3 Riccati Design for Synchronization of Continuous-Time Systems

Transcription

1 3 Rccat Desgn for Synchronzaton of Contnuous-Tme Systems Ths chapter studes cooperatve trackng control of mult-agent dynamcal systems nterconnected by a fxed communcaton graph topology. Each agent or node s mathematcally modeled by dentcal contnuous lnear tme-nvarant (LTI) systems, whch ncludes the sngle ntegrator and double ntegrator as specal cases. The communcaton network among the agents s descrbed by a drected graph. A command generator or leader node generates the desred trackng trajectory to whch all agents should synchronze. Only a few nodes are aware of nformaton from the leader node. A local Rccat desgn approach s ntroduced here to synthesze the dstrbuted cooperatve control protocols. A framework for cooperatve trackng control s proposed, ncludng full state feedback control protocols, observer desgn, and dynamc output regulator control. The classcal system theory noton of dualty s extended to networked cooperatve systems on graphs. It s shown that the local Rccat desgn method guarantees synchronzaton of mult-agent systems regardless of graph topology, as long as certan connectvty propertes hold. Ths s formalzed through the noton of synchronzaton regon. It s shown that the Rccat desgn method yelds unbounded synchronzaton regons and so acheves synchronzaton on arbtrary dgraphs contanng a spannng tree. 3.. Dualty, Stablty, and Optmalty for Cooperatve Control In system theory, the relatons between state varable feedback desgn, observer desgn, and output regulator desgn are well known and provde a beautful overall theory that reveals deep structural relatons for dynamcal systems n terms of dualty theory and other well developed notons. In nter-communcatng cooperatve control systems, however, the desgn of feedback control protocols s complcated by the fact that the communcaton graph topology can severely restrct what can be accomplshed by usng dstrbuted feedback control protocols, where the control polcy of each agent s allowed to depend only on ts own nformaton and

2 local nformaton avalable from ts neghbors n the graph. In ths chapter we show that deas of state feedback, observer desgn, and output regulator desgn can be extended to the case of cooperatve control on graphs n a natural by usng locally optmal desgn n terms of local Rccat equatons. The relatons between stablty and optmalty have long been debated n the communty and are now for the most part understood for sngle-agent systems. However, relatons of stablty and optmalty n mult-agent cooperatve control systems are only now begnnng to be clarfed. There are more ntrgung relatons between stablty and optmalty n cooperatve control than appear n the sngle-agent case, snce local stablty and global team stablty are not the same, and local optmalty and global team optmalty are not the same. In ths chapter we dscuss the desgn of cooperatve control protocols that are dstrbuted, n the sense that each agent s allowed to use only nformaton from tself and ts neghbors n the prescrbed communcaton graph topology. The objectve s for the states of all agents to synchronze to the state of a leader node, whch can be vewed as a command generator exosystem that generates a desred trajectory. Only a few agents have access to nformaton from the leader node. Frst we consder the case of cooperatve control usng full state varable feedback, then cooperatve observer desgn n the case of output feedback or ncomplete state measurements, then cooperatve output regulator protocol desgn. In Secton 3. we assume that the full state of each agent s avalable for feedback control desgn by tself and ts neghbors n the graph. Locally optmal desgn n terms of solvng local Rccat equatons s shown to guarantee synchronzaton on arbtrary graph topologes that have a spannng tree. The dea of synchronzaton regon s gven n Secton 3.3. In Secton 3.4 we desgn cooperatve state observers based on output feedback, or reduced state nformaton. A dualty theory for cooperatve control systems on graphs s gven n Secton 3.5, and t nvolves the reverse graph. In Secton 3.6 we show three methods for desgnng cooperatve dynamc regulators that guarantee synchronzaton usng only output feedback. These three methods depend on dfferent methods of exchangng nformaton between neghbors n the graph. The results n ths chapter come from [3]. 3.. State Feedback Desgn of Cooperatve Control Protocols In ths secton we desgn cooperatve controllers usng state varable feedback (SVFB), assumng that the full state of each agent s avalable for feedback control desgn by tself and ts neghbors n the communcaton graph. We requre that any control protocol be dstrbuted n the sense that the control for agent depends only on ts own nformaton and that from ts neghbors n the graph. It s shown that locally optmal desgn n terms of a local Rccat equaton, along wth selec-

3 3 ton of a sutable dstrbuted control protocol, guarantees synchronzaton on arbtrary communcaton graphs that have a spannng tree. 3.. Synchronzaton of Mult-agent Systems on Graphs Much attenton was pad early on n the cooperatve control lterature to the leaderless consensus problem of sngle ntegrator and double ntegrator dynamcs [],[5],[7]. Under proper selecton of a local neghborhood votng protocol, t was shown that f the graph has a spannng tree all nodes reach a consensus value that s a weghted average of the ntal condtons of the agent dynamcs. In ths chapter, we shall study the more general case where each agent has dynamcs of a contnuous lnear tme-nvarant (LTI) system. LTI systems form an mportant class of systems; they nclude the sngle ntegrator, double ntegrator, and hgher-order ntegrator dynamcs as specal cases. A large class of engneerng systems can be modeled by LTI systems, such as mechancal systems (e.g. masssprng-damper systems), electrcal systems (e.g. RLC crcuts, op-amp crcuts), and electromechancal systems (e.g. armature controlled drect current servomotor wth a load) [],[6]. We call the case where consensus s sought among agents when there s no leader node the cooperatve regulator problem. Here, we consder the cooperatve trackng problem, where there are dentcal follower agents or nodes and one leader node. Each follower node has dentcal dynamcs and s modeled by an LTI system x Ax Bu, y Cx, (3.) n m p where x s the state, u s the nput, y s the measured output, and {,,, }. The trple (A, B, C) s assumed to be stablzable and detectable. These follower nodes can communcate n a certan way, and the communcaton network s represented by a graph {, } wth nodes { v, v,, v }, and a set of edges or arcs. Let the assocated adjacency matrx be [ a j ], where aj s the weght for edge ( vj, v ). If there s an edge from node j to node (.e. ( vjv) ), aj. Ths means that node j s a neghbor of node, and node can get nformaton from node j. The neghbor set of node s denoted as { ja }. We assume there s no self loop,.e., a. The dynamcs of a leader or control node, labeled, s gven by j x Ax, y Cx (3.)

4 4 n p where x s the state and y s the measured output. The leader node can be consdered as a command generator exosystem that generates the desred target trajectory. It s an autonomous system and s not affected by any follower node. The objectve of the cooperatve trackng problem s to desgn local dstrbuted controllers u for all follower nodes ( ), such that all follower nodes track the state trajectory of the leader node,.e., lm t( x() t x()) t,. Then we say that the states of all agents synchronze to the state of the command generator. Remark 3.. Though the leader dynamcs (3.) s autonomous and has no control nput, t can generate a large class of useful command trajectores. The nterestng cases are when matrx A s not stable. The types of trajectores that can be generated by the leader nclude unt step (poston command), the ramp (useful n velocty trackng systems, e.g., satellte antenna pontng), snusodal waveforms (useful, e.g., n hard dsk drve control), and more. In ths chapter, A can be ether stable, margnally stable, or even unstable, as long as (A,B) s stablzable. Defne the augmented graph as {, }, where { vv,, v } s the node set, ncludng the leader node and all follower nodes, and s the edge set. ote that. For the cooperatve trackng problem, we need the followng assumpton on the graph topology. Assumpton 3.. The augmented graph contans a spannng tree wth the root node beng the leader node. In other words, there s a drected path (not necessarly unque) from the leader node to every follower node. Assumpton 3. s a necessary condton for solvng the cooperatve trackng problem. Ths s llustrated n the followng remark n an ntutve way. Remark 3.. Suppose Assumpton 3. does not hold and there s a set of k ( k ) nodes n graph,.e. Sk { n n,, nk} and Sk, whch do not have drected paths from the leader node. Ths further mples that the set of nodes S k do not have access to nformaton of the rest of the nodes S k {, } Sk. It s obvous that there must exst some node (nodes) n the set S k, whch s (are) ether solated (.e. solated sngle node or solated subgroup), or has an edge (have edges) to at least one node n S k. These cases are depcted n Fg. 3.. Obvously, n these cases, synchronzaton to the leader node cannot be acheved.

5 5 leader Fg. 3.. Graph topologes dscussed n Remark 3. Remark 3.3. Assumpton 3. ncludes the followng classes of graph topologes (see Fg. 3.) as specal cases. (a) Graph s strongly connected and at least one follower node n graph can get nformaton from the leader node [4],[5]. (b) Graph contans a spannng tree and at least the root node can get access to the leader node [4],[3]. (c) The augmented graph s tself a spannng tree wth the root beng the leader node [3]. (d) The augmented graph has a herarchcal structure [6]. (e) The augmented graph has a spannng tree, but graph s dsconnected and each separated subgroup ether s a sngle node or contans a spannng tree. The statement that a graph s a spannng tree (e.g. Fg. 3. (c)) s dfferent from the statement that a graph has (or contans) a spannng tree (e.g. Fg. 3. (a)-(e)). The former s a specal case of the latter. A graph s a spannng tree f t has a spannng tree, and each node has exactly one parent node, except for the root node whch does not have parent node.

6 (a) (b) (c) (d) (e) Fg. 3.. Fve classes of graph topologes descrbed n Remark Cooperatve SVFB Control In ths Secton 3., we assume that full state varable of each node s measurable, and desgn a cooperatve state varable feedback (SVFB) control law of dstrbuted form usng a Rccat desgn approach [3],[]. Ths s the case C=I n (3.). The case when C I and the measured output only contans reduced state nformaton s addressed n Sectons 3.4, 3.5, 3.6. We assume that the leader node can only be observed by a small subset of the follower nodes. Ths s often the case n practce when follower nodes are sparsely populated and lmted sensng ablty s taken nto account. If node observes the leader, an edge ( v v ) s sad to exst wth a weght g. The weghts g have been called pnnng gans n [7],[4],[] and f g then node s sad to be pnned to the leader. To make the controller fully dstrbuted, the control law of each agent must respect the graph topology and can only use the local neghborhood nformaton of that agent. Defne the local neghborhood trackng error of node as []

7 7 a ( x x ) g ( x x ) (3.3) j j j Ths can also be wrtten n the form ( d g) x ax gx d g j j j (3.4) where d a s the n-degree of node. Whle (3.3) can be nterpreted as a j j weghted sum of dfferences between the states of node and ts neghbors, (3.4) descrbes the dfference between the state of node and the weghted average center of ts neghbors states. Consder a statc SVFB control protocol for each node gven as u ck (3.5) mn wth couplng gan c and feedback gan matrx K. These controllers are dstrbuted n the sense that they are mplemented at each node usng only the local neghborhood trackng error nformaton. The task s to fnd sutable c and K such that all agents synchronze to the leader node for the gven graph topology. The closed-loop system of node s x Ax cbk aj( x j x) g( x x) j For ease of analyss, wrte the overall global closed-loop dynamcs as ( ) ( ( ) ) x I Ac LG BK x c LG BK x n where the global state x col( xx,, x ) s the columnwse concatenaton n of local state vectors x,, x, and x col( xx,, x). The Kronecker product s. The dentty matrx s I and G dag( g, g,, g ) s the dagonal matrx of pnnng gans whch descrbes the connectons between the leader node and follower nodes. The Laplacan matrx assocated wth graph s defned by L [ l ] D, wth D dag( dd,, d ) the dagonal matrx of ndegrees. It s apparent that j l j a, j j, a, j. k k

8 8 Defne the global system and control nput matrces Ac I Ac( LG) BK B c( LG) BK. c Matrx A c reflects the local agent closed-loop matrx A BK as modfed on the graph structure L G. ote that the graph structure L+G appears on the left-hand sde of the Kronecker product, whle the local state nformaton appears on the rght. Denote the dsagreement error wth respect to the leader for each node as x x. Defne the global dsagreement vector [5] as n col(,, ) x x (3.6) Then the cooperatve trackng problem s solved f lm t ( t). The global dsagreement error dynamcs s gven by x x A (3.7) c ote that the dsagreement error x x s not avalable to node unless t s pnned to the leader node (that s, g ), whereas the local neghborhood trackng error (3.3) s known to each node. Before movng on, we need the followng lemmas. In these lemmas, the notatons A and a stand for a general matrx A and ts entres, and are not related to j the system matrx A or the graph adjacency matrx entres a. j nn Lemma 3.. Geršgorn Dsc Theorem [9]. Let a matrx A[ a j ]. Then all egenvalues of A resde n the unon of the n dscs n n C a aj j j nn Lemma 3.. [] Let a matrx A[ a j ] and n J {,, n} a aj j j If for each J, there s a sequence of nonzero elements of A of the form a a a wth j J. Then A s nonsngular. l j

9 9 Lemma 3.3. Under Assumpton 3., the matrx L G s nonsngular. Moreover, all ts egenvalues are located n the open rght-half plane. Proof. Under Assumpton 3., at least one node n can get nformaton drectly from the leader node,.e., g for at least one. Wthout loss of generalty, we assume there are two nodes r and r such that g and g. Snce l, j j l and lj, l l. j j j Then for matrx L G, l g l j j j for all and strct nequalty holds for { r r}. Assumpton 3. mples that, for any other node whch does not have drect access to the leader node (.e., { r r }), there must be a drect path ether orgnated from node r or node r. Accordng to Lemma 3., L G s nonsngular. By Geršgorn dsc theorem, all egenvalues of L G le wthn the unon { Re( ) } {}. The fact that L G s nonsngular further mples all ts egenvalues are located n the open rght-half plane. The next result (cf. [8]) provdes a necessary and suffcent condton for asymptotc stablty of the dsagreement error dynamcs (3.7). Lemma 3.4. Let ( ) be the egenvalues of L G, whch may or may not be dstnct. Then system (3.7) s asymptotcally stable f and only f all the matrces r r A c BK,,, (3.8) are asymptotcally stable. Proof. The zero state of system (3.7) s asymptotcally stable f and only f A c s Hurwtz,.e., all egenvalues of A c le n the open left-half plane. There exsts a nonsngular matrx S, such that ( ) J n ( ) J n ( ), J n ( ) k k S L G S J where n n nk and J,,, n J n J n are Jordan blocks of szes n k, n,... ote that the egenvalues of L+G need not be dstnct. Smlarty transformaton of matrx A gves a block trangular matrx c

10 A S I S I c ( n) Ac( n) ( S In) ( I A c( L G) BK)( S In) I AcJ BK (3.9) Denote the dagonal entres of J as {,,, },.e. [,,, ] [,,,, k,, k]. Snce the egenvalues of a block trangular n nk matrx are the unon of the sets of egenvalues of the dagonal blocks, A c s Hurwtz f and only f all the dagonal entres A c BK (,,, ) are Hurwtz. Hence under ths condton A s Hurwtz. Ths completes the proof. c If matrx A s stable, then even wth a couplng gan of c= all nodes synchronze to the zero state. If the command generator matrx A s not stable (see Remark 3.), then t s nferred from Lemma 3.4 that all egenvalues of L+G must be nonzero. By Lemma 3.3, f the graph has a spannng tree wth the leader as the root node, then all egenvalues of L+G have postve real parts. Then, proper desgn of c, K to satsfy condton (3.8) guarantees synchronzaton. Lemma 3.4 mples that an arbtrary SVFB control gan K that stablzes the sngle-agent dynamcs A BK may fal to stablze the global dynamcs A c for a prescrbed graph structure. That s, though ndvdual agents may have stable dynamcs n solaton, when they are connected together by a communcaton graph, ther stablty may be destroyed. Ths s llustrated n Example 3. n Secton 3.3. The condton (3.8) s dffcult to guarantee and has been a stumblng block to the desgn of dstrbuted control protocols that guarantee synchronzaton on a gven graph topology Local Rccat Desgn of Synchronzng Protocols Our objectve now s to overcome the couplng between the local agent feedback control desgn and the graph topologcal propertes that s dsplayed n Lemma 3.4, and to provde a desgn method for feedback matrx K n protocol (3.5) that s ndependent of the graph topology. That s, we wsh to decouple the desgn of the control protocol from the graph propertes, and show how to desgn a protocol that guarantees synchronzaton on any drected graph wth sutable propertes. The next result shows how to select SVFB control gan K to guarantee stablty on arbtrary dgraphs satsfyng Assumpton 3. by usng a local Rccat desgn approach [3] and proper choce of the couplng gan c. Theorem 3.. Consder local dstrbuted control protocols (3.5). Suppose (A,B) s nn mm stablzable and let desgn matrces Q and R be postve defnte. Desgn the SVFB control gan K as

11 K T R B P (3.) where P s the unque postve defnte soluton of the control algebrac Rccat equaton (ARE) T T A PPAQPBR B P (3.) Then under Assumpton 3., the global dsagreement error dynamcs (3.7) s asymptotcally stable f the couplng gan satsfes c mn Re( ) (3.) wth ( ) the egenvalues of L G. Then, all agents synchronze to the leader node state trajectory. Proof. Let the egenvalues of L G be j, where,. Then by Lemma 3.3,,. Consderng (3.) and (3.), straghtforward computaton gves the Lyapunov equaton ( A c BK) P P( Ac BK) Q (c ) K T RK, where the superscrpt denotes conjugate transpose of a complex matrx. Snce P and Q, by Lyapunov theory [] matrx A c BK s Hurwtz f c /,. Then Lemma 3.4 completes the proof. The mportance of ths result s that t decouples the local SVFB gan desgn at each agent from the detals of the nterconnectng communcaton graph structure. The SVFB gan s desgned usng the control ARE (3.), and then the graph topology comes nto the choce of the couplng gan c through condton (3.). ote that the SVFB gan n Theorem 3. s the optmal gan [3] that mnmzes the local performance ndex J xqx urudt (3.3) T T subject to the local agent dynamcs (3.). Therefore, the theorem shows that locally optmal SVFB control desgn by each agent guarantees the synchronzaton of all agents on any graph that has a spannng tree wth the leader as the root node. The locally optmal desgn method n Theorem 3. s scalable, snce t only depends on the local agent dynamcs, whch have state space of dmenson n, and does not depend on the sze of the graph.

12 It s stressed that the locally optmal desgn that mnmzes the local agent performance ndex (3.3) does not guarantee the global optmalty of the moton of all the agents as a team. Globally optmal desgn for mult-agent systems s dscussed n Chapter Regon of Synchronzaton From Lemma 3.4, one sees that wthout usng Rccat desgn, one can stll fnd sutable couplng gan c and control gan K such that asymptotc trackng can be acheved by determnng a SVFB gan that stablzes (3.8) for all egenvalues of L G. The Rccat desgn n Theorem 3. decouples the desgn of the local agent feedback protocols from the graph propertes and so provdes a smple way of fndng such a gan. Ths secton shows another pont of vew of ths ssue by descrbng the concept of synchronzaton regon [7],[4],[]. Ths concept provdes a natural way to evaluate the performance of synchronzaton control laws by revealng how synchronzablty depends on structural parameters of the communcaton graph. Defnton 3.. Synchronzaton Regon []. Gven system (A,B) and feedback matrx K, the synchronzaton regon s the regon n the complex plane defned as S { s AsBK shurwtz}. Our applcaton of ths defnton s n regards to the dstrbuted control protocol (3.5). By Lemma 3.4 and Defnton 3., synchronzaton s acheved f c fall nto the synchronzaton regon S for all. Clearly, the synchronzaton regon depends on both the control matrx K and the couplng gan c. It s shown n [7] that, for unsutable choces of K the synchronzaton regon may be dsconnected. A synchronzaton regon that forms an connected unbounded rght-half plane s a desrable property of a consensus protocol [4], snce, accordng to Lemma 3.3, such a regon contans all possble graph egenvalues for approprate choce of couplng gan c. By Lemma 3.4, an unbounded synchronzaton regon mples that the same control gan K guarantees synchronzaton for any dgraph contanng a spannng tree. Corollary 3.. For protocol (3.5) wth the Rccat desgn-based control gan (3.), the synchronzaton regon s unbounded. More specfcally, a conservatve estmate for the synchronzaton regon s S { j [/, ), (, )}. Proof. Let s j. Usng the same development as n Theorem 3., we have T ( A sbk ) P P( A sbk ) Q ( ) K RK

13 3 Snce P, Q and R, A sbk s Hurwtz f and only f Q( ) K T RK. A suffcent condton s /. Ths completes the proof. The followng example shows that a random stablzng control gan K may yeld a bounded synchronzaton regon, whle the Rccat desgn-based control gan (3.) renders an unbounded synchronzaton regon. Example 3.. Consder the agent dynamcs (3.) wth matrces [4] A, B The feedback gan K [.5,.5] stablzes A-BK. The synchronzaton regon for x x x ths gan matrx s S x jy x ; y, whch s shadowed 8 n Fg If, for a gven graph, the egenvalues of the Laplacan matrx L fall wthn the regon S, synchronzaton occurs. However, ths s a severe restrcton on the allowable forms of communcaton between the agents. ote n partcular that f the communcaton graph s a cycle, the egenvalues of L are unformly dstrbuted about a crcle n the rght-half plane (Chapter ), and so would n lkelhood not fall wthn ths synchronzaton regon. ow consder the Rccat desgn-based feedback gan K [.544,.89] provded by Theorem 3. wth Q I and R. The synchronzaton regon s S x jy x x y x x.544 ;( ).36(.54 ), whch s unbounded as shadowed n Fg Lemma 4 n [4] s used n computng the synchronzaton regon. ote that the regon ndcated n Fg. 3.4 s the actual synchronzaton regon determned from solvng the nequalty gven. The regon gven by Corollary 3. s the regon { x jy x[/, ), y(, )}, whch s a conservatve estmate of S.

14 4 y Synchronzaton Regon x Fg Bounded synchronzaton regon for arbtrary stablzng SVFB gan 5 y 5 Synchronzaton Regon x Fg Unbounded synchronzaton regon for Rccat-based SVFB gan 3.4. Cooperatve Observer Desgn In Secton 3., a cooperatve trackng controller s desgned by assumng that the full state nformaton of all nodes can be measured. Unfortunately, nature seldom cooperates so eagerly. In practce, only lmted output nformaton can be measured. In ths case, for sngle-agent systems t s well understood how to desgn a dynamcal output regulator that stablzes a system or enforces a desred trackng

15 5 behavor. Some of these regulator desgns are based on state observers [], whch take the avalable output measurements and provde estmates of the full system state. In ths secton, we nvestgate cooperatve observer desgn for networked mult-agent systems, where the graph topology nteracts wth the local agent observer desgn [],[3]. In Secton 3.6 we show how to use the state feedback protocols n Secton 3. along wth these cooperatve observers to desgn dynamc regulators that guarantee synchronzaton usng only the avalable output measurements. Consder system (3.) for the -th agent, that s x Ax Bu, y Cx, (3.4) n Denote xˆ as the estmate of the state x, yˆ ˆ Cx as the consequent estmate of the output y. Defne the state estmaton error x ˆ x x and the output estmaton error y ˆ y y for node. Standard observer desgn for a sngle system often uses the output estmaton error nformaton y y yˆ. Smlarly, for cooperatve observer desgn, takng the graph topology nto account, we defne the local neghborhood output estmaton error for node as a ( y y ) g ( y y ) (3.5) j j j whch depends on the output estmaton errors of node and ts neghbors. The cooperatve observer for each node ( ) s desgned n the form xˆ Axˆ Bu cf (3.6) np where c s the couplng gan, and F s the observer gan. These observers are completely dstrbuted n the sense that each observer only requres ts own output estmaton error nformaton and that of ts neghbors. The cooperatve observer desgn problem s to select the couplng gan c and the observer gan F so that the state estmaton error x ˆ x xof all agents go to zero. Snce the leader node acts as a command generator, t s reasonable to assume that the leader node knows ts own state,.e., ˆx x and y. Then the global cooperatve observer dynamcs s x ˆ Axˆ ( I Bu ) c[( LG) F] y, where xˆ col( xˆ ˆ ˆ, x,, x ), y col( y, y,, y ), u col( u, u,, u ) and

16 6 A I Ac( LG) FC Matrx A reflects the local agent observer matrx A FC as modfed by the graph structure L G. ote that graph topology nformaton appears on the lefthand sde of the Kronecker product whle local agent nformaton appears on the rght. Let the global state estmaton error be x x xˆ (3.7) Then straghtforward computaton gves the dynamcs of the state estmaton error x Ax (3.8) ow the task for cooperatve observer desgn s to fnd a sutable couplng gan c and observer gan F, such that the matrx A s Hurwtz. Then the state estmaton error goes to zero. The next result s dual to Lemma 3.4 and provdes a necessary and suffcent condton for asymptotcal stablty of the global cooperatve observer dynamcs. Lemma 3.5. Let ( ) be the egenvalues of ( L G). Then the state estmaton error dynamcs (3.8) s asymptotcally stable f and only f all the matrces A c FC,, (3.9) are Hurwtz,.e., asymptotcally stable. Proof. See the proof of Lemma 3.4. If the command generator matrx A s not stable (see Remark 3.), then t s nferred from Lemma 3.5 that all egenvalues of L+G must be nonzero. By Lemma 3.3, f the graph has a spannng tree wth the leader as the root node, then all egenvalues of L+G have postve real parts. Then, proper desgn of c, F to satsfy condton (3.9) guarantees that all state estmaton errors converge to zero usng the cooperatve observers (3.6) Lemma 3.5 shows how the graph topology nterferes wth the desgn of the local agent observers. Ths s undesrable. We now wsh to devse a desgn method that decouples the local agent observer desgn from the graph propertes. Smlar to Theorem 3., the next result shows how to select the observer gan F to guarantee that (3.9) s satsfed on arbtrary dgraphs satsfyng Assumpton 3. by usng a local Rccat desgn approach [] and proper choce of the couplng gan c.

17 7 nn Theorem 3.. Suppose (A,C) s detectable and let desgn matrces Q and R pp be postve defnte. Desgn the observer gan F as F T PC R (3.) where P s the unque postve defnte soluton of the observer ARE T T APPA QPC R CP. (3.) Then the estmaton error dynamcs (3.8) are asymptotcally stable f the couplng gan satsfes c mn Re( ) (3.) wth ( ) be the egenvalues of ( L G). Proof. The development s the same as n Theorem 3.. Except that here we have * ( FC P P A cfc ( T A c ) ( ) Q c ) FRF The mportance of ths theorem s that t decouples the local observer gan desgn at each agent from the detals of the nterconnectng communcaton graph structure. The observer gan s desgned usng the observer ARE (3.) [], and then the graph topology comes nto the choce of the couplng gan c through condton (3.). The local Rccat desgn method n Theorem 3. s scalable, snce t only depends on the local agent dynamcs, whch have state space of dmenson n, and does not depend on the sze of the graph Dualty for Cooperatve Systems on Graphs As s well known n classcal control theory, controller desgn and observer desgn are dual problems []. One outcome of dualty s that these two problems are equvalent to the same algebrac problem. Formally, gven system (3.), (3.4), whch we denote as (A,B,C), the dual system s defned to be x Ax T Cu T, y Cx T (3.3)

18 8 T T T whch we denote as ( A C B ). A standard result s that system (3.4) s reachable (whch depends on the propertes of the nput-couplng par (A,B)) f and only f system (3.3) s detectable (whch depends on the propertes of the outputcouplng par ( A, B ). Moreover, replacng (A,B) n (3.) by ( T T T T A C ) results n (3.), so that the control ARE (3.) s dual to the observer ARE (3.). Therefore, the observer gan F n Theorem 3. can be found by computng the SVFB gan K n Theorem 3., wth (A,B) replaced by ( T T T A C ) and then settng F K []. Another remarkable nsght provded by dualty theory for standard sngle systems s that control desgn s dual to observer desgn and estmaton f the tme s reversed. Thus, optmal control s fundamentally a backwards-n-tme problem [3], as captured by the dea of dynamc programmng, whch proceeds backwards from a desred goal state. On the other hand, optmal estmaton s fundamentally a forwards-n-tme problem []. A complete theory of dualty s provded n the theory of electrc crcuts, where such notons are formally captured. Unfortunately, n nter-communcatng cooperatve mult-agent systems, the desgn of cooperatve feedback control protocols and cooperatve observers s complcated by the fact that the communcaton graph topology can severely restrct what can be accomplshed by usng dstrbuted protocols, where the polcy of each agent s allowed to depend only on ts own nformaton and local nformaton avalable from ts neghbors n the graph. Therefore, the theory of dualty has not been fully developed for cooperatve control systems. In ths secton, we extend the mportant concept of dualty to networked cooperatve lnear systems. We show that cooperatve SVFB controller desgn and cooperatve observer desgn are also dual problems under sutable defntons. The next defnton captures the noton of tme reversal n cooperatve systems on graphs. Defnton 3.. Reverse Dgraph. Gven a dgraph, the reverse dgraph s the graph wth the same nodes as dgraph, and wth the drectons of all edges reversed. Thus, f the adjacency matrx of dgraph s [ a j ], then the adjacency matrx of ts reverse dgraph s [ a j T ]. Ths means that the row sums n the reverse graph correspond to column sums n the orgnal graph, so that the roles of n-neghbors and out-neghbors are exchanged. Lemma 3.6. If the dgraph s balanced, then the transpose of ts Laplacan matrx L s the same as the Laplacan matrx L of ts reverse dgraph,.e. T L L.

19 9 Theorem 3.3. Dualty of cooperatve systems on graphs. Consder a networked system of dentcal lnear dynamcs ( ABC,, ) on a balanced communcaton graph. Suppose the SVFB gan K stablzes the synchronzaton error dynamcs (3.7). Then the observer gan K T stablzes the state estmaton error dynamcs (3.8) for a networked dual system of dentcal lnear dynamcs T T T ( A C B ) on the reverse graph. Proof. Consder the networked system wth ( ABC,, ) and graph. Under SVFB gan K, the synchronzaton error dynamcs (3.7) s [ I Ac( LG) BK] A (3.4) T T T For the networked dual system wth ( A C B ) and reverse graph, under the T observer gan K, the state estmaton error dynamcs (3.8) s [( T ) ( ) ( T T x I A c L G K B )] x A x Snce G dag( g ), t s follows that A ( A ) T c. Ths proves the dualty. The equvalence between these dual problems s summarzed as follows c o A A T B C T C B T L L Dualty of SVFB control and output feedback (OPFB) control s shown n [] for leaderless consensus on the same graph. Ths s dfferent from the dualty presented n Theorem 3.3,.e., dualty of SVFB control desgn and observer desgn for cooperatve trackng problems on reverse graphs. Theorem 3.3 extends the classcal concept of dualty n control theory [] to networked systems on graphs Cooperatve Dynamc Regulators for Synchronzaton One approach for the desgn of controllers based on reduced state or output feedback (OPFB) s to ntegrate an observer wth a state feedback controller. Then, the controller employs state estmate nformaton nstead of actual measured state nformaton. Snce ths state estmate nformaton s generated by an observer,

20 whch s a dynamcal system usng only system nput and output nformaton, ths ntegrated controller s called a dynamc output feedback regulator or tracker. The dynamcs of the regulator are provded by the observer. In ths secton, we propose three cooperatve dynamc output regulators to solve the cooperatve trackng problem. The Rccat desgn-based control gan (3.) and/or Rccat desgn based observer gan (3.) are used n these three dynamc output feedback trackng controllers, all of whch yeld unbounded synchronzaton regons. Ths hghlghts the mportance of local Rccat desgn approach n cooperatve control of networked systems. It also hghlghts the ncreased rchness of structure nherent n cooperatve control on graphs, whch admt three natural structures of regulators nstead of the sngle standard regulator based on SVFB and full observer desgn n the sngle system case. The work n ths secton comes from [3]. Varous papers have desgned varants of the three regulators descrbed n ths secton ncludng [7][4] eghborhood Controller and eghborhood Observer Recall the full SVFB cooperatve control protocol (3.5), that s u ck. Recall the local neghborhood output estmaton error (3.5) for node, re-wrtten here as a ( y y ) g ( y y ) (3.5) j j j Defne the local neghborhood state estmaton trackng error as j xˆ j xˆ xˆ xˆ (3.6) j ˆ a ( ) g ( ) These two local neghborhood errors represent the complete nformaton avalable to node based on measured output nformaton from tself and ts neghbors n the graph. In the frst regulator protocol, both controller and observer are desgned usng ths complete avalable local neghborhood nformaton. Ths cooperatve regulator s desgned usng OPFB as u ckˆ (3.7) xˆ Axˆ Bu cf (3.8) Then, the closed-loop dynamcs of node s

21 x Ax cbk ( ˆ j ˆ) ( ˆ ˆ aj x x g x x) j xˆ Axˆ Bu cf aj( y j y) g( y y) j Assume xˆ x for the leader node. Consderng the dentty ( L BK) x whch roots n the fact L, wth the -vector of s, the global closedloop dynamcs can be wrtten as x ( I A) xc[( LG) BK]( xˆ x ) (3.9) x ˆ [ I Ac( LG) FC] xˆ c[( LG) F] y c[( LG) BK] ( xˆ x ) (3.3) Theorem 3.4. Let (A,B,C) be stablzable and detectable and the graph have a spannng tree wth the leader node as the root. Let the cooperatve dynamc OPFB trackng control law be (3.7) and (3.8). Desgn the SVFB gan K and couplng gan c accordng to Theorem 3. and the observer gan F accordng to Theorem 3.. Then all nodes,,, synchronze to the leader node asymptotcally and the state estmaton error x n (3.7) approaches zero asymptotcally. Proof. Equaton (3.9) can be wrtten as x AxB( x x). c c Further computaton gves A B x c c The global state estmaton error dynamcs (3.8) s Then one has x xx ˆ [ I Ac( LG) FC] x A x A B c c x A o x o

22 The egenvalues of a block trangular matrx are the combned egenvalues of ts dagonal blocks. Theorem 3. and Theorem 3. guarantee A c and A o are Hurwtz, respectvely. Ths completes the proof. ext, we analyze the performance of ths proposed OPFB cooperatve regulator usng the concept of synchronzaton regon. The synchronzaton regon for ths dynamc output feedback control law can be defned as follows. Defnton 3.3. Consder the OPFB cooperatve regulator (3.7) and (3.8). The synchronzaton regon s the complex regon defned as S { sc A sbk s Hurwtz} { sc A sfc s Hurwtz} OPFB Corollary 3.. Consder the OPFB cooperatve regulator (3.7) and (3.8). When the control gan K and observer gan F are desgned as n Theorem 3. and Theorem 3., the synchronzaton regon s an unbounded rght-half plane. More specfcally, a conservatve synchronzaton regon s S { j [ ) ( )}. OPFB Proof. Let s j. Usng the same development as n Theorem 3. and Theorem 3., we have T ( A sbk) P P( A sbk) Q ( ) K RK T ( A sfc) P P( A sfc) Q ( ) FRF Therefore, P, Q and R, / guarantees that both A sbk and A sfc are Hurwtz eghborhood Controller and Local Observer The OPFB cooperatve regulator just presented uses the complete neghborhood nformaton avalable for both the controller desgn (3.7) and the observer desgn (3.8). In cooperatve mult-agent systems on graphs, to acheve synchronzaton t s not necessary to use neghbor nformaton n both the control protocol and the observer protocol, as long as one or the other contans neghbor nformaton. Ths reveals a rchness of structure not appearng n the case of standard sngle agent systems. The OPFB cooperatve regulator desgned n ths secton uses the neghborhood estmate trackng error nformaton ˆ for the controller desgn and only the local agent output estmaton error nformaton y for the observer desgn. Here by local, we mean the nformaton only about node tself.

23 3 For node, the controller uses complete neghbor nformaton and s desgned as u ˆ ck ck aj( xˆ j xˆ) g( xˆ xˆ ) (3.3) j whereas the observer use only local agent nformaton and s desgned as The closed-loop dynamcs of node s x ˆ Axˆ Bu cfy (3.3) x Ax cbk j( ˆ j ˆ) ( ˆ ˆ a x x g x x ) j x ˆ ˆ ˆ Ax Bu cf y y Here, the second equaton s that of a local agent observer not coupled to nformaton from any neghbors. The global closed-loop dynamcs can be wrtten as x ( I ) [( ) ]( ˆ A xc LG BK xx) x ˆ [ I ( AcFC)] xˆc[( LG) BK]( xˆ x ) ( I cfc) x It can be shown that Ac Bc x I ( AcFC) x (3.33) Theorem 3.5. Let (A,B,C) be stablzable and detectable and the graph have a spannng tree wth the leader node as the root. Let the OPFB cooperatve dynamc trackng control law be (3.3) and (3.3). Desgn the SVFB gan K and couplng gan c accordng to Theorem 5.. Desgn the observer gan F such that the local agent observer ( A cfc) s Hurwtz. Then all nodes,,, synchronze to the leader node asymptotcally and the state estmaton error x approaches zero asymptotcally. Proof. Smlar to Theorem 3.4, thus omtted. Snce ( A, C ) s detectable, an F can be selected such that ( A cfc) s Hurwtz. Also, F can be desgned usng the local Rccat equaton as n Theorem, except that

24 4 F T PC R A smlar synchronzaton regon analyss for ths OPFB cooperatve regulator can be carred out as that n Secton Local Controller and eghborhood Observer Ths thrd OPFB cooperatve regulator desgn uses only the local agent state estmate for controller desgn and the complete neghborhood nformaton for observer desgn. For node, the controller porton s desgned usng local agent feedback based on a sort of estmate of the leaders state u K( xˆ xˆ ) (3.34) x ˆ ( A BK)ˆ x (3.35) and usng complete neghbor nformaton n the observer dynamcs x ˆ ( A BK) xˆcf aj ( y j y) g ( y Cxˆ y ) (3.36) j Ths thrd desgn s somewhat surprsng snce the controller uses no nformaton about the neghbors, yet the followng result shows that synchronzaton s reached usng ths rather odd OPFB cooperatve regulator. Theorem 3.6. Let (A,B,C) be stablzable and detectable and the graph have a spannng tree wth the leader node as the root. Let the cooperatve dynamc OPFB trackng control law be (3.34), (3.35) and (3.36). Desgn the SVFB gan K such that A BK s Hurwtz. Desgn the observer gan F and the couplng gan c accordng to Theorem 3.. Then all nodes,,, synchronze to the leader node asymptotcally. Proof. Defne col(,, ) wth x x, col(,, ) wth x x, and col( ). Then asymptotc stablty of mples that ˆ ˆ lmt( x( t) x( t)) and lmt ( xˆ ( t) xˆ ( t)),. Let xˆ ( t ) xˆ j( t ), then xˆ () t xˆ j() t,, j and t t. Straghtforward computaton yelds the dynamcs of as I A I BK A (3.37) cl ( G) FC I BK A o

25 5 Matrx A s smlar to the matrx In fact, T I n In I n. I ( ABK) I BK A ( ) I ABK I BK T AT, where A o o In T and I I n Therefore, s asymptotcally stable f and only f both I ( A BK) and A are Hurwtz, whch are equvalent to the condtons that A BK and A c FC, are Hurwtz (see Lemma 3.5). Theorem 3. guarantees stablty of A c FC. A sutable K can be selected snce ( AB, ) s stablzable. Smlar synchronzaton regon analyss can be carred out as n Secton ote that the dervaton of (3.37) requres the condtons xˆ ( t ) xˆ j( t ) j. A specal case s when xˆ ( t ),. Then xˆ ( t ),, t t and the control law (3.34) and (3.36) are smplfed as n u Kxˆ x ˆ ( ABK) xˆ cf aj ( y j y) g ( y y) j Remark 3.4. The observer (3.36) was proposed n [4]. It s not a true observer, n the sense that lm ( ˆ t x x) and lm ( ˆ t x x ). The varables x ˆ and xˆ only take roles of ntermedate varables n the controller desgn Smulaton Examples A large scope of ndustral applcatons can be modeled as the mass-sprng system, ncludng vbraton n mechancal systems, anmaton of deformable objects, etc. In ths example, we consder a cooperatve trackng problem wth one leader node and 6 follower nodes. Each node s a two-mass-sprng system wth a sngle force nput, except for the leader node, whch s unforced. The system s shown n Fg. 3.5, where m and m are masses, k and k are sprng constants, u s the force nput for mass, and y and, y are the dsplacements of the two masses.,

26 6 T Defne the state vector for node as x [ x xx3x4] [ y y yy ] T and the measured output nformaton as y [ ] T y y. Then, ths two-masssprng system can be modeled by x Ax Bu (3.38) y Cx (3.39) wth A kk k m m k k m m, and C. m, B y y,, u k k m m Fg Two-mass-sprng system Let the command generator leader node consst of an unforced two-mass-sprng system, producng a desred state trajectory. Sx two-mass-sprng systems act as follower nodes and these nodes receve state or output nformaton from ther neghbors, accordng to the communcaton graph topology descrbed n Fg The edge weghts of the lnks are shown n the fgure Fg Communcaton graph topology

27 7 The objectve of the cooperatve trackng control problem s to desgn dstrbuted controllers u for the follower nodes, such the dsplacements for both of the two masses synchronze to that of the leader node,.e. lmt( y y ) and lmt( y y) for,, 6. The dsplacement y of mass s dffcult to control usng only the force nput to mass. In the followng, we shall verfy the proposed cooperatve control algorthms, namely, the SVFB control law from Theorem 3. and the three types of dynamc OPFB cooperatve trackers n Secton 3.6. In the smulatons, m kg, m 9 kg, k.5 /m and k /m. The couplng gan c, whch satsfes the condton (3.), (3.) for ths graph topology. The ntal values x (,, 6) are random and are generated by the Matlab functon randn. a. SVFB Cooperatve tracker Frst, we assume full SVFB, so that the dsplacements and veloctes of both mass and mass are measurable for all agents. The dstrbuted control law s (3.5) wth c, K found for each agent by usng the Rccat equaton desgn method from Theorem. The trackng performances are depcted n Fg Fg These fgures show that dsplacements y y (,, 6) synchronze to the dsplacements y y of the leader node wthn a few seconds. Velocty trackng s also acheved and the fgures are omtted here to avod redundancy. b. OPFB Dynamc Cooperatve Trackers When the veloctes of mass and mass are not measurable, but only poston feedback s possble, we desgn three types of dynamc OPFB cooperatve controllers. We use the three desgn methods for dynamc cooperatve trackers gven n Secton 3.6. For convenence, we label the three cases as Case : OPFB wth neghborhood controller and neghborhood observer (Secton 3.6.). Case : OPFB wth neghborhood controller and local observer (Secton 3.6.). Case 3: OPFB wth local controller and neghborhood observer (Secton 3.6.3). In the smulaton, for Case, the observer gan s taken as F , whch makes A cfc Hurwtz. For Case 3, the control gan s taken as K , whch makes A BK Hurwtz.

28 8 Usng these types of dynamc OPFB cooperatve trackers, the dsplacement trackng performance and estmaton performance are llustrated, respectvely, n Fg. 3.9-Fg ote that Fg. 3.7 shows that, for Case 3, x ˆ s not a true observer for x, nor x ˆ an observer for x. These varables smply act as ntermedate varables n the control protocol and are part of the dynamcs of the OPFB cooperatve tracker. Trajectores of dsplacements of mass node node node node 3 node 4 node 5 node tme (second) Fg Cooperatve control usng SVFB and local Rccat desgn. Profles of the dsplacements y (,,6), of mass

29 9 Trajectores of dsplacements of mass node node node node 3 node 4 node 5 node tme (second) Fg Cooperatve control usng SVFB and local Rccat desgn. Profles of the dsplacements y (,,6), of mass Trajectores of dsplacements of mass 4 3 node node node node 3 node 4 node 5 node tme (second) Fg Dynamc OPFB cooperatve tracker of Case. Profles of the dsplacements y (,, 6 ) of mass

30 3 Trajectores of dsplacements of mass 4 3 node node node node 3 node 4 node 5 node tme (second) Fg. 3.. Dynamc OPFB cooperatve tracker of Case. Profles of the dsplacements y (,, 6 ) of mass 4 3 state estmaton errors tme (second) Fg. 3.. Dynamc OPFB cooperatve tracker of Case. Profles of the state estmaton errors (,, 6 ; j,, 4 ) x j

31 3 Trajectores of dsplacements of mass 4 3 node node node node 3 node 4 node 5 node tme (second) Fg. 3.. Dynamc OPFB cooperatve tracker of Case. Profles of the dsplacements y (,, 6 ) of mass Trajectores of dsplacements of mass 3 3 node node node node 3 node 4 node 5 node tme (second) Fg Dynamc OPFB cooperatve tracker of Case. Profles of the dsplacements y (,, 6 ) of mass

32 3 5 4 state estmaton errors tme (second) Fg Dynamc OPFB cooperatve tracker of Case. Profles of the state estmaton errors x (, j,, 6; j,, 4 ) Trajectores of dsplacements of mass 4 3 node node node node 3 node 4 node 5 node tme (second) Fg Dynamc OPFB cooperatve tracker of Case 3. Profles of the dsplacements y, (,,6 ) of mass

33 33 Trajectores of dsplacements of mass node node node node 3 node 4 node 5 node tme (second) Fg Dynamc OPFB cooperatve tracker of Case 3. Profles of the dsplacements y (,,,6 ) of mass

34 34 x ˆx tme (second) 4 x ˆx tme (second) 4 ˆx ˆx tme (second) Fg Dynamc OPFB cooperatve tracker of Case 3. Profles of x ˆ, j x,, j x ˆ, j x and, j xˆ ˆ, j x (, j,, 6 ; j,,4 )

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