Feedback Passivation of Linear Systems with Fixed-Structured Controllers

F eedbac k P assiv ation of Linear Systems with Fixed-Structured Con trollers ? Lanlan Su a , Vija y Gupta a , and P anos An tsaklis a a Dep artment of Ele ctric al Engine ering, University of Notr e Dame, Notr e Dame, IN 46556 USA Abstract This pap er addresses the problem of designing an optimal output feedback con troller with a sp ecified controller structure for linear time-in v ariant (L TI) systems to maximize the passivity lev el for the closed-loop system, in both con tinuous-time (CT) and discrete-time (DT). Specifically , the set of con trollers under consideration is linearly parameterized with constrained parameters. Both input feedforward passivit y (IFP) and output feedback passivit y (OFP) indices are used to capture the lev el of passivity . Given a set of stabilizing con trollers, a necessary and sufficien t condition is prop osed for the existence of such fixed-structured output feedbac k con trollers that can passiv ate the closed-loop system. Moreo ver, it is sho wn that the condition can b e used to obtain the con troller that maximizes the IFP or the OFP index by solving a con vex optimization problem. Key wor ds: Output feedbac k passiv ation, Fixed-structued controller, Passivit y indices, SDP. 1 In tro duction P assivit y pro vides a ph ysically meaningful in terpre- tation of the energy dissipation of a system from the input-output p ersp ective (Willems (1972)). Notions of input and output passivit y indices giv e a widely used measure of the level of passivity for a system (Kotten- stette et al. (2014)). When exploited prop erly , passivity indices pro vide a means to design feedback con trollers via the process of comp ensating for the lac k of passivit y in one subsystem of a feedback configuration with pas- sivit y surplus in the other (V an Der Schaft 2000, Bao & Lee 2007, Antsaklis et al. 2013). F eedbac k passiv ation of plants that ma y not be pas- siv e is a widely studied problem (Larsen & Kokoto vic (2001), Zhu et al. (2014), Zhao & Gupta (2016)). In the existing works, the con troller can be c hosen without an y constrain t on its structure. In this pap er, w e study the problem of designing a con troller to maximize the closed- lo op passivity level (as measured b y a passivit y index) when the controller has to satisfy a fixed structure. Re- construction of such a controller can b e performed by solving a semidefinite programming (SDP). The results in this pap er can b e utilized to improv e the robust stabil- it y margins of in terconnected systems as measured from the p ersp e ctiv e of passivity . ? This pap er w as not presen ted at an y IF AC meeting. Email addr esses: lanlansu.work@gmail.com (Lanlan Su), vgupta2@nd.edu (Vija y Gupta), antsaklis.1@nd.edu (P anos Antsaklis). Our problem setup inv olves linearly parameterized sets of con trollers with constrained parameters. It is generally kno wn that synthesis of suc h controllers is a notoriously difficult problem. A naiv e application of the Kalman-Y akub o vic h-P op ov (KYP) lemma to tac kling the problem w ould result in bilinear matrix inequal- ities, whic h are intractable in general. Our prop osed approac h establishes and exploits relationships betw een the passivity of SISO systems and sum-of-square (SOS) p olynomials, whic h are amenable to con vex optimiza- tion form ulations. The introduction of linearly parame- terized sets of con trollers is motiv ated b y its ubiquit y in practical engineering applications, the most common of whic h consists of proportional-integral-deriv ative (PID) con trollers, where the parameters app ear linearly . By tuning the linear parameters of the con troller in order to optimize the lev el of passivity in a feedback is of in- terest in view of the popularity of such controllers. Suc h parameterized controllers b elong to a broader class of the so-called fixed-structured controllers. See, for in- stance, Saeki (2006) which considers fixed-structured PID con troller design for H ∞ con trol problems with linear constrain ts on the control structure; Malik et al. (2008) wherein a set of stabilizing fixed-structure and fixed-order con trollers is constructed; and Bazanella et al. (2011) whic h studies mo del-free fixed-structure con troller syn thesis. T o the b est of the authors’ knowl- edge, the problem of optimizing the passivity level of a system with a fixed-structure controller considered in this pap er has not b een considered elsewhere. Preprin t submitted to Automatica 31 July 2019 The rest of this pap er is organized as follo ws. Section 2 introduces some preliminaries and states the problem form ulation. Section 3 presen ts the main results. The main results are illustrated b y t w o examples in Section IV. Some final remarks and future w ork are describ ed in Section V. 2 Preliminaries and Problem F orm ulation 2.1 Notation The notation used in the paper is as follo ws. The sets of real and complex num b ers are denoted by R and C , and the imaginary unit is denoted as j . The notation Re ( λ ) and | λ | denote the real part and the magnitude of a complex num b er λ . A − 1 , A 0 and A ∗ denote the inv erse, the transp ose and the conjugate transp ose of matrix A , resp ectiv ely . Giv en a Hermitian matrix A = A ∗ , the notation λ ( A ) denotes the minimum eigen v alue of A . F or Hermitian matrices A, B , the notation A − B ≥ 0 denotes A − B is positive semidefinite. The degree of a polynomial p ( · ) is denoted by deg( p ( · )). The symbol ⊗ denotes the Kronec ker pro duct. The function f T is the truncation of f to the in terv al [0 , T ]. The operator < f , g > T is defined as the inner pro ducts of signal f and g o ver [0 , T ]. L 2 e denotes the extended L 2 signal space and || · || denotes the L 2 norm. F or briefness, the notation ? denotes the symmetric entries in a symmetric matrix. 2.2 Sum of squar e (SOS) matrix p olynomial Let us briefly introduce the class of SOS matrix p olynomials, see e.g., Chesi (2010) for details. A symmetric matrix polynomial F : R r → R n × n is said to b e SOS if and only if there exist matrix p oly- nomials F 1 , . . . , F k : R r → R n × n suc h that F ( s ) = P k i =1 F i ( s ) T F i ( s ) . SOS matrix p olynomials are p ostive- semidefinite, and it turns out that one can establish whether a symmetric matrix polynomial is SOS via an LMI feasibility test. Indeed, let d b e a nonnegativ e in teger such that 2 d ≥ deg( F ). By extending the Gram matrix metho d or SMR for scalar p olynomials to the representation of matrix p olynomials, F ( s ) can b e written as F ( s ) = ( b ( s ) ⊗ I ) T ( M + L ( α )) ( b ( s ) ⊗ I ) (1) where b ( s ) : R r → R σ ( r,d ) is a v ector con taining all the monomials of degree less than or equal to d in s with σ ( r, d ) = ( r + d )! r ! d ! , and M ∈ R nσ ( r,d ) × nσ ( r,d ) is a symmetric matrix satisfying F ( s ) = ( b ( s ) ⊗ I ) T M ( b ( s ) ⊗ I ) , L ( α ) : R ω ( r, 2 d,n ) → R nσ ( r,d ) × nσ ( r,d ) is a linear parametrization of the linear set L = { ˜ L = ˜ L T : ( b ( s ) ⊗ I ) T ˜ L ( b ( s ) ⊗ I ) = 0 } , (2) and α ∈ R ω ( r, 2 d,n ) is a free v ector with ω ( r, 2 d, n ) = 1 2 n ( σ ( r, d )( nσ ( r, d ) + 1) − ( n + 1) σ ( r, 2 d )) . It follows that F ( s ) is a SOS matrix p olynomial if and only if there exists α satisfying the LMI M + L ( α ) ≥ 0 . When n = 1 is considered, the abov e results are reduced to SOS p olynomials. 2.3 Strictly Input and Output Passive Systems W e start b y introducing the definitions of passiv- it y and p ositive realness for L TI systems, follo wed by a lemma revealing their relation. Definition 1 (Passivity (V an Der Schaft 2000, Kotten- stette et al. 2014)) Consider a CT or DT L TI system H : u ∈ L 2 e → y ∈ L 2 e . Then the system H is • p assive if ther e exists a c onstant β such that h H u, u i T ≥ β , ∀ u ∈ L 2 e , ∀ T ≥ 0 . (3) • strictly input p assive (SIP) if ther e exist ν > 0 and β such that h H u, u i T ≥ ν || u T || 2 2 + β , ∀ u ∈ L 2 e , ∀ T ≥ 0 , (4) and the lar gest ν > 0 satisfying (4) is c al le d the Input F e e dforwar d Passivity (IFP) index, denote d as IFP( ν ). • strictly output p assive (SOP) if ther e exist ξ > 0 and β such that h H u, u i T ≥ ξ || H u T || 2 2 + β , ∀ u ∈ L 2 e , ∀ T ≥ 0 , (5) and the lar gest ξ > 0 satisfying (5) is c al le d the Output F e e db ack Passivity (OFP) index, denote d as OFP( ξ ). The IFP and OFP indices, defined in terms of an excess of passivt y , are in tro duced to quan tify the degree of passivity . Definition 2 (Positive r e alness (Khalil & Grizzle 2002)) A squar e, pr op er and r ational tr ansfer function G ( s ) (or G ( z ) for DT c ase) is said to b e p ositive r e al if • G ( s ) is analytic in R e ( s ) > 0 in CT c ase; G ( z ) is analytic in | z | > 1 in DT c ase; • G ( j w ) + G ∗ ( j w ) ≥ 0 , ∀ ω ∈ R for which j ω is not a p ole of G ( s ) in CT c ase; G ( e j ω ) + G ∗ ( e j ω ) > 0 , ∀ ω ∈ [0 , 2 π ] for which e j ω is not a p ole of G ( z ) in DT c ase; • Any pur e imaginary p ole j ω o of G ( s ) is a simple p ole, and the asso ciate d r esidue G o , lim s → j ω o ( s − j ω o ) G ( s ) satisfies G o = G ∗ o ≥ 0 in CT c ase; If e j ω o is a p ole of G ( z ) it is at most a simple p ole, and the asso- ciate d r esidue G o , lim z → j ω o ( z − e j ω o ) G ( s ) satisfies G o = G ∗ o ≥ 0 in DT c ase. F or a stable 1 L TI system with transfer function G , the following lemma states the relation b etw een the passivit y and p ositive realness. Lemma 3 (Bao & L e e (2007))A stable L TI system H : u ∈ L 2 e → y ∈ L 2 e is p assive if and only if its tr ansfer function G is p ositive r e al. F or a stable L TI system with the tr ansfer function G ( s ) (or G ( z ) for DT c ase) that is strictly input p assive, its IFP index, ν , is given as ν =      1 2 min ω ∈ R λ ( G ( j ω ) + G ∗ ( j ω )) C T case 1 2 min ω ∈ [0 , 2 π ] λ  G ( e j ω ) + G ∗ ( e j ω )  D T case F or a minimum phase L TI system G ( s ) (or G ( z ) for DT c ase) that is strictly output p assive, its OFP index, 1 In this w ork, a L TI system is said to be stable if the system is asymptotically stable. 2 Fig. 1. F eedback control system ξ , is given as ξ =      1 2 min ω ∈ R λ  G − 1 ( j ω ) + [ G − 1 ( j ω )] ∗  C T case 1 2 min ω ∈ [0 , 2 π ] λ  G − 1 ( e j ω ) + [ G − 1 ( e j ω )] ∗  D T case 2.4 Pr oblem F ormulation In this work, we consider the problem of output feedbac k passiv ation of a single-input single-output (SISO) linear system through a fixed-structured con- troller (depicted in Figure 1). Particularly , the ob jective is to design an output feedback fixed-structure con- troller with parameter ρ ∗ , whic h maximizes the IFP or the OFP index for the closed-lo op system. The SISO plant with transfer function G 0 can b e either CT or DT systems. The set of the controllers whic h can b e implemented has a sp ecified con troller structure represen ted b y C = { C ( s, ρ ) : ρ ∈ P } for the CT case and C = { C ( z , ρ ) : ρ ∈ P } for the DT case, where P ⊆ R p is a set of admissible v alues of the con troller parameter vector ρ . W e assume that the controllers are linearly parameterized, i.e., C ( s, ρ ) = ρ T ¯ C ( s ) , C ( z , ρ ) = ρ T ¯ C ( z ) (6) where ρ is the parameter vector and ¯ C is the predefined parameter indep endent v ector of transfer functions. It is also assumed that all entries in ¯ C are selected to ha ve stable p oles. A t ypical class of controllers with linear pa- rameterization is PID controllers. The linearity mak es the resulting design problem more amendable to analy- sis. Moreo ver, it is sho wn in Bazanella et al. (2011) that an y parameteter-dep endent transfer function can be ap- pro ximated to an y degree of accuracy desired b y a trans- fer function of the form (6) with sufficiently large p . As it is often required to restrict the admissible con troller pa- rameters to some desired b ounded sets, we assume that the admissible set of ρ is describ ed by 2 P = { ρ ∈ R p : ρ i ≤ ρ i ≤ ρ i , i = 1 , . . . , p } . (7) The problems addressed in this work are as follows. Problem 4 F or a given set of c ontr ol lers, C = { C ( ρ ) : ρ ∈ P } , establish whether the close d-lo op system is stable for al l ρ ∈ P . With the set of stabilizing controllers in hand, we further in v estigate the following problem. Problem 5 Establish whether ther e exists a c ontr ol ler C 2 As it will be explained in Remark 19, the prop osed metho dology can b e used also to design feedbac k controllers with an y con vex set P . in the set C that c an p assivate the system G 0 . If the answer is p ositive, determine the c ontr ol ler C ∗ that maximizes the IFP index and the OFP index r esp e ctively for the close d-lo op system. It is well-kno wn that a necessary condition for a lin- ear system to b e feedback passiv ated is that the system should hav e a relative degree less than 2 and is weakly minim um phase (i.e., it should not hav e zeros on righ t side in s-plane or outside the unit circle in z-plane). Thus, w e assume throughout this work the follo wing assump- tion. Assumption 6 The plant G 0 has a r elative de gr e e less than 2 , and has al l its zer os in the close d left half of the s-plane in CT c ase (in DT c ase, r esp e ctively, inside or on the unit cir cle of the z-plane). A slightly more restrictive assumption is made when the optimal OFP controller design is considered. Assumption 7 The plant G 0 has a r elative de gr e e less than 2 , and has al l its zer os in the op en left half of the s- plane in CT c ase (in DT c ase, r esp e ctively, strictly inside the unit cir cle of the z-plane). 3 Main Results 3.1 Stability A nalysis Let us start by addressing Problem 4, which is to establish the robust stabilit y of the closed-lo op system for all parameter ρ ∈ P . First, let us observe that the controller set (7) can b e equiv alently describ ed as P = { ρ ∈ R p : c i ≥ 0 , i = 1 , . . . , p } (8) with c i = ( ρ i − ρ )( ρ i − ρ i ). F or CT case, let us denote the transfer function of the plant as G 0 ( s ) = N 0 ( s ) D 0 ( s ) , and denote the i -th com- p onen t in the v ector ¯ C ( s ) as ¯ C i ( s ) = N i ( s ) D i ( s ) . It follo ws that the closed-lo op system as shown in Figure 1 is rep- resen ted as G ( s, ρ ) = G 0 ( s ) 1 + G 0 ( s ) C ( s, ρ ) = N 0 ( s ) Q p i =1 D i ( s ) D 0 ( s ) Q p i =1 D i ( s )+ N 0 ( s ) P p i =1 ρ i N i ( s )  Q j 6 = i D j ( s )  , p N ( s ) p D ( s, ρ ) (9) where the p olynomials p N ( s ) and p D ( s, ρ ) denote the n umerator and denominator of the closed-loop transfer function resp ectively . Under Assumption 6 or 7 , it can be observ ed that there is no unstable zero-pole cancellation in the ab ov e closed-lo op transfer function. Rewrite the denominator p olynomial p D ( s, ρ ) as p D ( s, ρ ) = a n ( ρ ) s n + a n − 1 ( ρ ) s n − 1 + . . . + a 1 ( ρ ) s + a 0 ( ρ ) wherein the co efficients a 0 , . . . , a n are linear functions of the v ector v ariable ρ . In order to analyze the stability of the closed-lo op system, it is necessary and sufficien t to c hec k whether all the ro ots of the polynomial p D ( s, ρ ) ha v e negativ e real parts for all ρ ∈ P . T o this end, let us exploit the mo dified Routh-Hurwitz table for the p oly- 3 nomial p D ( s, ρ ). By multiplying eac h comp onent by their denominator in the classical Routh-Hurwitz table, we can obtain the modified Routh-Hurwitz table defined as a n ( ρ ) a n − 2 ( ρ ) a n − 4 ( ρ ) · · · a n − 1 ( ρ ) a n − 3 ( ρ ) a n − 5 ( ρ ) · · · a 31 ( ρ ) a 32 ( ρ ) a 33 ( ρ ) · · · . . . . . . . . . . . . (10) where the n umber of rows is n + 1 and the ij -th comp o- nen t is a ij ( ρ ) = a i − 1 , 1 ( ρ ) a i − 2 ,j +1 ( ρ ) − a i − 1 ,j +1 ( ρ ) a i − 2 , 1 ( ρ ) i = 3 , . . . , n + 1 , j = 1 , 2 , . . . (11) It can b e v erified that all the ro ots of p D ( s, ρ ) ha v e nega- tiv e real parts for all ρ ∈ P if and only if the p olynomials in the first column of the mo dified Routh-Hurwitz table (10) are p ositive for all ρ ∈ P . No w let us further consider a discrete-time transfer function of the plant as G 0 ( z ) = N 0 ( z ) D 0 ( z ) , which is in closed- lo op with the linearly parameterized controller C ( z , ρ ) = P p i =1 ρ i N i ( z ) D i ( z ) . With similar argument of the CT case, the closed-lo op system is represented as G ( z , ρ ) = p N ( z ) p D ( z , ρ ) (12) with p D ( z , ρ ) = a n ( ρ ) z n + a n − 1 ( ρ ) z n − 1 + . . . + a 1 ( ρ ) z + a 0 ( ρ ) wherein the co efficients a 0 , . . . , a n dep end linearly on the v ector v ariable ρ . Similarly , in order to establish the stabilit y of the closed-lo op system for all ρ ∈ P , it is necessary and sufficien t to c heck whether all the ro ots of the p olynomial p D ( z , ρ ) hav e magnitude less than 1. By multiplying the o dd ro ws b y their denominator and remo ving the ev en ro ws in the traditional Jury table, we define the mo dified Jury table where a ij ( ρ ) = a i − 1 ,j ( ρ ) a i − 1 , 1 ( ρ ) − a i − 1 ,n +4 − i − j ( ρ ) a i − 1 ,n +3 − i ( ρ ) i = 3 , . . . , n + 1 , j = 1 , 2 . . . It can b e v erified that all the ro ots of p D ( z , ρ ) ha ve mag- nitude less than 1 for all ρ ∈ P if and only if all the p oly- nomials in the first column of the mo dified Jury table are p ositive for all ρ ∈ P . Let us denote the entries of the first column in the mo dified Routh-Hurwitz table for CT case or in the mo d- ified Jury table for DT case as f i ( ρ ) , i = 1 , . . . , n + 1 where f i ( ρ ) denotes the i -th en try in the column. Based on the previous analysis, we hav e the following lemma. Lemma 8 The close d-lo op system is stable for al l ρ ∈ P define d in (8) if and only if f i ( ρ ) > 0 , i = 1 , . . . , n + 1 ∀ ρ ∈ P . (13) No w let us define the p olynomials g i ( ρ ) = f i ( ρ ) − p X j =1 s ij ( ρ ) c j ( ρ ) i = 1 , . . . , n + 1 (14) where s ij ( ρ ) are auxiliary p olynomials. Theorem 9 The close d-lo op system is stable for al l ρ ∈ P if and only if θ ∗ > 0 (15) wher e θ ∗ = max θ,s ij θ s.t.              g i ( ρ ) − θ is SOS s ij ( ρ ) is SOS ∀ i = 1 , . . . , n + 1 ∀ j = 1 , . . . , p (16) Pro of. ” ⇒ ” It can b e observed that the set P defined in (8) is a compact set, and c 1 , . . . c p are p olynomials of ev en degree and their highest degree forms do not ha ve common zeros except zero. Given an arbitrarily small scalar θ > 0, it follows from Theorem 7 in Chesi (2010) that f i ( ρ ) > θ , ∀ ρ ∈ P holds if and only if there exist SOS polynomials s ij ( ρ ) suc h that g i ( ρ ) − θ is SOS poly- nomial. Therefore, the condition (16) is satisfied with θ > 0, and hence the condition (15) holds. ” ⇐ ” Let us supp ose that (15)-(16) hold. Then, one has that g i ( ρ ) − θ and s ij ( ρ ) are nonnegative. Since c j ( ρ ) ≥ 0 whenever ρ ∈ P , it follows from (14) that f i ( ρ ) > 0 , i = 1 , . . . n + 1 for all ρ ∈ P .  Remark 10 The or em 9 shows that one c an establish the p ositivity of the p olynomials in the first c olumn in the mo difie d tables for al l ρ ∈ P by solving the optimization pr oblem (16) . It is worth mentioning that the c ondition for p olynomials which dep end on some de cision variables line arly to b e SOS p olynomials c an b e solve d e quivalently via LMIs b ase d on the Gr am matrix metho d as describ e d in Se ction 2.2. Ther efor e, for any chosen de gr e es of p oly- nomials s ij ( ρ ) , this the or em pr ovides a sufficient c ondi- tion solvable thr ough LMIs, which is also ne c essary when the de gr e es ar e lar ge enough. Since fixed-structured controllers, including PID con trol as a t ypical example, are so widely used in indus- trial applications, it is important to develop a metho d- ology to characterize the set of stabilizing controllers b efore carrying out the optimal control design. Theorem 9 provides a metho d to establish whether a giv en set of con trollers is stabilizing. In the next subsection, we will design the con troller b y choosing its parameter ρ from the set P to reach the maximized passivit y level for the closed-lo op system. Indeed, most existing mo dern optimal control techniques are incapable of accommo- dating constraints on the con troller order or structure in to their design metho ds, and consequen tly cannot b e used for designing optimal or robust controllers. 3.2 F e e db ack Passivation Giv en a set of stabilizing controllers C , w e proceed to address Problem 5 in this subsection. W e first consider the CT case, which is then extended to the DT case. 4 3.2.1 CT c ase Recall that the n umerator and denominator of the closed-lo op transfer function (9) are denoted as p olyno- mials p N ( s ) and p D ( s ), resp ectively , wherein the co effi- cien ts of p D ( s ) depends linearly on the vector v ariable ρ . By substituting s = j ω , p N and p D can b e rewritten via even-odd decomp osition as p N ( j ω ) = p e N ( w ) + j p o N ( w ) p D ( j ω , ρ ) = p e D ( ω , ρ ) + j p o D ( ω , ρ ) (17) where p e N , p o N , p e D , p o D are all real p olynomials in ω , and p e D ( ω , ρ ) and p o D ( ω , ρ ) dep end linearly on ρ . The fre- quency resp onse of the closed-lo op system (9) can b e expressed as G ( j ω , ρ ) = p e N ( w ) + j p o N ( w ) p e D ( ω , ρ ) + j p o D ( ω , ρ ) (18) whic h yields that G ( j ω , ρ ) + G ∗ ( j ω , ρ ) = 2 p e N ( w ) p e D ( ω , ρ ) + 2 p o N ( w ) p o D ( ω , ρ ) p e D ( ω , ρ ) 2 + p o D ( ω , ρ ) 2 . (19) Lemma 11 Ther e exists a c ontr ol ler C ( s, ρ ) in the c on- tr ol ler set C that c an fe e db ack p assivate the plant G o ( s ) if and only if ther e exists a ve ctor ρ ∈ P and a sc alar  ≥ 0 such that p e N ( w ) p e D ( ω , ρ ) + p o N ( w ) p o D ( ω , ρ ) −  is SOS . (20) Pro of. Since the con troller set C is stabilizing, it follo ws from Lemma 3 that a controller C ( s, ρ ) can feedback passiv ate the plant if and only if the closed-lo op system is positive real. F or a stable closed-lo op system, the first and the third condition in Definition 2 are trivially sat- isfied. Therefore, the closed-loop system is p ositive real if and only if G ( j ω , ρ ) + G ∗ ( j w , ρ ) ≥ 0 , ∀ ω ∈ R , whic h, according to (19), is equiv alent to p e N ( w ) p e D ( ω , ρ ) + p o N ( w ) p o D ( ω , ρ ) ≥ 0 , ∀ ω ∈ R . Therefore, there exists a controller in the set C that can feedbac k passiv ate the plant if and only if there exists a v ector ρ ∈ P and a scalar  ≥ 0 such that p e N ( w ) p e D ( ω , ρ ) + p o N ( w ) p o D ( ω , ρ ) −  ≥ 0 , ∀ ω ∈ R , Since there is no gap b etw een nonnegativ e p olynomials and SOS polynomials when the p olynomial is univ ariate, the ab ov e condition is is equiv alent to p e N ( w ) p e D ( ω , ρ ) + p o N ( w ) p o D ( ω , ρ ) −  is SOS , whic h completes the pro of.  Note that c hecking the feasibility of (20) can b e solv ed b y a SDP . Sp ecifically , the condition in (20) can b e rewritten based on Section 2.2 as M ( ρ,  ) + L ( α ) ≥ 0 where M ( ρ,  ) is a matrix dep ending linearly on ( ρ,  ) while L ( α ) a linear parametrization of the set defined in (2). Moreo ver, to take the constrain t ρ ∈ P in to account, the feasibilit y problem in (20) can b e equiv alently solved b y chec king the p ositivit y of  ∗ , which is the optimal solution of the following SDP:  ∗ = max ρ,α,               M ( ρ,  ) + L ( α ) ≥ 0 ρ i − ρ i 0 0 ρ i − ρ i ! ≥ 0 i = 1 , . . . , p (21) If the condition in (20) is feasible, i.e.,  ∗ > 0, let us further address the second part of Problem 5. Consider the problem of desgining an optimal IFP con troller. Ac- cording to Lemma 3 and (19), the problem can be equiv- alen tly rephrased in the following mathematical form max ρ ∈P ν s.t. p e N ( w ) p e D ( ω , ρ ) + p o N ( w ) p o D ( ω , ρ ) p e D ( ω , ρ ) 2 + p o D ( ω , ρ ) 2 ≥ ν , ∀ ω ∈ R . (22) Theorem 12 If the c ondition in (20) is fe asible, the maximum IFP index ν ∗ that c an b e achieve d by the fe e d- b ack c ontr ol ler set C is given by ν ∗ = ( γ ∗ ) 2 with γ ∗ de- fine d as γ ∗ = max ρ ∈P ,γ γ s.t.     p e N ( w ) p e D ( ω , ρ ) + p o N ( w ) p o D ( ω , ρ ) ? ? γ p e D ( ω , ρ ) 1 0 γ p o D ( ω , ρ ) 0 1     is SOS (23) and the c orr esp onding c ontr ol ler is given by the optimal solution ρ ∗ . Pro of. Supp ose the condition in (20) is feasible, it fol- lo ws that there exists ¯ ρ ∈ P suc h that p e N ( w ) p e D ( ω , ¯ ρ ) + p o N ( w ) p o D ( ω , ¯ ρ ) ≥ 0 , ∀ ω ∈ R and hence, p e N ( w ) p e D ( ω , ¯ ρ ) + p o N ( w ) p o D ( ω , ¯ ρ ) p e D ( ω , ¯ ρ ) 2 + p o D ( ω , ¯ ρ ) 2 ≥ 0 , ∀ ω ∈ R . Therefore, a low er b ound of the optimal ν ∗ in (22) is zero. Next, let us observe that the constraint in (22) can b e rewritten as p e N ( w ) p e D ( ω , ρ ) + p o N ( w ) p o D ( ω , ρ ) − ν p e D ( ω , ρ ) 2 − ν p o D ( ω , ρ ) 2 ≥ 0 . Since ν ≥ 0 and b y exploiting the Sch ur complement lemma, the ab ov e inequality can b e further equiv alently rewritten as     p e N ( w ) p e D ( ω , ρ ) + p o N ( w ) p o D ( ω , ρ ) ? ? √ ν p e D ( ω , ρ ) 1 0 √ ν p o D ( ω , ρ ) 0 1     ≥ 0 . According to Theorem 4 in Chesi (2010), we ha ve that a univ ariate matrix p olynomial is p ositive semidefinite if and only if it is SOS. Therefore, b y replacing √ ν with γ , the optimization problem in (22) can b e equiv alently solv ed by (23), which completes the pro of.  5 Remark 13 The or em 12 pr ovides a metho d via solving a c onvex optimization pr oblem to design the c ontr ol ler in the set C that maximizes the IFP index for the close d- lo op system. Particularly, the maximum γ ∗ in the c onvex optimization pr oblem (23) c an b e obtaine d by bise ction algorithm (i.e., at e ach step of the bise ction algorithm, fix the value of γ and che ck the fe asibility of (23) ). T o che ck the fe asibility of (23) with fixe d value of γ , let us observe that the matrix in (23) dep ends line arly on the de cision variables ρ , and the c onstr aint ρ ∈ P c an b e im- p ose d by adding extr a LMI c onstr aints as done in (21) . Similar to the sc alar p olynomial c ase in (20) , the c on- dition for a matrix p olynomial which dep ends on some de cision variables line arly to b e SOS p olynomials c an b e solve d e quivalently via a SDP, as shown in Se ction 2.2. Next, w e consider the optimal OFP con troller de- sign. T o this end, let us observe that the zeros of the closed-lo op transfer function (9) ha ve negativ e real part under Assumption 7 and the stable controller base ¯ C . Therefore, the closed-lo op system G ( s, ρ ) is minimum phase system. Now, we are ready the presen t the follow- ing theorem. Theorem 14 The maximum OFP index ξ ∗ that c an b e achieve d by the fe e db ack c ontr ol ler set C is given by ξ ∗ define d as ξ ∗ = max ρ ∈P ξ s.t. p e N ( w ) p e D ( ω , ρ ) + p o N ( w ) p o D ( ω , ρ ) − ξ p e N ( ω ) 2 − ξ p o N ( ω ) 2 is SOS . (24) and the c orr esp onding c ontr ol ler is given by the optimal solution ρ ∗ . Pro of. Since the closed-lo op system G ( s, ρ ) is minimum phase, its inv erse exists. F rom (18), we hav e that G − 1 ( j ω , ρ ) = p e D ( ω , ρ ) + j p o D ( ω , ρ ) p e N ( w ) + j p o N ( w ) , and G − 1 ( j ω , ρ ) + [ G − 1 ( j ω , ρ )] ∗ = 2 p e D ( ω , ρ ) p e N ( w ) + 2 p o D ( ω , ρ ) p o N ( w ) p e N ( w ) 2 + p o N ( w ) 2 It follo ws from Lemma 3 that the maximum OFP index that can b e reached is ξ ∗ = max ρ ∈P ξ s.t. p e D ( ω , ρ ) p e N ( w ) + p o D ( ω , ρ ) p o N ( w ) p e N ( w ) 2 + p o N ( w ) 2 ≥ ξ , ∀ ω ∈ R , whic h can b e rewritten into (24).  Similar to the optimization problem (20), since the p olynomial in (24) dep ends linearly on decision v ariables ρ and ξ , it can b e solved by a SDP . Remark 15 An alternative appr o ach to addr ess dir e ctly Pr oblem 5 without assuming that the set of C is stabi- lizing is to solve the SDP pr esente d in The or em 12, and then che ck the stability of the close d-lo op system with the derive d c ontr ol ler C ( ρ ∗ ) . Se e Example 2 in Se ction 4 for mor e details. 3.2.2 DT c ase In the end, we consider Problem 5 for the discrete- time systems (12). In order to establish whether a given stable closed-lo op system is passive, we need to chec k the p ositivit y of the real part of the transfer function G ( z , ρ ) ov er the complex unit circle { z ∈ C : | z | = 1 } . Let y ∈ R b e an auxiliary v ariable, and define the rational function as φ : R → C as φ ( y ) = 1 − y 2 + j 2 y 1+ y 2 . Note that the complex unit circle | z | = 1 is parameterized by the v ariable y ∈ R (Chesi (2019)). Consequently , one has that G ( z , ρ ) + G ∗ ( z , ρ ) ≥ 0 , ∀| z | = 1 (25) is equiv alent to G ( φ ( y ) , ρ ) + G ∗ ( φ ( y ) , ρ ) ≥ 0 , ∀ y ∈ R . (26) Let us denote the numerator and denominator of the transfer function in (12) as the p olynomials p N ( z ) = P d N i =0 q i z i and p D ( z , ρ ) = P d D i =0 b i ( ρ ) z i , re- sp ectiv ely , wherein the co efficients b i ( ρ ) , i = 1 , . . . , d D dep end linearly on the vector v ariable ρ . By substituting z = φ ( y ), w e hav e p N ( φ ( y )) = P d N i =0 q i  1 − y 2 + j 2 y 1+ y 2  i p D ( φ ( y ) , ρ ) = P d D i =0 b i ( ρ )  1 − y 2 + j 2 y 1+ y 2  i . (27) By even-odd decomp osition, it follo ws that p N ( φ ( y )) and p D ( φ ( y ) , ρ ) can b e expressed as p N ( φ ( y )) = p 1 ( y ) + j p 2 ( y ) (1 + y 2 ) d N p D ( φ ( y ) , ρ ) = p 3 ( y , ρ ) + j p 4 ( y , ρ ) (1 + y 2 ) d D where p 1 , p 2 , p 3 , p 4 are all real p olynomials in y with co efficien ts of p 3 , p 4 dep ending linearly on ρ . No w it is ready to see G ( φ ( y ) , ρ ) = (1 + y 2 ) d D − d N p 1 ( y ) + j p 2 ( y ) p 3 ( y , ρ ) + j p 4 ( y , ρ ) whic h follows that G ( φ ( y ) , ρ ) + G ∗ ( φ ( y ) , ρ ) = 2(1 + y 2 ) d D − d N p 1 ( y ) p 3 ( y , ρ ) + p 2 ( y ) p 4 ( y , ρ ) p 2 3 ( y , ρ ) + p 2 4 ( y , ρ ) . (28) Since the giv en set of controllers C is stabilizing, it follo ws from Lemma 3 and Definition 2 that the closed- lo op system is passive if and only if the condition (25) holds. Based on similar reasoning of Lemma 11, we can obtain the following result. Lemma 16 Ther e exists a c ontr ol ler C ( z ) in the c on- tr ol ler set C that c an fe e db ack p assivate the plant G o ( z ) if and only if ther e exists a ve ctor ρ ∈ P and a sc alar  ≥ 0 such that p 1 ( y ) p 3 ( y , ρ ) + p 2 ( y ) p 4 ( y , ρ ) −  is SOS . (29) Lemma 16 provides, for the DT case, a necessary and sufficient condition for determining the existence of a controller C ( z ) in the set C suc h that the closed- lo op system (12) is passive. Similar to the CT case, this condition can b e v erified by solving a SDP with the same 6 form in (21). When the condition (29) is satisfied, the next step is to determine the con troller C ∗ in the set C that can ac hiev e the maxim um IFP index ν ∗ for the closed-loop system (28). Corollary 17 If the c ondition in (29) is fe asible, the maximum IFP index ν ∗ that c an b e achieve d by the fe e d- b ack c ontr ol ler set C is given by ν ∗ = ( γ ∗ ) 2 with γ ∗ de- fine d as γ ∗ = max ρ ∈P ,γ γ s.t.     ¯ p ( y , ρ ) γ p 3 ( y , ρ ) γ p 4 ( y , ρ ) γ p 3 ( y , ρ ) 1 0 γ p 4 ( y , ρ ) 0 1     is SOS (30) wher e ¯ p ( y , ρ ) = (1 + y 2 ) d D − d N ( p 1 ( y ) p 3 ( y , ρ ) + p 2 ( y ) p 4 ( y , ρ )) . By taking the inv erse of G ( φ, ρ ), it is obtained that G − 1 ( φ ( y ) , ρ ) + [ G − 1 ( φ ( y ) , ρ )] ∗ = 2(1 + y 2 ) d N − d D p 1 ( y ) p 3 ( y , ρ ) + p 2 ( y ) p 4 ( y , ρ ) p 2 1 ( y ) + p 2 2 ( y ) . (31) Corollary 18 The maximum OFP index ξ ∗ that c an b e achieve d by the fe e db ack c ontr ol ler set C is given by ξ ∗ define d as ξ ∗ = max ρ ∈P ξ s.t. (1 + y 2 ) d N − d D ( p 1 ( y ) p 3 ( y , ρ ) + p 2 ( y ) p 4 ( y , ρ )) − ξ p 2 1 ( y ) − ξ p 2 2 ( y ) is SOS . (32) and the c orr esp onding c ontr ol ler is given by the optimal solution ρ ∗ . Remark 19 It c an b e e asily se en that the pr op ose d metho dolo gy in this subse ction c an b e use d not only for a hyp er-r e ctangle set P as define d in (7) , but also any c onvex set P . Inde e d, this c an b e achieve d by r eplacing the LMI the se c ond c onstr aint in (21) with appr opriate LMI c orr esp onding to the set ρ ∈ P . 4 Numerical Examples In this section, we pro vide tw o examples to illustrate the prop osed metho dologies. The computations are done b y Matlab with to olb ox SOSTOOLS and SeDuMi. 4.1 Example 1 Let us b egin with considering a DT plan t with trans- fer function G 0 ( z ) = z z − 2 , and the con troller set C de- scrib ed b y C ( z , ρ ) = ρ 1 + ρ 2 1 z − 0 . 5 with the parameter ρ ∈ P = [0 . 1 , 1] × [1 , 2]. Note that the plant is unstable since its pole has magnitude larger than 1. The closed- lo op system (12) is derived as G ( z , ρ ) = 2 z 2 − z (2 ρ 1 + 2) z 2 + (2 ρ 2 − ρ 1 − 5) z + 2 . The first problem is to establish whether the closed- lo op system is stable for all ρ ∈ P . T o address this, w e first compute the mo dified Jury table for the denomina- tor p N ( z , ρ ), and the first column of the table is obtained as f 1 = 2 ρ 1 + 2 , f 2 = 4 ρ 2 1 + 8 ρ 1 , f 3 = 12 ρ 4 1 + 16 ρ 3 1 ρ 2 + 24 ρ 3 1 − 16 ρ 2 1 ρ 2 2 + 80 ρ 2 1 ρ 2 − 36 ρ 2 1 . Next, we examine the p ositivit y of these p olynomials ov er the set ρ ∈ P based on Theorem 9. It is ob vious that f 1 ( ρ ) > 0 and f 2 ( ρ ) > 0 for all ρ ∈ P , so we just need to solve the SOS program in (16) for i = 3. By choosing the the degrees of the auxil- iary p olynomials s 31 ( ρ ) , s 32 ( ρ ) as 2, w e find the optimal solution as θ ∗ = 0 . 32, whic h guaran tees the positivity of f 3 ( ρ ) o ver ρ ∈ P . Therefore, it can be concluded that the closed-lo op system G ( z , ρ ) is stable for all ρ ∈ P . With this set of stabilizing controllers, w e further consider optimal IFP controller design in Problem 5. The first step is to determine the existence of con trollers in the set C that can feedback passiv ate the plant. This can b e done by solving the SDP in (29). Sp ecifically , by replacing z with φ ( y ) = 1 − y 2 + j 2 y 1+ y 2 , we hav e G ( φ ( y ) , ρ ) + G ∗ ( φ ( y ) , ρ ) = p 1 ( y ) p 3 ( y , ρ ) + p 2 ( y ) p 4 ( y , ρ ) p 2 3 ( y , ρ ) + p 2 4 ( y , ρ ) p 1 ( y ) = 3 y 4 − 12 y 2 + 1 p 2 ( y ) = − 10 y 3 + 6 y p 3 ( y ) = (3 ρ 1 − 2 ρ 2 + 9) y 4 − (12 ρ 1 + 8) y 2 + ρ 1 + 2 ρ 2 − 1 p 4 ( y ) = ( − 10 ρ 1 + 4 ρ 2 − 18) y 3 + (6 ρ 1 + 4 ρ 2 − 2) y . Then, we solv e the SOS program in (29), whic h is con v erted to solving a SDP in the form of (21), and it is obtained that the optimal solution of  in (21) is p ositiv e. Therefore, it can b e concluded that there exists a controller in the set C that can feedback passiv ate the plan t G 0 . The next step is to deriv e the controller C ∗ in the set C that maximizes IFP ( ν ) for the closed-loop system. This is accomplished b y solving the SDP (30) at eac h step of the bisection algorithm, which leads to the maximum ν as ν ∗ = 0 . 48 with the optimal solution ρ ∗ 1 = 0 . 1 , ρ ∗ 2 = 1 . 5. T o v erify the resulting IFP index ν ∗ , one can trans- form the closed-loop transfer function G ( z , ρ ∗ ) to a state space system ( A, B , C , D ), and then exploits the nec- essary and sufficien t LMI condition for dissipativit y to obtain the IFP index for the closed-lo op system. (See Lemma 2 in Kottenstette et al. (2014) for details) T o b e sp ecific, the closed-lo op system G ( z , ρ ∗ ) can b e rewrit- ten as the state space system as follows A = 0 . 955 0 . 91 1 0 ! , B = 1 0 ! , C =  0 . 413 − 0 . 826  , D = 0 . 91 . It can b e verified by Lemma 3 in Kottenstette et al. (2014) that the IFP index for this state space system is obtained as 0 . 48 as exp ected. 7 4.2 Example 2 In this example, we consider a CT plan t with the transfer function G 0 ( s ) = ( s +2)( s +3) ( s − 1)( s − 2) , and the controller set C is c hosen to b e the class of PI controllers, describ ed as C ( s, ρ ) = ρ 1 + ρ 2 1 s +1 with the parameter ρ ∈ P = [0 , 1] × [0 , 1]. The problem is to directly determine the controller C ∗ in the set C that maximize the IFP index and the OFP index for the closed-lo op system, resp ectively . Let us observe that the plan t is unstable since it has poles { 1 , 2 } . First, by substituting s = j ω , one can express the closed-lo op system (18) as G ( j ω , ρ ) = p e N ( w ) + j p o N ( w ) p e D ( ω , ρ ) + j p o D ( ω , ρ ) p e N ( w ) = − 6 ω 2 + 6 p o N ( w ) = − ω 3 + 11 ω p e D ( ω , ρ ) = (2 − 6 ρ 1 − ρ 2 ) ω 2 + (2 + 6 ρ 1 + 6 ρ 2 ) p o D ( ω , ρ ) = ( − 1 − ρ 1 ) ω 3 + ( − 1 + 11 ρ 1 + 5 ρ 2 ) ω . Then, we solv e the SOS progam in (20), which is con v erted to solving the SDP (21), and it is obtained that the optimal solution of  in (21) is p ositive. T o design the optimal IFP controller, we consider the optimization problem in (22). By solving the SDP (23) at each step of bisection algorithm, we obtain that the maxim um ν as ν ∗ = 0 . 658 with the solution ρ ∗ 1 = 0 . 516 , ρ ∗ 2 = 0 . 669. T o design the optimal OFP controller, w e solv e the optimization problem in (24), and obtain that the maximum ξ as ξ ∗ = 0 . 542 with the solution ρ 1 = ρ 2 = 1. In the end, we need to chec k the stability of the closed-lo op system (9) with the derived ρ ∗ . F or b oth the optimal IFP controller ρ ∗ 1 = 0 . 516 , ρ ∗ 2 = 0 . 669 and the optimal OFP controller ρ 1 = ρ 2 = 1, the closed- lo op system G ( s, ρ ∗ ) = G 0 ( s ) 1+ G 0 ( s ) C ( s,ρ ∗ ) can b e easily ver- ified via Routh-Hurwitz stability criterion or calculat- ing the p oles that the closed-loop system G ( s, ρ ∗ ) is sta- ble. Therefore, based on Lemma 3, one has that the maxim um IFP index that the closed-lo op system can ac hiev e is ν ∗ = 0 . 658, and the corresp onding controller is C ( s, ρ ∗ ) = 0 . 516 + 0 . 669 s +1 , and the maxim um OFP index that the closed-lo op system can achiev e is ξ ∗ = 0 . 542 and the corresp onding controller is C ( s, ρ ∗ ) = 1 + 1 s +1 . Similar to the previous example, the resulting IFP index ν ∗ and OFP index ξ ∗ can b e verified by trans- forming the closed-lo op transfer function G ( s, ρ ∗ ) to state space system ( A, B , C , D ), and then exploit the necessary and sufficient LMI condition for dissipativity (Lemma 2 in Kottenstette et al. (2014)) to obtain the IFP or OFP index for the closed-lo op system. 5 Conclusion This paper has considered feedbac k passiv ation of SISO L TI systems with linearly parameterized con- troller with the ob jective of maximizing the passivity lev el for the closed-lo op systems. First, we hav e pro- p osed a metho d to test whether a given set of controllers is stabilizing. Second, we hav e shown that given a set of stabilizing controllers, the optimal controller in the sense of maxim um IFP or OFP index can b e obtained b y solving a SDP . The prop osed results also pro vide an alternativ e metho d without assuming the set of con- trollers to be stabilizing. F uture w ork will consider ex- tensions to the multi-input multi-output (MIMO) case. References An tsaklis, P . J., Go o dwine, B., Gupta, V., McCourt, M. J., W ang, Y., W u, P ., Xia, M., Y u, H. & Zhu, F. (2013), ‘Con trol of cyberphysical systems using passivit y and dis- sipativit y based metho ds’, Eur op e an Journal of Contr ol 19 (5), 379–388. Bao, J. & Lee, P . L. (2007), Pr o c ess c ontrol: the p assive systems appr o ach , Springer Science & Business Media. Bazanella, A. S., Camp estrini, L. & Eckhard, D. (2011), Data-driven c ontr ol ler design: the H2 appr o ach , Springer Science & Business Media. Chesi, G. (2010), ‘LMI tec hniques for optimization ov er p oly- nomials in control: a survey’, IEEE T r ansactions on Au- tomatic Contr ol 55 (11), 2500–2510. Chesi, G. (2019), ‘Stability test for complex matrices ov er the complex unit circumference via LMIs and applications in 2d systems’, IEEE T r ansactions on Cir cuits and Systems I: R e gular Pap ers . Khalil, H. K. & Grizzle, J. W. (2002), Nonline ar systems , V ol. 3, Prentice hall Upp er Saddle River, NJ. Kottenstette, N., McCourt, M. J., Xia, M., Gupta, V. & An tsaklis, P . J. (2014), ‘On relationships among passiv- it y , p ositive realness, and dissipativity in linear systems’, Automatic a 50 (4), 1003–1016. Larsen, M. & Kokoto vic, P . V. (2001), ‘On passiv ation with dynamic output feedback’, IEEE T r ansactions on A uto- matic Contr ol 46 (6), 962–967. Malik, W. A., Darbha, S. & Bhattacharyy a, S. P . (2008), ‘A linear programming approac h to the syn thesis of fixed- structure con trollers’, IEEE T r ansactions on Automatic Contr ol 53 (6), 1341–1352. Saeki, M. (2006), ‘Fixed structure PID controller design for standard h con trol problem’, Automatic a 42 (1), 93–100. V an Der Sc haft, A. (2000), L 2 -gain and passivity te chniques in nonline ar c ontr ol , V ol. 2, Springer. Willems, J. C. (1972), ‘Dissipative dynamical systems part i: General theory’, Ar chive for r ational me chanics and anal- ysis 45 (5), 321–351. Zhao, Y. & Gupta, V. (2016), ‘F eedback passiv ation of discrete-time systems under comm unication constraints’, IEEE T r ansactions on A utomatic Contr ol 61 (11), 3521– 3526. Zh u, F., Xia, M. & Antsaklis, P . J. (2014), Passivit y anal- ysis and passiv ation of feedbac k systems using passivity indices, in ‘2014 American Control Conference’, IEEE, pp. 1833–1838. 8