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Distributed LQR-based Suboptimal Control for Coupled Linear Systems⁎
7th 7th IFAC IFAC Workshop Workshop on on Distributed Distributed Estimation Estimation and and Control Networked 7th IFACin Workshop onSystems Distributed Estimation Availableand online at www.sciencedirect.com Control in Networked Systems 7th IFAC Workshop on Distributed Estimation and Chicago, IL, USA, September 16-17, 2019 Control inIL, Networked Systems16-17, 2019 Chicago, USA, September Control in Networked Systems Chicago, IL, USA, September 16-17, 2019 Chicago, IL, USA, September 16-17, 2019
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IFAC PapersOnLine 52-20 (2019) 109–114
Distributed LQR-based Suboptimal Distributed LQR-based Suboptimal Distributed LQR-based Suboptimal Distributed LQR-based Suboptimal Control for Coupled Linear Systems Control for Coupled Linear Systems Control for Coupled Linear Systems Control for Coupled Linear Systems ∗ ∗∗
Eleftherios E. Vlahakis ∗ Leonidas Leonidas D. D. Dritsas Dritsas ∗∗ Eleftherios E. Vlahakis ∗ ∗ D. Dritsas ∗∗ Vlahakis Eleftherios E.George Leonidas ∗ D. ∗ Halikias ∗∗ George D. Halikias Eleftherios E.George Vlahakis Leonidas ∗ D. Dritsas D. Halikias ∗ George D. Halikias ∗ ∗ School of Mathematics, Computer Science and Engineering, City, of Mathematics, Computer Science and Engineering, City, ∗ School School of Mathematics, Science and Engineering, City, of (e-mail: ∗University University of London, London, UK UKComputer (e-mail: [email protected], [email protected], School of Mathematics, Computer Science and Engineering, City, University of London, [email protected]) UK (e-mail: [email protected], [email protected]) University of London, UK (e-mail: [email protected], ∗∗ [email protected]) ∗∗ Department of Electrical & & Electronic Electronic Engineering Engineering Educators, Educators, [email protected]) ∗∗ Department of Electrical Department of Electrical & Electronic Engineering Educators, School of Pedagogical & Technological Education, ASPETE, Athens, ∗∗ School of Pedagogical & Technological Education, ASPETE, Athens, Department of Electrical & Electronic Engineering Educators, School of Pedagogical & Technological Education, ASPETE, Athens, Greece (e-mail: [email protected]) Greece (e-mail: [email protected]) School of Pedagogical & Technological Education, ASPETE, Athens, Greece (e-mail: [email protected]) Greece (e-mail: [email protected]) Abstract: Abstract: A A well-established well-established distributed distributed LQR LQR method method for for decoupled decoupled systems systems is is extended extended to to the the Abstract: Acoupled well-established distributed LQRsystems methodare for dynamically decoupled systems is extended the dynamically case where the open-loop dependent. First, aatofully dynamically coupled case where the open-loop systems are dynamically dependent. First, fully Abstract: A well-established distributed LQR method for decoupled systems is extended to the dynamically coupled case where the open-loop systems are dynamically dependent. First, a fully centralized controller is designed which is subsequently substituted by a distributed statecentralized controller is designed which is subsequently substituted dependent. by a distributed statedynamically coupled case where the open-loop systems are dynamically First, a fully centralizedgain controller is designed which subsequently distributed feedback with structure. The control scheme is by aoptimizing optimizing an stateLQR by an LQR feedback gain with sparse sparse structure. Theis control scheme substituted is obtained obtained by centralized controller is adesigned which subsequently substituted distributed feedback gain with with sparse structure. Theiscontrol scheme is obtained by by aoptimizing an stateLQR performance index with tuning parameter utilized to emphasize/de-emphasize emphasize/de-emphasize relative state performance index a tuning parameter utilized to relative state feedback gain with with sparse structure. The control scheme is obtained by optimizing an LQR performance index a tuning parameter utilized to emphasize/de-emphasize state difference between interconnected systems. Overall stability is guaranteed via aarelative simple test difference between interconnected systems. Overall stability is guaranteed via simple test performance index with a tuning parameter utilized to emphasize/de-emphasize relative state difference between interconnected systems. Overall stability is guaranteed via a simple test applied to a convex combination of Hurwitz matrices, the validity of which leads to stable applied to a convex combination of Hurwitz matrices, the validity of which leads to stable difference between interconnected systems. Overall stability is shown guaranteed vialeads a simple test applied to a convex combination of Hurwitz matrices, the validity of which to stable global operation for a class of interconnection schemes. It is also that the suboptimality global operation for acombination class of interconnection schemes. Itthe is also shown that theleads suboptimality applied to a convex of Hurwitz matrices, validity of which to stable global operation for a class of interconnection schemes. It is also shown that the suboptimality of the the method method can can be be assessed assessed by by measuring measuring aa certain certain distance distance between between two two positive positive definite definite of global forbe a class of interconnection is also shown that suboptimality of the operation method can assessed by measuring aschemes. certain It distance between twothe positive definite matrices which can obtained by solving two Lyapunov equations. matrices which can be obtained by solving two Lyapunov equations. of the method certain distance between two positive definite matrices which can can be be assessed obtained by by measuring solving twoa Lyapunov equations. matrices which be obtained by by solving two Lyapunov Copyright © 2019. can The Authors. Published Elsevier Ltd. All rights equations. reserved. Keywords: Keywords: distributed distributed LQR, LQR, coupled coupled linear linear systems, systems, multi-agent multi-agent control, control, networked networked control. control. Keywords: distributed LQR, coupled linear systems, multi-agent control, networked control. Keywords: distributed LQR, coupled linear systems, multi-agent control, networked control. 1. INTRODUCTION INTRODUCTION stabilizing 1. stabilizing distributed distributed LQR-based LQR-based controller controller for for networks networks 1. INTRODUCTION stabilizing distributed LQR-based controller for networks formed of of identical identical agents with dynamical dynamical couplings. formed agents with couplings. 1. INTRODUCTION stabilizing distributed LQR-based controller for networks of identical agents with dynamical couplings. Networked systems systems have have attracted attracted attention attention from from the the formed Networked Over the past few years, there has been a renewal of formed of identical agents with dynamical couplings. Networked systems in have attracted attention frombroad the Over the past few years, there has been a renewal of control community community in recent years due due to their their broad control recent years to Over the few years, there has been renewal of interest in past control of networks networks composed of aa alarge large number Networked systems have attracted attention from the interest in control of composed of number control recent yearsare to referred their broad the past few years, there has been alarge renewal of range of of community applications.inSuch Such schemes aredue often referred to as as Over range applications. schemes often to interest in control of networks composed of a number of interacting systems. Rigorous methods for cooperative controlof community inSuch recent years due to referred their broad of interacting systems. Rigorous methods for cooperative range applications. schemes are often to as interest in control of networks composed of a large number multi-agent networks, with each agent having autonomous multi-agent networks,Such with schemes each agent autonomous interacting Rigorous methods cooperative control design for systems with distributed or range of applications. arehaving often referred to as of control design systems. for multi-agent multi-agent systems withfor distributed or multi-agent networks, agent having autonomous of interacting systems. Rigorous methods for cooperative actuation capacity. Thewith needeach for forming forming networks in many many control design for multi-agent systems with distributed or actuation capacity. The need for networks in decentralized pattern have been provided in Olfati-Saber multi-agent networks, with each agent having autonomous decentralized pattern have been provided indistributed Olfati-Saber actuation capacity. The need for forming networks in many control design for multi-agent systems with or cases arises from the fact that some problems may not cases arises from the fact that some problems may not decentralized pattern have been provided in Olfati-Saber (2006); Fax and Murray (2002); Wang et al. (2018). actuation capacity. Thethe need for forming networks in many (2006); Fax and Murray (2002); Wang etinal. (2018). A A cases arises from the fact that some problems may not decentralized pattern have been provided Olfati-Saber admit a solution at individual system level. Thus admit a solution at the individual system level. Thus (2006); Fax and Murray (2002); Wang et al. controllers (2018). A thorough procedure for designing distributed cases arises from the fact that some problems may not thorough procedure for designing distributed controllers admit solution the individual systemhave level. Faxofand Murray (2002); Wang et al. controllers (2018). A agents adespite despite theiratindependent independent operation have alsoThus the (2006); agents their operation also the thorough designing distributed for systems based on admit adespite the individual level. for aa class classprocedure of coupled coupledfor systems based on aa decomposition decomposition agents theiratindependent alsoThus the thorough procedure for designing distributed controllers ability tosolution cooperate with certainoperation of system their have counterparts for a class of coupled systems based on a decomposition ability to cooperate with certain of their counterparts has been in Massioni and Verhaegen agents the despite theirtowards independent operation have alsoother the approach approach has been presented presented inbased Massioni Verhaegen ability to cooperate with certain of their counterparts a class offundamental coupled systems on a and decomposition within network common objective. In within the network towards aa common objective. In other for approach has been presented in of Massioni and (2009). The works Borrelli and Keviczky ability to cooperate with certain of their counterparts (2009). The fundamental works of Borrelli andVerhaegen Keviczky within the network towards a common objective. In other approach has been presented in Massioni and Verhaegen cases, the topology of the network may be imposed by cases, the topology of the network may be imposed by (2009). The fundamental works of Borrelli and Keviczky (2008); Deshpande et al. (2012) discuss distributed LQR within the network towards a common objective. In other (2008); Deshpande et al. (2012) discuss distributed LQR cases, the links topology may bewhere imposed by (2009). The fundamental works of Borrelli and Keviczky structural links suchofas asthe in network power systems systems where agents structural such in power agents (2008); Deshpande et al. (2012) discuss distributed LQR design for a set of identical decoupled dynamical systems. cases, the topology of the network may be imposed by design for a set of identical decoupled dynamical systems. structural links such as in power systems where agents (2008); Deshpande et al. (2012) discuss distributed LQR denote power power generators generators and and interconnections interconnections represent represent design for a set of identical decoupled dynamical systems. denote Unfortunately, there is no documented distributed LQRstructural links such as in power systems where agents Unfortunately, there is no documented distributed LQRdenote power generators and interconnections represent design for a set of identical decoupled dynamical systems. power transmission lines. power lines. and interconnections represent Unfortunately, is no documented LQRbased approach to systems with denote transmission power generators based approachthere to networked networked systemsdistributed with dynamical dynamical power transmission lines. Unfortunately, there is no documented distributed LQRbased approach to networked systems with dynamical couplings in the context of relevant literature. In this transmission paper, we we focus focus on multi-agent multi-agent networks networks composed composed couplings power lines. in the context of relevant literature. In this paper, on based approach to networked systems with dynamical in the context of relevant literature. In this paper, we focus oncoupled multi-agent composed of identical identical dynamically coupled linearnetworks time-invariant sys- couplings of dynamically linear time-invariant sysIn we follow aa relevant top-down method couplings in the In identical thisWe paper, we focus on multi-agent networks composed In this this work, work, wecontext follow of top-downliterature. method to to approxapproxof dynamically coupled linear time-invariant systems. consider that these dynamical couplings can be tems. We consider that these dynamical couplings can be In this work, we follow a top-down method toby approximate a centralized LQR optimal controller a of identical dynamically coupled linear time-invariant sysimate a centralized LQR optimal controller by a disdistems. We consider that these dynamical couplings can be thisawork, wescheme. follow a top-down method tobynetwork approxexpressed in in aa state-space state-space form form of of aa certain certain structure. structure. In In In expressed imate centralized LQR optimal controller a distributed control It is shown that overall tems. We consider that these dynamical couplings can be tributed control scheme. It is shown controller that overallbynetwork expressed in a state-space form of a certain structure. In imate a centralized LQR optimal a dis-a our case each system representing an agent can produce acour case each representing canstructure. produce acscheme.via It isaa shown that overall network stability control is guaranteed guaranteed via stability test applied to expressed in asystem state-space form ofan aagent certain In tributed stability is stability test applied to a our case each system representing an agent can produce actributed control scheme. It is shown that overall network tuation signals independently and is dynamically coupled tuation signals independently andanisagent dynamically coupled stability is guaranteed via a stability testThe applied to of a convex combination of Hurwitz matrices. validity our case each system representing can produce acconvex combination of Hurwitz matrices. The validity of tuation signals independently is guaranteed via awith stability testThe applied to ofa and is dynamically coupled stability with certain certain number of its its peers peers referred to as as neighboring neighboring with number of referred to convex combination of Hurwitz matrices. validity this condition is consistent the stability of a class tuation signals independently and is dynamically coupled condition is consistent withmatrices. the stability of a class with certain number of its peers referred to as neighboring convex combination of Hurwitz The validity of agents with whom whom it can can exchange state information. information. Effec- this agents with it exchange state Effecthis condition is consistent with the stability of a class of interconnection structures which with certain number of the its peers referred to as neighboring of network network interconnection structures which is is identified. identified. agents with whom it can exchange state information. Effecthis condition is consistent with the stability of a class tively, we assume that topology of physical couplings tively, we assume that the topology of physical couplings of network interconnection structures which is identified. Sufficient condition condition for for stability stability of of convex convex combination combination agents with whom that it exchange state Effec- Sufficient tively, assume the of network interconnection structures which is identified. topology of information. physical couplings and thewe topology of can information exchange among agents and the topology of information exchange among agents Sufficient condition for stability ofin convex combination of Hurwitz matrices can be found Bia llas (2004). tively, we assume that the topology of physical couplings of Hurwitz matrices can be found in Bia as (2004). The The and the topology of information exchange among agents Sufficient condition for stability of convex combination coincide and are described by the same graph. Network coincide and are described by theexchange same graph. Network of Hurwitz matrices can beby found in Biaan las LQR (2004). The control scheme is obtained optimizing perforand the topology of information among agents control scheme is obtained by optimizing an LQR perforcoincide and are described by the same graph. Network of Hurwitz matrices can be found inwhich Biaan lascan (2004). The stabilization is is one one of of the the most most challenging challenging problems problems in in control scheme is obtained by optimizing LQR perforstabilization mance index with a tuning parameter be used to coincide and are described by the same graph. Network mance index with tuning parameter which be used to stabilization is one ofcontrol. the most challenging problems scheme is aaobtained by optimizing ancan LQR performulti-agent network network control. In this this work, we we propose in a control multi-agent In work, propose a mance index with tuning parameter which can be used to emphasize/de-emphasize relative state difference between stabilization is one of the most challenging problems in emphasize/de-emphasize relative state difference between multi-agent network control. In this work, we propose a mance index with a tuning parameter which can be used to emphasize/de-emphasize relative state difference between interconnected systems. Our approach was inspired by the multi-agent network control. In this work, we propose a interconnected systems. Our approach was inspired by the This emphasize/de-emphasize relative state difference between This work work was was supported supported by by a a City, City, University University of of London London scholarscholarinterconnected systems. Our approach was inspired by the powerful results proposed in Borrelli and Keviczky (2008). powerful results proposed in Borrelli and Keviczky (2008). ship held by Eleftherios Vlahakis. This work was supported by a City, University of London scholarinterconnected systems. Our approach was inspired by the ship held by was Eleftherios Vlahakis. powerful results proposed in Borrelli and Keviczky (2008). This work supported by a City, University of London scholarship held by Eleftherios Vlahakis. powerful results proposed in Borrelli and Keviczky (2008). ship held by Eleftherios Vlahakis.
2405-8963 Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved. Copyright © 2019 109 Copyright © under 2019 IFAC IFAC 109 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 109 10.1016/j.ifacol.2019.12.139 Copyright © 2019 IFAC 109
2019 IFAC NecSys 110 Eleftherios E. Vlahakis et al. / IFAC PapersOnLine 52-20 (2019) 109–114 Chicago, IL, USA, September 16-17, 2019
Therein, the subsystems constituting the network are dynamically decoupled, and the stability of the distributed scheme designed relies on the stability margins of LQR control. A complementary distributed LQR method has also been proposed in Deshpande et al. (2012), which consists of a bottom-up approach in which optimal interactions between self-stabilizing agents are defined so as to minimize an upper bound of the global LQR criterion. Our definition of multi-agent networks with dynamical couplings has been motivated by the structure of a multiarea power system. The proposed method has been successfully applied to multi-area power system control design numerous simulations of which can be found in Vlahakis et al. (2019). A major assumption of our work is that the dynamical models of each system are identical. Although this assumption may be unrealistic in practice, it simplifies the design problem considerably, which is especially hard due to the coupling terms appearing in the model. Future work will attempt to eliminate or relax this assumption. Preliminary results in this direction can be found in Vlahakis and Halikias (2018a,b). The remaining of the paper is organized in four sections. In Section 2 notation and some preliminaries on graph theory are presented. The main results of our work are presented in Section 3 and 4 where LQR properties of coupled linear systems and the distributed control algorithm are derived, respectively. Section 5 summarizes the main conclusions of the work where a discussion of the main results and suggestions for future work are also included. 2. NOTATION AND GRAPH THEORY PRELIMINARIES The field of real and complex numbers are denoted by R and C, respectively. Rn denotes the n-dimensional vector space over the field R. Rn×m denotes the set of n × m real matrices. ξ and Ξ denote the transpose of ξ and Ξ, respectively. Matrix Ξ ∈ Rn×n is called symmetric if Ξ = Ξ. The identity matrix of dimension m×m is denoted by Im ∈ Rm×m . The n×m zero matrix is denoted by 0n,m . C = {s ∈ C : Re(s) < 0}. C = {s ∈ C : Re(s) ≤ 0}.
Let A ∈ Rn×m and B ∈ Rq×p . Then, the Kronecker product of A and B is denoted by A ⊗ B and defined as a11 B · · · a1m B .. ∈ Rnq×mp , .. A ⊗ B = ... . . an1 B · · · ann B where aij is the (i, j)-th entry of A, with i = 1, · · · , n and j = 1, · · · , m.
Let λi (Ξ) denote the i-th eigenvalue of Ξ ∈ Rn×n , i = 1, · · · , n. Then, the spectrum of Ξ is denoted by S(Ξ) = {λ1 (Ξ), · · · , λn (Ξ)}. Definition 1. Matrix Ξ ∈ Rn×n is called Hurwitz (or stable) if all its eigenvalues have negative real part, i.e., λi (Ξ) ∈ C , i = 1, · · · , n. Proposition 2. Let A1 = aIm and A2 ∈ Rm×m . Then λi (A1 + A2 ) = a + λi (A2 ), i = 1, · · · , m. Proof. Let λi (A2 ) be any eigenvalue of A2 with corresponding eigenvector vi ∈ Cm . Then (A1 + A2 )vi = A1 vi + A2 vi = avi + λi (A2 )vi = (a + λi (A2 ))vi . 110
Proposition 3. Consider matrices A1 , A2 ∈ Rm×m and Ξ ∈ Rn×n and let A¯1 = In ⊗ A1 and A¯2 = Ξ ⊗ A2 with A¯1 , A¯2 ∈ Rnm×nm . Then S(A¯1 + A¯2 ) = ∪ni=1 S(A1 + λi (Ξ)A2 ) where λi (Ξ) represents the i-th eigenvalue of Ξ. Proof. Let v ∈ Cn be an eigenvector of Ξ associated with eigenvalue λ(Ξ) and u ∈ Cm be an eigenvector of M = A1 +λ(Ξ)A2 associated with eigenvalue λ(M ). Define the vector v ⊗ u ∈ Cnm and consider (A¯1 + A¯2 )(v ⊗ u) = v ⊗ A1 u + Ξu ⊗ A2 u = v ⊗ A1 u + λ(Ξ)v ⊗ u = v ⊗ (A1 u + λ(Ξ)A2 u). Since (A1 + λ(Ξ)A2 )u = λ(Ξ)u, we get (A¯1 + A¯2 )(v ⊗ u) = λ(Ξ)(v ⊗ u). A graph G is defined as the ordered pair G = (V, E), where V is the set of nodes (or vertices) V = {1, · · · , N } and E ⊆ V × V the set of edges (i, j) with i ∈ V, j ∈ V. The degree dj of a graph vertex j is the number of edges which start from j. Let dmax (G) denote the maximum vertex degree of the graph G. We denote by A(G) the (0, 1) adjacency matrix of the graph G. In particular, the (ij)th element of A, Aij = 1 if (i, j) ∈ E ∀ i, j = 1, · · · , N , i = j and zero otherwise. Let j ∈ Ni if (i, j) ∈ E and i = j. We call Ni the neighborhood of node i. The adjacency matrix A(G) of undirected graphs is symmetric. We define the Laplacian matrix as L(G) = D(G) − A(G), where D(G) is the diagonal matrix of vertex degrees di (also called the valence matrix). Let S(L(G)) = {λ1 (L(G)), · · · , λN (L(G))} be the spectrum of the Laplacian matrix L associated with an undirected graph G arranged in nondecreasing semi-order. The following Proposition is derived straight forward from Proposition 3 Proposition 4. Let A, B be matrices of appropriate dimensions and L be Laplacian matrix of graph G with spectrum S(L) = {λ1 (L), · · · , λN (L)}. Then, S(IN ⊗ A + L ⊗ B) = ∪i∈[1:N ] S(A + λi (L)B), with λi (L) ∈ S(L).
The following result is standard (Mohar (1991)) and is stated without proof. Proposition 5. Let G be a complete graph (with all possible edges) with NL vertices and L(G) be the corresponding Laplacian matrix. Then the following hold: L(G)p = NLp−1 L(G) and S(L(G)) = {0, NL , · · · , NL }. 3. LARGE-SCALE LQR FOR DYNAMICALLY COUPLED SYSTEMS Consider a network of NL dynamically coupled LTI systems referred to as agents. At local level the dynamics of the i-th agent is represented in state-space form as: NL x˙ i = A1 xi + A2 (xi − xj ) + Bui , x0,i = xi (0) (1) j=1,j=i
where xi ∈ Rn , ui ∈ Rm denote states and inputs of the i-th system, respectively. A complete graph (with all possible edges) G = (V, E) with Laplacian matrix Lc is utilized to model the topology of the physical links among the agents. Node i ∈ V of G corresponds to local state xi while edge (i, j) ∈ E corresponds to the xi − xj term
2019 IFAC NecSys Eleftherios E. Vlahakis et al. / IFAC PapersOnLine 52-20 (2019) 109–114 Chicago, IL, USA, September 16-17, 2019
in (1). Now construct the aggregate state x ˜ ∈ RnNL and mNL input vector u ˜ ∈ R by stacking all state and input vectors, respectively, of all NL systems taken in ascending order depending on their label in graph G. The aggregate state-space of the network becomes: ˜x + B ˜u x ˜˙ = A˜ ˜, x ˜0 = x ˜(0)
(2)
˜ = IN ⊗ B A˜ = INL ⊗ A1 + Lc ⊗ A2 , B L
(3)
(A1 + (NL − 1)A2 ) P˜ii − A 2 + P˜ii (A1 + (NL − 1)A2 ) −
with
Consider now LQR control problem for the network of NL coupled systems: ˜x + B ˜u min J(˜ u, x ˜0 ) s.t. x ˜˙ = A˜ ˜, x ˜0 = x ˜(0) u ˜
where the cost function J(˜ u, x ˜0 ) = with
∞
(4)
111
−
NL
k=1
NL
P˜ij
j=1 j=i
NL
P˜ij A2
j=1 j=i
P˜ik X P˜ik + Q1 + (NL − 1)Q2 = 0
(9)
for i = 1, · · · , N . Note that P˜ij = P˜ji due to symmetry of (7). Now let NL Fii = P˜ii + (10) P˜ij . i=1 j=i
˜x + u ˜ u dt x ˜ Q˜ ˜ R˜
(5)
Substituting (10) to (9) gives:
0
˜ = IN ⊗ R. ˜ = IN ⊗ Q1 + Lc ⊗ Q2 and R Q L L
(6)
≥ 0 and Q2 = Q 2 ≥ 0 state difference xi − xj
(NL − 1)(A 2 Fii + Fii A2 ) − NL (A 2
Under Assumption 6,7, problem (4) has a unique stabi˜x lizing solution u ˜ = K ˜, which gives finite performance ˜ ˜0 . The optimal state-feedback gain index (5) equal to x ˜0 P x ˜ P˜ , where P˜ is the symmetric positive defi˜ = −R ˜ −1 B K nite (s.p.d.) solution to the (large-scale) Algebraic Riccati Equation (ARE): ˜R ˜ −1 B ˜ P˜ + Q ˜ = 0. A˜ P˜ + P˜ A˜ − P˜ B
(7)
˜ and P˜ retain certain Due to special formulation of (4), K structure, which will prove essential for designing stabilizing distributed controllers in the next section. The specific structure of these matrices is proved in Theorem 8. In the following, we set X = BR−1 B . Theorem 8. Assume P˜ is the s.p.d solution to (7) associated with the optimal solution to (4). Let P˜ ∈ RnNL ×nNL be partitioned into NL2 blocks of dimension n × n, each denoted by P˜ij and referred to as the (i, j)-block of P˜ . Then, the following hold: NL ˜ Pij = P where I. For i = 1, · · · , NL , j=1
+ A 1 (Fii −
NL
−
NL
j=1 j=i
P˜ij ) + (Fii −
k=1
NL
P˜ij )A1
j=1 j=i
(NL − 1)(A 2 P˜ij + P˜ij A2 ) − A 2 (Fii − − (Fii −
NL
k=1 k=i
P˜ik )A2 − A 2
+ A 1 P˜ij + P˜ij A1 −
NL l=1 l=i l=j
NL
k=1
P˜il −
NL
P˜ik )
k=1 k=i
NL
P˜il A2
(12a)
l=1 l=i l=j
P˜ik X P˜kj − Q2 = 0
(12b)
Summing up (12a) for all j = i block-wise and adding this summation to (11a) gives (NL − 1)(A 2 Fii + Fii A2 ) − (NL − 1)A 2 Fii − Fii (NL − 1)A2 − NL A 2
(8)
+ (NL − 1)(A 2
II. P˜ij = P˜kl = P˜2 for all i, j, k, l = 1, · · · , NL with j = i and l = k, and P˜2 symmetric.
NL
− (NL − 1)(A 2
111
P˜ij A2 )
j=1 j=i
Using (10) the equations corresponding to off-diagonal blocks P˜ij , i = j of P˜ are
NL
Proof. First, we prove part I of the theorem. The equations corresponding to diagonal blocks P˜ii of P˜ in (7) are:
NL
P˜ik X P˜ik + Q1 + (NL − 1)Q2 = 0. (11b)
+ (NL − 1)(A 2
A 1 P + P A1 − P XP + Q1 = 0.
j=1 j=i
P˜ij −
(11a)
Q 1
Here Q1 = penalize local state and relative between the nodes i, j ∈ V, respectively. The matrix R = R > 0 weighs inputs of each subsystem. The following stabilizability and observability assumptions guarantee a unique stabilizing solution to LQR problem (4). Assumption 6. Let C1 C1 = Q1 . The pair (A1 , B) is stabilizable and (A1 , C1 ) is observable. Assumption 7. Let C12 C12 = Q1 + NL Q2 . The pair (A1 + NL A2 , B) is stabilizable and (A1 + NL A2 , C12 ) is observable.
NL
P˜ij +
j=1 j=i
l=1 l=i l=j
j=1 j=i
NL
P˜ij −
NL
P˜ij NL A2
j=1 j=i
P˜ij A2 )
j=1 j=i
P˜ik +
k=1 k=i NL
NL
NL
P˜ik A2 )
k=1 k=i
P˜il −
NL l=1 l=i l=j
P˜il A2 ) = 0
(13)
2019 IFAC NecSys 112 Eleftherios E. Vlahakis et al. / IFAC PapersOnLine 52-20 (2019) 109–114 Chicago, IL, USA, September 16-17, 2019
where all terms associated with A2 cancel out. By summing up now (12) for all j = i block-wise and adding this summation to (11) gives A 1 Fii + Fii A1 − Fii XFii +
NL
k=1 k =i
P˜ik X Fii − Fkk + Q1 = 0. (14)
Eq. (14) has been established in Theorem 1 of Borrelli and Keviczky (2008). It is true also here due to (13). Summing up (14) for all i = 1, · · · , NL we get NL A 1 Fii + Fii A1 − Fii XFii + Q1 = 0 (15) i=1
which is sum of NL identical ARE’s, i.e.,
(16) A 1 Fii + Fii A1 − Fii XFii + Q1 = 0. NL ˜ Equation (10) implies Fii = i=1 Pij which, along with (16), proves part I. The ARE (7) has a repetitive structure and essentially can be decomposed into NL identical ˜ R ˜ are block diagonal and A, ˜ Q ˜ have equations since B, repetitive pattern. This implies that matrices P˜ij with j = i are all equal and symmetric. This proves part II of the theorem. The following corollary of Theorem 8 is stated without proof due to lack of space. Corollary 9. Let P˜ij ∈ Rn×n , i, j = 1, · · · , NL , denote the (i, j)-block of P˜ in (7) associated with the optimal solution to (4). Then, the following hold: I. P˜ii = P − (NL − 1)P˜2 , for all i = 1, · · · , NL where P is the s.p.d solution to ARE (8). II. P˜ij = P˜2 , for i, j = 1, · · · , NL and i = j, where P˜2 is symmetric matrix associated with node-level ARE: (A1 + NL A2 ) (P − NL P˜2 ) + (P − NL P˜2 )(A1 +NL A2 ) − (P − NL P˜2 )X(P − NL P˜2 ) +Q1 + NL Q2 = 0. (17)
˜ and B ˜ are selected block By assumption, the matrices R ˜ = diagonal. Consequently, the state-feedback gain K ˜ −1 B ˜ P˜ associated with the optimal solution to (4) −R retains the same structure with P˜ . This leads to the following Corollary. ˜ P˜ is the optimal ˜ = −R ˜ −1 B Corollary 10. Assume K state-feedback gain obtained from the solution to (4) ˜0 with P˜ which gives minimum performance index x ˜ 0 P˜ x mNL ×nNL ˜ and being the s.p.d solution to (7). Let K ∈ R P˜ ∈ RnNL ×nNL be partitioned into NL2 blocks of dimension ˜ ij and P˜ij , respectively m × n and n × n, denoted by K each referred to as (i, j)-block of the respective matrix. Then, the following are true; I. P˜ = INL ⊗ P − Lc ⊗ P˜2 . N L ˜ −1 B P for i = 1, · · · , NL . II. j=1 Kij = −R −1 ˜ ii = −R B P + (NL − 1)R−1 B P˜2 for i = III. K 1, · · · , NL . ˜ ij = −R−1 B P˜2 for i, j = 1, · · · , NL and j = i. IV. K ˜ = −IN ⊗ R−1 B P + Lc ⊗ R−1 B P˜2 . V. K L 112
Theorem 8 along with the results stated in Corollary 9 suggest that due to special formulation of the cost function (5) and the structure of the aggregate state-space form (2), the large-scale LQR problem (4) under Assumption 6,7 can effectively be reduced to finding the solution of two node-level ARE’s. This feature may be highly beneficial for problems involving networks, the topology of which is modeled by graph with an excessively large number of vertices (NL ). Applying the stabilizing optimal state-feedback control ˜x u ˜ = K ˜ to (2) results in a closed-loop matrix, which is Hurwitz and is written as: (18) Acl = INL ⊗ (A1 − XP ) + Lc ⊗ (A2 + X P˜2 ). Due to Proposition 3 the spectrum of Acl can be decomposed into: S(Acl ) =
N L
i=0
S A1 + BK + λc,i (A2 − BK2 )
(19)
where λc,i ∈ S(Lc ). Remark 11. From Proposition 5, the matrix A1 + BK + αNL (A2 − BK2 ) is Hurwitz for α = 0 and α = 1. In the sequel we require that: Condition 12. The matrix A1 + BK + αNL (A2 − BK2 ) is Hurwitz for all α ∈ [0, 1]. Condition 12 states that all convex combinations of two Hurwitz matrices (20) µA¯1 + (1 − µ)A¯2 with µ ∈ [0, 1] ¯ are Hurwitz, where A1 = A1 + BK + NL (A2 − BK2 ) and A¯2 = A1 + BK. Sufficient conditions for Hurwitz stability of convex combination of Hurwitz matrices can be found in Theorem 2.2 in Bia£las (2004). In essence, Condition 12 characterizes a class of LQR problems (4) which admit of solutions for which the Condition 12 holds. This will be used later for the design of distributed stabilizing controllers. For a given selection of weighting matrices (Q1 , Q2 , R) of the LQR problem (4), the validity of Condition 12 can be verified by searching for a symmetric positive definite matrix P¯ for which the following Linear Matrix Inequality (LMI), 0n×n 0n×n −(A¯ 1 P¯ + P¯ A¯1 ) 0n×n −(A¯ 2 P¯ + P¯ A¯2 ) 0n×n > 0, (21) 0n×n 0n×n P¯ is feasible. Obviously, if matrix P¯ exists then √ premultiply√ ing and postmultiplying (21) by [ µIn 1 − µIn 0n×n ] √ √ and [ µIn 1 − µIn 0n×n ], respectively, for µ ∈ [0, 1] leads to Lyapunov inequality: (µA¯1 + (1 − µ)A¯2 ) P¯ + P¯ (µA¯1 + (1 − µ)A¯2 ) < 0, (22) which admits of a solution P¯ = P¯ > 0. This demonstrates that µA¯1 + (1 − µ)A¯2 is a Hurwitz matrix for all µ ∈ [0, 1]. Alternatively, the stability of µA¯1 + (1 − µ)A¯2 can be examined via a simple graphical test by plotting the eigenvalue with the maximum real part of the matrix µA¯1 + (1 − µ)A¯2 for µ ∈ [0, 1]. The following arguments are given without proof due to lack of space. Consider now the aggregate input matrix ˜ = IN ⊗ B1 + Lc ⊗ B2 (23) B L
2019 IFAC NecSys Eleftherios E. Vlahakis et al. / IFAC PapersOnLine 52-20 (2019) 109–114 Chicago, IL, USA, September 16-17, 2019
where B1 ∈ Rn×m , B2 ∈ Rn×m represent input channels corresponding to local inputs and relative input difference ui − uj between the node i, j ∈ V with i = j, respectively. In this case the state-space form at node level is written as: NL x˙ i = A1 xi + A2 (xi − xj ) + B1 ui j=1,j =i
+ B2
NL
(ui − uj ), x0,i = xi (0). (24)
j=1,j =i
˜ as Then solving LQR problem (4) with A˜ as given in (3), B ˜ ˜ given in (23) and weighting matrices (Q, R) as selected in (6), results in optimal solution P˜ in (7). This has structure as defined in Theorem 8 and Corollary 9, i.e., P˜ = INL ⊗ P − Lc ⊗ P˜2 where P is the solution to ARE (8) and P˜2 is symmetric matrix associated with ARE (17). The ˜ can then be derived optimal state-feedback controller K −1 ˜ ˜ ˜ ˜ by K = −R B P = INL ⊗ K − Lc ⊗ K2 with (25) K = −R−1 B1 P, −1 ˜ ˜ (26) K2 = −R (B1 P2 + B2 P − NL B2 P2 )
where the property L2c = NL Lc (see Proposition 5) was used. Proof of closed-loop stability of this control scheme along with derivation of all equations will be included in an extended work of this paper. In the sequel, we consider multi-agent systems with dynamical couplings in their states only. 4. DISTRIBUTED CONTROL DESIGN FOR DYNAMICALLY COUPLED SYSTEMS
Let a network be formed of N identical and dynamically coupled LTI systems. The couplings among the systems are modelled by undirected graph G = (V, E) with Laplacian matrix L. The neighborhood of the i-th system is denoted by Ni ⊂ V. Let the dynamics at local level of the i-th system be (xi − xj ) + Bui , x0,i = xi (0) (27) x˙ i = A1 xi + A2 j∈Ni
where xi ∈ Rn and ui ∈ Rm . The aggregate state-space of the network becomes ˆx + B ˆu x ˆ˙ = Aˆ ˆ, x ˆ0 = x ˆ(0) (28) nN mN where x ˆ∈R ,u ˆ∈R and ˆ = IN ⊗ B. Aˆ = IN ⊗ A1 + L ⊗ A2 , B (29)
Note that the Laplacian matrix L in (29) does not necessarily correspond to a complete graph in contrast to (3) and generically the matrix Aˆ in (29) is sparse. We note here that the aggregate state-space forms (28) and (2) differ in number of subsystems and the structure of the Laplacian matrix L. We denote aggregate state-space as in (2) when referring to centralized control problems with NL subsystems and we use aggregate state-space (28) when referring to distributed control problems with N subsystems. Similarly, tilded matrices are referred to centralized problems while hatted matrices to distributed problems. A stabilizing distributed controller for (28) is constructed in the following Theorem. 113
113
Theorem 13. Consider a network of N coupled systems with dynamics described in (27) modelled by graph GN with Laplacian matrix LN . Let λN be the maximum eigenvalue of LN and denote by dmax the smallest integer which is greater than or equal to λN . Consider LQR problem (4) for NL = dmax , define P and P˜2 via (8) and (17), respectively, and assume Condition 12 is valid. Define also: K = −R−1 B P , K2 = −R−1 B P˜2 and ˆ = IN ⊗ K − LN ⊗ K2 . (30) K Then, the closed-loop matrix Acl = IN ⊗ (A1 + BK) + LN ⊗ (A2 − BK2 ) is Hurwitz.
(31)
Proof. Consider the spectrum S(Acl ) = S(IN ⊗ (A1 + BK) + LN ⊗ (A2 − BK2 )). Let VN ⊗ In be state-space transformation where VN ∈ RN ×N is an orthogonal matrix whose columns consist of the eigenvectors of LN . In the transformed coordinates, A¯cl = IN ⊗ (A1 + BK) + ΛN ⊗ (A2 − BK2 ) where ΛN = diag(0, λ2 , · · · , λN ) with λN ≤ dmax . The spectrum of A¯cl is S(A¯cl ) =
N
i=1
S(A1 + BK + λi (A2 − BK2 ))
(32)
where λi for i = 1, · · · , N are the eigenvalues of LN . Condition 12 holds, hence (A1 +BK)+αdmax (A2 −BK2 ) is Hurwitz for all α ∈ [0, 1]. Consequently A¯cl is also Hurwitz since λi ∈ [0, dmax ] for all i = 1, · · · , N . This proves the Theorem. Remark 14. For a time-varying graph G(t) = (V, E(t)) with fixed number of vertices (N ) and time-varying edges the maximum eigenvalue of the time-varying Laplacian matrix L(t) is bounded by 2N . Consequently, solving (4) for NL = 2N and assuming Condition 12 holds leads to a ˆ which stabilizes the network for distributed controller K all possible couplings among the N systems. Naturally, this does not imply stability of switching between stable network topologies. 4.1 Measure of suboptimality At this point, we wish to employ a suboptimality measure which can be cast as a performance loss index of the distributed scheme proposed above. It is necessary so, to define a reference performance index with which the suboptimal scheme is compared. Such an index has been introduced in Borrelli and Keviczky (2008) and is also outlined here. Let matrix Ξ ∈ RmN ×nN be partitioned into N 2 blocks of dimension m × n, each referred to as (i, j)-block of Ξ and denoted by Ξij ∈ Rm×n , with i, j = 1, · · · , N . In particular, the (i, j)-block can be written as: Ξij = Ξ[(i − 1)m + 1 : im, (j − 1)n + 1 : jn]. The class of structured N (G) is now defined as follows: matrices Kn,m N (G) = {Ξ ∈ RmN ×nN | Ξij = 0m,n if (i, j) ∈ / E, Km,n Ξij = Ξ[(i − 1)m + 1 : im, (j − 1)n + 1 : jn], i, j = 1, · · · , N }.
Consider now the following optimal control problem:
2019 IFAC NecSys 114 Eleftherios E. Vlahakis et al. / IFAC PapersOnLine 52-20 (2019) 109–114 Chicago, IL, USA, September 16-17, 2019
min J(ˆ u, x ˆ0 ) = u ˆ
∞
ˆx + u ˆ u) dt (ˆ x Qˆ ˆ Rˆ
0
ˆx + B ˆu subject to: x ˆ˙ = Aˆ ˆ, x ˆ0 = x ˆ(0) ˆx u ˆ=K ˆ
ˆ
ˆ ∈ K ˆ∈ Q
(33a)
N Km,n (G) N Km,n (G), ˆ ˆ
(33b) (33c) (33d)
ˆ = IN ⊗ R R
(33e)
ˆ = Q ≥ 0 and R = R > 0. We note that in the where Q absence of constraint (33d), the optimal control problem (33) if feasible, yields a centralized optimal control u∗ = ˆ −1 B ˆ P ∗ and P ∗ is the symmetric K ∗x ˆ where K ∗ = −R positive definite solution to: ˆ = 0. ˆR ˆ −1 B ˆ P ∗ + Q (34) Aˆ P ∗ + P ∗ Aˆ − P ∗ B Note that, since constraint (33d) is not included in the optimization, K ∗ and P ∗ are centralized solutions, and thus, the value x ˆ 0 P ∗ x ˆ0 is the minimum achievable performance index for a given x ˆ0 . Assume now that constraint (33d) is in force. Then, the ˆ as constructed distributed state-feedback controller K in (30) can be considered as a suboptimal solution to ˆ B) ˆ as given in (29) and Q ˆ = I N ⊗ Q1 + (33) with (A, LN ⊗ Q2 , with LN as defined in Theorem 13. Then, a performance index for this suboptimal distributed scheme can be computed as ˆx ˆ 0 Pˆ x (35) ˆ0 , J(K ˆ, x ˆ0 ) = x ˆ where P is the positive definite solution to the following Lyapunov equation: ˆ K) ˆ Pˆ + Pˆ (Aˆ + B ˆ K) ˆ + (Q ˆ+K ˆ R ˆ K) ˆ = 0. (36) (Aˆ + B ∗ ∗ ˆ Since P is optimal, J(K x ˆ, x ˆ0 ) ≤ J(K x ˆ, x ˆ0 ) for all x ˆ0 and thus ∆P = Pˆ − P ∗ is a positive semidefinite matrix ˆ = K ∗ . Any norm of ∆P can which is equal to zero if K be cast as a measure of suboptimality of the distributed ˆ controller K. 5. CONCLUSIONS Stabilizing distributed state-feedback controller for networks of coupled identical systems was proposed based on a large-scale LQR optimal problem. This method has originally been proposed in Borrelli and Keviczky (2008) for the decoupled case and was extended here to include couplings between the subsystems representing a multiagent network. An effective condition for stability is also provided which admits of network stable operation for a class of interconnections. The control scheme was obtained by optimizing an LQR performance index with
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a tuning parameter utilized to emphasize/de-emphasize relative state difference between interconnected agents. The assumption of identical dynamics is clearly restrictive but simplifies the design problem considerably and leads to the derivation of a stability condition which can be easily tested. Attempts to eliminate or relax this assumption will be the topic of future work. Preliminary results in this direction can be found in Vlahakis and Halikias (2018a,b). The control scheme has also been successfully applied to a multi-area power grid numerous simulations for which can be found in Vlahakis et al. (2019). REFERENCES Bia£las, S. (2004). A sufficient condition for Hurwitz stability of the convex combination of two matrices. Control Cybern., 33(1). Borrelli, F. and Keviczky, T. (2008). Distributed LQR design for identical dynamically decoupled systems. IEEE Trans. Autom. Contr., 53(8), 1901–1912. Deshpande, P., Menon, P.P., Edwards, C., and Postlethwaite, I. (2012). Sub-optimal distributed control law with H2 performance for identical dynamically coupled linear systems. IET Control Theory Appl., 6(16), 2509– 2517. Fax, J.A. and Murray, R.M. (2002). Information Flow and Cooperative Control of Vehicle Formations. IFAC Proc. Vol., 35(1), 115–120. Massioni, P. and Verhaegen, M. (2009). Distributed control for identical dynamically coupled systems: A decomposition approach. IEEE Trans. Automat. Contr., 54(1), 124–135. Mohar, B. (1991). The Laplacian spectrum of graphs. Graph theory Comb. Appl. Olfati-Saber, R. (2006). Flocking for Multi-Agent Dynamic Systems:Algorithms and Theory. IEEE Trans. Autom. Contr., 51(3), 1–20. Vlahakis, E., Dritsas, L., and Halikias, G. (2019). Distributed LQR Design for a Class of Large-Scale MultiArea Power Systems. Energies, 12(14), 2664. Vlahakis, E.E. and Halikias, G.D. (2018a). Distributed LQR Methods for Networks of Non-Identical Plants. In 2018 IEEE Conf. Decis. Control, 6145–6150. IEEE. Vlahakis, E.E. and Halikias, G.D. (2018b). ModelMatching type-methods and Stability of Networks consisting of non-Identical Dynamic Agents. IFACPapersOnLine. Wang, Q., Yu, C., Gao, H., and Liu, F. (2018). A distributed control law with guaranteed convergence rate for identically coupled linear systems.
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IFAC PapersOnLine 52-20 (2019) 109–114
Distributed LQR-based Suboptimal Distributed LQR-based Suboptimal Distributed LQR-based Suboptimal Distributed LQR-based Suboptimal Control for Coupled Linear Systems Control for Coupled Linear Systems Control for Coupled Linear Systems Control for Coupled Linear Systems ∗ ∗∗
Eleftherios E. Vlahakis ∗ Leonidas Leonidas D. D. Dritsas Dritsas ∗∗ Eleftherios E. Vlahakis ∗ ∗ D. Dritsas ∗∗ Vlahakis Eleftherios E.George Leonidas ∗ D. ∗ Halikias ∗∗ George D. Halikias Eleftherios E.George Vlahakis Leonidas ∗ D. Dritsas D. Halikias ∗ George D. Halikias ∗ ∗ School of Mathematics, Computer Science and Engineering, City, of Mathematics, Computer Science and Engineering, City, ∗ School School of Mathematics, Science and Engineering, City, of (e-mail: ∗University University of London, London, UK UKComputer (e-mail: [email protected], [email protected], School of Mathematics, Computer Science and Engineering, City, University of London, [email protected]) UK (e-mail: [email protected], [email protected]) University of London, UK (e-mail: [email protected], ∗∗ [email protected]) ∗∗ Department of Electrical & & Electronic Electronic Engineering Engineering Educators, Educators, [email protected]) ∗∗ Department of Electrical Department of Electrical & Electronic Engineering Educators, School of Pedagogical & Technological Education, ASPETE, Athens, ∗∗ School of Pedagogical & Technological Education, ASPETE, Athens, Department of Electrical & Electronic Engineering Educators, School of Pedagogical & Technological Education, ASPETE, Athens, Greece (e-mail: [email protected]) Greece (e-mail: [email protected]) School of Pedagogical & Technological Education, ASPETE, Athens, Greece (e-mail: [email protected]) Greece (e-mail: [email protected]) Abstract: Abstract: A A well-established well-established distributed distributed LQR LQR method method for for decoupled decoupled systems systems is is extended extended to to the the Abstract: Acoupled well-established distributed LQRsystems methodare for dynamically decoupled systems is extended the dynamically case where the open-loop dependent. First, aatofully dynamically coupled case where the open-loop systems are dynamically dependent. First, fully Abstract: A well-established distributed LQR method for decoupled systems is extended to the dynamically coupled case where the open-loop systems are dynamically dependent. First, a fully centralized controller is designed which is subsequently substituted by a distributed statecentralized controller is designed which is subsequently substituted dependent. by a distributed statedynamically coupled case where the open-loop systems are dynamically First, a fully centralizedgain controller is designed which subsequently distributed feedback with structure. The control scheme is by aoptimizing optimizing an stateLQR by an LQR feedback gain with sparse sparse structure. Theis control scheme substituted is obtained obtained by centralized controller is adesigned which subsequently substituted distributed feedback gain with with sparse structure. Theiscontrol scheme is obtained by by aoptimizing an stateLQR performance index with tuning parameter utilized to emphasize/de-emphasize emphasize/de-emphasize relative state performance index a tuning parameter utilized to relative state feedback gain with with sparse structure. The control scheme is obtained by optimizing an LQR performance index a tuning parameter utilized to emphasize/de-emphasize state difference between interconnected systems. Overall stability is guaranteed via aarelative simple test difference between interconnected systems. Overall stability is guaranteed via simple test performance index with a tuning parameter utilized to emphasize/de-emphasize relative state difference between interconnected systems. Overall stability is guaranteed via a simple test applied to a convex combination of Hurwitz matrices, the validity of which leads to stable applied to a convex combination of Hurwitz matrices, the validity of which leads to stable difference between interconnected systems. Overall stability is shown guaranteed vialeads a simple test applied to a convex combination of Hurwitz matrices, the validity of which to stable global operation for a class of interconnection schemes. It is also that the suboptimality global operation for acombination class of interconnection schemes. Itthe is also shown that theleads suboptimality applied to a convex of Hurwitz matrices, validity of which to stable global operation for a class of interconnection schemes. It is also shown that the suboptimality of the the method method can can be be assessed assessed by by measuring measuring aa certain certain distance distance between between two two positive positive definite definite of global forbe a class of interconnection is also shown that suboptimality of the operation method can assessed by measuring aschemes. certain It distance between twothe positive definite matrices which can obtained by solving two Lyapunov equations. matrices which can be obtained by solving two Lyapunov equations. of the method certain distance between two positive definite matrices which can can be be assessed obtained by by measuring solving twoa Lyapunov equations. matrices which be obtained by by solving two Lyapunov Copyright © 2019. can The Authors. Published Elsevier Ltd. All rights equations. reserved. Keywords: Keywords: distributed distributed LQR, LQR, coupled coupled linear linear systems, systems, multi-agent multi-agent control, control, networked networked control. control. Keywords: distributed LQR, coupled linear systems, multi-agent control, networked control. Keywords: distributed LQR, coupled linear systems, multi-agent control, networked control. 1. INTRODUCTION INTRODUCTION stabilizing 1. stabilizing distributed distributed LQR-based LQR-based controller controller for for networks networks 1. INTRODUCTION stabilizing distributed LQR-based controller for networks formed of of identical identical agents with dynamical dynamical couplings. formed agents with couplings. 1. INTRODUCTION stabilizing distributed LQR-based controller for networks of identical agents with dynamical couplings. Networked systems systems have have attracted attracted attention attention from from the the formed Networked Over the past few years, there has been a renewal of formed of identical agents with dynamical couplings. Networked systems in have attracted attention frombroad the Over the past few years, there has been a renewal of control community community in recent years due due to their their broad control recent years to Over the few years, there has been renewal of interest in past control of networks networks composed of aa alarge large number Networked systems have attracted attention from the interest in control of composed of number control recent yearsare to referred their broad the past few years, there has been alarge renewal of range of of community applications.inSuch Such schemes aredue often referred to as as Over range applications. schemes often to interest in control of networks composed of a number of interacting systems. Rigorous methods for cooperative controlof community inSuch recent years due to referred their broad of interacting systems. Rigorous methods for cooperative range applications. schemes are often to as interest in control of networks composed of a large number multi-agent networks, with each agent having autonomous multi-agent networks,Such with schemes each agent autonomous interacting Rigorous methods cooperative control design for systems with distributed or range of applications. arehaving often referred to as of control design systems. for multi-agent multi-agent systems withfor distributed or multi-agent networks, agent having autonomous of interacting systems. Rigorous methods for cooperative actuation capacity. Thewith needeach for forming forming networks in many many control design for multi-agent systems with distributed or actuation capacity. The need for networks in decentralized pattern have been provided in Olfati-Saber multi-agent networks, with each agent having autonomous decentralized pattern have been provided indistributed Olfati-Saber actuation capacity. The need for forming networks in many control design for multi-agent systems with or cases arises from the fact that some problems may not cases arises from the fact that some problems may not decentralized pattern have been provided in Olfati-Saber (2006); Fax and Murray (2002); Wang et al. (2018). actuation capacity. Thethe need for forming networks in many (2006); Fax and Murray (2002); Wang etinal. (2018). A A cases arises from the fact that some problems may not decentralized pattern have been provided Olfati-Saber admit a solution at individual system level. Thus admit a solution at the individual system level. Thus (2006); Fax and Murray (2002); Wang et al. controllers (2018). A thorough procedure for designing distributed cases arises from the fact that some problems may not thorough procedure for designing distributed controllers admit solution the individual systemhave level. Faxofand Murray (2002); Wang et al. controllers (2018). A agents adespite despite theiratindependent independent operation have alsoThus the (2006); agents their operation also the thorough designing distributed for systems based on admit adespite the individual level. for aa class classprocedure of coupled coupledfor systems based on aa decomposition decomposition agents theiratindependent alsoThus the thorough procedure for designing distributed controllers ability tosolution cooperate with certainoperation of system their have counterparts for a class of coupled systems based on a decomposition ability to cooperate with certain of their counterparts has been in Massioni and Verhaegen agents the despite theirtowards independent operation have alsoother the approach approach has been presented presented inbased Massioni Verhaegen ability to cooperate with certain of their counterparts a class offundamental coupled systems on a and decomposition within network common objective. In within the network towards aa common objective. In other for approach has been presented in of Massioni and (2009). The works Borrelli and Keviczky ability to cooperate with certain of their counterparts (2009). The fundamental works of Borrelli andVerhaegen Keviczky within the network towards a common objective. In other approach has been presented in Massioni and Verhaegen cases, the topology of the network may be imposed by cases, the topology of the network may be imposed by (2009). The fundamental works of Borrelli and Keviczky (2008); Deshpande et al. (2012) discuss distributed LQR within the network towards a common objective. In other (2008); Deshpande et al. (2012) discuss distributed LQR cases, the links topology may bewhere imposed by (2009). The fundamental works of Borrelli and Keviczky structural links suchofas asthe in network power systems systems where agents structural such in power agents (2008); Deshpande et al. (2012) discuss distributed LQR design for a set of identical decoupled dynamical systems. cases, the topology of the network may be imposed by design for a set of identical decoupled dynamical systems. structural links such as in power systems where agents (2008); Deshpande et al. (2012) discuss distributed LQR denote power power generators generators and and interconnections interconnections represent represent design for a set of identical decoupled dynamical systems. denote Unfortunately, there is no documented distributed LQRstructural links such as in power systems where agents Unfortunately, there is no documented distributed LQRdenote power generators and interconnections represent design for a set of identical decoupled dynamical systems. power transmission lines. power lines. and interconnections represent Unfortunately, is no documented LQRbased approach to systems with denote transmission power generators based approachthere to networked networked systemsdistributed with dynamical dynamical power transmission lines. Unfortunately, there is no documented distributed LQRbased approach to networked systems with dynamical couplings in the context of relevant literature. In this transmission paper, we we focus focus on multi-agent multi-agent networks networks composed composed couplings power lines. in the context of relevant literature. In this paper, on based approach to networked systems with dynamical in the context of relevant literature. In this paper, we focus oncoupled multi-agent composed of identical identical dynamically coupled linearnetworks time-invariant sys- couplings of dynamically linear time-invariant sysIn we follow aa relevant top-down method couplings in the In identical thisWe paper, we focus on multi-agent networks composed In this this work, work, wecontext follow of top-downliterature. method to to approxapproxof dynamically coupled linear time-invariant systems. consider that these dynamical couplings can be tems. We consider that these dynamical couplings can be In this work, we follow a top-down method toby approximate a centralized LQR optimal controller a of identical dynamically coupled linear time-invariant sysimate a centralized LQR optimal controller by a disdistems. We consider that these dynamical couplings can be thisawork, wescheme. follow a top-down method tobynetwork approxexpressed in in aa state-space state-space form form of of aa certain certain structure. structure. In In In expressed imate centralized LQR optimal controller a distributed control It is shown that overall tems. We consider that these dynamical couplings can be tributed control scheme. It is shown controller that overallbynetwork expressed in a state-space form of a certain structure. In imate a centralized LQR optimal a dis-a our case each system representing an agent can produce acour case each representing canstructure. produce acscheme.via It isaa shown that overall network stability control is guaranteed guaranteed via stability test applied to expressed in asystem state-space form ofan aagent certain In tributed stability is stability test applied to a our case each system representing an agent can produce actributed control scheme. It is shown that overall network tuation signals independently and is dynamically coupled tuation signals independently andanisagent dynamically coupled stability is guaranteed via a stability testThe applied to of a convex combination of Hurwitz matrices. validity our case each system representing can produce acconvex combination of Hurwitz matrices. The validity of tuation signals independently is guaranteed via awith stability testThe applied to ofa and is dynamically coupled stability with certain certain number of its its peers peers referred to as as neighboring neighboring with number of referred to convex combination of Hurwitz matrices. validity this condition is consistent the stability of a class tuation signals independently and is dynamically coupled condition is consistent withmatrices. the stability of a class with certain number of its peers referred to as neighboring convex combination of Hurwitz The validity of agents with whom whom it can can exchange state information. information. Effec- this agents with it exchange state Effecthis condition is consistent with the stability of a class of interconnection structures which with certain number of the its peers referred to as neighboring of network network interconnection structures which is is identified. identified. agents with whom it can exchange state information. Effecthis condition is consistent with the stability of a class tively, we assume that topology of physical couplings tively, we assume that the topology of physical couplings of network interconnection structures which is identified. Sufficient condition condition for for stability stability of of convex convex combination combination agents with whom that it exchange state Effec- Sufficient tively, assume the of network interconnection structures which is identified. topology of information. physical couplings and thewe topology of can information exchange among agents and the topology of information exchange among agents Sufficient condition for stability ofin convex combination of Hurwitz matrices can be found Bia llas (2004). tively, we assume that the topology of physical couplings of Hurwitz matrices can be found in Bia as (2004). The The and the topology of information exchange among agents Sufficient condition for stability of convex combination coincide and are described by the same graph. Network coincide and are described by theexchange same graph. Network of Hurwitz matrices can beby found in Biaan las LQR (2004). The control scheme is obtained optimizing perforand the topology of information among agents control scheme is obtained by optimizing an LQR perforcoincide and are described by the same graph. Network of Hurwitz matrices can be found inwhich Biaan lascan (2004). The stabilization is is one one of of the the most most challenging challenging problems problems in in control scheme is obtained by optimizing LQR perforstabilization mance index with a tuning parameter be used to coincide and are described by the same graph. Network mance index with tuning parameter which be used to stabilization is one ofcontrol. the most challenging problems scheme is aaobtained by optimizing ancan LQR performulti-agent network network control. In this this work, we we propose in a control multi-agent In work, propose a mance index with tuning parameter which can be used to emphasize/de-emphasize relative state difference between stabilization is one of the most challenging problems in emphasize/de-emphasize relative state difference between multi-agent network control. In this work, we propose a mance index with a tuning parameter which can be used to emphasize/de-emphasize relative state difference between interconnected systems. Our approach was inspired by the multi-agent network control. In this work, we propose a interconnected systems. Our approach was inspired by the This emphasize/de-emphasize relative state difference between This work work was was supported supported by by a a City, City, University University of of London London scholarscholarinterconnected systems. Our approach was inspired by the powerful results proposed in Borrelli and Keviczky (2008). powerful results proposed in Borrelli and Keviczky (2008). ship held by Eleftherios Vlahakis. This work was supported by a City, University of London scholarinterconnected systems. Our approach was inspired by the ship held by was Eleftherios Vlahakis. powerful results proposed in Borrelli and Keviczky (2008). This work supported by a City, University of London scholarship held by Eleftherios Vlahakis. powerful results proposed in Borrelli and Keviczky (2008). ship held by Eleftherios Vlahakis.
2405-8963 Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved. Copyright © 2019 109 Copyright © under 2019 IFAC IFAC 109 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 109 10.1016/j.ifacol.2019.12.139 Copyright © 2019 IFAC 109
2019 IFAC NecSys 110 Eleftherios E. Vlahakis et al. / IFAC PapersOnLine 52-20 (2019) 109–114 Chicago, IL, USA, September 16-17, 2019
Therein, the subsystems constituting the network are dynamically decoupled, and the stability of the distributed scheme designed relies on the stability margins of LQR control. A complementary distributed LQR method has also been proposed in Deshpande et al. (2012), which consists of a bottom-up approach in which optimal interactions between self-stabilizing agents are defined so as to minimize an upper bound of the global LQR criterion. Our definition of multi-agent networks with dynamical couplings has been motivated by the structure of a multiarea power system. The proposed method has been successfully applied to multi-area power system control design numerous simulations of which can be found in Vlahakis et al. (2019). A major assumption of our work is that the dynamical models of each system are identical. Although this assumption may be unrealistic in practice, it simplifies the design problem considerably, which is especially hard due to the coupling terms appearing in the model. Future work will attempt to eliminate or relax this assumption. Preliminary results in this direction can be found in Vlahakis and Halikias (2018a,b). The remaining of the paper is organized in four sections. In Section 2 notation and some preliminaries on graph theory are presented. The main results of our work are presented in Section 3 and 4 where LQR properties of coupled linear systems and the distributed control algorithm are derived, respectively. Section 5 summarizes the main conclusions of the work where a discussion of the main results and suggestions for future work are also included. 2. NOTATION AND GRAPH THEORY PRELIMINARIES The field of real and complex numbers are denoted by R and C, respectively. Rn denotes the n-dimensional vector space over the field R. Rn×m denotes the set of n × m real matrices. ξ and Ξ denote the transpose of ξ and Ξ, respectively. Matrix Ξ ∈ Rn×n is called symmetric if Ξ = Ξ. The identity matrix of dimension m×m is denoted by Im ∈ Rm×m . The n×m zero matrix is denoted by 0n,m . C = {s ∈ C : Re(s) < 0}. C = {s ∈ C : Re(s) ≤ 0}.
Let A ∈ Rn×m and B ∈ Rq×p . Then, the Kronecker product of A and B is denoted by A ⊗ B and defined as a11 B · · · a1m B .. ∈ Rnq×mp , .. A ⊗ B = ... . . an1 B · · · ann B where aij is the (i, j)-th entry of A, with i = 1, · · · , n and j = 1, · · · , m.
Let λi (Ξ) denote the i-th eigenvalue of Ξ ∈ Rn×n , i = 1, · · · , n. Then, the spectrum of Ξ is denoted by S(Ξ) = {λ1 (Ξ), · · · , λn (Ξ)}. Definition 1. Matrix Ξ ∈ Rn×n is called Hurwitz (or stable) if all its eigenvalues have negative real part, i.e., λi (Ξ) ∈ C , i = 1, · · · , n. Proposition 2. Let A1 = aIm and A2 ∈ Rm×m . Then λi (A1 + A2 ) = a + λi (A2 ), i = 1, · · · , m. Proof. Let λi (A2 ) be any eigenvalue of A2 with corresponding eigenvector vi ∈ Cm . Then (A1 + A2 )vi = A1 vi + A2 vi = avi + λi (A2 )vi = (a + λi (A2 ))vi . 110
Proposition 3. Consider matrices A1 , A2 ∈ Rm×m and Ξ ∈ Rn×n and let A¯1 = In ⊗ A1 and A¯2 = Ξ ⊗ A2 with A¯1 , A¯2 ∈ Rnm×nm . Then S(A¯1 + A¯2 ) = ∪ni=1 S(A1 + λi (Ξ)A2 ) where λi (Ξ) represents the i-th eigenvalue of Ξ. Proof. Let v ∈ Cn be an eigenvector of Ξ associated with eigenvalue λ(Ξ) and u ∈ Cm be an eigenvector of M = A1 +λ(Ξ)A2 associated with eigenvalue λ(M ). Define the vector v ⊗ u ∈ Cnm and consider (A¯1 + A¯2 )(v ⊗ u) = v ⊗ A1 u + Ξu ⊗ A2 u = v ⊗ A1 u + λ(Ξ)v ⊗ u = v ⊗ (A1 u + λ(Ξ)A2 u). Since (A1 + λ(Ξ)A2 )u = λ(Ξ)u, we get (A¯1 + A¯2 )(v ⊗ u) = λ(Ξ)(v ⊗ u). A graph G is defined as the ordered pair G = (V, E), where V is the set of nodes (or vertices) V = {1, · · · , N } and E ⊆ V × V the set of edges (i, j) with i ∈ V, j ∈ V. The degree dj of a graph vertex j is the number of edges which start from j. Let dmax (G) denote the maximum vertex degree of the graph G. We denote by A(G) the (0, 1) adjacency matrix of the graph G. In particular, the (ij)th element of A, Aij = 1 if (i, j) ∈ E ∀ i, j = 1, · · · , N , i = j and zero otherwise. Let j ∈ Ni if (i, j) ∈ E and i = j. We call Ni the neighborhood of node i. The adjacency matrix A(G) of undirected graphs is symmetric. We define the Laplacian matrix as L(G) = D(G) − A(G), where D(G) is the diagonal matrix of vertex degrees di (also called the valence matrix). Let S(L(G)) = {λ1 (L(G)), · · · , λN (L(G))} be the spectrum of the Laplacian matrix L associated with an undirected graph G arranged in nondecreasing semi-order. The following Proposition is derived straight forward from Proposition 3 Proposition 4. Let A, B be matrices of appropriate dimensions and L be Laplacian matrix of graph G with spectrum S(L) = {λ1 (L), · · · , λN (L)}. Then, S(IN ⊗ A + L ⊗ B) = ∪i∈[1:N ] S(A + λi (L)B), with λi (L) ∈ S(L).
The following result is standard (Mohar (1991)) and is stated without proof. Proposition 5. Let G be a complete graph (with all possible edges) with NL vertices and L(G) be the corresponding Laplacian matrix. Then the following hold: L(G)p = NLp−1 L(G) and S(L(G)) = {0, NL , · · · , NL }. 3. LARGE-SCALE LQR FOR DYNAMICALLY COUPLED SYSTEMS Consider a network of NL dynamically coupled LTI systems referred to as agents. At local level the dynamics of the i-th agent is represented in state-space form as: NL x˙ i = A1 xi + A2 (xi − xj ) + Bui , x0,i = xi (0) (1) j=1,j=i
where xi ∈ Rn , ui ∈ Rm denote states and inputs of the i-th system, respectively. A complete graph (with all possible edges) G = (V, E) with Laplacian matrix Lc is utilized to model the topology of the physical links among the agents. Node i ∈ V of G corresponds to local state xi while edge (i, j) ∈ E corresponds to the xi − xj term
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in (1). Now construct the aggregate state x ˜ ∈ RnNL and mNL input vector u ˜ ∈ R by stacking all state and input vectors, respectively, of all NL systems taken in ascending order depending on their label in graph G. The aggregate state-space of the network becomes: ˜x + B ˜u x ˜˙ = A˜ ˜, x ˜0 = x ˜(0)
(2)
˜ = IN ⊗ B A˜ = INL ⊗ A1 + Lc ⊗ A2 , B L
(3)
(A1 + (NL − 1)A2 ) P˜ii − A 2 + P˜ii (A1 + (NL − 1)A2 ) −
with
Consider now LQR control problem for the network of NL coupled systems: ˜x + B ˜u min J(˜ u, x ˜0 ) s.t. x ˜˙ = A˜ ˜, x ˜0 = x ˜(0) u ˜
where the cost function J(˜ u, x ˜0 ) = with
∞
(4)
111
−
NL
k=1
NL
P˜ij
j=1 j=i
NL
P˜ij A2
j=1 j=i
P˜ik X P˜ik + Q1 + (NL − 1)Q2 = 0
(9)
for i = 1, · · · , N . Note that P˜ij = P˜ji due to symmetry of (7). Now let NL Fii = P˜ii + (10) P˜ij . i=1 j=i
˜x + u ˜ u dt x ˜ Q˜ ˜ R˜
(5)
Substituting (10) to (9) gives:
0
˜ = IN ⊗ R. ˜ = IN ⊗ Q1 + Lc ⊗ Q2 and R Q L L
(6)
≥ 0 and Q2 = Q 2 ≥ 0 state difference xi − xj
(NL − 1)(A 2 Fii + Fii A2 ) − NL (A 2
Under Assumption 6,7, problem (4) has a unique stabi˜x lizing solution u ˜ = K ˜, which gives finite performance ˜ ˜0 . The optimal state-feedback gain index (5) equal to x ˜0 P x ˜ P˜ , where P˜ is the symmetric positive defi˜ = −R ˜ −1 B K nite (s.p.d.) solution to the (large-scale) Algebraic Riccati Equation (ARE): ˜R ˜ −1 B ˜ P˜ + Q ˜ = 0. A˜ P˜ + P˜ A˜ − P˜ B
(7)
˜ and P˜ retain certain Due to special formulation of (4), K structure, which will prove essential for designing stabilizing distributed controllers in the next section. The specific structure of these matrices is proved in Theorem 8. In the following, we set X = BR−1 B . Theorem 8. Assume P˜ is the s.p.d solution to (7) associated with the optimal solution to (4). Let P˜ ∈ RnNL ×nNL be partitioned into NL2 blocks of dimension n × n, each denoted by P˜ij and referred to as the (i, j)-block of P˜ . Then, the following hold: NL ˜ Pij = P where I. For i = 1, · · · , NL , j=1
+ A 1 (Fii −
NL
−
NL
j=1 j=i
P˜ij ) + (Fii −
k=1
NL
P˜ij )A1
j=1 j=i
(NL − 1)(A 2 P˜ij + P˜ij A2 ) − A 2 (Fii − − (Fii −
NL
k=1 k=i
P˜ik )A2 − A 2
+ A 1 P˜ij + P˜ij A1 −
NL l=1 l=i l=j
NL
k=1
P˜il −
NL
P˜ik )
k=1 k=i
NL
P˜il A2
(12a)
l=1 l=i l=j
P˜ik X P˜kj − Q2 = 0
(12b)
Summing up (12a) for all j = i block-wise and adding this summation to (11a) gives (NL − 1)(A 2 Fii + Fii A2 ) − (NL − 1)A 2 Fii − Fii (NL − 1)A2 − NL A 2
(8)
+ (NL − 1)(A 2
II. P˜ij = P˜kl = P˜2 for all i, j, k, l = 1, · · · , NL with j = i and l = k, and P˜2 symmetric.
NL
− (NL − 1)(A 2
111
P˜ij A2 )
j=1 j=i
Using (10) the equations corresponding to off-diagonal blocks P˜ij , i = j of P˜ are
NL
Proof. First, we prove part I of the theorem. The equations corresponding to diagonal blocks P˜ii of P˜ in (7) are:
NL
P˜ik X P˜ik + Q1 + (NL − 1)Q2 = 0. (11b)
+ (NL − 1)(A 2
A 1 P + P A1 − P XP + Q1 = 0.
j=1 j=i
P˜ij −
(11a)
Q 1
Here Q1 = penalize local state and relative between the nodes i, j ∈ V, respectively. The matrix R = R > 0 weighs inputs of each subsystem. The following stabilizability and observability assumptions guarantee a unique stabilizing solution to LQR problem (4). Assumption 6. Let C1 C1 = Q1 . The pair (A1 , B) is stabilizable and (A1 , C1 ) is observable. Assumption 7. Let C12 C12 = Q1 + NL Q2 . The pair (A1 + NL A2 , B) is stabilizable and (A1 + NL A2 , C12 ) is observable.
NL
P˜ij +
j=1 j=i
l=1 l=i l=j
j=1 j=i
NL
P˜ij −
NL
P˜ij NL A2
j=1 j=i
P˜ij A2 )
j=1 j=i
P˜ik +
k=1 k=i NL
NL
NL
P˜ik A2 )
k=1 k=i
P˜il −
NL l=1 l=i l=j
P˜il A2 ) = 0
(13)
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where all terms associated with A2 cancel out. By summing up now (12) for all j = i block-wise and adding this summation to (11) gives A 1 Fii + Fii A1 − Fii XFii +
NL
k=1 k =i
P˜ik X Fii − Fkk + Q1 = 0. (14)
Eq. (14) has been established in Theorem 1 of Borrelli and Keviczky (2008). It is true also here due to (13). Summing up (14) for all i = 1, · · · , NL we get NL A 1 Fii + Fii A1 − Fii XFii + Q1 = 0 (15) i=1
which is sum of NL identical ARE’s, i.e.,
(16) A 1 Fii + Fii A1 − Fii XFii + Q1 = 0. NL ˜ Equation (10) implies Fii = i=1 Pij which, along with (16), proves part I. The ARE (7) has a repetitive structure and essentially can be decomposed into NL identical ˜ R ˜ are block diagonal and A, ˜ Q ˜ have equations since B, repetitive pattern. This implies that matrices P˜ij with j = i are all equal and symmetric. This proves part II of the theorem. The following corollary of Theorem 8 is stated without proof due to lack of space. Corollary 9. Let P˜ij ∈ Rn×n , i, j = 1, · · · , NL , denote the (i, j)-block of P˜ in (7) associated with the optimal solution to (4). Then, the following hold: I. P˜ii = P − (NL − 1)P˜2 , for all i = 1, · · · , NL where P is the s.p.d solution to ARE (8). II. P˜ij = P˜2 , for i, j = 1, · · · , NL and i = j, where P˜2 is symmetric matrix associated with node-level ARE: (A1 + NL A2 ) (P − NL P˜2 ) + (P − NL P˜2 )(A1 +NL A2 ) − (P − NL P˜2 )X(P − NL P˜2 ) +Q1 + NL Q2 = 0. (17)
˜ and B ˜ are selected block By assumption, the matrices R ˜ = diagonal. Consequently, the state-feedback gain K ˜ −1 B ˜ P˜ associated with the optimal solution to (4) −R retains the same structure with P˜ . This leads to the following Corollary. ˜ P˜ is the optimal ˜ = −R ˜ −1 B Corollary 10. Assume K state-feedback gain obtained from the solution to (4) ˜0 with P˜ which gives minimum performance index x ˜ 0 P˜ x mNL ×nNL ˜ and being the s.p.d solution to (7). Let K ∈ R P˜ ∈ RnNL ×nNL be partitioned into NL2 blocks of dimension ˜ ij and P˜ij , respectively m × n and n × n, denoted by K each referred to as (i, j)-block of the respective matrix. Then, the following are true; I. P˜ = INL ⊗ P − Lc ⊗ P˜2 . N L ˜ −1 B P for i = 1, · · · , NL . II. j=1 Kij = −R −1 ˜ ii = −R B P + (NL − 1)R−1 B P˜2 for i = III. K 1, · · · , NL . ˜ ij = −R−1 B P˜2 for i, j = 1, · · · , NL and j = i. IV. K ˜ = −IN ⊗ R−1 B P + Lc ⊗ R−1 B P˜2 . V. K L 112
Theorem 8 along with the results stated in Corollary 9 suggest that due to special formulation of the cost function (5) and the structure of the aggregate state-space form (2), the large-scale LQR problem (4) under Assumption 6,7 can effectively be reduced to finding the solution of two node-level ARE’s. This feature may be highly beneficial for problems involving networks, the topology of which is modeled by graph with an excessively large number of vertices (NL ). Applying the stabilizing optimal state-feedback control ˜x u ˜ = K ˜ to (2) results in a closed-loop matrix, which is Hurwitz and is written as: (18) Acl = INL ⊗ (A1 − XP ) + Lc ⊗ (A2 + X P˜2 ). Due to Proposition 3 the spectrum of Acl can be decomposed into: S(Acl ) =
N L
i=0
S A1 + BK + λc,i (A2 − BK2 )
(19)
where λc,i ∈ S(Lc ). Remark 11. From Proposition 5, the matrix A1 + BK + αNL (A2 − BK2 ) is Hurwitz for α = 0 and α = 1. In the sequel we require that: Condition 12. The matrix A1 + BK + αNL (A2 − BK2 ) is Hurwitz for all α ∈ [0, 1]. Condition 12 states that all convex combinations of two Hurwitz matrices (20) µA¯1 + (1 − µ)A¯2 with µ ∈ [0, 1] ¯ are Hurwitz, where A1 = A1 + BK + NL (A2 − BK2 ) and A¯2 = A1 + BK. Sufficient conditions for Hurwitz stability of convex combination of Hurwitz matrices can be found in Theorem 2.2 in Bia£las (2004). In essence, Condition 12 characterizes a class of LQR problems (4) which admit of solutions for which the Condition 12 holds. This will be used later for the design of distributed stabilizing controllers. For a given selection of weighting matrices (Q1 , Q2 , R) of the LQR problem (4), the validity of Condition 12 can be verified by searching for a symmetric positive definite matrix P¯ for which the following Linear Matrix Inequality (LMI), 0n×n 0n×n −(A¯ 1 P¯ + P¯ A¯1 ) 0n×n −(A¯ 2 P¯ + P¯ A¯2 ) 0n×n > 0, (21) 0n×n 0n×n P¯ is feasible. Obviously, if matrix P¯ exists then √ premultiply√ ing and postmultiplying (21) by [ µIn 1 − µIn 0n×n ] √ √ and [ µIn 1 − µIn 0n×n ], respectively, for µ ∈ [0, 1] leads to Lyapunov inequality: (µA¯1 + (1 − µ)A¯2 ) P¯ + P¯ (µA¯1 + (1 − µ)A¯2 ) < 0, (22) which admits of a solution P¯ = P¯ > 0. This demonstrates that µA¯1 + (1 − µ)A¯2 is a Hurwitz matrix for all µ ∈ [0, 1]. Alternatively, the stability of µA¯1 + (1 − µ)A¯2 can be examined via a simple graphical test by plotting the eigenvalue with the maximum real part of the matrix µA¯1 + (1 − µ)A¯2 for µ ∈ [0, 1]. The following arguments are given without proof due to lack of space. Consider now the aggregate input matrix ˜ = IN ⊗ B1 + Lc ⊗ B2 (23) B L
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where B1 ∈ Rn×m , B2 ∈ Rn×m represent input channels corresponding to local inputs and relative input difference ui − uj between the node i, j ∈ V with i = j, respectively. In this case the state-space form at node level is written as: NL x˙ i = A1 xi + A2 (xi − xj ) + B1 ui j=1,j =i
+ B2
NL
(ui − uj ), x0,i = xi (0). (24)
j=1,j =i
˜ as Then solving LQR problem (4) with A˜ as given in (3), B ˜ ˜ given in (23) and weighting matrices (Q, R) as selected in (6), results in optimal solution P˜ in (7). This has structure as defined in Theorem 8 and Corollary 9, i.e., P˜ = INL ⊗ P − Lc ⊗ P˜2 where P is the solution to ARE (8) and P˜2 is symmetric matrix associated with ARE (17). The ˜ can then be derived optimal state-feedback controller K −1 ˜ ˜ ˜ ˜ by K = −R B P = INL ⊗ K − Lc ⊗ K2 with (25) K = −R−1 B1 P, −1 ˜ ˜ (26) K2 = −R (B1 P2 + B2 P − NL B2 P2 )
where the property L2c = NL Lc (see Proposition 5) was used. Proof of closed-loop stability of this control scheme along with derivation of all equations will be included in an extended work of this paper. In the sequel, we consider multi-agent systems with dynamical couplings in their states only. 4. DISTRIBUTED CONTROL DESIGN FOR DYNAMICALLY COUPLED SYSTEMS
Let a network be formed of N identical and dynamically coupled LTI systems. The couplings among the systems are modelled by undirected graph G = (V, E) with Laplacian matrix L. The neighborhood of the i-th system is denoted by Ni ⊂ V. Let the dynamics at local level of the i-th system be (xi − xj ) + Bui , x0,i = xi (0) (27) x˙ i = A1 xi + A2 j∈Ni
where xi ∈ Rn and ui ∈ Rm . The aggregate state-space of the network becomes ˆx + B ˆu x ˆ˙ = Aˆ ˆ, x ˆ0 = x ˆ(0) (28) nN mN where x ˆ∈R ,u ˆ∈R and ˆ = IN ⊗ B. Aˆ = IN ⊗ A1 + L ⊗ A2 , B (29)
Note that the Laplacian matrix L in (29) does not necessarily correspond to a complete graph in contrast to (3) and generically the matrix Aˆ in (29) is sparse. We note here that the aggregate state-space forms (28) and (2) differ in number of subsystems and the structure of the Laplacian matrix L. We denote aggregate state-space as in (2) when referring to centralized control problems with NL subsystems and we use aggregate state-space (28) when referring to distributed control problems with N subsystems. Similarly, tilded matrices are referred to centralized problems while hatted matrices to distributed problems. A stabilizing distributed controller for (28) is constructed in the following Theorem. 113
113
Theorem 13. Consider a network of N coupled systems with dynamics described in (27) modelled by graph GN with Laplacian matrix LN . Let λN be the maximum eigenvalue of LN and denote by dmax the smallest integer which is greater than or equal to λN . Consider LQR problem (4) for NL = dmax , define P and P˜2 via (8) and (17), respectively, and assume Condition 12 is valid. Define also: K = −R−1 B P , K2 = −R−1 B P˜2 and ˆ = IN ⊗ K − LN ⊗ K2 . (30) K Then, the closed-loop matrix Acl = IN ⊗ (A1 + BK) + LN ⊗ (A2 − BK2 ) is Hurwitz.
(31)
Proof. Consider the spectrum S(Acl ) = S(IN ⊗ (A1 + BK) + LN ⊗ (A2 − BK2 )). Let VN ⊗ In be state-space transformation where VN ∈ RN ×N is an orthogonal matrix whose columns consist of the eigenvectors of LN . In the transformed coordinates, A¯cl = IN ⊗ (A1 + BK) + ΛN ⊗ (A2 − BK2 ) where ΛN = diag(0, λ2 , · · · , λN ) with λN ≤ dmax . The spectrum of A¯cl is S(A¯cl ) =
N
i=1
S(A1 + BK + λi (A2 − BK2 ))
(32)
where λi for i = 1, · · · , N are the eigenvalues of LN . Condition 12 holds, hence (A1 +BK)+αdmax (A2 −BK2 ) is Hurwitz for all α ∈ [0, 1]. Consequently A¯cl is also Hurwitz since λi ∈ [0, dmax ] for all i = 1, · · · , N . This proves the Theorem. Remark 14. For a time-varying graph G(t) = (V, E(t)) with fixed number of vertices (N ) and time-varying edges the maximum eigenvalue of the time-varying Laplacian matrix L(t) is bounded by 2N . Consequently, solving (4) for NL = 2N and assuming Condition 12 holds leads to a ˆ which stabilizes the network for distributed controller K all possible couplings among the N systems. Naturally, this does not imply stability of switching between stable network topologies. 4.1 Measure of suboptimality At this point, we wish to employ a suboptimality measure which can be cast as a performance loss index of the distributed scheme proposed above. It is necessary so, to define a reference performance index with which the suboptimal scheme is compared. Such an index has been introduced in Borrelli and Keviczky (2008) and is also outlined here. Let matrix Ξ ∈ RmN ×nN be partitioned into N 2 blocks of dimension m × n, each referred to as (i, j)-block of Ξ and denoted by Ξij ∈ Rm×n , with i, j = 1, · · · , N . In particular, the (i, j)-block can be written as: Ξij = Ξ[(i − 1)m + 1 : im, (j − 1)n + 1 : jn]. The class of structured N (G) is now defined as follows: matrices Kn,m N (G) = {Ξ ∈ RmN ×nN | Ξij = 0m,n if (i, j) ∈ / E, Km,n Ξij = Ξ[(i − 1)m + 1 : im, (j − 1)n + 1 : jn], i, j = 1, · · · , N }.
Consider now the following optimal control problem:
2019 IFAC NecSys 114 Eleftherios E. Vlahakis et al. / IFAC PapersOnLine 52-20 (2019) 109–114 Chicago, IL, USA, September 16-17, 2019
min J(ˆ u, x ˆ0 ) = u ˆ
∞
ˆx + u ˆ u) dt (ˆ x Qˆ ˆ Rˆ
0
ˆx + B ˆu subject to: x ˆ˙ = Aˆ ˆ, x ˆ0 = x ˆ(0) ˆx u ˆ=K ˆ
ˆ
ˆ ∈ K ˆ∈ Q
(33a)
N Km,n (G) N Km,n (G), ˆ ˆ
(33b) (33c) (33d)
ˆ = IN ⊗ R R
(33e)
ˆ = Q ≥ 0 and R = R > 0. We note that in the where Q absence of constraint (33d), the optimal control problem (33) if feasible, yields a centralized optimal control u∗ = ˆ −1 B ˆ P ∗ and P ∗ is the symmetric K ∗x ˆ where K ∗ = −R positive definite solution to: ˆ = 0. ˆR ˆ −1 B ˆ P ∗ + Q (34) Aˆ P ∗ + P ∗ Aˆ − P ∗ B Note that, since constraint (33d) is not included in the optimization, K ∗ and P ∗ are centralized solutions, and thus, the value x ˆ 0 P ∗ x ˆ0 is the minimum achievable performance index for a given x ˆ0 . Assume now that constraint (33d) is in force. Then, the ˆ as constructed distributed state-feedback controller K in (30) can be considered as a suboptimal solution to ˆ B) ˆ as given in (29) and Q ˆ = I N ⊗ Q1 + (33) with (A, LN ⊗ Q2 , with LN as defined in Theorem 13. Then, a performance index for this suboptimal distributed scheme can be computed as ˆx ˆ 0 Pˆ x (35) ˆ0 , J(K ˆ, x ˆ0 ) = x ˆ where P is the positive definite solution to the following Lyapunov equation: ˆ K) ˆ Pˆ + Pˆ (Aˆ + B ˆ K) ˆ + (Q ˆ+K ˆ R ˆ K) ˆ = 0. (36) (Aˆ + B ∗ ∗ ˆ Since P is optimal, J(K x ˆ, x ˆ0 ) ≤ J(K x ˆ, x ˆ0 ) for all x ˆ0 and thus ∆P = Pˆ − P ∗ is a positive semidefinite matrix ˆ = K ∗ . Any norm of ∆P can which is equal to zero if K be cast as a measure of suboptimality of the distributed ˆ controller K. 5. CONCLUSIONS Stabilizing distributed state-feedback controller for networks of coupled identical systems was proposed based on a large-scale LQR optimal problem. This method has originally been proposed in Borrelli and Keviczky (2008) for the decoupled case and was extended here to include couplings between the subsystems representing a multiagent network. An effective condition for stability is also provided which admits of network stable operation for a class of interconnections. The control scheme was obtained by optimizing an LQR performance index with
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a tuning parameter utilized to emphasize/de-emphasize relative state difference between interconnected agents. The assumption of identical dynamics is clearly restrictive but simplifies the design problem considerably and leads to the derivation of a stability condition which can be easily tested. Attempts to eliminate or relax this assumption will be the topic of future work. Preliminary results in this direction can be found in Vlahakis and Halikias (2018a,b). The control scheme has also been successfully applied to a multi-area power grid numerous simulations for which can be found in Vlahakis et al. (2019). REFERENCES Bia£las, S. (2004). A sufficient condition for Hurwitz stability of the convex combination of two matrices. Control Cybern., 33(1). Borrelli, F. and Keviczky, T. (2008). Distributed LQR design for identical dynamically decoupled systems. IEEE Trans. Autom. Contr., 53(8), 1901–1912. Deshpande, P., Menon, P.P., Edwards, C., and Postlethwaite, I. (2012). Sub-optimal distributed control law with H2 performance for identical dynamically coupled linear systems. IET Control Theory Appl., 6(16), 2509– 2517. Fax, J.A. and Murray, R.M. (2002). Information Flow and Cooperative Control of Vehicle Formations. IFAC Proc. Vol., 35(1), 115–120. Massioni, P. and Verhaegen, M. (2009). Distributed control for identical dynamically coupled systems: A decomposition approach. IEEE Trans. Automat. Contr., 54(1), 124–135. Mohar, B. (1991). The Laplacian spectrum of graphs. Graph theory Comb. Appl. Olfati-Saber, R. (2006). Flocking for Multi-Agent Dynamic Systems:Algorithms and Theory. IEEE Trans. Autom. Contr., 51(3), 1–20. Vlahakis, E., Dritsas, L., and Halikias, G. (2019). Distributed LQR Design for a Class of Large-Scale MultiArea Power Systems. Energies, 12(14), 2664. Vlahakis, E.E. and Halikias, G.D. (2018a). Distributed LQR Methods for Networks of Non-Identical Plants. In 2018 IEEE Conf. Decis. Control, 6145–6150. IEEE. Vlahakis, E.E. and Halikias, G.D. (2018b). ModelMatching type-methods and Stability of Networks consisting of non-Identical Dynamic Agents. IFACPapersOnLine. Wang, Q., Yu, C., Gao, H., and Liu, F. (2018). A distributed control law with guaranteed convergence rate for identically coupled linear systems.