Optimal Structured Static State-Feedback Control Design with Limited Model Information for Fully-Actuated Systems

Optimal Structured Static State-F eedbac k Con trol Design with Limited Mo del Information for F ully-Actuated Systems ∗ † F arhad F arokhi ‡ , C ´ edric Langb ort § , and Karl H. Johansson ‡ Abstract W e i ntroduce the family of li mited mod el information con trol design metho d s, whic h construct controllers by ac cessing the plant’s model in a constrained wa y , according to a given design graph. W e investi gate the closed-loop p erformance ac hiev able by such control design metho ds for fully-actuated discrete-time linear time-inv ariant systems, under a sepa- rable quadratic cost. W e restrict our stud y to control design meth ods whic h pro duce structured static state feedback controllers, where each sub control ler can at least access the state measurements of th ose subsys- tems that aﬀect its corresp onding subsystem. W e compute the optimal contro l design strategy (in terms o f the competitive ratio and dominatio n metrics) when the con trol designer has access to the local m o del informa- tion and the global interc onnection structure of the plant-to-b e-control led. Lastly , w e study the trade-oﬀ b etw een the amount of model informati on exploited b y a con trol design method and the b est closed-loop p erformance (in terms of t he competitive ratio) of control lers it can pro duce. 1 In tro duction Many mo dern control systems, such as aircr aft and satellite for mation [2, 3], a u- tomated highw ays and other shar e d infrastructure [4, 5], ﬂexible structures [6], and supply chains [7], co ns ist o f a la rge n um ber o f subsystems coupled through their p erfor mance g oals or sys tem dynamics. When reg ulating this kind of pla nt, it is o ften a dv antageous to ado pt a distributed control architecture, in which the controller itself is comp osed of interconnected sub controllers, each of whic h accesses a strict subset of the pla n t’s output. Se veral control synthesis metho ds ∗ An early version of this pap er is accepted for presen tation at the Am erican Con trol Con- ference, 2011 [1]. † The work of F. F arokhi and K . H. Johansson were supp orted by gran ts from the Swed ish Researc h Council and the Knut and Alice W allen b erg F oundation. The work of C. Langb ort wa s supp orted, in part, by the US Air F orce Oﬃce of Scientiﬁc Researc h (AFOSR) under gran t num ber M URI F A 9550-10-1-0573. ‡ F. F arokhi and K. H. Johansson ar e with ACCESS Linnaeus Center, School of Electrical Engineering, KTH-Roy al Institute of T ec hnology , SE-100 44 Stockholm, Sweden. E-mails: { farokhi,k allej } @ee.kth.se § C. Langbort is wi th the Department of Aerospace Engineering and the Co ordinated Sci- ence Lab oratory , Universit y of Illinois at Ur bana-Champaign, Illinois, USA. E-mail: lang- bort@i llinois.edu 1 hav e b een prop os e d ov er the past decades tha t res ult in distributed controllers of this form, with v ar ious t ypes of clo sed-lo op stabilit y and per formance guar- antees (e.g., [8–16]). Most recently , the to ols pr esented in [17] and [1 8] r e vealed how to explo it the sp eciﬁc interconnection of class es of plants (the so-ca lled quadratically in v ariant systems) to for mu late co n vex optimization pro blems for the design of structured H ∞ - and H 2 - o ptimal controllers. A common threa d in this part of the liter ature is the assumption that, even though the co ntroller is structured, its desig n can b e per formed in a centralized fashion, with full knowl- edge of the plant mo del. How e ver, in some applicatio ns (describ ed in mor e detail in the next para graph), this ass umption is no t alwa y s w arranted, a s the des ign of each sub controller may need to b e carr ied out by a diﬀerent control designe r , with no a ccess to the g lobal mo de l of the plant, a lthough its interconnection structure and the c o mmon clos ed-lo op cost function to b e minimized are public knowledge. This class o f problems, which we refer to as “ limited mo del infor- mation co n trol des ign pro blems”, is the main ob ject of interest in the present pap er. Limited mo del information control design o ccurs naturally in co n texts where the subsys tems b elong to diﬀeren t ent ities, which may co nsider their model in- formation priv ate and may th us b e relucta n t to sha re it with others . In this case, the designe r s may hav e to res ort to “ communication-less” strateg ies in which sub controller K i depe nds solely on the description o f subsystem i ’s mo del. This ca se is w ell illustrated b y supply chains, where the economic incen tives of comp eting companies might limit the exchange of mo del information (such as, inv entory volume, transp ortation eﬃciency , r aw material source s , and dec is ion pro cess) inside a layer of the chain. Another r e ason for using communication-less strategies in more g eneral des ig n situatio ns, even when the circula tion of plan t information is no t r estricted a prio ri, is that the resulting sub controller K i do es not need to be mo diﬁed if the characteristics of a particular subsystem, which is no t directly connected to subsystem i , v ar y . F or instance, co ns ider a chemical plant in the pr o cess indus tr y , with thousands of lo cal controllers. In s uch a large-s cale sy stem, the tuning of ea ch lo cal controller should no t requir e model parameters from other parts of the system so as to simplify ma in tenance and limit controller c o mplexity . Note that engineers often implement thes e lar ge- scale systems as a whole using commer cially av ailable pre - designed mo dules . These mo dules ar e designed, in a dv ance, with no prior knowledge o f their p os- sible use or future o per ating condition. This la ck o f av a ilability of the co mplete mo del of the plant, at the time o f the design, constra ins the designer to only use its o wn mo del pa r ameters in eac h mo dule’s control des ig n. Control design bas ed on uncer tain plant mo del information is a cla ssic topic in the robust control liter ature [19–22]. How ever, designing a n optimal c ontroller without a global mo del is diﬀerent from a ro bust control problem. In optimal control design with limited mo del information, subsys tems do not hav e any prior information ab out the other subsystems’ mo del; i.e ., there is no nominal mo del for the desig n pro cedure and there is no bo und on the mo del uncertain- ties. There hav e b een so me interesting a pproaches for tackling this problem. F or instance, refer ences [23–26] introduced metho ds for designing s ub- optimal decentralized controllers without a glo bal dynamical mo del o f the sys tem. In these pap ers, the author s assume that the larg e-scale system to b e c o nt rolled consists o f an interconnection of weakly coupled s ubsystems. They des ig n an optimal co nt roller for e a ch subsystem using only the co rresp onding lo cal mo del, 2 and co nnect the obta ine d sub controllers to construct a global controller. The y show tha t, when coupling is neg lig ible, this latter co n troller is sa tis fa ctory in terms of close d- lo op s tabilit y and p erforma nce . How ever, as coupling strength increases, even closed-lo op stability guarantees are lost. O ther appro aches such as [5, 7] are based on receding horizo n co ntrol and use decomp osition metho ds to solve each step’s optimization problem in a decentralized manner with only limited information exchange b et ween subsystems. What is missing from the literature, howev er, is a rig orous characterizatio n of the be st clos ed-lo op p er- formance that can b e attained through limited mo del informatio n design and, a study of the trade oﬀ b etw een the closed-lo o p p erforma nce and the amo un t of exchanged information. W e tackle this question in the pres ent pap er fo r a particular c la ss of systems (namely , the set o f fully-actuated discrete-time lin- ear time- inv ariant dynamica l sy stems) and a pa rticular class of control laws (namely , the set of structured linear static state feedback controllers where each sub c ontroller can at least access the s tate measurements of those subsystems that aﬀect its corre spo nding subsystem). In this pap er, w e study the pr op erties of limited model informatio n control design metho ds. W e inv estigate the relationship betw een the a mount of plant information av ailable to the designer s, the nature o f the plant interconnection graph, and the quality (measured by the closed- lo op control goal) of controllers that can b e construc ted using their knowledge. T o do so , w e lo o k at limited mo del information a nd communication-less control design metho ds a s b elong ing to a specia l clas s of maps b etw een the plant a nd con troller sets, and make use of the comp etitive ratio a nd domination metrics introduced in [27] to charac- terize their intrinsic limitations. T o the b est of our k nowledge, there a re no other metrics sp eciﬁca lly tuned to co ntrol design metho ds. W e address muc h more general clas ses o f subsystems and of limitatio ns on the mo del information av aila ble to the desig ner than is done in [27]. Sp eciﬁca lly , we co nsider limited mo del information structured static state-feedback control design for intercon- nections of fully-a ctuated (i.e., with inv ertible B - matrix) discrete-time linear time-inv aria nt subsys tems with quadratic separable (i.e., with blo ck dia gonal Q - and R -matrices) cost function. Our choice of such a cost function is motiv ated by our interest in applications such a s p ow er grids [28–3 1] a nd [5, Chs. 5,1 0], supply chains [7, 32], a nd water level c ontrol [5 , Ch. 18 ], which hav e b een s hown to b e well-modeled by dynamically-coupled but co st-decoupled interconnected systems. W e show in the last section of the pap er that the assumption on the B -ma trix ca n b e partia lly remov ed for the sinks (i.e., subsystems that cannot aﬀect a ny other s ubsystem) in the plant gr aph. W e inv estiga te the b est clo sed-lo op per formance achiev able by s tructured static state feedback controllers constructed by limited mo del informa tion design strategies. W e show that the r esult depends cruc ia lly on the plant gra ph and the control gra ph. In the case where the plant g raph contains no s ink and the control graph is a sup ergr aph o f the plant graph, we extend the fact pr ov en in [2 7] that the deadb eat strategy is the bes t communication-less control desig n metho d. How ever, the deadb eat control desig n strateg y is do minated when the plant graph has sinks, and we exhibit a better , undo minated, communication-less control design metho d, which, a lthough having the same comp etitive r atio as the deadb eat co n trol desig n stra tegy , takes adv antage of the knowledge of the sinks’ lo cation to achieve a b etter clo s ed-lo op p erformanc e in av era ge. W e characterize the amount of mo del infor mation nee ded to achieve better comp etitive r atio than 3 the deadb eat control design strategy . This amount of information is expresse d in terms of prop er ties of the des ig n graph; a directed gra ph which indicates the dep endency of each subsystem’s controller on diﬀerent par ts of the global dynamical mo del. This pap er is org anized as follows. After formulating the problem of interest and deﬁning the p erformance metrics in Section 2, we character ize the b est communication-less control de s ign metho d accor ding to b oth comp etitive ratio and domination metrics in Section 3. In Section 4, we s how that a chieving a strictly b etter comp etitive ra tio than these control desig n metho ds requir es a complete design graph when the plan t graph is itself complete. Finally , we end with a discussion on extens io ns in Section 5 a nd the conc lus ions in Section 6. 1.1 Notation Sets will b e denoted by calligraphic letters, such as P and A . If A is a subset of M then A c is the complemen t of A in M , i.e., M \ A . Matrices ar e denoted by ca pital roman letters s uch as A . A j will denote the j th row of A . A ij denotes a sub-ma trix of matrix A , the dimension and the po sition of whic h will b e deﬁned in the text. The entry in the i th row and the j th column o f the matrix A is a ij . Let S n ++ ( S n + ) b e the se t of symmetric p ositive deﬁnite (p os itive semideﬁnite) matrices in R n × n . A > ( ≥ )0 means that the s y mmetric matrix A ∈ R n × n is po sitive deﬁnite (po sitive semideﬁnite) and A > ( ≥ ) B mea ns that A − B > ( ≥ )0. λ ( Y ) a nd ¯ λ ( Y ) denote the s mallest a nd the la rgest eig env alues of the matrix Y , r esp ectively . Similar ly , σ ( Y ) and ¯ σ ( Y ) deno te the smallest and the lar gest singular v a lues of the matrix Y , r esp ectively . V ector e i denotes the column- vector with all entries zero exce pt the i th ent ry , which is equal to one. All g raphs conside r ed in this pap er ar e direc ted, p ossibly with self-lo ops, with vertex set { 1 , ..., q } for some p os itive integer q . If G = ( { 1 , ..., q } , E ) is a directed graph, we say that i is a sink if there do es not exist j 6 = i such that ( i, j ) ∈ E . A lo op of length t in G is a set of distinct vertices { i 1 , ..., i t } such that ( i t , i 1 ) ∈ E and ( i p , i p +1 ) ∈ E for all 1 ≤ p ≤ t − 1. W e will sometimes refer to this lo op as ( i 1 → i 2 → · · · → i t → i 1 ). The adjacency matrix S of graph G is the q × q matrix whose en tr ies satisfy s ij =  1 if ( j, i ) ∈ E 0 otherwise. Since the set of vertices is ﬁx ed here, a subgraph of G is a graph whose edg e set is a subs et of the edge set of G a nd a sup erg raph of G is a graph of whic h G is a subgraph. W e use the nota tio n G ′ ⊇ G to indicate that G ′ is a super graph of G . 2 Con trol Design with Limited Mo del In forma- tion In this section, we intro duce the system model a nd the problem under consid- eration, but ﬁrst, we present a simple illustrative example. 4 2.1 Illustrativ e Example Consider a discrete-time linear time-in v ariant dynamical s ystem comp osed of three subsystems represented in state-space form as   x 1 ( k + 1) x 2 ( k + 1) x 3 ( k + 1)   =   a 11 a 12 0 a 21 a 22 a 23 0 a 32 a 33     x 1 ( k ) x 2 ( k ) x 3 ( k )   +   b 11 u 1 ( k ) b 22 u 2 ( k ) b 33 u 3 ( k )   , where, for each subsystem i , x i ( k ) ∈ R is the state and u i ( k ) ∈ R is the con- trol signa l. This system, which is illus trated in Figure 1, is a simple net worked control sys tem. Net work e d control systems have several impo rtant character- istics. First, they are often distributed g e ographica lly . Therefor e, it is natural to assume that a given subsystem can only inﬂuence its neig h bo ring subsys- tems. W e capture this fa c t using a direc ted g raph called the plant gra ph like the one pres en ted in Figure 2( a ) for this exa mple. This sta r graph corresp onds to applications like unmanned aerial vehicles formation, plato on of vehicles, and comp osite forma tions of p ow er systems [33, 34]. Second, a ny communication medium that we use to transmit the sensor measurements and actuation signals in netw orked co n trol systems bring s some limitations. F or insta nce, every communication netw o rk has ba nd-limited chan- nels. Therefore, when designing sub controllers, it mig h t not make sense to assume that it can instan taneously a ccess full state meas urements of the plant. The sta te measurement av ailability in this example is   u 1 ( k ) u 2 ( k ) u 3 ( k )   =   k 11 k 12 0 k 21 k 22 k 23 0 k 32 k 33     x 1 ( k ) x 2 ( k ) x 3 ( k )   . W e use a con trol g raph to characteriz e the co n troller structure. Control graph G K in Figure 2( b ) re pr esents the s ta te-measurement av ailability in this exam- ple. It corres po nds to the ca s e where neighbor ing s ubsystems tra nsmit their state-measure men ts to eac h other, whic h is common for unmanned aerial vehi- cles formation, autono mous g round vehicles plato ons , and biological system of particles [2, 3, 35, 36]. Finally , in larg e -scale dy na mical systems, it might b e ex tremely diﬃcult (if not imp ossible ) to iden tify all system parameters and up date them globally . One can only hop e that the des igner ha s access to the lo cal par ameter v ariations and up date the corr esp onding sub controller based on them. Ther efore, it makes sense to assume that ea ch lo cal co nt roller only ha s access to mo del infor ma tion from its co rresp onding subsystem; i.e., desig ner of sub controller i uses only { a i 1 , a i 2 , a i 3 } in the design pr o cedure [ k i 1 k i 2 k i 3 ] = Γ i ([ a i 1 a i 2 a i 3 ] , b ii ) , where Γ i : R 3 × R → R 3 is the control design map. Note that the blo ck-diagra m in Figure 1 do es not sp ecify Γ. W e will use a directed g r aph called the desig n graph to capture structural prop erties of Γ. In the rest of this section, we formalize the ab ov e notions for more g eneral design pr oblems. 5 2.2 Plan t Mo del Let a gr aph G P = ( { 1 , ..., q } , E P ) b e giv en, with adjac ency matrix S P ∈ { 0 , 1 } q × q . W e deﬁne the following set o f matrices a s so ciated with S P : A ( S P ) = { A ∈ R n × n | A ij = 0 ∈ R n i × n j for all 1 ≤ i, j ≤ q s uch that ( s P ) ij = 0 } , (1) where fo r e ach 1 ≤ i ≤ q , in teger n um ber n i is the dimensio n o f s ubsystem i . Implicit in these deﬁnitions is the fact that P q i =1 n i = n . Also, for a g iven scalar ǫ > 0, w e let B ( ǫ ) = { B ∈ R n × n | σ ( B ) ≥ ǫ, B ij = 0 ∈ R n i × n j for all 1 ≤ i 6 = j ≤ q } . (2) The set B ( ǫ ) deﬁned in (2) is made of in vertible blo ck-diagonal square matrices since σ ( B ) ≥ ǫ > 0 for each matrix B ∈ B ( ǫ ) ⊆ R n × n . With these deﬁnitions, we can introduce the set P of plants of interest as the space of all discrete-time linear time-inv ariant dynamical systems of the form x ( k + 1 ) = Ax ( k ) + B u ( k ) ; x (0) = x 0 , (3) with A ∈ A ( S P ), B ∈ B ( ǫ ), and x 0 ∈ R n . Clea rly P is isomor ph to A ( S P ) × B ( ǫ ) × R n and, slightly abusing no tation, we will thus identify a plan t P ∈ P with the corresp onding triple ( A, B , x 0 ). A plant P ∈ P can b e thought of as the in terconnection of q subsystems, with the structur e of the interconnection sp eciﬁed by the graph G P (i.e., subsystem j ’s output feeds into subsy stem i o nly if ( j, i ) ∈ E P ). As a consequence, we refer to G P as the “plant g raph”. W e will denote the or dered set of state indices per taining to subsystem i as I i ; i.e., I i := (1 + P i − 1 j =1 n j , . . . , n i + P i − 1 j =1 n j ). F or subsystem i , state vector and input v ector ar e de ﬁned as x i =  x ℓ 1 · · · x ℓ n i  T , u i =  u ℓ 1 · · · u ℓ n i  T where the order ed set of indices ( ℓ 1 , . . . , ℓ n i ) ≡ I i , and its dynamics is sp eciﬁed by x i ( k + 1) = q X j =1 A ij x j ( k ) + B ii u i ( k ) . According to the sp eciﬁc s tr ucture of B ( ǫ ) given in (2 ), ea ch subsy stem is fully- actuated, with as many input as sta tes, and co nt rollable in one time-step. Pos- sible ge ner alization o f the results to a (r estricted) family of under-actuated sy s - tems is discussed in Sectio n 5. Figure 2( a ) shows an example o f a pla nt g raph G p . Each no de represents a subsystem of the s ystem. F or instanc e , the second subsy stem in this example may a ﬀect the ﬁr st subsystem and the third subsystem; i.e., sub-matr ices A 12 and A 32 can b e no nz e r o. The self-lo op for the seco nd subsystem shows that A 22 may be non-ze r o. The plant g raph G P in Figure 2( a ) do es not contain any sink. In contrast, the ﬁrs t subsys tem of the plant graph G ′ P in Fig ure 2( a ′ ) is a sink. The co n trol graph G K is intro duced in the next subse ction. 6 ( ) P ( ) K ( ) ! " #$ % $ " $# % # " ## % # & # # % $ ' ! % ! ! & ! " !! % ! % $ ' $ % # ' # % ! $ " $$ % $ & $ " !$ % $ " $! % ! % # % $ Figure 1: P hysical in terconnection b et ween diﬀeren t subsys tems and controllers corres p onding to G P and G K in Figur es 2( a ) and 2( b ), r esp ectively . ( ) a P G ( ) b K G ( ) c C G ( ) a P G ( ) b K G ( ) c C G 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 Figure 2: G P and G ′ P are examples o f plan t gra phs, G K and G ′ K are examples of co n trol graphs, and G C and G ′ C are exa mples of desig n graphs. 2.3 Con t r oller Mo del Let a control g raph G K be given, with adjacency matrix S K . The control laws of in terest in this paper are linear static s ta te-feedback control laws of the form u ( k ) = K x ( k ) , where K ∈ K ( S K ) = { K ∈ R n × n | K ij = 0 ∈ R n i × n j for all 1 ≤ i, j ≤ q such that ( s K ) ij = 0 } . (4) In particular , when G K is a complete graph, K ( S K ) = R n × n , while, if G K is to- tally disconnected with self-lo ops, K ( S K ) repre s ent s the set of fully-decentralized controllers. When adjacency matrix S K is not re le v ant or ca n b e deduced from context, we refer to the set of controllers as K . An example of a control gra ph G K is given in Figure 2( b ). Ea ch no de repre- sents a subsystem-controller pair of the ov erall system. F o r instance, Figure 2( b ) shows that the ﬁr st subsystem’s controller can use state measur e ments of the second s ubsystem be sides its own state measur emen ts. Figure 2( b ′ ) shows a complete gra ph, which indicates that ea ch subsystem has access to full state measurements of all other subsystems; i.e ., K ( S K ) = R n × n . 2.4 Linear State F eedbac k Control Design Metho ds A control de s ign metho d Γ is a map from the set o f plants P to the set o f controllers K . Just like pla nts and c o nt rollers, a control design metho d c an 7 exhibit structure which, in turn, can b e captured by a design gr aph. Let a control desig n metho d Γ be partitioned ac c ording to subsys tems dimensions a s Γ =    Γ 11 · · · Γ 1 q . . . . . . . . . Γ q 1 · · · Γ qq    (5) and a graph G C = ( { 1 , ..., q } , E C ) be given, w ith a djacency matrix S C . Each blo ck Γ ij represents a map A ( S P ) × B ( ǫ ) → R n i × n j . Co n trol design metho d Γ can b e further partitioned in the for m Γ =    γ 11 · · · γ 1 n . . . . . . . . . γ n 1 · · · γ nn    , where ea ch γ ij is a map A ( S P ) × B ( ǫ ) → R . W e say that Γ has structure G C if, for all i , the map [Γ i 1 · · · Γ iq ] is only a function of { [ A j 1 · · · A j q ] , B j j | ( s C ) ij 6 = 0 } . (6) In words, a co ntrol design metho d has structure G C if and o nly if, for all i , the sub controller of s ubsystem i is constructed with knowledge of the plant mo del of only thos e subsystems j such that ( j, i ) ∈ E C . The s et of all control design methods with str ucture G C will be denoted by C . In the particular case where G C is the tota lly disconnected gra ph with self-lo o ps (meaning that every no de in the gra ph ha s a se lf-lo op; i.e, S C = I q ), we say that a control desig n metho d in C is “ communication-less”, so as to capture the fact that subsystem i ’s s ubco n troller is cons tructed with no information coming fr om (and, henc e , no communication with) any o ther subsystem j , j 6 = i . Therefore, the design graph indicates knowledge (or lack ther e of ) of entire blo ck rows in the a ggreg ate system matrix. When G C is not a complete graph, we refer to Γ ∈ C a s b eing “a limited model information co ntrol design metho d”. Note that C can b e cons ide r ed a s a subset of the set of functions from A ( S P ) × B ( ǫ ) to K ( S K ), since a desig n metho d with structure G C is no t a function o f initial state x 0 . Hence, when Γ ∈ C we will wr ite Γ( A, B ) ins tead of Γ( P ) for plant P = ( A, B , x 0 ) ∈ P . An example of a design g r aph G C is g iven in Figure 2 ( c ). Ea ch no de repre- sents a subsy stem-controller pair o f the ov er all sy stem. F or ins tance, G C shows that the third subsystem’s mo del is av ailable to the desig ner o f the second sub- system’s controller but no t the ﬁr s t subsys tem’s mo del. Figur e 2( c ′ ) shows a fully disco nnected design gr aph with se lf- lo ops G ′ C . A lo ca l de s igner in this case can only r ely on the mo del of its corresp onding subsystem; i.e., the design strategy is commun ication-less. 2.5 P erformance Metrics The goal of this pap er is to inv es tig ate the inﬂuence o f the plant and design graph on the pr op erties of controllers c onstructed b y limited model info r mation control design methods . T o this end, w e will use tw o pe rformance metrics for control desig n methods. These p erfor mance metrics are adapted from the 8 notions of co mpetitive ra tio and domination int ro duced in [27], so as to take plant, controller, and control des ign structures into account. F ollowing the approach in [27], we start by asso c ia ting a c losed-lo op p erformance cr iterion to each plant P = ( A, B , x 0 ) ∈ P and controller K ∈ K . As explained in the int ro duction, we ar e particularly interested in dynamica lly-coupled but cost- decoupled systems in this pa per , hence, w e use a c ost of the form J P ( K ) = ∞ X k =1 x ( k ) T Qx ( k ) + ∞ X k =0 u ( k ) T Ru ( k ) , (7) where Q ∈ S n ++ and R ∈ S n ++ are blo ck diag o nal ma trices, with each dia gonal blo ck entry belo nging to S n i ++ . No te that the summation in the ﬁrst ter m on the right-hand s ide of (7) starts from k = 1. This is without lo ss of generality as the removed term x (0) T Qx (0) is not a function of the controller. W e make the following tw o sta nding assumptions: Assumption 2 .1. Q = R = I . This is without loss of genera lit y beca use the change of v ariables ( ¯ x , ¯ u ) = ( Q 1 / 2 x, R 1 / 2 u ) transfor ms the perfor mance criterio n and state space repr esen- tation into J P ( K ) = ∞ X k =1 ¯ x ( k ) T ¯ x ( k ) + ∞ X k =0 ¯ u ( k ) T ¯ u ( k ) , (8) and ¯ x ( k + 1) = Q 1 / 2 AQ − 1 / 2 ¯ x ( k ) + Q 1 / 2 B R − 1 / 2 ¯ u ( k ) = ¯ A ¯ x ( k ) + ¯ B ¯ u ( k ) , resp ectively , without aﬀecting the plant, c ontrol, or des ign g raph (due to the blo ck diag onal structure o f Q a nd R ). Assumption 2.2. The set of matric es B ( ǫ ) is r eplac e d with the set of diag onal matric es with diagonal entries gr e ater t han or e qual to ǫ . This assumption is without lo ss of g enerality . Indeed, consider a plant P = ( A, B , x 0 ) ∈ P . Every sub-system’s B ii matrix has a sing ular v alue decom- po sition B ii = U ii Σ ii V T ii with Σ ii ≥ ǫI n i × n i . Combining these singular v alue decomp ositions together results in a singula r v alue decomp ositio n for matrix B = U Σ V T where U = diag( U 11 , U 22 , · · · , U qq ), Σ = diag (Σ 11 , Σ 22 , · · · , Σ qq ), and V = diag( V 11 , V 22 , · · · , V qq ). Deﬁning ¯ x ( k ) = U T x ( k ) and ¯ u ( k ) = V T u ( k ) results in ¯ x ( k + 1) = U T AU ¯ x ( k ) + U T B V ¯ u ( k ) , where U T B V is diagonal. Beca use of the blo ck dia gonal structure of ma trices U and V , the change of v ar iables ( A, B , x 0 ) 7→ ( U T AU, U T B V , U T x 0 ) do es not aﬀect the plant, control, or design gra ph. In addition, the cos t function b ecomes J P ( K ) = ∞ X k =1 ¯ x ( k ) T U T U ¯ x ( k ) + ∞ X k =0 ¯ u ( k ) T V T V ¯ u ( k ) = ∞ X k =1 ¯ x ( k ) T ¯ x ( k ) + ∞ X k =0 ¯ u ( k ) T ¯ u ( k ) , 9 which is of the for m (8), beca use b oth U and V are unitary matr ices. W e a re now r e ady to deﬁne the p erfor ma nce metrics o f int erest in this pap er. Deﬁnition 2.3. (Comp etitive R atio) L et a plant gr aph G P , c ontr ol gr aph G K and c onstant ǫ > 0 b e given. Assume that, for every plant P ∈ P , t her e exist s an optimal c ontr ol 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 ,x 0 ) ∈P J P (Γ( A, B )) J P ( K ∗ ( P )) , with the c onvention that “ 0 0 ” e quals one. Note that the mapping K ∗ : P → K ∗ ( P ) is not itself requir ed to lie in the set C , as every comp onent o f the optimal c o nt roller may dep end on all en tries of the model matrices A and B . Deﬁnition 2 . 4. (Domination) A c ontr ol design met ho d Γ is said to dominate another c ontr ol design metho d Γ ′ if J P (Γ( A, B )) ≤ J P (Γ ′ ( A, B )) , ∀ P = ( A, B , x 0 ) ∈ P , (9) 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 that satisﬁes (9), we say that Γ ′ is undomi- nate d in C for plants in P . 2.6 Problem F orm ulation With the deﬁnitions o f the previous subsections in ha nd, we can refor m ulate the main question of this paper r egarding the connection b et ween closed-lo o p per formance, plant structure, and limited mo de l information control design as follows. F or a given plant graph, control gra ph, and desig n gr aph, we would like to determine arg min Γ ∈C r P (Γ) . (10) Since several design methods may achiev e this minim um, we are int erested in determining which ones o f these s trategies are undominate d . In [27], this pr oblem was so lved in the case when G P and G K are co mplete graphs, G C is a totally disconnected gra ph with s elf-lo ops (i.e., S C = I q ), and B ( ǫ ) is repla ced with sing leton { I n } . In this pap er, we inv estig a te the role of more ge ne r al plant a nd design graphs. W e a lso extend the results in [27] for scalar subsy s tems to subsystems of arbitr ary order n i ≥ 1 , 1 ≤ i ≤ q . 3 Plan t Graph Inﬂuence on Ac h iev able P erfor- mance In this section, we study the relatio ns hip b e tw een the plant graph and the achiev a ble closed-lo op p erfor mance in terms of the co mpetitive ra tio and dom- ination. 10 Deﬁnition 3. 1. The de adb e at c ontr ol design metho d Γ ∆ : A ( S P ) × B ( ǫ ) → K is deﬁne d as Γ ∆ ( A, B ) = − B − 1 A, for al l P = ( A, B , x 0 ) ∈ P . This control design metho d is communication-less; i.e., the control design for the subsystem i is a function of the mo del o f subsys tem i only , b ecause subsystem i ’s controller g ain  Γ ∆ i 1 ( A, B ) · · · Γ ∆ iq ( A, B )  equals to B − 1 ii [ A i 1 · · · A iq ]. The name “deadb eat” comes from the fact that the closed- lo op system obtained b y applying con troller Γ ∆ ( A, B ) to plan t P = ( A, B , x 0 ) reaches the orig in in just one time-step [37]. Remark 3.2. Note that for the c ase wher e the c ontro l gr aph G K is a c omplete gr aph; i.e., K = R n × n , t her e exist s a c ontr ol ler K ∗ ( P ) satisfying t he assump- tions of D eﬁnition 2.3 for al l P ∈ P , namely, the optimal line ar quadr atic r e gulator which is indep endent of the initial c ondition of the plant. F or inc om- plete c ontr ol gr aphs, the optimal c ontr ol design str ate gy K ∗ ( P ) (if exists) m ight b e c ome a function of t he initial c ondition [38]. Henc e, we wil l use K ∗ ( A, B ) inste ad of K ∗ ( P ) when the c ont r ol gr aph G K is a c omplete gr aph for e ach plant P = ( A, B , x 0 ) ∈ P to emphasi ze t his fact. F rom Deﬁnition 2.3, the notation K ∗ ( P ) is reserved for the optimal con- trol design strategy for a n y given cont rol graph G K . In contrast, when G K is not the co mplete gr aph, we will refer to the optima l unstru ct ur e d co ntroller a s K ∗ C ( A, B ). Lemma 3.3 . L et the c ont r ol gr aph G K b e a c omplete gr aph. The c ost of the optimal c ontr ol design st r ate gy K ∗ is lower-b ounde d by J P ( K ∗ ( A, B )) ≥  σ 2 ( B ) σ 2 ( B ) + 1  J P (Γ ∆ ( A, B )) , for al l plants P = ( A, B , x 0 ) ∈ P . Pro of: See App endix A. Theorem 3.4 . L et t he 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 met ho d Γ ∆ is r P (Γ ∆ ) = 1 + 1 /ǫ 2 . Pro of: Irresp ective o f the control g raph G K and for all pla n ts P ∈ P , it is true that J P ( K ∗ C ( A, B )) ≤ J P ( K ∗ ( P )). Therefor e , we g e t J P (Γ ∆ ( A, B )) J P ( K ∗ ( P )) ≤ J P (Γ ∆ ( A, B )) J P ( K ∗ C ( A, B )) . (11) Now, using Lemma 3.3, w e know tha t J P (Γ ∆ ( A, B )) J P ( K ∗ C ( A, B )) ≤ 1 + 1 σ 2 ( B ) , (12) for all P = ( A, B , x 0 ) ∈ P . Combining (12) and (11) results in r P (Γ ∆ ) = sup P ∈P J P (Γ ∆ ( A, B )) J P ( K ∗ ( P )) ≤ 1 + 1 ǫ 2 . 11 T o show that this upp er bound is attained, let us pick i 1 ∈ I i and j 1 ∈ I j where 1 ≤ i 6 = j ≤ q and ( s P ) ij 6 = 0 (such indices i a nd j exist beca use plant gra ph G P has no iso lated node by a ssumption). Consider the system A = e i 1 e T j 1 and B = ǫI . The unique po sitive deﬁnite solution of the discrete algebraic Riccati equation A T X A − A T X B ( I + B T X B ) − 1 B T X A = X − I , (13) is X = I +[1 / (1+ ǫ 2 )] e j 1 e T j 1 . Cons equently , the centralized c ontroller K ∗ C ( A, B ) = − ǫ/ (1 + ǫ 2 ) e i 1 e T j 1 belo ngs to the set K ( S K ) b ecause G K ⊇ G P . Th us, we get J ( A,B ,e j 1 ) ( K ∗ ( A, B , e j 1 )) ≤ J ( A,B ,e j 1 ) ( K ∗ C ( A, B )) (14) since K ∗ ( P ) has a low er co st than a ny other co n troller in K ( S K ). O n the other hand, it is evident that J ( A,B ,e j 1 ) ( K ∗ C ( A, B )) ≤ J ( A,B ,e j 1 ) ( K ∗ ( A, B , e j 1 )) (15) bec ause the cent ralized controller ha s access to more state measurements. Us- ing (14) and (15) simultaneously res ults in J ( A,B ,e j 1 ) ( K ∗ ( A, B , e j 1 )) = J ( A,B ,e j 1 ) ( K ∗ C ( A, B )) = 1 / (1 + ǫ 2 ) . On the o ther hand Γ ∆ ( A, B ) = − [1 /ǫ ] e i 1 e T j 1 and J ( A,B ,e j 1 ) (Γ ∆ ( A, B )) = 1 /ǫ 2 . Therefore, r P (Γ ∆ ) = 1 + 1 /ǫ 2 . Remark 3. 5 . Consider the limite d mo del information design pr oblem given by the plant gr aph G P in Figur e 2( a ) and the c ontr ol gr aph G ′ K in Figur e 2( b ′ ). The or em 3.4 shows that, if we apply the de adb e at c ontr ol design str ate gy t o this p articular pr oblem, the p erformanc e of the de adb e at c ont r ol design str ate gy, at most, c an b e 1 + 1 /ǫ 2 times the c ost of the optimal c ontr ol design str ate gy K ∗ . F or instanc e, when B = { I } as in [27], we have 1 + 1 /ǫ 2 = 2 sinc e in this c ase ǫ = 1 . Ther efor e, the de adb e at c ontr ol design str ate gy is never worse than twic e the optimal c ontr ol ler in this c ase. Remark 3.6. Ther e is no loss of gener ality in assuming that t her e is no isolate d no de in the plant gr aph G P , s inc e it is always p ossible to design a c ontr ol ler for an isolate d su bsyst em without any mo del information ab out the other su bsystems and without imp acting c ost (7). In p articular, this implies that ther e ar e q ≥ 2 vertic es in the gr aph b e c ause for q = 1 t he only subsystem that ex ists is an isolate d no de in the plant gr aph. Remark 3.7. F or implementation of the de adb e at c ontr ol design stra te gy in e ach no de, we only n e e d the state me asur ements of t he neighb ors of that no de. F or the implementation of the optimal c ontr ol design str ate gy K ∗ when the c ontr ol gr aph has many m or e links than t he plant gr aph, the c ontr ol ler gain K ∗ ( P ) is not ne c essarily a sp arse m atrix. With this c ha racteriza tion of Γ ∆ in hand, w e are now ready to tac k le prob- lem (10). 12 3.1 First case: plan t graph G P with no sink In this subsection, we show that the deadb eat control metho d Γ ∆ is undomi- nated by c ommu nication-less control des ign methods for plants in P , when G P contains no sink. W e a ls o show that Γ ∆ exhibits the smallest p ossible compe ti- tive r a tio amo ng such control desig n metho ds. First, w e state the following tw o lemmas. Lemma 3.8. L et the plant gr aph G P c ontain no isolate d n o de, t he design gr aph G C b e a total ly disc onne cte d gr aph with self-lo ops, and G K ⊇ G P . A c ont ro l de- sign metho d Γ ∈ C has b ounde d c omp etitive r atio only if the fol lowing implic ation holds for al l 1 ≤ i ≤ q and al l j : a ℓj = 0 for al l ℓ ∈ I i ⇒ γ ℓj ( A, B ) = 0 for al l ℓ ∈ I i , wher e I i is the s et of indic es r elate d to su bsystem i ; i.e., I i = (1 + P i − 1 z =1 n z , . . . , n i + P i − 1 z =1 n z ) . Pro of: See App endix B. Lemma 3.9. L et the plant gr aph G P c ontain no isolate d n o de, t he design gr aph G C b e a total ly disc onne cte d gr aph with self-lo ops, and G K ⊇ G P . Assume the plant gr aph G P has at le ast one lo op. Then, r P (Γ) ≥ 1 + 1 /ǫ 2 (16) for al l limite d mo del information c ontr ol design metho d Γ in C . Pro of: See App endix C. Using these t wo lemmas, we ar e rea dy to state and prove one o f the main theorems in this paper and, as a result, ﬁnd the s olution to pro blem (10) when the plant graph G P contains no sink. Theorem 3. 10. L et the plant gr aph G P c ontain no isolate d no de and no sink, the design gr aph G C b e a total ly disc onne cte d gr aph with self-lo ops, and G K ⊇ G P . Then the c omp etitive r atio of any c ontr ol design st r ate gy Γ ∈ C satisﬁes r P (Γ) ≥ 1 + 1 /ǫ 2 . Pro of: F r o m Lemma 1 .4 .23 in [39], we know that a directed g raph with no sink m ust ha ve at le ast one lo o p. Hence G P m ust cont ain a lo o p. The result then follows from Lemma 3.9. Remark 3.11. The or em 3.10 shows that r P (Γ) ≥ r P (Γ ∆ ) for any c ontr ol design str ate gy Γ ∈ C , and as a r esult the de adb e at c ontr ol design metho d Γ ∆ b e c omes a minimizer of the c omp etitive r atio function r P over the set of c ommu nic ation- less design metho ds. W e now turn our attent ion to domination prop erties of the deadb eat control design stra tegy . Lemma 3.12. L et t he 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 with self-lo ops, and G K ⊇ G P . The de adb e at c ontr ol design s t r ate gy Γ ∆ is undominate d, if t her e is no sink in the plant gr aph G P . 13 Pro of: See App endix D. The following theor e m shows that the dea dbe a t control desig n stra tegy is undominated by co mmunication-less design metho ds if a nd only if the plant graph G P has no sink . It thus provides a g o o d tra de-oﬀ b etw een worst-case and av er age p er formance. Theorem 3.13. L et the plant gr aph G P c ontain n o isolate d no de, the design gr aph G C b e a total ly disc onne cte d gr aph with s elf-lo ops, and G K ⊇ G P . Then the de adb e at c ontr ol design metho d Γ ∆ is undominate d in C for plants in P if and only if the plant gr aph G P has no sink. Pro of: Pr o of of the “if ” par t of the theorem, is giv en by Lemma 3.12. F or ea se of notation in this pro of, we use [Γ] i = [Γ i 1 · · · Γ iq ] and [ A ] i = [ A i 1 · · · A iq ]. In or de r to prove the “only if ” par t of the theorem, we need to s how that if the plant gra ph has a s ink (i.e., if there exists j such that ( s P ) ij = 0 for every i 6 = j ), then there exis ts a control design method Γ which dominates the deadb eat control design metho d. W e exhibit such a strategy . Without lo ss of g enerality , we can assume that ( s P ) iq = 0 for all i 6 = q , in which case every matrix A in A ( S P ) has the structure A =      A 11 · · · A 1 ,q − 1 0 . . . . . . . . . . . . A q − 1 , 1 · · · A q − 1 ,q − 1 0 A q 1 · · · A q,q − 1 A qq      . Deﬁne ¯ x 0 = [ x 1 (0) · · · x q − 1 (0) ] T , and le t control design strategy Γ b e deﬁned by      − B − 1 11 A 11 · · · − B − 1 11 A 1 ,q − 1 0 . . . . . . . . . . . . − B − 1 q − 1 ,q − 1 A q − 1 , 1 · · · − B − 1 q − 1 ,q − 1 A q − 1 ,q − 1 0 K q 1 ( A, B ) · · · K q,q − 1 ( A, B ) K qq ( A, B )      for all P = ( A, B , x 0 ) ∈ P , with ¯ K ( A, B ) : =  K q 1 ( A, B ) · · · K q,q − 1 ( A, B ) K qq ( A, B )  = − ( I + B T qq X qq B qq ) − 1 B T qq X qq [ A ] q , where X qq is the unique p os itiv e deﬁnite solution to the discr ete a lgebraic Riccati equation A T qq X qq B qq ( I + B T qq X qq B qq ) − 1 B T qq X qq A qq − A T qq X qq A qq + X qq − I = 0 . (17) In words, control desig n stra tegy Γ applies the dea dbea t strategy to subsystems 1 to q − 1 while, on subs y stem q , it uses the same s ubco nt roller as in the optimal controller for the pla nt ˆ x ( k + 1) = ˆ A ˆ x ( k ) + ˆ B ˆ u ( k ) , (18) 14 with cos t function J (2) ( A,B ,x 0 ) ( ¯ K ) = ∞ X k =1 ˆ x ( k ) T Q ˆ x ( k ) + ∞ X k =0 ˆ u ( k ) T ˆ u ( k ) , where Q = diag(0 , . . . , 0 , I n q × n q ), the matrix ˆ A is deﬁned as [ ˆ A ] q = [ A ] q and [ ˆ A ] z = 0 for all z 6 = q , a nd furthermo re, the matrix ˆ B is deﬁned a s ˆ B = diag(0 , . . . , 0 , B qq ). Note that Γ is indeed communication-less since ¯ K ( A, B ) de- ﬁned ab ov e can be computed with the sole knowledge o f the q th low er blo ck o f A and B . B ecause of the str ucture of matrices in A ( S P ) a nd this character ization of Γ, w e ha ve J ( A,B ,x 0 ) (Γ( A, B )) = J (1) ( A,B ,x 0 ) + J (2) ( A,B ,x 0 ) ( ¯ K ( A, B )) , where J (1) ( A,B ,x 0 ) = ¯ x T 0 ¯ A T ¯ B − T ¯ B − 1 ¯ A ¯ x 0 , with ¯ A =    A 11 · · · A 1 ,q − 1 . . . . . . . . . A q − 1 , 1 · · · A q − 1 ,q − 1    , and ¯ B = diag( B 11 , . . . , B q − 1 ,q − 1 ) and J (2) ( A,B ,x 0 ) ( ¯ K ( A, B )) is the clos ed-lo op cost for sys tem (18). Since ¯ K ( A, B ) is the optimal controller for this cost, J (2) ( A,B ,x 0 ) ( ¯ K ( A, B )) = x T 0 ˆ A T W ˆ Ax 0 , wher e W = diag(0 , . . . , 0 , X qq − X qq B qq ( I + B T qq X qq B qq ) − 1 B T qq X qq ) . Using part 2 of Subsection 3.5.2 in [40], we have the matrix inv ersion iden tit y X − X Y ( I + Z X Y ) − 1 Z X = ( X − 1 + Y Z ) − 1 , which results in W qq = X qq − X qq B qq ( I + B T qq X qq B qq ) − 1 B T qq X qq = ( X − 1 qq + B qq B T qq ) − 1 < B − T qq B − 1 qq . Note that X − 1 qq exists b ecause X qq ≥ I which fo llows from the discrete algebr aic Riccati eq uation in (17 ). This inequality implies that ˆ A T W ˆ A < ˆ A T ( ˆ B † ) T ˆ B † ˆ A where ˆ B † = dia g(0 , . . . , 0 , B − 1 qq ). Thus J ( A,B ,x 0 ) (Γ( A, B )) = J (1) ( A,B ,x 0 ) + J (2) ( A,B ,x 0 ) ( ¯ K ( A, B )) < J ( A,B ,x 0 ) (Γ ∆ ( A, B )) , for all P = ( A, B , x 0 ) ∈ P s uch that the q th low er blo ck of A is not zero, unless the J ( A,B ,x 0 ) (Γ( A, B )) = J ( A,B ,x 0 ) (Γ ∆ ( A, B )). Th us, control design metho d Γ dominates the deadbea t control design metho d Γ ∆ . 15 Remark 3 .14. Consider the limite d mo del information design pr oblem given by the plant gr aph G P in Figur e 2( a ), t he c ontr ol gr aph G ′ K in Figur e 2( b ′ ), and the design gr aph G ′ C in Figur e 2( c ′ ). The or ems 3.1 0 and 3.13 show that the de adb e at c ontr ol design st r ate gy Γ ∆ is the b est c ontr ol design st r ate gy that one c an pr op ose b ase d on the lo c al mo del of subsystems and the plant gr aph, b e c aus e the de adb e at c ontr ol design s t r ate gy is the minimizer of the c omp etitive r atio and it is undominate d. Remark 3 .15. It should b e n ote d that, the pr o of of the “only if ” p art of the The or em 3.13 is c onstruct ive. We u se this c onstruction to build a c ontr ol design str ate gy for the plant gr aphs with sinks in nex t subse ction. 3.2 Second case: plan t graph G P with at least one sink In this section, we consider the ca se where plant g raph G P has c ≥ 1 s inks. Accordingly , its adjacency matrix S P is of the form S P =  ( S P ) 11 0 ( q − c ) × ( c ) ( S P ) 21 ( S P ) 22  , (19) where ( S P ) 11 =    ( s P ) 11 · · · ( s P ) 1 ,q − c . . . . . . . . . ( s P ) q − c , 1 · · · ( s P ) q − c , q − c    , ( S P ) 21 =    ( s P ) q − c + 1 , 1 · · · ( s P ) q − c + 1 , q − c . . . . . . . . . ( s P ) q, 1 · · · ( s P ) q,q − c    , and ( S P ) 22 =    ( s P ) q − c + 1 , q − c +1 · · · 0 . . . . . . . . . 0 · · · ( s P ) qq    , where we assume, without loss o f g enerality , that the vertices are num b ered such that the sinks are labele d q − c + 1 , . . . , q . With this no ta tion, let us now int ro duce the con trol design method Γ Θ deﬁned by Γ Θ ( A, B ) = − diag( B − 1 11 , . . . , B − 1 q − c , q − c , W q − c + 1 ( A, B ) , . . . , W q ( A, B )) A (20) for all ( A, B ) ∈ A ( S P ) × B ( ǫ ), where W i ( A, B ) = ( I + B T ii X ii B ii ) − 1 B T ii X ii (21) for all q − c + 1 ≤ i ≤ q and X ii is the unique p ositive deﬁnite solution of the discrete alg ebraic Riccati equa tion A T ii X ii B ii ( I + B T ii X ii B ii ) − 1 B T ii X ii A ii − A T ii X ii A ii + X ii − I = 0 . (22) The cont rol design method Γ Θ applies the de a dbea t s trategy to every subsystem that is no t a sink and, for every sink, applies the same optimal co nt rol law as if the no de were deco upled from the rest of the graph. W e will s how that when the 16 plant graph contains sinks , Γ Θ has, in worst ca se, the sa me co mpetitive ratio as the deadb eat stra tegy . Unlike the deadb eat strategy , it has the additional prop erty of being undominated by commun ication-less methods for plants in P when the plan t graph G P has sinks. Lemma 3.16. L et t he 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 with self-lo ops, and G K ⊇ G P . L et Γ b e a c ontr ol design str ate gy in C . Su pp ose that ther e exist i and j 6 = i such that ( s P ) ij 6 = 0 and that no de i is not a sink. The c omp etitive r atio of Γ is b ounde d only if A ij + B ii Γ ij ( A, B ) = 0 , for al l P = ( A, B , x 0 ) ∈ P . Pro of: See App endix E. Remark 3.17. L emma 3.16 shows that a ne c essary c ondition for a b oun de d c omp etitive r atio is t o de c ouple t he no des t hat ar e n ot sinks fr om the r est of the network. Now, w e a r e ready to compute the co mpetitive ra tio of the newly deﬁned control design strateg y Γ Θ . This is done at ﬁrst for the c ase that the control graph G K is a complete gr aph. Theorem 3 .18. L et the plant gr aph G P c ontain no isolate d no de and at le ast one sink, and the c ontr ol gr aph G K b e a c omplete gr aph. Then the c omp etitive r atio of the c ommunic ation-less design metho d Γ Θ intr o duc e d in (20) is r P (Γ Θ ) =  1 , if ( S P ) 11 = 0 and ( S P ) 22 = 0 , 1 + 1 / ǫ 2 , otherwise. Pro of: B a sed o n Theo r em 3 .4 we know that, for every plant P = ( A, B , x 0 ) ∈ P J ( A,B ,x 0 ) ( K ∗ ( A, B )) ≥ ǫ 2 1 + ǫ 2 x T 0 A T B − T B − 1 Ax 0 , (23) In addition, pro cee ding as in the pr o of of the “ only if ” pa rt of the Theorem 3.13, we know that J ( A,B ,x 0 ) (Γ ∆ ( A, B )) ≥ J ( A,B ,x 0 ) (Γ Θ ( A, B )) . (24) Plugging equa tio n (24) into equatio n (23) results in J ( A,B ,x 0 ) (Γ Θ ( A, B )) J ( A,B ,x 0 ) ( K ∗ ( A, B )) ≤ 1 + 1 ǫ 2 , ∀ P = ( A, B , x 0 ) ∈ P . As a res ult, r P (Γ Θ ) ≤ 1 + 1 /ǫ 2 . T o show that this upper -b ound is tig h t, we now exhibit plants for which it is a tta ine d. W e use a diﬀerent construction depe nding on matrices ( S P ) 11 and ( S P ) 22 . If ( S P ) 11 6 = 0, tw o situations can o ccur. Case 1: ( S P ) 11 6 = 0 and it is not diagonal. There exist 1 ≤ i 6 = j ≤ q − c such that ( s P ) ij 6 = 0. I n this case, cho ose indices i 1 ∈ I i and j 1 ∈ I j and deﬁne A = e i 1 e T j 1 and B = ǫI . Then, fo r x 0 = e j 1 , we ﬁnd that J ( A,B ,x 0 ) (Γ Θ ( A, B )) J ( A,B ,x 0 ) ( K ∗ ( A, B )) = 1 /ǫ 2 1 / (1 + ǫ 2 ) = 1 + 1 ǫ 2 17 bec ause the con trol design Γ Θ acts like the deadb eat con trol design method on this plant. Case 2: ( S P ) 11 6 = 0 and it is diagonal . Ther e exists 1 ≤ i ≤ q − c such that ( s P ) ii 6 = 0. Pick an index i 1 ∈ I i . In that cas e , cons ider A ( r ) = re i 1 e T i 1 and B = ǫI . F or x 0 = e i 1 , the o ptimal cos t is J ( A ( r ) ,B ,x 0 ) ( K ∗ ( A ( r ) , B )) = √ r 4 + 2 r 2 ǫ 2 − 2 r 2 + ǫ 4 + 2 ǫ 2 + 1 + r 2 − ǫ 2 − 1 2 ǫ 2 , which results in lim r → 0 J ( A,B ,x 0 ) (Γ Θ ( A, B )) J ( A,B ,x 0 ) ( K ∗ ( A, B )) = 1 + 1 ǫ 2 . Now supp ose that ( S P ) 11 = 0 . Again, t wo diﬀerent situations ca n occur . Case 3: ( S P ) 11 = 0 and ( S P ) 22 6 = 0 . There exists q − c + 1 ≤ i ≤ q such that ( s P ) ii 6 = 0. F rom the assumption that the plant g raph co n tains no is olated no de, we know that ther e must exist 1 ≤ j ≤ q − c such that ( s P ) ij 6 = 0. Accordingly , let us pic k i 1 ∈ I i and j 1 ∈ I j and co nsider the 2-parameter family of matrices A ( r , s ) in A ( S P ) with all en tr ies equal to zero except a i 1 i 1 , which is equal to r , and a i 1 j 1 , which is equal to s . Let B = ǫI . F or any initia l condition x 0 , the corres p onding clo sed-lo op p erformance is J ( A ( r,s ) ,B,x 0 ) (Γ Θ ( A ( r , s ) , B )) = β Θ x T 0 a ( r , s ) a ( r, s ) T x 0 , where we hav e le t a ( r , s ) = A ( r , s ) T i 1 and β Θ is β Θ = √ r 4 + 2 r 2 ǫ 2 − 2 ar 2 + ǫ 4 + 2 ǫ 2 + 1 + r 2 − ǫ 2 − 1 2 ǫ 2 r 2 . Besides, the optimal closed-lo o p p erformance can be computed as J ( A ( r,s ) ,B,x 0 ) ( K ∗ ( A ( r , s ) , B )) = β K ∗ x T 0 a ( r , s ) a ( r, s ) T x 0 , where β K ∗ is β K ∗ = ǫ 2 s 2 + r 2 (1 + ǫ 2 ) − ( ǫ 2 + 1 ) 2 + √ c + c − 2 ǫ 2 ( ǫ 2 + 1 )( s 2 + r 2 ) , c ± = ( ǫ 2 s 2 + ( r 2 ± 2 r )( ǫ 2 + 1 ) + ( ǫ 2 + 1 ) 2 ) . Then, r P (Γ Θ ) ≥ lim r →∞ , s r →∞ J ( A ( r,s ) ,B,x 0 ) (Γ Θ ( A ( r , s ) , B )) J ( A ( r,s ) ,B,x 0 ) ( K ∗ ( A ( r , s ) , B )) = 1 + 1 ǫ 2 Case 4: ( S P ) 11 = 0 and ( S P ) 22 = 0 . Then, every matrix A ∈ A ( S P ) ha s the form  0 0 ∗ 0  and, in particular, is nilp otent of deg ree 2; i.e., A 2 = 0 . In this case, the Riccati equation yielding the optimal control gain K ∗ ( A, B ) can be readily solved, and we ﬁnd that K ∗ ( A, B ) = − ( I + B T B ) − 1 B T A for all ( A, B ). As a r esult, K ∗ ( A, B ) = Γ Θ ( A, B ) for all plant P = ( A, B , x 0 ) ∈ P (since 18 W i ( A, B ) = ( I + B T ii B ii ) − 1 B T ii for all q − c + 1 ≤ i ≤ q ), whic h implies that the comp etitive ratio of Γ Θ against plants in P is equal to one. In Theorem 3 .18, the co nt rol gr aph G K is assumed to b e a complete graph. W e needed this assumption to calculate the co st of the optima l control design strategy K ∗ ( P ) when ( S P ) 11 = 0 and ( S P ) 22 6 = 0 which is not an easy tas k when the control graph G K is incomplete. How e ver, more c a n b e sa id if ( S P ) 11 6 = 0 . Corollary 3.19 . L et t he plant gr aph G P c ontain no isolate d no de and at le ast one sink and G K ⊇ G P . Then r P (Γ Θ ) =  1 , if ( S P ) 11 = 0 and ( S P ) 22 = 0 , 1 + 1 / ǫ 2 , if ( S P ) 11 6 = 0 . Pro of: Accor ding to Theorem 3 .18, for ( S P ) 11 6 = 0, w e get r P (Γ Θ ) = s up P ∈P J ( A,B ,x 0 ) (Γ Θ ( A, B )) J ( A,B ,x 0 ) ( K ∗ ( P )) ≤ s up P ∈P J ( A,B ,x 0 ) (Γ Θ ( A, B )) J ( A,B ,x 0 ) ( K ∗ C ( A, B )) = 1 + 1 ǫ 2 . Case 1: ( S P ) 11 6 = 0 and it is not diago nal. F o r the s pecia l plant in tro duced in Case 1 in the pro of of Theorem 3.18, we hav e J ( A,B ,e j 1 ) ( K ∗ C ( A, B )) = J ( A,B ,e j 1 ) ( K ∗ ( A, B , e j 1 )) since A = e i 1 e T j 1 is a nilp otent matrix. The rest of the pro o f is similar to Case 1 in the pro of of Theor em 3.18. Case 2: ( S P ) 11 6 = 0 and it is diagonal. Note that, for the spec ial plant intro- duced Cas e 2 in the pro of of Theorem 3.18, we have K ∗ C ( A, B ) = − √ r 4 + 2 r 2 ǫ 2 − 2 r 2 + ǫ 4 + 2 ǫ 2 + 1 + r 2 − ǫ 2 − 1 2 ǫr 2 A which shows K ∗ C ( A, B ) ∈ K ( S K ) and similar to the pr o of of Theorem 3.4, we get J ( A,B ,e i 1 ) ( K ∗ C ( A, B )) = J ( A,B ,e i 1 ) ( K ∗ ( A, B , e i 1 )). The rest of the pr o of is similar to Case 2 in the pro of of Theorem 3.18. Case 3: ( S P ) 11 = 0 and ( S P ) 22 = 0 . Then, every A ∈ A ( S P ) is nilp otent matrix which results in J P ( K ∗ ( P )) = J P ( K ∗ C ( A, B )). The rest o f the pr o of is similar to Case 4 in the pro of of Theorem 3.18. Now that we hav e computed the comp etitive ratio of the control design strategy Γ Θ in the presenc e of sinks , we pr esent a theorem to show that the comp etitive ratio of any other communication-less control desig n stra tegy is low er- bo unded by the comp etitive ra tio of Γ Θ when the control g raph G K is a complete gr aph. Therefore, the c ontrol desig n strategy Γ Θ is a minimizer o f the c o mpetitive ratio ov er the set of limited mo del information control design strategies. Theorem 3 .20. L et the plant gr aph G P c ontain no isolate d no de and at le ast one sink, the c ontr ol gr aph G K b e a c omplete gr aph, and the design gr aph G C b e a total ly disc onne ct e d gr aph with self-lo ops. Then the c omp etitive r atio of any c ontr ol design str ate gy Γ ∈ C s atisﬁes r P (Γ) ≥ 1 + 1 /ǫ 2 , if either ( S P ) 11 is not diago nal or ( S P ) 22 6 = 0 . 19 Pro of: Case 1: ( S P ) 11 6 = 0 and it is not diagonal. Then, there ex ist 1 ≤ i, j ≤ q − c and i 6 = j such that ( s P ) ij 6 = 0. Cho ose indices i 1 ∈ I i and j 1 ∈ I j and consider the matrix A deﬁned by A = e i 1 e T j 1 and B = ǫ I . F r om Lemma 3.16, we know that a communication-less metho d Γ has a b ounded comp etitive ra tio only if Γ( A, B ) = − B − 1 A (bec ause no de i is a par t o f ( S P ) 11 and it is not a sink). Therefore r P (Γ) ≥ J ( A,B ,e j 1 ) (Γ( A, B )) J ( A,B ,e j 1 ) ( K ∗ ( A, B )) = 1 + 1 ǫ 2 for any such metho d. Case 2: ( S P ) 22 6 = 0 . There thus exists q − c + 1 ≤ i ≤ q such that ( s P ) ii 6 = 0. Note that, ther e exists 1 ≤ j ≤ q − c such that ( s P ) ij 6 = 0, since there is no isolated node in the plant graph. Cho ose indices i 1 ∈ I i and j 1 ∈ I j . Co nsider A deﬁned as A = r e i 1 e T j 1 + se i 1 e T i 1 and B = ǫI . As indica ted in the pro of o f Theorem 3.18, control design strategy Γ Θ yields the glo bally o ptimal c o nt roller with limited mo de l informa tio n for plan ts in this family . Hence, we kno w that r P (Γ) ≥ 1 + 1 /ǫ 2 for every communication-less strategy Γ. In Theorem 3 .20, we assume the control gra ph G K is a complete g raph. In the next corollar y , we generalize this r esult to the case where G K is a super graph of G P when ( S P ) 11 is not diagonal. Corollary 3.21 . L et t he plant gr aph G P c ontain no isolate d no de and at le ast one sink, the design gr aph G C b e a t otal ly disc onne ct e d gr aph with self-lo ops, and G K ⊇ G P . Then the c omp etitive ra tio of any c ontr ol design str ate gy Γ ∈ C satisﬁes r P (Γ) ≥ 1 + 1 /ǫ 2 , if ( S P ) 11 is not diagonal. Pro of: Considering that for the nilp otent matr ix A = e i 1 e T j 1 , we get J ( A,B ,e j 1 ) ( K ∗ ( A, B , e j 1 )) = J ( A,B ,e j 1 ) ( K ∗ C ( A, B )), the rest of the pro o f is s imila r to Case 1 in the pro of of Theorem 3.20. Remark 3.22. Combining The or ems 3.18 and 3.20 implies that if either ( S P ) 11 is not diagonal or ( S P ) 22 6 = 0 , c ontr ol design metho d Γ Θ exhibits t he same c omp etitive r atio as the de adb e at c ontr ol str ate gy, which is the smal lest r atio achieva ble by a c ommunic ation-less c ontr ol metho d. Ther efor e, it is a solution to pr oblem (10). F urthermor e, if ( S P ) 11 and ( S P ) 22 ar e b oth zer o, then Γ Θ is e qual to K ∗ , which shows that Γ Θ is a solut ion to pr oblem (10), in this c ase to o. Remark 3. 23. The c ase wher e ( S P ) 11 is diagonal and ( S P ) 22 = 0 is stil l op en. The next theore m shows that Γ Θ is a more desir a ble control design metho d than the deadb eat con trol design strategy when the pla n t gr aph G P has sinks, since it is then undo minated by communication-less design metho ds. Theorem 3 .24. L et the plant gr aph G P c ontain no isolate d no de and at le ast one sink, the design gr aph G C b e a t otal ly disc onne ct e d gr aph with self-lo ops, and G K ⊇ G P . The c ontr ol design metho d Γ Θ is undominate d by any c ontr ol design metho d Γ ∈ C . 20 Pro of: See App endix F. Remark 3.2 5. Consider the limite d mo del information design pr oblem given by the plant gr aph G ′ P in Figur e 2( a ′ ), the c ontro l gr aph G ′ K in Figur e 2( b ′ ), and t he design gr aph G ′ C in Figur e 2( c ′ ). The or ems 3.18, 3.20, and 3.24 to gether show that, the c ont r ol design str ate gy Γ Θ is the b est c ontr ol design str ate gy t hat one c an pr op ose b ase d on the lo c al mo del information and the plant gr aph, b e c ause the c ontr ol design stra te gy Γ Θ is a minimizer of the c omp etitive r atio and it is undominate d. Remark 3. 26. F or gener al weight matric es Q and R app e aring in the p erfor- manc e c ost, t he c omp etitive r atio of b oth the de adb e at c ont r ol design str ate gy Γ ∆ and the c ontr ol design str ate gy Γ Θ is 1 + ¯ σ ( R ) / ( σ ( Q ) ǫ 2 ) . In p articular, the c omp et itive r atio has a limit e qual to one as ¯ σ ( R ) /σ ( Q ) go es to zer o. We thus r e c over the wel l-known observation (e.g., [41 ]) that, for discr ete-time lin- e ar t ime-invariant systems, the optimal line ar quadr atic r e gulator appr o aches the de adb e at c ontr ol ler in t he limit of “che ap c ontr ol”. 4 Design G raph Inﬂuence on Ac hiev able P erfor- mance In the previous s ection, we hav e shown that communicat-ion-less control des ign metho ds (i.e., G C is totally disco nnected with self-lo ops ) have intrinsic perfor - mance limitations , and w e hav e character ized minimal elements fo r b oth the comp etitive ratio and domination metrics. A na tural ques tion is “ given plan t graph G P , which desig n graph G C is necessary to ensure the e x istence of Γ ∈ C with b etter compe titiv e r atio than Γ ∆ and Γ Θ ?”. W e tackle this question in this sectio n. Theorem 4 .1. L et t he plant gr aph G P and the design gr aph G C b e given and G K ⊇ G P . If one of the fol lowing c onditions is satisﬁe d then r P (Γ) ≥ 1 + 1 /ǫ 2 for al l Γ ∈ C : (a) G P c ontains the p ath k → i → j with distinct no des i , j , and k while ( j, i ) / ∈ E C . (b) Ther e exist i 6 = j such that n i ≥ 2 and ( i, j ) ∈ E P while ( j, i ) / ∈ E C . Pro of: W e pr ove the case when condition (a ) holds. The pro of for co ndi- tion (b) is similar . Let i , j , and k b e three distinct no des suc h tha t ( s P ) ik 6 = 0 and ( s P ) j i 6 = 0 (i.e., the pa th k → i → j is contained in the plant graph G P ). Let us pick i 1 ∈ I i , j 1 ∈ I j and k 1 ∈ I k and co ns ider the 2 -parameter family of matrices A ( r , s ) in A ( S P ) with all entries equal to zero except a i 1 k 1 , which is equal to r , and a j 1 i 1 , which is equal to s . Let B = ǫ I and let Γ ∈ C b e a limited mo del information with design gra ph G C . F or x 0 = e k 1 , we hav e J ( A ( r,s ) ,B,e k 1 ) (Γ( A ( r , s ) , B )) ≥ ( r + ǫγ i 1 k 1 ( A, B )) 2 [ γ 2 j 1 i 1 + ( s + ǫγ j 1 i 1 ( A, B )) 2 ] 21 where γ i 1 k 1 cannot be a function of s beca use ( j, i ) / ∈ E C . Note that, ir resp ective of the c hoice of γ j 1 i 1 ( A, B ), we have J ( A ( r,s ) ,B,e k 1 ) (Γ( A ( r , s ) , B )) ≥ ( r + ǫ γ i 1 k 1 ( A, B )) 2 s 2 1 + ǫ 2 . The co st of the deadbeat control desig n on this pla nt s atisﬁes J ( A ( r,s ) ,B,e k 1 ) (Γ ∆ ( A ( r , s ) , B )) = r 2 /ǫ 2 , and thus r P (Γ) = sup P ∈P J P (Γ( A, B )) J P ( K ∗ ( P )) = s up P ∈P  J P (Γ( A, B )) J P (Γ ∆ ( A, B )) J P (Γ ∆ ( A, B )) J P ( K ∗ ( P ))  ≥ s up P ∈P J P (Γ( A, B )) J P (Γ ∆ ( A, B )) , ≥ lim s →∞ ǫ 2 ( r + ǫγ i 1 k 1 ( A, B )) 2 s 2 (1 + ǫ 2 ) r 2 . (25) This shows that r P (Γ) is unbounded unless r + ǫγ i 1 k 1 ( A ( r , s ) , B ) = 0 for a ll r , s . Now cons ider the 1-para meter family of matr ic e s ¯ A ( r ) with all entries equal to zero except a i 1 k 1 , which is equal to r . Because of ( j, i ) / ∈ E C , w e know that Γ z ( ¯ A ( r ) , B ) = Γ z ( A ( r , s ) , B ) for all z ∈ I i . Th us J ( ¯ A ( r ) ,B ,e k 1 ) (Γ( ¯ A ( r ) , B )) ≥ r 2 /ǫ 2 . On the other hand, similar to the pro of of Theo rem 3.4, we ca n co mpute the optimal controller for sy stems in this 1 − par ameter family and ﬁnd J ( ¯ A ( r ) ,B ,e k 1 ) ( K ∗ ( ¯ A ( r ) , B , e k 1 )) = J ( ¯ A ( r ) ,B ,e k 1 ) ( K ∗ C ( ¯ A ( r ) , B )) = r 2 / (1 + ǫ 2 ) , As a result, we g e t r P (Γ) ≥ r 2 /ǫ 2 r 2 / (1 + ǫ 2 ) = 1 + 1 ǫ 2 , which concludes the proo f for this case. Remark 4. 2 . Consider the limite d mo del information design pr oblem given by the plant gr aph G P in Figur e 2( a ), t he c ontr ol gr aph G ′ K in Figur e 2( b ′ ), and the design gr aph G C in Figur e 2( c ). The or em 4.1 shows t hat, b e c aus e t he plant gr aph G P c ontains t he p ath 3 → 2 → 1 but the design gr aph G C do es n ot c ontain 1 → 2 , the c omp etitive r atio of any c ontr ol design str ate gy Γ ∈ C would b e gr e ater than or e qual to 1 + 1 /ǫ 2 . Corollary 4. 3. L et b oth t he plant gr aph G P and t he c ontr ol gr aph G K b e c om- plete gr aphs. If the design gr aph G C is not e qual t o G P , then r P (Γ) ≥ 1 + 1 /ǫ 2 for al l Γ ∈ C . 22 Pro of: T he pro of is a dir ect applicatio n of Theorem 4.1 with condition (a) fulﬁlled. Remark 4.4. Cor ol lary 4.3 shows that, when G P is a c omplete gr aph, achiev- ing a b et ter c omp etitive r atio than the de adb e at design s tr ate gy r e qu ir es e ach subsystem to have ful l know le dge of the plant mo del when c onst ructing e ach sub c ont ro l ler. 5 Extensions to Und er-Actuated Sinks In the pr evious sectio ns , we gave an ex plic it solution to the problem in (10) under the assumption that all the subsystems ar e fully-actuated; i.e., a ll the matrices B ∈ B ( ǫ ) a re s quare inv er tible matrices . Note that this assumption stems from the fact that the subsystems that are not sinks in the plant gr a ph are required to decouple themselves fro m the res t of the plant to av oid inﬂuencing highly sensitive (and p otentially har d to control) subsy stems in order to keep the c o mpetitive ratio ﬁnite (see Le mma 3.16). Therefor e, w e ass ume these sub- systems ar e fully-actuated to ea sily decouple them from the r est of the system. As a future dir ection for improvemen t, one can try to repla ce this assumption with other conditions (e.g., g eometric conditions ) to ensure that the s ubsystems can decouple themselves. F rom the same a rgument, it should b e ex pected that the assumption o f a squa re in vertible B-matr ix is disp ensable for sink no des. In this section, we brieﬂy discuss a n extension o f our results to the s lightly mo r e general, but still restricted, class of plan ts whose sinks are under-actuated. Consider the limited mo del information control design problem g iven with the pla nt graph G P , the co ntrol gra ph G K , and the desig n graph G C given in Figure 3. The state space re pr esentation of the system is given as  x 1 ( k + 1 ) x 2 ( k + 1 )  = A  x 1 ( k ) x 2 ( k )  + B  u 1 ( k ) u 2 ( k )  , where A =  A 11 0 A 21 A 22  , B =  B 11 0 0 B 22  , with x 1 ( k ) ∈ R n 1 , x 2 ( k ) ∈ R n 2 , u 1 ( k ) ∈ R n 1 , and u 2 ( k ) ∈ R m 2 for some given int egers n 1 ≥ 1, n 2 > m 2 ≥ 1. Thus, for the seco nd subsystem the matrix B 22 ∈ R n 2 × m 2 is a non-square ma trix, and as a result the second subsystem is an under-actua ted subsystem. Let us a ssume that the matrices A 21 , A 22 , B 22 satisfy the “ma tc hing co nditio n”; i.e., the pair ( A 22 , B 22 ) is c o nt rollable a nd span( A 21 ) ⊆ span( B 22 ) [42]. B e sides, a s sume that for all ma trices B , we hav e σ ( B ) ≥ ǫ for some ǫ > 0. F or this case, we have Γ Θ ( A, B ) = − diag( B − 1 11 , W 2 ( A 22 , B 22 )) A, where W 2 ( A 22 , B 22 ) is deﬁned in (21). Note that we do not requir e the matrix B 22 to b e squa r e inv ertible. Under so me additional conditions and following a simila r appr oach as ab ov e, it ca n b e shown that the co ntrol design stra tegy Γ Θ bec omes an undominated minimizer of the compe titiv e ratio ov er the set of limited mo del information control desig n stra tegies. This result can b e gener al- ized to ca ses w ith higher num b er of subsystems as long a s the sinks in the pla n t graph G P are the only under-a c tua ted subsystems [4 3]. 23 ( ) a P G ( ) b K G ( ) c C G 2 1 2 1 2 1 Figure 3 : Pla nt gr aph G P , co nt rol gr a ph G K , a nd des ign gr aph G C used to illustrate an extension to under-actuated systems. 6 Conclusion W e presented a framework for the study of c o nt rol desig n under limited model information, a nd in vestigated the co nnection b etw een the quality o f controllers pro duced by a design metho d and the amount o f pla n t mo del informa tion av a il- able to it. 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A thans, “On the determination of the optimal constant output feedback gains for line a r m ultiv ariable systems,” Automatic Contr ol, IEEE T r ansactions on , vol. 1 5, no. 1 , pp. 44 – 48, 19 7 0. [39] D. B. W est, Intro duction to Gr aph The ory . P rentice Hall, 2001. [40] H. L ¨ utkepohl, Handb o ok of matric es . Wiley , 1996. [41] J. O ’Reilly , “The discrete linear time inv ar iant time-optimal control problem–an overview,” Automatic a , vol. 1 7, no. 2, pp. 3 63 – 37 0, 198 1. [42] D. D. ˇ Siljak, D e c entr alize d c ontr ol of c omplex systems . No. 1 8 4, Academic Press, 19 91. [43] F. F aro khi and K. H. Johansso n, “Dynamic control design based on lim- ited mo del information,” in Communic ation, Contr ol, and Computing, 49th Annual Al lerton Confer enc e on , pp. 15 76 – 15 83, 2011 . [44] N. K omaroﬀ, “I ter ative matrix b ounds and co mputatio nal so lutions to the discrete a lgebraic R icca ti eq uation,” Automatic Contr ol, IEEE T r ansac- tions on , vol. 39, no. 8, pp. 16 76 – 16 78, 1994 . A Pro of of Lemma 3.3 F or a n y plant P = ( A, B , x 0 ) ∈ P , the optimal controller K ∗ ( P ) exists (b e- cause the plan t is controllable since B is inv er tible by assumption) and can be computed using the unique p os itiv e deﬁnite solution to the discrete a lgebraic Riccati eq uation X = A T X A − A T X B ( I + B T X B ) − 1 B T X A + I . (26 ) The corresp onding cost is J P ( K ∗ ( A, B )) = x T 0 ( X − I ) x 0 . Inserting the pro duct B B − 1 befo re every matrix A and B − T B T after every matrix A T in (26) results in X − I = A T B − T B T X B B − 1 A − A T B − T B T X B ( I + B T X B ) − 1 B T X B B − 1 A. (27) Naming B T X B as Y simpliﬁes (27) in to X − I = A T B − T [ Y − Y ( I + Y ) − 1 Y ] B − 1 A. (28) Note that Y is a p ositive deﬁnite matrix beca us e X is p ositive deﬁnite and B is full rank. L e t us deno te the right-hand side of (2 8 ) by A T B − T g ( Y ) B − 1 A . Then we can make the following tw o cla ims reg arding the r ational function g ( · ). Claim 1: The function y 7→ g ( y ) = y / (1 + y ) is a monotonic a lly increasing ov er R + . Claim 2: Let Y ∈ S n ++ and D , T b e diagonal and unitary matrice s , resp ec- tively , such that Y = T T D T . Then g ( Y ) = T T diag( g ( d ii )) T , where d ii are the diagonal elements of D (and the eigen v alues of Y ). 27 Claim 1 is prov ed by computing the deriv ative of g ov er R + , while Claim 2 follows from the fact that all matrice s involv ed in the co mputation of g ( Y ) can be diagonaliz ed in the sa me basis yielding g ( Y ) = Y − Y ( I + Y ) − 1 Y = T T D T − T T D T ( I + T T D T ) − 1 T T D T = T T ( D − D ( I + D ) − 1 D ) T = T T g ( D ) T . Using these tw o claims, we ﬁnd that, for a ll Y with eigenv alues denoted by λ 1 ( Y ) , . . . , λ n ( Y ) X − I = A T B − T g ( Y ) B − 1 A = A T B − T T T diag( g ( λ i ( Y ))) T B − 1 A ≥ ( g ( λ ( Y ))) A T B − T B − 1 A, (29) where λ ( Y ) is a po sitive num b er b ecause matrix Y is a po sitive deﬁnite matrix. Now, acc ording to [44 ], λ ( X ) ≥ λ ( A T ( I + B B T ) − 1 A + I ) ≥ σ 2 ( A ) 1 + ¯ σ 2 ( B ) + 1 . (30) Using (30) in inequality λ ( Y ) ≥ σ 2 ( B ) λ ( X ) gives λ ( Y ) ≥ σ 2 ( B ) σ 2 ( A ) 1 + ¯ σ 2 ( B ) + σ 2 ( B ) , (31) and, b eca us e of the claim 1 a nd the inequa lit y in (31), we will hav e g ( λ ( Y )) ≥ σ 2 ( B )[ σ 2 ( A ) + ¯ σ 2 ( B ) + 1] 1 + ¯ σ 2 ( B ) + σ 2 ( B )[ σ 2 ( A ) + ¯ σ 2 ( B ) + 1] ≥ σ 2 ( B ) σ 2 ( B ) + 1 . (32) Combining (29) and (32) results in X − I ≥ σ 2 ( B ) σ 2 ( B ) + 1 A T B − T B − 1 A, and, ther efore J P ( K ∗ ( A, B )) = x T 0 ( X − I ) x 0 ≥  σ 2 ( B ) σ 2 ( B ) + 1  x T 0 ( A T B − T B − 1 A ) x 0 =  σ 2 ( B ) σ 2 ( B ) + 1  J P (Γ ∆ ( A, B )) . 28 B Pro of of Lemma 3.8 Let Γ ∈ C and assume that the implica tio n do e s not ho ld, i.e., that there exists a matrix A and indices i , j with ℓ 0 ∈ I i such that a ℓj = 0 for a ll ℓ ∈ I i but γ ℓ 0 j ( A, B ) 6 = 0. Consider ma trix ¯ A such that ¯ A ℓ = A ℓ for all ℓ ∈ I i and ¯ A z = 0 for a ll z / ∈ I i . Ba s ed on the deﬁnition of limited-mo del-infor mation control design methods , we know that Γ ℓ ( ¯ A, B ) = Γ ℓ ( A, B ) for all ℓ ∈ I i and Γ z ( ¯ A, B ) = 0 for all z / ∈ I i (beca use Γ z ( A, B ) = Γ z (0 , B ) for all z / ∈ I i and, as shown in [27], it is neces sary that Γ(0 , B ) = 0 for a ﬁnite comp etitive r atio). F or x = e j , we hav e J ( ¯ A,B ,e j ) (Γ( ¯ A, B )) ≥ X ℓ ∈I i γ ℓj ( ¯ A, B ) 2 = X ℓ ∈I i γ ℓj ( A, B ) 2 ≥ γ ℓ 0 j ( A, B ) 2 > 0 . Using (25), w e get r P (Γ) ≥ J ( ¯ A,B ,e j ) (Γ( ¯ A, B )) J ( ¯ A,B ,e j ) (Γ ∆ ( ¯ A, B )) = ∞ , since J ( ¯ A,B ,e j ) (Γ ∆ ( ¯ A, B )) = 0. This proves the claim by co n trap ositive. C Pro of of Lemma 3.9 Clearly , it is enough to pro ve ineq ua lit y (16) for control design methods with a ﬁnite comp etitive ratio . W e pro ce e d in three steps. First, using Lemma 3.8, we c haracter iz e the design str ategies lea ding to a ﬁnite comp etitive ratio. Then, we arg ue tha t the controllers pro duced by such strategies must b e stabilizing for a ll plants, and use the fact that every closed-lo o p characteristic p olyno mial is Sch ur to construct a seq ue nc e of rea l num b er s with sp eciﬁc prop erties for each control design stra tegy . W e then use this sequence to co nstruct a sequence o f plants allowing us to low er b ound the co mpetitive ra tio of e ach control design strategy . Let G P hav e a lo op and Γ ∈ C hav e ﬁnite comp etitive ratio. Without lo ss of gener ality , let us ass ume tha t the no des of g raph G P are num b ered such that it a dmits the following lo o p of length ℓ : 1 → 2 → · · · → ℓ → 1. Le t us choose indices i 1 ∈ I 1 , i 2 ∈ I 2 , . . . , i ℓ ∈ I ℓ and co ns ider the one- parameter family o f matrices { A ( r ) } deﬁned b y a i 2 i 1 ( r ) = r , a i 3 i 2 ( r ) = r , . . . , a i ℓ i ℓ − 1 ( r ) = r , a i 1 i ℓ ( r ) = r , a nd all other e n tries equal to zero, for a ll r . Let B = ǫI . Because of Lemma 3.8, the controller gain entries γ j 2 i 1 ( A ( r ) , B ) for a ll j 2 ∈ I 2 , γ j 3 i 2 ( A ( r ) , B ) for all j 3 ∈ I 3 , . . . , γ j ℓ i ℓ − 1 ( A ( r ) , B ) for all j ℓ ∈ I ℓ , γ j 1 i ℓ ( A ( r ) , B ) for all j 1 ∈ I 1 can b e non-ze ro, but all other entries o f the controller ga in Γ( A ( r ) , B ) are z ero for all r . As a result, the characteristic p olyno mia l o f matrix A ( r ) + B Γ( A ( r ) , B ) can b e computed as: λ n − ℓ [ λ ℓ − ( − 1) ℓ ( r + ǫ γ i 2 i 1 ( A ( r ) , B ))( r + ǫγ i 3 i 2 ( A ( r ) , B )) × · · · × ( r + ǫγ i ℓ i ℓ − 1 ( A ( r ) , B ))( r + ǫγ i 1 i ℓ ( A ( r ) , B ))] . Now, note that b ecause Γ ha s a b ounded comp etitive ra tio agains t P by as- sumption, this p olyno mial should b e sta ble fo r all r . (Indeed, Γ ca n have a 29 ﬁnite comp etitive ratio o nly if A + B Γ( A, B ) is stable for all matr ic es A , oth- erwise it would yield an inﬁnite c o st for s o me plants while the corres po nding optimal c o st remains bounded since the pa ir ( A, B ) is c o nt rollable for all plant in P ). As a result, we must hav e | ( r + ǫγ i 2 i 1 ( A ( r ) , B )) · · · ( r + ǫγ i 1 i ℓ ( A ( r ) , B )) | = | r + ǫγ i 2 i 1 ( A ( r ) , B ) | · · · | r + ǫγ i 1 i ℓ ( A ( r ) , B ) | < 1 (33) for all r . Let { r z } ∞ z =1 be a sequence of r eal num ber s with the prop erty that r z go es to inﬁnit y as z go es to inﬁnity . F rom (33), w e know that there exis ts an index ¯ m such that ∀ N , ∃ z > N such that | r z + ǫ γ i ¯ m ⊕ 1 i ¯ m ( A ( r z ) , B ) | < 1 , (34) where “ ⊕ ” des ig nated addition mo dulo ℓ ; i.e., i ⊕ j = ( i + j ) − ⌊ ( i + j ) /ℓ ⌋ ℓ where ⌊ x ⌋ = ma x { y ∈ Z | y ≤ x } for all x ∈ R . Indeed, if this is not the case, it is true that ∀ m, ∃ N m such that | r z + ǫ γ i m ⊕ 1 i m ( A ( r z ) , B ) | ≥ 1 , ∀ z > N m . Then, for all z > max m N m and all m , | r z + ǫ γ i m ⊕ 1 i m ( A ( r z ) , B ) | ≥ 1 which contradicts (33). Without loss of generality (since this just amounts to renum b ering the nodes in the plant g raph), we a ssume that ¯ m = 1. Using (34), we can then cons tr uct a subseque nc e { r φ ( z ) } of { r z } with the prop erty that | r φ ( z ) + ǫ γ i 2 i 1 ( A ( r φ ( z ) ) , B ) | < 1 for a ll z . Now introduce the sequence of matrice s { ¯ A ( z ) } ∞ z =1 deﬁned by ¯ A i 2 i 1 ( z ) = r φ ( z ) for all z and every other row equal to zer o. F or large enough z (and hence, la rge enough r φ ( z ) ), we get J ( ¯ A ( z ) ,B ,e i 1 ) (Γ( ¯ A ( z ) , B )) ≥ γ i 2 i 1 ( ¯ A ( z ) , B ) 2 = γ i 2 i 1 ( A ( r φ ( z ) ) , B ) 2 ≥ ( | r φ ( z ) | − 1 ) 2 ǫ 2 , and thus J ( ¯ A ( z ) ,B ,e i 1 ) (Γ( ¯ A ( z ) , B )) J ( ¯ A ( z ) ,B ,e i 1 ) ( K ∗ ( ¯ A ( z ) , B , e i 1 )) ≥ ( | r φ ( z ) | − 1) 2 /ǫ 2 r 2 φ ( z ) / (1 + ǫ 2 ) . This, in particular, implies that r P (Γ) ≥ lim z →∞ J ( ¯ A ( z ) ,B ,e i 1 ) (Γ( ¯ A ( z ) , B )) J ( ¯ A ( z ) ,B ,e i 1 ) ( K ∗ ( ¯ A ( z ) , B , e i 1 )) ≥ 1 + 1 /ǫ 2 . Note that ¯ A ( z ) is a nilp otent matrix for all z , a nd th us J ( ¯ A ( z ) ,B ,e i 1 ) ( K ∗ ( ¯ A ( z ) , B , e i 1 )) = J ( ¯ A ( z ) ,B ,e i 1 ) ( K ∗ C ( ¯ A ( z ) , B )) similar to the pro of o f Theorem 3.4, a nd therefor e J ( ¯ A ( z ) ,B ,e i 1 ) ( K ∗ C ( ¯ A ( z ) , B )) = r 2 φ ( z ) / (1 + ǫ 2 ) using the unique p ositive-deﬁnite solution of discr ete alge braic Riccati equation in (13). 30 D Pro of of Lemma 3.12 W e prov e that if there is no sink in the plant g r aph (i.e., accor ding to [39], if ∀ j ∃ k , k 6 = j , such that ( s P ) kj 6 = 0 ) then the dea dbea t cont rol design method is undominated. F or proving this claim, we are going to prove tha t for any control desig n Γ ∈ C \{ Γ ∆ } , ther e exists a plant P = ( A, B , x 0 ) ∈ P suc h that J P (Γ( A, B )) > J P (Γ ∆ ( A, B )) = x T 0 [ A T B − T B − 1 A ] x 0 . W e will pro ceed in several steps, which require us to pa rtition the set of limited mo del informa tion control desig n metho ds C as follows C = L c ∪ W 1 ∪ W 2 ∪ { Γ ∆ } , where L := { Γ ∈ C |∃ Λ j : R n j × n × R n j × n j → R n j × n j , [Γ( A, B )] j = Λ j ([ A ] j , B j j )[ A ] j , for all j = 1 , · · · , q } , W 1 := { Γ ∈ L|∃ j, i 6 = j and A ij ∈ R n i × n j nonzero s.t. I + B ii Λ i ([0 · · · 0 A ij 0 · · · 0] , B ii ) 6 = 0 } , and W 2 := { Γ ∈ L \ W 1 |∃ i ∈ { 1 , · · · , q } , [ A ] i ∈ R n i × n , with appropria te s tructure s.t. I + B ii Λ i ([ A ] i , B ii ) 6 = 0 } . In words, L is the set o f all control design metho ds for which sub- controller K j can b e wr itten as a linear com bination o f vectors in { A i , i ∈ I j } fo r all j . Sets W 1 and W 2 are subsets of L which put further c o nstraints on map Γ. Using diﬀerent low er b ounds on closed-lo op perfo rmance in each case, we show that Γ ∆ is undomina ted by control stra tegies in ea ch of L c , W 1 , and W 2 . First, we pr ov e that the deadbe a t c o nt rol design metho d is undo minated by control design str ategies in L c . L e t Γ ∈ L c and let j b e such that ∃ j 1 ∈ I j which Γ j 1 ( ¯ A, B ) T cannot b e written as a linear combination of vectors in the set { ¯ A T i , ∀ i ∈ I j } for some ma tr ix ¯ A and matr ix B . Let a T i = ¯ A i for all i ∈ I j and co nsider matrix A such that the r ow A i = a T i for all i ∈ I j and A i = 0 for all i ∈ I c j . If Γ(0 , B ) 6 = 0, then Γ cannot dominate Γ ∆ (since Γ ∆ (0 , B ) = 0 for all x 0 ) and, thus, there is no loss o f generality in ass uming that Γ(0 , B ) = 0 fo r all x 0 , and, in turn that Γ i ( A, B ) = 0 for a ll i ∈ I c j . Let us also denote Γ( A, B ) by K and Γ i ( A, B ) = Γ i ( ¯ A, B ) by K T i for all i ∈ I j . F or a ll x 0 , J ( A,B ,x 0 ) (Γ( A, B )) ≥ x T 0 [ K T K + ( A + B K ) T ( A + B K )] x 0 , and J ( A,B ,x 0 ) (Γ( A, B )) − J ( A,B ,x 0 ) (Γ ∆ ( A, B )) ≥ x T 0 [ A T ( I − B − T B − 1 ) A + A T B K + K T B T A + K T ( I + B T B ) K ] x 0 . (35) W e know that null( A ) = spa n { A T i , ∀ i ∈ I j } ⊥ 6 = { 0 } , beca use n j < n . On the other hand, we k now that there exists a n j 1 ∈ I j such that K j 1 / ∈ span { A T i , ∀ i ∈ I j } which shows that span { A T i , ∀ i ∈ I j } span { A T i , ∀ i ∈ I j } + spa n { K T i , ∀ i ∈ I j } , 31 Thu s, we ca n choose an initial condition x 0 ∈ n ull( A ) such that K x 0 6 = 0 . Using this x 0 in (35) results in J ( A,B ,x 0 ) (Γ( A, B )) − J ( A,B ,x 0 ) (Γ ∆ ( A, B )) ≥ x T 0 [ K T ( I + B T B ) K ] x 0 > 0 . (36) Therefore, the control desig n strategies in L c cannot do minate the deadb eat control desig n strategy Γ ∆ . Second, we prove that the deadbea t control desig n strategy is undominated by co n trol design metho ds in W 1 . Let Γ ∈ W 1 and let j b e such that ( I + B ii Λ i (  0 · · · 0 ¯ A ij 0 · · · 0  , B ii )) 6 = 0 for some i 6 = j . It means that there exists at lea st i 1 ∈ I i and j 1 ∈ I j such that ¯ a i 1 j 1 6 = 0 and ¯ a i 1 j 1 + b i 1 i 1 γ i 1 j 1 ( ¯ A, B ) 6 = 0. Using the structure matrix, w e know that there exits a ℓ 6 = i such that ( s P ) ℓi 6 = 0. Cho ose an index ℓ 1 ∈ I ℓ . Co nsider the matrix A deﬁned by [ A ] i = [ ¯ A ] i , a ℓ 1 i 1 = r a nd all other entries equal to zer o. Then, [Γ( A, B )] i = Λ i ([ A ] i , B ii )[ A ] i , [Γ( A, B )] ℓ = Λ ℓ ([ A ] ℓ , B ℓℓ )[ A ] ℓ (beca use Γ ∈ L ), and [Γ( A, B )] z = 0 for all z 6 = i , ℓ . Denote Γ( A, B ) b y K . W e ha ve J ( A,B ,x 0 ) (Γ( A, B )) ≥ x T 0 [( A + B K ) T K T K ( A + B K ) +(( A + B K ) 2 ) T ( A + B K ) 2 ] x 0 . Using x 0 = e j 1 results in J ( A,B ,e j 1 ) (Γ( A, B )) − J ( A,B ,e j 1 ) (Γ ∆ ( A, B )) ≥ [ k 2 ℓ 1 i 1 + ( r + b ℓ 1 ℓ 1 k ℓ 1 i 1 ) 2 ]( a i 1 j 1 + b i 1 i 1 k i 1 j 1 ) 2 − X z ∈I i a 2 z j 1 b 2 z z . (37) Note that, irresp ective of the choice o f the con troller gain k ℓ 1 i 1 , k 2 ℓ 1 i 1 + ( r + b ℓ 1 ℓ 1 k ℓ 1 i 1 ) 2 ≥ r 2 / (1 + b 2 ℓ 1 ℓ 1 ) , and as a result, lim r → + ∞ [ k 2 ℓ 1 i 1 + ( r + b ℓ 1 ℓ 1 k ℓ 1 i 1 ) 2 ]( a i 1 j 1 + b i 1 i 1 k i 1 j 1 ) 2 = ∞ , bec ause a i 1 j 1 + b i 1 i 1 k i 1 j 1 6 = 0 . Hence, we can a lw ays construct A with a ppropriate choice of index ℓ and a scalar r large enoug h to make the right ha nd s ide of the expression (37) positive. As a result, Γ ∈ W 1 cannot dominate Γ ∆ . Third, we prov e that the deadb eat control design str ategy is undominated by co n trol design metho ds in W 2 . Let Γ ∈ W 2 and index i and vector [ ¯ A ] i be such that I + Λ i ([ ¯ A ] i , B ii ) 6 = 0. Thus we know tha t there exists at least i 1 ∈ I i such that ¯ A i 1 6 = 0 and ¯ A i 1 + b i 1 i 1 Γ i 1 ( ¯ A, B ) 6 = 0. Base d on the structure matrix we know that there exits ℓ 6 = i such that ( s P ) ℓi 6 = 0 . Cho ose an index ℓ 1 ∈ I ℓ . Consider the matrix A deﬁned by [ A ] i = [ ¯ A ] i and a ℓ 1 i 1 = r a nd all other entries of A equal to zero. Then [ A ] i + B ii [Γ( A, B )] i = ( I + B ii Λ i ([ A ] i , B ii ))[ A ] i and [ A ] j + B j j [Γ( A, B )] j = 0 for all j 6 = i (and, in particular, j = ℓ since Γ do e s not belo ng to W 1 ). Ag a in, K will stand for Γ( A, B ). W e hav e K T K + ( A + B K ) T K T K ( A + B K ) − A T B − T B − 1 A ≥ ( A i 1 + b i 1 i 1 Γ i 1 ( A, B )) T ( A i 1 + b i 1 i 1 Γ i 1 ( A, B )) × r 2 /b 2 ℓ 1 ℓ 1 − X z ∈I i A T z A z /b 2 z z , 32 and hence, s ince A i 1 + b i 1 i 1 Γ i 1 ( A, B ) 6 = 0, we can choose r large enough to ensure that this matrix has a s tr ictly po sitive eigenv alue. Thus, the control design stra tegy Γ ∈ W 2 cannot dominate Γ ∆ . E Pro of of L emma 3.16 The pro of is by contrap ositive. Let Γ b e communication-less and assume that there e x ist matrices A and B and indices i 1 ∈ I i and j 1 ∈ I j such that a i 1 j 1 + b i 1 i 1 γ i 1 j 1 ( A, B ) 6 = 0. Cho ose an index k 1 ∈ I k . Consider the one-par ameter family of matr ic e s ¯ A ( r ) deﬁned by [ ¯ A ( r )] i = [ A ] i , ¯ a k 1 i 1 = r , and all other ent ries of ¯ A ( r ) b eing equal to zero for all r . W e know that [Γ( ¯ A ( r ) , B )] i = [Γ( A, B )] i and Γ ¯ k ( ¯ A ( r ) , B ) = γ ¯ ki 1 ( r ) e T i 1 for all ¯ k ∈ I k (beca use o f Lemma 3.8), [Γ( ¯ A ( r ) , B )] z = 0 for all z 6 = i, k . F or x 0 = e j 1 , we hav e J ( ¯ A ( r ) ,B ,e j 1 ) (Γ( ¯ A ( r ) , B )) ≥ ( a i 1 j 1 + b i 1 i 1 γ i 1 j 1 ( A, B )) 2 × [ γ k 1 i 1 ( r ) 2 + ( r + b k 1 k 1 γ k 1 i 1 ( r )) 2 ] . The minimum v alue of function y 7→ [ y 2 + ( r + b k 1 k 1 y ) 2 ] is r 2 / (1 + b 2 k 1 k 1 ). Hence, irresp ective of function γ k 1 i 1 , J ( ¯ A ( r ) ,B ,e j 1 ) (Γ( ¯ A ( r ) , B )) ≥ ( a i 1 j 1 + b i 1 i 1 γ i 1 j 1 ( A, B )) 2 r 2 / (1 + b 2 k 1 k 1 ) . Note that the term ( a i 1 j 1 + b i 1 i 1 γ i 1 j 1 ( A, B )) 2 is indep endent from r b eca use Γ is communication-less. In addition, J ( ¯ A ( r ) ,B ,e j 1 ) (Γ ∆ ( ¯ A ( r ) , B )) = X z ∈I i ¯ a 2 z j 1 b 2 z z = X z ∈I i a 2 z j 1 b 2 z z for a ll r a nd, thus, J ( ¯ A ( r ) ,B ,e j 1 ) (Γ ∆ ( ¯ A ( r ) , B )) is also indep endent from r . Then, pro ceeding a s in (25), w e deduce that r P (Γ) ≥ ( a i 1 j 1 + b i 1 i 1 γ i 1 j 1 ( A, B )) 2 (1 + b 2 k 1 k 1 ) J ( ¯ A ( r ) ,B ,e j 1 ) (Γ ∆ ( ¯ A ( r ) , B )) lim r →∞ r 2 . Since ( a i 1 j 1 + b i 1 i 1 γ i 1 j 1 ( A, B )) 6 = 0 b y assumption, we then deduce that Γ has an unbounded competitive ra tio , which proves the lemma by contrapositive. F Pro of of T heorem 3.24 W e prov e that for any control design method Γ ∈ C \{ Γ Θ } , there exis ts a plant P = ( A, B , x 0 ) ∈ P such that J P (Γ( A, B )) > J P (Γ Θ ( A, B )). L ike in the pro of of Theo rem 3.6, we pa rtition the set of limited mo del informatio n control design metho ds C as follo ws C = L c ∪ W 0 ∪ W 1 ∪ W 2 ∪ { Γ Θ } , where L := { Γ ∈ C |∃ Λ i : R n i × n × R n i × n i → R n i × n i , [Γ( A, B )] i = Λ i ([ A ] i , B ii )[ A ] i , for all i = 1 , · · · , q } , 33 W 0 := { Γ ∈L , ∃ i ∈ { q − c + 1 , . . . , q } such that Λ i ([ A ] i , B ii ) 6 = W i ([ A ] i , B ii ) } , with W i deﬁned as in equation (21), W 1 := { Γ ∈ L \ W 0 |∃ i ∈ { 1 , · · · , q − c } , ∃ j 6 = i and A ij ∈ R n i × n j nonzero such that I + B ii Λ i ([0 · · · 0 A ij 0 · · · 0] , B ii ) 6 = 0 } , and W 2 := { Γ ∈ L \ W 0 ∪ W 1 |∃ i ∈ { 1 , · · · , q − c } , [ A ] i ∈ R n i × n , with appr o priate structure s uch tha t I + B ii Λ i ([ A ] i , B ii ) 6 = 0 } . First, we prov e that Γ Θ is undominated by con trol design metho ds in L c . Let Γ ∈ L c and let i be s uc h that there exists a plant with matrix ¯ A with the pr op erty that sub c ontroller [Γ] i ([ ¯ A ] i , B ii ) T do es not belo ng to the linea r subspace spanned by the columns of [ ¯ A ] T i . If 1 ≤ i ≤ q − c then, pro ceeding as in the pro o f o f Theorem 3.1 3, we ca n ﬁnd matrices A , B and initial condition x 0 such that J P (Γ( P )) > J P (Γ ∆ ( P )) = J P (Γ Θ ( P )) for P = ( A, B , x 0 ) (with the last equality following fro m the structure of matrix A ). Hence, without loss of generality , we assume that q − c + 1 ≤ i ≤ q . Cons ider matrix A deﬁned as [ A ] i = [ ¯ A ] i and [ A ] j = 0 for a ll j 6 = i . F or this par ticular matrix A and a n y B , x 0 we know from the pro of of the “only if ” part of the Theorem 3.1 3 that Γ Θ ( A, B , x 0 ) is the globally o ptimal controller. Hence, every o ther control design metho d in C lea ds to a controller with gre a ter pe r formance criterio n than Γ Θ for this particular type of plants. Therefore, the control design Γ Θ is undominated by control desig n metho ds in L c . The same r easoning shows that Γ Θ is also undominated by control design metho ds in W 0 . W e now prov e that Γ Θ is undominated by control design strategies in W 1 . Let Γ ∈ W 1 and let 1 ≤ i ≤ q − c b e such that ( I + B ii Λ i ([ ¯ A ] i , B ii )) 6 = 0 where [ ¯ A ] i =  0 · · · 0 ¯ A ij 0 · · · 0  for some j 6 = i . This means that ther e exists at least one i 1 ∈ I i and j 1 ∈ I j such that ¯ a i 1 j 1 6 = 0 and ¯ a i 1 j 1 + b i 1 i 1 γ i 1 j 1 ( A, B ) 6 = 0. Because subsystem i is not a sink (since 1 ≤ i ≤ q − c ), we know that there exists a z 6 = i suc h that ( s P ) z i 6 = 0. If 1 ≤ z ≤ q − c w e can aga in pro ceed as in the pro of of Theorem 3.1 3 to co nstruct a plant P for which J P (Γ( P )) > J P (Γ Θ ( P )). Thu s, without lo ss of genera lit y , we assume that q − c + 1 ≤ z ≤ q . Cho ose an index z 1 ∈ I z and consider the ma tr ix A deﬁned by [ A ] i = [ ¯ A ] i , a z 1 i 1 = r and all other entries equal to zero. Then, [Γ( A, B )] i = Λ i ([ A ] i , B ii )[ A ] i , [Γ( A, B )] z = − b z 1 z 1 / (1 + b 2 z 1 z 1 )[ A ] z (beca use Γ / ∈ W 0 ∪ L c ), and [Γ( A, B )] t = 0 for all t 6 = i, z . Denoting Γ( A, B ) by K , we see that J ( A,B ,x 0 ) (Γ( A, B )) ≥ x T 0 [( A + B K ) T K T K ( A + B K ) +(( A + B K ) 2 ) T ( A + B K ) 2 ] x 0 for all B ∈ B ( ǫ ) and x 0 . T aking x 0 = e j 1 then res ults in J ( A,B ,e j 1 ) (Γ( A, B )) − J ( A,B ,e j 1 ) (Γ Θ ( A, B )) ≥ [ k 2 z 1 i 1 + ( r + b z 1 z 1 k z 1 i 1 ) 2 ]( a i 1 j 1 + b i 1 i 1 k i 1 j 1 ) 2 − X t ∈I i a 2 tj 1 b 2 tt . (38) Note that, irresp ective of the choice o f the con troller gain k z 1 i 1 , k 2 z 1 i 1 + ( r + b z 1 z 1 k z 1 i 1 ) 2 ≥ r 2 / (1 + b 2 z 1 z 1 ) , 34 and as a result, lim r → + ∞ [ k 2 z 1 i 1 + ( r + b z 1 z 1 k z 1 i 1 ) 2 ]( a i 1 j 1 + b i 1 i 1 k i 1 j 1 ) 2 = + ∞ , bec ause a i 1 j 1 + b i 1 i 1 k i 1 j 1 6 = 0 . Hence, we can a lw ays construct A with a ppropriate choice of index z and a scala r r large enough to make the cost diﬀerence p ositive. As a result, Γ ca nnot dominate Γ Θ . Finally , w e prove that Γ Θ is undominated by control desig n metho ds in W 2 . Let Γ ∈ W 2 and index 1 ≤ i ≤ q − c and model sub-matrices [ ¯ A ] i and B ii such that I + Λ i ([ ¯ A ] i , B ii ) 6 = 0. Ther efore, we know that there exists at least one index i 1 ∈ I i such that ¯ A i 1 6 = 0 and ¯ A i 1 + b i 1 i 1 Γ i 1 ( ¯ A, B ) 6 = 0. Based on the fact that node i is not a sink, we know that there exis ts z 6 = i such that ( s P ) z i 6 = 0. F o r the sa me r easons as b efore we again restrict o ur selves, without lo ss of generality , to the ca se where q − c + 1 ≤ z ≤ q . Consider the matrix A deﬁned by [ A ] i = [ ¯ A ] i and a z 1 i 1 = r and a ll other entries of A equal to zer o. Then, [ A ] i + [Γ( A, B )] i = ( I + Λ i ([ A ] i , B ii ))[ A ] i and [Γ( A, B )] z = − b z 1 z 1 / (1 + b 2 z 1 z 1 )[ A ] z (beca use Γ / ∈ W 0 ∪ L c ). Again, K will sta nd fo r Γ( A, B ). Then, for a ll B ∈ B ( ǫ ) and x 0 J ( A,B ,x 0 ) (Γ( A, B )) − J ( A,B ,x 0 ) (Γ Θ ( A, B )) ≥ x T 0 ( A i 1 + b i 1 i 1 Γ i 1 ( A, B )) T ( A i 1 + b i 1 i 1 Γ i 1 ( A, B )) x 0 × r 2 b 2 z 1 z 1 / (1 + b 2 z 1 z 1 ) 2 − X t ∈I i x T 0 A T t A t x 0 /b 2 tt , and hence, s ince A i 1 + b i 1 i 1 Γ i 1 ( A, B ) 6 = 0, we can choose r large enough to ensure that this diﬀerence is strictly p ositive for some x 0 ∈ R n since the inner matrix will hav e a str ictly pos itiv e eigenv alue for large v alues of r . Thus, the control desig n strategy Γ ∈ W 2 cannot dominate the control design Γ Θ . 35

Optimal Structured Static State-Feedback Control Design with Limited Model Information for Fully-Actuated Systems

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