Optimal Disturbance Accommodation with Limited Model Information

Optimal Disturb anc e Accommo dation with Limited Mo del Information ∗ F arhad F arokhi † , C ´ edric Langb o rt ‡ , and Karl H. Johansson † Octob er 8, 2018 Abstract The design of optimal dynamic disturb ance accommodation controller with limited mod el information is considered. W e adapt th e family of limited mo del information control design strategies, defined earlier by the authors, to handle dynamic con trollers. This family of limi ted mod el in- formation design strategies construct sub con trollers d istributively by ac- cessing only local plant mo del information. The clo sed- loop p erformance of the dynamic controllers that they can prod uce are stu died using a p er- formance metric called the comp etitive ratio which is the worst case ratio of the cost a control design strategy to th e cost of t h e optimal control design with full mo del info rmation. 1 In tro du ction Recent adv ances in netw orked control engineering have op ened new do ors to- ward co ntrolling larg e - scale systems. These larg e-scale systems are natur a lly comp osed of man y smaller unit that are coupled to eac h other [1–4]. F or these lar ge-scale interconnected systems, we ca n either design a centralized o r a decentralized co n tr oller. Contrary to a cent r alized controller, each sub co n- troller in a decentralized controller only obser ves a lo cal subset o f the state- measurements (e.g ., [5 – 7]). When designing these controllers, genera lly , it is assumed that the global mo del of the system is av ailable to each sub controller’s designer. Howev er, ther e a re sev er al reaso ns why such plant mo del information would not b e glo ba lly known. One reason could b e that the s ubsystems con- sider their mo del informatio n pr iv ate, and therefore, they a re reluctant to share information with other subsystems. This case can be well illus tr ated b y supply chains or p ower netw ork s where the economic incentiv es o f comp eting co mpa- nies might limit the ex change o f mo del information b etw een the companies. It ∗ The work of F. F arokhi and K. H. Johansson was supp orted by gr an ts from the Swedish Researc h Council and the Knu t and Alice W allen b erg F oundation. The w ork of C. Lang - bort w as supp orted, in part, by the 2010 A F OSR MURI “Mul ti-Lay er and M ulti-Resolution Net works of Int eracting Agen ts in Adv ers arial Environmen ts”. † F. F arokhi and K. H. Johansson ar e with ACCESS Linnaeus Center, School of Electrical Engineering, KTH-Roy al I nstitute of T ec hnology , SE-100 44 Stockholm, Swe den. E-mails: { farokhi,k allej } @ee.kth.se ‡ C. Langbort i s with the Departmen t of A erospace Engineering, Universit y of Illi nois at Urbana-Champaign, Illinois, USA. E-mail : langbort@il linois.edu 1 might also b e the case that the full mo del is not av ailable at the moment, or the designer would like to not mo dify a par ticular s ubco ntroller, if the mo del of a subsy s tem changes. F o r instance, in the cas e of co op era tive driving, ea ch vehicle con tro lle r simply cannot b e desig ne d based on mo del infor mation of all po ssible v ehicles that it may in teract with in future. T he r efore, we are interested in finding control design strategies w hich construct sub controllers distributively for plants made o f int er connected subsystems without the globa l mo del of the system. The interconnection str ucture a nd the co mmon clos ed-lo op cos t to b e minimized are ass umed to b e public knowledge. W e ident ify these control design metho ds by “limited mo del information” control des ign strategies [8, 9]. Multi-v ariable ser vomec hanism and disturbance a ccommo dation con trol de- sign is o ne of the oldest problems in control engineering [10]. W e adapt the pro- cedure intro duced in [10 , 11] to desig n optimal disturbance accommo da tio n con- trollers for discrete-time linea r time-inv a riant plan ts under a sepa rable quadratic per formance measure . The c hoice of the cost function is motiv ated first by the optimal dis turbance acco mmo dation litera ture [10, 11], and seco nd by our in- terest in dyna mically-coupled but co s t-decoupled plants and their applica tions in supply chains and sha red infrastructures [3, 4]. Then, we focus on the dis- turbance a ccommo dation design pro blem under limited mo del information. W e inv estigate the achiev able closed- lo op p er formance of the dynamic controllers that the limited mo del infor mation control design s trategies can pro duce using the comp etitive ra tio, that is, the worst case ratio of the cos t a control design strategy to the co st of the optimal co nt r ol design with full mo del information. W e find a minimizer of the comp etitive ra tio ov er the set of limited mo del in- formation control design s tr ategies. Since this minimizer may not b e unique we prov e that it is undomina ted, that is, ther e is no other control design metho d that acts b e tter while exhibiting the sa me w or st-case ratio. This pap er is org anized as follows. W e mathematically for mulate the prob- lem in Section 2. In Section 3, we introduce t wo useful con trol des ig n stra teg ies and study their prop erties . W e c har acterize the b est limited mo del information control design metho d a s a function of the subsystems interconnection pattern in Section 4. In Section 5, we study the tra de-off betw ee n the amount of the information av ailable to each subsystem and the qua lity of the controllers that they can pro duce. Finally , we end with conclusions in Section 6. 1.1 Notation The s et of real num ber s and complex num b ers are denoted by R and C , respec- tively . All other sets a re denoted by calligr aphic letters such as P and A . The notation R denotes the set of prop er real ratio nal functions. Matrices ar e denoted by capital r oman letters suc h as A . A j will deno te the j th row of A . A ij denotes a s ub-matrix of ma trix A , the dimens io n and the po sition o f which will b e defined in the text. The entry in the i th row and the j th column of the matrix A is a ij . Let S n ++ ( S n + ) b e the se t o f symmetric po sitive definite (p ositive s emidefinite) matrices in R n × n . A > ( ≥ )0 mea ns that the sy mmetr ic matrix A ∈ R n × n is po sitive definite (p ositive semidefinite) and A > ( ≥ ) B means A − B > ( ≥ )0. σ ( Y ) a nd σ ( Y ) denote the sma llest a nd the larges t singular v alues of the matrix Y , resp ectively . V ector e i denotes the column-vector w ith a ll ent r ies zero except the i th ent r y , which is equa l to one. 2 All graphs co nsidered in this pap er ar e direc ted, p os sibly with self- lo ops, with vertex set { 1 , . . . , n } for some p ositive integer n . W e say i is a sink in G = ( { 1 , . . . , n } , E ), if there do es no t exist j 6 = i such that ( i, j ) ∈ E . T he adjacency matrix S ∈ { 0 , 1 } n × n of graph G is a matrix w ho se en trie s are defined as s ij = 1 if ( j, i ) ∈ E and s ij = 0 otherwise. Since the set of vertices is fixe d here, a subgraph of G is a graph whose edge se t is a subset of the edge set of G and a supe r graph of G is a graph o f which G is a subgr aph. W e use the notation G ′ ⊇ G to indicate that G ′ is a sup ergra ph o f G . 2 Mathematical F orm ulati on 2.1 Plan t Model Consider the discrete-time linear time-inv ariant dynamical system des crib ed in state-space representation by x ( k + 1) = Ax ( k ) + B ( u ( k ) + w ( k )) ; x (0) = x 0 , (1) where x ( k ) ∈ R n is the state vector, u ( k ) ∈ R n is the control input, and w ( k ) ∈ R n is the distur bance vector. In addition, assume that w ( k ) is a dynamic disturbance mo deled as w ( k + 1) = D w ( k ) ; w (0) = w 0 . (2) Let a plant gra ph G P with a djacency matr ix S P be given. W e define the fol- lowing se t of matrices A ( S P ) = { ¯ A ∈ R n × n | ¯ a ij = 0 for all 1 ≤ i, j ≤ n such tha t ( s P ) ij = 0 } . Also, let us define B ( ǫ b ) = { ¯ B ∈ R n × n | σ ( ¯ B ) ≥ ǫ b , ¯ b ij = 0 for all 1 ≤ i 6 = j ≤ n } , for a given sca lar ǫ b > 0 and D = { ¯ D ∈ R n × n | ¯ d ij = 0 for all 1 ≤ i 6 = j ≤ n } . W e ca n introduce the set of plants of interest P as the space of a ll discrete- time linear time-inv ariant systems of the form (1) a nd (2) with A ∈ A ( S P ), B ∈ B ( ǫ b ), D ∈ D , x 0 ∈ R n , and w 0 ∈ R n . Since P is isomorph to A ( S P ) × B ( ǫ b ) × D × R n × R n , we ident ify a plant P ∈ P with its co rresp onding tuple ( A, B , D , x 0 , w 0 ) with a slight abuse of nota tion. W e can think o f x i ∈ R , u i ∈ R , and w i ∈ R as the state, input, and disturbance of scala r subsystem i with its dynamic given as x i ( k + 1) = n X j =1 a ij x j ( k ) + b ii ( u i ( k ) + w i ( k )) . W e call G P the plant graph since it illustrates the interconnection structure betw ee n different subsys tems , that is, subsystem j can affect subsystem i only if ( j, i ) ∈ E P . In this pap er , we a s sume that ov era ll system is fully-ac tua ted, that is, any B ∈ B ( ǫ b ) is a squar e inv ertible matr ix. This assumption is mo tiv ated by the fact that we w ant all the s ubsystems to b e directly controllable. 3 2.2 Con troller The control laws of in teres t in this pap er are discrete-time linea r time-inv ar iant dynamic state-feedback con trol laws o f the form x K ( k + 1) = A K x K ( k ) + B K x ( k ) ; x K (0) = 0 , u ( k ) = C K x K ( k ) + D K x ( k ) . Each controller can also be r epresented by its transfer function K ,  A K B K C K D K  = C K ( z I − A K ) − 1 B K + D K , where z is the symbol for o ne time-step forward s hift op erator. Let a control graph G K with adjacency matrix S K be g iven. E a ch controller K must b elong to K ( S K ) = { ¯ K ∈ R n × n | ¯ k ij = 0 for all 1 ≤ i, j ≤ n such tha t ( s K ) ij = 0 } . When adjacency matrix S K is not relev an t or can b e deduced from context, we refer to the set of controllers as K . Since it makes sense for each s ubsystem’s controller to hav e access to at least its own state-mea surements, we make the standing assumption that in each c o ntrol gr aph G K , all the self- lo ops are present. Finding the optimal s tructured controller is difficult (n umer ically in tr a ctable) for g eneral G K and G P even when the global mode l is known. Therefore, in this pap er, a s a sta rting p oint, we only co ncentrate on the cas es wher e the control graph G K is a sup ergra ph o f the plant gra ph G P . 2.3 Con trol Design Methods A control design metho d Γ is a mapping from the se t of plants P to the set of controllers K . W e can write the cont r ol design method Γ as Γ =    γ 11 · · · γ 1 n . . . . . . . . . γ n 1 · · · γ nn    where each entry γ ij represents a map A ( S P ) × B ( ǫ b ) × D → R . Let a design graph G C with adjacency ma tr ix S C be given. The control design strategy Γ has structure G C if, for all i , the map Γ i = [ γ i 1 · · · γ in ] is only a function of { [ a j 1 · · · a j n ] , b j j , d j j | ( s C ) ij 6 = 0 } . Consequently , for each i , sub controller i is constructed with mo del information of only those subsystems j that ( j, i ) ∈ E C . W e ar e only interested in thos e control design strategies that are neither a function of the initia l state x 0 nor of the initial disturbanc e w 0 . The set of all control design stra teg ies w ith the design gr aph G C is denoted by C . Since it makes sense for the designer of eac h subsystem’s con tro lle r to have a ccess to at least its own model parameters, w e make the standing assumption that in each design graph G C , all the self-lo ops are present. F or simplicity o f notation, let us assume that a ny control design s trategy Γ ∈ C has a state-space realiza tion o f the form Γ( A, B , D ) =  A Γ ( A, B , D ) B Γ ( A, B , D ) C Γ ( A, B , D ) D Γ ( A, B , D )  , 4 where matrices A Γ ( A, B , D ), B Γ ( A, B , D ), C Γ ( A, B , D ), a nd D Γ ( A, B , D ) are of appropria te dimension for each plan t P = ( A, B , D , x 0 , w 0 ) ∈ P . The ma trices A Γ ( A, B , D ) and C Γ ( A, B , D ) are blo ck diagonal matrice s s ince different sub- controllers sho uld not shar e state v ariables. This rea lization is not nec essarily a minimal realizatio n. 2.4 P erformance Metr ic s W e need to introduce perfor mance metr ics to compa re the control des ig n meth- o ds. These p erfor mance metrics ar e adapted fro m earlier definitions in [8, 12 ]. Let us start with in tr o ducing the closed-lo op p erfor ma nce criterion. T o e ach plant P = ( A, B , D , x 0 , w 0 ) ∈ P and co nt r oller K ∈ K , we a sso ciate the pe r formance criterio n J P ( K ) = ∞ X k =0 [ x ( k ) T Qx ( k ) + ( u ( k ) + w ( k )) T R ( u ( k ) + w ( k ))] where Q ∈ S n ++ and R ∈ S n ++ are diag onal matrices. W e make the standing assumption tha t Q = R = I . This is without los s of generality b ecause of the change of v ariables ( ¯ x, ¯ u, ¯ w ) = ( Q 1 / 2 x, R 1 / 2 u, R 1 / 2 w ) that trans forms the state-space representation into ¯ x ( k + 1) = Q 1 / 2 AQ − 1 / 2 ¯ x ( k )+ Q 1 / 2 B R − 1 / 2 ( ¯ u ( k )+ ¯ w ( k )) = ¯ A ¯ x ( k ) + ¯ B ( ¯ u ( k ) + ¯ w ( k )) , and the per formance criterio n into J P ( K ) = ∞ X k =0 [ ¯ x ( k ) T ¯ x ( k ) + ( ¯ u ( k ) + ¯ w ( k )) T ( ¯ u ( k ) + ¯ w ( k ))] . (3) This change o f v ar ia ble w ould no t affect the plant, control, or design gr aph since bo th Q and R are diago nal matrices. Definition 2. 1 (Competitive Ratio) L et a plant gr aph G P and a c onstant ǫ b > 0 b e given. A ssume t hat, for every plant P ∈ P , ther e exists an opti- mal c ontro l ler K ∗ ( P ) ∈ K such that J P ( K ∗ ( P )) ≤ J P ( K ) , ∀ K ∈ K . The c omp etitive r atio of a c ontr ol design metho d Γ is define d as r P (Γ) = sup P =( A, B ,D ,x 0 ,w 0 ) ∈P J P (Γ( A, B , D )) J P ( K ∗ ( P )) , with t he c onvention that “ 0 0 ” e quals one. Definition 2. 2 (Domination) A c ontr ol design metho d Γ is said t o dominate another c ontr ol design metho d Γ ′ if for al l plants P = ( A, B , D , x 0 , w 0 ) ∈ P J P (Γ( A, B , D )) ≤ J P (Γ ′ ( A, B , D )) , (4) with st rict ine quality holding for at le ast one plant in P . When Γ ′ ∈ C and no c ontr ol design metho d Γ ∈ C exist s t hat dominates it, we say that Γ ′ is undominate d in C . 5 2.5 Problem F orm ulation F or a given plan t gr aph G P , control graph G K , and des ign graph G C , we w a n t to solve the pr o blem arg min Γ ∈C r P (Γ) . (5) Because the solution to this problem migh t not b e unique, w e a lso wan t to determine which one s of thes e minimizers are undominated. 3 Preliminary Results In o rder to g ive the main results o f the pape r , we need to in tro duce t wo control design strategies and study their prop er ties . 3.1 Optimal Centralized Control Design Str ategy In this subsectio n, we find the optimal centralized cont r ol design strategy K ∗ C ( P ) for a ll plants P ∈ P ; i.e., the o ptima l control design str ategy when the control graph G K is a co mplete gr aph. Note that we use the notation K ∗ C ( P ) to de- note the centralized optimal control design strateg y as the notation K ∗ ( P ) is reserved for the optimal control design strategy for a given control gra ph G K . W e adapt the pro cedure g iven in [10 , 11] fo r co nstant input-disturbance rejection in contin uous-time systems to our framework. First, le t us define the a ux iliary v ariables ξ ( k ) = u ( k ) + w ( k ) and ¯ u ( k ) = u ( k + 1 ) − D u ( k ). It is evident that ξ ( k + 1) = D ξ ( k ) + ¯ u ( k ) . (6) Augment ing (6) with the s ystem state-spac e represe ntation in (1) r esults in  x ( k + 1) ξ ( k + 1)  =  A B 0 D   x ( k ) ξ ( k )  +  0 I  ¯ u ( k ) . (7) In addition, we c a n write the perfo r mance measure in (3) as J P ( K ) = ∞ X k =0  x ( k ) ξ ( k )  T  x ( k ) ξ ( k )  . (8) T o g uarantee existence and uniqueness of the o ptimal controller K ∗ C ( P ) for an y given plant P ∈ P , we need the following lemma to hold [13]. Lemma 3.1 The p air ( ˜ A, ˜ B ) with ˜ A =  A B 0 D  , ˜ B =  0 I  , (9) is c ont ro l lable for any given P = ( A, B , D , x 0 , w 0 ) ∈ P . Pr o of: The pair ( ˜ A, ˜ B ) is controllable if and only if  ˜ A − λI ˜ B  =  A − λI B 0 0 D − λI I  6 is full-ra nk for all λ ∈ C . This condition is always satis fied since all the ma trices B ∈ B ( ǫ b ) are full-rank matrices. Now, the pr oblem of minimizing the cost function in (8) sub ject to pla nt dynamics in (7) be comes a state-feedback linear quadratic optimal control design with a unique solution of the form ¯ u ( k ) = G 1 x ( k ) + G 2 ξ ( k ) where G 1 ∈ R n × n and G 2 ∈ R n × n . Therefore, we have u ( k + 1 ) = D u ( k ) + ¯ u ( k ) = D u ( k ) + G 1 x ( k ) + G 2 ξ ( k ) . (10) Using ξ ( k ) = B − 1 ( x ( k + 1 ) − Ax ( k )) in (10), we get u ( k + 1 ) = D u ( k ) + G 1 x ( k ) + G 2 B − 1 ( x ( k + 1) − Ax ( k )) . (11) Putting a control sig nal of the form u ( k ) = x K ( k ) + D K x ( k ) in (11) results in x K ( k + 1) = D x K ( k )+( D D K + G 1 − G 2 B − 1 A ) x ( k ) + ( G 2 B − 1 − D K ) x ( k + 1 ) . Now, b ecause o f the form o f the control laws of interest introduced earlie r in Subsection 2.2, we have to enfor ce G 2 B − 1 − D K = 0 . Therefor e , the optimal controller K ∗ C ( P ) b e c omes x K ( k + 1) = D x K ( k ) + [ G 1 + DG 2 B − 1 − G 2 B − 1 A ] x ( k ) , u ( k ) = x K ( k ) + G 2 B − 1 x ( k ) , with the initial condition x K (0) = 0 a gain b eca use of the form o f the control laws o f interest. Lemma 3.2 L et the c ont r ol gr aph G K b e a c omplete gr aph. The n , the c ost of the optimal c ontr ol design str ate gy K ∗ C for e ach plant P ∈ P is lower-b ounde d as J P ( K ∗ C ( P )) ≥  x 0 B w 0  T  V 11 V 12 V T 12 V 22   x 0 B w 0  , wher e V 11 = W + D 2 B − 2 + DW D , (12) V 12 = − D ( W + B − 2 ) , (13) V 22 = W + B − 2 , (14) with t he matrix W define d as W = A T ( I + B 2 ) − 1 A + I . (15) Pr o of: T o make the pro of easier, let us define ¯ J P ( K, ρ ) = ∞ X k =0  x ( k ) ξ ( k )  T  x ( k ) ξ ( k )  + ρ ¯ u ( k ) T ¯ u ( k ) ! , 7 and ¯ K ∗ ρ ( P ) = arg min K ∈ K ¯ J P ( K, ρ ) . Using Lemma 3.1, we know that ¯ K ∗ ρ ( P ) uniquely exists. W e can find ¯ J P ( ¯ K ∗ ρ ( P ) , ρ ) using X ( ρ ) as the unique positive definite solution o f the discr ete algebraic Ric- cati equation ˜ A T X ( ρ ) ˜ B ( ρI + ˜ B T X ( ρ ) ˜ B ) − 1 ˜ B T X ( ρ ) ˜ A − ˜ A T X ( ρ ) ˜ A + X ( ρ ) − I = 0 , (16) with ˜ A and ˜ B defined in (9). According to [14], we have X ( ρ ) ≥ ˜ A T ( X − 1 1 + (1 /ρ ) ˜ B ˜ B T ) − 1 ˜ A + I = ˜ A T ( X 1 − X 1 ˜ B ( ρI + ˜ B T X 1 ˜ B ) − 1 ˜ B T X 1 ) ˜ A + I , where X 1 = ˜ A T ( I + (1 /ρ ) ˜ B ˜ B T ) − 1 ˜ A + I . Basic algebra ic calcula tions show that lim ρ → 0 + X 1 − X 1 ˜ B ( ρI + ˜ B T X 1 ˜ B ) − 1 ˜ B T X 1 =  W 0 0 0  where W is defined in (15). According to [15], we know lim ρ → 0 + ¯ J P ( ¯ K ∗ ρ ( P ) , ρ ) = J P ( K ∗ C ( P )) and as a result X = lim ρ → 0 + X ( ρ ) ≥  A B 0 D  T  W 0 0 0   A B 0 D  + I . Equiv alently , we ge t  X 11 X 12 X T 12 X 22  ≥  A T W A + I A T W B B W A B W B + I  . (17) Now, we can calculate the cost of the optimal control desig n stra tegy as J P ( K ∗ C ( P )) =  x 0 ξ (0)  T  X 11 X 12 X T 12 X 22   x 0 ξ (0)  (18) where ξ (0) = G 2 B − 1 x 0 + w 0 = − ( X − 1 22 X T 12 + DB − 1 ) x 0 + w 0 . (19) If we put (19) in (1 8 ) and use the sub-Riccati equation X 22 − I = B X 11 B − B X 12 X − 1 22 X T 12 B , that is extracted from the Riccati equa tion in (16) when ρ = 0 , we can simplify J P ( K ∗ ( P )) in (18) to  x 0 − ( X − 1 22 X T 12 + D B − 1 ) x 0 + w 0  T  X 11 X 12 X T 12 X 22   x 0 − ( X − 1 22 X T 12 + D B − 1 ) x 0 + w 0  =  x 0 w 0  T  X 11 − X 12 X − 1 22 X T 12 + B − 1 DX 22 DB − 1 − B − 1 DX 22 − X 22 DB − 1 X 22   x 0 w 0  =  x 0 w 0  T  B − 1 ( X 22 + D X 22 D − I ) B − 1 − B − 1 DX 22 − X 22 DB − 1 X 22   x 0 w 0  . (20) 8 Now, using (17) it is evident that X 22 ≥ B W B + I , and as a result J P ( K ∗ C ( P )) ≥  x 0 w 0  T  V 11 V 12 B B V T 12 B V 22 B   x 0 w 0  . where V 11 , V 12 , and V 22 are in tro duced in (12)- (14). The res t is o nly a straight forward matr ix manipulation (factoring the matrix B ). 3.2 Deadb eat Con tr ol Design Strategy In this subsection, we int r o duce the deadb eat control desig n stra tegy and give a useful lemma ab out its comp etitive ratio. Definition 3. 1 The de adb e at c ontr ol design st ra te gy Γ ∆ : A ( S P ) × B ( ǫ b ) × D → K is define d as Γ ∆ ( A, B , D ) =  D − B − 1 D 2 I − B − 1 ( A + D )  . Using this c ont ro l design str ate gy, irr esp e ctive of the value of the initial state x 0 and the initial disturb anc e w 0 , the close d-lo op s ystem r e aches the origin just in two time-steps. Note that the de adb e at c ontr ol design str ate gy is a limite d mo del information c ontr ol design met ho d sinc e Γ ∆ i ( A, B , D ) = − ( z − d ii ) − 1 b − 1 ii d 2 ii e T i − b − 1 ii ( A i + D i ) for e ach 1 ≤ i ≤ n . The c ost of the de adb e at c ontr ol design str ate gy Γ ∆ for any P = ( A, B , D , x 0 , w 0 ) ∈ P is J P (Γ ∆ ( A, B , D )) =  x 0 B w 0  T  Q 11 Q 12 Q T 12 Q 22   x 0 B w 0  , wher e Q 11 = I + D 2 ( I + B − 2 ) + A T B − 2 A + D A T B − 2 AD + A T B − 2 D + D B − 2 A, (21 ) Q 12 = − D − A T B − 2 − DB − 2 − DA T B − 2 A, (22) Q 22 = A T B − 2 A + B − 2 + I . (23) The close d-lo op system with de adb e at c ontr ol design str ate gy is shown in Fig- ur e 1(a). This fe e db ack lo op c an b e r e-arr ange d as the one in Figur e 1(b) which has two sep ar ate c omp onents. One c omp onent is a static-de adb e at c ontr ol design str ate gy [8] for r e gulating the state of t he plant and t he other one is the de adb e at observer for c anc eling the disturb anc e effe ct. Lemma 3.3 L et the plant gr aph G P c ontain no isolate d no de and G K ⊇ G P . Then, the c omp etitive r atio of the de adb e at c ontr ol design m etho d Γ ∆ satisfies r P (Γ ∆ ) ≤ (2 ǫ 2 b + 1 + p 4 ǫ 2 b + 1) / (2 ǫ 2 b ) . Pr o of: First, let us define the set of all rea l num be r s that a re greater than or equal to r P (Γ ∆ ) as M =  ¯ β ∈ R     J P (Γ ∆ ( A, B , D )) J P ( K ∗ ( P )) ≤ ¯ β ∀ P ∈ P  . 9 It is evident that J P ( K ∗ C ( P )) ≤ J P ( K ∗ ( P )) fo r ea ch plant P ∈ P , irresp ective of the control g raph G K , and as a result J P (Γ ∆ ( A, B , D )) J P ( K ∗ ( P )) ≤ J P (Γ ∆ ( A, B , D )) J P ( K ∗ C ( P )) . (24) Using E quation (2 4), Definition 3.1 , and Lemma 3.2, we get tha t β b elongs to the set M if  x 0 B w 0  T  Q 11 Q 12 Q T 12 Q 22   x 0 B w 0   x 0 B w 0  T  V 11 V 12 V T 12 V 22   x 0 B w 0  ≤ β , (25) for a ll A ∈ A ( S P ), B ∈ B ( ǫ b ), D ∈ D , x 0 ∈ R n , and w 0 ∈ R n where Q 11 , Q 12 , and Q 22 are defined in (21 )- (23) and V 11 , V 12 , and V 22 are defined in (12 )- (14). The condition in (25) is satisfied if and only if  β V 11 − Q 11 β V 12 − Q 12 β V T 12 − Q T 12 β V 22 − Q 22  ≥ 0 , for all A ∈ A ( S P ), B ∈ B ( ǫ b ), and D ∈ D . Now, using Sch ur complement [16], we can show that β b elongs to the set M if b o th c o nditions Z = β V 22 − Q 22 = A T ( β ( I + B 2 ) − 1 − B − 2 ) A + ( β − 1)( B − 2 + I ) ≥ 0 , (26) and β V 11 − Q 11 − [ β V 12 − Q 12 ][ β V 22 − Q 22 ] − 1 [ β V T 12 − Q T 12 ] ≥ 0 , (27) are satisfied fo r all ma trices A ∈ A ( S P ), B ∈ B ( ǫ b ), and D ∈ D . W e can go further and simplify the condition in (27) to β ( W + DW D + D 2 B − 2 ) − Q 11 −  − D Z + A T B − 2  Z − 1  − Z D + B − 2 A  ≥ 0 , (28) where Z is in tro duced in (26). F or all β ≥ 1 + 1 /ǫ 2 b , we know that Z ≥ ( β − 1)( B − 2 + I ) ≥ 0 a nd, as a result the condition ( β − 1) I + A T  β ( I + B 2 ) − 1 − B − 2 − ( β − 1) − 1 B − 2 ( B − 2 + I ) − 1 B − 2  A ≥ 0 (29) bec omes a sufficient condition for the condition in (28 ) to be satisfied. Conse- quently , β b elongs to the set M , if it is greater than or equa l to 1 + 1 / ǫ 2 b and it satisfies the condition in (29). Thus, w e get  β | β ≥ (2 ǫ 2 b + 1 + q 4 ǫ 2 b + 1) / (2 ǫ 2 b )  ⊆ M . This concludes the pro of. 10  ݔ ௄ ሺ ݇ ൅ ͳ ሻ ൌ ܦ ݔ ௄ ሺ ݇ ሻ െ ܤ ିଵ ܦ ଶ ݔ ሺ ݇ ሻ ݑ ሺ ݇ ሻ ൌ ݔ ௄ ሺ ݇ ሻ െ ܤ ିଵ ሺ ܣ ൅ ܦ ሻ ݔ ሺ ݇ ሻ ݑ ሺ ݇ ሻ ݓ ሺ ݇ ሻ ݔ ሺ ݇ ሻ + + ሺ ܽ ሻ ݑ ଵ ሺ ݇ ሻ ൌ െ ܤ ିଵ ܣݔ ሺ ݇ ሻ ݔ ா ሺ ݇ ൅ ͳ ሻ ൌ ܦ ݔ ா ሺ ݇ ሻ െ ܤ ିଵ ܦ ଶ ݔ ሺ ݇ ሻ ݑ ଶ ሺ ݇ ሻ ൌ ݔ ா ሺ ݇ ሻ െ ܤ ିଵ ܦݔ ሺ ݇ ሻ +  ݔ ሺ ݇ ሻ ݓ ሺ ݇ ሻ + ݓ + ሺ ܾ ሻ ݑ ଵ ሺ ݇ ሻ ݑ ଶ ሺ ݇ ሻ + Figure 1: The closed-lo op system with ( a ) the deadbeat con tro l design strategy Γ ∆ and ( b ) r earra nging this control desig n strategy as a static dea dbea t control design and a deadb eat observer design. 4 Plan t Graph Infl uence on Ac hiev able P erfor- mance First, we need to give the following lemmas to make pr o of easier. Lemma 4.1 L et the plant gr aph G P c ontain no isolate d no de and G K ⊇ G P . L et P = ( A, B , D , x 0 , w 0 ) ∈ P b e a plant such that A is a nilp otent matrix of de gr e e two. Then, J P ( K ∗ ( P )) = J P ( K ∗ C ( P )) . Pr o of: When matr ix A is nilp otent, based on the unique p ositive-definite solution of the discr ete algebraic Riccati equation in (16) when ρ = 0, the optimal cent r alized controller K ∗ C ( P ) b e c omes K ∗ C ( P ) =  D D ( I + B 2 ) − 1 B − 1 A − B − 1 D 2 I − ( I + B 2 ) − 1 B A − B − 1 D  . Thu s , K ∗ C ( P ) ∈ K ( S K ) b ecause G K ⊇ G P . Now, b ecause K ∗ ( P ) is the global optimal decent r alized controller, it ha s a low er cost than a ny other decen tra lized controller K ∈ K ( S K ), and in particula r J P ( K ∗ ( P )) ≤ J P ( K ∗ C ( P )) . (30) On the other hand, it is evident tha t J P ( K ∗ C ( P )) ≤ J P ( K ∗ ( P )) . (31) The rest of the pro of is a direct use of (30) and (31) simultaneously . Lemma 4.2 Fix r e al nu mb ers a ∈ R and b ∈ R . F or any x ∈ R , we hav e x 2 + ( a + bx ) 2 ≥ a 2 / (1 + b 2 ) . 11 Pr o of: Consider the function x 7→ x 2 + ( a + bx ) 2 . Since this function is bo th contin uously differentiable and strictly conv ex, we can find its unique minimizer as ¯ x = − ab/ (1 + b 2 ) b y putting its deriv ative equal to zero. As a result, we g et x 2 + ( a + bx ) 2 ≥ ¯ x 2 + ( a + b ¯ x ) 2 = a 2 / (1 + b 2 ). Lemma 4.3 L et the plant gr aph G P c ontain no isolate d no de, the design gr aph G C b e a total ly disc onne cte d gr aph, and G K ⊇ G P . F urthermor e, assu m e that no de i is not a sink in the plant gr aph G P . Then, the c omp etitive r atio of c ontr ol design str ate gy Γ ∈ C is b ounde d only if a ij + b ii ( d Γ ) ij ( A, B , D ) = 0 for all j 6 = i and al l matric es A ∈ A ( S P ) , B ∈ B ( ǫ b ) , and D ∈ D . Pr o of: The pro o f is by co ntrapositive. Assume that the matrices ¯ A ∈ A ( S P ), B ∈ B ( ǫ b ), D ∈ D , and indices i and j exist such that i 6 = j and ¯ a ij + b ii ( d Γ ) ij ( ¯ A, B , D ) 6 = 0 for some control desig n strategy Γ ∈ C . Let 1 ≤ ℓ ≤ n be an index such that ℓ 6 = i and ( s P ) ℓi 6 = 0 (such an index exists b e c ause no de i is not a s ink in the plan t graph). Define ma trix A such that A i = ¯ A i , A ℓ = r e T i , and A t = 0 for a ll t 6 = i, ℓ . It is ev ident that Γ i ( ¯ A, B , D ) = Γ i ( A, B , D ) since the design gra ph is a to ta lly disco nnected gr aph. Using the s tructure of the c ost function in (3) and plant dynamics in (1), the cost o f the co n tr o l design stra tegy Γ when w 0 = e j and x 0 = 0 sa tisfies J P (Γ( A, B , D )) ≥ ( u ℓ (2) + w ℓ (2)) 2 + x ℓ (3) 2 = ( u ℓ (2) + w ℓ (2)) 2 + ( rx i (2) + b ℓℓ ( u ℓ (2) + w ℓ (2))) 2 . With the help of Le mma 4.2 and the fac t that x i (2) = ( a ij + b ii ( d Γ ) ij ( A, B , D )) b j j (see Figure 2), we get J P (Γ( A, B , D )) ≥ r 2 x i (2) 2 / (1 + b 2 ℓℓ ) = ( a ij + b ii ( d Γ ) ij ( A, B , D )) 2 b 2 j j r 2 / (1 + b 2 ℓℓ ) . The cost of the deadb eat control des ign strategy is J P (Γ ∆ ( A, B , D )) = e T j B T ( A T B − 2 A + B − 2 + I ) B e j = b 2 j j + 1 + a 2 ij b 2 j j /b 2 ii . Using the inequality r P (Γ) = sup P ∈P J P (Γ( A, B , D )) J P ( K ∗ ( P )) = s up P ∈P  J P (Γ( A, B , D )) J P (Γ ∆ ( A, B , D )) J P (Γ ∆ ( A, B , D )) J P ( K ∗ ( P ))  ≥ s up P ∈P J P (Γ( A, B , D )) J P (Γ ∆ ( A, B , D )) , gives r P (Γ) ≥ ( a ij + b ii ( d Γ ) ij ( A, B , D )) 2 b 2 j j (1 + b 2 ℓℓ )( b 2 j j + 1 + a 2 ij b 2 j j /b 2 ii ) lim r →∞ r 2 = ∞ . This prov es the statemen t by con tra p ositive. Now, we ar e r eady to tackle the pro blem (5). As the main results of the pap er crucially depends on the pro p e rties o f the pla n t graph, w e s plit these results to tw o differen t subse c tio ns. 12 4.1 Plan t Graphs without Sinks In this section, w e assume that there is no sink in the plant graph, a nd we try to find the b est co n tr o l design stra teg y in terms of the comp etitive ratio a nd the domination. Theorem 4. 4 L et the plant gr aph G P c ontain n o isolate d no de and no sink, the design gr aph G C b e a t otal ly disc onne cte d gr aph, and G K ⊇ G P . Then, the fol lowing st atements hold: (a) The c omp etitive r atio of any c ontr ol design str ate gy Γ ∈ C satisfies r P (Γ) ≥ r P (Γ ∆ ) = (2 ǫ 2 b + 1 + p 4 ǫ 2 b + 1) / (2 ǫ 2 b ) . (b) The c ontr ol design str ate gy Γ ∆ is undominate d, if and only if, ther e is no sink in t he plant gr aph G P . Pr o of: First, let us pr ov e statement ( a ). It is alwa ys p ossible to pick indices j 6 = i such that ( s P ) j i 6 = 0 since there is no isola ted no de in the plant graph G P . Let us define a one-para meter family of matrices { A ( r ) } wher e A ( r ) = re j e T i for each r ∈ R . In addition, let B = ǫ b I and D = I . Acco rding to Lemma 4.3 , r P (Γ) is bo unded o nly if r + ǫ b ( d Γ ) j i ( r ) = 0. Therefore, there is no loss of gener ality in assuming that ( d Γ ) j i ( r ) = − r /ǫ b bec ause o ther wise r P (Γ) is infinit y and the inequa lit y r P (Γ) ≥ r P (Γ ∆ ) is trivially satisfied (co nsidering that using Lemma 3.3 we know r P (Γ ∆ ) is b ounded). F or each r ∈ R , the matrix A ( r ) is a nilp otent matrix of degree tw o. T hus, using Lemma 4.1, we get J P ( K ∗ ( P )) = J P ( K ∗ C ( P )) for this sp ecial plant. The unique po sitive definite solution of the discrete a lg ebraic Riccati eq uation in (16) for a fixed r (when ρ = 0) is X =  A ( r ) T A ( r ) ǫ b A ( r ) T ǫ b A ( r ) ǫ 2 b / (1 + ǫ 2 b ) A ( r ) T A ( r ) + ǫ 2 b I  + I . Thu s , the cos t of the optimal cont r ol design strategy for x 0 = ( ǫ 2 b + 1)( p 4 ǫ 2 b + 1 + 1 ) 2 ǫ b r e i , (32) and w 0 = ( ǫ 2 b + 1)( p 4 ǫ 2 b + 1 + 1 ) 2 ǫ 2 b r e i − e j , (33) is equal to J P ( K ∗ ( P )) = ǫ 2 b p 4 ǫ 2 b + 1 + 5 ǫ 2 b + 4 ǫ 4 b + p 4 ǫ 2 b + 1 + 1 2 ǫ 2 b + (2 ǫ 2 b + p 4 ǫ 2 b + 1 + 1 ) p 4 ǫ 2 b + 1 2 ǫ 2 b r 2 , On the other hand, for eac h r ∈ R , the co st o f the control design strategy Γ fo r x 0 and w 0 given in (32 ) and (33) is lo wer-b ounded b y J P (Γ( A, B , D )) ≥ ( u j (0) + w j (0)) 2 + x j (1) 2 = ( ǫ 2 b + 1)(3 ǫ 2 b p 4 ǫ 2 b + 1 + 5 ǫ 2 b + 4 ǫ 4 b + p 4 ǫ 2 b + 1 + 1 ) 2 ǫ 4 b . 13 ࢝ ሺ ૙ ሻ ൌ ࢝ ૙ ࢞ ሺ ૙ ሻ ൌ ૙ ࢞ ࡷ ሺ ૙ ሻ ൌ ૙ ࢛ ሺ ૙ ሻ ൌ ૙ ࢝ ሺ ૚ ሻ ൌ ࡰ࢝ ૙ ࢞ ሺ ૚ ሻ ൌ ࡮࢝ ૙ ࢞ ࡷ ሺ ૚ ሻ ൌ ૙ ࢛ ሺ ૚ ሻ ൌ ࡰ ડ ࡮࢝ ૙ ࢝ ሺ ૛ ሻ ൌ ࡰ ૛ ࢝ ૙ ࢞ ሺ ૛ ሻ ൌ ሺ ࡭ ൅ ࡮ࡰ ડ ൅ ࡰ ሻ ࡮࢝ ૙ ࢞ ࡷ ሺ ૛ ሻ ൌ ࡮ ડ ࡮࢝ ૙ ࢛ ሺ ૛ ሻ ൌ ࡯ ડ ࡮ ડ ࢞ ሺ ૚ ሻ ൅ ࡰ ડ ࢞ ሺ ૛ ሻ ... Figure 2: Sta te evolution o f the closed-lo op system when x 0 = 0 . Therefore, for any Γ ∈ C , we hav e r P (Γ) ≥ lim r →∞ J P (Γ( A, B , D )) J P ( K ∗ ( P )) = 2 ǫ 2 b + 1 + p 4 ǫ 2 b + 1 2 ǫ 2 b . (34) Considering the fact that Γ ∆ also be lo ngs to C , the rest is a s imple c ombination of (34) and Lemma 3.3. Now, we can prove statement ( b ). The “if ” par t of the pr o of is done by constructing plants P = ( A, B , D , x 0 , w 0 ) ∈ P that satisfy J P (Γ( A, B , D )) > J P (Γ ∆ ( A, B , D )) for a ny cont r ol des ig n metho d Γ ∈ C \ { Γ ∆ } . F or the “only if ” part, we show tha t Γ Θ int r o duced later in (36) dominates Γ ∆ when G P has at least one sink. See [9, p.124] for the detailed pro of. Theorem 4.4 s hows tha t the deadbe at control design method Γ ∆ is an un- dominated minimizer of the comp etitive ratio r P ov er the se t of limited mo del information design metho ds C . 4.2 Plan t Graphs with Sinks In this section, w e study the case where there a re c ≥ 1 sinks in the plant g raph G P . By r enum ber ing the sinks as subsystems num b er n − c + 1 , . . . , n , the ma tr ix S P can be written as S P =  ( S P ) 11 0 ( q − c ) × ( c ) ( S P ) 21 ( S P ) 22  , (35) where ( S P ) 11 =    ( s P ) 11 · · · ( s P ) 1 ,n − c . . . . . . . . . ( s P ) n − c, 1 · · · ( s P ) n − c,n − c    , ( S P ) 21 =    ( s P ) n − c +1 , 1 · · · ( s P ) n − c +1 ,n − c . . . . . . . . . ( s P ) n, 1 · · · ( s P ) n,n − c    , and ( S P ) 22 = diag (( s P ) n − c +1 ,n − c +1 , . . . , ( s P ) nn ). F rom now on, without los s of g e nerality , we assume that the s tructure matrix is the one defined in (35). F or all plants P ∈ P , cont r ol design method Γ Θ is defined as Γ Θ ( A, B , D ) =  D B − 1 D F ( A, B ) A − B − 1 D 2 I B − 1 ( F ( A, B ) − I ) A − B − 1 D  (36) 14 where F ( A, B ) = diag (0 , . . . , 0 , f n − c +1 ( A, B ) , . . . , f n ( A, B )) , and f ( A, B ) = 2 b 2 ii + a 2 ii + 1 + p ( a 2 ii + b 2 ii ) 2 + 2( b 2 ii − a 2 ii ) + 1 for all n − c + 1 ≤ i ≤ n . The con tr o l desig n strateg y Γ Θ applies the dea dbe a t to every s ubsystem that is not a sink and, for every sink, applies the same o ptimal control la w as if the no de were decoupled fr om the rest of the graph. Lemma 4.5 L et the plant gr aph G P c ontain no isolate d no de and at le ast one sink and G K ⊇ G P . Then, t he c omp etitive r atio of the design metho d Γ Θ intr o- duc e d in (36) is r P (Γ Θ ) =  (2 ǫ 2 b + 1 + p 4 ǫ 2 b + 1) / (2 ǫ 2 b ) , if ( S P ) 11 is n ot diagonal , 1 , if ( S P ) 11 = 0 & ( S P ) 22 = 0 . Pr o of: Based the pr o of of the “only if ” part o f s tatement ( b ) of Theore m 4 .4, we know that J P (Γ Θ ( A, B , D )) ≤ J P (Γ ∆ ( A, B , D )) , for all P = ( A, B , D , x 0 , w 0 ) ∈ P and as a result r P (Γ Θ ) = sup P ∈P J P (Γ Θ ( A, B , D )) J P ( K ∗ ( P )) ≤ s up P ∈P J P (Γ ∆ ( A, B , D )) J P ( K ∗ ( P )) ≤ 2 ǫ 2 b + 1 + p 4 ǫ 2 b + 1 2 ǫ 2 b . Now if ( S P ) 11 has an o ff-diagonal entry , then there exist 1 ≤ i , j ≤ n − c and i 6 = j such that ( s P ) j i 6 = 0 . Using the second part o f the pro of of Theorem 4.4, it is easy to see r P (Γ Θ ) ≥ 2 ǫ 2 b + 1 + p 4 ǫ 2 b + 1 2 ǫ 2 b , bec ause the control design Γ Θ acts like the deadb eat con trol design strateg y on that part of the system. Using b oth these inequalities pr ov es the statement. If ( S P ) 11 = 0 and ( S P ) 22 = 0, every matrix A with struc tur e matrix S P bec omes a nilpo tent matrix of degree tw o. Thus, a ccording to Le mma 4.1, we get that J P ( K ∗ ( P )) = J P ( K ∗ C ( P )), and based on the unique solution of the ass o ciated discre te algebra ic Riccati equa tion, for this pla nt, the optimal centralized c ontrol design is K ∗ C ( P ) =  D D ( I + B 2 ) − 1 B − 1 A − B − 1 D 2 I − ( I + B 2 ) − 1 B A − B − 1 D  , which is exa ctly equal to Γ Θ ( A, B , D ). Th us, r P (Γ Θ ) = 1 . Theorem 4. 6 L et t he plant gr aph G P c ontain no isolate d no de and c ontain at le ast one sink, the design gr aph G C b e a total ly disc onn e cte d gr aph, and G K ⊇ G P . Then, the fol lowing statements hold: 15 (a) The c omp etitive r atio of any c ontr ol design str ate gy Γ ∈ C satisfies r P (Γ) ≥ (2 ǫ 2 b + 1 + p 4 ǫ 2 b + 1) / (2 ǫ 2 b ) , if ( S P ) 11 is n ot diagonal. (b) The c ontr ol design metho d Γ Θ is undominate d by al l limite d mo del informa- tion c ontr ol design metho ds in C . Pr o of: Firs t, we prove statement ( a ). Suppos e tha t ( S P ) 11 6 = 0 and ( S P ) 11 is not a dia gonal ma tr ix, then there exist 1 ≤ i , j ≤ n − c and i 6 = j such that ( s P ) j i 6 = 0. Consider the family of matrices A ( r ) defined by A ( r ) = re j e T i . Based on Lemma 4.3, if we wan t to hav e a b ounded comp etitive ratio, the control des ign strategy should satisfy r + b j j ( d Γ ) j i ( A ( r ) , B , D ) = 0 (b ecause no de 1 ≤ j ≤ n − c is not a sink). The rest o f the pr o of is similar to the pr o of of Theorem 4.4. See [9, p.130] for the detailed pro of of statement ( b ). Combining Lemma 4.5 and Theorem 4.6 illustrates that if ( S P ) 11 6 = 0 is not dia g onal, the co nt r ol design metho d Γ Θ has the smalle st ratio achiev a ble by limited mo del infor mation control metho ds. Thus, it is a so lution to the problem (5). F urthermo re, if ( S P ) 11 and ( S P ) 22 are both zer o, then Γ Θ bec omes equal to K ∗ . This shows that Γ Θ is a solution to the problem (5) in this case to o. The rest of the cases are still op en. 5 Design G raph Influence on Ac hiev able P erfor- mance In the previo us sectio n, we solved the o ptimal control design under limited mo del information when G C is a totally disconnected graph. In this section, w e study the necess ary a mount of information nee ded in ea ch subsystem to ensure the existence o f a limited mo del informa tio n control desig n strategy with a b etter comp etitive r atio than Γ ∆ and Γ Θ . Theorem 5. 1 L et the plant gr aph G P and t he design gr aph G C b e given and G K ⊇ G P . Then, we have r P (Γ) ≥ (2 ǫ 2 b + 1 + p 4 ǫ 2 b + 1) / (2 ǫ 2 b ) for al l Γ ∈ C if G P c ontains the p ath i → j → ℓ with distinct no des i , j , and ℓ while ( ℓ, j ) / ∈ E C . Pr o of: See [9, p.132] for the detailed pro of. 6 Conclusions W e studied the design of o ptimal dynamic disturba nce acc ommo dation c o n- trollers under limited plant mo del informatio n. 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