- Email: [email protected]
Adaptive H∞ Consensus Control for Distributed Parameter Systems of Hyperbolic Type on Directed Graph
7th IFAC Workshop on Distributed Estimation and 7th IFAC Workshop on Distributed Estimation and Control Networked 7th IFACin onSystems Distributed Estimation Control inWorkshop Networked Systems Availableand online at www.sciencedirect.com 7th IFACinWorkshop onSystems Distributed Groningen, NL, August 27-28, 2018Estimation and Control Networked Groningen, NL, August 27-28, 2018 Control in Networked Systems Groningen, NL, August 27-28, 2018 Groningen, NL, August 27-28, 2018
ScienceDirect
IFAC PapersOnLine 51-23 (2018) 307–312
of of of of
Adaptive H Consensus Control ∞ Adaptive H Control ∞ Consensus Adaptive H Consensus Control ∞ for Distributed Parameter Systems Adaptive H∞ Consensus Control for Distributed Parameter Systems for Distributed Parameter Systems Hyperbolic Type on Directed Graph for Distributed Parameter Systems Hyperbolic Type on Directed Hyperbolic Type on Directed Graph Graph HyperbolicYoshihiko Type Miyasato on Directed Graph ∗ ∗
Yoshihiko Miyasato ∗ Yoshihiko Miyasato ∗ Yoshihiko Miyasato ∗ ∗ The Institute of Statistical Mathematics, Tachikawa, Tokyo, 190-8562 Institute of Statistical Mathematics, Tachikawa, Tokyo, 190-8562 ∗ The JAPAN (Tel: +81-50-5533-8429; e-mail: [email protected]). The Institute of Statistical Mathematics, Tachikawa, Tokyo, 190-8562 JAPAN (Tel: +81-50-5533-8429; e-mail: [email protected]). ∗ The Institute of Statistical Mathematics, Tachikawa, Tokyo, 190-8562 JAPAN (Tel: +81-50-5533-8429; e-mail: [email protected]). JAPAN (Tel: +81-50-5533-8429; e-mail: [email protected]). Abstract: A A design design method method of of adaptive adaptive H H∞ consensus control of multi-agent systems composed Abstract: ∞ consensus control of multi-agent systems composed consensus controlgraphs, of multi-agent systems composed Abstract: A design method of adaptive H of a class of infinite-dimensional systems on directed network is in paper. ∞ of a class ofAinfinite-dimensional systemsH on directed network graphs, is presented presented in this this paper. consensus control of multi-agent systems composed Abstract: design method of isadaptive ∞directed The proposed control scheme derived as a solution of certain H control problems, where of a class of infinite-dimensional systems on network graphs, is presented in this ∞ control problems, paper. The proposed control scheme is derived as a solution of certain H where ∞is presented in this paper. of a class of infinite-dimensional systems on directed network graphs, the effects of neglected infinite-dimensional modes are regarded as external disturbances to the The proposed control scheme is derived as a solution of certain H control problems, where ∞ the effects of neglected infinite-dimensional modes are regarded as external disturbances to the Theeffects proposed controlthat scheme is derived consensus as modes a solution of certain Hexternal problems,via where ∞ control process. It is shown the desirable tracking is achieved approximately the of neglected infinite-dimensional are regarded as disturbances to the process. It is shown that the desirable consensus tracking is achieved approximately via the the effects ofis neglected infinite-dimensional modes are regarded as external disturbancesvia to the process. It shown that the desirable consensus tracking is achieved approximately the finite dimensional adaptive controller. finite dimensional adaptive controller. process. It is shown that the desirable consensus tracking is achieved approximately via the finite dimensional adaptive controller. © 2018, IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. finite dimensional adaptiveFederation controller. Keywords: Keywords: Adaptive Adaptive control, control, consensus consensus control, control, distributed distributed parameter parameter system, system, multi-agent multi-agent Keywords: Adaptive control, consensus control, distributed parameter system, system, H control ∞ control control, consensus control, distributed parameter system, multi-agent system, H∞Adaptive Keywords: multi-agent system, H∞ control system, H∞ control 1. sus 1. INTRODUCTION INTRODUCTION sus tracking tracking on on aa directed directed graph graph is is achieved achieved approximately approximately 1. INTRODUCTION sus tracking on a directed graph is achieved approximately via adaptation schemes. via adaptation schemes. 1. INTRODUCTION sus tracking on schemes. a directed graph is achieved approximately via adaptation Among The problems of consensus Among cooperative cooperative control control problems problems of of multi-agent multi-agent syssys- via adaptation schemes. The problems of consensus observers observers and and control control for for disdistems, distributed consensus tracking of multi-agent Among cooperative control problems sysThe problems of consensus observers and control for distributed parameter systems with finite dimensional inputs tems, distributed consensus tracking of multi-agent systributed parameter systems with finite dimensional inputs Among cooperative control problems of multi-agent systems, distributed consensus tracking of multi-agent sysThe problems of consensus observers and control for distems with limited communication networks, has been a baand outputs, have been investigated actively in the miletributed parameter systems with finite dimensional inputs tems with limited communication networks, has been asysba- and outputs, have been investigated actively in the miletems, distributed consensus tracking of multi-agent parameter systems with finiteactively dimensional inputs sic and important topic, and various research tems with limited communication networks, hasresults been ahave ba- tributed stone works Demetriou (2013), Demetriou (2018) together and outputs, have been investigated in the milesic and important topic, and various research results have works Demetriou (2013), Demetriou (2018) together tems with limited communication networks, hasresults been ahave ba- stone and outputs, haveofbeen investigated actively inofthe milebeen proposed such as et (2005), sic and important and various with the notion strictly positive realness infinitestone works Demetriou (2013), Demetriou (2018) together been proposed suchtopic, as Kingston Kingston et al. al.research (2005), Olfati-Saber Olfati-Saber with the notion of strictly positive realness of infinitesic and important topic, and various research results have stone works Demetriou (2013), Demetriou (2018) together et al. (2007), Cao and Ren (2011), Mei and Ren (2011), been proposed such as Kingston et al. (2005), Olfati-Saber with the notion of strictly positive realness of infinitedimensional systems, but the control issue with finiteet al.proposed (2007), Cao and Ren (2011), and Olfati-Saber Ren (2011), dimensional systems, but the control issue with finitebeen suchHowever, as Kingston etresults al.Mei (2005), et al.et (2007), Cao and Ren those (2011), Mei and Ren (2011), the notion of strictly positive realness of infiniteMei al. (2014). are restricted to controllers did not seem be in dimensional systems, but with finiteMei et al. (2014). However, those results are restricted to with dimensional controllers didthe notcontrol seem to toissue be discussed discussed in et al.et(2007), Cao However, andsystems, Ren those (2011), Mei and Ren (2011), dimensional systems, but the control issue with finitesimple and low-order or finite-dimensional meMei al. (2014). results are restricted to detail. The present work is an attempt to extend the dimensional controllers did not seem to be discussed in simple and low-order systems, or finite-dimensional medetail. The present work is an attempt to extend the Mei et al. (2014). However, those results are restricted to dimensional controllers didisnot seem to be discussed in chanical systems (mobile robots), those approaches do simple and low-order systems, orand finite-dimensional meconsensus control issue itself on a directed network graph detail. The present work an attempt to extend the chanical systems (mobile robots), and those approaches do consensus control issue itselfis on a attempt directed network graph simple and low-order systems, orand finite-dimensional medetail. The present work an to extend the not have been applied to the control of infinite-dimensional chanical systems (mobile robots), those approaches do to infinite-dimensional systems via finite-dimensional comconsensus control issue itself on a directed network graph not have systems been applied to the control of those infinite-dimensional finite-dimensional comchanical (mobile robots), and approaches do to infinite-dimensional not have been applied to the control of infinite-dimensional control issuesystems itself onvia aa directed networktograph (or high-order) systems via finite-dimensional (or pensators, and it would provide useful strategy deal to infinite-dimensional systems via finite-dimensional com(or high-order) systems viacontrol finite-dimensional (or lowlow- consensus pensators, and it would provide a useful strategy to deal not have been applied to the of infinite-dimensional infinite-dimensional systems via finite-dimensional comorder) compensators. Furthermore, the case of multi-agent (or high-order) systems via finite-dimensional (or low- to with the coordinate control of large flexible structures. pensators, and it would provide a useful strategy to deal order) compensators. Furthermore, the case of multi-agent with the coordinate control of large flexible structures. (or high-order) systems via finite-dimensional (or lowpensators, and control it would provide a useful strategy to deal systems with unknown and different system on order) compensators. Furthermore, the case parameters of multi-agent The formation version and the consensus control with the coordinate control of large flexible structures. systems with unknown and different system parameters on The formation controlcontrol versionofand theflexible consensus control order) compensators. Furthermore, the case not of multi-agent with the coordinate large structures. directed information network graphs, does have been systems with unknown and different system parameters on version on an undirected graph with the same policy, were The formation control version and the consensus control directed information graphs, does not have been version on an undirected graph and withthe the consensus same policy, were systems unknownnetwork and different system parameters on directed with information network graphs, does works. not have been formation control version control discussed in the related This is also discussed in Miyasato (2012) Miyasato (2013), version on an undirected graph withand the same policy, were discussed in detail detail in innetwork the previous previous related works. This is The also discussed in Miyasato (2012) and Miyasato (2013), directed information graphs, does not have been on an undirected graph withand the same policy, were because symmetric and definite structures of rediscussed in detail in thepositive previous related works. This is version respectively. also discussed in Miyasato (2012) Miyasato (2013), because symmetric and positive definite structures of rediscussed in detail in thepositive previous related works. This is respectively. also discussed in Miyasato (2012) and Miyasato (2013), lated matrices in the adaptation mechanisms on indirected because symmetric and definite structures of rerespectively. lated matrices in theand adaptation mechanisms on indirected because symmetric positive definiteinstructures of re- respectively. networks can no longer be maintained the of lated matrices in the adaptation mechanisms oncase indirected networks can no longer be maintained in the of the the 2. lated matrices in the adaptation mechanisms oncase indirected 2. MULTI-AGENT MULTI-AGENT SYSTEM SYSTEM AND AND INFORMATION INFORMATION multi-agent control on directed networks. Thus, it is not so networks can no longer be maintained in the case of the multi-agent control on directed networks.inThus, it is not so 2. MULTI-AGENT SYSTEM AND INFORMATION NETWORK GRAPH networks can no longer be maintained the case of the NETWORK GRAPH easy to extend adaptive consensus strategies on indirected multi-agent control on directed networks. Thus, it is not so 2. MULTI-AGENT SYSTEMGRAPH AND INFORMATION easy to extend adaptive consensus strategies on it indirected NETWORK multi-agent control onones directed networks. Thus, is not so networks to adaptive on directed networks directly. easy to extend adaptive consensus strategies on indirected NETWORK GRAPH networks to adaptive ones on directed networks directly. Mathematical preliminaries on information easy to extend adaptive consensus strategies on indirected Mathematical preliminaries on information network network graph graph networks to adaptive ones on directed networks directly. of multi-agent systems are summarized briefly (Ren and The purpose of the present paper is to extend consenMathematical preliminaries on information network graph networks to adaptive ones on paper directed multi-agent systems are summarized briefly (Ren and The purpose of the present is networks to extenddirectly. consen- of Mathematical preliminaries on information network graph Cao (1996), J. Mei and Chen (2014), Mei et al. (2014)). As sus control to complicated processes on directed network of multi-agent systems are summarized briefly (Ren and The purpose of the present paper is to extend consenCao (1996), J. Mei and Chen (2014), Mei et al. (2014)). As sus control to complicated processes on directed network of multi-agent systems are summarized briefly (Ren and The purpose of the present paper isin toMiyasato extend network consenCao (1996), J. Mei and Chen (2014), Mei et al. (2014)). As sus control to complicated processes on directed a model of interaction among agents, a weighted directed graphs by applying the procedure (2015), a model of interaction among agents, a weighted directed graphs by applying the procedure in Miyasato (2015), Cao (1996), J. Mei and Chen (2014), Mei et{1, al. ·(2014)). As sus control to complicated processes on directedconsensus network graph G = (V, E) is considered, where V = · · , N } is and to present a design method of adaptive a model of interaction among agents, a weighted directed graphs by applying the procedure in Miyasato (2015), G= (V, E) is considered, where Va = {1, · · · ,directed N } is a a and to by present a design method of inadaptive consensus agraph model of interaction among agents, weighted graphs applying thesystems procedure Miyasato (2015), graph node set corresponding to a set of agents, and E ⊆ V × V is control of multi-agent composed of distributed G = (V, E) is considered, where V = {1, · · · , N } is a and to present a design method of adaptive consensus node set corresponding to a set of agents, and E ⊆ V × V is control of multi-agent systems composed of distributed graph G set. = (V, E)edge is considered, where V =that {1, E·agent ·⊆ · , VN× } Vcan is is a and to of present a design method of adaptive consensus an edge An (i, j) ∈ E indicates j parameter systems of hyperbolic type (a class of infinitenode set corresponding to a set of agents, and control multi-agent systems composed of distributed edge set. An edge (i,toj)a ∈set E of indicates that agent j Vcan parameter systems of hyperbolic type (a class of infinite- an node set corresponding agents, and E ⊆ V × is control of multi-agent systems composed of distributed an edge set. An edge (i, j) ∈ E indicates that agent j can parameter systems of hyperbolic type (a class of infiniteobtain information from i. the edge j), dimensional systems) with inputs and obtain information from i. In In edge (i, (i,that j), ii is is a a parent parent dimensional systems) with finite-dimensional finite-dimensional and an edge set. An edge (i, j) ∈and Ethe indicates jnode can parameter systems of hyperbolic type (a proposed class inputs of infinitenode and jj is aa child node, the in-degree of outputs on directed network graphs. The control information from i. In the edge (i, j), iagent is the a parent dimensional systems) with finite-dimensional inputs and obtain node and is child node, and the in-degree of the node outputs on directed network graphs. The proposed control obtain information from i. In the edge (i, j), i is a parent dimensional systems) with finite-dimensional inputs and i is the number of its parents, and the out-degree of ii is strategy is composed of finite dimensional compensators, node and j is a child node, and the in-degree of the node outputs on directed network graphs. The proposed control inode is the of itsnode, parents, and out-degree ofnode is strategy is directed composednetwork of finite dimensional compensators, andnumber j is aitschild andagent the the in-degree ofno theparent outputs on graphs. The proposed control the number of children. An which has and is derived as a solution of certain H control problem i is the number of its parents, and the out-degree of i is strategy is composed of finite dimensional compensators, ∞ the number of its children. An agent which has no parent and is derived as a solution of certain H control problem ∞ i is with the number its parents, and as theaout-degree of i is strategy is composed of finite dimensional compensators, the number itsofchildren. agent hasA parent and is derived as aof solution of certain H∞ control problem the in-degree 0), called directed where the neglected infinite-dimensional modes (or with theof 0), is isAn aswhich a root. root. Ano directed where the effects effects ofsolution neglected infinite-dimensional modes (or the number ofin-degree its children. Ancalled agent which has no parent and is process derived as aregarded of external certain Hdisturbances ∞ control problem path is a sequence of edges in the form (i , i ), (i of the are as to the (or with the in-degree 0), is called as a root. A directed where the effects of neglected infinite-dimensional modes is athe sequence of edges in the as form (i11 , i22A), directed (i22 ,, ii33 ), ), of the process are regarded as external disturbances modes to the path (or with in-degree 0), isA called atree root. where the It effects of neglected infinite-dimensional · · · (∈ E), where i ∈ V. directed is a directed processes. is shown that the resulting control systems path is a sequence of edges in the form (i , i ), (i2 , i3 ), of the process are regarded as external disturbances to the j 1 2 · · · (∈ E), where i processes. It is shown that the resulting control systems ∈ V. A directed tree is a directed j awhich sequence edges the form , i2a), directed (iexcept of the process are regarded asthe external disturbances to the graph 1is 2 , i3 ), every exactly are robust to uncertain system parameters and ·path · · (∈isin where ij of∈node V. has A indirected tree(iparent processes. shown that resulting control systems are robust It to is system and neglected neglected graph inE), every node exactly one one except · · · a(∈unique E),which where iand V. has A directed tree parent ispaths a directed processes. It isuncertain shown that the parameters resulting control systems j ∈ are robust to uncertain system parameters and neglected graph in which every node has exactly one parent except for root, the root has directed to infinite-dimensional modes, and that the desirable conseninfinite-dimensional modes, and parameters that the desirable consen- graph for a unique root, andnode the rootexactly has directed pathsexcept to all all in which every one parent are robust to uncertain system and neglected root, and the has root has directed paths to all infinite-dimensional modes, and that the desirable consen- for a unique for a unique root, and the root has directed paths to all infinite-dimensional modes, and that the desirable consen2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 307 Copyright © under 2018 IFAC 307 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 307 10.1016/j.ifacol.2018.12.053 Copyright © 2018 IFAC 307
IFAC NecSys 2018 308 Groningen, NL, August 27-28, 2018
Yoshihiko Miyasato / IFAC PapersOnLine 51-23 (2018) 307–312
other node. A directed spanning tree GS = (VS , ES ) of the directed graph G = (V, E) is a subgraph of G such that GS is a directed tree and VS = V.
Associated with E, a weighted adjacency matrix A = [aij ] ∈ RN ×N is introduced, and the entry aij of it is defined such as aij > 0 (when (j, i) ∈ E) and aij = 0 (otherwise). For the adjacency matrix A = [aij ], the Laplacian matrix L = [lij ] ∈ RN ×N is defined by lii = ∑N (i ̸= j). Laplacian matrix j = 1 aij and lij = −aij , j ̸= i
has a simple 0 eigenvalue with the associated eigenvector 1 = [1 · · · 1]T , and all other eigenvalues have positive real parts, if and only if the corresponding directed graph has a directed spanning tree.
In this manuscript, consensus control of leader-follower type is considered. Let y0 be a leader which each agent i ∈ V (follower) should follow. Then, ai0 is defined such as ai0 > 0 (when leader’s information is available to follower i), and ai0 = 0 (otherwise), and from ai0 and L, the matrix M ∈ RN ×N is defined by M = L + diag (a10 · · · aN 0 ). Then, −M is a Hurwitz matrix, if and only if 1. at least one ai0 (1 ≤ i ≤ N ) is positive, and 2. the graph G has a directed spanning tree with the root i = 0. Hereafter, it is assumed that the conditions 1 and 2 are satisfied, and no further assumption is added to the graph G. In the manuscript, two adjacency matrices A = [aij ], C = [cij ] ∈ RN ×N are introduced for a directed graph G, and the corresponding matrices are denoted as La , Lc (Laplacian matrices), and Ma , Mc , respectively. 3. PROBLEM STATEMENT A multi-agent system composed of distributed parameter systems of hyperbolic type is considered. For each agent, let Ωi be a bounded open domain in a finite dimensional Euclidian space, and L2 (Ωi ) is defined as the Hilbert space of all square integrable functions with the inner ∫ product (ui , vi ) = Ωi ui (xi )vi∗ (xi )dxi , where vi∗ is a complex conjugate of vi . A single-input, single-output distributed parameter system of hyperbolic type in L2 (Ωi ) (Sakawa (1984)) described by (1), (2), is considered (i = 1, · · · , N ). d2 d ui (t) + 2αi Ai ui (t) + Ai ui (t) = bi fi (t), (1) 2 dt dt (2) yi (t) = (ci , ui (t)) ≡ Ci ui (t),
where ui (t) (∈ L2 (Ωi )) is a state, fi (t) (an input) and yi (t) (an output) are scalar functions on t ∈ [0, ∞), bi (∈ L2 (Ωi )) is an input influence function, and ci (∈ L2 (Ωi )) is a sensor influence function. αi is a small damping constant (0 < αi ≪ 1). The operator Ai is a self-adjoint, positive definite, and unbounded operator with compact resolvent whose eigenvalues λij (0 < λi1 < λi2 < · · · < λij < · · · , (limj→∞ λij = ∞)) are simple. The domain D(Ai ) is dense in L2 (Ωi ). The normalized eigenfunctions of Ai are denoted by ϕij satisfying the following relations. Ai ϕij = λij ϕij , (j = 1, 2, · · ·). (3)
The set ϕij (j = 1, 2, · · ·) forms a complete orthonormal system in L2 (Ωi ). For each controlled process (1), (2), d yi (t) are only the input fi (t) and the output yi (t), and dt
308
available for measurement, but the state ui (t) and the systems parameters in Ai , bi , ci , and αi are unknown. The control objective is to design an adaptive consensus control system for a swarm of infinite-dimensional systems (1), (2) in which consensus tracking (4) is achieved via finite dimensional adaptation schemes. yi → yj → y0 , y˙ i → y˙ j → y˙ 0 , (i, j = 1, · · · , N ).
(4)
4. MATHEMATICAL PRELIMINARY In this section, mathematical preliminaries on the solution of the distributed parameter systems of hyperbolic type (1) is summarized. The next assumption is introduced. Assumption 1 αi and λij satisfy the following conditions. αi2 λ2ij − λij ̸= 0, (j ≥ 1), 1 (5) . αi λi1 < 2αi Based on Assumption 1, gi (Ai ) is defined by ( 2 2 )1/2 , gi (λij ) ≡ αi λij − λij ∞ ∑ gi (λij )(·, ϕij )ϕij . gi (Ai ) ≡
(6)
j=1
Since gi (λij ) ∼ αi λij as j → ∞, gi (Ai ) is an unbounded operator and D(g(Ai )) = D(Ai ). Furthermore, (5) shows that gi (Ai )−1 is a bounded operator. By utilizing gi (Ai ), the solution of the process (1) is given by Lemma 1. Lemma 1 (Sakawa (1984)) The next evolution equations in L2 (Ωi ) are considered, −1 d ξi (t) = A+ bi fi (t), i ξi (t) + gi (Ai ) dt (7) d −1 ηi (t) = A− bi fi (t), i ηi (t) − gi (Ai ) dt A± (8) i ≡ −αi Ai ± gi (Ai ).
Then, the unique solution ui (t) of (1) is described as follows: ξi (t) + ηi (t) ui (t) = , (9) 2 where initial conditions ξi (0), ηi (0) of (7) are determined d ui (0) of (1) uniquely from initial conditions ui (0), dt The operator A± i have eigenfunctions ϕij and corresponding eigenvalues µ± ij as follows: ± A± i ϕij = µij ϕij , ( ) 1 + + , µij = −αi λij + gi (λij ), lim µij = − 2α)i (j→∞ lim µ+ µ− ij = −αi λij − gi (λij ), ij = −∞ .
(10) (11)
j→∞
Then, ξi (t), ηi (t) and ui (t) are rewritten into the following eigenfunction expansion forms: ∞ ∑ ξi (t) = ξij (t)ϕij , j=1 (12) ∞ ∑ η (t) = η (t)ϕ , ij ij i j=1
IFAC NecSys 2018 Groningen, NL, August 27-28, 2018
Yoshihiko Miyasato / IFAC PapersOnLine 51-23 (2018) 307–312
−1 d ξij (t) = µ+ bij fi (t), ij ξij (t) + gi (λij ) dt d −1 ηij (t) = µ− bij fi (t), ij ηij (t) − gi (λij ) dt ∞ ∑ ξij (t) + ηij (t) ui (t) = ϕij , 2 j=1
Assumption 2 The finite dimensional subsystem [Si1 ] (C¯iN , A¯iN , ¯biN ) is completely controllable and observable, that is, cij ̸= 0, bij ̸= 0, (1 ≤ j ≤ Ni )
(13)
Then, on Assumption 2, a finite dimensional observer can ′ be constructed for [Si1 ], which is denoted by [Si1 ]. d ¯ iN yiN (t), ˆ¯iN (t) + ¯biN fi (t) + K ˆ¯iN (t) = A¯iN K u u (25) dt where A¯iN K (∈ C2Ni ×2Ni ) is a stable matrix defined by ¯ iN C¯iN , (26) A¯iN K = A¯iN − K 2Ni ×1 ¯ and KiN (∈ C ) is an observer gain matrix selected properly such that the following relations hold for λif > 0.
(14)
where ξij (t) = (ξi (t), ϕij ), ηij (t) = (ηi (t), ϕij ), bij = (bi , ϕij ), respectively. 5. SYSTEM REPRESENTATION In the present section, we obtain an input-output representation of each process (1), (2) of the multi-agent system, which is compatible to the finite dimensional controller design (Miyasato (2006), Miyasato (2012), Miyasato (2013)). ˜ iN (> 0) be a given damping constant. An integer Ni Let λ 1 ˜ iN ≤ −ℜ(µ± ˜ is chosen such that λ iNi +1 ), 0 < λiN < 2αi . ˜ iN < −ℜ(µ± ), (j ≥ Ni + 1). By Then, it follows that λ ij utilizing Ni , orthogonal projection operators PiN and QiN is defined. Ni ∑ ( ·, ϕij )ϕij , PiN · = j=1 (15) ∞ ∑ QiN · = (I − PiN ) · = ( ·, ϕ )ϕ . ij ij
∥ exp(A¯iN K t)∥C2Ni ≤ const. exp(−λif t), (27) ˆ¯iN (t)∥C2Ni ∼ exp(−λif t) → 0. (28) ∥¯ uiN (t) − u ˆ¯iN (t) and Ci u ˜iN (t), the input-output By combining Ci u representation is given such that d2 ¯ iN C¯iN A¯iN K )u ˆ¯iN (t) yi (t) = (C¯iN A¯2iN K + C¯iN K dt2 ¯ iN + (C¯iN K ¯ iN )2 }yiN (t) +{C¯iN A¯iN K ∞ ∑ cij − + {h(µ+ ij )ξij (t) + h(µij )ηij (t)} 2 j=Ni +1
j=Ni +1
Then, ui (t) and yi (t) of (1), (2) are expressed by { ui (t) = PiN ui (t) + QiN ui (t) ≡ uiN (t) + u ˜iN (t), yi (t) = Ci {uiN (t) + u ˜iN (t)} ≡ yiN (t) + y˜iN (t), and those are expanded into the following forms. Ni ∑ ξij (t) + ηij (t) u (t) = ϕij , iN 2 j=1 [Si1 ] Ni ∑ ξij (t) + ηij (t) yiN (t) = cij , 2 j=1 ∞ ∑ ξij (t) + ηij (t) ϕij , (t) = u ˜ iN 2 j=Ni +1 [Si2 ] ∞ ∑ ξij (t) + ηij (t) cij , (t) = y ˜ iN 2
(16)
(17)
(18)
j=Ni +1
where cij ≡ (ci , ϕij ) = Ci ϕij . Thus, each controlled process (1), (2) of the multi-agent system is divided into two subsystems [Si1 ]:(uiN (t), yiN (t)) and [Si2 ]:(˜ uiN (t), y˜iN (t)), respectively. [Si1 ] is a finite dimensional (2Ni ) system, and is represented in the state space form. du ¯iN (t) = A¯iN u ¯iN (t) + ¯biN fi (t), dt [Si1 ] (19) ¯iN (t), y (t) = C¯iN u iN T¯ ¯iN (t) ϕiN , uiN (t) = u u ¯iN (t) = [ξi1 (t), ηi1 (t), · · · , ξiNi (t), ηiNi (t)]T , ϕ¯iN = [ϕi1 , ϕi1 , · · · , ϕiN , ϕiN ]T , i
i
− − + A¯iN = diag (µ+ i1 , µi1 , · · · , µiNi , µiNi ), ¯biN = [gi (λi1 )−1 bi1 , −gi (λi1 )−1 bi1 , · · · , gi (λiNi )−1 biNi , −gi (λiNi )−1 biNi ]T ,
C¯iN = [ci1 /2, ci1 /2, · · · , ciNi /2, ciNi /2]. For the subsystem [Si1 ], we assume that
(20)
(21) (22) (23) (24)
309
θi0
+θi0 f (t) + ϵi (t), ∫ ∞ ∑ ≡ cij bij = ci (xi )bi (xi )dxi . j=1
(29)
(30)
Ωi
Hereafter, all exponentially decaying terms are denoted by ϵi (t). Next, fif (t) is introduced such as d fif (t) = −ai0 fif (t) + fi (t), (31) dt ˆ¯iN (t) where ai0 is a positive constant. The substitution of u and yiN (t) = y(t) − y˜iN (t) into (29) yields d2 T T v¯iN 2 (t) + θi3 fif (t) + θi4 yi (t) yi (t) = θi1 v¯iN 1 (t) + θi2 dt2 (32) +θi0 fi (t) + δi (t) + ϵi (t), where v¯iN 1 (t) and v¯iN 2 (t) are finite dimensional state variable filters (2Ni dimension) such as d v¯iN 1 (t) = F¯iN v¯iN 1 (t) + g¯iN fif (t), dt (33) d v¯ (t) = F¯ v¯ (t) + g¯ y (t). iN 2 iN iN 2 iN i dt ¯ (FiN , g¯iN ) (F¯iN ∈ R2Ni ×2Ni , g¯iN ∈ R2Ni ) is controllable, and F¯iN is chosen such that det(sI − F¯iN ) = det(sI − ¯ iN A¯iN K ). Since (C¯iN , A¯iN ) is observable, there exists K satisfying the relation for an arbitrary stable matrix F¯iN (∈ R2Ni ×2Ni ). θi1 , θi2 (∈ R2Ni ) are vectors satisfying the following relation (34) (since (F¯iN , g¯iN ) is controllable, there exists θi1 , θi2 (∈ R2Ni ) satisfying (34) (Ioannou and Sun (1996)). ¯ iN C¯iN A¯iN K ) · (C¯iN A¯2iN K + C¯iN K ∫t ¯ · (AiN K + ai0 I) {exp A¯iN K (t − τ )} · ¯biN fif (τ )τ 0 ∫t ¯ iN yi (τ )dτ + {exp A¯iN K (t − τ )} · K 0
T T v¯iN 1 (t) + θi2 v¯iN 2 (t) + ϵi (t). = θi1
309
(34)
IFAC NecSys 2018 310 Groningen, NL, August 27-28, 2018
Yoshihiko Miyasato / IFAC PapersOnLine 51-23 (2018) 307–312
Furthermore, θi3 , θi4 (∈ R), δi (t) are defined by ¯ iN C¯iN A¯iN K )¯biN , θi3 = (C¯iN A¯2iN K + C¯iN K ¯ iN + (C¯iN K ¯ iN )2 }, θi4 = {C¯iN A¯iN K
(35) (36)
¯ iN C¯iN A¯iN K ) · δi (t) = −(C¯iN A¯2iN K + C¯iN K ∫t ¯ iN y˜iN (τ )dτ · {exp A¯iN K (t − τ )} · K
(37)
Therefore, the input-output representation of each process of the multi-agent system is deduced such as (32), and is T T composed of θi1 v¯iN 1 (t) + θi2 v¯iN 2 (t) + θi3 fif (t) + θi4 yi (t) + θi0 fi (t) and δi (t). The former term is considered as a primal part for the finite dimensional controller design. On the contrary, the latter term δi (t) comes from the infinite dimensional system [Si2 ], and is seen as a residual part for the design of the control system, and the interferences from the infinite dimensional system [Si2 ] are to be taken into consideration to stabilize the total system hereafter. Next, the residual part δi (t) is to be evaluated. For that purpose, state variable filters whose dimensions are 1, are ˜ introduded by utilizing design parameters λiN and λif . ˜ iN wi1 (t) + |fif (t)|, d wi1 (t) = −λ dt (38) d wi2 (t) = −λif wi2 (t) + wi1 (t). dt The sensor influence function ci and the input influence function bi are assumed to be smooth in the following sense.
(39)
j=1
j=1
Then, δi (t) of (37) is evaluated by Lemma 2 (Miyasato (2006), Miyasato (2012), Miyasato (2013)). Lemma 2 On Assumption 3, δi (t) is evaluated as follows: |δi (t)| ≤ giδ (t)T diδ + |ϵi (t)|,
(40) T
giδ = [ |fif (t)|, wi1 (t), wi2 (t) ] ,
(41)
diδ = [ Mi1 , Mi2 , Mi3 ] ,
(42)
T
0 < Mi1 ∼ Mi3 < ∞, ˜ iN t −λ
ϵi (t) ∼ e
Assumption 4 θi0 ̸= 0, and sgn θi0 is known. In the following, it is assumed that θi0 > 0 without loss of generality. There exist {Mif 0 and� Mif 1 such �} that �d � |fif (t)| ≤ Mif 0 + Mif 1 sup |yi (τ )|, �� yi (τ )�� , dτ 0≤τ ≤t
Assumption 5
j=Ni +1
Assumption 3 The following inequalities hold. ∞ ∞ ∑ ∑ 2 λ2k |λkij cij bij | < ∞, ij cij < ∞, (k = 1, 2).
6.1 Assumptions By utilizing the representation (43), the proposed adaptive H∞ consensus control scheme is to be constructed via finite dimensional compensators. The next assumptions are introduced.
0
¯ iN + (C¯iN K ¯ iN )2 }˜ yiN (t) −{C¯iN A¯iN K ∞ ∑ cij − + {h(µ+ ij )ξij (t) + h(µij )ηij (t)}. 2
6. ADAPTIVE H∞ CONSENSUS CONTROL
, e−λif t , e−ai0 t → 0.
Hereafter, the input-output representation of each agent is written in the following form. d2 yi (t) = ΘT (43) i ωi (t) + θi0 fi (t) + δi (t) + ϵi (t), dt2 [ T T ]T (44) , θi2 , θi3 , θi4 , Θi = θi1 [ ]T T T (45) ωi (t) = viN 1 (t) , viN 2 (t) , fif (t), yi (t) ,
where ΘT i ωi (t) + θi0 fi (t) is utilized for the finite dimensional controller design, and the evaluation of δi (t) (Lemma 2) is to be applied to stabilize the total system.
310
(0 ≤ Mf 0 < ∞, 0 < Mf 1 < ∞).
(46)
Remark Assumption 4 states that the relative degree of each agent is 2, and Assumption 5 asserts that the process has a stable inverse. 6.2 Control Law and Error Equation First, an estimation procedure of y˙0 (the leader’s information) is constructed via available data from the follower i. A similar estimator was given in Mei and Ren (2011) for second-order Euler-Lagrange systems. N ∑ cij {ˆ zi (t) − zˆj (t)} zˆ˙ i (t) = −β j=1 j ̸= i
zi (t) − y˙ 0 (t)} + ni0 y¨0 (t), (47) −βci0 {ˆ where zˆi is a current estimate of y˙ 0 , and is synthesized from the data available to the follower i. cij (1 ≤ i ≤ N, 0 ≤ j ≤ N ) is defined as the entry of the adjacency matrix C and ai0 , and β > 0 is a design parameter. Concerned with ci0 , ni0 is defined as ni0 = 1 (when ci0 > 0), ni0 = 0 (otherwise). By utilizing zˆi , the following control law is employed. N ∑ aij {yi (t) − yj (t)}, (48) y˙ ri (t) = zˆi (t) − α j=0 j ̸= i
si (t) = y˙ i (t) − y˙ ri (t), [ ] ˆ i (t)T ωi (t) + y¨ri (t) + vi (t) fi (t) = pˆi (t) −Θ
(49)
(50) ≡ pˆi (t)fi0 (t) + vi (t), where aij (1 ≤ i ≤ N, 0 ≤ j ≤ N ) is defined from the entry of the adjacency matrix A and ai0 , and α > 0 is ˆ is denoted as a current estimate a design parameter. (·) of (·), and pi is defined by pi = 1/bi . Furthermore, vi is a stabilizing signal which is to be determined later based on H∞ control criterion. An estimation error between the leader y˙ 0 and the estimate zˆi is defined such as (51) z˜i (t) ≡ zˆi (t) − y˙ 0 (t), and the following relations are given for si and z˜i . N ∑ ˙z˜i (t) = −β cij {˜ zi (t) − z˜j (t)} j=1 j ̸= i
IFAC NecSys 2018 Groningen, NL, August 27-28, 2018
Yoshihiko Miyasato / IFAC PapersOnLine 51-23 (2018) 307–312
−βci0 z˜i (t) + (ni0 − 1)¨ y0 (t), (52) T ˆ i (t)} ωi (t) + θi0 {ˆ pi (t) − pi }fi0 (t) s˙ i (t) = {Θi − Θ +θi0 vi (t) + δi (t) + ϵi (t).
(53)
Then, the total representations of the multi-agent system are deduced as follows: z˜˙ (t) = −βMc z˜(t) + (N0 − 1)¨ y0 (t), (54) ˆ p(t) − p} s(t) ˙ = Ω(t){Θ − Θ(t)} + F0 (t)Θ0 {ˆ +Θ0 v(t) + δ(t) + ϵ(t),
(55)
where T T T T z˜ = [˜ z1T , · · · , z˜N ] , s = [sT 1 , · · · , sN ] ,
T T T ), Θ = [ΘT Ω = block diag (ω1T , · · · , ωN 1 , · · · , ΘN ] ,
F0 = diag (f10 , · · · , fN 0 ), Θ0 = diag (θ10 , · · · , θN 0 ), p = [p1 , · · · , pN ]T , N0 = [n10 , · · · , nN 0 ]T ,
1 = [1, · · · , 1]T , v = [v1 , · · · , vN ]T ,
δ = [δ1 , · · · , δN ]T , ϵ = [ϵ1 , · · · , ϵN ]T .
Especially, owing to the assumption of the network graph G, the matrix −Mc is shown to be Hurwitz.
For that purpose, the following Hamilton-Jacobi-Isaacs (HJI) equation { 2 and its solution W0 are introduced } 1 ∑ ∥Lg1i W0 ∥2 − (Lg2 W0 )R−1 (Lg2 W0 )T L f W0 + 4 i=1 γi2
+q = 0, (62) 1 T (63) W0 = s s, 2 where q and R are a positive function and a positive definite matrix respectively. Those are derived from HJI equation based on inverse optimality (Krsti´c and Deng (1998)) for the given solution W0 and the positive constants γ1 , γ2 . The substitution of the solution W0 (63) into HJI equation (62) yields ( ) I 1 T G δ GT −1 ˆ T δ ˆ (64) s + 2 − Θ0 R Θ0 s + q = 0. 4 γ12 γ2 Then, R and q are derived such as )−1 ( ˆ −T ˆ −T ˆ −1 Θ ˆ −1 Gδ GT Θ Θ Θ 0 δ 0 + 0 20 +K , (65) R= γ12 γ2
6.3 Adaptive H∞ Consensus Control A positive function W is defined by { } }T 1 {ˆ 1 ˆ Θ(t) −Θ Θ(t) − Θ Γ−1 W (t) = s(t)T s(t) + 1 2 2 1 T p(t) − p} Θ0 Γ−1 p(t) − p} + {ˆ 2 {ˆ 2 } }T { 1 {ˆ (56) θˆ0 (t) − θ0 , θ0 (t) − θ0 Γ−1 + 3 2 where Γ1 , Γ2 and Γ3 are diagonal and positive definite matrices, and θ0 is defined as follows: (57) θ0 = [θ10 , · · · , θN 0 ]T . ˆ ˆ The tuning by laws of Θ,{ pˆ, θ0 are given } ˙ T ˆ Θ(t) = Pr{ Γ1 Ω(t) s(t) ,} (58) pˆ˙ (t) = Pr −Γ2 F0 (t)T s(t) , ˙ { } T ˆ θ0 (t) = Pr Γ3 V (t) s(t) , V = diag (v1 , · · · , vN ), (59)
where Pr(·) are projection operations in which tuning ˆ pˆ and θˆ0 are constrained to bounded regions parameters Θ, deduced from upper-bounds of ∥Θ∥ and upper-bounds and lower-bounds of each element of p and θ0 , respectively (Ioannou and Sun (1996)). Then, the time derivative of W along its trajectory is derived as follows: ˆ 0 (t)v(t). (60) ˙ (t) ≤ s(t)T δ(t) + s(t)T ϵ(t) + s(t)T Θ W ˙ (60), we introduce the next From the evaluation of W virtual system, z˜˙ = f + g11 d1 + g12 d2 + g2 v, (61) ˆ f = 0, g11 = Gδ , g12 = I, g2 = Θ0 , T T Gδ = block diag (gδ1 , · · · , gδN ),
d1 =
[dT δ1 , · · · ,
T dT δN ] ,
311
d2 = ϵ.
and are to stabilize this virtual system via a control input v by utilizing H∞ criterion, where d1 , d2 are considered as external disturbances to the process (Miyasato (2000)). 311
1 ˆ ˆT q = sT Θ (66) 0 K Θ0 s, 4 where K is a diagonal positive definite matrix (a design parameter). From R, v is deduced as a solution of the corresponding H∞ control problem as follows: 1 1 ˆT (67) v = − R−1 (Lg2 W0 )T = − R−1 Θ 0 s, 2 2 ˆ 0 are constructed from the elements where the entries of Θ of θˆ0 (58). Then, the time derivative of W is evaluated by ˙ ≤ −q − v T Rv W ( )T ( ) 1 −1 ˆ T 1 −1 ˆ T + v + R Θ0 s R v + R Θ0 s 2 2 �2 N � ∑ � |si | gδi � 2 2 2 � � +γ1 ∥d1 ∥ − γ1 �dδi − 2γ 2 � 1 i=1 )2 N ( ∑ |si | 2 2 2 +γ2 ∥d2 ∥ − γ2 , (68) |ϵi | − 2 2γ2 i=1
ˆ pˆ, and θˆ0 are bounded for the and it follows that s, Θ, tuning laws (58) and the stabilizing signal v (67). Next, for stability analysis of the estimation error z˜, a positive function V1 is introduced such as (69) V1 = z˜T Pc z˜, (70) Pc Mc + McT Pc = I, (Pc = PcT > 0). There exists a positive definite and symmetric matrix Pc satisfying (70), since −Mc is Hurwitz. Then, the time derivative of V1 along its trajectory is evaluated as follows: 2 β z ∥2 + ∥Pc (N0 − 1)¨ y0 ∥2 , V˙ 1 ≤ − ∥˜ 2 β and it is shown that z˜ is bounded for bounded y¨0 .
(71)
Finally, for stability analysis of the control error yi − y0 , y˜i and y˜ are defined by y˜i = yi − y0 , (72)
T T ] . y˜ = [˜ y1T , · · · , y˜N Then, the following relation holds
(73)
IFAC NecSys 2018 312 Groningen, NL, August 27-28, 2018
Yoshihiko Miyasato / IFAC PapersOnLine 51-23 (2018) 307–312
y˜˙ = s + z˜ − αMa y˜. (74) −Ma is shown to be Hurwitz because of the assumption of the network graph G. From that property, a positive function V2 is defined by V2 = y˜T Pa y˜, (75) (76) Pa Ma + MaT Pa = I, (Pa = PaT > 0). Similarly to the previous case, there exists a positive definite and symmetric matrix Pa satisfying (76), since −Ma is Hurwitz. Then, the time derivative of V2 along its trajectory is evaluated as follows: α 4 y ∥2 + ∥Pa ∥2 (∥s∥2 + ∥˜ z ∥2 ), (77) V˙ 2 ≤ − ∥˜ 2 α and it follows that y˜ is bounded for bounded s, z˜. From the three stages of stability analysis (the evaluations ˙ , V˙ 1 , V˙ 2 ), the next theorem is obtained. of W Theorem 1 In the proposed adaptive control system, the stabilizing signal v is a sub-optimal control input minimizing the upper bound on the cost functional J. t ∫ {q + v T Rv}dτ + W (t) J(t) ≡ sup d1 ,d2 ∈L2
−
2 ∑
γi2
i=1
0
∫t 0
∥di ∥2 dτ .
(78)
∥di ∥2 dτ + W (0).
(79)
Also the next inequality holds. ∫t {q + v T Rv}dτ + W (t) 0
≤
2 ∑ i=1
γi2
∫t 0
Theorem 2 The adaptive control system is uniformly bounded, and the next relations hold. ∫T 1 ∥˜ y (t)∥2 dt ≤ const · (γ12 + γ22 ) lim sup T →∞ T 0 } { 1 2 , (80) ∥(N − 1)¨ y ∥ +const · sup 0 0 α2 β 2 ∫T 1 lim sup ∥y˜˙ (t)∥2 dt ≤ const · (γ12 + γ22 ) T →∞ T 0 } { 1 2 ∥(N0 − 1)¨ y0 ∥ . (81) +const · sup β2 Remark The magnitude of the tracking error is evaluated according to the direct availability of y¨0 ((N0 −1) = 0 or not), or the value of y¨0 . The total adaptive control scheme is constructed in a distributed fashion. 7. CONCLUDING REMARKS A design procedure of finite-dimensional adaptive H∞ consensus control of multi-agent systems composed of distributed parameter systems of hyperbolic type on directed network graphs, has been provided in the paper. 312
The present work is an attempt to extend the consensus control issue itself to infinite-dimensional systems, and the proposed method would provide a basic and useful strategy to deal with the coordinate control of large flexible structures on a directed network graph via low-order (or finite dimensional) controllers. REFERENCES Cao, Y. and Ren, W. (2011). Distributed coordinate tracking via a variable structure approach - part i : Consensus tracking. Proc. of 2010 ACC, 4744–4749. Demetriou, M. (2013). Synchronization and consensus controllers for a class of parabolic distributed parameter systems. Systems & Control Letters, 70–76. Demetriou, M. (2018). Design of adaptive output feedback synchronizing controllers for networked pdes with boundary and in-domain structured perturbations and disturbances. Automatica, 220–229. Ioannou, P. and Sun, J. (1996). Robust Adaptive Control. PTR Prentice-Hall, New Jersey. J. Mei, W.R. and Chen, J. (2014). Consensus of secondorder heterogeneous multi-agent systems under a directed graph. Proc. of 2014 ACC, 802–807. Kingston, D., Ren, W., and Beard, R. (2005). Consensus algorithms are input-to-state stable. Proc. of 2005 ACC, 1686–1690. Krsti´c, M. and Deng, H. (1998). Stabilization of Nonlinear Uncertain Systems. Springer. Mei, J. and Ren, W. (2011). Distributed coordinate tracking with a dynamic leader for multiple euler-lagrange systems. IEEE Transactions on Robotics and Automation, 1415–1421. Mei, J., Ren, W., Chen, J., and Anderson, B. (2014). Consensus of linear multi-agent systems with fully distributed control gains under a general directed graph. Proc. of the 53rdt IEEE CDC, 2993–2998. Miyasato, Y. (2000). Adaptive nonlinear h∞ control for processes with bounded variations of parameters – general relative degree case –. Proc. of the 39th IEEE CDC, 1453–1458. Miyasato, Y. (2006). Model reference adaptive h∞ control for distributed parameter systems of hyperbolic type by finite-dimensional controller. Proc. of the 45th IEEE CDC, 459–464. Miyasato, Y. (2012). Adaptive h∞ formation control for infinite-dimensional systems. Proc. of the 51st IEEE CDC, 6071–6076. Miyasato, Y. (2013). Adaptive h∞ consensus control for distributed parameter systems of hyperbolic type. Proc. of 2013 IEEE Multi-Conference on Systems and Control, 1018–1023. Miyasato, Y. (2015). Adaptive h∞ consensus control of multi-agent systems on directed graph. Proc. of the 41st IEEE CDC, 7592–7597. Olfati-Saber, R., Fax, J.A., and Murray, R.M. (2007). Consensus and cooperation in networked multi-agent systems. Proc. of the IEEE, 215–233. Ren, W. and Cao, Y. (1996). Distributed Coordination of Multi-Agent Networks. Springer. Sakawa, Y. (1984). Feedback control of second order evolution equations with damping. SIAM J. Control and Optimization, 22, 343–361.
ScienceDirect
IFAC PapersOnLine 51-23 (2018) 307–312
of of of of
Adaptive H Consensus Control ∞ Adaptive H Control ∞ Consensus Adaptive H Consensus Control ∞ for Distributed Parameter Systems Adaptive H∞ Consensus Control for Distributed Parameter Systems for Distributed Parameter Systems Hyperbolic Type on Directed Graph for Distributed Parameter Systems Hyperbolic Type on Directed Hyperbolic Type on Directed Graph Graph HyperbolicYoshihiko Type Miyasato on Directed Graph ∗ ∗
Yoshihiko Miyasato ∗ Yoshihiko Miyasato ∗ Yoshihiko Miyasato ∗ ∗ The Institute of Statistical Mathematics, Tachikawa, Tokyo, 190-8562 Institute of Statistical Mathematics, Tachikawa, Tokyo, 190-8562 ∗ The JAPAN (Tel: +81-50-5533-8429; e-mail: [email protected]). The Institute of Statistical Mathematics, Tachikawa, Tokyo, 190-8562 JAPAN (Tel: +81-50-5533-8429; e-mail: [email protected]). ∗ The Institute of Statistical Mathematics, Tachikawa, Tokyo, 190-8562 JAPAN (Tel: +81-50-5533-8429; e-mail: [email protected]). JAPAN (Tel: +81-50-5533-8429; e-mail: [email protected]). Abstract: A A design design method method of of adaptive adaptive H H∞ consensus control of multi-agent systems composed Abstract: ∞ consensus control of multi-agent systems composed consensus controlgraphs, of multi-agent systems composed Abstract: A design method of adaptive H of a class of infinite-dimensional systems on directed network is in paper. ∞ of a class ofAinfinite-dimensional systemsH on directed network graphs, is presented presented in this this paper. consensus control of multi-agent systems composed Abstract: design method of isadaptive ∞directed The proposed control scheme derived as a solution of certain H control problems, where of a class of infinite-dimensional systems on network graphs, is presented in this ∞ control problems, paper. The proposed control scheme is derived as a solution of certain H where ∞is presented in this paper. of a class of infinite-dimensional systems on directed network graphs, the effects of neglected infinite-dimensional modes are regarded as external disturbances to the The proposed control scheme is derived as a solution of certain H control problems, where ∞ the effects of neglected infinite-dimensional modes are regarded as external disturbances to the Theeffects proposed controlthat scheme is derived consensus as modes a solution of certain Hexternal problems,via where ∞ control process. It is shown the desirable tracking is achieved approximately the of neglected infinite-dimensional are regarded as disturbances to the process. It is shown that the desirable consensus tracking is achieved approximately via the the effects ofis neglected infinite-dimensional modes are regarded as external disturbancesvia to the process. It shown that the desirable consensus tracking is achieved approximately the finite dimensional adaptive controller. finite dimensional adaptive controller. process. It is shown that the desirable consensus tracking is achieved approximately via the finite dimensional adaptive controller. © 2018, IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. finite dimensional adaptiveFederation controller. Keywords: Keywords: Adaptive Adaptive control, control, consensus consensus control, control, distributed distributed parameter parameter system, system, multi-agent multi-agent Keywords: Adaptive control, consensus control, distributed parameter system, system, H control ∞ control control, consensus control, distributed parameter system, multi-agent system, H∞Adaptive Keywords: multi-agent system, H∞ control system, H∞ control 1. sus 1. INTRODUCTION INTRODUCTION sus tracking tracking on on aa directed directed graph graph is is achieved achieved approximately approximately 1. INTRODUCTION sus tracking on a directed graph is achieved approximately via adaptation schemes. via adaptation schemes. 1. INTRODUCTION sus tracking on schemes. a directed graph is achieved approximately via adaptation Among The problems of consensus Among cooperative cooperative control control problems problems of of multi-agent multi-agent syssys- via adaptation schemes. The problems of consensus observers observers and and control control for for disdistems, distributed consensus tracking of multi-agent Among cooperative control problems sysThe problems of consensus observers and control for distributed parameter systems with finite dimensional inputs tems, distributed consensus tracking of multi-agent systributed parameter systems with finite dimensional inputs Among cooperative control problems of multi-agent systems, distributed consensus tracking of multi-agent sysThe problems of consensus observers and control for distems with limited communication networks, has been a baand outputs, have been investigated actively in the miletributed parameter systems with finite dimensional inputs tems with limited communication networks, has been asysba- and outputs, have been investigated actively in the miletems, distributed consensus tracking of multi-agent parameter systems with finiteactively dimensional inputs sic and important topic, and various research tems with limited communication networks, hasresults been ahave ba- tributed stone works Demetriou (2013), Demetriou (2018) together and outputs, have been investigated in the milesic and important topic, and various research results have works Demetriou (2013), Demetriou (2018) together tems with limited communication networks, hasresults been ahave ba- stone and outputs, haveofbeen investigated actively inofthe milebeen proposed such as et (2005), sic and important and various with the notion strictly positive realness infinitestone works Demetriou (2013), Demetriou (2018) together been proposed suchtopic, as Kingston Kingston et al. al.research (2005), Olfati-Saber Olfati-Saber with the notion of strictly positive realness of infinitesic and important topic, and various research results have stone works Demetriou (2013), Demetriou (2018) together et al. (2007), Cao and Ren (2011), Mei and Ren (2011), been proposed such as Kingston et al. (2005), Olfati-Saber with the notion of strictly positive realness of infinitedimensional systems, but the control issue with finiteet al.proposed (2007), Cao and Ren (2011), and Olfati-Saber Ren (2011), dimensional systems, but the control issue with finitebeen suchHowever, as Kingston etresults al.Mei (2005), et al.et (2007), Cao and Ren those (2011), Mei and Ren (2011), the notion of strictly positive realness of infiniteMei al. (2014). are restricted to controllers did not seem be in dimensional systems, but with finiteMei et al. (2014). However, those results are restricted to with dimensional controllers didthe notcontrol seem to toissue be discussed discussed in et al.et(2007), Cao However, andsystems, Ren those (2011), Mei and Ren (2011), dimensional systems, but the control issue with finitesimple and low-order or finite-dimensional meMei al. (2014). results are restricted to detail. The present work is an attempt to extend the dimensional controllers did not seem to be discussed in simple and low-order systems, or finite-dimensional medetail. The present work is an attempt to extend the Mei et al. (2014). However, those results are restricted to dimensional controllers didisnot seem to be discussed in chanical systems (mobile robots), those approaches do simple and low-order systems, orand finite-dimensional meconsensus control issue itself on a directed network graph detail. The present work an attempt to extend the chanical systems (mobile robots), and those approaches do consensus control issue itselfis on a attempt directed network graph simple and low-order systems, orand finite-dimensional medetail. The present work an to extend the not have been applied to the control of infinite-dimensional chanical systems (mobile robots), those approaches do to infinite-dimensional systems via finite-dimensional comconsensus control issue itself on a directed network graph not have systems been applied to the control of those infinite-dimensional finite-dimensional comchanical (mobile robots), and approaches do to infinite-dimensional not have been applied to the control of infinite-dimensional control issuesystems itself onvia aa directed networktograph (or high-order) systems via finite-dimensional (or pensators, and it would provide useful strategy deal to infinite-dimensional systems via finite-dimensional com(or high-order) systems viacontrol finite-dimensional (or lowlow- consensus pensators, and it would provide a useful strategy to deal not have been applied to the of infinite-dimensional infinite-dimensional systems via finite-dimensional comorder) compensators. Furthermore, the case of multi-agent (or high-order) systems via finite-dimensional (or low- to with the coordinate control of large flexible structures. pensators, and it would provide a useful strategy to deal order) compensators. Furthermore, the case of multi-agent with the coordinate control of large flexible structures. (or high-order) systems via finite-dimensional (or lowpensators, and control it would provide a useful strategy to deal systems with unknown and different system on order) compensators. Furthermore, the case parameters of multi-agent The formation version and the consensus control with the coordinate control of large flexible structures. systems with unknown and different system parameters on The formation controlcontrol versionofand theflexible consensus control order) compensators. Furthermore, the case not of multi-agent with the coordinate large structures. directed information network graphs, does have been systems with unknown and different system parameters on version on an undirected graph with the same policy, were The formation control version and the consensus control directed information graphs, does not have been version on an undirected graph and withthe the consensus same policy, were systems unknownnetwork and different system parameters on directed with information network graphs, does works. not have been formation control version control discussed in the related This is also discussed in Miyasato (2012) Miyasato (2013), version on an undirected graph withand the same policy, were discussed in detail detail in innetwork the previous previous related works. This is The also discussed in Miyasato (2012) and Miyasato (2013), directed information graphs, does not have been on an undirected graph withand the same policy, were because symmetric and definite structures of rediscussed in detail in thepositive previous related works. This is version respectively. also discussed in Miyasato (2012) Miyasato (2013), because symmetric and positive definite structures of rediscussed in detail in thepositive previous related works. This is respectively. also discussed in Miyasato (2012) and Miyasato (2013), lated matrices in the adaptation mechanisms on indirected because symmetric and definite structures of rerespectively. lated matrices in theand adaptation mechanisms on indirected because symmetric positive definiteinstructures of re- respectively. networks can no longer be maintained the of lated matrices in the adaptation mechanisms oncase indirected networks can no longer be maintained in the of the the 2. lated matrices in the adaptation mechanisms oncase indirected 2. MULTI-AGENT MULTI-AGENT SYSTEM SYSTEM AND AND INFORMATION INFORMATION multi-agent control on directed networks. Thus, it is not so networks can no longer be maintained in the case of the multi-agent control on directed networks.inThus, it is not so 2. MULTI-AGENT SYSTEM AND INFORMATION NETWORK GRAPH networks can no longer be maintained the case of the NETWORK GRAPH easy to extend adaptive consensus strategies on indirected multi-agent control on directed networks. Thus, it is not so 2. MULTI-AGENT SYSTEMGRAPH AND INFORMATION easy to extend adaptive consensus strategies on it indirected NETWORK multi-agent control onones directed networks. Thus, is not so networks to adaptive on directed networks directly. easy to extend adaptive consensus strategies on indirected NETWORK GRAPH networks to adaptive ones on directed networks directly. Mathematical preliminaries on information easy to extend adaptive consensus strategies on indirected Mathematical preliminaries on information network network graph graph networks to adaptive ones on directed networks directly. of multi-agent systems are summarized briefly (Ren and The purpose of the present paper is to extend consenMathematical preliminaries on information network graph networks to adaptive ones on paper directed multi-agent systems are summarized briefly (Ren and The purpose of the present is networks to extenddirectly. consen- of Mathematical preliminaries on information network graph Cao (1996), J. Mei and Chen (2014), Mei et al. (2014)). As sus control to complicated processes on directed network of multi-agent systems are summarized briefly (Ren and The purpose of the present paper is to extend consenCao (1996), J. Mei and Chen (2014), Mei et al. (2014)). As sus control to complicated processes on directed network of multi-agent systems are summarized briefly (Ren and The purpose of the present paper isin toMiyasato extend network consenCao (1996), J. Mei and Chen (2014), Mei et al. (2014)). As sus control to complicated processes on directed a model of interaction among agents, a weighted directed graphs by applying the procedure (2015), a model of interaction among agents, a weighted directed graphs by applying the procedure in Miyasato (2015), Cao (1996), J. Mei and Chen (2014), Mei et{1, al. ·(2014)). As sus control to complicated processes on directedconsensus network graph G = (V, E) is considered, where V = · · , N } is and to present a design method of adaptive a model of interaction among agents, a weighted directed graphs by applying the procedure in Miyasato (2015), G= (V, E) is considered, where Va = {1, · · · ,directed N } is a a and to by present a design method of inadaptive consensus agraph model of interaction among agents, weighted graphs applying thesystems procedure Miyasato (2015), graph node set corresponding to a set of agents, and E ⊆ V × V is control of multi-agent composed of distributed G = (V, E) is considered, where V = {1, · · · , N } is a and to present a design method of adaptive consensus node set corresponding to a set of agents, and E ⊆ V × V is control of multi-agent systems composed of distributed graph G set. = (V, E)edge is considered, where V =that {1, E·agent ·⊆ · , VN× } Vcan is is a and to of present a design method of adaptive consensus an edge An (i, j) ∈ E indicates j parameter systems of hyperbolic type (a class of infinitenode set corresponding to a set of agents, and control multi-agent systems composed of distributed edge set. An edge (i,toj)a ∈set E of indicates that agent j Vcan parameter systems of hyperbolic type (a class of infinite- an node set corresponding agents, and E ⊆ V × is control of multi-agent systems composed of distributed an edge set. An edge (i, j) ∈ E indicates that agent j can parameter systems of hyperbolic type (a class of infiniteobtain information from i. the edge j), dimensional systems) with inputs and obtain information from i. In In edge (i, (i,that j), ii is is a a parent parent dimensional systems) with finite-dimensional finite-dimensional and an edge set. An edge (i, j) ∈and Ethe indicates jnode can parameter systems of hyperbolic type (a proposed class inputs of infinitenode and jj is aa child node, the in-degree of outputs on directed network graphs. The control information from i. In the edge (i, j), iagent is the a parent dimensional systems) with finite-dimensional inputs and obtain node and is child node, and the in-degree of the node outputs on directed network graphs. The proposed control obtain information from i. In the edge (i, j), i is a parent dimensional systems) with finite-dimensional inputs and i is the number of its parents, and the out-degree of ii is strategy is composed of finite dimensional compensators, node and j is a child node, and the in-degree of the node outputs on directed network graphs. The proposed control inode is the of itsnode, parents, and out-degree ofnode is strategy is directed composednetwork of finite dimensional compensators, andnumber j is aitschild andagent the the in-degree ofno theparent outputs on graphs. The proposed control the number of children. An which has and is derived as a solution of certain H control problem i is the number of its parents, and the out-degree of i is strategy is composed of finite dimensional compensators, ∞ the number of its children. An agent which has no parent and is derived as a solution of certain H control problem ∞ i is with the number its parents, and as theaout-degree of i is strategy is composed of finite dimensional compensators, the number itsofchildren. agent hasA parent and is derived as aof solution of certain H∞ control problem the in-degree 0), called directed where the neglected infinite-dimensional modes (or with theof 0), is isAn aswhich a root. root. Ano directed where the effects effects ofsolution neglected infinite-dimensional modes (or the number ofin-degree its children. Ancalled agent which has no parent and is process derived as aregarded of external certain Hdisturbances ∞ control problem path is a sequence of edges in the form (i , i ), (i of the are as to the (or with the in-degree 0), is called as a root. A directed where the effects of neglected infinite-dimensional modes is athe sequence of edges in the as form (i11 , i22A), directed (i22 ,, ii33 ), ), of the process are regarded as external disturbances modes to the path (or with in-degree 0), isA called atree root. where the It effects of neglected infinite-dimensional · · · (∈ E), where i ∈ V. directed is a directed processes. is shown that the resulting control systems path is a sequence of edges in the form (i , i ), (i2 , i3 ), of the process are regarded as external disturbances to the j 1 2 · · · (∈ E), where i processes. It is shown that the resulting control systems ∈ V. A directed tree is a directed j awhich sequence edges the form , i2a), directed (iexcept of the process are regarded asthe external disturbances to the graph 1is 2 , i3 ), every exactly are robust to uncertain system parameters and ·path · · (∈isin where ij of∈node V. has A indirected tree(iparent processes. shown that resulting control systems are robust It to is system and neglected neglected graph inE), every node exactly one one except · · · a(∈unique E),which where iand V. has A directed tree parent ispaths a directed processes. It isuncertain shown that the parameters resulting control systems j ∈ are robust to uncertain system parameters and neglected graph in which every node has exactly one parent except for root, the root has directed to infinite-dimensional modes, and that the desirable conseninfinite-dimensional modes, and parameters that the desirable consen- graph for a unique root, andnode the rootexactly has directed pathsexcept to all all in which every one parent are robust to uncertain system and neglected root, and the has root has directed paths to all infinite-dimensional modes, and that the desirable consen- for a unique for a unique root, and the root has directed paths to all infinite-dimensional modes, and that the desirable consen2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 307 Copyright © under 2018 IFAC 307 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 307 10.1016/j.ifacol.2018.12.053 Copyright © 2018 IFAC 307
IFAC NecSys 2018 308 Groningen, NL, August 27-28, 2018
Yoshihiko Miyasato / IFAC PapersOnLine 51-23 (2018) 307–312
other node. A directed spanning tree GS = (VS , ES ) of the directed graph G = (V, E) is a subgraph of G such that GS is a directed tree and VS = V.
Associated with E, a weighted adjacency matrix A = [aij ] ∈ RN ×N is introduced, and the entry aij of it is defined such as aij > 0 (when (j, i) ∈ E) and aij = 0 (otherwise). For the adjacency matrix A = [aij ], the Laplacian matrix L = [lij ] ∈ RN ×N is defined by lii = ∑N (i ̸= j). Laplacian matrix j = 1 aij and lij = −aij , j ̸= i
has a simple 0 eigenvalue with the associated eigenvector 1 = [1 · · · 1]T , and all other eigenvalues have positive real parts, if and only if the corresponding directed graph has a directed spanning tree.
In this manuscript, consensus control of leader-follower type is considered. Let y0 be a leader which each agent i ∈ V (follower) should follow. Then, ai0 is defined such as ai0 > 0 (when leader’s information is available to follower i), and ai0 = 0 (otherwise), and from ai0 and L, the matrix M ∈ RN ×N is defined by M = L + diag (a10 · · · aN 0 ). Then, −M is a Hurwitz matrix, if and only if 1. at least one ai0 (1 ≤ i ≤ N ) is positive, and 2. the graph G has a directed spanning tree with the root i = 0. Hereafter, it is assumed that the conditions 1 and 2 are satisfied, and no further assumption is added to the graph G. In the manuscript, two adjacency matrices A = [aij ], C = [cij ] ∈ RN ×N are introduced for a directed graph G, and the corresponding matrices are denoted as La , Lc (Laplacian matrices), and Ma , Mc , respectively. 3. PROBLEM STATEMENT A multi-agent system composed of distributed parameter systems of hyperbolic type is considered. For each agent, let Ωi be a bounded open domain in a finite dimensional Euclidian space, and L2 (Ωi ) is defined as the Hilbert space of all square integrable functions with the inner ∫ product (ui , vi ) = Ωi ui (xi )vi∗ (xi )dxi , where vi∗ is a complex conjugate of vi . A single-input, single-output distributed parameter system of hyperbolic type in L2 (Ωi ) (Sakawa (1984)) described by (1), (2), is considered (i = 1, · · · , N ). d2 d ui (t) + 2αi Ai ui (t) + Ai ui (t) = bi fi (t), (1) 2 dt dt (2) yi (t) = (ci , ui (t)) ≡ Ci ui (t),
where ui (t) (∈ L2 (Ωi )) is a state, fi (t) (an input) and yi (t) (an output) are scalar functions on t ∈ [0, ∞), bi (∈ L2 (Ωi )) is an input influence function, and ci (∈ L2 (Ωi )) is a sensor influence function. αi is a small damping constant (0 < αi ≪ 1). The operator Ai is a self-adjoint, positive definite, and unbounded operator with compact resolvent whose eigenvalues λij (0 < λi1 < λi2 < · · · < λij < · · · , (limj→∞ λij = ∞)) are simple. The domain D(Ai ) is dense in L2 (Ωi ). The normalized eigenfunctions of Ai are denoted by ϕij satisfying the following relations. Ai ϕij = λij ϕij , (j = 1, 2, · · ·). (3)
The set ϕij (j = 1, 2, · · ·) forms a complete orthonormal system in L2 (Ωi ). For each controlled process (1), (2), d yi (t) are only the input fi (t) and the output yi (t), and dt
308
available for measurement, but the state ui (t) and the systems parameters in Ai , bi , ci , and αi are unknown. The control objective is to design an adaptive consensus control system for a swarm of infinite-dimensional systems (1), (2) in which consensus tracking (4) is achieved via finite dimensional adaptation schemes. yi → yj → y0 , y˙ i → y˙ j → y˙ 0 , (i, j = 1, · · · , N ).
(4)
4. MATHEMATICAL PRELIMINARY In this section, mathematical preliminaries on the solution of the distributed parameter systems of hyperbolic type (1) is summarized. The next assumption is introduced. Assumption 1 αi and λij satisfy the following conditions. αi2 λ2ij − λij ̸= 0, (j ≥ 1), 1 (5) . αi λi1 < 2αi Based on Assumption 1, gi (Ai ) is defined by ( 2 2 )1/2 , gi (λij ) ≡ αi λij − λij ∞ ∑ gi (λij )(·, ϕij )ϕij . gi (Ai ) ≡
(6)
j=1
Since gi (λij ) ∼ αi λij as j → ∞, gi (Ai ) is an unbounded operator and D(g(Ai )) = D(Ai ). Furthermore, (5) shows that gi (Ai )−1 is a bounded operator. By utilizing gi (Ai ), the solution of the process (1) is given by Lemma 1. Lemma 1 (Sakawa (1984)) The next evolution equations in L2 (Ωi ) are considered, −1 d ξi (t) = A+ bi fi (t), i ξi (t) + gi (Ai ) dt (7) d −1 ηi (t) = A− bi fi (t), i ηi (t) − gi (Ai ) dt A± (8) i ≡ −αi Ai ± gi (Ai ).
Then, the unique solution ui (t) of (1) is described as follows: ξi (t) + ηi (t) ui (t) = , (9) 2 where initial conditions ξi (0), ηi (0) of (7) are determined d ui (0) of (1) uniquely from initial conditions ui (0), dt The operator A± i have eigenfunctions ϕij and corresponding eigenvalues µ± ij as follows: ± A± i ϕij = µij ϕij , ( ) 1 + + , µij = −αi λij + gi (λij ), lim µij = − 2α)i (j→∞ lim µ+ µ− ij = −αi λij − gi (λij ), ij = −∞ .
(10) (11)
j→∞
Then, ξi (t), ηi (t) and ui (t) are rewritten into the following eigenfunction expansion forms: ∞ ∑ ξi (t) = ξij (t)ϕij , j=1 (12) ∞ ∑ η (t) = η (t)ϕ , ij ij i j=1
IFAC NecSys 2018 Groningen, NL, August 27-28, 2018
Yoshihiko Miyasato / IFAC PapersOnLine 51-23 (2018) 307–312
−1 d ξij (t) = µ+ bij fi (t), ij ξij (t) + gi (λij ) dt d −1 ηij (t) = µ− bij fi (t), ij ηij (t) − gi (λij ) dt ∞ ∑ ξij (t) + ηij (t) ui (t) = ϕij , 2 j=1
Assumption 2 The finite dimensional subsystem [Si1 ] (C¯iN , A¯iN , ¯biN ) is completely controllable and observable, that is, cij ̸= 0, bij ̸= 0, (1 ≤ j ≤ Ni )
(13)
Then, on Assumption 2, a finite dimensional observer can ′ be constructed for [Si1 ], which is denoted by [Si1 ]. d ¯ iN yiN (t), ˆ¯iN (t) + ¯biN fi (t) + K ˆ¯iN (t) = A¯iN K u u (25) dt where A¯iN K (∈ C2Ni ×2Ni ) is a stable matrix defined by ¯ iN C¯iN , (26) A¯iN K = A¯iN − K 2Ni ×1 ¯ and KiN (∈ C ) is an observer gain matrix selected properly such that the following relations hold for λif > 0.
(14)
where ξij (t) = (ξi (t), ϕij ), ηij (t) = (ηi (t), ϕij ), bij = (bi , ϕij ), respectively. 5. SYSTEM REPRESENTATION In the present section, we obtain an input-output representation of each process (1), (2) of the multi-agent system, which is compatible to the finite dimensional controller design (Miyasato (2006), Miyasato (2012), Miyasato (2013)). ˜ iN (> 0) be a given damping constant. An integer Ni Let λ 1 ˜ iN ≤ −ℜ(µ± ˜ is chosen such that λ iNi +1 ), 0 < λiN < 2αi . ˜ iN < −ℜ(µ± ), (j ≥ Ni + 1). By Then, it follows that λ ij utilizing Ni , orthogonal projection operators PiN and QiN is defined. Ni ∑ ( ·, ϕij )ϕij , PiN · = j=1 (15) ∞ ∑ QiN · = (I − PiN ) · = ( ·, ϕ )ϕ . ij ij
∥ exp(A¯iN K t)∥C2Ni ≤ const. exp(−λif t), (27) ˆ¯iN (t)∥C2Ni ∼ exp(−λif t) → 0. (28) ∥¯ uiN (t) − u ˆ¯iN (t) and Ci u ˜iN (t), the input-output By combining Ci u representation is given such that d2 ¯ iN C¯iN A¯iN K )u ˆ¯iN (t) yi (t) = (C¯iN A¯2iN K + C¯iN K dt2 ¯ iN + (C¯iN K ¯ iN )2 }yiN (t) +{C¯iN A¯iN K ∞ ∑ cij − + {h(µ+ ij )ξij (t) + h(µij )ηij (t)} 2 j=Ni +1
j=Ni +1
Then, ui (t) and yi (t) of (1), (2) are expressed by { ui (t) = PiN ui (t) + QiN ui (t) ≡ uiN (t) + u ˜iN (t), yi (t) = Ci {uiN (t) + u ˜iN (t)} ≡ yiN (t) + y˜iN (t), and those are expanded into the following forms. Ni ∑ ξij (t) + ηij (t) u (t) = ϕij , iN 2 j=1 [Si1 ] Ni ∑ ξij (t) + ηij (t) yiN (t) = cij , 2 j=1 ∞ ∑ ξij (t) + ηij (t) ϕij , (t) = u ˜ iN 2 j=Ni +1 [Si2 ] ∞ ∑ ξij (t) + ηij (t) cij , (t) = y ˜ iN 2
(16)
(17)
(18)
j=Ni +1
where cij ≡ (ci , ϕij ) = Ci ϕij . Thus, each controlled process (1), (2) of the multi-agent system is divided into two subsystems [Si1 ]:(uiN (t), yiN (t)) and [Si2 ]:(˜ uiN (t), y˜iN (t)), respectively. [Si1 ] is a finite dimensional (2Ni ) system, and is represented in the state space form. du ¯iN (t) = A¯iN u ¯iN (t) + ¯biN fi (t), dt [Si1 ] (19) ¯iN (t), y (t) = C¯iN u iN T¯ ¯iN (t) ϕiN , uiN (t) = u u ¯iN (t) = [ξi1 (t), ηi1 (t), · · · , ξiNi (t), ηiNi (t)]T , ϕ¯iN = [ϕi1 , ϕi1 , · · · , ϕiN , ϕiN ]T , i
i
− − + A¯iN = diag (µ+ i1 , µi1 , · · · , µiNi , µiNi ), ¯biN = [gi (λi1 )−1 bi1 , −gi (λi1 )−1 bi1 , · · · , gi (λiNi )−1 biNi , −gi (λiNi )−1 biNi ]T ,
C¯iN = [ci1 /2, ci1 /2, · · · , ciNi /2, ciNi /2]. For the subsystem [Si1 ], we assume that
(20)
(21) (22) (23) (24)
309
θi0
+θi0 f (t) + ϵi (t), ∫ ∞ ∑ ≡ cij bij = ci (xi )bi (xi )dxi . j=1
(29)
(30)
Ωi
Hereafter, all exponentially decaying terms are denoted by ϵi (t). Next, fif (t) is introduced such as d fif (t) = −ai0 fif (t) + fi (t), (31) dt ˆ¯iN (t) where ai0 is a positive constant. The substitution of u and yiN (t) = y(t) − y˜iN (t) into (29) yields d2 T T v¯iN 2 (t) + θi3 fif (t) + θi4 yi (t) yi (t) = θi1 v¯iN 1 (t) + θi2 dt2 (32) +θi0 fi (t) + δi (t) + ϵi (t), where v¯iN 1 (t) and v¯iN 2 (t) are finite dimensional state variable filters (2Ni dimension) such as d v¯iN 1 (t) = F¯iN v¯iN 1 (t) + g¯iN fif (t), dt (33) d v¯ (t) = F¯ v¯ (t) + g¯ y (t). iN 2 iN iN 2 iN i dt ¯ (FiN , g¯iN ) (F¯iN ∈ R2Ni ×2Ni , g¯iN ∈ R2Ni ) is controllable, and F¯iN is chosen such that det(sI − F¯iN ) = det(sI − ¯ iN A¯iN K ). Since (C¯iN , A¯iN ) is observable, there exists K satisfying the relation for an arbitrary stable matrix F¯iN (∈ R2Ni ×2Ni ). θi1 , θi2 (∈ R2Ni ) are vectors satisfying the following relation (34) (since (F¯iN , g¯iN ) is controllable, there exists θi1 , θi2 (∈ R2Ni ) satisfying (34) (Ioannou and Sun (1996)). ¯ iN C¯iN A¯iN K ) · (C¯iN A¯2iN K + C¯iN K ∫t ¯ · (AiN K + ai0 I) {exp A¯iN K (t − τ )} · ¯biN fif (τ )τ 0 ∫t ¯ iN yi (τ )dτ + {exp A¯iN K (t − τ )} · K 0
T T v¯iN 1 (t) + θi2 v¯iN 2 (t) + ϵi (t). = θi1
309
(34)
IFAC NecSys 2018 310 Groningen, NL, August 27-28, 2018
Yoshihiko Miyasato / IFAC PapersOnLine 51-23 (2018) 307–312
Furthermore, θi3 , θi4 (∈ R), δi (t) are defined by ¯ iN C¯iN A¯iN K )¯biN , θi3 = (C¯iN A¯2iN K + C¯iN K ¯ iN + (C¯iN K ¯ iN )2 }, θi4 = {C¯iN A¯iN K
(35) (36)
¯ iN C¯iN A¯iN K ) · δi (t) = −(C¯iN A¯2iN K + C¯iN K ∫t ¯ iN y˜iN (τ )dτ · {exp A¯iN K (t − τ )} · K
(37)
Therefore, the input-output representation of each process of the multi-agent system is deduced such as (32), and is T T composed of θi1 v¯iN 1 (t) + θi2 v¯iN 2 (t) + θi3 fif (t) + θi4 yi (t) + θi0 fi (t) and δi (t). The former term is considered as a primal part for the finite dimensional controller design. On the contrary, the latter term δi (t) comes from the infinite dimensional system [Si2 ], and is seen as a residual part for the design of the control system, and the interferences from the infinite dimensional system [Si2 ] are to be taken into consideration to stabilize the total system hereafter. Next, the residual part δi (t) is to be evaluated. For that purpose, state variable filters whose dimensions are 1, are ˜ introduded by utilizing design parameters λiN and λif . ˜ iN wi1 (t) + |fif (t)|, d wi1 (t) = −λ dt (38) d wi2 (t) = −λif wi2 (t) + wi1 (t). dt The sensor influence function ci and the input influence function bi are assumed to be smooth in the following sense.
(39)
j=1
j=1
Then, δi (t) of (37) is evaluated by Lemma 2 (Miyasato (2006), Miyasato (2012), Miyasato (2013)). Lemma 2 On Assumption 3, δi (t) is evaluated as follows: |δi (t)| ≤ giδ (t)T diδ + |ϵi (t)|,
(40) T
giδ = [ |fif (t)|, wi1 (t), wi2 (t) ] ,
(41)
diδ = [ Mi1 , Mi2 , Mi3 ] ,
(42)
T
0 < Mi1 ∼ Mi3 < ∞, ˜ iN t −λ
ϵi (t) ∼ e
Assumption 4 θi0 ̸= 0, and sgn θi0 is known. In the following, it is assumed that θi0 > 0 without loss of generality. There exist {Mif 0 and� Mif 1 such �} that �d � |fif (t)| ≤ Mif 0 + Mif 1 sup |yi (τ )|, �� yi (τ )�� , dτ 0≤τ ≤t
Assumption 5
j=Ni +1
Assumption 3 The following inequalities hold. ∞ ∞ ∑ ∑ 2 λ2k |λkij cij bij | < ∞, ij cij < ∞, (k = 1, 2).
6.1 Assumptions By utilizing the representation (43), the proposed adaptive H∞ consensus control scheme is to be constructed via finite dimensional compensators. The next assumptions are introduced.
0
¯ iN + (C¯iN K ¯ iN )2 }˜ yiN (t) −{C¯iN A¯iN K ∞ ∑ cij − + {h(µ+ ij )ξij (t) + h(µij )ηij (t)}. 2
6. ADAPTIVE H∞ CONSENSUS CONTROL
, e−λif t , e−ai0 t → 0.
Hereafter, the input-output representation of each agent is written in the following form. d2 yi (t) = ΘT (43) i ωi (t) + θi0 fi (t) + δi (t) + ϵi (t), dt2 [ T T ]T (44) , θi2 , θi3 , θi4 , Θi = θi1 [ ]T T T (45) ωi (t) = viN 1 (t) , viN 2 (t) , fif (t), yi (t) ,
where ΘT i ωi (t) + θi0 fi (t) is utilized for the finite dimensional controller design, and the evaluation of δi (t) (Lemma 2) is to be applied to stabilize the total system.
310
(0 ≤ Mf 0 < ∞, 0 < Mf 1 < ∞).
(46)
Remark Assumption 4 states that the relative degree of each agent is 2, and Assumption 5 asserts that the process has a stable inverse. 6.2 Control Law and Error Equation First, an estimation procedure of y˙0 (the leader’s information) is constructed via available data from the follower i. A similar estimator was given in Mei and Ren (2011) for second-order Euler-Lagrange systems. N ∑ cij {ˆ zi (t) − zˆj (t)} zˆ˙ i (t) = −β j=1 j ̸= i
zi (t) − y˙ 0 (t)} + ni0 y¨0 (t), (47) −βci0 {ˆ where zˆi is a current estimate of y˙ 0 , and is synthesized from the data available to the follower i. cij (1 ≤ i ≤ N, 0 ≤ j ≤ N ) is defined as the entry of the adjacency matrix C and ai0 , and β > 0 is a design parameter. Concerned with ci0 , ni0 is defined as ni0 = 1 (when ci0 > 0), ni0 = 0 (otherwise). By utilizing zˆi , the following control law is employed. N ∑ aij {yi (t) − yj (t)}, (48) y˙ ri (t) = zˆi (t) − α j=0 j ̸= i
si (t) = y˙ i (t) − y˙ ri (t), [ ] ˆ i (t)T ωi (t) + y¨ri (t) + vi (t) fi (t) = pˆi (t) −Θ
(49)
(50) ≡ pˆi (t)fi0 (t) + vi (t), where aij (1 ≤ i ≤ N, 0 ≤ j ≤ N ) is defined from the entry of the adjacency matrix A and ai0 , and α > 0 is ˆ is denoted as a current estimate a design parameter. (·) of (·), and pi is defined by pi = 1/bi . Furthermore, vi is a stabilizing signal which is to be determined later based on H∞ control criterion. An estimation error between the leader y˙ 0 and the estimate zˆi is defined such as (51) z˜i (t) ≡ zˆi (t) − y˙ 0 (t), and the following relations are given for si and z˜i . N ∑ ˙z˜i (t) = −β cij {˜ zi (t) − z˜j (t)} j=1 j ̸= i
IFAC NecSys 2018 Groningen, NL, August 27-28, 2018
Yoshihiko Miyasato / IFAC PapersOnLine 51-23 (2018) 307–312
−βci0 z˜i (t) + (ni0 − 1)¨ y0 (t), (52) T ˆ i (t)} ωi (t) + θi0 {ˆ pi (t) − pi }fi0 (t) s˙ i (t) = {Θi − Θ +θi0 vi (t) + δi (t) + ϵi (t).
(53)
Then, the total representations of the multi-agent system are deduced as follows: z˜˙ (t) = −βMc z˜(t) + (N0 − 1)¨ y0 (t), (54) ˆ p(t) − p} s(t) ˙ = Ω(t){Θ − Θ(t)} + F0 (t)Θ0 {ˆ +Θ0 v(t) + δ(t) + ϵ(t),
(55)
where T T T T z˜ = [˜ z1T , · · · , z˜N ] , s = [sT 1 , · · · , sN ] ,
T T T ), Θ = [ΘT Ω = block diag (ω1T , · · · , ωN 1 , · · · , ΘN ] ,
F0 = diag (f10 , · · · , fN 0 ), Θ0 = diag (θ10 , · · · , θN 0 ), p = [p1 , · · · , pN ]T , N0 = [n10 , · · · , nN 0 ]T ,
1 = [1, · · · , 1]T , v = [v1 , · · · , vN ]T ,
δ = [δ1 , · · · , δN ]T , ϵ = [ϵ1 , · · · , ϵN ]T .
Especially, owing to the assumption of the network graph G, the matrix −Mc is shown to be Hurwitz.
For that purpose, the following Hamilton-Jacobi-Isaacs (HJI) equation { 2 and its solution W0 are introduced } 1 ∑ ∥Lg1i W0 ∥2 − (Lg2 W0 )R−1 (Lg2 W0 )T L f W0 + 4 i=1 γi2
+q = 0, (62) 1 T (63) W0 = s s, 2 where q and R are a positive function and a positive definite matrix respectively. Those are derived from HJI equation based on inverse optimality (Krsti´c and Deng (1998)) for the given solution W0 and the positive constants γ1 , γ2 . The substitution of the solution W0 (63) into HJI equation (62) yields ( ) I 1 T G δ GT −1 ˆ T δ ˆ (64) s + 2 − Θ0 R Θ0 s + q = 0. 4 γ12 γ2 Then, R and q are derived such as )−1 ( ˆ −T ˆ −T ˆ −1 Θ ˆ −1 Gδ GT Θ Θ Θ 0 δ 0 + 0 20 +K , (65) R= γ12 γ2
6.3 Adaptive H∞ Consensus Control A positive function W is defined by { } }T 1 {ˆ 1 ˆ Θ(t) −Θ Θ(t) − Θ Γ−1 W (t) = s(t)T s(t) + 1 2 2 1 T p(t) − p} Θ0 Γ−1 p(t) − p} + {ˆ 2 {ˆ 2 } }T { 1 {ˆ (56) θˆ0 (t) − θ0 , θ0 (t) − θ0 Γ−1 + 3 2 where Γ1 , Γ2 and Γ3 are diagonal and positive definite matrices, and θ0 is defined as follows: (57) θ0 = [θ10 , · · · , θN 0 ]T . ˆ ˆ The tuning by laws of Θ,{ pˆ, θ0 are given } ˙ T ˆ Θ(t) = Pr{ Γ1 Ω(t) s(t) ,} (58) pˆ˙ (t) = Pr −Γ2 F0 (t)T s(t) , ˙ { } T ˆ θ0 (t) = Pr Γ3 V (t) s(t) , V = diag (v1 , · · · , vN ), (59)
where Pr(·) are projection operations in which tuning ˆ pˆ and θˆ0 are constrained to bounded regions parameters Θ, deduced from upper-bounds of ∥Θ∥ and upper-bounds and lower-bounds of each element of p and θ0 , respectively (Ioannou and Sun (1996)). Then, the time derivative of W along its trajectory is derived as follows: ˆ 0 (t)v(t). (60) ˙ (t) ≤ s(t)T δ(t) + s(t)T ϵ(t) + s(t)T Θ W ˙ (60), we introduce the next From the evaluation of W virtual system, z˜˙ = f + g11 d1 + g12 d2 + g2 v, (61) ˆ f = 0, g11 = Gδ , g12 = I, g2 = Θ0 , T T Gδ = block diag (gδ1 , · · · , gδN ),
d1 =
[dT δ1 , · · · ,
T dT δN ] ,
311
d2 = ϵ.
and are to stabilize this virtual system via a control input v by utilizing H∞ criterion, where d1 , d2 are considered as external disturbances to the process (Miyasato (2000)). 311
1 ˆ ˆT q = sT Θ (66) 0 K Θ0 s, 4 where K is a diagonal positive definite matrix (a design parameter). From R, v is deduced as a solution of the corresponding H∞ control problem as follows: 1 1 ˆT (67) v = − R−1 (Lg2 W0 )T = − R−1 Θ 0 s, 2 2 ˆ 0 are constructed from the elements where the entries of Θ of θˆ0 (58). Then, the time derivative of W is evaluated by ˙ ≤ −q − v T Rv W ( )T ( ) 1 −1 ˆ T 1 −1 ˆ T + v + R Θ0 s R v + R Θ0 s 2 2 �2 N � ∑ � |si | gδi � 2 2 2 � � +γ1 ∥d1 ∥ − γ1 �dδi − 2γ 2 � 1 i=1 )2 N ( ∑ |si | 2 2 2 +γ2 ∥d2 ∥ − γ2 , (68) |ϵi | − 2 2γ2 i=1
ˆ pˆ, and θˆ0 are bounded for the and it follows that s, Θ, tuning laws (58) and the stabilizing signal v (67). Next, for stability analysis of the estimation error z˜, a positive function V1 is introduced such as (69) V1 = z˜T Pc z˜, (70) Pc Mc + McT Pc = I, (Pc = PcT > 0). There exists a positive definite and symmetric matrix Pc satisfying (70), since −Mc is Hurwitz. Then, the time derivative of V1 along its trajectory is evaluated as follows: 2 β z ∥2 + ∥Pc (N0 − 1)¨ y0 ∥2 , V˙ 1 ≤ − ∥˜ 2 β and it is shown that z˜ is bounded for bounded y¨0 .
(71)
Finally, for stability analysis of the control error yi − y0 , y˜i and y˜ are defined by y˜i = yi − y0 , (72)
T T ] . y˜ = [˜ y1T , · · · , y˜N Then, the following relation holds
(73)
IFAC NecSys 2018 312 Groningen, NL, August 27-28, 2018
Yoshihiko Miyasato / IFAC PapersOnLine 51-23 (2018) 307–312
y˜˙ = s + z˜ − αMa y˜. (74) −Ma is shown to be Hurwitz because of the assumption of the network graph G. From that property, a positive function V2 is defined by V2 = y˜T Pa y˜, (75) (76) Pa Ma + MaT Pa = I, (Pa = PaT > 0). Similarly to the previous case, there exists a positive definite and symmetric matrix Pa satisfying (76), since −Ma is Hurwitz. Then, the time derivative of V2 along its trajectory is evaluated as follows: α 4 y ∥2 + ∥Pa ∥2 (∥s∥2 + ∥˜ z ∥2 ), (77) V˙ 2 ≤ − ∥˜ 2 α and it follows that y˜ is bounded for bounded s, z˜. From the three stages of stability analysis (the evaluations ˙ , V˙ 1 , V˙ 2 ), the next theorem is obtained. of W Theorem 1 In the proposed adaptive control system, the stabilizing signal v is a sub-optimal control input minimizing the upper bound on the cost functional J. t ∫ {q + v T Rv}dτ + W (t) J(t) ≡ sup d1 ,d2 ∈L2
−
2 ∑
γi2
i=1
0
∫t 0
∥di ∥2 dτ .
(78)
∥di ∥2 dτ + W (0).
(79)
Also the next inequality holds. ∫t {q + v T Rv}dτ + W (t) 0
≤
2 ∑ i=1
γi2
∫t 0
Theorem 2 The adaptive control system is uniformly bounded, and the next relations hold. ∫T 1 ∥˜ y (t)∥2 dt ≤ const · (γ12 + γ22 ) lim sup T →∞ T 0 } { 1 2 , (80) ∥(N − 1)¨ y ∥ +const · sup 0 0 α2 β 2 ∫T 1 lim sup ∥y˜˙ (t)∥2 dt ≤ const · (γ12 + γ22 ) T →∞ T 0 } { 1 2 ∥(N0 − 1)¨ y0 ∥ . (81) +const · sup β2 Remark The magnitude of the tracking error is evaluated according to the direct availability of y¨0 ((N0 −1) = 0 or not), or the value of y¨0 . The total adaptive control scheme is constructed in a distributed fashion. 7. CONCLUDING REMARKS A design procedure of finite-dimensional adaptive H∞ consensus control of multi-agent systems composed of distributed parameter systems of hyperbolic type on directed network graphs, has been provided in the paper. 312
The present work is an attempt to extend the consensus control issue itself to infinite-dimensional systems, and the proposed method would provide a basic and useful strategy to deal with the coordinate control of large flexible structures on a directed network graph via low-order (or finite dimensional) controllers. REFERENCES Cao, Y. and Ren, W. (2011). Distributed coordinate tracking via a variable structure approach - part i : Consensus tracking. Proc. of 2010 ACC, 4744–4749. Demetriou, M. (2013). Synchronization and consensus controllers for a class of parabolic distributed parameter systems. Systems & Control Letters, 70–76. Demetriou, M. (2018). Design of adaptive output feedback synchronizing controllers for networked pdes with boundary and in-domain structured perturbations and disturbances. Automatica, 220–229. Ioannou, P. and Sun, J. (1996). Robust Adaptive Control. PTR Prentice-Hall, New Jersey. J. Mei, W.R. and Chen, J. (2014). Consensus of secondorder heterogeneous multi-agent systems under a directed graph. Proc. of 2014 ACC, 802–807. Kingston, D., Ren, W., and Beard, R. (2005). Consensus algorithms are input-to-state stable. Proc. of 2005 ACC, 1686–1690. Krsti´c, M. and Deng, H. (1998). Stabilization of Nonlinear Uncertain Systems. Springer. Mei, J. and Ren, W. (2011). Distributed coordinate tracking with a dynamic leader for multiple euler-lagrange systems. IEEE Transactions on Robotics and Automation, 1415–1421. Mei, J., Ren, W., Chen, J., and Anderson, B. (2014). Consensus of linear multi-agent systems with fully distributed control gains under a general directed graph. Proc. of the 53rdt IEEE CDC, 2993–2998. Miyasato, Y. (2000). Adaptive nonlinear h∞ control for processes with bounded variations of parameters – general relative degree case –. Proc. of the 39th IEEE CDC, 1453–1458. Miyasato, Y. (2006). Model reference adaptive h∞ control for distributed parameter systems of hyperbolic type by finite-dimensional controller. Proc. of the 45th IEEE CDC, 459–464. Miyasato, Y. (2012). Adaptive h∞ formation control for infinite-dimensional systems. Proc. of the 51st IEEE CDC, 6071–6076. Miyasato, Y. (2013). Adaptive h∞ consensus control for distributed parameter systems of hyperbolic type. Proc. of 2013 IEEE Multi-Conference on Systems and Control, 1018–1023. Miyasato, Y. (2015). Adaptive h∞ consensus control of multi-agent systems on directed graph. Proc. of the 41st IEEE CDC, 7592–7597. Olfati-Saber, R., Fax, J.A., and Murray, R.M. (2007). Consensus and cooperation in networked multi-agent systems. Proc. of the IEEE, 215–233. Ren, W. and Cao, Y. (1996). Distributed Coordination of Multi-Agent Networks. Springer. Sakawa, Y. (1984). Feedback control of second order evolution equations with damping. SIAM J. Control and Optimization, 22, 343–361.