Output Synchronization on Strongly Connected Graphs

Transcription

1 Output Synchronization on Strongly Connected Graphs Nikhil Chopra and Mark W. Spong Abstract In this paper we study output synchronization of networked multiagent systems. The agents, modeled as nonlinear systems with relative degree one, exchange information over a network described by an interagent communication graph. The agents are said to output synchronize if their outputs are pairwise convergent. Inspired by the recent results in [14], we extend our earlier results on output synchronization to include the case of strongly connected graphs. We first demonstrate output synchronization for input-output passive systems communicating over strongly connected graphs and include the practical case of constant time delays in communication. It is well known [4] that weakly minimum phase systems with relative degree one are feedback equivalent to a passive system with a positive definite storage function. We exploit this feedback equivalence to develop control laws for output synchronization of such systems, exchanging outputs on strongly connected graphs, and in the presence of communication delays. Simulations results are also presented to validate the proposed results. I. INTRODUCTION The problem of coordination and control of multi-agent systems is important in numerous practical applications, such as sensor networks, unmanned air vehicles, and robot networks. Thus there has recently been considerable research devoted to the analysis and control of the coordinated behavior of such systems. The goal is to generate a desired collective behavior by local interaction among the agents. Consensus and agreement behavior has been studied in [38], [31], [37], [2], [16], [23], [28], [25], [32], [33]. Group coordination and formation stability problems have been recently addressed in [1], [17], [20], [12], [22], [21], [36] among others. We refer the readers to [30], [24] for surveys on these research efforts. The concepts of passivity and dissipativity have been used for analyzing synchronization behavior in [26], [27], [35], [8], [5]. In this paper we study output synchronization of networked multiagent systems. We treat the case where each agent in the network can be modeled as a nonlinear dynamical system that is input/output passive. The goal then is to drive the outputs of the agents to each other asymptotically. The assumption of passivity is a natural one for the types of problems considered. Many of the existing results in the literature model the agents as velocity-controlled particles; in This research was partially supported by a faculty startup support from the University of Maryland, by the National Science Foundation under grant NSF ECCS and the Office of Naval Research under grant ONR grant N Nikhil Chopra is with the Department of Mechanical Engineering and the Institute for Systems Research, University of Maryland, College Park, MD ( nchopra@umd.edu) Mark W. Spong is with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL USA ( mspong@uiuc.edu) other words, as first-order integrators, which are the simplest type of passive systems. On the other hand, quite complicated Lagrangian and Hamiltonian systems, such as n-degree-offreedom robots, satisfy a natural and well-studied passivity property [34]. It was shown in [8], [6] it is possible to output synchronize nonlinear passive systems, provided the storage function is positive definite and the interagent communication graph is balanced. It was also demonstrated that the output synchronization property is robust to communication delays and switching interconnection topologies. Furthermore, for the case of nonlinear Lagrangian systems the output synchronization results in [8] led to algorithms for state synchronization of Lagrangian systems. The results were also used to develop synchronization algorithms for bilateral teleoperators [7], [11], [9], [18]. Output synchronization for kinematic particle in SE(3) was studied by [14]. By employing a novel weighted sum of the storage functions for the individual agents as the Lyapunov function candidate, the authors demonstrated output synchronization for strongly connected communication graphs, under constant time delays and brief connectivity losses. Recently in [10], the output synchronization results were extended to nonlinear systems in the normal form with relative degree one. In this paper, inspired by the recent motion coordination results in SE(3) [14], we extend our synchronization results in [7], [8], [6] to demonstrate output synchronization for strongly connected communication graphs. We next provide a brief background on passivity and graph theory, followed by our output synchronization results for the delay free case in Section II. We then address output synchronization with time delays in Section III, present simulations in Section IV, and finally summarize the results in Section V. A. Passivity Consider a control affine nonlinear system of the form { ẋ(t) = f(x) + g(x)u(t) Σ = (1) y = h(x) where x R n, u R m, and y R m. The functions f( ) R n, g( ) R n m, and h( ) R m are assumed to be sufficiently smooth, the admissible inputs are taken to be piecewise continuous and locally square integrable. We assume, for simplicity, that f(0) = 0 and h(0) = 0 and we note that the dimensions of the input and output are the same. We will often abbreviate the statement that the equilibrium solution x = 0 of (1) is stable to the statement that the system Σ is stable when there is no confusion in doing so.

2 Definition The nonlinear system Σ is said to be passive if there exists a C 1 storage function V (x) 0, V (0) = 0 and a function S(x) 0 such that for all t 0: V (x) = u T (t)y(x) S(x) (2) The system Σ is strictly passive if S(x) > 0 and lossless if S(x) = 0. B. Graph Theory and Communication Topology Information exchange between agents can be represented as a communication graph. We give here some basic terminology and definitions from graph theory [13] sufficient to follow the subsequent development. Definition By a graph G we mean a finite set V(G) = {v i,..., v N }, whose elements are called nodes or vertices, together with set E(G) V V, whose elements are called edges. An edge is therefore an ordered pair of distinct vertices. If, for all (v i, v j ) E(G), the edge (v j, v i ) E(G) then the graph is said to be undirected. Otherwise, it is called a directed graph. An edge (v i, v j ) is said to be incoming with respect to v j and outgoing with respect to v i and can be represented as an arrow with vertex v i as its tail and vertex v j as its head. The in-degree of a vertex v G is the number of edges that have this vertex as a head. Similarly, the out-degree of a vertex v G is the number of edges that have this vertex as the tail. If the in-degree equals the out-degree for all vertices v V(G), then the graph is said to be balanced. A path of length r in a directed graph is a sequence v 0,..., v r of r + 1 distinct vertices such that for every i {0,..., r 1}, (v i, v i+1 ) is an edge. A weak path is a sequence v 0,..., v r of r + 1 distinct vertices such that for each i {0,..., r 1} either (v i, v i+1 ) or (v i+1, v i ) is an edge. A directed graph is strongly connected if any two vertices can be joined by a path and is weakly connected if any two vertices can be joined by a weak path. Given a directed graph G, the weighted Laplacian Lw(G) is defined as Lw ij = if i = j = if j N i = 0 otherwise We quote the following result from [29] which is important in our subsequent analysis. Lemma 1.1: If the communication graph is strongly connected and the weights are positive, then there exists a vector γ (with positive elements) satisfying γ T Lw = 0 II. OUTPUT SYNCHRONIZATION: THE DELAY FREE CASE In this section we show that, when the interagent communication graph is strongly connected, diffusive coupling among the agents is sufficient to achieve output synchronization. Recall that each agent s (passive) dynamics can be written, for i = 1,..., N, as ẋ i = f i (x i ) + g i (x i )u i y i = h i (x i ) Definition Suppose we have a network of N agents as above. In the absence of communication delays, the agents are said to output synchronize if lim y i(t) y j (t) = 0 i, j = 1,..., N (4) t where denotes the Euclidean norm of the enclosed signal. We first analyze the case when there is no time delay in communication. Suppose that the agents are coupled together using the control u i (t) = (y j (t) y i (t)), i = 1,..., N (5) where is a positive constant and N i is the set of agents transmitting their outputs to the i th agent. Theorem 2.1: Consider the dynamical system described by (3) with the control (5). If the agents dynamics are inputoutput passive with a radially unbounded positive definite storage function and the communication graph is strongly connected, then the coupled nonlinear system (3),(5) is globally stable and the agents output synchronize. Proof: As the interagent communication graph is assumed to be strongly connected, there exists a vector γ = [γ 1 γ 2... γ N ] such that γ T Lw = 0 (see Lemma 1.1). Consider a weighted positive definite Lyapunov function for the N agent system as (3) V (x 1,..., x N ) = 2(γ 1 V 1 (x 1 ) + + γ N V N (x N )) (6) where V i (x i ), are the storage function and the weighting factor, respectively for the i th agent. The derivative of the Lyapunov function along trajectories of (3) and (5) is given as V = 2 ( S i (x i ) + yi T u i ) = 2 S i (x i ) + 2 yi T (y j y i ) (y j y i ) T (y j y i ) (yi T y i yj T y j ) (y j y i ) T (y j y i ) γ T Lw(Y T Y ) (y j y i ) T (y j y i ) 0

3 where Y T Y = y T 1 y 1 y T 2 y 2. y T N y N and γt Lw(Y T Y ) 0 from Lemma 1.1. Thus the solution to (3) and (5) is globally stable and all signals are bounded. Consider the set E = {x i R n 1, i = 1,..., N V 0}. The set E is characterized by all trajectories such that {S i (x i ) 0, (y i (t) y j (t)) T (y i (t) y j (t)) 0 j N i, i = 1,..., N}. Using Lasalle s Invariance Principle [19], all bounded solutions of the dynamical system given by (3) and (5) converge to M as t, where M is the largest invariant set contained in E. Strong connectivity of the communication graph then implies output synchronization of (3). Corollary 2.2: Let M be the largest invariant set in the set E defined above. Suppose that the vector fields f i (x i ) vanish on M. Then lim t y i (t) y ci i, where y ci is a constant. Proof: Under the above assumption, we have that, on the set M, ẋ i 0, i = 1,..., N (7) Since x i (t) is bounded, it follows that lim t x i = x ci where x ci is a constant. Thus y i converges to y ci = h i (x ci ). In particular, Corollary 2.2 implies that, in systems without drift (f i (x i ) = 0), the outputs will converge to a common constant value. If the agents are identical linear passive systems with a positive definite storage function, then we have the following result Corollary 2.3: Consider a multi-agent system where the identical agent dynamics are given by, ẋ i = Ax i + Bu i y i = Cx i i = 1,..., N (8) and let the coupling control u i (t) be given by (5). If the pair (A, C) is observable then lim t x i (t) x j (t) = 0 i, j. Proof: Using (8) and noting that the agents are identical y i y j = C(x i x j ) i, j Using Theorem 2.1, all trajectories converge to the set where y i y j 0 i, j Differentiating the above equation n 1 times, and noting (using (8)) that on the largest invariant set ẋ i = Ax i i, we get y i y j C ẏ i ẏ j CA ÿ i ÿ j = CA 2 (x. i x j ) (9). y n 1 i y n 1 j. CA n 1 If the pair (A, C) is observable, the observability matrix O = [C CA CA 2... CA n 1 ] T is full rank. Hence from (9) lim t x i (t) x j (t) = 0 i, j. III. OUTPUT SYNCHRONIZATION WITH TIME DELAY In this section, we study the problem of output synchronization when there are communication delays in the network. The delays are assumed to be constant and bounded. As there can be multiple paths between two agents, T k ij denotes the communication delay along the k th path from the i th agent to the j th agent. We only impose the restriction that delays along all paths of length one are unique, i.e. the one-hop transmission delay between two agents is uniquely defined. Definition The agents are said to output synchronize if lim y i(t Tij) k y j (t) = 0 i, j k (10) t Let the agents be coupled using the control u i = (y j (t T ji ) y i (t)) i = 1,..., N. Then the following result holds Theorem 3.1: Consider the dynamical system described by (3) with the aforementioned coupling control. If the agents dynamics are input-output passive with a radially unbounded positive definite storage function and the interagent communication graph is strongly connected, then the nonlinear system (3) is globally stable and the agents output synchronize in the sense of (10). Proof: As before, under the assumption of a strongly connected information graph, there exists a vector γ such that γ T Lw = 0. Consider a weighted positive definite Lyapunov- Krasovskii functional for N agent system as V (x X N ) = 2(γ 1 V 1 (x 1 ) + + γ N V N (x N )) t + yj T (s)y j (s)ds t T ji where V i (x i ) is the storage function for agent i. The derivative of this functional along trajectories of the system is V = 2 ( S i (x i ) + yi T u i ) = 2 S i (x i ) + 2 yi T (y j (t T ji ) y i ) ( ) + yj T y j yj T (t T ji )y j (t T ji ) (y j (t T ji ) y i ) T (y j (t T ji ) y i ) (yi T y i yj T y j ) (y j (t T ji ) y i ) T (y j (t T ji ) y i ) γ T Lw(Y T Y ) (y j (t T ji ) y i ) T (y j (t T ji ) y i ) 0

4 where Y T Y = y T 1 y 1 y T 2 y 2. yn T y N Therefore, as V 0 and V 0, the zero solution of (3) and (5) is globally stable and lim t V (x1 (t),..., x N ) exists and is finite. Integrating the expression for V, we have V (x(0)) t 0 (y j (s T ji ) y i (s)) 2 ds Letting t, j N i, i, y j (t T ji ) y i (t) L 2 [0, ]. As all signals are bounded, ẏ j (t T ji ) ẏ i (t) L. It is well known that a square integrable signal with a bounded derivative converges to the origin [19]. Therefore, lim t y j (t T ji ) y i (t) = 0, j N i, i. As the communication graph is strongly connected, the agents delay output synchronize in the sense of (10). We next address the case when the agents are not inputoutput passive but are feedback equivalent to a passive system [15]. If the following conditions hold [3] H1 The matrix L g h(x) is nonsingular for each x R n H2 The vector fields g 1 (x),..., g m (x) are complete where [ g 1 (x)... g m (x)] = g(x)[l g h(x)] 1. H3 The vector fields g 1 (x),..., g m (x) commute. then there exists a globally defined diffeomorphism that transforms Σ into the celebrated normal form [15] ż = q(z, y) ẏ = b(z, y) + a(z, y)u where the matrix a(z, y) is non-singular and (n m) real value functions z 1 (x),..., z n m (x), along with the m-dimensional output y = h(x), define the new set of coordinates. We assume that all agents above satisfy assumptions H1 H3 and therefore can be transformed into the globally defined normal form ż i = q i (z i, y i ) (11) ẏ = b i (z i, y i ) + a i (z i, y i )u i (12) As a i (z i, y i ) is nonsingular, the following preliminary feedback law u i = a i (z i, y i ) 1 ( b i (z i, y i ) + v i ) (13) is well defined and the resultant dynamical system can be written as ż i = q i (z i, y i ) ẏ i = v i i = 1,..., N (14) where the zero-dynamics are given as ż i = q i (z i, 0). The control term v i is used to couple the agents to achieve output synchronization based on the interconnection graph among the individual agents. We note that first equation in the transformed agent dynamics (14) can be rewritten as ż i = q i (z i, 0) + p(z i, y i )y i (15) Assuming that the individual system dynamics (14) are globally weakly minimum phase, there exists a C 2 positive definite, radially unbounded function W i (z i ) such that L qi (z i,0)w i 0. Let z s = [z 1... z N ] T and y s = [y 1... y N ] T Define the control law for each agent as v i (z i, y s ) = (y j (t T ji ) y i (t)) (G) ( L pi(z i,y i)w i ) T (16) Then the next result follows Theorem 3.2: Consider the dynamical system (14) along with the coupling control (16). If all multiagent systems are globally weakly minimum phase and the interconnection graph is strongly connected, then (14) output synchronizes in the sense of (10). Proof: Consider a positive definite, radially unbounded Lyapunov function for the N-agent system as V p (z s, y s ) = + (2W i (z i ) + yi T y i ) t t T ji y T j (s)y j (s)ds (17) The derivative of V (z s, y s ) along trajectories generated by (14) and (16) is given as V (z s, y s ) = + = (2 Ẇ i + 2 yi T v i ) (yj T y j yj T (t T ji )y j (t T ji )) (2L qi (z i,0)w i + 2L pi (z i,y i )W i y i ) 2 yi T ( ) T Lpi(z i,y i)w i +2 yi T (y j (t T ji ) y i ) + (yj T y j yj T (t T ji )y j (t T ji )) 2 yi T (y j (t T ji ) y i ) ( ) + yj T y j yj T (t T ji )y j (t T ji ) Using the proof of Theorem 3.1, we get V (z s, y s ) y j (t T ji ) y i ) 2 0

5 y Therefore, the dynamical system (14) and (16) is globally stable and output synchronization follows from the proof of Theorem 3.1. IV. SIMULATIONS For the sake of brevity and clarity, we consider the following agent dynamics that are already in the normal form (14) y 1 y 2 y 3 ż i = z i + z 2 i y i ẏ i = v i i = 1, 2, 3 where z, y R. The above system is globally minimum phase with W (z) = 1 2 z2 as Ẇ (z) is negative definite along the zero dynamics. Therefore, the coupling control v i for each agent is given as v i = ( ) T L pi (z i,y i )W i + (y j (t T ji ) y i ) = z 3 + (y j (t T ji ) y i ) The agents are assumed to be interconnected using a timeinvariant topology as shown in Figure 1. Therefore the closed Time Fig. 2. The agents output synchronize independent of the time delay. V. CONCLUSIONS In this paper we studied output synchronization of networked multiagent systems exchanging outputs on strongly connected communication graphs. Considering both the case of input-output passive agents and systems in the normal form, we demonstrated output synchronization for strongly connected graphs communication graphs. It was also demonstrated that output synchronization is robust to constant time delays. Simulations results were presented to verify the proposed results. VI. ACKNOWLEDGMENTS The authors would like to thank Professor M. Fujita, Y. Igarishi, and T. Hatanaka for useful discussions on this topic. REFERENCES Fig. 1. The interconnection graph for the three agents. loop dynamical system is given as ż i = z i + z 2 i y i i = 1, 2, 3 ẏ 1 = z (y 3 (t T 31 ) y 1 ) ẏ 2 = z (y 1 (t T 12 ) y 2 ) + 5(y 3 (t T 12 ) y 2 ) ẏ 3 = z (y 2 (t T 23 ) y 3 ) The weighted Laplacian for this interconnection graph is given as Lw = It can be easily shown that for this choice coupling gains (link weights), γ = [ ] is the left eigenvector for the zero eigenvalue of Lw. Therefore, using the weighted Lyapunov function (with the elements of the vector γ as the weights), it can be demonstrated (see Theorem 3.2) that the agents delay-output synchronize. In the simulation, the time delays in the simulation were set as T 31 =.3, T 12 =.5, T 23 = 1, and T 32 =.4. As seen in Figure 2, the agents delay-output synchronize. [1] Murat Arcak. Passivity as a design tool for group coordination. In Proc. ACC, Minneapolis, MN, June [2] V. D. Blondel, J. M. Hendrickx, A. Olshevsky, and J. N. Tsitsiklis. Convergence in multiagent coordination, consensus, and flocking. In Proceedings of the Joint 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC 05), Seville, Spain, December [3] C.I. Byrnes and A. Isidori. Asymptotic stabilization of minimum phase nonlinear systems. IEEE Transactions on Automatic Control, 36(10): , [4] C.I. Byrnes, A. Isidori, and J.C. Willems. Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Transactions on Automatic Control, 36(11): , [5] N. Chopra. Output Synchronization of Networked Passive Systems. Ph.d. thesis, University of Illinois at Urbana-Champaign, Department of Industrial and Enterprise Systems Engineering, [6] N. Chopra and M. W. Spong. Output synchronization of nonlinear systems with time delay in communication. IEEE Conference on Decision and Control, [7] N. Chopra and M.W. Spong. Synchronization of networked passive systems with applications to bilateral teleoperation. In Society of Instrumentation and Control Engineering of Japan Annual Conference, Japan, August [8] N. Chopra and M.W. Spong. Passivity-based control of multi-agent systems. In Sadao Kawamura and Mikhail Svinin, editors, Advances in Robot Control: From Everyday Physics to Human-Like Movements, pages Springer Verlag, [9] N. Chopra and M.W. Spong. Adaptive synchronization of bilateral teleoperators with time delay. In Manuel Ferre et al., editor, Advances in Telerobotics, pages Springer Verlag, [10] N. Chopra and M.W. Spong. Output synchronization of nonlinear systems with relative degree one. In Hidenori Kimura Vincent D. Blondel, Stephen P. Boyd, editor, Recent Advances in Learning and Control, pages Springer Verlag, 2008.

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