T V T 0 ∗ − Q 0 0 B ∗ ∗ − γ 1 0 D ∗ ∗ ∗ − Q 0 ∗ ∗ ∗ ∗ − γ 1       < 0 . (4. 1 2 ) 57 2. There exist P ∈ S n , W ∈ R n × n , and γ ∈ R > 0 , wh ere P > 0 , such that       − ( W + W T ) W T A + P W T B W T 0 ∗ − P 0 0 C T ∗ ∗ − γ 1 0 D T ∗ ∗ ∗ − P 0 ∗ ∗ ∗ ∗ − γ 1       < 0 . (4.13) Pr oof. Identical to the proof of ( 4.12 ) in [ 5 , p. 156], except starting w i th the Bounded Real Lemma in the form  A Q + QA T + 1 γ QC T CQ B + 1 γ QC T D ∗ − γ 1 + 1 γ D T D  , which requires Φ =  − 1 A B 1 0 0 C D 0 − γ 1  . When D = 0 , t hen th e inequali ty k G k ∞ > γ holds if and on l y if there exist Z 11 ∈ S n , Z 12 ∈ R n × m , and Z 22 ∈ S m such that [ 14 ] Z 11 A T + AZ 11 + Z 12 B T + BZ T 12 = 0 ,  Z 11 Z 12 ∗ Z 22  ≥ 0 , tr ( Z 22 ) = 1 , tr  CZ 11 C T  > γ . 4.2.2 Discrete- Time Bounded Real Lemma Consider a dis crete-time L TI system, G : ℓ 2 e → ℓ 2 e , wi th state-space realization ( A d , B d , C d , D d ) , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , and D d ∈ R p × m . The H ∞ norm of G is k G k ∞ = sup u ∈ ℓ 2 , u 6 = 0 k G u k 2 k u k 2 . The inequality k G k ∞ < γ h o lds under any of the following equiva lent necessary and sufﬁ cient conditions. 1. [ 70 ] There exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that   A T d P A d − P A T d PB d C T d ∗ B T d PB d − γ 1 D T d ∗ ∗ − γ 1   < 0 . 2. [ 164 ] There exist Q ∈ S n and γ ∈ R > 0 , wh ere Q > 0 , s uch that   A d QA T d − Q B d A d QC T d ∗ − γ 1 D T d ∗ ∗ C d QC T d − γ 1   < 0 . 58 3. [ 165 ] There exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that     P A d P B d 0 ∗ P 0 PC T d ∗ ∗ γ 1 D T d ∗ ∗ ∗ γ 1     > 0 . 4. [ 166 , 167 ] There exist Q ∈ S n and γ ∈ R > 0 , wh ere Q > 0 , such that     Q QA d QB d 0 ∗ Q 0 C T d ∗ ∗ γ 1 D T d ∗ ∗ ∗ γ 1     > 0 . 5. [ 70 ] There exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that     P − 1 A d B d 0 ∗ P 0 C T d ∗ ∗ γ 1 D T d ∗ ∗ ∗ γ 1     > 0 . (4.14) 6. [ 165 ] There exist P ∈ S n , X ∈ R n × n , and γ ∈ R > 0 , where P > 0 and X has full rank, such that     P A d X B d 0 ∗ X T P − 1 X 0 XC T d ∗ ∗ 1 D T d ∗ ∗ ∗ γ 2 1     > 0 . 7. There exist P ∈ S n , X ∈ R n × n , and γ ∈ R > 0 , wh ere P > 0 and X has full rank, such that     X T P − 1 X XA d XB d 0 ∗ P 0 C T d ∗ ∗ 1 D T d ∗ ∗ ∗ γ 2 1     > 0 . (4.15) Pr oof. Apply the congruence transformatio n W = diag { X T , 1 , 1 , 1 } to ( 4.14 ), where W h as full rank since X has ful l rank. 8. [ 165 , 168 ] There exist P ∈ S n , X ∈ R n × n , and γ ∈ R > 0 , wh ere P > 0 , such that     P A d X B d 0 ∗ X + X T − P 0 XC T d ∗ ∗ 1 D T d ∗ ∗ ∗ γ 2 1     > 0 . (4.16) 59 9. There exist Q ∈ S n , X ∈ R n × n , and γ ∈ R > 0 , wh ere Q > 0 , such that     X + X T − Q XA d XB d 0 ∗ Q 0 C T d ∗ ∗ 1 D T d ∗ ∗ ∗ γ 2 1     > 0 . (4.17) Pr oof. Same as the proof of ( 4.16 ) in [ 165 ], by which it is shown th at ( 4.17 ) is equivalent to ( 4.15 ). 10. [ 169 , p p . 48–49] There exist P ∈ S n , F 1 , F 2 ∈ R n × n , and γ ∈ R > 0 , where P > 0 , such that     − P + A d F 1 + F T 1 A T d A d F 2 − F T 1 F T 1 C T d B d ∗ P −  F 2 + F T 2  F T 2 C T d 0 ∗ ∗ − γ 1 D d ∗ ∗ ∗ − γ 1     < 0 . 11. [ 169 , pp. 48–49] There exist P ∈ S n , F 1 , F 2 , X 1 , X 2 , X 3 ∈ R n × n , and γ ∈ R > 0 , where P > 0 , such that       − P + X 1 F 1 + F T 1 X T 1 X 1 F 2 + F T 1 X T 2 A d − X 1 + F T 1 X T 3 B d 0 ∗ P + X 2 F 2 + F T 2 X T 2 − 1 − X 2 + F T 2 X T 3 0 0 ∗ ∗ −  X 3 + X T 3  0 C T d ∗ ∗ ∗ − γ 1 D T d ∗ ∗ ∗ ∗ − γ 1       < 0 . 12. [ 169 , pp. 48– 4 9 ] There exist P ∈ S n , Y 1 , Y 2 , Y 3 , X 1 , X 2 , X 3 ∈ R n × n , and γ ∈ R > 0 , where P > 0 , such that       − P + X 1 Y 1 + Y T 1 X T 1 X 1 Y 2 + Y T 1 X T 2 A d + X 1 Y 3 + Y T 1 X T 3 B d 0 ∗ P + X 2 Y 2 + Y T 2 X T 2 − 1 + X 2 Y 3 + Y T 2 X T 3 0 0 ∗ ∗ X 3 Y 3 + Y T 3 X T 3 0 C T d ∗ ∗ ∗ − γ 1 D T d ∗ ∗ ∗ ∗ − γ 1       < 0 . The H ∞ norm of G is t h e mini mum value of γ ∈ R > 0 that s atisﬁes any of the above con- ditions. If ( A d , B d , C d , D d ) is a mini mal realization, then the matrix inequalities can be non- strict [ 163 ], [ 170 ]. 4.2.3 Descriptor System Bounded Real Lemma Consider a descriptor s y s tem, G : L 2 e → L 2 e , described by E ˙ x = Ax + Bu , y = Cx , 60 where E , A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and it i s assumed that the system is regular . The H ∞ norm of G is k G k ∞ = sup u ∈L 2 , u 6 = 0 k G u k 2 k u k 2 . The descriptor sys tem i s admissible and t h e inequality k G k ∞ < γ holds und er any of the following equiv alent necessary and sufﬁcient cond itions. 1. [ 148 ] There exist X ∈ R n × n and γ ∈ R > 0 , s u ch that E T X = X T E ≥ 0 and  X T A + A T X + C T C X T B ∗ − γ 2 1  < 0 . 2. [ 148 , 171 ] There exist Y ∈ R n × n and γ ∈ R > 0 , s u ch that YE T = EY T ≥ 0 and  A Y T + Y A T + BB T YC T ∗ − γ 1  < 0 . 3. [ 148 ] There exist X ∈ R n × n and γ ∈ R > 0 , s u ch that E T X = X T E ≥ 0 and   X T A + A T X X T B C T ∗ − γ 1 0 ∗ ∗ − γ 1   < 0 . 4. [ 148 ] There exist Y ∈ R n × n and γ ∈ R > 0 , s u ch that YE T = EY T ≥ 0 and   A Y T + Y A T YC T B ∗ − γ 1 0 ∗ ∗ − γ 1   < 0 . 5. [ 150 ] There exist P ∈ S n , X ∈ R ( n − n e ) × n , Z ∈ R n × ( n − n e ) , and γ ∈ R > 0 , where n e = rank ( E ) and P > 0 , satisfying E T Z = 0 and  A T ( PE + ZX ) + ( PE + ZX ) T A + C T C C ( PE + ZX ) T B ∗ − γ 2 1  < 0 . 6. [ 171 ] Th ere exist P ∈ S n , S ∈ R ( n − n e ) × ( n − n e , U , V ∈ R n × ( n − n e ) , and γ ∈ R > 0 , where n e = rank ( E ) , R ( U ) = N ( E T ) , R ( V ) = N ( E ) , and P > 0 , satisfying  A  PE T + VSU T  +  PE T + VSU T  T A T + BB T  PE T + VSU T  T C T ∗ − γ 2 1  < 0 . 7. [ 150 ] There exist P ∈ S n , X ∈ R ( n − n e ) × n , Z ∈ R n × ( n − n e ) , and γ ∈ R > 0 , where n e = rank ( E ) and P > 0 , satisfying EZ = 0 and  A ( PE + ZX ) + ( PE + ZX ) T A T + BB T ( PE + ZX ) T C T ∗ − γ 2 1  < 0 . 61 8. [ 150 ] There exist P ∈ S n , X ∈ R ( n − n e ) × n , Z ∈ R ( n + m ) × ( n − n e ) , F , G ∈ R ( n + m ) × ( n + m ) , and γ ∈ R > 0 , wh ere n e = rank ( E ) and P > 0 , satisfying ¯ E T Z = 0 and  ¯ A T G T + G ¯ A + ¯ C T ¯ C  ¯ P ¯ E + Z ¯ X  T + ¯ A T F T − G ∗ −  F + F T   < 0 , where ¯ A =  A B 0 − 1  , ¯ E =  E 0 0 1 2 γ 2 1  , ¯ C =  C 0  , ¯ P =  P 0 0 1  , ¯ X =  X 0  . 9. [ 150 ] There exist P ∈ S n , X ∈ R ( n − n e ) × n , Z ∈ R ( n + p ) × ( n − n e ) , F , G ∈ R ( n + p ) × ( n + p ) , and γ ∈ R > 0 , wh ere n e = rank ( E ) and P > 0 , satisfying ¯ EZ = 0 and " ¯ AG + G T ¯ A T + ¯ B ¯ B T  ¯ P ¯ E T + Z ¯ X  T + ¯ AF − G T ∗ −  F + F T  # < 0 , where ¯ A =  A 0 C − 1  , ¯ E =  E 0 0 1 2 γ 2 1  , ¯ B =  B 0  , ¯ P =  P 0 0 1  , ¯ X =  X 0  . 4.2.4 Discrete- Time Descriptor System Bound ed Real Lemma Consider a discrete-time descriptor system, G : ℓ 2 e → ℓ 2 e , describ ed by E d x k +1 = A d x k + B d u k , y k = C d x k + D d u k , where E d , A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , and D d ∈ R p × m . The H ∞ norm of G is k G k ∞ = sup u ∈ ℓ 2 , u 6 = 0 k G u k 2 k u k 2 . The descriptor sys tem i s admissible and t h e inequality k G k ∞ < γ holds und er any of the following equiv alent necessary and sufﬁcient cond itions. 1. [ 152 ] There exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that E T d PE d ≥ 0 and   A T d P A d − E T d PE d A T d PB d C T d ∗ B T d PB d − γ 1 D T d ∗ ∗ − γ 1   < 0 . 2. [ 172 ] There exist Q ∈ S n and γ ∈ R > 0 , wh ere Q > 0 , s uch that E d QE T d ≥ 0 and   A d QA T d − E d QE T d A d QC T d B d ∗ C d QC T d − γ 1 D d ∗ ∗ − γ 1   < 0 . 62 3. [ 154 , 155 ] There exist P ∈ S n , X ∈ S ( n − n e ) , Z ∈ R n × ( n − n e ) , and γ ∈ R > 0 , where n e = rank ( E d ) and P > 0 , satis fyi ng E T d Z = 0 and  A T d  P − ZXZ T  A d − E T d PE d + C T d C d A T d  P − ZXZ T  B d + C T d D d ∗ B T d  P − ZXZ T  B d − γ 2 1 + D T d D d  < 0 . 4. [ 155 ] There exist P ∈ S n , X ∈ S ( n − n e ) , Z ∈ R ( n + p ) × ( n + p − m − n e ) , F ∈ R ( n + p ) × ( n + p ) , G ∈ R ( n + m ) × ( n + p ) , and γ ∈ R > 0 , where m ≤ p , n e = rank ( E d ) and P > 0 , such th at ¯ E T Z = 0 and  ¯ E T ¯ P ¯ E + G ¯ A + ¯ A T G T − G + ¯ A T F T ∗ ¯ P − Z ¯ XZ T −  F + F T   < 0 , where ¯ A =  A d B d C d D d  , ¯ E =   E d 0 0 γ  1 m × m 0 p × m    , ¯ P =  P 0 0 1  , ¯ X =  X 0 0 0  . 5. [ 155 ] There exist P ∈ S n , X ∈ S ( n − n e ) , Z ∈ R ( n + m ) × ( n − n e ) , F ∈ R ( n + m ) × ( n + m ) , G ∈ R ( n + p ) × ( n + m ) , and γ ∈ R > 0 , where m ≤ p , n e = rank ( E d ) and P > 0 , such that ¯ EZ = 0 and  ¯ E ¯ P ¯ E T + G ¯ A T + ¯ AG T − G + ¯ AF T ∗ ¯ P − ZXZ T −  F + F T   < 0 , where ¯ A =  A d B d C d D d  , ¯ E =  E d 0 0 γ 1  , ¯ P =  P 0 0 1  . 4.3 H 2 Norm 4.3.1 Continuous-T ime H 2 Norm Consider a conti nuous-time L TI system, G : L 2 e → L 2 e , wi th state-space realization ( A , B , C , 0 ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and A is Hurwitz. Th e H 2 norm of G is k G k 2 = p tr( CWC T ) = p tr( B T MB ) , where W , M ∈ S n , W > 0 , M > 0 , and A W + W A T + BB T = 0 , MA + A T M + C T C = 0 . The inequality k G k 2 < µ ho l ds un d er any of the foll owing equiva lent necessary and sufﬁcient conditions. 1. [ 3 , p . 77] There exist X ∈ S n and µ ∈ R > 0 , wh ere X > 0 , such that AX + XA T + BB T < 0 , tr  CXC T  < µ 2 . 63 2. [ 3 , p. 77] There exist Y ∈ S n and µ ∈ R > 0 , wh ere Y > 0 , such that A T Y + Y A + C T C < 0 , tr  B T YB  < µ 2 . 3. [ 3 , p. 77], [ 73 ] There exist Y ∈ S n , Z ∈ S p , and µ ∈ R > 0 , w h ere Y > 0 and Z > 0 , such t h at  A T Y + Y A YB ∗ − µ 1  < 0 ,  Y C T ∗ Z  > 0 , tr( Z ) < µ. 4. [ 3 , p. 77] T h ere exist X ∈ S n , Z ∈ S p , and µ ∈ R > 0 , wh ere X > 0 and Z > 0 , such that  XA T + AX XC T ∗ − µ 1  < 0 ,  X B ∗ Z  > 0 , tr( Z ) < µ. 5. [ 173 ] There exist Y ∈ S n , Z ∈ S m , F , G ∈ R n × n , and µ ∈ R > 0 , where Y > 0 and Z > 0 , such that   F + F T G − F T + Y A 0 ∗ − ( G + G T ) C T ∗ ∗ − 1   < 0 ,  Y YB ∗ Z  > 0 , tr( Z ) < µ 2 . 6. [ 173 ] There exist X ∈ S n , Z ∈ S m , F , G ∈ R n × n , and µ ∈ R > 0 , where X > 0 and Z > 0 , such that   F + F T G − F T + XA T 0 ∗ − ( G + G T ) B ∗ ∗ − 1   < 0 ,  X XC T ∗ Z  > 0 , tr( Z ) < µ 2 . 64 7. [ 173 ] There exist X ∈ S n , Z ∈ S m , F , G ∈ R n × n , and µ ∈ R > 0 , where X > 0 and Z > 0 , such that   F + F T G − F T + AX 0 ∗ − ( G + G T ) XC T ∗ ∗ − 1   < 0 ,  X B ∗ Z  > 0 , tr( Z ) < µ 2 . 8. [ 173 ] There exist Y ∈ S n , Z ∈ S m , F , G ∈ R n × n , and µ ∈ R > 0 , where Y > 0 and Z > 0 , such that   F + F T G − F T + A T Y 0 ∗ − ( G + G T ) XB ∗ ∗ − 1   < 0 ,  Y C T ∗ Z  > 0 , tr( Z ) < µ 2 . 9. [ 173 ] There exist X ∈ S n , Z ∈ S m , F , G ∈ R n × n , and µ ∈ R > 0 , where X > 0 and Z > 0 , such that   AF + F T A T X − F T + A G F T C T ∗ − ( G + G T ) G T C T ∗ ∗ − 1   < 0 ,  X B ∗ Z  > 0 , tr( Z ) < µ 2 . 10. [ 173 ] There exist Y ∈ S n , Z ∈ S m , F , G ∈ R n × n , and µ ∈ R > 0 , where Y > 0 and Z > 0 , such that   A T F + F T A Y − F T + A T G F T B ∗ − ( G + G T ) G T B ∗ ∗ − 1   < 0 ,  Y C T ∗ Z  > 0 , tr( Z ) < µ 2 . 11. [ 73 ] There exist X ∈ S n , Z ∈ S p , V ∈ R n × n , and µ ∈ R > 0 , where X > 0 and Z > 0 , such 65 that     −  V + V T  V T A + X V T B V T ∗ − X 0 0 ∗ ∗ − µ 2 1 0 ∗ ∗ ∗ − X     < 0 , (4.18)  X C T ∗ Z  > 0 , tr( Z ) < 1 . 12. [ 73 ] There exist X ∈ S n , Z ∈ S m , V ∈ R n × n , and µ ∈ R > 0 , where X > 0 and Z > 0 , such that     −  V + V T  V T A T + X V T C T V T ∗ − X 0 0 ∗ ∗ − µ 2 1 0 ∗ ∗ ∗ − X     < 0 , ( 4.19 )  X B ∗ Z  > 0 , tr( Z ) < 1 . 13. [ 92 ] There exist X ∈ S n , Z ∈ S m , Γ ∈ R n × n , and µ ∈ R > 0 , where X > 0 and Z > 0 , such that   0 − X 0 ∗ 0 0 ∗ ∗ − 1   + He (   A 1 C   Γ  1 − ǫ 1 0  ) < 0 ,  Z B T ∗ X  > 0 , tr( Z ) < µ 2 . The H 2 norm of G is the m inimum value of µ ∈ R > 0 that satisﬁes any of the above condi tions. 4.3.2 Discrete- Time H 2 Norm Without Feedthr ough Consider a dis crete-time L TI system, G : ℓ 2 e → ℓ 2 e , wi th state-space realization ( A d , B d , C d , 0 ) , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , and A d is Schur . The H 2 norm of G is k G k 2 = q tr( C d WC T d ) = q tr( B T d MB d ) , where W , M ∈ S n , W > 0 , M > 0 , and A d W A T d − W + B d B T d = 0 , A T d MA d − M + C T d C d = 0 . The inequalit y k G k 2 < µ holds under any of following equivalent necessary and sufﬁcient condi - tions. 66 1. There exist P ∈ S n and µ ∈ R > 0 , wh ere P > 0 , such that A d P A T d − P + B d B T d < 0 , tr  C d PC T d  < µ 2 . 2. There exist Q ∈ S n and µ ∈ R > 0 , wh ere Q > 0 , such that A T d QA d − Q + C T d C d < 0 , tr  B T d QB d  < µ 2 . 3. [ 165 ] There exist P ∈ S n , Z ∈ S p , and µ ∈ R > 0 , wh ere P > 0 and Z > 0 , such that   P A d P B d ∗ P 0 ∗ ∗ 1   > 0 , (4.20)  Z C d P ∗ P  > 0 , (4.21) tr( Z ) < µ 2 . 4. There exist Q ∈ S n , Z ∈ S m , and µ ∈ R > 0 , where Q > 0 and Z > 0 , such that   Q A T d Q C T d ∗ Q 0 ∗ ∗ 1   > 0 , (4.22)  Z B T d Q ∗ Q  > 0 , (4.23) tr( Z ) < µ 2 . 5. There exist Q ∈ S n , Z ∈ S p , and µ ∈ R > 0 , where Q > 0 and Z > 0 , such that   Q QA d QB d ∗ Q 0 ∗ ∗ 1   > 0 , (4.24)  Z C d ∗ Q  > 0 , tr( Z ) < µ 2 . Pr oof. Apply the congruence t ransformation W 1 = diag { Q , Q , 1 } t o ( 4.20 ) and W 2 = diag { 1 , Q } t o ( 4.21 ), where Q = P − 1 . 6. There exist P ∈ S n , Z ∈ S m , and µ ∈ R > 0 , wh ere P > 0 and Z > 0 , such that   P P A T d PC T d ∗ P 0 ∗ ∗ 1   > 0 , (4.25)  Z B T d ∗ P  > 0 , tr( Z ) < µ 2 . 67 7. [ 165 ] There exist P ∈ S n , Z ∈ S p , X ∈ R n × n , and µ ∈ R > 0 , where P > 0 , Z > 0 , and X has full rank, such that   P A d X B d ∗ X T P − 1 X 0 ∗ ∗ 1   > 0 ,  Z C d X ∗ X T P − 1 X  > 0 , tr( Z ) < µ 2 . 8. There exist Q ∈ S n , Z ∈ S m , X ∈ R n × n , and µ ∈ R > 0 , where Q > 0 and Z > 0 , and X has full rank, such that   Q A T d X C T d ∗ X T Q − 1 X 0 ∗ ∗ 1   > 0 ,  Z B T d X ∗ X T Q − 1 X  > 0 , tr( Z ) < µ 2 . Pr oof. Apply the cong ruence transformati on W 1 = diag { 1 , X T Q − 1 , 1 } to ( 4.22 ) and the congruence transformation W 2 = diag { 1 , X T Q − 1 } to ( 4.23 ), where W 1 and W 2 hav e full rank si nce X h as full rank. 9. There exist Q ∈ S n , Z ∈ S p , X ∈ R n × n , and µ ∈ R > 0 , where Q > 0 , Z > 0 , and X has full rank, su ch that   X T Q − 1 X X T A d X T B d ∗ Q 0 ∗ ∗ 1   > 0 , (4.26)  Z C d ∗ Q  > 0 , tr( Z ) < µ 2 . Pr oof. Apply the congruence transformation W = diag { X T Q − 1 , 1 , 1 } to ( 4.24 ), where W has ful l rank since X has full rank. 10. There exist P ∈ S n , Z ∈ S m , X ∈ R n × n , and µ ∈ R > 0 , where P > 0 and Z > 0 , and X has 68 full rank, such that   X T P − 1 X X T A T d X T C T d ∗ P 0 ∗ ∗ 1   > 0 , (4.27)  Z B T d ∗ P  > 0 , tr( Z ) < µ 2 . Pr oof. Apply the cong ruence transformatio n W = diag { X T P − 1 , 1 , 1 } to ( 4.25 ), where W has ful l rank since X has full rank. 11. [ 165 ] There exist P ∈ S n , Z ∈ S p , X ∈ R n × n , and µ ∈ R > 0 , where P > 0 and Z > 0 , such that   P A d X B d ∗ X + X T − P 0 ∗ ∗ 1   > 0 , (4.28)  Z C d X ∗ X + X T − P  > 0 , (4.29) tr( Z ) < µ 2 . (4.30) 12. There exist Q ∈ S n , Z ∈ S m , X ∈ R n × n , and µ ∈ R > 0 , where Q > 0 and Z > 0 , and X has full rank, such that   Q A T d X C T d ∗ X + X T − Q 0 ∗ ∗ 1   > 0 ,  Z B T d X ∗ X + X T − Q  > 0 , tr( Z ) < µ 2 . Pr oof. Same as the proof of ( 4.28 ), ( 4.29 ), ( 4.30 ) in [ 165 ]. 13. There exist Q ∈ S n , Z ∈ S p , X ∈ R n × n , and µ ∈ R > 0 , wh ere Q > 0 and Z > 0 , such that   X + X T − Q X T A d X T B d ∗ Q 0 ∗ ∗ 1   > 0 , (4.31)  Z C d ∗ Q  > 0 , tr( Z ) < µ 2 . 69 Pr oof. Same as t h e proof of ( 4.2 8 ), ( 4.2 9 ), ( 4.30 ) in [ 165 ], by which it i s s hown that ( 4.31 ) is equiva lent to ( 4.26 ). 14. There exist P ∈ S n , Z ∈ S m , X ∈ R n × n , and µ ∈ R > 0 , where P > 0 and Z > 0 , and X has full rank, such that   X + X T − P X T A T d X T C T d ∗ P 0 ∗ ∗ 1   > 0 , (4.32)  Z B T d ∗ P  > 0 , tr( Z ) < µ 2 . Pr oof. Same as t h e proof of ( 4.28 ), ( 4.29 ), ( 4.30 ) in [ 165 ], by which it is shown that ( 4.32 ) is equiva lent to ( 4.27 ). 15. [ 169 , pp. 53–54] There exist P ∈ S n , Z ∈ S p , F 1 , F 2 , F 5 ∈ R n × n , F 4 ∈ R n × p , and µ ∈ R > 0 , where P > 0 and Z > 0 , such that   − P + A d F 1 + F T 1 A T d A d F 2 − F T 1 B d ∗ P −  F 2 + F T 2  0 ∗ ∗ − γ 1   < 0 ,  − Z + C d F 4 + F T 4 C T d C d F 5 − F T 4 ∗ P −  F 5 + F T 5   < 0 , tr( Z ) < µ 2 . 16. [ 169 , pp. 53–54] There exist P ∈ S n , Z ∈ S p , F 1 , F 2 , F 5 , X 1 , X 2 , X 3 , X 5 , X 6 ∈ R n × n , F 4 ∈ R n × p , X 4 ∈ R p × n , and µ ∈ R > 0 , wh ere P > 0 and Z > 0 , such that     − P + X 1 F 1 + F T 1 X T 1 X 1 F 2 + F T 1 X T 2 A d − X 1 + F T 1 X T 3 B d ∗ P + X 2 F 2 + F T 2 X T 2 − 1 − X 2 + F T 2 X T 3 0 ∗ ∗ −  X 3 + X T 3  0 ∗ ∗ ∗ − 1     < 0 ,   − Z + X 4 F 4 + F T 4 X T 4 X 4 F 5 + F T 4 X T 5 C d − X 4 + F T 4 X T 6 ∗ P + X 5 F 5 + F T 5 X T 5 − 1 − X 5 + F T 5 X T 6 ∗ ∗ −  X 6 + X T 6    < 0 , tr( Z ) < µ 2 . 17. [ 169 , pp. 53– 5 4] There exist P ∈ S n , Z ∈ S p , Y 1 , Y 2 , Y 3 , Y 5 , Y 6 , X 1 , X 2 , X 3 , X 5 , X 6 ∈ R n × n , 70 Y 4 ∈ R n × p , X 4 ∈ R p × n , and µ ∈ R > 0 , wh ere P > 0 and Z > 0 , such that     − P + X 1 Y 1 + Y T 1 X T 1 X 1 Y 2 + Y T 1 X T 2 A d + X 1 Y 3 + Y T 1 X T 3 B d ∗ P + X 2 Y 2 + Y T 2 X T 2 − 1 + X 2 Y 3 + Y T 2 X T 3 0 ∗ ∗ X 3 Y 3 + Y T 3 X T 3 0 ∗ ∗ ∗ − 1     < 0 ,   − Z + X 4 Y 4 + Y T 4 X T 4 X 4 Y 5 + Y T 4 X T 5 C d + X 4 Y 6 + Y T 4 X T 6 ∗ P + X 5 Y 5 + Y T 5 X T 5 − 1 + X 5 Y 6 + Y T 5 X T 6 ∗ ∗ X 6 Y 6 + Y T 6 X T 6   < 0 , tr( Z ) < µ 2 . The H 2 norm of G is the m inimum value of µ ∈ R > 0 that satisﬁes any of the above condi tions. 4.3.3 Discrete- Time H 2 Norm With Feedthr ough Consider a dis crete-time L TI system, G : ℓ 2 e → ℓ 2 e , wi th state-space realization ( A d , B d , C d , D d ) , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , D d ∈ R p × m , and A d is Schur . The H 2 norm of G is k G k 2 = q tr( C d WC T d + D d D T d ) = q tr( B T d MB d + D T d D d ) , where W , M ∈ S n , W > 0 , M > 0 , and A d W A T d − W + B d B T d = 0 , A T d MA d − M + C T d C d = 0 . The inequalit y k G k 2 < µ holds under any of following equivalent necessary and sufﬁcient condi - tions. 1. [ 174 ] There exist Q ∈ S n and µ ∈ R > 0 , wh ere Q > 0 , s uch that A T d QA d − Q + C T d C d < 0 , tr  B T d QB d + D T d D d  < µ 2 . 2. [ 175 ] There exist P ∈ S n and µ ∈ R > 0 , wh ere P > 0 , such that A d P A T d − P + B d B T d < 0 , tr  C d PC T d + D d D T d  < µ 2 . 3. [ 175 ] There exist Q ∈ S n , Z ∈ S m , and µ ∈ R > 0 , wh ere Q > 0 and Z > 0 , such that  QA T d QA d C T d ∗ 1  > 0 , (4.33)  Z − D T d D d B T d Q ∗ Q  > 0 , (4.34) tr( Z ) < µ 2 . 71 4. [ 175 ] There exist P ∈ S n , Z ∈ S p , and µ ∈ R > 0 , wh ere P > 0 and Z > 0 , such that  P − A d P A T d B d ∗ 1  > 0 , (4.35)  Z − D d D T d C d P ∗ P  > 0 , (4.36) tr( Z ) < µ 2 . 5. [ 176 , p. 25] There exist Q ∈ S n , Z ∈ S m , and µ ∈ R > 0 , wh ere Q > 0 and Z > 0 , su ch that   Q A d Q C T d ∗ Q 0 ∗ ∗ 1   > 0 , (4.37)   Z B T d Q D T d ∗ Q 0 ∗ ∗ 1   > 0 , (4.38) tr( Z ) < µ 2 . (4.3 9 ) Pr oof. Applyin g t h e Schur complement to ( 4.33 ) and ( 4.34 ) yields ( 4.37 ) and ( 4.38 ). 6. [ 176 , p. 26 ] There exist P ∈ S n , Z ∈ S p , and µ ∈ R > 0 , wh ere P > 0 and Z > 0 , such that   P A T d P B d ∗ P 0 ∗ ∗ 1   > 0 , (4.40)   Z C d P D d ∗ P 0 ∗ ∗ 1   > 0 , (4.41) tr( Z ) < µ 2 . (4.42) Pr oof. Applyin g t h e Schur complement to ( 4.35 ) and ( 4.36 ) yields ( 4.40 ) and ( 4.41 ). 7. There exist P ∈ S n , Z ∈ S m , and µ ∈ R > 0 , wh ere P > 0 and Z > 0 , such that   P P A d PC T d ∗ P 0 ∗ ∗ 1   > 0 ,   Z B T d D T d ∗ P 0 ∗ ∗ 1   > 0 , tr( Z ) < µ 2 . Pr oof. Apply the congruence transform ation W 1 = dia g { P , P , 1 } to ( 4.37 ) and W 2 = diag { 1 , P , 1 } to ( 4.3 8 ), where P = Q − 1 . 72 8. [ 177 ] There exist Q ∈ S n , Z ∈ S p , and µ ∈ R > 0 , wh ere Q > 0 and Z > 0 , su ch that   Q Q A T d QB d ∗ Q 0 ∗ ∗ 1   > 0 ,   Z C d D d ∗ Q 0 ∗ ∗ 1   > 0 , tr( Z ) < µ 2 . 9. [ 175 ] There exist P ∈ S n , Z ∈ S p , X ∈ R n × n , and µ ∈ R > 0 , where P > 0 and Z > 0 , such that   P A d X B d ∗ X + X T − P 0 ∗ ∗ 1   > 0 ,   Z C d X D d ∗ X + X T − P 0 ∗ ∗ 1   > 0 , tr( Z ) < µ 2 . 10. [ 176 , pp. 26– 2 7 ] There exist Q ∈ S n , Z ∈ S m , X ∈ R n × n , and µ ∈ R > 0 , where Q > 0 and Z > 0 , such that   P A T d X C T d ∗ X + X T − P 0 ∗ ∗ 1   > 0 ,   Z B T d X D T d ∗ X + X T − P 0 ∗ ∗ 1   > 0 , tr( Z ) < µ 2 . 11. [ 169 , pp. 53– 5 4] There exist P ∈ S n , Z ∈ S p , F 1 , F 2 , F 5 ∈ R n × n , F 4 ∈ R n × p , and µ ∈ R > 0 , where P > 0 and Z > 0 , such that   − P + A d F 1 + F T 1 A T d A d F 2 − F T 1 B d ∗ P −  F 2 + F T 2  0 ∗ ∗ − γ 1   < 0 ,   − Z + C d F 4 + F T 4 C T d C d F 5 − F T 4 D d ∗ P −  F 5 + F T 5  0 ∗ ∗ − 1   < 0 , tr( Z ) < µ 2 . 73 12. [ 169 , pp. 53–54] There exist P ∈ S n , Z ∈ S p , F 1 , F 2 , F 5 , X 1 , X 2 , X 3 , X 5 , X 6 ∈ R n × n , F 4 ∈ R n × p , X 4 ∈ R p × n , and µ ∈ R > 0 , wh ere P > 0 and Z > 0 , such that     − P + X 1 F 1 + F T 1 X T 1 X 1 F 2 + F T 1 X T 2 A d − X 1 + F T 1 X T 3 B d ∗ P + X 2 F 2 + F T 2 X T 2 − 1 − X 2 + F T 2 X T 3 0 ∗ ∗ −  X 3 + X T 3  0 ∗ ∗ ∗ − 1     < 0 ,     − Z + X 4 F 4 + F T 4 X T 4 X 4 F 5 + F T 4 X T 5 C d − X 4 + F T 4 X T 6 D d ∗ P + X 5 F 5 + F T 5 X T 5 − 1 − X 5 + F T 5 X T 6 0 ∗ ∗ −  X 6 + X T 6  0 ∗ ∗ ∗ − 1     < 0 , tr( Z ) < µ 2 . 13. [ 169 , pp. 53– 5 4] There exist P ∈ S n , Z ∈ S p , Y 1 , Y 2 , Y 3 , Y 5 , Y 6 , X 1 , X 2 , X 3 , X 5 , X 6 ∈ R n × n , Y 4 ∈ R n × p , X 4 ∈ R p × n , and µ ∈ R > 0 , wh ere P > 0 and Z > 0 , such that     − P + X 1 Y 1 + Y T 1 X T 1 X 1 Y 2 + Y T 1 X T 2 A d + X 1 Y 3 + Y T 1 X T 3 B d ∗ P + X 2 Y 2 + Y T 2 X T 2 − 1 + X 2 Y 3 + Y T 2 X T 3 0 ∗ ∗ X 3 Y 3 + Y T 3 X T 3 0 ∗ ∗ ∗ − 1     < 0 ,     − Z + X 4 Y 4 + Y T 4 X T 4 X 4 Y 5 + Y T 4 X T 5 C d + X 4 Y 6 + Y T 4 X T 6 D d ∗ P + X 5 Y 5 + Y T 5 X T 5 − 1 + X 5 Y 6 + Y T 5 X T 6 0 ∗ ∗ X 6 Y 6 + Y T 6 X T 6 0 ∗ ∗ ∗ − 1     < 0 , tr( Z ) < µ 2 . The H 2 norm of G is the m inimum value of µ ∈ R > 0 that satisﬁes any of the above condi tions. 4.3.4 Descriptor System H 2 Norm Consider a descriptor s y s tem, G : L 2 e → L 2 e , described by E ˙ x = Ax + Bu , y = Cx , where E , A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and it is assumed that the syst em is regular . The H 2 norm of G is [ 178 , 179 ] k G k 2 = r tr  ˆ CEW ˆ C T  = r tr  ˆ B T ME ˆ B  , where ˆ C ∈ R p × n , ˆ B ∈ R n × m , W , M ∈ R n × n , C = ˆ CE , B = E ˆ B , W E T = EW T > 0 , E T M = M T E > 0 , and A W T + W A T + BB T = 0 , M T A + A T M + C T C = 0 . The descriptor system is admissible and the inequality k G k 2 < µ h o lds under any of the fol- lowing equiv alent necessary and s u f ﬁcient cond itions. 74 1. [ 180 ] The descriptor state-space m atrices sati s fy R ( B ) ⊆ R ( E ) and there exist Q ∈ S n , S ∈ R ( n − n e ) × ( n − n e , U , V ∈ R n × ( n − n e ) , and µ ∈ R > 0 , w h ere n e = rank ( E ) , R ( U ) = N ( E T ) , R ( V ) = N ( E ) , and Q > 0 , satisfyi ng A T  QE + USV T  +  QE + USV T  T A + C T C < 0 , tr  B T QB  < µ 2 . 2. [ 180 ] The descriptor state-space matrices satisfy N ( E ) ⊆ N ( C ) and th ere exist P ∈ S n , S ∈ R ( n − n e ) × ( n − n e , U , V ∈ R n × ( n − n e ) , and µ ∈ R > 0 , w h ere n e = rank ( E ) , R ( U ) = N ( E T ) , R ( V ) = N ( E ) , and P > 0 , satisfying A  PE T + VSU T  +  PE T + VSU T  T A T + BB T < 0 , tr  CPC T  < µ 2 . 3. The descripto r state-space matrices satisfy R ( B ) ⊆ R ( E ) and there exist Q ∈ S n , X ∈ R ( n − n e ) × n , Z ∈ R n × ( n − n e ) , and µ ∈ R > 0 , where n e = rank ( E ) and Q > 0 , sat i sfying E T Z = 0 and A T ( QE + ZX ) + ( QE + ZX ) T A + C T C < 0 , tr  B T QB  < µ 2 . 4. [ 181 ] The descriptor state-space matrices satisfy N ( E ) ⊆ N ( C ) and th ere exist P ∈ S n , X ∈ R ( n − n e ) × n , Z ∈ R n × ( n − n e ) , and µ ∈ R > 0 , where n e = rank ( E ) and P > 0 , satisfying EZ = 0 and A  PE T + ZX  +  PE T + ZX  T A T + BB T < 0 , tr  CPC T  < µ 2 . 4.3.5 Discrete- Time Descriptor System H 2 Norm Consider a discrete-time descriptor system, G : ℓ 2 e → ℓ 2 e , describ ed by E d x k +1 = A d x k + B d u k , y k = C d x k + D d u k , where E d , A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , and D d ∈ R p × m . The H 2 norm o f G is [ 182 , pp. 87–88 ], [ 183 ] k G k 2 = q tr  C d WC T d + D d D T d  = q tr  B T d MB d + D T d D d  , where W , M ∈ R n × n , W > 0 , M > 0 , A d W A T d − E d WE T d + B d B T d = 0 , A T d MA d − E T d ME d + C T d C d = 0 . The descriptor system is admissible and the inequality k G k 2 < µ h o lds under any of the fol- lowing equiv alent necessary and s u f ﬁcient cond itions. 75 1. [ 184 ] There exist Q ∈ S n , Z ∈ S m , and γ ∈ R > 0 , wh ere Q > 0 , s uch that E T d QE d ≥ 0 , A T d QA d − E T d QE d + C T d C d < 0 , (4.43) B T d QB d + D T d D d − Z < 0 , (4.44) tr( Z ) < µ 2 . Note that in [ 184 ], ( 4.43 ) is missing the − E T d PE d term. Pr oof. The proof follows from the deﬁnition of t he H 2 norm using an approach similar to that in [ 2 , p p. 201-211, Proposition 6.13], where tr  B T d QB d + D T d D d  < µ 2 is equiva lent to B T d QB d + D T d D d − Z < 0 and tr( Z ) < µ 2 . 2. There exist P ∈ S n , Z ∈ S p , and γ ∈ R > 0 , wh ere P > 0 , such that E d PE T d ≥ 0 , A d s P A T d − E d PE T d + B d B T d < 0 , (4.45) C d PC T d + D d D T d − Z < 0 , (4.46) tr( Z ) < µ 2 . Pr oof. The proof follows from the deﬁnition of t he H 2 norm using an approach similar to that in [ 2 , p p. 20 1 -211, Propositi o n 6. 1 3], where tr  C d PC T d + D d D T d  < µ 2 is equiva lent to C d PC T d + D d D T d − Z < 0 and tr( Z ) < µ 2 . 3. There exist Q ∈ S n , Z ∈ S m , and γ ∈ R > 0 , wh ere Q > 0 , such that E T d QE d ≥ 0 ,   E T d QE d A d Q C T d ∗ Q 0 ∗ ∗ 1   > 0 , (4. 4 7)   Z B T d Q D T d ∗ Q 0 ∗ ∗ 1   > 0 , (4. 4 8) tr( Z ) < µ 2 . Pr oof. Applyin g t h e Schur complement to ( 4.43 ) and ( 4.44 ) yields ( 4.47 ) and ( 4.48 ). 4. There exist P ∈ S n , Z ∈ S o , and γ ∈ R > 0 , wh ere P > 0 , such that E d PE T d ≥ 0 ,   E d PE T d A T d P B d ∗ P 0 ∗ ∗ 1   > 0 , (4.49)   Z C d P D d ∗ P 0 ∗ ∗ 1   > 0 , (4.50) tr( Z ) < µ 2 . Pr oof. Applyin g t h e Schur complement to ( 4.45 ) and ( 4.46 ) yields ( 4.49 ) and ( 4.50 ). 76 4.4 Generalized H 2 Norm (Induced L 2 - L ∞ Norm) Consider a conti nuous-time L TI system, G : L 2 e → L 2 e , wi th state-space realization ( A , B , C , 0 ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and A is Hurwitz. Th e generalized H 2 norm of G is k G k 2 , ∞ = sup u ∈L 2 , u 6 = 0 k G u k ∞ k u k 2 . The inequality k G k 2 , ∞ < µ holds u n der any of following equiv alent necessary and suf ﬁcient conditions. 1. [ 3 , p. 79], [ 1 8 5 ] There exist P ∈ S n and µ ∈ R > 0 , where P > 0 , such that  A T P + P A PB ∗ − µ 1  < 0 ,  P C T ∗ µ 1  > 0 . 2. [ 186 ] There exist Q ∈ S n and µ ∈ R > 0 , wh ere Q > 0 , s uch that  QA T + A Q B ∗ − µ 1  < 0 ,  Q Q C T ∗ µ 1  > 0 . 3. There exist P ∈ S n , V ∈ R n × n , and µ ∈ R > 0 , wh ere P > 0 , such that     −  V + V T  V T A + P V T B V T ∗ − P 0 0 ∗ ∗ − µ 1 0 ∗ ∗ ∗ − P     < 0 ,  P C T ∗ µ 1  > 0 . Pr oof. Identical t o the proof in [ 73 ] u s ed to obt ain the dilated matrix inequality in ( 4.18 ). The g eneralized H 2 norm of G is the m i nimum value of µ ∈ R > 0 that sati s ﬁes any of the above conditions. 4.5 Peak-to-P eak Norm (Induced L ∞ - L ∞ Norm) [ 3 , p. 80], [ 185 ] Consider a conti nuous-time L TI system, G : L 2 e → L 2 e , wi th state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , D ∈ R p × m , and A is H u rwitz. The peak-to-peak n o rm o f G is k G k ∞ , ∞ = sup u ∈L ∞ , u 6 = 0 k G u k ∞ k u k ∞ . The i n equality k G k ∞ , ∞ < µ holds un d er any of the following equiv alent sufﬁcient condi tions. 77 1. There exist P ∈ S n and λ , ǫ , µ ∈ R > 0 , wh ere P > 0 , such that  A T P + P A + λ P PB ∗ − ǫ 1  < 0 ,   λ P 0 C T ∗ ( µ − ǫ ) 1 D T ∗ ∗ µ 1   > 0 . 2. There exist Q ∈ S n and λ , ǫ , µ ∈ R > 0 , where Q > 0 , such that  QA T + A Q + λ Q B ∗ − ǫ 1  < 0 ,   λ Q 0 QC T ∗ ( µ − ǫ ) 1 D T ∗ ∗ µ 1   > 0 . The peak-to-peak norm of G is small er than any µ ∈ R > 0 that s atisﬁes either of t h e above conditions. 4.6 Kalman-Y akubo vich-Popov ( KYP) Lemma 4.6.1 KYP Lemma for Q SR Diss ipative Systems [ 138 , 163 , 187 ] Consider a continuous-ti me L TI system, G : L 2 e → L 2 e , wi t h minim al state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and D ∈ R p × m . The system G is QSR dissipative [ 188 , 189 ] if Z T 0  y T ( t ) Qy ( t ) + 2 y T ( t ) Su ( t ) + u T ( t ) Ru ( t )  d t ≥ 0 , ∀ u ∈ L 2 e , ∀ T ∈ R ≥ 0 , where u ( t ) i s the input t o G , y ( t ) is t he out p ut of G , Q ∈ S p , S ∈ R p × m , and R ∈ S m . The sys t em G is also QSR dissipative if and only if there exists P ∈ S n , where P > 0 , such that  P A + A T P − C T QC PB − C T S − C T QD ∗ − D T QD −  D T S + S T D  − R  ≤ 0 . Note th at the Bounded Real Lemm a (Section 4.2.1 ) i s a special case of the KYP Lemma for QSR d i ssipative s y stems with Q = − 1 , S = 0 , and R = γ 2 1 . 4.6.2 Discrete- Time K YP Lemma for QSR Dissipative Systems [ 187 ], [ 190 , p. 49 5] Consider a discrete-time L TI system, G : ℓ 2 e → ℓ 2 e , with minimal s t ate-space realization ( A d , B d , C d , D d ) , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , and D d ∈ R p × m . The sy s tem G is QSR d i ssipative [ 188 , 189 ] if k X i =0  y T i Qy i + 2 y T i Su i + u T i Ru i  ≥ 0 , ∀ u ∈ ℓ 2 e , ∀ k ∈ Z ≥ 0 , 78 where u k is the input to G , y k is the output of G , Q ∈ S p , S ∈ R p × m , and R ∈ S m . The system G is also QSR dis sipative if and only if there exists P ∈ S n , wh ere P > 0 , such that  A T d P A d − P − C T d QC d A T d PB d − C T d S − C T d QD d ∗ B T d PB d − D T d QD d −  D T d S + S T D d  − R  ≤ 0 . Note that the Discrete-T ime Bounded Real Lemma (Section 4.2.2 ) is a sp ecial case of the Discrete-T ime KY P Lemma for QSR dissipative sy s tems with Q = − 1 , S = 0 , and R = γ 2 1 . 4.6.3 KYP (Positive Real) Lemma Without F eedthr ough [ 191 , p. 219], [ 19 2 ], [ 193 , p. 14] Consider a square, continuous-tim e L TI system , G : L 2 e → L 2 e , with mini mal st ate-space realization ( A , B , C , 0 ) , where A ∈ R n × n , B ∈ R n × m , and C ∈ R m × n . The system G is positive real (PR) under eit her of the following equiv alent necessary and sufﬁcient conditions. 1. There exists P ∈ S n , wh ere P > 0 , such that P A + A T P ≤ 0 , PB = C T . 2. There exists Q ∈ S n , wh ere Q > 0 , such that A Q + QA T ≤ 0 , B = QC T . This is a special case of the KYP Lemma for QSR dissip ativ e systems with Q = 0 , S = 1 2 · 1 , and R = 0 . The sy stem G i s st rictly positive real (SPR) under either of the following necessary and suf ﬁ- cient conditions. 1. There exists P ∈ S n , wh ere P > 0 , such that P A + A T P < 0 , PB = C T . 2. There exists Q ∈ S n , wh ere Q > 0 , such that A Q + QA T < 0 , B = QC T . This is a special case of the KYP Lem m a for Q SR dissipative system s with Q = ǫ · 1 , S = 1 2 · 1 , and R = 0 , where ǫ ∈ R > 0 . 79 4.6.4 KYP (Positiv e Real) Lemma With Feedthr ough [ 1 , p. 25], [ 191 , p. 218], [ 19 2 ], [ 194 , pp. 79–80] Consider a square, continuous-tim e L TI system , G : L 2 e → L 2 e , with mini mal st ate-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R m × n , and D ∈ R m × m . The system G is posi tiv e real (PR) under either of the following equiv alent necessary and sufﬁcient conditions. 1. There exists P ∈ S n , wh ere P > 0 , such that  P A + A T P PB − C T ∗ −  D + D T   ≤ 0 . 2. There exists Q ∈ S n , wh ere Q > 0 , such that  A Q + QA T B − Q C T ∗ −  D + D T   ≤ 0 . This is a special case of the KYP Lemma for QSR dissip ativ e systems with Q = 0 , S = 1 2 · 1 , and R = 0 . The system G is strictly positive real (SPR) under either of the following equiv alent necessary and s u f ﬁcient cond itions. 1. There exists P ∈ S n , wh ere P > 0 , such that  P A + A T P PB − C T ∗ −  D + D T   < 0 . 2. There exists Q ∈ S n , wh ere Q > 0 , such that  A Q + QA T B − QC T ∗ −  D + D T   < 0 . This is a special case of t he KYP Lemma for Q SR d i ssipative system s wi th Q = ǫ 1 , S = 1 2 · 1 , and R = 0 , where ǫ ∈ R > 0 . 4.6.5 Discrete- Time KYP (Positive R eal ) Lemma With F eedthr ough [ 194 , pp. 171–172], [ 195 ], [ 196 ] Consider a square, discrete-time L TI system, G : ℓ 2 e → ℓ 2 e , with minimal state-space realiza- tion ( A d , B d , C d , D d ) , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R m × n , and D d ∈ R m × m . The system G is positive real (PR) u n der any of t he foll o win g equivalent necessary and sufﬁcient conditions. 1. [ 197 ] There exists P ∈ S n , wh ere P > 0 , such that  A T d P A d − P A T d PB d − C T d ∗ B T d PB d −  D d + D T d   ≤ 0 . 80 2. There exists Q ∈ S n , wh ere Q > 0 , such that  A d QA T d − Q A d QC T d − B d ∗ C d QC T d −  D d + D T d   ≤ 0 . 3. [ 167 ] There exists P ∈ S n , wh ere P > 0 , such that   P P A d PB d ∗ P C T d ∗ ∗ D d + D T d   ≥ 0 . 4. There exists Q ∈ S n , wh ere Q > 0 , such that   Q A d Q B d ∗ Q QC T d ∗ ∗ D d + D T d   ≥ 0 . This is a special case of the Discrete-T ime KYP Lemma for QSR dissipative systems wi t h Q = 0 , S = 1 2 · 1 , and R = 0 . The system G is strictly positive real (SPR) under any of the following equiv alent necessary and s u f ﬁcient cond itions. 1. There exists P ∈ S n , wh ere P > 0 , such that  A T d P A d − P A T d PB d − C T d ∗ B T d PB d −  D d + D T d   < 0 . 2. There exists Q ∈ S n , wh ere Q > 0 , such that  A d QA T d − Q A d QC T d − B d ∗ C d QC T d −  D d + D T d   < 0 . 3. There exists P ∈ S n , wh ere P > 0 , such that   P P A d PB d ∗ P C T d ∗ ∗ D d + D T d   > 0 . 4. There exists Q ∈ S n , wh ere Q > 0 , such that   Q A d Q B d ∗ Q QC T d ∗ ∗ D d + D T d   > 0 . This i s a sp ecial case of the Discrete-Time KYP Lemma for QSR dissipative systems wi th Q = ǫ 1 , S = 1 2 · 1 , and R = 0 , where ǫ ∈ R > 0 . 81 4.6.6 KYP Lemma for Descriptor Systems [ 194 , pp. 91–93], [ 198 ] Consider a square, L TI descriptor system given by E ˙ x = Ax + Bu , y = Cx + Du , where E , A ∈ R n × n , B ∈ R n × m , C ∈ R m × n , and D ∈ R m × m . The system i s extended stri ctly positive real (ESPR) if and on ly if t here exist X ∈ R n × n and W ∈ R n × m such that E T X = X T E ≥ 0 , E T W = 0 , and  X T A + A T X A T W + X T B − C T ∗ W T B + B T W −  D + D T   < 0 . The s y s tem is also ESPR if there exists X ∈ R n × n such that E T X = X T E ≥ 0 and [ 199 ]  X T A + A T X X T B − C T ∗ −  D + D T   < 0 . 4.6.7 Discrete- Time K YP Lemma for Descriptor Systems [ 200 , 201 ] Consider a square, discrete-time L TI descriptor system giv en by E d x k +1 = A d x k + B d u k , y k = C d x k + D d u k , where E d , A d ∈ R n × n , B d ∈ R n × m , C d ∈ R m × n , and D d ∈ R m × m . The system is extended strictl y positive real (ESPR) if and only if there exists X ∈ S n such that E T XE ≥ 0 and  A T d XA d − E T d XE d A T d XB d − C T d ∗ −  D d + D T d − B T d XB d   < 0 . 4.6.8 QSR Dissipativity-Related Prop erties 1. [ 202 ] Consider a QSR-dissip ative continuous-time L TI system, G : L 2 e → L 2 e , wi th minimal state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and D ∈ R p × m . The H ∞ norm of G is less than γ (i.e., k G k ∞ < γ ) if there exist α , γ ∈ R > 0 such that 1 + α Q < 0 and  1 + α Q α S ∗ α R − γ 2 1  ≤ 0 . 4.7 Conic Sectors 4.7.1 Conic Sector Lemma Consider a square, continuous-tim e L TI system , G : L 2 e → L 2 e , with mini mal st ate-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R m × n , and D ∈ R m × m . The sy stem G is inside the cone [ a, b ] , where a , b ∈ R , and a 0 , such that  P A + A T P + C T C PB − a + b 2 C T + C T D ∗ D T D − a + b 2  D + D T  + ab 1  ≤ 0 . (4.51) Note that the matrix i n equality of ( 4.51 ) does not allo w for the case w h ere the upper bou nd b is inﬁnite. 2. [ 204 , p. 28] There exists P ∈ S n , wh ere P > 0 , such that  P A + A T P + 1 b C T C PB − 1 2  a b + 1  C T + 1 b C T D ∗ 1 b D T D − 1 2  a b + 1   D + D T  + a 1  ≤ 0 . 3. [ 205 ] There exists P ∈ S n , wh ere P > 0 , such that   P A + A T P PB C T ∗ − ( a − b ) 2 4 b 1 D T − a + b 2 1 ∗ ∗ − b 1   ≤ 0 . 4. There exists Q ∈ S n , wh ere Q > 0 , such that   A Q + QA T B QC T ∗ − ( a − b ) 2 4 b 1 D T − a + b 2 1 ∗ ∗ − b 1   ≤ 0 . The system G is i nside the cone o f radius r centered at c , where r ∈ R > 0 and b ∈ R , under any of th e following equivalent necessary and sufﬁcient conditi ons. 1. [ 206 ], [ 207 , pp. 23– 24] There e xis ts P ∈ S n , wh ere P > 0 , such that  P A + A T P + C T C PB − c C T + C T D ∗ D T D − c  D + D T  + ( c 2 − r 2 ) 1  ≤ 0 . (4.52) Note that the matrix i n equality of ( 4.52 ) does not allo w for the case w h ere the upper bou nd b is inﬁnite. The Conic Sector Lemma is a special case of the KYP Lemma for QSR dissipative sys t ems with Q = − 1 , S = a + b 2 1 = c 1 , and R = − ab 1 = ( r 2 − c 2 ) 1 . 4.7.2 Exterior Conic Sector Lemma Consider a sq uare, cont i nuous-time L TI system, G : L 2 e → L 2 e , with state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R m × n , and D ∈ R m × m . The system G is in the exterior cone of radius r centered at c (i.e., G ∈ excone r ( c ) ), where r ∈ R > 0 and c ∈ R , under either o f the foll owing equiv alent necessary and sufﬁcient conditions . 1. [ 208 ] There exists P ∈ S n , wh ere P ≥ 0 , such that  P A + A T P − C T C PB − C T ( D − c 1 ) ∗ r 2 1 − ( D − c 1 ) T ( D − c 1 )  ≤ 0 . (4.53) 83 2. There exists P ∈ S n , wh ere P ≥ 0 , s uch that   P A + A T P − C T C PB − C T ( D − c 1 ) 0 ∗ − ( D − c 1 ) T ( D − c 1 ) r 1 ∗ ∗ − 1   ≤ 0 . (4.54) Pr oof. Applyin g t h e Schur complement lem ma to t he r 2 1 t erm in ( 4.53 ) gives ( 4.54 ). 4.7.3 Modiﬁed Exterior Conic Sector Lemma Consider a sq uare, cont i nuous-time L TI system, G : L 2 e → L 2 e , with state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R m × n , and D ∈ R m × m . The system G is in the exterior cone of radius r centered at c (i.e., G ∈ excone r ( c ) ), where r ∈ R > 0 and c ∈ R , under either o f the foll owing equiv alent sufﬁcient conditions. 1. There exists P ∈ S n , wh ere P ≥ 0 , s uch that  P A + A T P PB − C T ( D − c 1 ) ∗ r 2 1 − ( D − c 1 ) T ( D − c 1 )  ≤ 0 . (4.55) Pr oof. The t erm − C T C in ( 4.53 ) makes the m atrix inequality “more” negati ve deﬁnite. Therefore,  P A + A T P − C T C PB − C T ( D − c 1 ) ∗ r 2 1 − ( D − c 1 ) T ( D − c 1 )  ≤  P A + A T P PB − C T ( D − c 1 ) ∗ r 2 1 − ( D − c 1 ) T ( D − c 1 )  , and ( 4.55 ) imp lies ( 4.53 ). 2. There exists P ∈ S n , wh ere P ≥ 0 , s uch that   P A + A T P PB − C T ( D − c 1 ) 0 ∗ − ( D − c 1 ) T ( D − c 1 ) r 1 ∗ ∗ − 1   ≤ 0 . (4.56) Pr oof. Applyin g t h e Schur complement lem ma to t he r 2 1 t erm in ( 4.55 ) gives ( 4.56 ). A sy stem satisfying the Modiﬁed Exterior Conic Sector Lemma is L yapunov stable if the additional restriction P > 0 is m ade, which is not necessarily true for a syst em sati sfying the Exterior Conic Sector Lemm a. The system G is also i n t he exterior cone o f radius r centered at c , where r ∈ R > 0 and c ∈ R , under eit her of the following equiv alent sufﬁcient cond itions. 1. There exists Q ∈ S n , wh ere Q > 0 , such that  A Q + QA T B − Q C T ( D − c 1 ) ∗ r 2 1 − ( D − c 1 ) T ( D − c 1 )  ≤ 0 . 2. There exists Q ∈ S n , wh ere Q > 0 , such that   A Q + QA T B − QC T ( D − c 1 ) 0 ∗ − ( D − c 1 ) T ( D − c 1 ) r 1 ∗ ∗ − 1   ≤ 0 . 84 4.7.4 Generalized K Y P (GKYP) Lemma for Conic Sectors Consider a sq uare, cont i nuous-time L TI system, G : L 2 e → L 2 e , with state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R m × n , and D ∈ R m × m . Also consider Π c ( a, b ) ∈ S m , wh i ch is deﬁned as Π c ( a, b ) =  1 b 1 − 1 2  1 + a b  1 ∗ a 1  , where a ∈ R , b ∈ R > 0 , and a < b . The foll owing generalized KYP Lemmas give conditio n s for G to be inside th e cone [ a, b ] within ﬁnite frequency band w i dths. 1. ( Low F r equency Range [ 209 ]) T he system G is inside the cone [ a, b ] for al l ω ∈ { ω ∈ R | | ω | < ω 1 , det( j ω 1 − A ) 6 = 0 } , where ω 1 ∈ R > 0 , a ∈ R , b ∈ R > 0 , and a 0 , where Q ≥ 0 , such that  A B 1 0  T  − Q P ∗ ( ω 1 − ¯ ω 1 ) 2 Q   A B 1 0  +  C D 0 1  T Π c ( a, b )  C D 0 1  < 0 . (4.57) If ω 1 → ∞ , P > 0 , and Q = 0 , then the traditional Conic Sector Lemm a is recovered [ 210 ]. The parameter ¯ ω 1 is inclu ded in ( 4.57 ) to effecti vely transform | ω | ≤ ( ω 1 − ¯ ω 1 ) into t he strict inequality | ω | < ω 1 . 2. ( Intermediate F r equency Range [ 210 – 212 ]) The system G is ins i de the cone [ a, b ] for all ω ∈ { ω ∈ R | ω 1 ≤ | ω | < ω 2 , det( j ω 1 − A ) 6 = 0 } , where ω 1 , ω 2 ∈ R > 0 , a ∈ R , b ∈ R > 0 , and a 0 , and ˆ ω 2 = ( ω 1 + ( ω 2 − ¯ ω 2 )) / 2 , where P H = P , Q H = Q , and Q ≥ 0 , such that  A B 1 0  T  − Q P + j ˆ ω 2 Q P − j ˆ ω 2 Q − ω 1 ( ω 2 − ¯ ω − 2 ) Q   A B 1 0  +  C D 0 1  T Π c ( a, b )  C D 0 1  < 0 . (4.58) The parameter ¯ ω 2 is included in ( 4.58 ) to effecti vely transform ω 1 ≤ | ω | ≤ ( ω 2 − ¯ ω 2 ) into the s trict inequalit y ω 1 ≤ | ω | < ω 2 . 3. ( High F r equency Range [ 211 ]) The syst em G is insid e the cone [ a, b ] for all ω ∈ { ω ∈ R | ω 2 ≤ | ω | , det( j ω 1 − A ) 6 = 0 } , where ω 2 ∈ R > 0 , a ∈ R , b ∈ R > 0 , and a < b , if there exist P , Q ∈ S n , wh ere Q ≥ 0 , such that  A B 1 0  T  Q P ∗ − ω 2 2 Q   A B 1 0  +  C D 0 1  T Π c ( a, b )  C D 0 1  < 0 . (4.59) If ( A , B , C , D ) is a minimal realization , then the matrix inequalit ies in ( 4.57 ), ( 4.58 ), and ( 4.59 ) can be no n s trict [ 209 ]. 4.8 Minimum Gain 4.8.1 Minimum Gain Lemma Consider a conti nuous-time L TI system, G : L 2 e → L 2 e , wi th state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and D ∈ R p × m . The system G has minim um gain ν under any of the following equiv alent sufﬁcient conditions . 85 1. [ 213 ] There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P ≥ 0 , s uch that  P A + A T P − C T C PB − C T D ∗ ν 2 1 − D T D  ≤ 0 . 2. [ 214 ] There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P ≥ 0 , s uch that   P A + A T P − C T C PB − C T D 0 ∗ − D T D ν 1 ∗ ∗ − 1   ≤ 0 . If G is a square system (i.e., m = p ) or span ( C ) ⊆ span ( D ) , then t he preceding condit ions are necessary and sufﬁcient for G to ha ve m inimum gain ν ∈ R ≥ 0 [ 213 ]. The min i mum gain l em ma is a special case of t he exterior conic sector lemma with a = − ν and b = ν . The s y s tem G also has m inimum gain ν under any o f the following sufﬁcient conditions. 1. There e xist P ∈ S n , V 11 ∈ R n × n , V 12 ∈ R n × m , V 21 ∈ R p × n , V 22 ∈ R p × m , and ν ∈ R ≥ 0 , where P ≥ 0 , such that       − ( V 11 + V T 11 ) V T 11 A + V T 21 C + P V T 11 B + V T 21 D − V 12 V T 11 ν V T 21 ∗ − P C T V 22 + A T V 12 0 0 ∗ ∗ ν 1 + V 22 D + D T V T 22 + V T 12 B + B T V 12 V T 12 ν V T 22 ∗ ∗ ∗ − P 0 ∗ ∗ ∗ ∗ − ν 1       ≤ 0 . (4.60) Pr oof. Applyin g the congruence transformatio n W = diag { ν − 1 / 2 1 , ν − 1 / 2 1 } and deﬁning ¯ P = ν − 1 P , the matrix inequ ality of ( 2 ) can be re written as  ¯ PA + A T ¯ P − ν − 1 C T C ¯ PB − ν − 1 C T D ∗ ν 1 − ν − 1 D T D  ≤ 0 . (4.61) Using Property 3 from Section 2.4.3 and making t he assum ption that ¯ P i s in vertible, ( 4.61 ) is equiva lent to   ¯ PA + A T ¯ P − ¯ P − ν − 1 C T C ¯ PB − ν − 1 C T D ¯ P ∗ ν 1 − ν − 1 D T D 0 ∗ ∗ − ¯ P   ≤ 0 . which is rewritten as   A T 1 0 0 − ν − 1 C T B T 0 1 0 − ν − 1 D T 1 0 0 1 0         0 ¯ P 0 0 0 ∗ − ¯ P 0 0 0 ∗ ∗ ν 1 0 0 ∗ ∗ ∗ − ¯ P 0 ∗ ∗ ∗ ∗ − ν 1             A B 1 1 0 0 0 1 0 0 0 1 − ν − 1 C − ν − 1 D 0       ≤ 0 . (4.62) 86 Since ¯ P > 0 and ν ∈ R ≥ 0 , i t is also known that   − ¯ P 0 0 ∗ − ¯ P 0 ∗ ∗ − ν 1   ≤ 0 , which can be rewritten as   0 1 0 0 0 0 0 0 1 0 0 0 0 0 1         0 ¯ P 0 0 0 ∗ − ¯ P 0 0 0 ∗ ∗ ν 1 0 0 ∗ ∗ ∗ − ¯ P 0 ∗ ∗ ∗ ∗ − ν 1             0 0 0 1 0 0 0 0 0 0 1 0 0 0 1       ≤ 0 . (4.63) The matrix in equ al i ties in ( 4.62 ) and ( 4.63 ) are i n the form of the nonstrict proj ection lemma. Speciﬁcally , ( 4. 6 2 ) is in the form of N T G Φ N G ≤ 0 , where Φ =       0 ¯ P 0 0 0 ∗ − ¯ P 0 0 0 ∗ ∗ ν 1 0 0 ∗ ∗ ∗ − ¯ P 0 ∗ ∗ ∗ ∗ − ν 1       , N G =       A B 1 1 0 0 0 1 0 0 0 1 − ν − 1 C − ν − 1 D 0       . The m atrix inequalit y o f ( 4.63 ) is in th e form of N T H Φ N H < 0 , where N H =       0 0 0 1 0 0 0 0 0 0 1 0 0 0 1       . The no nstrict projection lemma states that ( 4.62 ) and ( 4.63 ) are equiv alent to Φ + GVH T + HV T G T , (4.64) where N ( G T ) = R ( N G ) , N ( H T ) = R ( N H ) , V ∈ R n × n , and R ( G ) , R ( H ) are lin early independent. Choosi n g G T =  − 1 A B 1 0 0 C D 0 ν 1  , H T =  1 0 0 0 0 0 0 1 0 0  , V =  V 11 V 12 V 21 V 22  , where R ( G ) and R ( H ) are in fact linearly independent, the matrix inequali ty of ( 4.64 ) can 87 be rewritten as       0 ¯ P 0 0 0 ∗ − ¯ P 0 0 0 ∗ ∗ ν 1 0 0 ∗ ∗ ∗ − ¯ P 0 ∗ ∗ ∗ ∗ − ν 1       +       − 1 0 A T C T B T D T 1 0 0 ν 1        V 11 V 12 V 21 V 22   1 0 0 0 0 0 0 1 0 0  +       1 0 0 0 0 1 0 0 0 0        V T 11 V T 21 V T 12 V T 22   − 1 A B 1 0 0 C D 0 ν 1  < 0 , or equivalently       − ( V 11 + V T 11 ) V T 11 A + V T 21 C + ¯ P V T 11 B + V T 21 D − V 12 V T 11 ν V T 21 ∗ − ¯ P C T V 22 + A T V 12 0 0 ∗ ∗ ν 1 + V 22 D + D T V T 22 + V T 12 B + B T V 12 V T 12 ν V T 22 ∗ ∗ ∗ − ¯ P 0 ∗ ∗ ∗ ∗ − ν 1       ≤ 0 . (4.65) Redeﬁning P = ¯ P , ( 4.65 ) is identical to ( 4.60 ). 2. There exist P ∈ S n , V 11 ∈ R n × n , and ν ∈ R ≥ 0 , where P ≥ 0 , such that     − ( V + V T ) V T A + P V T B V T ∗ − P − C T 0 ∗ ∗ 2 ν 1 − ( D + D T ) 0 ∗ ∗ ∗ − P     < 0 . (4.66) Pr oof. The m atrix inequali ty of ( 4.66 ) is derived from ( 4.60 ) with V 11 = V , V 12 = 0 , V 21 = 0 , and V 22 = − 1 . The dilation in ( 4.60 ) relies on the projection lemma and becomes only a sufﬁcient condition i n thi s case due t o the structure imposed on V 11 , V 12 , V 21 , and V 22 . 4.8.2 Modiﬁed Minimum Gain Lemma Consider a conti nuous-time L TI system, G : L 2 e → L 2 e , wi th state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and D ∈ R p × m . The system G has minim um gain ν under any of the following equiv alent sufﬁcient conditions . 1. [ 215 ] There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P ≥ 0 , s uch that  P A + A T P PB − C T D ∗ ν 2 1 − D T D  ≤ 0 . (4.67) 88 2. There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P ≥ 0 , such that   P A + A T P PB − C T D 0 ∗ − D T D ν 1 ∗ ∗ − 1   ≤ 0 . (4.68) Pr oof. Applyin g t h e Schur complement lem ma to t he ν 2 1 t erm in ( 4.67 ) gi ves ( 4.68 ). A sy stem satisfying the Modiﬁed Minim um Gain Lemma is L yapunov stable if t h e additional restriction P > 0 is made, which is not necessarily true for a system satisfying the Minimum Gain Lemma. The system G also has mi nimum gain ν under any of t he following equivalent sufﬁcient con- ditions. 1. There exist Q ∈ S n and ν ∈ R ≥ 0 , wh ere Q > 0 , s uch that  A Q + QA T B − Q C T D ∗ ν 2 1 − D T D  ≤ 0 . 2. There exist Q ∈ S n and ν ∈ R ≥ 0 , wh ere Q > 0 , s uch that   A Q + QA T B − QC T D 0 ∗ − D T D ν 1 ∗ ∗ − 1   ≤ 0 . 4.8.3 Discrete- Time Mi nimum Gain Lemma Consider a dis crete-time L TI system, G : ℓ 2 e → ℓ 2 e , wi th state-space realization ( A d , B d , C d , D d ) , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , and D d ∈ R p × m . The system G has minim um gain ν under any of the following equiv alent sufﬁcient conditions . 1. [ 216 , p. 30] There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P ≥ 0 , s uch that  A T d P A d − P − C T d C d A T d PB d − C T d D d ∗ B T d PB d + ν 2 1 − D T d D d  ≤ 0 . (4.69) 2. There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P ≥ 0 , such that   A T d P A d − P − C T d C d A T d PB d − C T d D d 0 ∗ B T d PB d − D T d D d ν 1 ∗ ∗ 1   ≤ 0 . (4.70) Pr oof. Applyin g t h e Schur complement lem ma to t he ν 2 1 t erm in ( 4.69 ) gi ves ( 4.70 ). The system G also has mi nimum gain ν under any of t he following equivalent sufﬁcient con- ditions. 89 1. There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P > 0 , such that   P P A d PB d ∗ P + C T d C d C T d D d ∗ ∗ D T d D d − ν 2 1   ≥ 0 . (4.71) Pr oof. Under the assumpti on that P > 0 , the nonstrict Schur complement lemma is applied to ( 4.69 ) to yield ( 4.7 1 ). 2. There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P > 0 , such that     P P A d PB d 0 ∗ P + C T d C d C T d D d 0 ∗ ∗ D T d D d ν 1 ∗ ∗ ∗ 1     ≥ 0 . (4.72) Pr oof. Applyin g t h e Schur complement lem ma to t he ν 2 1 t erm in ( 4.71 ) gi ves ( 4.72 ). 4.8.4 Discrete- Time Modiﬁed Minimum Gain Lemma Consider a dis crete-time L TI system, G : ℓ 2 e → ℓ 2 e , wi th state-space realization ( A d , B d , C d , D d ) , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , and D d ∈ R p × m . The system G has minim um gain ν under any of the following equiv alent sufﬁcient conditions . 1. There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P ≥ 0 , such that  A T d P A d − P A T d PB d − C T d D d ∗ B T d PB d + ν 2 1 − D T d D d  ≤ 0 . (4.73) Pr oof. The term − C T d C d in ( 4.69 ) makes the matrix inequality “more” n egative deﬁnite. Therefore,  A T d P A d − P − C T d C d A T d PB d − C T d D d ∗ B T d PB d + ν 2 1 − D T d D d  ≤  A T d P A d − P A T d PB d − C T d D d ∗ B T d PB d + ν 2 1 − D T d D d  , and ( 4.73 ) imp lies ( 4.69 ). 2. There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P > 0 , such that   A T d P A d − P A T d PB d − C T d D d 0 ∗ B T d PB d − D T d D d ν 1 ∗ ∗ 1   ≤ 0 . (4.74) Pr oof. Applyin g t h e Schur complement lem ma to t he ν 2 1 t erm in ( 4.73 ) gi ves ( 4.74 ). A system sat i sfying the Discrete-T ime Modiﬁed M inimum Gain Lemma is L yapunov stable if the additional restri cti on P > 0 is made, which i s not necessarily true for a sys tem satisfyi ng the Discrete-T ime M inimum Gain Lemma. The s y s tem G also has m inimum gain ν under any o f the following sufﬁcient conditions. 90 1. There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P > 0 , such that   P P A d PB d ∗ P C T d D d ∗ ∗ D T d D d − ν 2 1   ≥ 0 . (4.75) Pr oof. Under the assumpti on that P > 0 , the nonstrict Schur complement lemma is applied to ( 4.73 ) to yield ( 4.7 5 ). 2. There exist P ∈ S n and ν ∈ R ≥ 0 , wh ere P > 0 , such that     P P A d PB d 0 ∗ P C T d D d 0 ∗ ∗ D T d D d ν 1 ∗ ∗ ∗ 1     ≥ 0 . (4.76) Pr oof. Applyin g t h e Schur complement lem ma to t he ν 2 1 t erm in ( 4.75 ) gi ves ( 4.76 ). 3. There exist Q ∈ S n and ν ∈ R ≥ 0 , wh ere Q > 0 , s uch that   Q A d Q B d ∗ Q QC T d D d ∗ ∗ D T d D d − ν 2 1   ≥ 0 . 4. There exist Q ∈ S n and ν ∈ R ≥ 0 , wh ere Q > 0 , s uch that     Q A d Q B d 0 ∗ Q QC T d D d 0 ∗ ∗ D T d D d ν 1 ∗ ∗ ∗ 1     ≥ 0 . 4.9 Negative I m aginary Systems 4.9.1 Negative Imaginary Lemma [ 217 , 218 ] Consider a sq uare, cont i nuous-time L TI system, G : L 2 e → L 2 e , with state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R m × n , and D ∈ S m . The system G is negati ve imaginary un d er either o f the following equivalent necessary and sufﬁcient conditions. 1. There exists P ∈ S n , wh ere P ≥ 0 , s uch that  P A + A T P PB − A T C T ∗ −  CB + B T C T   ≤ 0 . (4.77) 2. There exists Q ∈ S n , wh ere Q ≥ 0 , such that  A Q + QA T B − QA T C T ∗ −  CB + B T C T   ≤ 0 . (4.78) The s y s tem G is s trictly negative im aginary if det( A ) 6 = 0 and either ( 4.77 ) is satisﬁed with P > 0 or ( 4.78 ) is satisﬁed with Q > 0 . 91 4.9.2 Discrete- Time Nega tive I ma ginary Lemma Consider a square, d iscrete-time L TI s y stem, G : ℓ 2 e → ℓ 2 e , with state-space realization ( A d , B d , C d , D d ) , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R m × n , D d ∈ R m × m , C d ( z 1 − A d ) − 1 B d + D d = B T d  z 1 − A T d  − 1 C T d + D T d , det ( 1 + A ) 6 = 0 , and det ( 1 − A ) 6 = 0 . The sys tem G is negati ve imaginary un d er either o f the following equivalent necessary and sufﬁcient conditions. 1. [ 219 , 220 ] There exists P ∈ S n , wh ere P > 0 , such that A T d P A d − P ≤ 0 , C d + B T d  A T d − 1  − 1 P ( A d + 1 ) = 0 . 2. [ 219 ] There exists Q ∈ S n , wh ere Q > 0 , s uch that A d QA T d − Q ≤ 0 , B d + ( A d − 1 ) − 1 Q  A T d + 1  C T d = 0 . 4.9.3 Generalized Negative Ima g inary Lemma Consider a sq uare, cont i nuous-time L TI system, G : L 2 e → L 2 e , with state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R m × n , and D ∈ S m . Also consider Π p ∈ S m , which is deﬁned as Π p =  0 1 1 0  , The following generalized KYP Lemmas give conditions for G t o be negati ve imaginary within ﬁnite frequency bandwid ths. 1. ( Low F r equency Range [ 221 ]) The system G is negati ve imaginary for all ω ∈ { ω ∈ R | | ω | < ω 1 , det ( j ω 1 − A ) 6 = 0 } , where ω 1 ∈ R > 0 , if D = D T and there exist P , Q ∈ S n and ¯ ω 1 ∈ R > 0 , wh ere Q ≥ 0 , such that  A B 1 0  T  − Q P ∗ ( ω 1 − ¯ ω 1 ) 2 Q   A B 1 0  −  CA CB 0 1  T Π p  CA CB 0 1  < 0 . (4.79) If ω 1 → ∞ , P > 0 , and Q = 0 , then t he traditional Negativ e Imaginary Lemma is recov- ered [ 221 ]. The parameter ¯ ω 1 is inclu ded in ( 4.79 ) to effecti vely transform | ω | ≤ ( ω 1 − ¯ ω 1 ) into t he strict inequality | ω | < ω 1 . 2. ( Intermediate F r equency Range ) The s y stem G i s n egati ve imaginary for all ω ∈ { ω ∈ R | ω 1 ≤ | ω | < ω 2 , det( j ω 1 − A ) 6 = 0 } , where ω 1 , ω 2 ∈ R > 0 , if D = D T and t h ere exist P , Q ∈ C n , ¯ ω 2 ∈ R > 0 , and ˆ ω 2 = ( ω 1 + ( ω 2 − ¯ ω 2 )) / 2 , where P H = P , Q H = Q , and Q ≥ 0 , such that  A B 1 0  T  − Q P + j ˆ ω 2 Q P − j ˆ ω 2 Q − ω 1 ( ω 2 − ¯ ω − 2 ) Q   A B 1 0  −  CA CB 0 1  T Π p  CA CB 0 1  < 0 . (4.80) The parameter ¯ ω 2 is included in ( 4.80 ) to effecti vely transform ω 1 ≤ | ω | ≤ ( ω 2 − ¯ ω 2 ) into the s trict inequalit y ω 1 ≤ | ω | < ω 2 . 92 3. ( High F r equency Range ) The system G is negati ve imaginary for all ω ∈ { ω ∈ R | ω 2 ≤ | ω | , det( j ω 1 − A ) 6 = 0 } , where ω 2 ∈ R > 0 , if D = D T and there exist P , Q ∈ S n , where Q ≥ 0 , such that  A B 1 0  T  Q P ∗ − ω 2 2 Q   A B 1 0  −  CA CB 0 1  T Π p  CA CB 0 1  < 0 . (4.81) 4.9.4 Negative Imaginary Sys tem DC Constraint [ 222 , 223 ], [ 224 , pp. 32–34] Consider an NI transfer matri x G 1 ( s ) and an SNI transfer matrix G 2 ( s ) = C 2 ( s 1 − A 2 ) − 1 B 2 + D 2 . The condition ¯ λ ( G 1 (0) G 2 (0)) < 1 is satisﬁed if and only if S T ( − C 2 A − 1 2 B 2 + D 2 ) S < 1 , where SS T = G 1 (0) . 4.10 Algebraic Riccati Inequalities 4.10.1 Algebraic Riccati Inequality [ 138 ] Consider A ∈ R n × n , B ∈ R n × m , P , Q ∈ S n , N ∈ R n × m , and R ∈ S m , where P > 0 , Q ≥ 0 , and R > 0 . T h e algebraic Riccati inequali ty given by A T P + P A −  PB + N T  R − 1  B T P + N  + Q ≥ 0 , can be re writt en using the Schur complement lemma as  A T P + P A + Q PB + N T ∗ R  ≥ 0 . 4.10.2 Discrete-T ime Algebraic Riccati Inequality [ 225 ] Consider A d ∈ R n × n , B d ∈ R n × m , P , Q ∈ S n , and R ∈ S m , where P > 0 , Q ≥ 0 , and R > 0 . The di screte-time algebraic Riccati inequalit y g iven by A T d P A d − A T d PB d  R + B T d PB d  − 1 B T d P A d + Q − P ≥ 0 , can be re writt en using the Schur complement lemma as  A T d P A d − P + Q A T d PB d ∗ R + B T d PB d  ≥ 0 . Equiv alently , this discrete-time algebraic Riccati inequ al i ty i s s at i sﬁed under any of the following necessary and sufﬁcient condi t ions. 1. There exist P , Q ∈ S n , and R ∈ S m , wh ere P > 0 , Q ≥ 0 , and R > 0 , such that     Q 0 A T d P P ∗ R B T d P 0 ∗ ∗ P 0 ∗ ∗ ∗ P     ≥ 0 . 93 2. There exist X , Q ∈ S n , and R ∈ S m , wh ere X > 0 , Q ≥ 0 , and R > 0 , such that     Q 0 A T d 1 ∗ R B T d 0 ∗ ∗ X 0 ∗ ∗ ∗ X     ≥ 0 . 4.11 Stabilizability 4.11.1 Continuous-T ime Stabilizability [ 5 , pp. 166–16 8 ] Consider a conti nuous-time L TI system, G : L 2 e → L 2 e , wi th state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and D ∈ R p × m . The system G is stabilizable if and only if t here exists P ∈ S n , wh ere P > 0 , such that AP + P A T − BB T < 0 . The matrix A + BK is Hurwitz with K = − 1 2 B T P − 1 . Equ ivalently , G is stabili zable if and only if there exist P ∈ S n and W ∈ R m × n , wh ere P > 0 , such that AP + P A T + BW + W T B T < 0 . The m atrix A + BK is Hurwitz with K = WP − 1 . 4.11.2 Discrete-T ime Stabilizability [ 5 , pp. 172–17 6 ] Consider a dis crete-time L TI system, G : ℓ 2 e → ℓ 2 e , wi th state-space realization ( A d , B d , C d , D d ) , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , and D d ∈ R p × m . The system G is stabili zable if and only if there exists P ∈ S n , wh ere P > 0 , such that  P P A T d ∗ P + B d B T d  > 0 . The matrix A d + B d K d is Schur with K d = −  2 1 + B T d P − 1 B d  − 1 B T d P − 1 A d . Equivalently , G is stabilizable i f and only if there exist P ∈ S n and W ∈ R m × n , wh ere P > 0 , such that  P A d P + B d W ∗ P  > 0 . The m atrix A d + B d K d is Schur with K d = W P − 1 . 4.12 Detectability 4.12.1 Continuous-T ime Detectability [ 5 , pp. 170–17 1 ] Consider a conti nuous-time L TI system, G : L 2 e → L 2 e , wi th state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and D ∈ R p × m . The system G is detectable if and only if t here exists P ∈ S n , wh ere P > 0 , such that P A + A T P − C T C < 0 . 94 The matrix A + LC i s Hurwitz wit h L = − 1 2 P − 1 C T . Equiv alently , G is detectable if and onl y if there exist P ∈ S n and W ∈ R p × n , wh ere P > 0 , such that P A + A T P + W T C + C T W < 0 . The m atrix A + LC is Hurwit z with L = − 1 2 P − 1 W T . 4.12.2 Discrete-T ime Detectab il i ty [ 5 , pp. 177–17 8 ] Consider a dis crete-time L TI system, G : ℓ 2 e → ℓ 2 e , wi th state-space realization ( A d , B d , C d , D d ) , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , and D d ∈ R p × m . The system G is detectable if and only if there exists P ∈ S n , wh ere P > 0 , such that  P P A d ∗ P + C T d C d  > 0 . The matrix A d + LC d is Schur with L = − A d P − 1 C T d  2 1 + C d P − 1 C T d  − 1 . E quiv alently , G is detectable i f and onl y if there exist P ∈ S n and W ∈ R m × n , wh ere P > 0 , such that  P A T d P + C T d W ∗ P  > 0 . The m atrix A d + LC d is Schur with L = P − 1 W . 4.13 Static Output Feed back Stabiliz ability 4.13.1 Continuous-T ime Static Output Feedb ack Stabilizability [ 226 , 227 ], [ 118 , p. 1 20] Consider a conti nuous-time L TI system, G : L 2 e → L 2 e , wi th state-space realization ( A , B , C , 0 ) , where A ∈ R n × n , B ∈ R n × m , and C ∈ R p × n . The s ystem G is static output feedback stabili zable under any of the following equiv alent necessary and sufﬁcient conditio n s. 1. There exist K ∈ R m × p and P ∈ S n , where P > 0 , such that  A T P + P A − PBB T P PB + C T K T ∗ − 1  < 0 . 2. There exist K ∈ R m × p and Q ∈ S n , wh ere Q > 0 , s uch that  QA T + A Q − QC T CQ BK + Q C T ∗ − 1  < 0 . 3. There exist K ∈ R m × p and Q ∈ S n , wh ere Q > 0 , s uch that  QA T + A Q − BB T B + Q C T K T ∗ − 1  < 0 . 95 4. There exist K ∈ R m × p and P ∈ S n , where P > 0 , such that  A T P + P A − C T C PBK + C T ∗ − 1  < 0 . 5. There exist K ∈ R m × p , P , X ∈ S n , wh ere P > 0 and X > 0 , su ch that  A T X + XA − PBB T X − XBB T P + XBB T X PB + C T K T ∗ − 1  < 0 . 6. There exist K ∈ R m × p and Q , X ∈ S n , wh ere Q > 0 and X > 0 , such that  QA T + A Q − QC T CX − XC T CQ + XC T CX BK + QC T ∗ − 1  < 0 . 4.13.2 Discrete-T ime Static Output Feedback Stabilizability Consider a dis crete-time L TI system, G : ℓ 2 e → ℓ 2 e , wi th state-space realization ( A d , B d , C d , 0 ) , where A d ∈ R n × n , B d ∈ R n × m , and C d ∈ R p × n . The s ystem G is static output feedback s t abiliz- able un d er any of the following equiv alent necessary and sufﬁcient cond itions. 1. There exist K d ∈ R m × p and P ∈ S n , wh ere P > 0 , such that  − P ( A d + B d K d C d ) P ∗ − P  < 0 . (4.82) 2. There exist K d ∈ R m × p and P ∈ S n , wh ere P > 0 , such that   − A d PP A T d A d P + B d K d C d A d P ∗ − 1 0 ∗ ∗ − P   < 0 . (4.8 3 ) Pr oof. Applyin g t h e reve rse Schur compl ement lemma to ( 4.82 ) yields ( A d + B d K d C d ) P ( A d + B d K d C d ) T − P < 0 . Multiply ing out this matrix inequality and adding 0 = A d PP A d − A d PP A d to the left-hand side gives A d P A T d − A d PP A T d + ( A d P + B d K d C d ) ( A d P + B d K d C d ) T < 0 . Applying the Schur com p lement lemm a twice gives ( 4.83 ). The sy stem G is also static out p u t feedback s t abilizable if t here exist K d ∈ R m × p and P , X ∈ S n , wh ere P > 0 and X > 0 , su ch that     − A d ( XP + PX ) A T d A d P + B d K d C d A d P A d X ∗ − 1 0 0 ∗ ∗ − P 0 ∗ ∗ ∗ − 1     < 0 . (4.84) 96 Pr oof. Using completion of the squ ares, it can be shown that − A d PP A T d ≤ − A d ( XP + PX ) A T d + A d XXA T d . (4.85) Substitutin g ( 4.85 ) in t o ( 4.83 ) and using the Schur complement lemm a yields ( 4.84 ). The ma- trix inequalit y in ( 4.84 ) is onl y a sufﬁcient condi tion for static o u tput feedback st abi lizability since ( 4. 8 5 ) is an inequ ali ty . 4.14 Strong Stabili zability 4.14.1 Continuous-T ime Stron g Stabilizability [ 228 ] Consider a conti nuous-time L TI system, G : L 2 e → L 2 e , wi th state-space realization ( A , B , C , 0 ) , where A ∈ R n × n , B ∈ R n × m , and C ∈ R p × n , and it i s assumed that ( A , B ) is stabil izable, ( A , C ) is detectable, and the transfer matrix G ( s ) = C ( s 1 − A ) − 1 B has no poles on the im agi nary axis. The s y s tem G is strongly stabil izable if there exist P ∈ S n , Z ∈ R n × p , and γ ∈ R > 0 , wh ere P > 0 , such that P A + A T P + ZC + C T Z T < 0 ,   P ( A + BF ) + ( A + BF ) T P + ZC + C T Z T − Z − XB ∗ − γ 1 0 ∗ ∗ − γ 1   < 0 , where F = − B T X and X ∈ S n , X ≥ 0 is t he sol u tion to th e L yapunov equati on given by XA + A T X − XBB T X = 0 . Moreover , a controller that strongly s tabilizes G i s given by the stat e-space realization ˙ x c =  A + BF + P − 1 ZC  x − P − 1 Zu , y c = − B T Xx . 4.14.2 Discrete-T ime Str ong Stabilizability Consider a dis crete-time L TI system, G : ℓ 2 e → ℓ 2 e , wi th state-space realization ( A d , B d , C d , 0 ) , where A d ∈ R n × n , B d ∈ R n × m , and C d ∈ R p × n , and i t is assumed th at ( A d , B d ) is stabilizable, ( A d , C d ) is detectable, and t he t ransfer matrix G ( z ) = C d ( z 1 − A d ) − 1 B d has no poles on th e uni t circle. The system G is strongl y stabilizable i f there exist P ∈ S n , Z ∈ R n × p , and γ ∈ R > 0 , where P > 0 , such that  A T d P A d − P − A T d ZC d − C T d Z T A d C T d Z T ∗ − P  < 0 , (4.86)     N 11 ( A d + B d F ) T Z XB d C T d Z T ∗ − γ 1 0 Z T ∗ ∗ − γ 1 0 ∗ ∗ ∗ − P     < 0 , (4.87) 97 where N 11 = ( A d + B d F ) T P ( A d + B d F ) − P + ( A d + B d F ) T ZC d + C T d Z T ( A d + B d F ) , F = − B T d X , X = Y , and Y ∈ S n , Y ≥ 0 is t he sol ution to t h e discrete-tim e L y apu nov equ at i on g iven by A d Y A T d − Y − B d B T d = 0 . Moreover , a discrete-time cont roller that strongl y stabili zes G is given by the state-space realizatio n x c,k +1 =  A d + B d F + P − 1 ZC d  x k − P − 1 Zu k , (4.88) y c,k = − B T d Xx k . (4.89) Pr oof. The proof follows t h e same procedure as in [ 228 ] for th e continuous-tim e case, where ( 4.86 ) ensures th at the feedback controll er deﬁned by ( 4.88 ) and ( 4.89 ) renders the closed-loop sy s tem asymptoticall y stable and ( 4.87 ) ensures that the feedback controller deﬁned by ( 4.88 ) and ( 4.89 ) has a ﬁnite H ∞ norm, and thus i s asym ptotically stable. 4.15 System Zeros 4.15.1 System Zer os without Feedth rough [ 229 ] Consider a continuous-ti me L TI system, G : L 2 e → L 2 e , wi t h minim al state-space realization ( A , B , C , 0 ) , where A ∈ R n × n , B ∈ R n × m , and C ∈ R p × n . The transmi ssion zeros of G ( s ) = C ( s 1 − A ) − 1 B are the eigen values of N AM , where N ∈ R q × n , M ∈ R n × q , CM = 0 , NB = 0 , and NM = 1 . Therefore, G ( s ) is minim um phase if and only if there exists P ∈ S q , where P > 0 , such that PNAM + M T A T N T P < 0 . 4.15.2 System Zer os with Feedthr ough Consider a continuous-ti me L TI system, G : L 2 e → L 2 e , wi t h minim al state-space realization ( A , B , C , D ) , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , D ∈ R p × m , m ≤ p , and D is full rank. The transmissio n zeros of G ( s ) = C ( s 1 − A ) − 1 B + D are the eigen v alues of A − B  D T D  − 1 D T C . 1. G ( s ) is m inimum phase if and only if there e xis ts P ∈ S n , wh ere P > 0 , such that P  A − B  D T D  − 1 D T C  +  A − B  D T D  − 1 D T C  T P < 0 . (4.90) If the system is square ( m = p ), then D full rank imp l ies D − 1 exists and ( 4.90 ) sim pliﬁes to P  A − BD − 1 C  +  A − BD − 1 C  T P < 0 . (4.91) Pr oof. The system G can be written in state-space form as ˙ x = Ax + Bu , (4.92) y = Cx + Du . (4.93) Left-multiply i ng ( 4.93 ) by D T and rearranging yields D T Du = − D T Cx + D T y . (4.94) 98 Since D is full rank,  D T D  − 1 exists. Therefore, left-m u ltiplying ( 4.94 ) by  D T D  − 1 giv es u = −  D T D  − 1 D T Cx +  D T D  − 1 D T y . (4.95) Substitutin g ( 4.95 ) into ( 4.9 2 ) gives the following state-sp ace representati on of the inv erted transfer matri x from y to u . ˙ x =  A − B  D T D  − 1 D T C  x + B  D T D  − 1 D T y , (4. 9 6) u = −  D T D  − 1 D T Cx +  D T D  − 1 D T y . (4.97) The transmissio n zeros of G ( s ) are the poles of the in verted transfer m atrix from y to u , which are the eigen v alues of  A − B  D T D  − 1 D T C  . Substituti ng this matrix into a L ya- punov i nequality gives the desired inequali ty in ( 4.90 ). If the system is square and D − 1 exists, then  D T D  − 1 D T = D − 1 and ( 4.90 ) simpliﬁes to ( 4.91 ). 2. The transfer matrix G ( s ) is also min imum phase if and only if there exist P ∈ S n and Q ∈ S n , where P > 0 and Q = P − 1 , such that M T  P A + A T P  M < 0 , (4.98) N  A Q + QA T  N T < 0 , (4.99) where N ∈ R q × n , M ∈ R n × q , R ( N T ) = N ( B T ) , and R ( M ) = N ( C ) . Pr oof. Applyin g t h e Strict Projection Lemm a to ( 4.90 ) yields ( 4.98 ) and ( 4.99 ). 4.15.3 Discrete-T ime System Zer os with Fe edthrou gh Consider a discrete-time L TI system, G : ℓ 2 e → ℓ 2 e , with minimal s t ate-space realization ( A d , B d , C d , D d ) , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , D d ∈ R p × m , m ≤ p , and D d is full rank. The transmissi o n zeros of G ( z ) = C d ( z 1 − A d ) − 1 B d + D d are the eigen v alues of A d − B d  D T d D d  − 1 D T d C d . Therefore, G ( z ) is minim u m p hase if and o nly if t here exists P ∈ S n , where P > 0 , such that " P  A d − B d  D T d D d  − 1 D T d C d  P ∗ P # > 0 . (4. 1 0 0) If the system is square ( m = p ), then D d full rank implies D − 1 d exists and ( 4.100 ) sim pliﬁes to  P  A d − B d D − 1 d C d  P ∗ P  > 0 . Pr oof. The proof fol lows the same procedure used in the p roo f o f the continuou s -time resul t i n Section 4.15.2 . 99 4.16 D -Stability 4.16.1 General LMI Region D -Stability [ 5 , pp. 107–10 8 ], [ 230 ] Consider A ∈ R n × n . The eigen va lues of a D -stable m atrix lie within the LMI region D of th e complex plane, whi ch is deﬁned as D = { z ∈ C : f D ( z ) < 0 } , where f D ( z ) := Λ + z Φ + z Φ T = [ λ k l + φ k l z + φ lk z ] 1 ≤ k , l ≤ m , Λ ∈ S m , Φ ∈ R m × m , and z is the complex conjugate of z . The matrix A is D -stable i f and only i f any of the following equivalent conditions are satis ﬁed. 1. There exists P ∈ S n , wh ere P > 0 , such that [ λ k l P + φ k l AP + φ lk P A T ] 1 ≤ k , l ≤ m < 0 , 2. There exists P ∈ S n , wh ere P > 0 , such that Λ ⊗ P + Φ ⊗ ( AP ) + Φ T ⊗  P A T  < 0 , (4.101) where ⊗ is t he Kroenecker produ ct . Alternative ly , consider the LMI region D of the com plex plane deﬁned by [ 3 , p . 70] D = { z ∈ C :  1 z 1  H  Q S S T Q   1 z 1  < 0 } , where Q , R ∈ S m and S ∈ R m × m . The matrix A is D -stable if and only if there exists P such that  1 A ⊗ 1  T  P ⊗ Q P ⊗ S P ⊗ S T P ⊗ R   1 A ⊗ 1  < 0 . 4.16.2 α -Stability Region Consider A ∈ R n × n and α ∈ R > 0 . The matrix A satisﬁes λ ( A ) ⊂ D ( α ) , where D ( α ) := { z ∈ C : Re ( z ) < − α } if and only if any of the foll owing equiv alent conditi ons are satis ﬁed. 1. [ 1 , pp. 66-67], [ 5 , p. 99], [ 230 – 232 ] T h ere exists P ∈ S n , wh ere P > 0 , such that AP + P A T + 2 α P < 0 . (4.102) 2. There exists P ∈ S n , wh ere P > 0 , such that  AP + P A T α P ∗ − 1 2 P  < 0 . (4.103) Pr oof. Equation ( 4.102 ) is re written as AP + P A T − ( α P )  − 1 2 α P  − 1 ( α P ) < 0 , which is equiv alent to ( 4.103 ) usi n g the Schur complem ent. 100 3. [ 92 ] There exist X ∈ S n , ǫ ∈ R > 0 , and F ∈ R n × n , wh ere X > 0 , such that   0 − X X ∗ 0 0 ∗ ∗ − 1 2 α − 1 X   + He      A 1 0   F  1 − ǫ 1 ǫ 1     < 0 . (4.104) Moreover , for eve ry X that satisﬁes ( 4.102 ), X and F = − ǫ − 1 ( A − ǫ − 1 1 ) − 1 X are s o l utions to ( 4.104 ). 4. [ 143 ] There exist P ∈ S n , Y 1 , Y 2 , Y 3 , X 1 , X 2 , X 3 ∈ R n × n , and γ ∈ R > 0 , wh ere P > 0 , such that   X 1 Y 1 + Y T 1 X T 1 P + X 1 Y 2 + Y T 1 X T 2 A T − α 1 + X 1 Y 3 + Y T 1 X T 3 ∗ X 2 Y 2 + Y T 2 X T 2 − γ 1 + X 2 Y 3 + Y T 2 X T 3 ∗ ∗ X 3 Y 3 + Y T 3 X T 3   < 0 . If λ ( A ) ⊂ D ( α ) , t h en the solu t ion to ˙ x = Ax , x (0) = x 0 satisﬁes k x ( t ) k 2 ≤ p κ ( P ) k x 0 k 2 e − αt , where κ ( P ) i s the condit i on number of P . This system is exponentially st able with exponential decay rate α . 4.16.3 V ertical Band [ 5 , p. 99], [ 230 – 2 32 ] Consider A ∈ R n × n and α , β ∈ R > 0 . Th e m at ri x A s at i sﬁes λ ( A ) ⊂ D ( α, β ) , where D ( α, β ) := { z ∈ C : − β < Re ( z ) < − α } if and only if there exists P ∈ S n , where P > 0 , such that AP + P A T + 2 α P < 0 , AP + P A T + 2 β P > 0 . If λ ( A ) ⊂ D ( α , β ) , then the sol ution to ˙ x = Ax , x (0 ) = x 0 satisﬁes k x ( t ) k 2 ≤ p κ ( P ) k x 0 k 2 e − αt , where κ ( P ) is th e condition number of P . This syst em is exponentially stable with exponential de- cay rate α . 4.16.4 Conic Sector Region Consider A ∈ R n × n and θ ∈ R > 0 . Th e matrix A satisﬁes λ ( A ) ⊂ D ( k ) , where D ( k ) := { z ∈ C : | Im ( z ) | < − tan( θ ) Re ( z ) , 0 < θ < π / 2 } , if and only i f any of the following equi valent conditions are satisﬁed. 1. [ 5 , pp. 105–106], [ 230 ] There exists P ∈ S n , wh ere P > 0 , such that  sin( θ )  AP + P A T  cos( θ )  AP − P A T  ∗ sin( θ )  AP + P A T   < 0 . 2. [ 92 ] There exists P ∈ S n , wh ere P > 0 , such that  k  AP + P A T  AP − P A T ∗ k  AP + P A T   < 0 , (4.105) where k = tan( θ ) . 101 3. [ 92 ] There exist X ∈ S n , ǫ ∈ R > 0 , and F ∈ R n × n , wh ere X > 0 , such that     0 − k X X 0 ∗ 0 0 − X ∗ ∗ 0 − k X ∗ ∗ ∗ 0     + He            A 0 1 0 0 1 0 A      F 0 0 F   k 1 − ǫk 1 ǫ 1 1 − 1 − ǫ 1 ǫk 1 k 1         < 0 , (4.106) where k = tan( θ ) . Moreover , for every X that satisﬁes ( 4.105 ), X and F = − ǫ − 1 ( A − ǫ − 1 1 ) − 1 X are solut ions to ( 4.106 ). 4.16.5 Circular Region Consider A ∈ R n × n , r ∈ R > 0 , and c ∈ R < 0 , where c < − r . The matrix A satisﬁes λ ( A ) ⊂ D ( c, r ) , where D ( c, r ) := { z ∈ C : ( Re ( z ) − c ) 2 + ( Im ( z )) 2 < r 2 } , if and only if any of t he following equiva lent cond itions are satisﬁed. 1. [ 5 , p. 101], [ 230 , 232 ] There exists P ∈ S n , wh ere P > 0 , such that  − r P − c P + AP ∗ − r P  < 0 . 2. [ 92 ] There exists P ∈ S n , wh ere P > 0 , such that AP + P A T − c 2 − r 2 c P − 1 c AP A T < 0 . (4.1 0 7) 3. [ 92 ] There exist X ∈ S n , ǫ ∈ R > 0 , and F ∈ R n × n , wh ere X > 0 , such that     0 − X X 0 ∗ 0 0 − X ∗ ∗ c c 2 − r 2 X 0 ∗ ∗ ∗ c X     + He            A 1 0 0     F  1 − ǫ 1 ǫ 1 1         < 0 . (4.1 0 8 ) Moreover , for eve ry X that satisﬁes ( 4.107 ), X and F = − ǫ − 1 ( A − ǫ − 1 1 ) − 1 X are s o l utions to ( 4.108 ). 4.16.6 Horizontal Band [ 231 ], [ 233 , p. 16 4], [ 234 , p. 48] Consider A ∈ R n × n and γ ∈ R > 0 . The matrix A satisﬁes λ ( A ) ⊂ D ( γ ) , where D ( γ ) := { z ∈ C : | Im ( z ) | < γ } , if and only if there e xis ts P ∈ S n , where P > 0 , such that  − 2 γ P AP − P A T ∗ − 2 γ P  < 0 . 102 4.16.7 Elliptic Region [ 235 , p. 31] Consider A ∈ R n × n , a , b ∈ R > 0 , and c ∈ R . The matri x A satisﬁes λ ( A ) ⊂ D ( γ ) , where D ( a, b, c ) := { z ∈ C :  Re ( z ) − c a  2 +  Im ( z ) b  2 < 1 } , if and only if there exists P ∈ S n , where P > 0 , such that  2 ab P ( a + b ) P A + ( b − a ) A T P ∗ 2 ab P  > 0 . The parameter c is the center of the ellipse region on the real axis, a is th e semi-major axis, and b is the semi-minor axis . 4.16.8 Hyperbolic Region [ 235 , p. 32] Consider A ∈ R n × n , a , b ∈ R > 0 , and c ∈ R . The matri x A satisﬁes λ ( A ) ⊂ D ( γ ) , where D ( a, b, c ) := { z ∈ C :  Re ( z ) − c a  2 −  Im ( z ) b  2 > 1 } , if and only if there exists P ∈ S n , where P > 0 , such that  2 bc P − b  P A + A T P  2 ab P + a  P A − A T P  ∗ 2 bc P − b  P A + A T P   > 0 . The parameter c is t he center of the hyperbol i c re gion on the real axis, a is the semi-major axis, and b is the semi-mino r axis. 4.17 D -Admissibility 4.17.1 General LMI Region D -Admissibility Consider A , E ∈ R n × n . The pair ( E , A ) is D -admis s ible if it is regular and causal, and the eigen va lues of ( E , A ) lie within the LM I region D of the complex plane, which is deﬁned as D = { z ∈ C : f D ( z ) < 0 } , where f D ( z ) := Λ + z Φ + z Φ T = [ λ k l + φ k l z + φ lk z ] 1 ≤ k , l ≤ m , Λ ∈ S m , Φ ∈ R m × m , and z is the complex conjugate of z . The pair ( E , A ) i s D -admiss i ble if and only if any of the following equiv alent conditions are satisﬁed. 1. [ 151 ] There exist P ∈ S n , S ∈ R ( n − n e ) × ( n − n e , and U , V ∈ R n × ( n − n e ) , where n e = rank ( E ) , R ( U ) = N ( E T ) , R ( V ) = N ( E ) , and P > 0 , satisfying [ λ k l EPE T + φ k l APE + φ lk E T P A T + A VSU T + US T V T A T ] 1 ≤ k , l ≤ m < 0 , 2. [ 236 ] There exist P , Q ∈ S n , wh ere P > 0 , sati sfying E T QE ≥ 0 and [ λ k l EPE T + φ k l APE + φ lk E T P A T + A T QA ] 1 ≤ k , l ≤ m < 0 , 3. [ 236 ] There exist P ∈ S n , S ∈ R ( n − n e ) × ( n − n e , U ∈ R n × ( n − n e ) , w h ere n e = rank ( E ) , UE = 0 , and P > 0 , satisfying [ λ k l EPE T + φ k l APE + φ lk E T P A T + A T U T SU A ] 1 ≤ k , l ≤ m < 0 , 103 4. [ 151 ] There exist P ∈ S n , S ∈ R ( n − n e ) × ( n − n e , and U , V ∈ R n × ( n − n e ) , where n e = rank ( E ) , R ( U ) = N ( E T ) , R ( V ) = N ( E ) , and P > 0 , satisfying Λ ⊗ EPE T + Φ ⊗ ( APE ) + Φ T ⊗  EP A T  + 1 mm ⊗  A VSU T + US T V T A T  < 0 , where ⊗ is t he Kroenecker produ ct and 1 mm is an m × m matrix ﬁlled wi th ones. 5. [ 236 ] There exist P , Q ∈ S n , wh ere P > 0 , sati sfying E T QE ≥ 0 and Λ ⊗ EPE T + Φ ⊗ ( APE ) + Φ T ⊗  EP A T  + 1 mm ⊗  A T QA  < 0 , where ⊗ is t he Kroenecker produ ct and 1 mm is an m × m matrix ﬁlled wi th ones. 6. [ 236 ] There exist P ∈ S n , S ∈ R ( n − n e ) × ( n − n e , U ∈ R n × ( n − n e ) , w h ere n e = rank ( E ) , UE = 0 , and P > 0 , satisfying Λ ⊗ EPE T + Φ ⊗ ( APE ) + Φ T ⊗  EP A T  + 1 mm ⊗  A T U T SU A  < 0 , where ⊗ is t he Kroenecker produ ct and 1 mm is an m × m matrix ﬁlled wi th ones. 4.17.2 Circular Region [ 157 ] Consider A , E ∈ R n × n , a , b ∈ R , and d ∈ R > 0 , where b 6 = 0 . The pair ( E , A ) is D -admi s sible with D = { z ∈ C : a + 2 b Re ( z ) + d | z | 2 < 0 } if and on l y i f there exist X ∈ R n × n and α ∈ R such that E T X = X T E ≥ 0 and  − a E T X − b  X T A + A T X  A T X ∗ d − 1 E T X + α  1 − E † E   > 0 , where E † is the pseudoin verse of E . The region D describes a circular region of the complex plane with radius r = p − a/d + b 2 /d 2 centered at ( c, 0) , where c = − b/d . 4.18 DC Gain of a T ransfer Matrix Consider γ ∈ R > 0 and a cont inuous-time L TI system, G : L 2 e → L 2 e , wit h transfer mat ri x G ( s ) = C ( s 1 − A ) − 1 B + D , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and D ∈ R p × m . The DC gain of G is strictly less than γ (i . e., ¯ σ ( G (0) ) < γ ) if and only if  γ 1 − CA − 1 B + D ∗ γ 1  > 0 , (4.109) or  γ 1 − B T A − T C T + D T ∗ γ 1  > 0 . (4.110) Pr oof. ¯ σ ( G (0)) < γ if and only if ¯ λ  G (0) G T (0)  < γ 2 , or equiv alently G (0) G T (0) − γ 2 1 < 0 G (0)( − γ − 1 1 ) G T (0) − γ 1 < 0 γ 1 − G (0)( γ − 1 1 ) G T (0) > 0  γ 1 G ( 0 ) ∗ γ 1  > 0 . (4.111) 104 Substitutin g G (0) = − CA − 1 B + D into ( 4.111 ) giv es ( 4.109 ). Starting with ¯ σ ( G (0)) < γ ⇐ ⇒ ¯ λ  G T (0) G (0)  < γ 2 in the ﬁrst step of t he proo f and foll owing the s am e steps yields ( 4.110 ). 4.19 T ransient Bounds 4.19.1 T ransient State Bound for A utonomous L TI Systems [ 1 , p. 88], [ 237 , 238 ] Consider t he conti nuous-time L TI s ystem with state-space realization ˙ x = Ax , where A ∈ R n × n and x (0) = x 0 . The Euclidean norm of th e state satisﬁes k x ( T ) k 2 ≤ γ k x 0 k 2 , ∀ T ∈ R ≥ 0 if t here exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.112)  P 1 ∗ γ 1  ≥ 0 , (4.113) P A + A T P ≤ 0 . ( 4.11 4 ) Pr oof. Deﬁne V = x T Px . Evaluating ˙ V and substi t uting in the matrix inequali ty from ( 4.114 ) results in ˙ V ≤ 0 . Integrating bo t h si des of this inequ al i ty from t = 0 to t = T , w h ere T ∈ R ≥ 0 giv es V ( T ) ≤ V (0 ) x T ( T ) Px ( T ) ≤ x T 0 Px 0 . (4.115) Using the non-strict Schur complement, ( 4.113 ) can be re written as γ − 1 1 ≤ P . Subst ituting this and ( 4.1 12 ) into ( 4.115 ) yiel d s γ − 1 x T ( T ) x ( T ) ≤ γ x T 0 x 0 k x ( T ) k 2 ≤ γ k x 0 k 2 . 4.19.2 T ransient State Bound for Discrete-T ime A utonomous L TI Systems Consider t he dis crete-time L TI sys t em with state-space realization x k +1 = A d x k , where A d ∈ R n × n . The Euclidean norm of th e state satisﬁes k x k k 2 ≤ γ k x 0 k 2 , ∀ k ∈ Z ≥ 0 105 if t here exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.116)  P 1 ∗ γ 1  ≥ 0 , (4.1 1 7 ) A T d P A d − P ≤ 0 . (4.118) Pr oof. Deﬁne V ( k ) = x T k Px k . Ev aluatin g V ( k + 1) − V ( k ) and s ubstitutin g i n th e matrix inequali ty from ( 4.118 ) results in V ( k + 1) ≤ V ( k ) x T k +1 Px k +1 ≤ x T k Px k . Using in duction, this inequality impli es x T k Px k ≤ x T 0 Px 0 . (4.119) Using the non-strict Schur complement, ( 4.117 ) can be re written as γ − 1 1 ≤ P . Subst ituting this and ( 4.116 ) int o ( 4.119 ) yields γ − 1 x T k x k ≤ γ x T 0 x 0 k x k k 2 ≤ γ k x 0 k 2 . 4.19.3 T ransient State Bound for Non-A utonomous L TI Systems [ 1 , p. 77–78 ] Consider t he conti nuous-time L TI s ystem with state-space realization ˙ x = Ax + Bu , where A ∈ R n × n , B ∈ R n × m and x ( 0 ) = x 0 . The Euclidean norm of the state satisﬁes k x ( T ) k 2 2 ≤ γ 2  k x 0 k 2 2 + k u k 2 2 T  , ∀ T ∈ R ≥ 0 if t here exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.120)  P 1 ∗ γ 1  ≥ 0 , (4.121)  P A + A T P P B ∗ − γ 1  ≤ 0 . (4.122) If x 0 = 0 and u is a unit-energy in p ut (i.e., k u k 2 T ≤ 1 , ∀ T ∈ R ≥ 0 ), then the preceding conditions ensure that k x ( T ) k 2 ≤ γ , ∀ T ∈ R ≥ 0 . 106 Pr oof. Deﬁne V = x T Px . Evaluating ˙ V results i n ˙ V =  x T u T   P A + A T P PB ∗ 0   x u  =  x T u T   P A + A T P P B ∗ − γ 1   x u  + γ u T u . (4.123) Substitutin g ( 4.122 ) into ( 4.123 ) give s ˙ V ≤ γ u T u . Integrating both sides of this inequality from t = 0 to t = T , where T ∈ R ≥ 0 yields x T ( T ) Px ( T ) ≤ x T 0 Px 0 + γ k u k 2 2 T . ( 4.12 4 ) Substitutin g ( 4.120 ) and ( 4.121 ) int o ( 4.124 ) resul ts in γ − 1 x T ( T ) x ( T ) ≤ γ x T 0 x 0 + γ k u k 2 2 T k x ( T ) k 2 2 ≤ γ 2  k x 0 k 2 2 + k u k 2 2 T  . 4.19.4 T ransient State Bound for Discrete-T ime Non-A utonomous L TI Systems Consider t he dis crete-time L TI sys t em with state-space realization x k +1 = A d x k + B d u k , where A d ∈ R n × n and B d ∈ R n × m . The Euclidean norm of th e state satisﬁes k x k k 2 2 ≤ γ 2  k x 0 k 2 2 + k u k 2 2 k  , ∀ k ∈ Z ≥ 0 if t here exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.125)  P 1 ∗ γ 1  ≥ 0 , (4.126)  A T d P A d − P A T d PB d ∗ B T d B d − γ 1  ≤ 0 . (4.127) If x 0 = 0 and u is a unit -ener gy i n put (i.e., k u k 2 k ≤ 1 , ∀ k ∈ Z ≥ 0 ), then the p receding conditions ensure th at k x k k 2 ≤ γ , ∀ k ∈ Z ≥ 0 . Pr oof. Deﬁne V ( k ) = x T k Px k . Evaluating V ( k + 1) − V ( k ) results in V ( k + 1) − V ( k ) =  x T k u T k   A T d P A d − P A T d PB d ∗ B T d B d   x k u k  =  x T k u T k   A T d P A d − P A T d PB d ∗ B T d B d − γ 1   x k u k  + γ u T k u k . (4.1 28) 107 Substitutin g i n ( 4.127 ) and using induction gives x T k Px k ≤ x T 0 Px 0 + γ k X i =0 u T i u i . (4.129) Substitutin g ( 4.1 25 ) and ( 4.126 ) into ( 4.129 ) yi elds γ − 1 x T k x k ≤ γ x T 0 x 0 + γ k u k 2 2 k k x k k 2 2 ≤ γ 2  k x 0 k 2 2 + k u k 2 2 k  . 4.19.5 T ransient Output Bound for A utonomous L TI Systems [ 1 , p. 88], [ 239 ] Consider t he conti nuous-time L TI s ystem with state-space realization ˙ x = Ax , y = Cx , where A ∈ R n × n , C ∈ R p × n and x ( 0 ) = x 0 . The Euclidean norm of th e outp u t s atisﬁes k y ( T ) k 2 ≤ γ k x 0 k 2 , ∀ T ∈ R ≥ 0 if t here exist P ∈ S p and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.130)  P C T ∗ γ 1  ≥ 0 , (4.131) P A + A T P ≤ 0 . Pr oof. The proo f fol l ows the same procedure as the proof in Section 4.19.1 , except the inequali t ies in ( 4.130 ) and ( 4.131 ) are substituted in to the inequality of ( 4.115 ). 4.19.6 T ransient Output Bound for Discrete- Time A utonomous L TI Systems Consider t he dis crete-time L TI sys t em with state-space realization x k +1 = A d x k , y k = C d x k , where A d ∈ R n × n and C d ∈ R p × n . The Euclidean norm of th e outp u t s atisﬁes k y k k 2 ≤ γ k x 0 k 2 , ∀ k ∈ Z ≥ 0 if t here exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.132)  P C T d ∗ γ 1  ≥ 0 , (4.1 3 3 ) A T d P A d − P ≤ 0 . Pr oof. The proo f fol l ows the same procedure as the proof in Section 4.19.2 , except the inequali t ies in ( 4.132 ) and ( 4.133 ) are substituted in to the inequality of ( 4.119 ). 108 4.19.7 T ransient Output Bound for Non-A utonomous L TI Systems Consider t he conti nuous-time L TI s ystem with state-space realization ˙ x = Ax + Bu , y = Cx , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and x (0 ) = x 0 . The Eucli d ean norm of the output satisﬁes k y ( T ) k 2 2 ≤ γ 2  k x 0 k 2 2 + k u k 2 2 T  , ∀ T ∈ R ≥ 0 if t here exist P ∈ S p and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.134)  P C T ∗ γ 1  ≥ 0 , (4.135)  P A + A T P P B ∗ − γ 1  ≤ 0 . If x 0 = 0 and u is a unit-energy in p ut (i.e., k u k 2 T ≤ 1 , ∀ T ∈ R ≥ 0 ), then the preceding conditions ensure that k y ( T ) k 2 ≤ γ , ∀ T ∈ R ≥ 0 . Pr oof. The proo f fol l ows the same procedure as the proof in Section 4.19.3 , except the inequali t ies in ( 4.134 ) and ( 4.135 ) are sub stituted in to the inequality of ( 4.124 ). 4.19.8 T ransient Output Bound for Discrete- Time Non-A utonomous L TI Systems Consider t he dis crete-time L TI sys t em with state-space realization x k +1 = A d x k + C d u k , y k = C d x k , where A d ∈ R n × n , B d ∈ R n × m , and C d ∈ R p × n . The Euclidean norm of t h e output sati sﬁes k y k k 2 2 ≤ γ 2  k x 0 k 2 2 + k u k 2 2 k  , ∀ k ∈ Z ≥ 0 if t here exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.136)  P C T d ∗ γ 1  ≥ 0 , (4.137)  A T d P A d − P A T d PB d ∗ B T d B d − γ 1  ≤ 0 . If x 0 = 0 and u is a unit -ener gy i n put (i.e., k u k 2 k ≤ 1 , ∀ k ∈ Z ≥ 0 ), then the p receding conditions ensure th at k y k k 2 ≤ γ , ∀ k ∈ Z ≥ 0 . Pr oof. The proo f fol l ows the same procedure as the proof in Section 4.19.4 , except the inequali t ies in ( 4.136 ) and ( 4.137 ) are substituted in to the inequality of ( 4.129 ). 109 4.19.9 T ransient Impulse Response Bound [ 185 ] Consider the sin g le-input mu lti-output continuous-tim e L TI system with state-space realization ˙ x = Ax + B u, y = Cx , where A ∈ R n × n , B ∈ R n × 1 , and C ∈ R p × n . Let z ( t ) = C e A t B be the unit im pulse response of the s ystem. The E uclidean norm of the impulse response satisﬁes k z ( T ) k 2 ≤ γ , ∀ T ∈ R ≥ 0 if t here exist P ∈ S p and γ ∈ R > 0 , wh ere P > 0 , such that  P PB ∗ γ  ≥ 0 , (4.138)  P C T ∗ γ 1  ≥ 0 , (4.139) P A + A T P ≤ 0 . Pr oof. The proof follows the same procedure as the proof in Section 4.19.5 , where the i nitial condition is chosen as x 0 = B . This yi el d s the result x T ( T ) Px ( T ) ≤ B T PB . (4.140) Using the non -strict Schur complement, the matrix inequality in ( 4.138 ) is equiv alent to B T PB ≤ γ . Subst ituting this and ( 4.139 ) into ( 4.140 ) giv es the desired result. 4.19.10 Discrete-T ime T ransient Impulse Response Bound Consider t he sin gle-input multi -output d i screte-time L TI sy stem with state-space realization x k +1 = A d x k + B d u k , y k = C d x k , where A d ∈ R n × n , B d ∈ R n × 1 , C d ∈ R p × n , and i t is assum ed that A d is in vertible. Let z k = C d A k − 1 d B d be the uni t impulse response of t he system. The Euclidean norm of the im pulse response satisﬁes k z k k 2 ≤ γ , ∀ k ∈ Z ≥ 0 if t here exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that  P P A − 1 d B d ∗ γ  ≥ 0 , (4.141)  P C T d ∗ γ 1  ≥ 0 , (4.142) A T d P A d − P ≤ 0 . 110 Pr oof. The proof follows the same procedure as the proof in Section 4.19.6 , where th e initial condition is chosen as x 0 = A − 1 d B d so that t he unit impulse respon s e matching t h e free respon s e z k = C d A k d x 0 . Thi s yields th e result x T k Px k ≤ B T d A − T d P A − 1 d B d . (4.143) Using the non-strict Schur complement, the matri x inequality in ( 4.141 ) is equiv alent to the in - equality B T d A − T d P A − 1 d B d ≤ γ . Substituti ng this and ( 4.142 ) into ( 4.143 ) giv es the d esired re- sult. 4.20 Output Energy Bounds 4.20.1 Output Ener gy Bound f or A utonomous L TI Systems [ 1 , pp. 85–86] Consider t he conti nuous-time L TI s ystem with state-space realization ˙ x = Ax , y = Cx , where A ∈ R n × n , C ∈ R p × n and x ( 0 ) = x 0 . The output satis ﬁes s Z T 0 y T y d t = k y k 2 T ≤ γ k x 0 k 2 , ∀ T ∈ R ≥ 0 if t here exist P ∈ S p and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.144)  P A + A T P C T ∗ − γ 1  ≤ 0 . (4.145) Pr oof. Deﬁne V = x T Px . Evaluating ˙ V results i n ˙ V = x T  P A + A T P  x = x T  P A + A T P + γ − 1 C T C  x − γ − 1 y T y . (4.146 ) Using the Schur compl em ent lemma and substitutin g ( 4.145 ) into ( 4.146 ) giv es ˙ V ≤ − γ − 1 y T y . Integrating both sides of this inequali ty from t = 0 to t = T , where T ∈ R ≥ 0 yields γ − 1 k y k 2 2 T ≤ − x T ( T ) Px ( T ) + x T 0 Px 0 ≤ x T 0 Px 0 (4.147) Substitutin g ( 4.144 ) int o ( 4.147 ) resul ts in γ − 1 k y k 2 2 T ≤ γ x T 0 x 0 k y k 2 T ≤ γ k x 0 k 2 . 111 4.20.2 Output Ener gy Bound f or Discrete-T ime A utonomous L TI Systems Consider t he dis crete-time L TI sys t em with state-space realization x k +1 = A d x k , y k = C d x k , where A d ∈ R n × n and C d ∈ R p × n . The output satis ﬁes k y k 2 k ≤ γ k x 0 k 2 , ∀ k ∈ Z ≥ 0 if t here exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.1 4 8 )  A T d P A d − P C T d ∗ − γ 1  ≤ 0 . (4.149) Pr oof. Deﬁne V ( k ) = x T k Px k . Evaluating V ( k + 1) − V ( k ) results in V ( k + 1) − V ( k ) = x T k  A T d P A d − P  x k = x T k  A T d P A d − P + γ − 1 C T d C d  x k − γ − 1 y T k y k . (4.150) Using th e Schur complement lemma, substitutin g ( 4.149 ) into ( 4.150 ), and usin g inductio n g ives γ − 1 k X i =0 y T i y i ≤ − x T k Px k + x T 0 Px 0 γ − 1 k y k 2 2 k ≤ − x T k Px k + x T 0 Px 0 ≤ x T 0 Px 0 (4.151) Substitutin g ( 4.148 ) int o ( 4.151 ) y i elds γ − 1 k y k 2 2 k ≤ γ x T 0 x 0 k y k 2 k ≤ γ k x 0 k 2 . 4.20.3 Output Ener gy Bound f or Non-A utonomous L TI Systems Consider t he conti nuous-time L TI s ystem with state-space realization ˙ x = Ax + Bu , y = Cx + Du , where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , D ∈ R p × m , and x (0) = x 0 . The output satis ﬁes Z T 0 y T y d t = k y k 2 2 T ≤ γ 2  k x 0 k 2 2 + k u k 2 2 T  , ∀ T ∈ R ≥ 0 112 if t here exist P ∈ S p and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.152)   P A + A T P P B C T ∗ − γ 1 D T ∗ ∗ − γ 1   ≤ 0 . (4.153) If x 0 = 0 , then th e preceding condition s match th e Bounded Real Lemma and ensure that k y k 2 T ≤ γ k u k 2 T , ∀ T ∈ R ≥ 0 . Pr oof. Deﬁne V = x T Px . Evaluating ˙ V results i n ˙ V =  x T u T   P A + A T P PB ∗ 0   x u  =  x T u T   P A + A T P + γ − 1 C T C P B + γ − 1 C T D ∗ − γ 1 + γ − 1 D T D   x u  + γ u T u − γ − 1 y T y . (4.154) Using t he Schur complement lemm a and sub stituting ( 4.153 ) into ( 4.154 ) gives ˙ V ≤ γ u T u − γ − 1 y T y . Integrating both sides of this i n equality from t = 0 to t = T , where T ∈ R ≥ 0 yields γ − 1 k y k 2 2 T ≤ − x T ( T ) Px ( T ) + x T 0 Px 0 + γ k u k 2 2 T ≤ x T 0 Px 0 + γ k u k 2 2 T (4.155) Substitutin g ( 4.152 ) int o ( 4.155 ) resul ts in γ − 1 k y k 2 2 T ≤ γ x T 0 x 0 + γ k u k 2 2 T k y k 2 2 T ≤ γ 2  k x 0 k 2 2 + k u k 2 2 T  . 4.20.4 Output Ener gy Bound f or Discrete-T ime Non-A utonomous L TI Systems Consider t he dis crete-time L TI sys t em with state-space realization x k +1 = A d x k + B d u k , y k = C d x k + D d u k , where A d ∈ R n × n , B d ∈ R n × m , C d ∈ R p × n , and D d ∈ R p × m . The output satisﬁes k X i =0 y T i y i = k y k 2 2 k ≤ γ 2  k x 0 k 2 2 + k u k 2 2 k  , ∀ k ∈ Z ≥ 0 if t here exist P ∈ S n and γ ∈ R > 0 , wh ere P > 0 , such that P − γ 1 ≤ 0 , (4.156)   A T d P A d − P A T d PB d C T d ∗ B T d B d − γ 1 D T d ∗ ∗ − γ 1   ≤ 0 . (4.157) If x 0 = 0 , then th e preceding condition s match th e Bounded Real Lemma and ensure that k y k 2 k ≤ γ k u k 2 k , ∀ k ∈ Z ≥ 0 . 113 Pr oof. Deﬁne V ( k ) = x T k Px k . Evaluating V ( k + 1) − V ( k ) results in V ( k + 1) − V ( k ) =  x T k u T k   A T d P A d − P A T d PB d ∗ B T d B d   x k u k  =  x T k u T k   A T d P A d − P + γ − 1 C T d C d A T d PB d + γ − 1 C T d D d ∗ B T d B d − γ 1 + γ − 1 D T d D d   x k u k  + γ u T k u k − γ − 1 y T k y k . (4.158) Using th e Schur complement lemma, substitutin g ( 4.157 ) into ( 4.158 ), and usin g inductio n g ives γ − 1 k X i =0 y T i y i ≤ − x T k Px k + x T 0 Px 0 + γ k X i =0 u T i u i γ − 1 k y k 2 2 k ≤ − x T k Px k + x T 0 Px 0 + γ k u k 2 2 k ≤ x T 0 Px 0 + γ k u k 2 2 k (4.159) Substitutin g ( 4.156 ) int o ( 4.159 ) y i elds γ − 1 k y k 2 2 k ≤ γ x T 0 x 0 + γ k u k 2 2 k k y k 2 2 k ≤ γ 2  k x 0 k 2 2 + γ k u k 2 2 k  . 4.21 Kharitonov-Bern stein-Haddad (KB H) Theor em [ 24 0 ] Consider t he set of matrices A = ( A =  0 ( n − 1) × 1 1 ( n − 1) × ( n − 1) − a 0 · · · − a n − 1  | ¯ a j ≤ a j ≤ ¯ a j , j = 0 , 1 , 2 , . . . , n − 1 ) . (4.16 0) Every matrix i n t he set A is Hurwitz if and only if there exist P i ∈ S n , i = 1 , 2 , 3 , 4 , where P i > 0 , i = 1 , 2 , 3 , 4 , such that P i A i + A T i P i < 0 , i = 1 , 2 , 3 , 4 , where A i =   0 ( n − 1) × 1 1 ( n − 1) × ( n − 1)  a i  , i = 1 , 2 , 3 , 4 , a 1 = −  ¯ a 0 ¯ a 1 ¯ a 2 ¯ a 3 · · · ¯ a n − 4 ¯ a n − 3 ¯ a n − 2 ¯ a n − 1  , a 2 = −  ¯ a 0 ¯ a 1 ¯ a 2 ¯ a 3 · · · ¯ a n − 4 ¯ a n − 3 ¯ a n − 2 ¯ a n − 1  , a 3 = −  ¯ a 0 ¯ a 1 ¯ a 2 ¯ a 3 · · · ¯ a n − 4 ¯ a n − 3 ¯ a n − 2 ¯ a n − 1  , a 4 = −  ¯ a 0 ¯ a 1 ¯ a 2 ¯ a 3 · · · ¯ a n − 4 ¯ a n − 3 ¯ a n − 2 ¯ a n − 1  . Equiv alently , ev ery matrix in the set A is Hurwit z if and o nly if there exist Q i ∈ S n , i = 1 , 2 , 3 , 4 , where Q i > 0 , i = 1 , 2 , 3 , 4 , such that A i Q i + Q i A T i < 0 , i = 1 , 2 , 3 , 4 . 114 4.22 Stability of Discrete-T ime System with Polytopic Uncertainty 4.22.1 Open-Loop Rob ust Stability [ 144 ] Consider t he set of matrices A = ( A d ( α ) ∈ R n × n | A d ( α ) = n X i =1 α i A d ,i , A d ,i ∈ R n × n , α i ∈ R ≥ 0 , n X i =1 α i = 1 ) . The discrete-time L TI syst em x k +1 = A d ( α ) x k is asymptotically stabl e for all A d ( α ) ∈ A i f there exist P i ∈ S n , i = 1 , . . . , n , and G ∈ R n × n , wh ere P i > 0 , i = 1 , . . . , n , such that  P i A T d ,i G T ∗ G + G T − P i  < 0 , i = 1 , . . . , n. 4.22.2 Closed-Loop R o b ust Stability [ 144 ] Consider t he set of matrices A = ( A d ( α ) ∈ R n × n | A d ( α ) = n X i =1 α i A d ,i , A d ,i ∈ R n × n , α i ∈ R ≥ 0 , n X i =1 α i = 1 ) . and B = ( B d ( β ) ∈ R n × m | B d ( β ) = p X i =1 β i B d ,i , B d ,i ∈ R n × m , β i ∈ R ≥ 0 , m X i =1 β i = 1 ) . The discrete-time L TI system x k +1 = A d ( α ) x k + B d ( β ) u k is asymp totically s tabilized by the state feedback control law u k = − LG − 1 u k for all A d ( α ) ∈ A and B d ( α ) ∈ B if there exist P ij ∈ S n , i = 1 , . . . , n , j = 1 , . . . , p , G ∈ R n × n , and L ∈ R m × n , where P ij > 0 , i = 1 , . . . , n , j = 1 , . . . , p and G is in vertible, such that  P ij A d ,i G − B d ,j L ∗ G + G T − P ij  < 0 , i = 1 , . . . , n, j = 1 , . . . , p. 4.23 Quadratic Stability 4.23.1 Continuous-T ime Quadratic Stability [ 5 , pp. 112–115] Consider t he uncertain continuous-tim e lin ear s ystem wit h state-space representatio n ˙ x = ( A 0 + ∆ A ( δ ( t ))) x , (4.161) where A 0 ∈ R n × n , ∆ A ( δ ( t )) = P k i =1 δ i ( t ) A i ∈ R n × n , δ i ∈ R , i = 1 , . . . , k , A i ∈ R n × n , i = 1 , . . . , k , δ T ( t ) =  δ 1 ( t ) δ 2 ( t ) · · · δ k ( t )  ∈ ∆ , and ∆ is the s et of perturbation parameters. The un certain system in ( 4.161 ) is quadratically stable i f there exists P ∈ S n , where P > 0 , such that ( A 0 + ∆ A ( δ ( t ))) T P + P ( A 0 + ∆ A ( δ ( t ))) < 0 , ∀ δ ( t ) ∈ ∆ . The fol l owing statements can be m ade for particular s ets of perturbations. 115 1. Consider the case where the set of perturbation parameters is deﬁned by a regular poly h edron as ∆ = { δ ( t ) =  δ 1 ( t ) δ 2 ( t ) · · · δ k ( t )  ∈ R k | δ i ( t ) , ¯ δ i , ¯ δ i ∈ R , ¯ δ i ≤ δ i ( t ) ≤ ¯ δ i ] } . The uncertain system in ( 4.161 ) is quadratically st abl e i f and only if there exists P ∈ S n , where P > 0 , such that ( A 0 + ∆ A ( δ ( t ))) T P + P ( A 0 + ∆ A ( δ ( t ))) < 0 , ∀ δ i ( t ) ∈ { ¯ δ i , ¯ δ i } , i = 1 , . . . , k . 2. Consider the case where the set of perturbation parameters is deﬁned by a p o lytope as ∆ = { δ ( t ) =  δ 1 ( t ) δ 2 ( t ) · · · δ k ( t )  ∈ R k | δ i ( t ) ∈ R ≥ 0 , k X i =1 δ i ( t ) = 1 } . The uncertain system in ( 4.161 ) is quadratically st abl e i f and only if there exists P ∈ S n , where P > 0 , such that ( A 0 + A i ) T P + P ( A 0 + A i ) < 0 , i = 1 , . . . , k . 4.23.2 Discrete-T ime Quadratic Stability [ 5 , pp. 116–118] Consider t he uncertain discrete-time linear system wi th state-space representation x k +1 = ( A d , 0 + ∆ A d ( δ ( t ))) x k , (4.162) where A d , 0 ∈ R n × n , ∆ A d ( δ ( t )) = P k i =1 δ i ( t ) A d ,i ∈ R n × n , δ i ∈ R , i = 1 , . . . , k , A d ,i ∈ R n × n , i = 1 , . . . , k , δ T ( t ) =  δ 1 ( t ) δ 2 ( t ) · · · δ k ( t )  ∈ ∆ , and ∆ is the s et of perturbation parameters. The un certain system in ( 4.161 ) is quadratically stable i f there exists P ∈ S n , where P > 0 , such that ( A d , 0 + ∆ A d ( δ ( t ))) T P ( A d , 0 + ∆ A d ( δ ( t ))) − P < 0 , ∀ δ ( t ) ∈ ∆ . The fol l owing statements can be m ade for particular s ets of perturbations. 1. Consider the case where the set of perturbation parameters is deﬁned by a regular poly h edron as ∆ = { δ ( t ) =  δ 1 ( t ) δ 2 ( t ) · · · δ k ( t )  ∈ R k | δ i ( t ) , ¯ δ i , ¯ δ i ∈ R , ¯ δ i ≤ δ i ( t ) ≤ ¯ δ i ] } . The uncertain system in ( 4.161 ) is quadratically st abl e i f and only if there exists P ∈ S n , where P > 0 , such that ( A d , 0 + ∆ A d ( δ ( t ))) T P ( A d , 0 + ∆ A d ( δ ( t ))) − P < 0 , ∀ δ i ( t ) ∈ { ¯ δ i , ¯ δ i } , i = 1 , 2 , . . . , k . 2. Consider the case where the set of perturbation parameters is deﬁned by a p o lytope as ∆ = { δ ( t ) =  δ 1 ( t ) δ 2 ( t ) · · · δ k ( t )  ∈ R k | δ i ( t ) ∈ R ≥ 0 , k X i =1 δ i ( t ) = 1 } . The uncertain system in ( 4.161 ) is quadratically st abl e i f and only if there exists P ∈ S n , where P > 0 , such that ( A d , 0 + A d ,i ) T P ( A d , 0 + A d ,i ) − P < 0 , i = 1 , 2 , . . . , k . 116 4.24 Stability of Time-Delay Systems Consider t he conti nuous-time linear time-delay system with state-space representation ˙ x ( t ) = Ax ( t ) + A d x ( t − d ) , (4.1 63) where A , A d ∈ R n × n , d , ¯ d ∈ R > 0 , and the initi al condi tion is gi ven by x ( t ) = φ ( t ) , t ∈ [ − d, 0] , where ¯ d is a k n o wn upper -bound on t he time-delay (i.e., 0 < d ≤ ¯ d ). 4.24.1 Delay-Independent Co ndition [ 5 , p. 126], [ 241 , pp. 18–19] The t ime-delay syst em in ( 4.163 ) is asy m ptotically st abl e if there exist P , S ∈ S n , w h ere P > 0 and S > 0 , such that  A T P + P A + S P A d ∗ − S  < 0 . 4.24.2 Delay-Dependent Condition The t ime-delay syst em in ( 4.163 ) is u n iformly asympto t ically stable under either o f th e follow- ing sufﬁcient conditions . 1. [ 5 , pp. 128–129] There exist X ∈ S n and β ∈ R > 0 , wh ere X > 0 and β < 1 , such t h at   X ( A + A d ) T + ( A + A d ) X + ¯ d A d A T d ¯ d XA T ¯ d XA T d ∗ − ¯ dβ 1 0 ∗ ∗ − ¯ d (1 − β ) 1   < 0 . 2. [ 241 , pp. 19–21] There exist X , Q 1 , Q 2 ∈ S n , wh ere X > 0 , Q 1 > 0 , and Q 2 > 0 , such that   X ( A + A d ) T + ( A + A d ) X + ¯ d ( Q 1 + Q 2 ) ¯ d XA d ¯ d XA d ∗ − Q 1 0 ∗ ∗ − Q 2   < 0 . 4.25 µ -Analysis [ 1 , p. 38–39], [ 242 ] Consider the m atrix A ∈ C n × n and the in vertible matrix D ∈ C n × n . The inequality ¯ σ ( D AD − 1 ) < γ holds if and only if there exist X ∈ C n × n and γ ∈ R > 0 , wh ere X = X H > 0 , satisfying A T XA − γ 2 X < 0 . (4.164) The i n equality ¯ σ ( DAD − 1 ) < γ holds for D satisfying X = D H D and X sati s fying ( 4.164 ). 4.26 Static Output Feed back Algebraic Loop [ 7 , p. 1284], [ 216 , pp. 39–4 0 ] Consider a continuous -time L TI system , G : L 2 e → L 2 e , wi th state-space realization ˙ x = Ax + B 1 w + B 2 u , (4.165) z = C 1 x + D 11 w + D 12 u , (4.166) y = C 2 x + D 21 w + D 22 u , 117 where x ( t ) ∈ R n x is the system stat e, z ( t ) ∈ R n z is the performance signal, y ( t ) ∈ R n y is the measurement signal, w ( t ) ∈ R n w is the exogenous signal, u ( t ) ∈ R n u is the control input s i gnal, and t he state-space matrices are real matrices with appropriate dim ensions. Additio n ally , con s ider a static output feedback con troller of the form u = K y , where K ∈ R n u × n y and it is assum ed that the feedback i n t erconnection is well-posed, that is, det( 1 − KD 22 ) 6 = 0 . The closed-loop system can be described by the following state-space realization. ˙ x =  A + B 2 ¯ KC 2  x +  B 1 + B 2 ¯ KD 21  w , (4.167) z =  C 1 + D 12 ¯ KC 2  x +  D 11 + D 12 ¯ KD 21  w , (4.1 6 8) where ¯ K = ( 1 − KD 22 ) − 1 K . The change of va riable ¯ K = ( 1 − KD 22 ) − 1 K allows for the simp liﬁcation of matrix inequali- ties in volving the closed-loop sy s tem. Pr oof. Substitu t ing the expression for y into u = K y gives u = K ( C 2 x + D 21 w + D 22 u ) . Bringing the terms wi th u to the left-hand-side of the equati o n, left-mult i plying by ( 1 − K D 22 ) − 1 , and d eﬁning ¯ K = ( 1 − KD 22 ) − 1 K yields ( 1 − K D 22 ) u = KC 2 x + KD 21 w u = ( 1 − K D 22 ) − 1 KC 2 x + ( 1 − K D 22 ) − 1 KD 21 w u = ¯ KC 2 x + ¯ KD 21 w . (4.169) Substitutin g ( 4.169 ) int o ( 4.165 ) and ( 4.166 ) giv es ( 4.167 ) and ( 4.168 ). 118 5 LMIs in O p timal Control This section presents controller synthesis methods using LMIs for a number of well-known optimal control prob l ems. The deri vation of the LMIs used for cont rol ler synthesis is p rovided in some cases, while longer deriv ations can be found in t he cited references. 5.1 The Generalized Plant 5.1.1 The Continuous-T ime Generalized Plant w P z K y u Figure 1: Block diagram of the generalized plant P with the controller K . Consider the g eneralized L TI plant P : L 2 e → L 2 e , shown in Figure 1 , wit h a min imal state-space realization [ 7 , pp. 1291–1292], [ 4 , Section 3.8], [ 243 , p. 141], [ 244 , pp . 14 – 16], [ 245 , pp. 809–8 1 7] ˙ x = Ax + B 1 w + B 2 u , z = C 1 x + D 11 w + D 12 u , y = C 2 x + D 21 w + D 22 u , where x ( t ) ∈ R n x is the system stat e, z ( t ) ∈ R n z is the performance signal, y ( t ) ∈ R n y is the measurement signal, w ( t ) ∈ R n w is the exogenous signal, u ( t ) ∈ R n u is the control input s i gnal, and the state-space matrices are real matrices with appropriate dimens ions. The generalized L TI plant can also be written in transfer m atrix form as  z ( s ) y ( s )  = P ( s )  w ( s ) u ( s )  , where the transfer m atrix P ( s ) ∈ C ( n z + n y ) × ( n w + n u ) is partitioned as P ( s ) =  P z w ( s ) P z u ( s ) P y w ( s ) P y u ( s )  =  C 1 ( s 1 − A ) − 1 B 1 + D 11 C 1 ( s 1 − A ) − 1 B 2 + D 12 C 2 ( s 1 − A ) − 1 B 1 + D 21 C 2 ( s 1 − A ) − 1 B 2 + D 22  . The generalized pl ant, also known as the standard control problem in [ 7 , pp. 1291–1292], [ 244 , pp. 14–16], [ 246 ], is useful, as it i s p o ssible to represent a number of L TI systems in this form, as shown i n the fol lowing example. 119 G p ( s ) K ( s ) y p ( s ) d ( s ) u c ( s ) r ( s ) − W d ( s ) W n ( s ) W r ( s ) n ( s ) Figure 2: Block diagram of the basic servo loop with plant G p ( s ) , controll er K ( s ) , and weight ing transfer matri ces W r ( s ) , W d ( s ) , and W n ( s ) . Example 5.1 (Basic Servo Loop T racking [ 216 , p. 18], [ 244 , p. 18], [ 246 ]) . Cons ider the basic servo loop shown in Figure 2 i n v olv i ng the L T I controller K ( s ) ∈ C n y c × n u c and t he plant G p ( s ) ∈ C n y p × n u p , where th e weig h ting transfer matrices are simply chosen as W r ( s ) = 1 , W d ( s ) = 1 , and W n ( s ) = 1 . Th e plant G p ( s ) has a minim al state-space realization ( A p , B p , C p , D p ) and t he state x p ( t ) . T h e p erformance variables are t h e t rue tracking error z 1 ( t ) = e ( t ) = r ( t ) − y p ( t ) and t h e control ef fort z 2 ( t ) = u c ( t ) , where z T ( t ) =  z T 1 ( t ) z T 2 ( t )  . The generalized plant can be formulated wit h minim al s tate-space representation ˙ x = A p x +  0 B p 0  w + B p u , z =  − C p 0  x +  1 − D p 0 0 0 0  w +  − D p 1  u , y = − C p x +  1 − D p − 1  w − D p u , where x ( t ) = x p ( t ) , w T ( t ) =  r T ( t ) d T ( t ) n T ( t )  , u ( t ) = u c ( t ) , and y ( t ) = r ( t ) − y p ( t ) − n ( t ) . Example 5.2 (Basic Serv o Loop Tracking wi th W eights [ 4 , Section 9.3.6], [ 216 , p. 19], [ 247 , pp. 169–1 7 0]) . Consid er the same basic serv o lo op s hown in Figure 2 in volving the L TI controller K ( s ) ∈ C n y c × n u c , the plant G p ( s ) ∈ C n y p × n u p , and the wei g hting t ransfer matrices W r ( s ) ∈ C n r × n r , W d ( s ) ∈ C n d × n d , and W n ( s ) ∈ C n n × n n . The plant G p ( s ) has a minim al state-space real- ization ( A p , B p , C p , D p ) and the weighting transfer matrices W r ( s ) , W d ( s ) , and W n ( s ) have mini - mal state-space realizati o ns ( A r , B r , C r , D r ) , ( A d , B d , C d , D d ) , and ( A n , B n , C n , D n ) , respectively . The performance variable is deﬁned as the weigh t ed true tracking error z 1 ( s ) = W e ( s ) e ( s ) = W e ( s ) ( W r ( s ) r ( s ) − y p ( s )) and th e weighted control eff ort z 2 ( s ) = W u ( s ) u c ( s ) , where z T ( s ) =  z T 1 ( s ) z T 2 ( s )  and W e ( s ) ∈ C n e × n e , W u ( s ) ∈ C n u × n u are weighting transfer matrices with mi n- imal state-space realizations ( A e , B e , C e , D e ) and ( A u , B u , C u , D u ) , respecti vely . The generalized 120 plant can be formu l ated with minimal state-space representation ˙ x =         A p 0 B p C d 0 0 0 0 A r 0 0 0 0 0 0 A d 0 0 0 0 0 0 A n 0 0 − B e C p B e C r − B e D p C d 0 A e 0 0 0 0 0 0 A u         x +         0 B p D d 0 B r 0 0 0 B d 0 0 0 B n B e D r − B e D p D d 0 0 0 0         w +         B p 0 0 0 − B e D p B u         u , z =  − D e C p D e C r − D e D p C d 0 C e 0 0 0 0 0 0 C u  x +  D e D r − D e D p D d 0 0 0 0  w +  − D e D p D u  u , y =  − C p C r − D p C d − C n 0 0  x +  D r − D p D d − D n  w − D p u , where x T ( t ) =  x T p ( t ) x T r ( t ) x T d ( t ) x T n ( t ) x T e ( t ) x T u ( t )  , w T ( t ) =  r T ( t ) d T ( t ) n T ( t )  , u ( t ) = u c ( t ) , y ( s ) = W r ( s ) r ( s ) − y p ( s ) − W n ( s ) n ( s ) , and x r ( t ) , x d ( t ) , x n ( t ) , x e ( t ) , and x u ( t ) are the states asso ci at ed wi t h the state-space realizations of the weightin g transfer matrices W r ( s ) , W d ( s ) , W n ( s ) , W e ( s ) , and W u ( s ) , respectively . 5.1.2 The Discr ete-T ime Generalized Plant The discrete-time generalized L TI plant P : ℓ 2 e → ℓ 2 e , shown in Figure 1 , is described by the state-space realization x k +1 = A d x k + B d1 w k + B d2 u k , z k = C d1 x k + D d11 w k + D d12 u k , y k = C d2 x k + D d21 w k + D d22 u k , where x k ∈ R n x is t he system state at tim e step k , z k ∈ R n z is the performance sig nal at tim e step k , y k ∈ R n y is the measurement signal at time step k , w k ∈ R n w is the exogenous signal at time st ep k , u k ∈ R n u is the control input sign al at time step k , and the state-space m atrices hav e appropriate di mensions. The generalized L TI plant can also be written in discrete-time transfer matrix form as  z ( z ) y ( z )  = P ( z )  w ( z ) u ( z )  , where the transfer m atrix P ( z ) ∈ C ( n z + n y ) × ( n w + n u ) is partitioned as P ( z ) =  P z w ( z ) P z u ( z ) P y w ( z ) P y u ( z )  =  C d1 ( z 1 − A d ) − 1 B d1 + D d11 C d1 ( z 1 − A d ) − 1 B d2 + D d12 C d2 ( z 1 − A d ) − 1 B d1 + D d21 C d2 ( z 1 − A d ) − 1 B d2 + D d22  . 5.2 H 2 -Optimal Contr ol The goal of H 2 -optimal control is to design a cont roller that minimi zes t he H 2 norm of the closed-loop t ransfer matrix from w to z . 121 5.2.1 H 2 -Optimal Full-State Feed back Contr ol [ 5 , pp. 257– 2 58] Consider t he conti nuous-time generalized L TI plant P with state-space realization ˙ x = Ax + B 1 w + B 2 u , (5.1) z = C 1 x + D 12 u , (5.2) y = x , where it is assumed that ( A , B 2 ) is stabili zable. A ful l -state feedback controller K = K ∈ R n u × n x (i.e., u = K x ) is to be designed to minimize the H 2 norm of the closed loop transfer matrix from the exogenous input w to the performance output z . Subst ituting the full-st ate feedback control ler into ( 5.1 ) and ( 5.2 ) yi elds ˙ x = ( A + B 2 K ) x + B 1 w , z = ( C 1 + D 12 K ) x , and a closed-loop transfer matrix T ( s ) = ( C 1 + D 12 K ) ( s 1 − ( A + B 2 K )) − 1 B 1 . Minimizi n g the H 2 norm of the t ransfer matrix T ( s ) is equiv alent to mi nimizing J ( µ ) = µ 2 subject to  ( A + B 2 K ) P + P ( A + B 2 K ) T P ( C 1 + D 12 K ) T ∗ − 1  < 0 , (5.3)  Z B T 1 ∗ P  > 0 , (5.4) tr( Z ) < µ 2 , (5.5) where P ∈ S n x , Z ∈ S n w , µ ∈ R > 0 , P > 0 , and Z > 0 . A change of va riables is performed with F = KP and ν = µ 2 , which transforms ( 5.3 ) and ( 5.5 ) i nto LMIs in the v ariables P , F , Z , and ν giv en by  AP + P A T + B 2 F + F T B T 2 PC T 1 + F T D T 12 ∗ − 1  < 0 , (5. 6 ) tr( Z ) < ν. (5.7) Synthesis Method 5.1. The H 2 -optimal full-state feedback controller is s y nthesized by solvin g for P ∈ S n x , Z ∈ S n w , F ∈ R n u × n x , and ν ∈ R > 0 that minimi ze J ( ν ) = ν subject to P > 0 , Z > 0 , ( 5.4 ), ( 5.6 ), and ( 5.7 ). The H 2 -optimal full-state feedback gain i s recovered by K = FP − 1 and t he H 2 norm of T ( s ) i s µ = √ ν . 5.2.2 Discrete- Time H 2 -Optimal Full-State Feed back Contr ol Consider t he dis crete-time generalized L TI plant P with s t ate-space realization x k +1 = A d x k + B d1 w k + B d2 u k , z k = C d1 x k + D d12 u k , y k = x k , 122 where it is assumed that ( A d , B d2 ) i s stabil i zable. A full-state feedback con t roller K = K d ∈ R n u × n x (i.e., u k = K d x k ) is to b e designed to minim ize the H 2 norm of the cl o sed loop transfer matrix from the exogenous input w k to the performance out put z k , given by T ( z ) = ( C d1 + D d12 K d ) ( z 1 − ( A d + B d2 K d )) − 1 B d1 . Synthesis Method 5.2. The discrete-time H 2 -optimal full-st at e feedback controller is synt hesized by solvi ng for P ∈ S n x , Z ∈ S n z , F d ∈ R n u × n x , and ν ∈ R > 0 that min imize J ( ν ) = ν subject to P > 0 , Z > 0 ,   P A d P + B d2 F d B d 1 ∗ P 0 ∗ ∗ 1   > 0 ,  Z C d 1 P + D d12 F d ∗ P  > 0 . tr( Z ) < ν. The H 2 -optimal full-state feedback gain is recove red by K d = F d P − 1 and the H 2 norm of T ( z ) is µ = √ ν . 5.2.3 H 2 -Optimal Dynamic Output Fe edback Control [ 185 , 248 ] Consider t he conti nuous-time generalized L TI plant P with minimal state-space realization ˙ x = Ax + B 1 w + B 2 u , z = C 1 x + D 11 w + D 12 u , y = C 2 x + D 21 w + D 22 u . A continuo u s-time dyn am ic output feedback L TI control ler with state-space realization ( A c , B c , C c , D c ) is to be designed to mini m ize th e H 2 norm of the closed-loop system transfer matrix from w to z , giv en by T ( s ) = C CL ( s 1 − A CL ) − 1 B CL + D CL , where A CL = " A + B 2 D c ˜ D − 1 C 2 B 2  1 + D c ˜ D − 1 D 22  C c B c ˜ D − 1 C 2 A c + B c ˜ D − 1 D 22 C c # , B CL = " B 1 + B 2 D c ˜ D − 1 D 21 B c ˜ D − 1 D 21 # , C CL = h C 1 + D 12 D c ˜ D − 1 C 2 D 12  1 + D c ˜ D − 1 D 22  C c i , D CL = D 11 + D 12 D c ˜ D − 1 D 21 , and ˜ D = 1 − D 22 D c . 123 Synthesis Method 5 . 3 . Solve for A n ∈ R n x × n x , B n ∈ R n x × n y , C n ∈ R n u × n x , D n ∈ R n u × n y , X 1 , Y 1 ∈ S n x , Z ∈ S n z , and ν ∈ R > 0 that mi n imize J ( ν ) = ν subject to X 1 > 0 , Y 1 > 0 , Z > 0 ,   A Y 1 + Y 1 A T + B 2 C n + C T n B T 2 A + A T n + B 2 D n C 2 B 1 + B 2 D n D 21 ∗ X 1 A + A T X 1 + B n C 2 + C T 2 B T n X 1 B 1 + B n D 21 ∗ ∗ − 1   < 0 ,   X 1 1 Y 1 C T 1 + C T n D T 12 ∗ Y 1 C T 1 + C T 2 D T n D T 12 ∗ ∗ Z   > 0 , D 11 + D 12 D n D 21 = 0 , (5.8)  X 1 1 ∗ Y 1  > 0 , tr( Z ) < ν. The cont roller is recovered by A c = A K − B c ( 1 − D 22 D c ) − 1 D 22 C c , B c = B K ( 1 − D 22 D c ) , C c = ( 1 − D c D 22 ) C K , D c = ( 1 + D K D 22 ) − 1 D K , where  A K B K C K D K  =  X 2 X 1 B 2 0 1  − 1  A n B n C n D n  −  X 1 A Y 1 0 0 0   Y T 2 0 C 2 Y 1 1  − 1 , and the matrices X 2 and Y 2 satisfy X 2 Y T 2 = 1 − X 1 Y 1 . If D 22 = 0 , then A c = A K , B c = B K , C c = C K , and D c = D K . Giv en X 1 and Y 1 , the m atrices X 2 and Y 2 can be found usi n g a matrix decomposition, s u ch as a LU decompositi on or a Cholesky decom position. If D 11 = 0 , D 12 6 = 0 , and D 21 6 = 0 , t hen i t is often si m plest to cho o se D n = 0 in order to s at i sfy the equ al i ty constraint o f ( 5.8 ). 5.2.4 Discrete- Time H 2 -Optimal Dynamic Output Feedback Control Consider t he dis crete-time generalized L TI plant P with s t ate-space realization x k +1 = A d x k + B d1 w k + B d2 u k , z k = C d1 x k + D d11 w k + D d12 u k , y k = C d2 x k + D d21 w k + D d22 u k , A discrete-time dynam ic output feedback L TI control ler with state-space realization ( A d c , B d c , C d c , D d c ) is to be designed to minimize the H 2 norm of the closed-loop s y stem transfer matrix from w k to z k , given by T ( z ) = C d CL ( z 1 − A d CL ) − 1 B d CL + D d CL , 124 where A d CL = " A d + B d2 D d c ˜ D − 1 d C d2 B d2  1 + D d c ˜ D − 1 d D d22  C d c B d c ˜ D − 1 d C d2 A d c + B d c ˜ D − 1 d D d22 C d c # , B d CL = " B d1 + B d2 D d c ˜ D − 1 d D d21 B d c ˜ D − 1 d D d21 # , C d CL = h C d1 + D d12 D d c ˜ D − 1 d C d2 D d12  1 + D d c ˜ D − 1 d D d22  C d c i , D d CL = D d11 + D d12 D d c ˜ D − 1 d D d21 , and ˜ D d = 1 − D d22 D d c . Synthesis Method 5.4. [ 165 ] Solve for A d n ∈ R n x × n x , B d n ∈ R n x × n y , C d n ∈ R n u × n x , D d n ∈ R n u × n y , X 1 , Y 1 ∈ S n x , Z ∈ S n z , G , H , J , S ∈ R n x × n x , and ν ∈ R > 0 that m inimize J ( ν ) = ν subject to X 1 > 0 , Y 1 > 0 , Z > 0 ,       X 1 J T HA d + B d n C d2 A d n HB d1 + B d n D d21 ∗ Y 1 A d + B d2 D d n C d2 A d G + B d2 C d n B d1 + B d2 D d n D d21 ∗ ∗ H + H T − X 1 1 + S − J T 0 ∗ ∗ ∗ G + G T − Y 1 0 ∗ ∗ ∗ ∗ 1       > 0 , (5.9)   Z C d1 + D d12 D d n C d2 C d1 G + D d12 C d n ∗ H + H T − X 1 1 + S − J T ∗ ∗ G + G T − Y 1   > 0 , (5.10) D d11 + D d12 D d n D d21 = 0 , (5.11) tr( Z ) < ν. The cont roller is recovered by A d c = A d K − B d c ( 1 − D d22 D d c ) − 1 D d22 C d c , B d c = B d K ( 1 − D d22 D d c ) , C d c = ( 1 − D d c D d22 ) C d K , D d c = ( 1 + D d K D d22 ) − 1 D d K , where  A d K B d K C d K D d K  =  Y − T 2 Y − T 2 HB d2 0 1   A d n B d n C d n D d n  −  HA d G 0 0 0   X − 1 2 0 − C d2 GX − 1 2 1  , and the m atrices X 2 and Y 2 satisfy X 2 Y T 2 = 1 − H G . If D d22 = 0 , th en A d c = A d K , B d c = B d K , C d c = C d K , and D d c = D d K . Giv en G and H , the matrices X 2 and Y 2 can be found using a m atrix decomposi tion, such as a LU d ecom position or a Cholesky decomposi tion. If D d11 = 0 , D d12 6 = 0 , and D d21 6 = 0 , then it is often simplest to choose D d n = 0 in order to satisfy th e equality constraint of ( 5.11 ). 125 The LMI in ( 5.9 ) is derived from the LMI in Theorem 7 of [ 165 ] by performing a congruence transformation in volving a multipl i cation on the left and right by t h e symmetric matrix W 1 = diag n  0 1 1 0  ,  0 1 1 0  , 1 o . Similarly , t he LMI in ( 5.10 ) is derived from the LMI in Theorem 7 o f [ 165 ] by performing a congruence transformation in volving a m ultiplicati o n on the left and right by th e symmetri c matrix W 2 = diag n 1 ,  0 1 1 0  o . Synthesis Method 5.5. Solve for A d n ∈ R n x × n x , B d n ∈ R n x × n y , C d n ∈ R n u × n x , D d n ∈ R n u × n y , X 1 , Y 1 ∈ S n x , Z ∈ S n z , and ν ∈ R > 0 that mi n imize J ( ν ) = ν subject to X 1 > 0 , Y 1 > 0 , Z > 0 ,       X 1 1 X 1 A d + B d n C d2 A d n X 1 B d1 + B d n D d21 ∗ Y 1 A d + B d2 D d n C d2 A d Y 1 + B d2 C d n B d1 + B d2 D d n D d21 ∗ ∗ X 1 1 0 ∗ ∗ ∗ Y 1 0 ∗ ∗ ∗ ∗ 1       > 0 ,   Z C d1 + D d12 D d n C d2 C d1 Y 1 + D d12 C d n ∗ X 1 1 ∗ ∗ Y 1   > 0 , (5.12) D d11 + D d12 D d n D d21 = 0 , (5.13)  X 1 1 ∗ Y 1  > 0 , (5.14) tr( Z ) < ν. The cont roller is recovered by A d c = A d K − B d c ( 1 − D d22 D d c ) − 1 D d22 C d c , B d c = B d K ( 1 − D d22 D d c ) , C d c = ( 1 − D d c D d22 ) C d K , D d c = ( 1 + D d K D d22 ) − 1 D d K , where  A d K B d K C d K D d K  =  X 2 X 1 B d2 0 1  − 1  A d n B d n C d n D d n  −  X 1 A d Y 1 0 0 0   Y T 2 0 C d2 Y 1 1  − 1 , and t he matri ces X 2 and Y 2 satisfy X 2 Y T 2 = 1 − X 1 Y 1 . If D d22 = 0 , then A d c = A d K , B d c = B d K , C d c = C d K , and D d c = D d K . Giv en X 1 and Y 1 , the m atrices X 2 and Y 2 can be found usi n g a matrix decomposition, s u ch as a LU decompositi on or a Cholesky decom position. If D d11 = 0 , D d12 6 = 0 , and D d21 6 = 0 , then it is often simplest to choose D d n = 0 in order to satisfy th e equality constraint of ( 5.13 ). 126 The LMIs in ( 5.12 ) and ( 5.13 ) are deriv ed from ( 5.9 ) and ( 5.1 0 ) using the change of v ariables S = J = 1 , H = X 1 , G = Y 1 . The LMI in ( 5.14 ) is added to ensure that 1 − X 1 Y 1 ≥ 0 in a sim i lar fashion to the approach used in [ 185 ]. An al t ernate formulati on of th is synth es i s m ethod in volves replacing ( 5.12 ) and ( 5.13 ) with     Z C d1 + D d12 D d n C d2 C d1 Y 1 + D d12 C d n D d11 + D d12 D d n D d21 ∗ X 1 1 0 ∗ ∗ Y 1 0 ∗ ∗ ∗ 1     > 0 . (5. 1 5) The matrix i n equality in ( 5.15 ) is deriv ed by performing the same procedure used i n [ 165 ] with the change of var iables S = J = 1 , H = X 1 , G = Y 1 , but i nstead starting with t he matrix inequality formulat i on of the H 2 that allows for a non-zero feedthrough term in [ 176 , p. 25] (sum - marized by ( 4.40 ), ( 4.41 ), and ( 4.42 )). In general, the matrix inequality in ( 5.15 ) is less conser - vati ve than ( 5.12 ) and ( 5.13 ), as it allows for the resulting closed-l o op system to have non -zero feedthrough, which, for a discrete-time system, is possible while maintaining a ﬁnite H 2 norm. 5.3 H ∞ -Optimal Control The goal of H ∞ -optimal control is to d es i gn a con t roller that m inimizes the H ∞ norm of the closed-loop t ransfer matrix from w to z . 5.3.1 H ∞ -Optimal Full-State Feed back Contr ol [ 5 , pp. 251–252] Consider t he conti nuous-time generalized L TI plant P with state-space realization ˙ x = Ax + B 1 w + B 2 u , (5.16) z = C 1 x + D 11 w + D 12 u , (5.17) y = x , where it is assumed that ( A , B 2 ) is stabili zable. A ful l -state feedback controller K = K ∈ R n u × n x (i.e., u = Kx ) i s to be designed t o minimize H ∞ norm of th e closed loop transfer matri x from the exogenous input w to the performance output z . Subst ituting the full-st ate feedback control ler into ( 5.16 ) and ( 5.17 ) yields ˙ x = ( A + B 2 K ) x + B 1 w , z = ( C 1 + D 12 K ) x + D 11 w , and a closed-loop transfer matrix T ( s ) = ( C 1 + D 12 K ) ( s 1 − ( A + B 2 K )) − 1 B 1 + D 11 . From the Bounded Real Lemma in Section 4.2.1 , the H ∞ of th e closed-loop system i s the m inimum value of γ ∈ R > 0 that satis ﬁes   P ( A + B 2 K ) + ( A + B 2 K ) T P PB 1 ( C 1 + D 12 K ) T ∗ − γ 1 D T 11 ∗ ∗ − γ 1   < 0 , (5.18) 127 where P ∈ S n x and P > 0 . A cong ru ence transformation is performed on ( 5.18 ) with W = diag { P − 1 , 1 , 1 } and a change of var iabl es is made with Q = P − 1 and F = KQ . This yields an LMI in the design variables Q , F , and γ , gi ven by   A Q + QA T + B 2 F + F T B T 2 B 1 QC T 1 + F T D T 12 ∗ − γ 1 D T 11 ∗ ∗ − γ 1   < 0 . (5.19) Synthesis Method 5.6. T h e H ∞ -optimal full-state feedback controller is synthesi zed by s o lving for Q ∈ S n x and F ∈ R n u × n x that m inimize J ( γ ) = γ subject to Q > 0 and ( 5.19 ). The H ∞ - optimal full-state feedback controll er gain is recovered by K = FQ − 1 and the H ∞ norm of T ( s ) i s γ . 5.3.2 Discrete- Time H ∞ -Optimal Full-State Feed back Contr ol Consider t he dis crete-time generalized L TI plant P with s t ate-space realization x k +1 = A d x k + B d1 w k + B d2 u k , z k = C d1 x k + D d12 u k , y k = x k , where it is assumed that ( A d , B d2 ) i s stabil i zable. A full-state feedback con t roller K = K d ∈ R n u × n x (i.e., u k = K d x k ) is to be d esi gned t o m i nimize the H ∞ norm of t h e closed loop transfer matrix from the exogenous input w k to the performance out put z k , given by T ( z ) = ( C d1 + D d12 K d ) ( z 1 − ( A d + B d2 K d )) − 1 B d1 . Synthesis Method 5.7. The discrete-time H ∞ -optimal full-state feedback controller is synth esized by solving for P ∈ S n x , F d ∈ R n u × n x , and γ ∈ R > 0 that mi n imize J ( γ ) = γ subject to P > 0 ,     P d A d P d + B d2 F d B d 1 0 ∗ P d 0 P d C T d 1 + F T d D T d12 ∗ ∗ γ 1 D T d 11 ∗ ∗ ∗ γ 1     > 0 . The H ∞ -optimal full-stat e feedback gain i s recov ered by K d = F d P − 1 and the H ∞ norm of T ( z ) is γ . 5.3.3 H ∞ -Optimal Dynamic Output Feedback Control Consider t he conti nuous-time generalized L TI plant P with minimal state-space realization ˙ x = Ax + B 1 w + B 2 u , z = C 1 x + D 11 w + D 12 u , y = C 2 x + D 21 w + D 22 u . 128 A continuo u s-time dyn am ic output feedback L TI control ler with state-space realization ( A c , B c , C c , D c ) is to be designed t o minimize the H ∞ norm of the closed-loo p s ystem transfer m atrix from w to z , giv en by T ( s ) = C CL ( s 1 − A CL ) − 1 B CL + D CL , where A CL = " A + B 2 D c ˜ D − 1 C 2 B 2  1 + D c ˜ D − 1 D 22  C c B c ˜ D − 1 C 2 A c + B c ˜ D − 1 D 22 C c # , B CL = " B 1 + B 2 D c ˜ D − 1 D 21 B c ˜ D − 1 D 21 # , C CL = h C 1 + D 12 D c ˜ D − 1 C 2 D 12  1 + D c ˜ D − 1 D 22  C c i , D CL = D 11 + D 12 D c ˜ D − 1 D 21 , and ˜ D = 1 − D 22 D c . T wo differe nt synthesi s m ethods for the H ∞ -optimal dynamic output feedback control problem are presented as fol l ows. Synthesis Method 5.8. [ 185 , 249 , 250 ] Solve for A n ∈ R n x × n x , B n ∈ R n x × n y , C n ∈ R n u × n x , D n ∈ R n u × n y , X 1 , Y 1 ∈ S n x , and γ ∈ R > 0 that mi n imize J ( γ ) = γ subject to X 1 > 0 , Y 1 > 0 ,     N 11 A + A T n + B 2 D n C 2 B 1 + B 2 D n D 21 Y T 1 C T 1 + C T n D T 12 ∗ X 1 A + A T X 1 + B n C 2 + C T 2 B T n X 1 B 1 + B n D 21 C T 1 + C T 2 D T n D T 12 ∗ ∗ − γ 1 D T 11 + D T 21 D T n D T 12 ∗ ∗ ∗ − γ 1     < 0 ,  X 1 1 ∗ Y 1  > 0 , where N 11 = A Y 1 + Y 1 A T + B 2 C n + C T n B T 2 . The controller is recovered by A c = A K − B c ( 1 − D 22 D c ) − 1 D 22 C c , B c = B K ( 1 − D 22 D c ) , C c = ( 1 − D c D 22 ) C K , D c = ( 1 + D K D 22 ) − 1 D K , where  A K B K C K D K  =  X 2 X 1 B 2 0 1  − 1  A n B n C n D n  −  X 1 A Y 1 0 0 0   Y T 2 0 C 2 Y 1 1  − 1 , and the matrices X 2 and Y 2 satisfy X 2 Y T 2 = 1 − X 1 Y 1 . If D 22 = 0 , then A c = A K , B c = B K , C c = C K , and D c = D K . Giv en X 1 and Y 1 , the m atrices X 2 and Y 2 can be found usi n g a matrix decomposition, s u ch as a LU decompositi on or a Cholesky decom position. 129 Synthesis Method 5.9. [ 70 ], [ 2 , pp . 224–232] The controller is sol ved for in the following two steps. 1. Solve for P , Q ∈ S n x and γ ∈ R > 0 , where P > 0 and Q > 0 , that min imize J ( γ ) = γ subject to  N o 0 0 1  T   P A + A T P PB 1 C T 1 ∗ − γ 1 D T 11 ∗ ∗ − γ 1    N o 0 0 1  < 0 ,  N c 0 0 1  T   A Q + QA T QC T 1 B 1 ∗ − γ 1 D 11 ∗ ∗ − γ 1    N c 0 0 1  < 0 ,  P 1 ∗ Q  ≥ 0 , (5.20) where R ( N o ) = N  C 2 D 21  and R ( N c ) = N  B T 2 D T 12  . Deﬁne P CL =  P P T 2 ∗ 1  , where P 2 P T 2 = P − Q − 1 . 2. Fix P CL and solve for A n ∈ R n x × n x , B n ∈ R n x × n y , C n ∈ R n u × n x , D n ∈ R n u × n y , and γ ∈ R > 0 that mi n imize J ( γ ) = γ subject to   P CL ¯ A + ¯ A T P CL P CL ¯ B ¯ C T ∗ − γ 1 D T 11 ∗ ∗ − γ 1   +   P CL B 0 D 12    A n B n C n D n   C D 21 0  +   C T D T 21 0    A n B n C n D n  T  B T P CL 0 D T 12  < 0 , where ¯ A =  A 0 0 0  , ¯ B =  B 1 − B 2 ¯ D c D 21 0  , ¯ C =  C 1 0  , C =  0 1 C 2 0  , B =  0 − B 2 1 0  , D 12 =  0 − D 12  , D 21 =  0 D 21  . The cont roller is recovered by A c = A n − B c ( 1 − D 22 D c ) − 1 D 22 C c , B c = B n ( 1 − D 22 D c ) , C c = ( 1 − D c D 22 ) C n , D c = ( 1 + D n D 22 ) − 1 D n . If D 22 = 0 , then A c = A n , B c = B n , C c = C n , and D c = D n . 130 Note that t h e purpose of th e matrix inequali ty  P 1 ∗ Q  ≥ 0 in ( 5.20 ) is t o ensure that t here exists P CL =  P P T 2 ∗ 1  > 0 and P − 1 CL =  Q − QP 2 ∗ P T 2 QP 2 + 1  . This fol l ows from Property 9 in Section 2.4.3 . 5.3.4 Discrete- Time H ∞ -Optimal Dynamic Output Feedback Control Consider t he dis crete-time generalized L TI plant P with m inimal state-space realization x k +1 = A d x k + B d1 w k + B d2 u k , z k = C d1 x k + D d11 w k + D d12 u k , y k = C d2 x k + D d21 w k + D d22 u k , A discrete-time dynam ic output feedback L TI control ler with state-space realization ( A d c , B d c , C d c , D d c ) is to be designed t o minimize the H ∞ norm of the closed-loo p s ystem transfer m atrix from w to z , giv en by T ( z ) = C d CL ( z 1 − A d CL ) − 1 B d CL + D d CL , where A d CL = " A d + B d2 D d c ˜ D − 1 d C d2 B d2  1 + D d c ˜ D − 1 d D d22  C d c B d c ˜ D − 1 d C d2 A d c + B d c ˜ D − 1 d D d22 C d c # , B d CL = " B d1 + B d2 D d c ˜ D − 1 d D d21 B d c ˜ D − 1 d D d21 # , C d CL = h C d1 + D d12 D d c ˜ D − 1 d C d2 D d12  1 + D d c ˜ D − 1 d D d22  C d c i , D d CL = D d11 + D d12 D d c ˜ D − 1 d D d21 , and ˜ D d = 1 − D d22 D d c . Synthesis Method 5.10. [ 165 ] Solve for A d n ∈ R n x × n x , B d n ∈ R n x × n y , C d n ∈ R n u × n x , D d n ∈ R n u × n y , X 1 , Y 1 ∈ S n x , G , H , J , S ∈ R n x × n x , and γ ∈ R > 0 that mini mize J ( γ ) = γ s ubject to X 1 > 0 , Y 1 > 0 ,         X 1 J T HA d + B d n C d2 A d n HB d1 + B d n D d21 0 ∗ Y 1 A d + B d2 D d n C d2 A d G + B d2 C d n B d1 + B d2 D d n D d21 0 ∗ ∗ H + H T − X 1 1 + S − J T 0 C T d1 + C T d2 D T d n D T d12 ∗ ∗ ∗ G + G T − Y 1 0 G T C T d1 + C T d n D T d12 ∗ ∗ ∗ ∗ γ 1 D T d11 + D T d21 D T d n D T d12 ∗ ∗ ∗ ∗ ∗ γ 1         > 0 . (5.21) 131 The cont roller is recovered by A d c = A d K − B d c ( 1 − D d22 D d c ) − 1 D d22 C d c , B d c = B d K ( 1 − D d22 D d c ) , C d c = ( 1 − D d c D d22 ) C d K , D d c = ( 1 + D d K D d22 ) − 1 D d K , where  A d K B d K C d K D d K  =  Y − T 2 Y − T 2 HB d2 0 1   A d n B d n C d n D d n  −  HA d G 0 0 0   X − 1 2 0 − C d2 GX − 1 2 1  , and the m atrices X 2 and Y 2 satisfy X 2 Y T 2 = 1 − H G . If D d22 = 0 , th en A d c = A d K , B d c = B d K , C d c = C d K , and D d c = D d K . Giv en G and H , the matrices X 2 and Y 2 can be found using a m atrix decomposi tion, such as a LU d ecom position or a Cholesky decomposi tion. The LMI in ( 5.21 ) is derived from t he LMI in Theorem 8 o f [ 165 ] by performi ng a congruence transformation in volving a multipl i cation on the left and right by t h e symmetric matrix W = diag n  0 √ γ 1 1 √ γ 1 0  ,  0 √ γ 1 1 √ γ 1 0  , √ γ 1 , 1 √ γ 1 o , followed by the change of v ariables γ = µ 2 , X 1 = γ H , Y 1 = γ − 1 P . Synthesis Method 5. 1 1 . Solve for A d n ∈ R n x × n x , B d n ∈ R n x × n y , C d n ∈ R n u × n x , D d n ∈ R n u × n y , X 1 , Y 1 ∈ S n x , and γ ∈ R > 0 that mi n imize J ( γ ) = γ s ubject to X 1 > 0 , Y 1 > 0 ,         X 1 1 X 1 A d + B d n C d2 A d n X 1 B d1 + B d n D d21 0 ∗ Y 1 A d + B d2 D d n C d2 A d Y 1 + B d2 C d n B d1 + B d2 D d n D d21 0 ∗ ∗ X 1 1 0 C T d1 + C T d2 D T d n D T d12 ∗ ∗ ∗ Y 1 0 Y 1 C T d1 + C T d n D T d12 ∗ ∗ ∗ ∗ γ 1 D T d11 + D T d21 D T d n D T d12 ∗ ∗ ∗ ∗ ∗ γ 1         > 0 , (5.22)  X 1 1 ∗ Y 1  > 0 . (5.23) The cont roller is recovered by A d c = A d K − B d c ( 1 − D d22 D d c ) − 1 D d22 C d c , B d c = B d K ( 1 − D d22 D d c ) , C d c = ( 1 − D d c D d22 ) C d K , D d c = ( 1 + D d K D d22 ) − 1 D d K , 132 where  A d K B d K C d K D d K  =  X 2 X 1 B d2 0 1  − 1  A d n B d n C d n D d n  −  X 1 A d Y 1 0 0 0   Y T 2 0 C d2 Y 1 1  − 1 , and t he matri ces X 2 and Y 2 satisfy X 2 Y T 2 = 1 − X 1 Y 1 . If D d22 = 0 , then A d c = A d K , B d c = B d K , C d c = C d K , and D d c = D d K . Giv en X 1 and Y 1 , the m atrices X 2 and Y 2 can be found usi n g a matrix decomposition, s u ch as a LU decompositi on or a Cholesky decom position. The LMI in ( 5.22 ) is deri ved from ( 5.21 ) using the change of variables S = J = 1 , H = X 1 , G = Y 1 . The LMI in ( 5.23 ) i s added to ensure that 1 − X 1 Y 1 ≥ 0 in a similar fashion t o the approach used in [ 185 ]. 5.4 Mixed H 2 - H ∞ -Optimal Control The goal of mi xed H 2 - H ∞ -optimal control i s to design a controller that m inimizes the H 2 norm of the closed-loop transfer matrix from w 1 to z 1 , while ensuring that the H ∞ norm of the closed-loop t ransfer function from w 2 to z 2 is below a speciﬁed bound. 5.4.1 Mixed H 2 - H ∞ -Optimal Full-State F eedback Contr ol [ 5 , pp. 329–330] Consider t he conti nuous-time generalized L TI plant P with state-space realization ˙ x = Ax +  B 1 , 1 B 1 , 2   w 1 w 2  + B 2 u ,  z 1 z 2  =  C 1 , 1 C 1 , 2  x +  0 D 11 , 12 D 11 , 21 D 11 , 22   w 1 w 2  +  D 12 , 1 D 12 , 2  u , y = x , where it is assumed th at ( A , B 2 ) i s st abilizable. A full-state feedback control l er K = K ∈ R n u × n x (i.e., u = Kx ) is to be designed to min i mize the H 2 norm of the closed-loo p transfer matrix T 11 ( s ) from t h e exogenous input w 1 to the performance outp ut z 1 while ensuring the H ∞ norm of the closed-loop transfer m atrix T 22 ( s ) from the exogenous input w 2 to t h e p erformance output z 2 is less than γ d , wh ere T 11 ( s ) = ( C 1 , 1 + D 12 , 1 K ) ( s 1 − ( A + B 2 K )) − 1 B 1 , 1 , T 22 ( s ) = ( C 1 , 2 + D 12 , 2 K ) ( s 1 − ( A + B 2 K )) − 1 B 1 , 2 + D 11 , 22 . Synthesis Method 5.12. The mixed H 2 - H ∞ -optimal full-state feedback controller is synt hesized by s olving for P ∈ S n x , Z ∈ S n w , F ∈ R n u × n x , and ν ∈ R > 0 that minimize J ( ν ) = ν subject to 133 P > 0 , Z > 0 ,  AP + P A T + B 2 F + F T B T 2 PC T 1 , 1 + F T D T 12 , 1 ∗ − 1  < 0 ,   AP + P A T + B 2 F + F T B T 2 B 1 , 2 PC T 1 , 2 + F T D T 12 , 2 ∗ − γ d 1 D T 11 , 22 ∗ ∗ − γ d 1   < 0 ,  Z B T 1 , 1 ∗ P  > 0 , tr( Z ) < ν. The H 2 -optimal full-st at e feedback gain is reco vered by K = FP − 1 , t h e H 2 norm of T 11 ( s ) is less than µ = √ ν , and the H ∞ norm of T 22 ( s ) is less than γ d . 5.4.2 Discrete- Time Mi xed H 2 - H ∞ -Optimal Full-State Feed back Contr ol Consider t he dis crete-time generalized L TI plant P with s t ate-space realization x k +1 = A d x k +  B d1 , 1 B d1 , 2   w 1 ,k w 2 ,k  + B d2 u k ,  z 1 ,k z 2 ,k  =  C d1 , 1 C d1 , 2  x k +  0 D d11 , 12 D d11 , 21 D d11 , 22   w 1 ,k w 2 ,k  +  D d12 , 1 D d12 , 2  u k , y k = x k , where it is assumed that ( A d , B d2 ) i s stabil i zable. A full-state feedback con t roller K = K d ∈ R n u × n x (i.e., u k = K d x k ) is to b e designed to minim ize the H 2 norm of the cl o sed loop transfer matrix T 11 ( z ) from the exogenous inp u t w 1 ,k to the performance output z 1 ,k while ensuring the H ∞ norm of the closed-loop transfer matrix T 22 ( z ) from t he exogenous input w 2 ,k to the performance output z 2 ,k is less than γ d , wh ere T 11 ( z ) = ( C d1 , 1 + D d12 , 1 K d ) ( z 1 − ( A d + B d2 K d )) − 1 B d1 , 1 , T 22 ( z ) = ( C d1 , 2 + D d12 , 2 K d ) ( z 1 − ( A d + B d2 K d )) − 1 B d1 , 2 + D d11 , 22 . Synthesis Method 5.13. T h e discrete-time m ixed H 2 - H ∞ -optimal full-state feedback controller is synthesized by solv i ng for P ∈ S n x , Z ∈ S n w , F d ∈ R n u × n x , and ν ∈ R > 0 that mi nimize J ( ν ) = ν subject to P > 0 , Z > 0 ,   P A d P + B d2 F d B d 1 , 1 ∗ P 0 ∗ ∗ 1   > 0 ,     P A d P + B d2 F d B d 1 , 2 0 ∗ P 0 PC T d 1 , 2 + F T d D T d12 , 2 ∗ ∗ γ d 1 D T d 11 , 22 ∗ ∗ ∗ γ d 1     > 0 ,  Z C d 1 , 1 P + D d12 , 1 F d ∗ P  > 0 . tr( Z ) < ν. 134 The H 2 -optimal full-state feedback gain is recover ed by K d = F d P − 1 , the H 2 norm of T 11 ( z ) is less than µ = √ ν , and the H ∞ norm of T 22 ( z ) is less t h an γ d . 5.4.3 Mixed H 2 - H ∞ -Optimal Dynamic Output Feedback Control [ 185 , 251 ] Consider t he conti nuous-time generalized L TI plant P with minimal state-space realization ˙ x = Ax +  B 1 , 1 B 1 , 2   w 1 w 2  + B 2 u ,  z 1 z 2  =  C 1 , 1 C 1 , 2  x +  D 11 , 11 D 11 , 12 D 11 , 21 D 11 , 22   w 1 w 2  +  D 12 , 1 D 12 , 2  u , y = C 2 x +  D 21 , 1 D 21 , 2   w 1 w 2  + D 22 u . A continuo u s-time dyn am ic output feedback L TI control ler with state-space realization ( A c , B c , C c , D c ) is to be designed to minim ize the H 2 norm of the cl o sed-loop transfer matrix T 11 ( s ) from the ex- ogenous input w 1 to the performance output z 1 while ensuring the H ∞ norm of the closed-loop transfer matrix T 22 ( s ) from the e xogeno us i n put w 2 to the performance output z 2 is less than γ d , where T 11 ( s ) = C CL1 , 1 ( s 1 − A CL ) − 1 B CL1 , 1 , T 22 ( s ) = C CL1 , 2 ( s 1 − A CL ) − 1 B CL1 , 2 + D CL11 , 22 , A CL = " A + B 2 D c ˜ D − 1 C 2 B 2  1 + D c ˜ D − 1 D 22  C c B c ˜ D − 1 C 2 A c + B c ˜ D − 1 D 22 C c # , B CL1 , 1 = " B 1 , 1 + B 2 D c ˜ D − 1 D 21 , 1 B c ˜ D − 1 D 21 , 1 # , B CL1 , 2 = " B 1 , 2 + B 2 D c ˜ D − 1 D 21 , 2 B c ˜ D − 1 D 21 , 2 # , C CL1 , 1 = h C 1 , 1 + D 12 , 1 D c ˜ D − 1 C 2 , 1 D 12 , 1  1 + D c ˜ D − 1 D 22  C c i , C CL1 , 2 = h C 1 , 2 + D 12 , 2 D c ˜ D − 1 C 2 , 2 D 12 , 2  1 + D c ˜ D − 1 D 22  C c i , D CL11 , 22 = D 11 , 22 + D 12 , 2 D c ˜ D − 1 D 21 , 2 , and ˜ D = 1 − D 22 D c . Synthesis Method 5.14. Solve for A n ∈ R n x × n x , B n ∈ R n x × n y , C n ∈ R n u × n x , D n ∈ R n u × n x , 135 X 1 , Y 1 ∈ S n x , Z ∈ S n z 1 , and ν ∈ R > 0 that minimize J ( ν ) = ν subject t o X 1 > 0 , Y 1 > 0 , Z > 0 ,   N 11 A + A T n + B 2 D n C 2 B 1 , 1 + B 2 D n D 21 , 1 ∗ X 1 A + A T X 1 + B n C 2 + C T 2 B T n X 1 B 1 , 1 + B n D 21 , 1 ∗ ∗ − 1   < 0 ,     N 11 A + A T n + B 2 D n C 2 B 1 , 2 + B 2 D n D 21 , 2 Y 1 C T 1 , 2 + C T n D T 12 , 2 ∗ X 1 A + A T X 1 + B n C 2 + C T 2 B T n X 1 B 1 , 2 + B n D 21 , 2 C T 1 , 2 + C T 2 D T n D T 12 , 2 ∗ ∗ − γ d 1 D T 11 , 22 + D T 21 , 2 D T n D T 12 , 2 ∗ ∗ ∗ − γ d 1     < 0 ,   Y 1 1 Y 1 C T 1 , 1 + C T n D T 12 , 1 ∗ X 1 C T 1 , 1 + C T 2 D T n D T 12 , 1 ∗ ∗ Z   > 0 ,  X 1 1 ∗ Y 1  > 0 , D 11 , 11 + D 12 , 1 D n D 21 , 1 = 0 , (5.24) tr( Z ) < ν, where N 11 = A Y 1 + Y 1 A T + B 2 C n + C T n B T 2 . The controller is recovered by A c = A K − B c ( 1 − D 22 D c ) − 1 D 22 C c , B c = B K ( 1 − D 22 D c ) , C c = ( 1 − D c D 22 ) C K , D c = ( 1 + D K D 22 ) − 1 D K , where  A K B K C K D K  =  X 2 X 1 B 2 0 1  − 1  A n B n C n D n  −  X 1 A Y 1 0 0 0   Y T 2 0 C 2 Y 1 1  − 1 , and the matrices X 2 and Y 2 satisfy X 2 Y T 2 = 1 − X 1 Y 1 . If D 22 = 0 , then A c = A K , B c = B K , C c = C K , and D c = D K . Giv en X 1 and Y 1 , the m atrices X 2 and Y 2 can be found usi n g a matrix decomposition, s u ch as a LU decompositi on or a Cholesky decom position. If D 11 , 11 = 0 , D 12 , 1 6 = 0 , and D 21 , 1 6 = 0 , then it is o ft en si mplest to choose D n = 0 in order to satisfy th e equality constraint of ( 5.24 ). 136 5.4.4 Discrete- Time Mi xed H 2 - H ∞ -Optimal Dynamic Output Feedback Control Consider t he dis crete-time generalized L TI plant P with m inimal state-space realization x k +1 = A d x k +  B d1 , 1 B d1 , 2   w 1 ,k w 2 ,k  + B d2 u k ,  z 1 ,k z 2 ,k  =  C d1 , 1 C d1 , 2  x k +  D d11 , 11 D d11 , 12 D d11 , 21 D d11 , 22   w 1 ,k w 2 ,k  +  D d12 , 1 D d12 , 2  u k , y k = C d2 x k +  D d21 , 1 D d21 , 2   w 1 ,k w 2 ,k  + D d22 u k . A discrete-time dynam ic output feedback L TI control ler with state-space realization ( A d c , B d c , C d c , D d c ) is to be designed to mi nimize the H 2 norm of the closed loop transfer m atrix T 11 ( z ) from the ex- ogenous in put w 1 ,k to the performance out put z 1 ,k while ensurin g t he H ∞ norm of the closed-loo p transfer matrix T 22 ( z ) from the exogenous input w 2 ,k to t h e performance ou tput z 2 ,k is l ess than γ d , wh ere T 11 ( z ) = C d CL 1 , 1 ( z 1 − A d CL ) − 1 B d CL 1 , 1 , T 22 ( z ) = C d CL 1 , 2 ( z 1 − A d CL ) − 1 B d CL 1 , 2 + D d CL 11 , 22 , A d CL = " A d + B d2 D d c ˜ D − 1 d C d2 B d2  1 + D d c ˜ D − 1 d D d22  C d c B d c ˜ D − 1 d C d2 A d c + B d c ˜ D − 1 d D d22 C d c # , B d CL 1 , 1 = " B d1 , 1 + B d2 D d c ˜ D − 1 d D d21 , 1 B d c ˜ D − 1 d D d21 , 1 # , B d CL 1 , 2 = " B d1 , 2 + B d2 D d c ˜ D − 1 d D d21 , 2 B d c ˜ D − 1 d D d21 , 2 # , C d CL 1 , 1 = h C d1 , 1 + D d12 , 1 D d c ˜ D − 1 d C d2 , 1 D d12 , 1  1 + D d c ˜ D − 1 d D d22  C d c i , C d CL 1 , 2 = h C d1 , 2 + D d12 , 2 D d c ˜ D − 1 d C d2 , 2 D d12 , 2  1 + D d c ˜ D − 1 d D d22  C d c i , D d CL 11 , 22 = D d11 , 22 + D d12 , 2 D d c ˜ D − 1 d D d21 , 2 , and ˜ D d = 1 − D d22 D d c . Synthesis Method 5. 1 5 . Solve for A d n ∈ R n x × n x , B d n ∈ R n x × n y , C d n ∈ R n u × n x , D d n ∈ R n u × n y , 137 X 1 , Y 1 ∈ S n x , Z ∈ S n z 1 , and ν ∈ R > 0 that minimize J ( ν ) = ν subject t o X 1 > 0 , Y 1 > 0 , Z > 0 ,       X 1 1 X 1 A d + B d n C d2 A d n X 1 B d1 , 1 + B d n D d21 , 1 ∗ Y 1 A d + B d2 D d n C d2 A d Y 1 + B d2 C d n B d1 , 1 + B d2 D d n D d21 , 1 ∗ ∗ X 1 1 0 ∗ ∗ ∗ Y 1 0 ∗ ∗ ∗ ∗ 1       > 0 ,         X 1 1 X 1 A d + B d n C d2 A d n X 1 B d1 , 2 + B d n D d21 , 2 0 ∗ Y 1 A d + B d2 D d n C d2 A d Y 1 + B d2 C d n B d1 , 2 + B d2 D d n D d21 , 2 0 ∗ ∗ X 1 1 0 C T d1 , 2 + C T d2 D T d n D T d12 , 2 ∗ ∗ ∗ Y 1 0 Y 1 C T d1 , 2 + C T d n D T d12 , 2 ∗ ∗ ∗ ∗ γ d 1 D T d11 , 22 + D T d21 , 2 D T d n D T d12 , 2 ∗ ∗ ∗ ∗ ∗ γ d 1         > 0 ,   Z C d1 , 1 + D d12 , 1 D d n C d2 C d1 , 1 Y 1 + D d12 , 1 C d n ∗ X 1 1 ∗ ∗ Y 1   > 0 , (5.25) D d11 , 11 + D d12 , 1 D d n D d21 , 1 = 0 , (5.26)  X 1 1 ∗ Y 1  > 0 , tr( Z ) < ν. The cont roller is recovered by A d c = A d K − B d c ( 1 − D d22 D d c ) − 1 D d22 C d c , B d c = B d K ( 1 − D d22 D d c ) , C d c = ( 1 − D d c D d22 ) C d K , D d c = ( 1 + D d K D d22 ) − 1 D d K , where  A d K B d K C d K D d K  =  X 2 X 1 B d2 0 1  − 1  A d n B d n C d n D d n  −  X 1 A d Y 1 0 0 0   Y T 2 0 C d2 Y 1 1  − 1 , and t he matri ces X 2 and Y 2 satisfy X 2 Y T 2 = 1 − X 1 Y 1 . If D d22 = 0 , then A d c = A d K , B d c = B d K , C d c = C d K , and D d c = D d K . Giv en X 1 and Y 1 , the m atrices X 2 and Y 2 can be found usi n g a matrix decomposition, s u ch as a LU decompositi on or a Cholesky decom position. If D d11 , 11 = 0 , D d12 , 1 6 = 0 , and D d21 , 1 6 = 0 , then it is often simplest to choose D d n = 0 in order to satis fy the equali ty constraint of ( 5.26 ). An al t ernate formulati on of th is synth es i s m ethod in volves replacing ( 5.25 ) and ( 5.26 ) with     Z C d1 , 1 + D d12 , 1 D d n C d2 C d1 , 1 Y 1 + D d12 , 1 C d n D d11 , 11 + D d12 , 1 D d n D d21 , 1 ∗ X 1 1 0 ∗ ∗ Y 1 0 ∗ ∗ ∗ 1     > 0 . (5.27) 138 The matrix i n equality in ( 5.27 ) is derived by performing the same procedure used i n [ 165 ] w i th the change of var iables S = J = 1 , H = X 1 , G = Y 1 , but i nstead starting with t he matrix inequality formulat i on of the H 2 that allows for a non-zero feedthrough term in [ 176 , p. 25] (sum- marized by ( 4.40 ), ( 4.4 1 ), and ( 4.42 )). In general, the matrix inequality in ( 5.27 ) is less conser- vati ve than ( 5.25 ) and ( 5.26 ), as it allows for th e resulting closed-loop sys t em to hav e non -zero feedthrough, which, for a discrete-time system, is possible while maintaining a ﬁnite H 2 norm. 139 6 LMIs in O p timal Estimatio n and Filtering This section presents controller synthesis methods using LMIs for a number of well-known optimal state-estimation and ﬁltering problems. The deriva ti o n of the LMIs used for synthesis is provided in so me cases, while lo nger deriv ations can be found in t h e cited reference s. 6.1 H 2 -Optimal State Estimation The goal of H 2 -optimal s tate esti mation is to design an observer that minimizes th e H 2 norm of th e closed-loo p t ransfer matrix from w to z . 6.1.1 H 2 -Optimal Observ er [ 5 , p. 296] Consider t he conti nuous-time generalized pl ant P wit h state-space realization ˙ x = Ax + B 1 w , y = C 2 x + D 21 w , where it is ass umed that ( A , C 2 ) is detectable. An observer of t he form ˙ ˆ x = A ˆ x + L ( y − ˆ y ) , ˆ y = C 2 ˆ x , is to be designed, where L ∈ R n x × n y is the observer gain. Deﬁning t he error st ate e = x − ˆ x , the error dynami cs are found to be ˙ e = ( A − LC 2 ) e + ( B 1 − LD 21 ) w , and t he performance output is deﬁned as z = C 1 e . The observer gain L is to be designed such that th e H 2 norm of the t ransfer matrix from w to z , giv en by T ( s ) = C 1 ( s 1 − ( A − LC 2 )) − 1 ( B 1 − LD 21 ) , is mini mized. Mini m izing the H 2 norm of th e transfer matrix T ( s ) is equ ivalent to min i mizing J ( µ ) = µ 2 subject to  P ( A − LC 2 ) + ( A − LC 2 ) T P P ( B 1 − LD 21 ) ∗ − 1  < 0 , (6.1)  P C T 1 ∗ Z  > 0 , (6.2) tr( Z ) < µ 2 , (6.3) 140 where P ∈ S n x , Z ∈ S n z , µ ∈ R > 0 , P > 0 , and Z > 0 . A change of v ariables is performed with G = PL and ν = µ 2 , which transforms ( 6.1 ) and ( 6.3 ) into LMIs i n th e variables P , G , Z , and ν giv en by  P A + A T P − G C 2 − C T 2 G T PB 1 − GD 21 ∗ − 1  < 0 , (6.4) tr( Z ) < ν. (6.5) Synthesis Method 6.1. The H 2 -optimal observer gain is synthesized by solving for P ∈ S n x , Z ∈ S n z , G ∈ R n x × n y , and ν ∈ R > 0 that mi n imize J ( ν ) = ν subject to P > 0 , Z > 0 , ( 6.2 ), ( 6.4 ), and ( 6.5 ). The H 2 -optimal observer gain is recovered by L = P − 1 G and th e H 2 norm of T ( s ) is µ = √ ν . 6.1.2 Discrete- Time H 2 -Optimal Observer Consider t he dis crete-time generalized L TI plant P with s t ate-space realization x k +1 = A d x k + B d1 w k , y k = C d2 x k + D d21 w k , where it is ass umed that ( A d , C d2 ) is detectable. An observer of t he form ˆ x k +1 = A d ˆ x k + L d ( y k − ˆ y k ) , ˆ y k = C d2 ˆ x k , is to be designed, where L d ∈ R n x × n y is the observer g ai n . Deﬁning the error state e k = x k − ˆ x k , the error dynamics are found to be e k +1 = ( A d − L d C d2 ) e k + ( B d1 − L d D d21 ) w k , and t he performance output is deﬁned as z k = C d1 e k . The o bserver gain L d is to be designed such that the H 2 of the transfer matrix from w k to z k , given by T ( z ) = C d1 ( z 1 − ( A d − L d C d2 )) − 1 ( B d1 − L d D d21 ) , is mi n imized. Synthesis Method 6.2. The discrete-time H 2 -optimal observer gain is synthesized by solving for P ∈ S n x , Z ∈ S n z , G d ∈ R n x × n y , and ν ∈ R > 0 that minimize J ( ν ) = ν subject t o P > 0 , Z > 0 ,   P P A d − G d C d2 PB d 1 − G d D d21 ∗ P 0 ∗ ∗ 1   > 0 ,  Z C d 1 ∗ P  > 0 . tr( Z ) < ν. The H 2 -optimal observer gain is recovered by L d = P − 1 G d and t he H 2 norm of T ( z ) is µ = √ ν . 141 6.2 H ∞ -Optimal State Estima tion The goal of H ∞ -optimal state estimation is to desig n an observer that mini m izes the H ∞ norm of th e closed-loo p t ransfer matrix from w to z . 6.2.1 H ∞ –Optimal O bserver [ 5 , p. 295] Consider t he conti nuous-time generalized pl ant P wit h state-space realization ˙ x = Ax + B 1 w , y = C 2 x + D 21 w , where it is ass umed that ( A , C 2 ) is detectable. An observer of t he form ˙ ˆ x = A ˆ x + L ( y − ˆ y ) , ˆ y = C 2 ˆ x , is to be designed, where L ∈ R n x × n y is the observer gain. Deﬁning t he error st ate e = x − ˆ x , the error dynami cs are found to be ˙ e = ( A − LC 2 ) e + ( B 1 − LD 21 ) w , and t he performance output is deﬁned as z = C 1 e + D 11 w . The ob s erver gain L is to be desi gned such t h at the H ∞ of th e transfer matrix from w to z , giv en by T ( s ) = C 1 ( s 1 − ( A − LC 2 )) − 1 ( B 1 − LD 21 ) + D 11 , is mi n imized. Synthesis Method 6 . 3 . The H ∞ -optimal observer gain is synthesized by solving for P ∈ S n x , G ∈ R n x × n y , and γ ∈ R > 0 that mi n imize J ( γ ) = γ subject to P > 0 and   P A + A T P − G C 2 − C T 2 G T PB 1 − GD 21 C T 1 ∗ − γ 1 D T 11 ∗ ∗ − γ 1   < 0 . The H ∞ -optimal observer gai n is recovered by L = P − 1 G and the H ∞ norm of T ( s ) i s γ . 6.2.2 Discrete- Time H ∞ –Optimal O bserver Consider t he dis crete-time L TI plant G wit h state-space realization x k +1 = A d x k + B d1 w k , y k = C d2 x k + D d21 w k , 142 where it is ass umed that ( A d , C d2 ) is detectable. An observer of t he form ˆ x k +1 = A d ˆ x k + L d ( y k − ˆ y k ) , ˆ y k = C d2 ˆ x k , is to be designed, where L d ∈ R n x × n y is the observer g ai n . Deﬁning the error state e k = x k − ˆ x k , the error dynamics are found to be e k +1 = ( A d − L d C d2 ) e k + ( B d1 − L d D d21 ) w k , and t he performance output is deﬁned as z k = C d1 e k + D d11 w k . The observer gain L d is to be desi gned such that the H ∞ of the transfer m atrix from w k to z k , g iv en by T ( z ) = C d1 ( z 1 − ( A d − L d C d2 )) − 1 ( B d1 − L d D d21 ) + D d11 , is mi n imized. Synthesis Method 6 . 4 . The H ∞ -optimal observer gain is synthesized by solving for P ∈ S n x , G d ∈ R n x × n y , and γ ∈ R > 0 that minimize J ( γ ) = γ sub j ect to P > 0 and     P P A d − G d C d2 PB d 1 − G d D d21 0 ∗ P 0 C T d 1 ∗ ∗ γ 1 D T d 11 ∗ ∗ ∗ γ 1     > 0 . The H ∞ -optimal observer gai n is recovered by L d = P − 1 G d and t he H ∞ norm of T ( z ) is γ . 6.3 Mixed H 2 - H ∞ -Optimal State Estima tion The g oal of mixed H 2 - H ∞ -optimal state estim ation is to design an ob server t hat mini mizes the H 2 norm of the closed-loo p t ransfer matrix from w 1 to z 1 , while ensuring t hat the H ∞ norm of the closed-loop t ransfer matrix from w 2 to z 2 is below a speciﬁed bound. 6.3.1 Mixed H 2 - H ∞ -Optimal Observer Consider t he conti nuous-time generalized pl ant P wit h state-space realization ˙ x = Ax + B 1 , 1 w 1 + B 1 , 2 w 2 , y = C 2 x + D 21 , 1 w 1 + D 21 , 1 w 2 , where it is ass umed that ( A , C 2 ) is detectable. An observer of t he form ˙ ˆ x = A ˆ x + L ( y − ˆ y ) , ˆ y = C 2 ˆ x , 143 is to be designed, where L ∈ R n x × n y is the observer gain. Deﬁning t he error st ate e = x − ˆ x , the error dynami cs are found to be ˙ e = ( A − LC 2 ) e + ( B 1 , 1 − LD 21 , 1 ) w 1 + ( B 1 , 2 − LD 21 , 2 ) w 2 , and t he performance output is deﬁned as  z 1 z 2  =  C 1 , 1 C 1 , 2  e +  0 D 11 , 12 D 11 , 21 D 11 , 22   w 1 w 2  . The ob server gain L is to b e design ed to m inimize t h e H 2 norm of the closed-loop t ransfer m atrix T 11 ( s ) from the exogenous i nput w 1 to the performance output z 1 while ensuring the H ∞ norm of the clo sed-loop transfer m atrix T 22 ( s ) from the exogenous inpu t w 2 to the performance outpu t z 2 is less than γ d , wh ere T 11 ( s ) = C 1 , 1 ( s 1 − ( A − LC 2 )) − 1 ( B 1 , 1 − LD 21 , 1 ) , T 22 ( s ) = C 1 , 2 ( s 1 − ( A − LC 2 )) − 1 ( B 1 , 2 − LD 21 , 2 ) + D 11 , 22 . Synthesis Method 6.5. The mi xed H 2 - H ∞ -optimal observer gain is synthesized by so lving for P ∈ S n x , Z ∈ S n z , G ∈ R n x × n y , and ν ∈ R > 0 that mi n imize J ( ν ) = ν subj ect to P > 0 , Z > 0 ,  P A + A T P − G C 2 − C T 2 G T PB 1 , 1 − GD 21 , 1 ∗ − 1  < 0 ,   P A + A T P − GC 2 − C T 2 G T PB 1 , 2 − GD 21 , 2 C T 1 , 2 ∗ − γ d 1 D T 11 , 22 ∗ ∗ − γ d 1   < 0 ,  P C T 1 , 1 ∗ Z  > 0 , tr( Z ) < ν. The mixed- H 2 - H ∞ -optimal observer gain is recovered by L = P − 1 G , t he H 2 norm of T 11 ( s ) is less than µ = √ ν , and the H ∞ norm of T 22 ( s ) is less than γ d . 6.3.2 Discrete- Time Mi xed H 2 - H ∞ -Optimal Observer Consider t he dis crete-time generalized L TI plant P with s t ate-space realization x k +1 = A d x k + B d1 , 1 w 1 ,k + B d1 , 1 w 1 ,k , y k = C d2 x k + D d21 , 1 w 1 ,k + D d21 , 2 w 2 ,k , where it is ass umed that ( A d , C d2 ) is detectable. An observer of t he form ˆ x k +1 = A d ˆ x k + L d ( y k − ˆ y k ) , ˆ y k = C d2 ˆ x k , is to be designed, where L d ∈ R n x × n y is the observer g ai n . Deﬁning the error state e k = x k − ˆ x k , the error dynamics are found to be e k +1 = ( A d − L d C d2 ) e k + ( B d1 , 1 − L d D d21 , 1 ) w 1 ,k + ( B d1 , 2 − L d D d21 , 2 ) w 2 ,k , 144 and t he performance output is deﬁned as  z 1 ,k z 2 ,k  =  C d1 , 1 C d1 , 2  e k +  0 D d11 , 12 D d11 , 21 D d11 , 22   w 1 ,k w 2 ,k  . The o bserver gain L d is to be designed to mi nimize the H 2 norm of the closed lo op transfer mat ri x T 11 ( z ) from the exogenous input w 1 ,k to the performance ou t put z 1 ,k while ensurin g the H ∞ norm of t he closed-loop transfer m atrix T 22 ( z ) from the exogenous input w 2 ,k to the performance output z 2 ,k is less than γ d , wh ere T 11 ( z ) = C d1 , 1 ( z 1 − ( A d − L d C d2 )) − 1 ( B d1 , 1 − L d D d21 , 1 ) , T 22 ( z ) = C d1 , 2 ( z 1 − ( A d − L d C d2 )) − 1 ( B d1 , 2 − L d D d21 , 2 ) + D d11 , 22 . Synthesis Method 6.6. The dis crete-time mixed- H 2 - H ∞ -optimal observer g ain is s ynthesized by solving for P ∈ S n x , Z ∈ S n z , G d ∈ R n x × n y , and ν ∈ R > 0 that mini mize J ( ν ) = ν sub ject to P > 0 , Z > 0 ,   P P A d − G d C d2 PB d 1 , 1 − G d D d21 , 1 ∗ P 0 ∗ ∗ 1   > 0 ,     P P A d − G d C d2 PB d 1 , 2 − G d D d21 , 2 0 ∗ P 0 C T d 1 , 2 ∗ ∗ γ d 1 D T d 11 , 22 ∗ ∗ ∗ γ d 1     > 0 ,  Z C d 1 , 1 ∗ P  > 0 . tr( Z ) < ν. The m ixed- H 2 - H ∞ -optimal observer g ain is recov ered by L d = P − 1 G d , t h e H 2 norm of T 11 ( z ) is less than µ = √ ν , and the H ∞ norm of T 22 ( z ) is less t h an γ d . 6.4 Continuous-Ti me and Discr ete-Time Optimal Filtering The goal of optimal ﬁltering is to design a ﬁlter that acts on the out put z of th e generalized plant and optimi zes the transfer m atrix from w t o the ﬁltered outp u t. Continuous-T ime Filtering : Consid er the continuous-t ime g eneralized L TI plant with mi ni- mal states-sp ace realization ˙ x = Ax + B 1 w , z = C 1 x + D 11 w , y = C 2 x + D 21 w , where it is assumed that A is Hurwitz. A continu o us-time dynamic L T I ﬁlter with state-space realization ˙ x f = A f x f + B f y , ˆ z = C f x f + D f y , 145 is to be designed to o ptimize the transfer fun ction from w to ˜ z = z − ˆ z , given by ˜ P ( s ) = ˜ C 1  s 1 − ˜ A  − 1 ˜ B 1 + ˜ D 11 , (6.6) where ˜ A =  A 0 B f C 2 A f  , ˜ B 1 =  B 1 B f D 21  , ˜ C 1 =  C 1 − D f C 2 − C f  , ˜ D 11 = D 11 − D f D 21 . This can alternative ly be formulated as a s p ecial case of synth esi zing a d y namic output “feedback” controller for the g eneralized plant g iven by ˙ x = Ax + B 1 w , z = C 1 x + D 11 w − u , y = C 2 x + D 21 w . The cont roller in this case is not truly a feedback controller , as it only appears as a feedthrough term in the performance channel. The synthesi s meth ods presented in this subsection take adva ntage of this fact, resulting in a si mpler formu lation than applyin g the controller syn t hesis m ethods in Section 5 . Discre te-Time Filtering : Consider the discrete-time generalized L TI plant with mi n imal states- space realization x k +1 = A d x k + B d1 w k , z k = C d1 x k + D d11 w k , y k = C d2 x k + D d21 w k , where it is assum ed that A d is Schur . A discrete-time dynamic L TI ﬁlt er with state-space realization x f ,k +1 = A f x f ,k + B f y k , ˆ z k = C f x f ,k + D f y k , is to be designed to o ptimize the transfer fun ction from w k to ˜ z k = z k − ˆ z k , given by ˜ P ( z ) = ˜ C d1  z 1 − ˜ A d  − 1 ˜ B d1 + ˜ D d11 , (6.7) where ˜ A d =  A d 0 B f C d2 A f  , ˜ B d1 =  B d1 B f D d21  , ˜ C d1 =  C d1 − D f C d2 − C f  , ˜ D d11 = D d11 − D f D d21 . This can alternative ly be formulated as a s p ecial case of synth esi zing a d y namic output “feedback” controller for the g eneralized plant g iven by x k +1 = A d x k + B d1 w k , z k = C d1 x k + D d11 w k − u k , y k = C d2 x k + D d21 w k . 146 6.4.1 H 2 -Optimal Filter An H 2 -optimal ﬁlter is d esi gned to minimize the H 2 norm of ˜ P ( s ) in ( 6.6 ). Synthesis Method 6.7. [ 5 , pp. 309–310] Solve for A n ∈ R n x × n x , B n ∈ R n x × n y , C f ∈ R n z × n x , D f ∈ R n z × n y , X , Y ∈ S n x , Z ∈ S n z , and ν ∈ R > 0 that m inimize J ( ν ) = ν s ubject to X > 0 , Y > 0 , Z > 0 ,   Y A + A T Y + B n C 2 + C T 2 B T n A n + C T 2 B T n + A T X YB 1 + B n D 21 ∗ A n + A T n XB 1 + B n D 21 ∗ ∗ − 1   < 0 ,   − Z C 1 − D f C 2 − C f ∗ − Y − X ∗ ∗ − X   < 0 , D 11 − D f D 21 = 0 , (6.8 ) Y − X > 0 , tr( Z ) < ν. The ﬁlter is recovered by t h e state-space matrices A f = X − 1 A n , B f = X − 1 B n , C f , and D f . If D 11 = 0 and D 21 6 = 0 , then it is often simp l est to choose D f = 0 in order to satisfy the equality cons t raint of ( 6.8 ). Synthesis Method 6.8. [ 5 , pp. 309–310] Solve for A n ∈ R n x × n x , B n ∈ R n x × n y , C f ∈ R n z × n x , D f ∈ R n z × n y , X , Y ∈ S n x , Z ∈ S n z , and ν ∈ R > 0 that m inimize J ( ν ) = ν s ubject to X > 0 , Y > 0 , Z > 0 ,   Y A + A T Y + B n C 2 + C T 2 B T n A n + C T 2 B T n + A T X C T 1 − C T 2 D T f ∗ A n + A T n − C T f ∗ ∗ − 1   < 0 ,   − Z B T 1 Y T + D T 21 B T n B T 1 X T + D T 21 B T n ∗ − Y − X ∗ ∗ − X   < 0 , D 11 − D f D 21 = 0 , (6.9) Y − X > 0 , tr( Z ) < ν. The ﬁlter is recovered by t h e state-space matrices A f = X − 1 A n , B f = X − 1 B n , C f , and D f . If D 11 = 0 and D 21 6 = 0 , then it is often simp l est to choose D f = 0 in order to satisfy the equality cons t raint of ( 6.9 ). 6.4.2 Discrete- Time H 2 -Optimal Filter Synthesis Method 6.9. [ 252 ] Consider the case where D d11 = 0 and D f = 0 . Solve for A d n ∈ R n x × n x , B d n ∈ R n x × n y , C d n ∈ R n u × n x , X , Y ∈ S n x , Z ∈ S n z , and ν ∈ R > 0 that minim ize 147 J ( ν ) = ν subject to X > 0 , Y > 0 , Z > 0 ,       X X XA d XA d XB d1 ∗ Y Y A d + B d n C d1 + A d n Y A d + B d n C d1 YB d1 + B d n D d21 ∗ ∗ X X 0 ∗ ∗ ∗ Y 0 ∗ ∗ ∗ ∗ 1       > 0 ,   Z C d1 C d1 − C d n ∗ Y X ∗ ∗ X   > 0 ,  Y X ∗ X  > 0 , tr( Z ) < ν. The ﬁlter is recovered by A f = − Y − 1 A d n ( 1 − Y − 1 X ) − 1 , B f = − Y − 1 B d n , and C f = C d n ( 1 − Y − 1 X ) − 1 . Synthesis Method 6.10. Solve for A d n ∈ R n x × n x , B d n ∈ R n x × n y , C d n ∈ R n u × n x , D f ∈ R n u × n y , X 1 , Y 1 ∈ S n x , Z ∈ S n z , and ν ∈ R > 0 that mi n imize J ( ν ) = ν subject to X 1 > 0 , Y 1 > 0 , Z > 0 ,       X 1 1 X 1 A d + B d n C d2 A d n X 1 B d1 + B d n D d21 ∗ Y 1 A d A d Y 1 B d1 ∗ ∗ X 1 1 0 ∗ ∗ ∗ Y 1 0 ∗ ∗ ∗ ∗ 1       > 0 ,   Z C d1 − D f C d2 C d1 Y 1 − C d n ∗ X 1 1 ∗ ∗ Y 1   > 0 , D d11 − D f D d21 = 0 , (6.10)  X 1 1 ∗ Y 1  > 0 , tr( Z ) < ν. The ﬁlter state-space matrices are recovered by A f = X − 1 2 ( A d n − X 1 A d Y 1 ) Y − T 2 , B f = X − 1 2 B d n , C f = C d n Y − T 2 , and D f , where the matrices X 2 and Y 2 satisfy X 2 Y T 2 = 1 − X 1 Y 1 . Gi ven X 1 and Y 1 , the m atrices X 2 and Y 2 can be found using a matrix decomposition, such as a LU decom position or a Cholesky decomposit ion. If D d11 = 0 and D d21 6 = 0 , then it is often sim plest to choose D f = 0 in order to satisfy the equality cons t raint of ( 6.10 ). This synthesis m ethod is deriv ed from the di screte-time H 2 -optimal dynami c output feedback controller synthesis m ethod in Synthesis Method 5.5 using the fact that H 2 -optimal ﬁlter synthesi s is a special case of t his problem . 6.4.3 H ∞ -Optimal Filter An H ∞ -optimal ﬁlter is d esi gned to minimize the H ∞ norm of ˜ P ( s ) in ( 6.6 ). 148 Synthesis Method 6.1 1. [ 5 , pp. 303–30 4 ] Solve for A n ∈ R n x × n x , B n ∈ R n x × n y , C f ∈ R n z × n x , D f ∈ R n z × n y , X , Y ∈ S n x , and γ ∈ R > 0 that mi n imize J ( γ ) = γ s ubject to X > 0 , Y > 0 ,     Y A + A T Y + B n C 2 + C T 2 B T n A n + C T 2 B T n + A T X YB 1 + B n D 21 C T 1 − C T 2 D T f ∗ A n + A T n XB 1 + B n D 21 − C T f ∗ ∗ − γ 1 D T 11 − D T 21 D T f ∗ ∗ ∗ − γ 1     < 0 , Y − X > 0 . The ﬁlter is recovered by A f = X − 1 A n and B f = X − 1 B n . 6.4.4 Discrete- Time H ∞ -Optimal Filter Synthesis Method 6.1 2. [ 252 ] Consider t he case where D d11 = 0 and D f = 0 . Sol ve for A d n ∈ R n x × n x , B d n ∈ R n x × n y , C d n ∈ R n u × n x , X , Y ∈ S n x , and γ ∈ R > 0 that mi nimize J ( γ ) = γ subject to X > 0 , Y > 0 ,         X X XA d XA d XB d1 0 ∗ Y Y A d + B d n C d1 + A d n Y A d + B d n C d1 YB d1 + B d n D d21 0 ∗ ∗ X X 0 C T d1 − C T d n ∗ ∗ ∗ Y 0 C T d1 ∗ ∗ ∗ ∗ 1 0 ∗ ∗ ∗ ∗ ∗ γ 1         > 0 ,  Y X ∗ X  > 0 . The ﬁlter is recovered by A f = − Y − 1 A d n ( 1 − Y − 1 X ) − 1 , B f = − Y − 1 B d n , and C f = C d n ( 1 − Y − 1 X ) − 1 . Synthesis Method 6. 1 3 . Solve for A d n ∈ R n x × n x , B d n ∈ R n x × n y , C d n ∈ R n u × n x , D d n ∈ R n u × n y , X 1 , Y 1 ∈ S n x , and γ ∈ R > 0 that mi n imize J ( γ ) = γ s ubject to X 1 > 0 , Y 1 > 0 ,         X 1 1 X 1 A d + B d n C d2 A d n X 1 B d1 + B d n D d21 0 ∗ Y 1 A d A d Y 1 B d1 0 ∗ ∗ X 1 1 0 C T d1 − C T d2 D T d n ∗ ∗ ∗ Y 1 0 Y 1 C T d1 − C T d n ∗ ∗ ∗ ∗ γ 1 D T d11 − D T d21 D T d n ∗ ∗ ∗ ∗ ∗ γ 1         > 0 ,  X 1 1 ∗ Y 1  > 0 . The ﬁlter state-space matrices are recovered by A f = X − 1 2 ( A d n − X 1 A d Y 1 ) Y − T 2 , B f = X − 1 2 B d n , C f = C d n Y − T 2 , and D f , where the matrices X 2 and Y 2 satisfy X 2 Y T 2 = 1 − X 1 Y 1 . Gi ven X 1 and Y 1 , the m atrices X 2 and Y 2 can be found using a matrix decomposition, such as a LU decom position or a Cholesky decomposit ion. This synthesis method i s d erived from the discrete-time H ∞ -optimal dynamic output feedback controller syn thesis method i n Synthesis M ethod 5.11 usi ng the fact that H ∞ -optimal ﬁlter synthe- sis is a special case of this problem. 149 A V ersion History A.1 Updates in V ersion 4 (Nov ember 27, 2024 ) The major structural up d ate in V ersion 4 is that Section 2 from V ersion 3 has b een split up in to two section (Section 2 and Section 3). This change helps separate the LMI properties and t ri cks that are prim aril y focused on reformulating BMIs as LMIs (Section 2) from other LMI properties (Section 3 ). Also, the s ubsections of Section 2 have been re-ordered sli g htly , so th at methods that typically reformul ate BMIs as equi valent LMIs are presented ﬁrst, foll owe d by method s th at are typically used to derive LM I conditions that imply the o ri g inal BMI cond i tions. The sectio n headings referred to in this update section correspond to those in V ersion 4. Section 1 Sec. 1.3.2 : Added missing “ Q < 0 ” to LMI in Exampl e 1.3 . Sec. 1.5. 1 : Ad ded new solver (STRIDE). Sec. 1.5.2 : Added ne w p arser (R OLMIP) and removed Scilab parser d ue to inaccessible code. Section 2 Sec. 2.3. 3 . 2 : Fixed typo. Tra nspo ses were swapped on some terms i n th e proof. Sec. 2.4. 3 . 9 : Fixed typo. Sec. 2.6. 4 : Ad ded Strict Petersen’ s lemma. Sec. 2.6. 5 : Ad ded Nons trict Petersen’ s lemma. Sec. 2.8.5 : Added con ve x-concav e d ecom position. Sec. 2.9 : Added penalized con ve x relaxation condition s. Sec. 2.10 : Added section on coordinate descent. Sec. 2.11 : The discussi on on ho w t o reformulate BMIs as LMIs was extended and rewritten. Section 3 Secs. 3.4.5 , 3.4. 6 : Fixed wrong ordering of weights. Sec. 3.10 : Fixed t ypo in Douglas-Fillmore-W illiams Lemma. Section 4 Sec. 4.2.1 : Added ne w resul ts. Sec. 4.3. 1 : Ad ded new results. Sec. 4.7.4 : Slight adj ustment in the presentation of t h e results . Sec. 4.16 .7 : Added ell iptic region to D -stability resul ts. Sec. 4.16 .8 : Added hy perbolic region to D -stability results. Sec. 4.24.2 : Added additional delay-dependent condition. Sec. 4.25 : Fixed t ypo in deﬁnition of va riables. 150 Section 5 Sec. 5.2.4 : Added alternativ e formul ation that allows for the closed-loop system to have non-zero feedthrough. Secs. 5.3 - 5.4 : Fixed typos regarding positive vs negative feedback. Sec. 5.4. 1 : Fixed typo in synthesis m ethod. Sec. 5.4.3 : Fixed typo in synth esi s method. Sec. 5.4.4 : Added alternative formulatio n that allows for th e closed-loop system to ha ve no n -zero feedthrough. Section 6 Sec. 6.1.2 : Fixed typo in synth esi s method. 151 A.2 Updates in V ersion 3 (A pril 2, 2021) Section 1 Sec. 1.3. 2 : Added m atri x var iable form of LMI deﬁnitio n. Sec. 1.4: Ne w section with improved discussion o n SDPs. Sec. 1.5: Edited LMI Solvers section t o include more details on s o lvers/parsers. Section 2 Sec. 2.3. 3 : Added Lin earization Lemma. Sec. 2.6: Updated Finsler’ s Lemma. Sec. 2.7: Fixed a typo and swapped variables to be consist ent wi th Y oung’ s Relation. Sec. 2.9: Slight adjus tment to S-Procedure. Sec. 2.10 : Added Duali zation Lemma. Sec. 2.11 : Added Frobenius norm and nuclear norm. Sec. 2.12: Added additional eigen value properties (sum, sum of absolute values, weight ed sum, weighted sum of abs o lute values). Sec. 2.14 : Added spectral radius. Sec. 2.15: Added more details on the trace of a sy mmetric matrix. Fixed t ypos. Added new fact from Duan and Y u [ 5 ]. Sec. 2.16 : Added fact o n th e range o f a symmetric matrix. Sec. 2.17 : Added th e Douglas-Fillm ore-W i lliams Lemma. Section 3 Sec. 3.1. 2 , 3.1.4: A d ded dilated results. Sec. 3.1. 5 , 3.1.6: A d ded descriptor s ystem admi ssibility . Sec. 3.2. 1 , 3.2.2: A d ded dilated results. Sec. 3.2. 3 , 3.2.4: A d ded descriptor s ystem Bounded Real Lemm a. Sec. 3.3. 1 -3.3.3: Added dilated results. Secs. 3.3.2, 3.3.3: Added ne w results for H 2 norm of discrete-time sy stems. Secs. 3.3.4, 3.3.5: Added ne w results for H 2 norm of descriptor sys tems. Sec. 3.4: Updated reference for Generalized H 2 Norm and added “(Induced L 2 - L ∞ Norm)” to T itle. Sec. 3.5: Updated reference for Peak-to-Peak Norm and added “(Induced L ∞ - L ∞ Norm)” to T itl e. Sec. 3.6. 7 : Added th e Discrete-T ime KYP Lemma for descriptor systems. Sec. 3.6. 8 : Added a QSR d issipativity-related property . Sec. 3.9. 2 : Added th e discrete-time N I Lemma. Sec. 3.9. 3 : Added th e negativ e im aginary system DC constraint . Sec. 3.15 : Added di s crete-time zeros condition. Sec. 3.16 : Improved organization o f D-Stability section. Sec. 3.17 : Added D-Adm issibilit y section. Sec. 3.19 : Added transient bounds on state and output for autonomo us and non -autonomous L TI systems. Also added transient bounds on unit im pulse response. Sec. 3.20 : Added out put energy bounds for autono mous and non-aut o nomous L TI systems 152 Section 4 Sec. 4.1. 1 : Fixed typo i n generalized plant of Ex am ple 4.2 . Secs. 4.2.3, 4.2.4, 4 .3.3, 4.3.4, 4.4.3, 4.4.4: Fixed t y pos in reformulation of B c . Sec. 4.2.4: Added a second synthesis meth o d for discrete-time H ∞ -optimal dynamic output feed- back cont rol . Sec. 4.3.4: Added a second synthesis method for discrete-time H 2 -optimal dynami c output feed- back cont rol . Section 5 Sec. 5.4: Added discrete-tim e optimal ﬁlterin g results. Secs. 5.2.1, 5.3.1: Fixed m i ssing transpose on matrices C 1 and C 1 , 2 153 References [1] S. Boyd, L. El Ghaoui, E . Feron, and V . Balakrishnan, Linear Ma t rix Inequalities in System and Contr ol Theory . Philadelphi a, P A: Society for Industrial and Applied M athematics, 1994. [2] G. E. Dullerud and F . Paganini, A Course in Rob ust Contr ol Theory: A Con ve x Appr oach , ser . T exts in Applied Math ematics. New Y ork, NY : Springer , 2000, no. 36. [3] C. Scherer and S. W eiland, “Linear m atrix i nequalities in control, ” January 2015. [Onli n e]. A vailable: https:/ / www .imng.u ni- stu ttgart.de/mst/ﬁles/LectureNotes.pdf [4] S. Skogestad and I. Postlethwaite, Multivariable F eedback Contr ol: Analysis and Design , 2nd ed. Hoboken, NJ: W iley , 2005. [5] G.-R. Duan and H.-H. Y u, LMIs in Contr ol Systems: Analysis, Design and Applicatio n s . Boca Raton, FL: CRC Press, 2013 . [6] R. A. Horn and C. R. Johnson, Matrix Analysi s , 2nd ed. Ne w Y ork, NY : Cambridge Univ ersity Press, 2013. [7] D. S. Bernstein, Scalar , V ector , and Matrix Math ematics: Theory , F acts, and F ormula s . Princeton, NJ: Princeton University Press, 2018. [8] S. Boyd and L. V andenberghe, Semideﬁnite Pr ogramming Relaxations of Non-Con ve x Pr ob- lems in Control and Combi natorial Optimization . Boston, MA: Springer US, 1 997, pp. 279–287. [9] L. El Ghaoui and S.-I. Niculescu, Advances in Linear Ma trix Inequality Met h ods in Con- tr ol , ser . Ad vances in Design and Control. Philadelphia, P A: Society for Industrial and Applied Mathematics, 2000, ch. Rob ust Decision Problems i n Eng ineering: A Linear Ma- trix Inequalit y Approach. [10] J . G. V anAnt werp and R. D. Braatz, “ A tutorial on linear and b ilinear m atrix inequalit ies, ” J ourn al of Pr ocess Contr ol , vol. 10, pp. 363–385, 20 0 0. [11] G. Herrmann, M. C. T urner , and I. Postlethwaite, “Li n ear matrix i nequalities in control , ” in Mathemat ical Methods for Robust and Nonlinear Contr ol: EPSRC Summer School , ser . Lecture Notes i n Control and Information Sciences, M . C. T urner and D. G. Bates, E ds. Berlin, Germany: Springer-V erlag, 2 0 07, vol. 367, pp. 123–14 2 . [12] K. Lange, Opt imization . Ne w Y ork, NY : Springer , 2013. [13] S. Boyd and L. V andenber ghe, Con vex Optimiz ation . Cambridge, UK: Cambridge Univer - sity Press, 2004. [14] V . Balakrishnan and L. V andenber ghe, “Semideﬁnite programming d u ality and linear time- in variant s ystems, ” IEEE T ransactions on Automatic Control , vol. 48, no. 1, p p . 30–41, 2003. 154 [15] — —, “Semideﬁnit e programming duality and linear tim e-inv ariant sys t ems, ” Department of Electrical and Computer En g ineering, Purdue University , W est Lafayette, IN, T ech. Rep. TR-ECE-02-02, 200 2 . [16] K. C. T oh, M. J. T odd , and R. H. T ¨ ut ¨ unc ¨ u, “ SDPT3 - a MATLAB software package for semideﬁnite prog ramming, ” Opti mization Method s and Soft war e , vol. 11 , no . 1– 4, p p. 5 4 5– 581, 1999 . [17] K. C. T oh, R. H. T ¨ ut ¨ unc ¨ u, and M. J. T odd, “SDPT 3 a matlab soft- ware package for semideﬁnite-quadratic-linear pgrogramming. ” [Online]. A vailable: http://www .math.cmu.edu/ ∼ reha/sdpt3.html [18] J . Strum, “Us i ng S eDuMi 1.02, a MATLAB toolbox for optimization over sy mmetric cones, ” Optimizati on Methods and Soft war e: Special Issue on Interior P oint Methods , v ol. 11, no. 1–4, pp. 625–653, 19 9 9. [19] “SeDuMi . ” [Online]. A vailable: http://s edu m i.ie.lehigh.edu/ [20] M OSEK ApS, “The mosek optimi zation software, ” Onli ne at http://www .mo sek.com , 2018. [21] B. Borchers, “CSDP, a C library for semid eﬁnit e programming, ” Optimiza tion Methods and Softwar e , vol. 11, no. 1, pp. 613– 6 23, 1999. [22] — —, “CSDP, ” 2018. [Online]. A vailable: https:// g ithub .com/coin- or/Csdp [23] M . Andersen, J. Dahl, Z. Liu, L. V andenberghe, S. Sra, S. Now ozin, and S. Wright, “Interior- point methods for large-scale cone programmin g , ” i n Optimizat ion for Machine Learning , S. Sra, S. No wozin, and S. J . Wright, E ds. Cambridge, M A: MIT Press, 2012, vol. 5583, ch. 3, pp. 55–83. [24] M . Andersen, J. Dahl, and L. V andenber ghe, “CVXOPT: Python s o ft ware for con vex optimizatio n , ” 2020. [Onl ine]. A vailable: http://cvxopt .org/inde x.ht ml [25] M . Karimi and L. T unc ¸ el, “Domai n -Dri ven Solver (DDS) V ersion 2.1: a M A TLAB-based software package for con vex opt imization problems in domain-driv en form, ” Mathematical Pr ogramming Computation , v ol. 16, no. 1, pp. 37–92, 202 4 . [26] M . Karimi and L. T unc ¸ el, “DDS users’ guide. ” [Onli ne]. A vailable: http://www .math.uwaterloo.ca/ ∼ m7karimi/DDS.ht ml [27] S. J. Benson and Y . Y e, “DSDP5: Software for semideﬁnite programmin g, ” ACM T ransac- tions on Mathematical Sof t war e , vol. 34, no . 3, p p . 16: 1–20, 2005. [28] “DSDP: Software for semideﬁnite programming, ” 200 6. [Onl ine]. A vailable: https:// w ww .mcs.anl.gov/hs/software/DSDP/ [29] P . Gahinet, A. Nemirovskii, A. J . Laub, and M. Chilali, “The LMI control toolbox, ” i n Pr oc. IEEE Confer ence on Decision and Cont r ol , Lake Buena V ista, FL, 1994, pp. 2038–2041. 155 [30] J . Fiala, M. K o ˇ cv ara, and M. Stin gl, “PENLAB: A MA TLAB solver for nonli n ear semideﬁ- nite opti mization, ” arXiv , 20 13. [Online]. A vailable: https://arxiv .org/abs/1311.5240 [31] M . K o ˇ cvara, “PENLAB, ” 2017. [Online]. A vailable: http://web .m at . b ham.ac.uk/kocv ara/penlab/ [32] B. O’Donoghue, E. Chu, N. Par ikh , and S. Boyd, “Conic optimi zation via operator splitting and homog eneou s s elf-dual em b edding, ” Journal of O p timization Theory and Ap p lications , vol. 16 9 , no. 3, pp. 1042– 1068, 2016 . [33] — —, “SCS: Split ting conic solver , version 2 . 1 .2, ” 2 0 19. [Onli n e]. A vailable: https:// g ithub .com/cvxgrp/s cs [34] M . Y amashita, K. Fuji saw a, and M. K ojim a, “Implementatio n and e valuation of SDP A 6.0 (SemiDeﬁnite Programm i ng Al gorithm 6.0), ” Optimiza tion Methods and Sof t war e , vol. 18, no. 4, pp. 491– 505, 2003. [35] M . Y amashita, K. Fujis aw a, K. Nakata, M. Nakata, M. Fukud a, K. K obayashi, and K. Goto, “ A high-performance software package for semideﬁnite programs: SDP A 7, ” Dept. of Math- ematical and Computing Science, T okyo Inst i tute of T echnolo g y , T okyo, Japan, T ech. Rep. B-460, 2010. [36] K. Fujisawa, M. Fukuda, Y . Futakata, K. K obayashi, M. Kojima, K. Nakata, M. Nakata, and M. Y amashita, “SDP A of ﬁcial page, ” 2020. [Online]. A vailable: http://s d pa.sourcefor ge.net/i n dex.html [37] M . S. Andersen, J. Dahl, and L. V and enb er ghe, “Implementati on of no n symmetric i n terior- point methods for l i near optimization over sparse matrix cones, ” Mathematical Pr ogram- ming Computa t ion , vol. 2, n o . 3–4, p p. 16 7 –201, 2010. [38] M . S. Andersen and L. V andenberghe, “SMCP - Python extension for sparse matrix cone programs, ” 2018. [Online]. A vailable: https:// s mcp.readthedocs.io/en/latest / [39] L . Q. Y ang, D. F . Sun, and K. C. T oh, “SDPN AL+: A majo ri zed sem i smooth Newton- CG augmented Lagrangian m ethod for semi deﬁnite programmi n g wit h nonnegative con- straints, ” Mathematical Pr ogramming Computation , vol. 7, pp. 331 – 366, 2015. [40] D. F . Sun and K. C. T oh, “SDPN ALpl us. ” [Online]. A vailable: https:// b log.nus.edu.sg/ m attohkc/softwares/sdpnalplus/ [41] H. Y ang, L. Liang, L. Carlone, and K.-C. T oh, “ An inexact projected gradient method wit h rounding and lifti ng by nonl inear programming for s olving rank-one semideﬁnite relaxation of polynomial optimization, ” Mathematical Pr ogramming , v ol. 201, no . 1-2, pp. 409–472, 2023. [42] H. Y ang and L. Liang , “STRIDE: SpecT rahedRal Inexact projected gradient Descent along vErtices, ” 2022. [Online]. A vailable: https:// g ithub .com/MIT - SP ARK/STRIDE 156 [43] H. D. Mitt el m ann, “ An in dependent benchmarking of SDP and SOCP s olvers, ” Mathemati - cal Pr ogramming , vol. 95, no. 2, pp. 40 7–430, 2002 . [44] D. Arzelier , D. Peaucelle, and D. Henrion, “Some notes on standard LMI solvers, ” 20 0 2. [Online]. A vailable: http://ho mepages.laas.fr/arzelier/publis/20 0 2/prague102.pdf [45] H. D. Mi t telmann, “Decision tree for optimizati o n software, ” 20 18. [Online]. A vailable: http://pl ato.la.asu.edu/bench.htm l [46] J . L ¨ oftberg, “ YALMIP : A toolbox for m o deling and opt i mization in MATL AB , ” i n IEE E International Symposium on Compu t er Aided Cont r ol Systems Design , 2004. [47] — —, “Y almip, ” 2020. [Online]. A vailable: https:// y almip.github.io/ [48] M . Grant and S. Boyd, “Graph imp lementations for nonsmooth conv ex programs, ” in Recent Advances in Learning and Contr ol , ser . Lecture Notes in Cont rol and Information Sciences, V . Blondel, S. Boyd, and H. Kimura, Eds. Springer-V erlag Limited, 2008 , pp. 95–110 . [49] — —, “CVX: Matl ab software for discipli n ed con vex programming , version 2.1, ” 2014. [Online]. A vailable: http://cvxr .com/cvx [50] C. M. Agulhari, A. Felipe, R. C. L. F . Oliveira, and P . L. D. Peres, “ Algorithm 998: The Robust LMI Parser - a toolb ox to construct LMI conditio n s for uncertain systems, ” A CM T ransactions on Mathematical Sof t war e , vol. 45 , no. 3, p. 36, 2 0 19. [51] — —, “Robust LMI parser , ” October 202 0. [Online]. A vailable: https:// rol mip.githu b.io/ [52] S. Diamond and S. Boyd, “CVXPY: A Python -embedded mo deling language for conv ex optimizatio n , ” Journal of Machine Learning Resear ch , vol. 17, no. 83, pp . 1–5 , 2016. [53] A. Agrawa l, R. V erschueren, S. Diamond, and S. Boyd, “ A rewriting system for con vex optimizatio n problems , ” J ournal of Contr ol and Decision , v ol. 5, no. 1, pp. 42–60, 201 8 . [54] S. Diamon d and A. Agrawa l, “W elcome to CVXPY 1.0, ” 2019. [Online]. A vailable: https:// w ww .cvxpy .org/index.html [55] G. Sagnol and M. Stahlberg, “ A Pyth o n interface to conic optimizatio n solvers, ” 2020 . [Online]. A vailable: https://picos - api.gitlab.io/picos/introduction.html [56] M . G h asemi, “Irene 1.2.3 documentat i on, ” 2017. [Online]. A vailable: https:// i rene.readthedocs.io/en/latest/i n dex.html [57] C. D. Sousa, “PyLMI-SDP 0.2, ” 2013. [Online]. A vailable: https:// pypi.o r g/proj ect/ PyLMI- SDP [58] M . Udell, K. Mohan, D. Zeng, J. Hong, S. Diamon d , and S. Boyd, “Con ve x optimization in Julia, ” in F irst W orkshop for High P erformance T echnical Computing in Dynamic Lan- guages , New Orleans, LA, 2014, p p. 18 –28. 157 [59] J . Hong, K. Mohan, M. Udell , and D . Zeng, “Con ve x.j l - con vex opt i mization in Ju lia, ” 2019. [Onli ne]. A vailable: ht tps://www .juliaopt.org/Con vex.jl/stable/ [60] I. Dunnin g , J. Huchette, and M. Lubin, “JuMP: A Modeling Language for M at h ematical Optimization , ” SIAM Rev iew , vol. 59, no. 2 , pp. 2 95–320, 2017. [61] — —, “JuMP. ” [Onli ne]. A vailable: https:// w ww .juliaopt.org/JuMP .jl/stable/ [62] J . P . Chancelier , P . V . Pakshin, and S. G. Soloviev , “LMI parse for NSP software package, ” IF AC P r oceedings V ol umes: 18th IF A C W orld Congr ess , v ol. 4 4 , no. 1, pp. 14 253 – 14 258, 2011. [63] J . P . Chancelier , “Nsp toolboxes, ” 2016. [Online]. A vailable: https:// cermi cs.enpc.fr/ ∼ jpc/nsp- tiddly/ [64] D. W . Gu, P . H. Petkov , and M. M. Konstantinov , Robust Contr ol Design with MA TLAB , 2nd ed. London , UK: Springer , 2013. [65] K. Gu , “P artial solution of LMI in st ability problem of tim e-delay systems, ” in Pr oc. IEEE Confer ence on Decision an d Contr ol , Phoenix, AZ, 1999 , pp. 2 27–232. [66] K. Gu , V . L. Kharitonov , and J. Chen, Sta b ility of T ime-Delay Systems . Boston, MA: Birkhauser Boston , 2003. [67] J . C. Geromel, “Rob ustness o f linear dynamic systems , ” August 2005. [Online]. A vailable: http://www .dt.fee.unicamp.br/ ∼ geromel/rob mult i.pdf [68] X. H. Chang and G. H. Y ang, “Ne w results on output feedback control for linear d i screte- time system s , ” IEEE T ransacti ons on Automatic Contr ol , vol. 59, no . 5, pp. 1355–1 3 59, 2013. [69] K. Gu, “ A further reﬁnement of dis cretized lyapunov fun ctional method for the stability of time-delay sy s tems, ” Int ernational Journal of Contr ol , vol. 74, no. 1 0, pp. 967–976, 2001. [70] P . Gahinet and P . Apkarian, “ A li n ear matrix inequality approach to H ∞ control, ” Interna- tional Journal of Rob ust a n d Nonlinear Contr ol , vol. 4, no. 4 , pp. 421–448, 19 94. [71] X. Zhan, Matrix Inequalities , ser . Lecture Notes in M athematics. Berlin, Germ any: Springer -V erlag, 2002, v ol. 1790. [72] A. Helmersson, “Methods for robust gain scheduling, ” Ph.D. di ssertation, Link ¨ oping Uni- versity , Link ¨ oping, Sweden, Nov . 199 5. [73] P . Ap karian, H. D. T uan, and J. Bernussou, “Continuous-tim e analysis, eigenst ructure as- signment, and H 2 synthesis with enhanced l inear matrix inequali t ies (LMI) characteriza- tions, ” IEEE T ransactions on A utoma tic Cont r ol , vol. 46, no . 12, pp. 1941– 1946, 2001. [74] X. H. Chang and G. H. Y ang, “ A descripto r representation approach to observer -based H ∞ control synthesis for discrete-time fuzzy systems, ” Fuzzy Sets and Syst ems , v ol. 18 5, no. 1, pp. 38–51 , 2011. 158 [75] F . Delmo tte, T . M. Guerra, and M. Ks antini, “Contin uous T akagi-Sugeno’ s models: Re- duction of the num ber of LMI conditions in va rious fuzzy cont rol design technics, ” IEE E T ransactions on Fuzzy Systems , vol. 15, no. 3 , pp. 4 26–438, 20 0 7. [76] X. H. Chang and G. H. Y ang, “Non fragil e H ∞ ﬁltering of con t inuous-time fuzzy system s, ” IEEE T ransactions on Signa l Pr ocessing , vol. 59, no. 4, pp. 15 2 8 –1538, 2011. [77] X. -H. Chang, Robust Out put F eedback H ∞ Contr ol and Filtering for Un certa in Linear Sys- tems . Berlin, G erm any: Springer , 2014. [78] P . Finsler , “ ¨ Uber das vorkommen deﬁniter und s emideﬁniter formen in scharen quadratis- cher formen, ” Commentarii Mathematici Helvetici , vol. 9, no. 1, pp. 1 8 8–192, 1936. [79] I. R. Petersen, “ A stabilization algorithm for a class of uncertain l i near syst ems, ” Syst ems & Contr ol Letters , vol. 8, n o. 4, pp. 3 5 1–357, 1987. [80] M . C. de Oliveira and R. E. Skelton, “Stabilit y tests for constrained l i near syst ems, ” in P erspectives in Ro bust Cont r ol , ser . Lecture Notes i n Control and Informati on Sciences, S. P . Moheim ani, E d . L o ndon, UK: Springer , 2001, vol. 268. [81] D. H. Jacobson, Extensions of Linear -Quadratic Contr ol, Optimi z ation an d Matrix Theory , ser . Mathem atics in Science and Engineering. New Y ork, NY : Academic Press, 1977, vol. 133. [82] R. E. Skelton, T . Iwasaki, and K. Grigoriadis, A Uniﬁed Algebraic Appr oach t o Linear Contr ol Design . London , UK: T ayl or & Francis, 1998 . [83] M . W u, Y . H e, and J. H. She, Stabil ity Analysis a nd Robust Cont r ol of T ime-Delay Systems . Berlin, Heidelberg: Springer , 2010. [84] L . X i e, M. Fu, and C. de Souza, “ H ∞ control and qu adrati c stabilization of syst ems wi th pa- rameter uncertainty via output feedback, ” IEEE T ransactions on Automatic Cont rol , vol. 37, no. 8, pp. 1253 –1256, 1992. [85] L . Xi e, “Outpu t feedback H ∞ control of sys t ems with parameter uncertainty , ” Internationa l J ourn al of Contr ol , v ol. 63, no. 4, pp. 741–750, 199 6 . [86] I. R. Petersen and C. V . Hollot, “ A Riccati equation approach to the stabil ization of uncertain linear system s , ” A utomat ica , vol. 22, no. 4 , pp. 397–411, 1 9 86. [87] A. Biso fﬁ, C. De Persis, and P . T esi, “Data-driven control via Petersen’ s lemma, ” A utoma t - ica , vol. 145, p. 110537 , November 2022. [88] M . V . Khlebnikov , “Quadratic stabilization of dis crete-tim e bilinear system s, ” Automation and Remote Contr ol , vol. 79, no. 7, pp. 1222–12 3 9, 2 018. [89] M . V . Khlebnikov and P . S. Shcherbako v , “Petersen’ s lemm a on matrix uncertainty and it s generalizations, ” Automation and Remote Contr ol , vol. 69, no. 11, pp. 1 9 32–1945, 2008. 159 [90] P . Shcherbak ov and M . T opunov , “Ext ensions of Petersen’ s lemm a on matrix uncertainty , ” IF AC Pr oceedings V olu mes , vol. 41, no. 2 , pp. 11 385–11 390, 2008. [91] M . V . Khlebnikov , “New generalizations of the Petersen lemma, ” A utoma tion and Remote Contr ol , vol. 75, no. 5, pp. 917–921, 2014. [92] Y . Ebihara and T . Hagiwara, “New dilated LMI characterizations for continuous-ti m e m u l - tiobjective controller synthesi s , ” A utomat ica , vol. 10, pp. 2 003–2009, 2004. [93] K. Zhou and P . P . Khargonekar , “Robust stabili zation of linear systems wi th norm-boun ded time-varying uncertainty , ” Systems & Control Letters , vol. 10, no. 1, pp. 1 7 –20, 1988. [94] A. Zemouche, R. Rajamani, B. Boulkroune, H. Rafaralahy , and M . Zasadzinski , “ H ∞ circle criterion observer design for Li p schitz nonlinear systems with enhanced LMI conditi o ns, ” in Pr oc. American Contr ol Confere nce , Boston, MA, 2016, pp. 131–136. [95] R. Merco, F . Ferrante, R. G. Sanfelice, and P . Pisu, “LMI-based output feedback control design in the presence of s poradic m easurem ent s, ” in Pr oc. American Contr ol Confer ence , Den ver , CO, 2020, p p . 333 1 –3336. [96] Y . Y . Cao, Y . X. Sun, and C. Cheng, “Delay-dependent robust stabilization of uncertain sys- tems with multipl e state delays, ” IEEE T ransactions o n Automatic Contr ol , vol. 4 3 , no. 11, pp. 1608– 1 612, 1998. [97] Y . W ang, L. Xi e, and C. E. de Souza, “Rob ust cont rol of a class of uncertaint nonlinear systems, ” Systems & Contr ol Letters , v ol. 19, no. 2, pp. 139–149, 1992. [98] F . T ahir and I. M. Jaimoukha, “Low-complexity polytopic inv ariant s et s for linear sy s tems subject to norm-bounded un certaint y , ” IEEE T ransa cti ons on Automatic Contr ol , vol. 60, no. 5, pp. 1416 –1421, 2015. [99] Q. T . Dinh, S. Gum ussoy , W . Michiels, and M. Diehl, “Combini n g con vex–conca ve d ecom - positions and linearization approaches for solving BMIs, with applicatio n to static output feedback, ” IEEE T ransactions on A utomat i c Control , vol. 57, no. 6, pp. 137 7–1390, 2011. [100] A. Priuli, S. T arbouriech, and L. Z accarian, “Static li n ear anti-windup desi gn with sign- indeﬁnite quadratic forms, ” IEEE Contr ol S yst ems Letters , vol. 6, pp. 3158 –3163, 2022. [101] E. C. W arner and J. T . Scruggs, “Control of v ibratory networks wit h passiv e and regenera tive systems, ” in Pr oc. American Contr ol Confere nce , Chicago, IL, 2015, p p. 55 0 2–5508. [102] ——, “Iterativ e conv ex overbounding algorith m s for BMI opti m ization problems, ” IF A C P apersOnline , vol. 50, no. 1, pp. 10 449–10 45 5 , 2017. [103] M. Kheirandishfard, F . Zohrizadeh, and R. Madani, “Con vex relaxation of bi linear matrix inequalities part i : Theoretical results, ” in IEEE Confer ence on Decision an d Contr ol , Mi- ami, FL, 2018, pp. 67–74. 160 [104] M. Kheirandish fard, F . Zo h rizadeh, M. Adil, and R. M adani, “Con vex relaxation of bil i near matrix inequalities part ii: App lications t o opt imal control synthesis , ” in IEEE Confere nce on Decision and Cont r ol , Miami, FL, 2018, pp. 75–82. [105] Y . W ang, A. Zem o uche, and R. Rajamani, “ A sequential LM I app roach t o design a BMI- based mult i-objective nonlinear observer , ” Eur opean Journal of Contr ol , vol. 44, pp. 50–57, 2018. [106] T . Iwasaki, “The du al iteratio n for ﬁxed-order control, ” IEEE T ransactions on Automatic Contr ol , vol. 44, no. 4, pp. 783–788, 1999. [107] J. C. Doyle and C.-C. Chu, “Matrix interpolation and H ∞ performance bounds, ” in Ameri - can Contr ol Confer ence , Boston, M A, 19 85, pp. 129–134. [108] J. C. Doyle, “Structured uncertainty in control system desi gn, ” in IEEE Confer ence on De- cision and Contr ol , Fort Lauderdale, FL, 1985 , pp. 26 0 –265. [109] ——, “Synthesis of robust controll ers and ﬁlt ers, ” in IEEE Confer ence on Decision and Contr ol , San Ant onio, TX, 1983, pp. 109–114. [110] J. Geromel, P . Peres, and S. Souza, “Outp ut feedback st abi lization o f un certain sy stems through a min/max problem, ” IF A C Proce eding s V o lumes , vol. 26 , no. 2, pp. 215–218 , 1 993. [111] M. A. Rotea and T . Iwasaki, “ An alternative to the DK it eration?” in American Contr ol Confer ence , vol. 1, Baltimore, MD, 1994, pp. 53–57. [112] T . Iwasaki and R. Skelton, “The XY-centring algorithm for th e dual LMI problem: a new approach t o ﬁxed-order control desi g n, ” Interna t ional Journal of Contr ol , vol. 62, no. 6 , pp . 1257–1272, 199 5 . [113] Y . Y amada and S. Hara, “ An LMI approach to local optim ization for cons tantly scaled H ∞ control probl ems, ” Internation a l Journal of Cont r ol , vol. 67, no . 2, p p . 233 – 250, 1997. [114] A. Doroudchi, S. Shiva kum ar , R. E. Fisher , H. Marvi, D. Au kes, X. He, S. Berman, and M. M. Peet, “Decentralized control of distributed actuation in a segmented soft robot arm, ” in IEEE Confer ence on Decision a n d Contr ol , Miam i , FL, 2018, pp. 7002–7009. [115] S. Dahdah and J. R. Forbes, “System norm regularization methods for K oopm an operator approximation, ” Pr oceedings of the Royal So ci et y A , vol. 478, no. 2 265, p. 2022 0 162, 2022 . [116] V . A. Y akubovich, “The S-procedure in non-linear control theory , ” V estni k Leningrad Uni- versity , Mat hematics , vol. 4, pp. 73–93, 1977. [117] U. T . J ¨ onsson, “ A lecture on the S-procedure, ” Lectur e Notes at the Royal Institute of T echnology , 2001. [Online]. A vailable: https:// p eople.kth.se/ ∼ uj/5B5746/Lecture.ps [118] M. Fathi and H. Bevrani, O p timization in El ectr i cal Engineering . Cham, Switzerland: Springer , 2019 . 161 [119] S. Lall, “Engr21 0a lecture 3: Singular v alues and LMIs, ” Aug u st 2001. [Online]. A vailable: https:// l all.stanford.edu/engr210a/l ectures/lecture3 2001 10 08 01.pdf [120] M. Fazel, H. Hindi, and S. P . Boyd, “ A rank minimi zati on heuristic with application to minimum order system approximati o n, ” in Pr oc. American Contr ol Confer ence , Arling t on, V A, 2001, pp. 4734–4739 . [121] B. Recht, M . Fazel, and P . A. Parrilo, “Guaranteed min i mum-rank soluti ons of lin ear matrix equations v ia nuclear norm minim ization, ” SIAM Review , vol. 52, no. 3, pp. 471– 501, 2010. [122] F . Alizadeh, “Interior poi n t methods in sem ideﬁnite programmi ng wi th applicati o ns to com- binatorial optim ization, ” SIAM Journal on Optimizatio n , vol. 5, no. 1, pp. 13–51, 1995. [123] F . Zhang, Matrix Theory: B a s ic Resu l s and T ec hniques , 2nd ed. New Y ork, NY : Springer , 2011. [124] J.-C. Bourin, “Some inequalit ies for norms on matrices and operators, ” Linear Al gebra and its App l ications , vol. 29 2, no . 1–3, pp. 139–154, 199 9 . [125] M. V . Tra vaglia, “On an inequality in volving power and contractio n matrices with and wi t h- out trace, ” Journal of Inequalities in Pur e an d Applied Mathemati cs , vol. 7, no. 2, p. 6 5, 2006. [126] R. G. Douglas, “On majorization, factorication, and range inclusion of operators on Hilbert space, ” Pr oc. American Mathematics Society , v ol. 17, n o . 2, pp. 413–415 , 1966. [127] P . A. Fillmore and J. P . W i lliams, “On operator ranges, ” Advances i n Mat h ematics , vol. 7, no. 3, pp. 254– 281, 1971. [128] H. Dym, Linear Al gebra in Acti on . Providence, RI: American Mathematical Society , 2006. [129] Y .-H. Au-Y eung, “Some inequalities for the rational power o f a no n negati ve deﬁnit e matrix, ” Linear Algebra and its Applicati ons , vol. 7 , no. 4, pp. 347–350, 1973. [130] N. N. Chan and M. K. Kwong, “Hermitian m atrix i nequalities and a conjecture, ” Ameri can Mathematical Month ly , vol. 92, no. 8, pp. 533–541 , 19 85. [131] R. Bhatia and F . Kittaneh, “On the singular values of a p rod uct o f operators, ” SIAM Journal on Matri x Analysis and Applicati ons , vol. 1 1 , no. 2, pp. 272–277, 1990. [132] J. S. Auj la and J.-C. Bourin, “Eigen value inequaliti es for con vex and log-con vex functio ns, ” Linear Algebra and its Applicati ons , vol. 4 24, no. 1, pp. 25–35, 2007. [133] J.-C. Bourin, “Re verse rearrangement i nequalities v i a matrix technics, ” J ournal of Inequal- ities in Pur e and Applied Mat hematics , vol. 7, n o . 2, p. 43, 2006. [134] J. K. Baksalary and F . Pukelsheim, “On t he L ¨ owner, minus, and star partial orderings of nonnegativ e deﬁnite matrices, ” Linear Algebra and its Applicat ions , vol. 151, pp. 135–141, June 1991. 162 [135] M. K. Kwong, “Some results o n m atrix monotone functions, ” Linear Al gebra and i t s A p pli- cations , vol. 118, pp. 129–1 5 3 , June 19 8 9. [136] R. Bellman, “Some inequalities for the square root of a po s itiv e deﬁnit e matrix, ” Linear Algebra and its Applicat ions , vol. 1, n o. 3, pp. 321–32 4 , 1968. [137] K. V . Bhagwat and R. Subramanian, “Inequaliti es between means of positive operators, ” Mathematical Pr oceedings of the Campbridge Phi losophical So ci ety , vol. 83, no. 3, pp . 393–401, 1978. [138] J. C. W illems , “Least squares s t ationary optimal control and the algebraic Riccati equat i on, ” IEEE T ransactions on Automatic Contr ol , vol. 16, no. 6, pp. 6 21–634, 197 1. [139] R. V enkataraman and P . Seiler , “Con vex LPV synthesis of estim ators and feedforwards using dualuty and integral quadrati c constraint s, ” International J ourna l of Ro bust and Nonli n ear Contr ol , vol. 28, no. 3, pp. 953–975, 2018. [140] Y . Ebihara, D. Peaucelle, and D. Arzelier , S -V ari able Appr oach to LMI-Based Robust Con- tr ol . Lo ndon, UK: Springer , 2015. [141] J. C. Geromel, M. C. de Oliveira, and L. Hsu , “LMI characterization of structu ral and robust stability , ” Linear Algebra and its Applicat ions , vol. 285, no. 1 – 3 , pp. 69–80, 199 8 . [142] A. Felipe, R. C. L. F . Ol iveira, and P . L. D. Peres, “ An iterative LMI based procedure for robust stabilization of conti nuous-time poly topic sys t ems, ” in Pr oc. A merican Contr ol Con- fer ence , Boston, MA, 2016, pp. 3 8 2 6–3831. [143] A. Felipe and R. C. L. F . Oliveira, “ An LMI-based algorithm to compute robust stabilizing feedback gain s directly as optimizatio n variables, ” IEEE T ransactions on Automatic Con- tr ol , 2020, in press. [144] M. C. De Oliveira , J . Bernussou, and J. C. Geromel, “ A new discrete-tim e robust st ability conditions, ” Systems & Control Letters , vol. 37, no. 4, pp. 26 1 –265, 1999. [145] M. C. De Oliv eira, J. C. Geromel, and L. Hsu, “LMI characterization of s tructural and robust stability: The discrete-time case, ” Linear Algebra and its Applicatio ns , vol. 29 6 , no. 1–3, pp. 27–38 , 1999. [146] A. Felipe, “Um algoritmo de busca local baseado em LMIs para comput ar ganhos de realimentac ¸ ˜ ao estabilizantes diretamente como vari ´ aveis de otimizac ¸ ˜ ao, ” Master’ s thesi s, Univ ersidade Estuadu al de Campinas, Campinas, Brazil, 2017. [147] A. Spagoll a, C. F . Morais, R. C. L. F . Oliveira, and P . L. D. Peres, “Realimentac ¸ ˜ ao est ´ atica de s a´ ıda de s i stemas LPV positiv os a tempo discreto, ” in Simp ´ osio Brasileir o de Automac ¸ ˜ ao Inteligente , Ou ro Preto, Brazil, 2019, pp. 774–779. [148] I. Masubuchi, Y . Kamitane, A. Ohara, and N. Suda, “ H ∞ control for d escriptor s y stems: A matrix in equ al i ties approach, ” Automatica , vol. 33, no. 4, pp. 669–67 3 , 1997. 163 [149] H.-S. W ang, C.-F . Y u n g, and F .-R. Chang, “Bounded real lemma and H ∞ control for de- scriptor systems, ” IEE Pr oceedings - Contr ol Theory and Application s , vol. 145, no. 3, pp. 316–322, 1998. [150] M. Chadli, P . Shi, Z. Feng, and J. Lam, “New bounded real lemma formul ation and H ∞ control for continuous-tim e descriptor systems, ” Asian J ournal of Contr ol , vol. 19, no. 6, pp. 2192– 2 198, 2017. [151] B. Marx, D. K oenig, and D. Georges, “Robust pole-clustering for descriptor s ystems a stri ct LMI characterization, ” in Pr oc. Eur opean Contr ol Confer ence , Cambridge, UK, 2003, pp. 1117–1122. [152] K.-L. Hsiung and L . Lee, “L yapunov inequalit y and bounded real lemma for dis crete-tim e descriptor s ystems, ” IEE Pr oceedings - Contr ol Theory and Application s , v ol. 146, no. 4, pp. 327–3 3 1, 1999. [153] S. Xu and C. Y ang, “Stabilization of discrete-tim e singular sys tems: A matrix inequalities approach, ” Automatica , vol. 35, no. 9 , pp. 1613–1617, 1999. [154] G. Zhang , Y . Xia, and P . Shi, “New bounded real lem m a for discrete-time sing ular sys t ems, ” Automatica , vol. 44, no. 3, pp. 88 6–890, 2008. [155] M. Chadli and M. Darouach, “Novel bounded real lemm a for discrete-time d escriptor s y s - tems: Application to H ∞ control desig n, ” A utoma t ica , vol. 48, no. 2, p p . 449–453, 2012. [156] S. Xu and J. Lam, “Robust stabi lity and stabilizatio n of discrete singular systems: An equiv- alent characterization, ” IEEE T ransactions on Automatic Contr ol , vol. 49, no. 4, pp. 568– 574, 2004 . [157] I. Masubuchi and Y . Ohta, “Stabili ty and stabilization of discrete-time descriptor s ystems with se veral extensions, ” i n Pr oc. Eur opean Contr ol Conf er ence , Z ¨ urich, Switzerland, 2013, pp. 3378– 3 383. [158] C. Scherer , “The Riccati inequality and state-space H ∞ -optimal control, ” Ph.D. diss ertation, Julius Maxim ilians University W ¨ urzb urg, W ¨ urzb urg, Germany , 1990. [159] W . Xi e, “ An equiv alent LMI representation of bounded real lemm a for cont i nuous-time systems, ” Journal of Inequa l ities and Applications , vol. 2008, p . 672905, 2008. [160] D. Krokavec and A. Filasov ´ a, “Equiv alent representations of bounded real lemma, ” in 18t h International Confer ence on Pr ocess Control , T atransk ´ a Lomniva, Slova kia, 2011, pp. 106– 110. [161] A. A. Lem ai re, “M ´ etodos iterativ os baseados em desigualdades matriciais lineares para con- trole de sistemas l ineares incertos positiv os cont ´ ınuos no tempo, ” Master’ s t hesis, Universi- dade Est uadual de Campin as, Campinas, Brazil, 2019. [162] B. D. O. Anderson and S. V ongpani tlerd, Network Analysis and Synt h esis: A Mod ern Sys- tems Theory App roac h , ser . Networks Series, R. W . Newcomb, E d. Englew ood Cliffs, NJ: Prentice-Hall, 197 3. 164 [163] A. Rantzer , “On the Kalman-Y akubovich-Popov lemma, ” Systems & Contr ol Letters , vol. 28 , no. 1 , pp. 7–10, 1996 . [164] L. Xie, C. E. de Souza, and Y . W ang, “Robust ﬁltering for a class o f discrete-tim e uncer- tain nonl i near system s: An H ∞ approach, ” Internati onal Journal of Robust and Nonlinear Contr ol , vol. 6, n o. 4, pp. 297–31 2 , 1996. [165] M. C. De Oli veira, J. C. Geromel, and J. Bernussou, “Extended H 2 and H ∞ norm charac- terization and controller parameterization for discrete-time systems, ” International Journal of Contr ol , vol. 75, no. 9, pp. 66 6–679, 200 2 . [166] I. Masubuchi, A. Ohara, and N. Suda, “LMI-based output feedback controller desig n , ” i n Pr oc. American Contr ol Confere nce , Seattle, W A, 1995, pp. 34 73–3477. [167] ——, “LMI-based controller synthesi s: A uniﬁed formulation and solution, ” Int ernational J ourn al of Robust and Nonlinear Contr ol , v ol. 8, no. 8, p p . 669–686, 19 9 8. [168] C. E. de Souza, K. A. Barbosa, and A. T . Neto, “Robust H ∞ ﬁltering for discrete-tim e linear systems with uncertain time-varying parameters, ” IEEE T ransactions on Si g nal Pr ocessing , vol. 54 , no. 6 , pp. 2110–2118, 2006. [169] A. Spagolla, “ An ´ alise de estabili dade e s ´ ın t ese de controle para si stemas lin eares positivos discretos no tem po po r meio de desigualdades matriciais l ineares, ” Master’ s t hesis, Univ er- sidade Estuadu al de Campinas, Campinas, Brazil, 2019. [170] P . P . V aidyanathan, “The di s crete-time bou nded-real lemm a in digital ﬁltering, ” IEEE T rans- actions on Cir cuits and Systems , vol. 32, no. 9 , pp. 918–924, 1985. [171] E. Uezato and M. Ikeda, “Strict LMI conditions for stability , robust stabilization, and H ∞ control of descriptor systems, ” in Pr oc. IEEE Confer ence on Decisi on and Contr ol , Phoenix, AZ, 1999, pp. 40 92–4097. [172] A. Rehm and F . Al lg ¨ ower , “ An LM I approach to wards H ∞ control of discrete-time descrip- tor system s, ” in Proc . American Contr ol Confer ence , Anchorage, A K, 2002, pp. 614–619. [173] A.-G. W u and G.-R. D u an, “Enhanced LMI representations for H 2 performance of poly- topic uncertaint systsems: Continu o us-time case, ” Internation al Journal of A utomat ion and Computing , vol. 3, pp. 304–308, 2006. [174] T . R. V . Steentjes, M. Lazar , and P . M . J. V an den Hof, “Distrib uted H 2 control for interconnected di screte-time s ystems: A dissipativity-based approach, ” arXiv , 2020. [Online]. A vailable: https:// arxiv .org/abs/2001.04875v1 [175] J. De Caigny, J. F . Camino, R. C. L. F . Oliv eira, P . L. D. Peres, and J. Swe vers, “Gain- scheduled H 2 and H ∞ control of discrete-time polytopic time-varying systems, ” IET Con- tr ol Theory a n d Applicati ons , vol. 4 , no. 3, pp . 362–380, 2010. [176] L. A. F . Santos, “Projeto de controladores e ﬁltros robustos para s i stemas lin eares di scretos com enriquecimento de di n ˆ amica, ” Ph.D. dissertation , Univ ersidade Estuadual de Campinas, Campinas, Brazil, 2017. 165 [177] J. C. Geromel, P . L. D. Peres, and S. R. Souza, “ H 2 guaranteed cost control for uncertain discrete-time linear systems, ” Internat ional Journal of Contr ol , vol. 5 7, no. 4, pp. 853–864, 1993. [178] K. T akaba and T . Katayama, “Robust H 2 performance o f uncertain d escriptor sys tem, ” in Pr oc. Eur opean Contr ol Confere nce , Brussels, Belgium, 1997, pp. 950– 955. [179] K. T akaba, “Rob ust H 2 control of descrip t or system with tim e-v arying uncertainty , ” Inter - national Journal of Control , vol. 71, no. 4, pp. 55 9–579, 1998. [180] M. Ikeda, T .-W . Lee, and E. Uezato, “ A st rict LMI condition for H 2 control of descriptor systems, ” in Pr oc. IEEE Confere nce on Decision and Contr ol , Sydney , Australia, 2000, pp. 601–604. [181] M. Y agoubi, “On multi objectiv e synthesis for parameter-dependent descript or systems, ” IET Contr ol Theory a n d Applicat ions , vol. 4, no. 5, pp. 81 7 –826, 2010. [182] A. A. Belov , O. G. And ri anova, and A. P . Kurdyukov , Cont r ol of Discr ete-T ime Descripto r Systems: An Anisot ropy-Based Appr oach , ser . Studies in Systems, Decisi on and Control. Cham, Switzerland: Springer , 2018, v ol. 157. [183] D. M. Y ang, Q. L. Zhang, B. Y ao, and C. M . Sha, “ H 2 performance analy s is and control for discrete-time descript o r s y stems, ” in Pr oc. W orld Congr ess on Intelligent Cont r ol and Automation , Shanghai, China, 2 0 02, pp. 3039–3043. [184] D. Kang, S. Li, and H. -M . Lee, “Robust H 2 state est imation for discrete-time descriptor systems, ” in Pr oc. Internati o n al Confer ence on Information and Communi cat ion T ec hno l - ogy Con ver gence , Jeju, South Kore a, 2018, pp. 1488–1490. [185] C. Scherer , P . Gahinet, and M. Chilali, “Multi objectiv e output-feedback control via LM I optimizatio n , ” IEEE T ransactions on A utomat ic Control , vol. 42, no. 7, pp. 8 9 6–911, 1997. [186] M. A. Rotea, “The g eneralized H 2 control problem, ” Automatica , vol. 29, n o. 2, p p. 373– 385, 1993 . [187] N. K ottenstette, M. J. McCourt, M. Xia, V . Gup t a, and P . J. Antsakl is, “On relati onships among passivity , p o sitive realness, and dissipativity in linear systems, ” Automatica , vol. 50, no. 4, pp. 1003 –1016, 2014. [188] J. C. W illem s, “Dissipative dynamical systems - part I: General theory , ” Ar chive Ration al Mechanics and Anal ysis , vol. 4 5 , no. 5, pp . 321–351, 1 972. [189] D. J. Hill and P . J. Moylan, “The stabili ty of nonlin ear dissipative systems , ” IEEE T ransac- tions on Automatic Control , vol. 21, no. 5, pp. 708 – 711, 1976. [190] G. C. Goo d win and K. S. Sin, Adaptive F iltering Pr ediction and Contr ol . Englew ood Cliff s, NJ: Prentice-Hall, 1984. [191] H. M arquez, Nonlinear Contr ol Systems: Analysis and Design . Ho boken, NJ: W iley , 2 003. 166 [192] B. D. O. Anderson, “ A s ystem theory criterion for positive real matrices, ” SIAM Journal on Contr ol , vol. 5, n o. 2, pp. 171–18 2 , 1967. [193] J. Bao and P . L. Lee, Process Cont rol: The P ass ive Systems Appr oach . London, UK : Springer -V erlag, 2007. [194] B. Brogli at o , R. Lozano, B. Maschke, and O. Eg eland, Dissipati ve Syst ems Ana l ysis and Contr ol: Theory and Applicati ons , 2nd ed. London, UK: Springer , 2 007. [195] L. Hitz and B. D. O. Anderson, “Discrete pos i tiv e-real functions and their application to system stabil i ty , ” Pr oceedings of the IEEE , vol. 116, no. 1, pp. 153–155, 1969. [196] W . H. Haddad and D. S. Ber nst ein, “Explicit construction of quadratic L yapunov functions for th e small gain, positivity , circle, and Popov theorems and their appl ication to rob ust sta- bility . P art II: Discrete-time th eory , ” International Journal of Ro b ust and Nonlinear Control , vol. 4, no. 2 , pp. 2 49–265, 1994. [197] S.-P . W u, S. Boyd, and L. V andenber ghe, “FIR ﬁlter design via semideﬁnite programming and spectral factorization, ” in Pr oc. IEEE Confer ence on Decisi on and Contr ol , Kobe, Japan, 1996, pp. 271–276. [198] I. Masubuchi, “Dis sipativity inequalities for continuo u s-time descriptor system s wi th appli- cations to synthesis of control gain s, ” Systems & Control Letters , vol. 55, no. 2, pp. 158–164, 2006. [199] R. W . Freund and F . Jarre, “ An extension of the posit iv e real lemma to descriptor sy stems, ” Optimizati on Methods and Software , vol. 1 9 , no. 1, pp. 69–87, 20 0 4. [200] L. Zhang, J. Lam, and S. Xu, “On pos i tiv e realness of d escriptor sy stems, ” IEEE T ransac- tions on Cir cuits and Systems , v ol. 49, no. 3, pp. 401–407, 2002. [201] L. Lee and J . L. Chen, “Strictly po s itiv e real lemma and absolute s t ability for discrete-time descriptor syst ems, ” IEEE T ransaction s o n Contr ol of Network Syst ems , v ol. 50, no. 6, pp. 788–794, 2003. [202] W . T ang and P . Daoutidis, “Input -ou tput data-driven cont rol through di ssipativity learning, ” in Pr oc. American Contr ol Confere nce , Philadelphia, P A, 2019, pp . 4217–42 2 2. [203] S. Gu pta and S. M . Joshi , “Some properties and s t ability results for s ector-bounded L TI systems, ” in Pr oc. IEEE Confere nce on Decision and Cont rol , Lake Buena V ista, FL, 1994 , pp. 2973– 2 978. [204] J. R. F orbes, “Extensions of inpu t-output stabili ty theory and th e control of aerospace sys- tems, ” Ph.D. dissertation, University of T oronto, T oronto, Canada, 2011. [205] L. J. Bridgem an and J. R. Forbes, “Conic-sector -based control to circum vent passivity vio- lations, ” Internation al J ourn al of Contr ol , vol. 87, no. 8, pp. 1 4 67–1477, 2014. 167 [206] S. M . Joshi and A. G. Kelkar , “Design of norm-bound ed and sector-bounded LQG con- trollers for uncertain s ystems, ” Journal o f Optimizatio n Theory and App lications , vol. 113, no. 2, pp. 269– 282, 2002. [207] L. Bridgeman, “Methods exploiting and extending the conic sector theorem, ” Ph.D. dis s er - tation, McGill Univ ersity , Montreal, Canada, 2016. [208] L. J . Bridgeman and J. R. Forbes, “The exterior conic sector lemm a, ” International Journal of Contr ol , vol. 88, no. 11, pp. 2 250–2263, 2015. [209] T . Iwasaki, S. Hara, and H . Y amauchi, “Dynamical syst em s design from a control per - spectiv e: Fini t e frequency posi t iv e-realness approach, ” IEEE T ransacti ons on Automatic Contr ol , vol. 48, no. 8, pp. 1337–135 4 , 20 0 3. [210] T . Iwasaki, S. Hara, and A. L. Fradkov , “Time do m ain i nterpretations of frequency do main inequalities on (semi)ﬁnite ranges, ” Systems & Contr ol Letters , vol. 54, no. 7, pp. 681–6 9 1, 2005. [211] S. Hara and T . Iwasaki, “Finite frequency characterization of easily controllable plant to- ward structure/control design integration, ” in Cont r ol and Modeling of Complex Systems: Cybernetics in the 21st Century , K. Hashimoto , Y . Oishi, and Y . Y amamoto , Ed s . Bost on, MA: Birkhauser , 2003, pp. 183–196. [212] T . Iwasaki and S. Hara, “Generalized KYP lemma: Uniﬁed frequenc y domain inequalities with design applications, ” IEEE T ransactions on Automatic Contr ol , vol. 50, no. 1, pp. 41– 59, 2005. [213] L. J. Bridgeman and J. R. Forbes, “The m inimum gain lemma, ” Internat i onal Journal of Robust and Nonlinear Contr ol , vol. 25, no. 14, pp. 2 515–2531, 2015. [214] R. J. Ca verly and J. R. Forbes, “ H ∞ -optimal parallel feedforward control using mi nimum gain, ” IEEE Control Systems Letters , vol. 2, no. 4, pp. 677–682 , 2018. [215] ——, “Robust controller design using the large gain theorem: The full-state feedback case, ” in Pr oc. American Contr ol Confere nce , Boston, MA, July 2016, pp. 3832–3837. [216] R. J. Cave rly , “Optim al o utput m odiﬁcation and rob ust control usin g m inimum gain and t h e lar ge gain theorem, ” Ph.D. dissertati on, Univ ersity of Michigan, Ann Arbor , MI, 2018. [217] A. Lanzon and I. R. Petersen, “Stabili ty robustness of a feedback int erconnectio n of sys tems with negativ e i maginary frequency resp o nse, ” IEEE T ransactions on Automatic Contr ol , vol. 53 , no. 4 , pp. 1042–1046, 2008. [218] Z. Song, A. Lanzon, S. Pitra, and I. R. Petersen, “ A negative -imagi nary lemma without m in- imality assum p t ions and robust state-feedback synth esis for uncertain negativ e-imaginary systems, ” Systems & Contr ol Letters , v ol. 61, no. 12, pp. 1269–127 6 , 20 1 2. [219] A. Ferrante, A. Lanzon, and L. Ntogramatzid i s, “Discrete-time n egative im agi nary sy s - tems, ” Automatica , vol. 79, pp. 1–10, M ay 20 1 7 . 168 [220] M. L i u and J. Xiong, “Properties and stability analys i s of dis crete-time negati ve im aginary systems, ” Automatica , vol. 8 3 , pp. 58–64, September 2017. [221] J. Xio n g, I. R. Petersen, and A. Lanzon, “Finite frequency negati ve imaginary systems, ” IEEE T ransactions on Automatic Contr ol , vol. 57, no. 11 , pp. 2917–2922, 2012. [222] R. J. Ca verly and M. Chakraborty , “Con vex synthesis o f strict l y negati ve imagin ary feed- back controllers, ” in Pr oc. IEEE Confer ence on Decision and Contr ol , Nice, France, 2019 , pp. 7578– 7 583. [223] K. Lee and J. R. Forbes, “Synthesis of strictly negative imaginary controllers using a H ∞ performance index, ” in Pr oc. American Contr ol Confer ence , Phil adelphia, P A, 2 019, pp. 497–502. [224] K. Lee, “Synthesis and appl ication of optimal st rictly negative imaginary controllers, ” Mas- ter’ s thesis, McGil l Univ ersity , Montreal, Canada, 2019. [225] Y . S. Hung and D. L. Chu, “Relationships betw een discrete-time and continuous-tim e al- gebraic Riccati inequaliti es, ” Linear Algebra and i ts Ap p lications , vol. 270, no. 1– 3, p p. 287–313, 1998. [226] Y . Y . Cao, J . Lam, and Y . X . Sun, “Static out p ut feedback st abi lization: An ILMI app roach, ” Automatica , vol. 34, no. 12, pp. 1 641–1645, 1998. [227] V . Kucera and C. E. de Souza, “ A necessary and sufﬁcient con d ition for output feedback stabilization, ” Automatica , vol. 31, no. 9, pp. 1357–13 5 9, 1 9 95. [228] S. G ¨ um ¨ us ¸soy and H. ¨ Ozbay , “Remarks on st ron g stabil i zation and stabl e H ∞ controller design, ” IEEE T ransactions on A utomat ic Contr ol , vol. 50, no. 12, pp. 2083 – 2 087, 2005. [229] B. Kouv aritakis and A. G. J. MacFarlane, “Geometric approach to analy s is and s ynthesis of system zeros: Part 1. sq uare system s, ” International Journal of Contr ol , vol. 23, no. 2, pp. 149–160, 1976. [230] M. Chilali and P . Gahinet, “ H ∞ design with pole placement constraints: An LMI approach, ” IEEE T ransactions on Automatic Contr ol , vol. 41, no. 3, pp. 3 58–367, 199 6. [231] A. Ohara, S. Nakazumia, and N. Suda, “Relations b etween a paramterizati o n of stabilizing state feedback gains and eigen v alue locations, ” Systems & Cont rol Letters , vol. 16, no. 4, pp. 261–2 6 6, 1991. [232] R. K. Y edav alli, “Robust root clustering for linear uncertain systems u s ing generalized L ya- punov t heory , ” Automatica , vol. 29, n o . 1, pp. 237–240 , 1993. [233] M. Chadli and P . Borne, Multi ple Models Approac h in A utoma tion: T akagi-Sugeno Fuzzy Systems . London, UK: John W il ey & Sons, Inc., 2013. [234] X. Xue, “Nov el rob ust and adapti ve dist ributed protocol for consens us-based control of un- certain multi-agent systems, ” Ph.D. dissertati on, North Carolina State Univ ersity , Raleigh, NC, 2019 . 169 [235] T . Iwasaki, Lectur e Notes: Mul t ivariable Contr ol , December 2007. [Online]. A vailable: https:// s ites.google.com/ g .ucla.edu/cyclab [236] C.-H. K uo and L. Lee, “Robust D -admissibil ity in generalized LMI re gio n s for descriptor systems, ” in Pr oc. Asian Control Confer ence , Melbourne, Aust rali a, 2004, p p . 105 8 –1065. [237] J. F . Whidborne and J . Mckernan, “On th e minim ization of maximum transient energy growth, ” IEEE T ransactions on A utomat ic Cont rol , vol. 52, no. 9, pp. 1 7 62–1767, 2007. [238] B. T . Polyak, A. A. Tremba, M. V . Khlebnikov , P . S. Shcherbakov , and G. V . Smirnov , “Lar ge d eviations in linear control systems w i th nonzero init ial condit i ons, ” Automation and Remote Contr ol , vol. 76, no. 6, pp. 957–976 , 2015. [239] A. Hayes, I. Nom pelis, R. J. Cav erly , J. M u eller , and D. Gebre-Egziabher , “Dynamic stabi l - ity analysis of a hypersonic entry vehicle with a non-linear aerodynamic model, ” i n Pr oc. Modeling and Si m u l ation T echnologies Confer ence, AIAA A viation , V irtual E vent, 2020, AIAA 2020-3201 . [240] D. S. Bernstein and W . M. Haddad, “Robust cont roller synthesis using Kharitonov’ s theo- rem, ” IEEE T ransactions on A utoma t ic Contr ol , vol. 37, no . 1, pp. 12 9–132, Jan. 1992. [241] R. Dey , G. Roy , and V . E. Balas, Sta bility and Stabiliz a tion of Linear and Fuzz y T im e-Delay Systems , ser . Intell igent Systems Reference Library . Cham, Switzerland: Springer , 2018, vol. 14 1 . [242] J. Doyle, A. Packard, and K. Zho u, “Revie w of LFTs, LMIs, and µ , ” i n Confer ence on Decision and Contr ol , Brighton, En gland, 1991, pp. 1 2 27–1232. [243] M. Green and D. J. N. Limebeer , Linear Ro bust Control . Mineaola, NY : Dove r , 20 1 2. [244] B. A. Francis, A Course in H ∞ Contr ol Theory , ser . Lecture Notes i n Control and Informa- tion Sciences, M. Thomas and A. W yner , Eds. Berlin, Germany: Springer-V erlag, 1987, vol. 88 . [245] K. Ogata, Modern Cont rol Engineering , 5th ed. Upp er Saddle River , NJ: Prentice Hall, 2010. [246] D. S. Bernstein, “Lecture not es for AER OSP 580 - linear feedback control system, ” 20 14. [247] K. Zh ou and J. C. Doyle, Ess entials of Rob ust Control . Upper Saddle Riv er , NJ: Prentice- Hall, 1998. [248] M. M. Peet, “Mo d ern O p timal Control l ecture 22 : H 2 , LQR and LQG, ” 2011. [Onl i ne]. A vailable: http://cont rol.asu.edu/Classes/MAE5 07/507Lecture22.pdf [249] ——, “Modern Opti mal Control lecture 2 1 : Opt imal o u tput feedback control, ” 2011. [Online]. A vailable: http://cont rol.asu.edu/Classes/MAE5 07/507Lecture21.pdf [250] S. Lall, “Engr210a lecture 16: H ∞ synthesis, ” November 2001. [Online]. A vailable: https:// l all.stanford.edu/engr210a/l ectures/lecture16 2001 11 25 04.pdf 170 [251] M. M. Peet, “LMI M ethods i n Opti m al and Robust Control lecture 11: Relationship between H 2 , LQG and LGR and LMIs for state and output feedback H 2 synthesis, ” 2016. [Online]. A vailable: http://control.asu .edu/Classes/MAE598/ 5 98Lecture11.pdf [252] J. C. Geromel, J. Bernussou , G. Garcia, and M . C. de Oliv eira, “ H 2 and H ∞ robust ﬁltering for discrete-time li near s ystems, ” SIAM Journal on Contr ol and Op t imization , vol. 38, no. 5, pp. 1353– 1 368, 2000. 171 Index algebraic lo o p, 117 basic s erv o loop, 120 bilinear matri x inequality (BMI) deﬁnition, 10 discussion, 37 block coordinat e descent, 36 bounded real lemma continuous-ti m e, 55 discrete-time, 58 change of variables, 16 completion of the squares, see Y oun g’ s relation 29 complex conjugate, 100 , 103 condition num ber , 43 , 101 congruence transform ation, 16 conic s ectors conic s ector lemm a, 82 exterior conic s ector lemma, 83 modiﬁed exterior conic sector lemma, 84 conjugate transpo s e, 7 con ve x objectiv e functions, 13 con ve x ov erboundi n g discussion, 39 iterativ e con vex overbounding, 34 con ve x-concave decomposi tion, 33 discussion, 39 coordinate descent, 36 DC gain, 104 deﬁniteness deﬁnition, 8 relativ e deﬁniteness , 11 descriptor syst ems, 82 D -admissib ility , 10 3 H 2 norm, 74 admissibil ity , 53 , 54 bounded real lemma, 60 discrete-time H 2 norm, 75 discrete-time bou n ded real lemma, 62 KYP lemm a, 82 detectability , 9 4 determinant, 4 6 dilation, 26 Douglas-Fillmore-W illi ams Lemma, 45 dualization l emma, 40 dynamic out put feedback H 2 -optimal, 123 H ∞ -optimal, 128 discrete-time H 2 -optimal, 124 discrete-time H ∞ -optimal, 131 discrete-time m i xed H 2 - H ∞ -optimal, 137 mixed H 2 - H ∞ -optimal, 135 eigen va lues, 8 , 98 – 100 , 103 maximum eigen value, 8 , 42 minimum eigen value, 8 , 42 sum of absolute v alue of largest eigen va lues, 43 sum of largest eigen values, 4 2 weighted sum of abs o lute value of lar gest eigen values, 43 weighted sum of l ar gest eigen values, 43 ener gy bound discrete-time out put energy bound, 111 – 113 output ener gy bound, 112 estimation H 2 -optimal, 140 H ∞ -optimal, 142 mixed H 2 - H ∞ -optimal, 143 extended strictly positiv e real (ESPR) , 82 ﬁltering H 2 -optimal, 147 H ∞ -optimal, 148 discrete-time H 2 -optimal, 147 discrete-time H ∞ -optimal, 149 Finsler’ s lemma lemma, 24 modiﬁed lemma, 25 full-state feedback H 2 -optimal, 122 H ∞ -optimal, 127 discrete-time H 2 -optimal, 122 discrete-time H ∞ -optimal, 128 172 discrete-time m i xed H 2 - H ∞ -optimal, 134 mixed H 2 - H ∞ -optimal, 133 generalized KYP Lemma (GKYP), 85 generalized plant , 119 Hermitian m atrix, 46 Hermitian trans p ose, 7 Hurwitz matrix , 8 , 5 0 , 94 , 95 , 114 identity matrix , 7 impulse response, 110 Kalman-Y akubovich-Popov (KYP) lem ma, 78 Kharitonov-Bernstein-Haddad (KBH) theorem, 114 Kroenecker product, 8 , 100 , 104 LMI concatenation, 12 con ve xit y , 12 deﬁnition, 10 , 11 nonstrict L MIs, 12 parsers, 15 region, 100 , 10 3 solvers, 15 strict LMIs , 12 logarithm, 45 L yapu n ov equation, 49 – 52 inequality , 49 stability , 49 , 51 matrix in equ al i ty deﬁnition, 10 minimum gain discrete-time m i nimum gain lemma, 89 discrete-time m o diﬁed minim um gain lemma, 90 minimum gain lemma, 85 modiﬁed mi nimum gain lemma, 88 minimum phase, 98 , 99 Moore-Penrose inv erse, 17 negati ve imagi n ary systems, 91 discrete-time negative imaginary l em ma, 92 generalized negative imaginary lem ma, 92 negati ve i maginary lemm a, 91 norm H 2 norm, 63 H ∞ norm, 55 discrete-time H 2 norm, 66 , 71 discrete-time H ∞ norm, 58 Euclidean norm, 105 – 110 Frobenius norm, 14 , 4 2 generalized H 2 norm, 77 induced L 2 - L 2 norm, 55 induced L 2 - L ∞ norm, 77 induced L ∞ - L ∞ norm, 77 nuclear norm , 42 peak-to-peak no rm , 77 weighted norm , 46 nullspace, 8 observer H 2 -optimal, 140 H ∞ -optimal, 142 discrete-time H 2 -optimal, 141 discrete-time m i xed H 2 - H ∞ -optimal, 144 discrete-time H ∞ -optimal, 142 mixed H 2 - H ∞ -optimal, 143 penalized conv ex relaxation, 35 sequential, 36 Petersen’ s lemma nonstrict Petersen’ s l emma, 26 strict Petersen’ s lemm a, 25 polytopic uncertainty , 115 positive real (PR), 79 , 80 projection lem ma nonstrict p rojection lemm a, 23 reciprocal projection lemma, 23 strict projecti o n lemma, 22 pseudoin verse, 55 , 10 4 QSR d i ssipative, 78 quadratic i nequality , 46 173 range, 8 , 45 rank, 16 , 21 , 33 , 34 , 48 , 54 , 5 5 , 59 , 61 – 63 , 68 – 70 , 75 , 98 , 99 , 103 , 104 S-procedure, 4 0 Schur compl ement nonstrict Schur complement lem m a, 17 Schur compl ement-based properties, 18 strict Schur complement l emma, 17 Schur matrix , 8 , 52 , 94 , 95 semideﬁnite program (SDP), 12 – 14 solvers, 15 singular value Frobenius norm, 42 maximum si ngular value, 8 , 4 1 , 117 minimum sing u lar value, 8 , 41 nuclear norm , 42 spectral radiu s, 44 stability α -stabili ty , 100 D -stabilit y , 100 asymptotic stabi lity , 50 , 52 exponential stability , 100 , 101 L yapu n ov stabil ity , 49 , 51 quadratic s tability , 115 stabilizabilit y , 94 static out p ut feedback stabilizability , 95 strong st abilizability , 97 state-space realization continuous ti m e, 8 discrete t ime, 8 static output feedback, 11 7 stabilizabilit y , 95 strictly p ositive real (SPR), 79 – 81 structured singular value, 117 submatrix, 46 time delay , 117 trace, 44 , 45 transient discrete-time im pulse response bound, 110 discrete-time out put bound , 108 , 109 discrete-time st at e bound, 105 , 107 impulse response bound, 110 output bound, 108 , 109 state bound, 105 , 106 transmissio n zeros, 98 , 99 triangle inequality , 47 unitary matri x, 47 Y ou n g’ s relation, 20 lemma, 29 reformulation, 2 9 special cases, 29 Y ou n g’ s relation-based properties, 32 174

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