RECENTLY, the study of cooperative control of multiagent

Transcription

1 24 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL., NO. 2, APRIL 24 Consensus Robust Output Regulation of Discrete-time Linear Multi-agent Systems Hongjing Liang Huaguang Zhang Zhanshan Wang Junyi Wang Abstract This paper deals with consensus robust output regulation of discrete-time linear multi-agent systems under a directed interaction topology. The digraph is assumed to contain a spanning tree. Every agent or subsystem is identical and uncertain, but subsystems have different external disturbances. Based on the internal model and general discrete-time algebraic Riccati equation, a distributed consensus protocol is proposed to solve the regulator problem. A numerical simulation demonstrates the effectiveness of the proposed theoretical results. Index Terms Cooperative control, discrete-time, uncertain multi-agent systems, robust output regulation. I. INTRODUCTION RECENTLY, the study of cooperative control of multiagent systems has been received much attention from many fields. As an important issue of cooperative control, the agents should be able to reach a common trajectory based on a consensus protocol under the shared information in the presence of information exchange and dynamically changing interaction topologies [ 2]. It is widely applied to flocks [ 2], vehicle formations [3] and attitude alignment [4]. The central problem is to design a distributed algorithm by which all the agents reach an agreement on their states or outputs, using local information exchange under the communication topologies. The consensus problem of multi-agent systems with leaderless was widely investigated in [5 6]. Pioneering contributions to various distributed strategies that achieved consensus were described by [, 3, 5]. They dealt with consensus problems for networks of dynamic agents under directed networks with fixed and switching topologies and undirected networks with communication time-delays and fixed topology. Diverse applications of consensus problems were discussed in [6] such as fast consensus in small-world networks, and Markov processes load balancing in networked dynamic systems. In leader-following problems, a subsystem named leader moves along a predefined trajectory, while other subsystems named followers have the same trajectories as the Manuscript received June 9, 23; accepted August 3, 23. This work was supported by National Basic Research Program of China 973 Program 29CB326, National High Technology Research and Development Program of China 863 Program 22AA44, and National Natural Science Foundation of China 59778, 6345, Recommended by Associate Editor Yiguang Hong Citation: Hongjing Liang, Huaguang Zhang, Zhanshan Wang, Junyi Wang. Consensus robust output regulation of discrete-time linear multi-agent systems. IEEE/CAA Journal of Automatica Sinica, 24, 2: Hongjing Liang is with the School of Information Science and Engineering, Northeastern University, Shenyang 89, China lianghongjing99@63.com. Huaguang Zhang is with the School of Information Science and Engineering, Northeastern University, Shenyang 89, Chinaand also with the State Key Laboratory of Synthetical Automation for Process Industries Northeastern University, Shenyang 89, China hgzhang@ieee.org. Zhanshan and Junyi Wang are with the School of Information Science and Engineering, Northeastern University, Shenyang 89, China wangzhanshan@ise.neu.edu.cn; wjyi68@26.com. leader [3 9]. Reference [5] introduced the optimal tracking problem based on the adaptive dynamic programming method. In many applications, there might exist leaders in the multi-agent networks. However, some variables of an active leader cannot be measured, [6] proposed an observer by inserting an integrator into the loop for each agent to estimate the leader s velocity. Reference [7] considered the leader-following formation problem of multi-robot systems with switching interconnection topologies using dynamic output feedback with the help of a canonical internal model. Kinematics equations of the robots are given in [9] and it uses adaptive control law to solve the robust leader-following formation problem. A further concept termed output regulation has been introduced to characterize the output consensus of leader-following multi-agent systems. Output regulation theory is to control one or more than one plant so that their outputs can track a reference signal and/or reject a disturbance produced by an exosystem. Well-known results about output regulation of linear systems and nonlinear systems have been posed and solved by [2 22]. In [23], the tracked signal or rejected disturbance is the same for all the nodes but only partial nodes can access the state information of exosystem and a synchronous manifold method was employed in it. But the majority of existing algorithms use the internal model principle method [24 27]. Based on it, a distributed control approach for robust output regulation problem was considered in [24]. A general result was given in [27] for linear uncertain multi-agent systems using distributed dynamic state feedback control law and distributed dynamic output feedback control law. Since not all the agents can access the exosystem information, [28 29] gave a dynamic compensator for the un-access part to solve the output regulation problem for heterogeneous agents. There have been a lot of results about output regulation problem of continuous-time linear or nonlinear multi-agent systems, but few papers dealt with discrete-time multi-agent systems. In view of these issues, this technical note presents a framework on output regulation of discrete-time multi-agent systems. Every subsystem is identical and uncertain. Each agent also has different disturbance. Therefore, we propose an internal model with n adjustable parameters. Then a distributed consensus protocol is given which not only makes the subsystem stable regardless of external disturbance but also regulates the errors to converge to zero. The outline of this paper is as follows: Section I introduces graph theory and mathematical preliminaries. Then the network models are established. Section II presents the consensus protocols and solves the robust output regulation problem of discrete-time multi-agent systems. An example is given in Section III. Finally, concluding remarks are drawn in Section IV. Some standard notations of matrix are used throughout this paper. R and C denote the sets of real numbers and complex

2 LIANG et al.: CONSENSUS ROBUST OUTPUT REGULATION OF DISCRETE-TIME LINEAR MULTI-AGENT SYSTEMS 25 numbers. The set of positive real numbers is denoted by R +. R m n is a set of matrices with m rows and n columns. For matrix A = [a ij ] R n n, A > means A is a positive definite matrix and the vector veca is defined by veca = a,, a n,, a n,, a nn T. Given λ C, Reλ, Imλ and λ are the real part, the imaginary part and the modulus of λ, respectively. n R n is the vector with all entries being. II. PRELIMINARIES AND NETWORK MODELS A. Basic Knowledge In this section, some basic concepts and results are introduced. For more details, please refer to [3 3]. Graph theory: A weighted digraph G = V, E, A of order n is composed of a vertex set V = v, v 2,, v n }, an edge set E = e ij = v i, v j } V V and a weighted adjacency matrix A = [a ij ] with nonnegative adjacent elements a ij. Moreover, a ij > if v i, v j E, a ij = if v i, v j E and a ii = for all i =,, n. v i, v j E if and only if ith agent can receive information from jth agent directly. N i = v j V : v i, v j E} is defined as the set of neighbors of node v i. In association with G, the in-degree and out-degree of node v i are defined as follows: deg in v i = n a ji, deg out v i = j= n a ij. j= The degree matrix D = diagd, d 2,, d n } is a diagonal matrix, whose diagonal elements are given as d i = deg out v i. The Laplacian matrix L = D A = [l ij ] is defined by l ij = a ij and l ii = n j= a ij. A directed path is a sequence of edges in a directed graph of the form v, v 2, v 2, v 3,, where v i V. A directed graph contains a directed spanning tree if there exists at least one agent which is called root node that has a directed path to every other agents. 2 Matrix theory: Lemma [3]. Let A R n n and λ, λ 2,, λ n be the eigenvalues of A. Then for c C, λ +c, λ 2 + c,, λ n + c are the eigenvalues of A + ci n. Lemma 2. For any real symmetric matrix Q >, there exists real symmetric matrix ˇQ > such that Q = ˇQ ˇQ. Proof. From the expression of Q, there is a real orthogonal matrix U such that Q = UΛ Q U T = U Λ Q U T U Λ Q U T, with λ qi >, i =, 2,, N being the eigenvalues of Q and Λ Q = diagλ q,, λ qn }, ΛQ = diag λ q,, λqn }. Let ˇQ = U ΛQ U T ; then the proof is completed. Lemma 3 [3] Geršgorin. Let A = [a ij ] R n n, and R ia n a ij, i n, j= j i and denote the deleted absolute row sums of A. Then all the eigenvalues of A are located in the union of n discs N z C : z a ii R ia} GA. i= B. Network Models Let the digraph Ḡ = V, Ē, Ā of order N + consist of a vertex set V = v, v,, v N and v is defined as the exosystem. a i > if agent i can receive information from the exosystem and else a i =. L is defined as the Laplacian matrix of subgraph Ḡ except node v. The adjacency matrix of digraph Ḡ is obtained as Ā =, A N Ā s in which A = diaga, a 2,, a N } and Ās is the adjacency matrix of subgraph Ḡs with vertex set Vs = v, v 2,, v N. Then the Laplacian matrix L of digraph Ḡ is L = A N A + L, 2 s in which L s = D s Ās is the Laplacian matrix of subgraph Ḡ s. Ds = diag d, d 2,, d N } and Ās are the degree matrix and adjacency matrix of subgraph Ḡs, respectively. Lemma 4 [32]. All the eigenvalues of H = A + L s defined in 2 have positive real parts if and only if the digraph Ḡ contains a spanning tree and node v as its root. A group of N discrete-time linear agents with uncertain dynamics are described by x i = Āx i + Bu i + Ēiω, x i = x i, y i = Cx i, i =, 2,, N, where x i R n is the state, y i R p is the output and xk+ is defined as x i, u i R q denotes the consensus protocol to be designed which depends on agent i and its neighbors, Ē i ω denotes the external disturbance of the ith agent. The matrices Ā, B, C and Ē i are uncertain and they could be written as follows: 3 Ā = A + A, B = B + B, C = C + C, Ē i = E i + E i, 4 where A, B, C and E i are the perturbations of nominal matrices A, B, C and E i, respectively. It is convenient to identify the system uncertainties with a vector = vec A, B, E,, E N vec C R nm+n+p+nq. System 3 with = is called a nominal system. In addition, the reference inputs and the disturbances can be lumped together as the exosystem: ω = Sω, ω = ω, 5 y r = F ω, with ω R s denoting ωk + and y r R p the reference signal. Thus, the consensus output regulation objectives may be prescribed in terms of the tracking errors e i R p as the following form: e i = y i y r = Cx i F ω, i =, 2,, N. 6

3 26 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL., NO. 2, APRIL 24 III. CONSENSUS PROTOCOL AND MAIN RESULTS Assume that each agent can communicate with its neighbor agents and exosystem by sharing their output states. The output-coupling variable relationship between agent i and its neighbors j N i is defined as follows: δ i = j N i a ij y i y j + a i y i y r, i =, 2,, N. The internal model technique which will be employed to synthesize protocols has important advantages that the parameters of the internal model are not sensitive to the controlled systems [33]. It is also necessary to solve the output regulation for uncertain multi-agent systems. Thus, the internal model servocompensator z i R psm under the network topology is built as follows: 7 z i = G z i + G 2 φ i δ i, i =, 2,, N, 8 with φ i R +, i =, 2,, N. The pair of matrices G, G 2 is said to incorporate a p-copy internal model of matrix S with G = block diagβ, β 2,, β p }, G 2 = block diagσ, σ 2,, σ p }, for all i =,, p, and β i is a constant square matrix. σ i is a constant column vector such that β i, σ i is controllable and the minimal polynomial of S divides the characteristic polynomial of β i. To solve the regulator problem, we give the following distributed consensus protocols: u i = K φ i a ij x i x j + a i x i + K 2 z i, 9 j N i with i =, 2,, N, K R q n, and K 2 R q psm being the gain matrices to be designed later. Let x = x T, x T 2,, x T N T, z = z T, z T 2,, z T N T and ω = N ω. Then using protocol 9, the uncertain discrete-time linear dynamics 3 and the internal model servocompensator 8 can be written as x = I N Ā + ΦH BK x + I N BK 2 z + Ē ω, z = ΦH G 2 Cx + IN G z ΦH G 2 F ω, where Φ = diagφ, φ 2,, φ N }, Ē = block diagē, Ē2,, ĒN}, ΦA G 2 F ω = ΦH G 2 F ω. Also, let ζ = x T, z T T. Then the closed loop system can be written in a compact form as ζ = Ācζ + Ēc ω, where IN Ā c = Ā + ΦH BK I N BK 2, ΦH G 2 C IN G Ē Ē c =. ΦH G 2 F Problem. The robust cooperative output regulation of discrete-time multi-agent systems is solved if there is the consensus protocol 9 such that The nominal closed-loop system matrix A c = IN A + ΦH BK I N BK 2 is Schur. ΦH G 2 C I N G 2 There exists an open neighborhood W of =. For the closed-loop system, the local tracking error e i k as k, for any initial conditions x i R n, ω R s. The following basic assumptions and lemmas are necessary to solve the output regulation problem: Assumption. The pair A, B is stabilizable. Assumption 2. S has no eigenvalues in the interior of the unit circle in the z-plane. Remark. If all the eigenvalues of S lay in the interior of unit circle in the z-plane, the trajectories of ω will decay exponentially to zero and not affect the output regulation. Without loss of generality, we assume that all the eigenvalues lay on or out of the unit circle. Assumption 3. For all λ σg, A λin B rank C = n + p. Assumption 4. Digraph Ḡ contains a spanning tree and node v as its root. Lemma 5 [33]. If Assumptions 3 hold, the pair G, G 2 incorporates a p-copy internal model of matrix S. Let A = A, B = G 2 C G B, then pair A, B is stabilizable. Lemma 6 [33]. Under Assumption 2, assume G, G 2 incorporates a p-copy internal model of S. If the matrix equation Π 2 S = G Π 2 + G 2 Ω has a solution Π 2, then Ω =. Lemma 7. If A, B is stabilizable and the gain matrix K is defined as K = B T P B B T P A, then ρ = λ max [ ˇQ A T P BB T P B B T P A ˇQ ], with P = P T >, Q = Q T > and Q = ˇQ ˇQ for symmetric matrix ˇQ >. Matrix A + sbk, s C is stable if and only if s lies in the stability region Ψ = s C : s 2 < ρ}. 2 Proof. Since A, B is stabilizable, for any Q = Q T >, the following discrete-time algebraic Riccati equation: A T P A P A T P BB T P B B T P A + Q = 3 has a unique solution P = P T >. Under Lyapunov stability theorem, for P >, A + sbk is stable for some K if and only if A + sbk P A + sbk P <

4 LIANG et al.: CONSENSUS ROBUST OUTPUT REGULATION OF DISCRETE-TIME LINEAR MULTI-AGENT SYSTEMS 27 Necessity. Since the gain matrix K is defined as K = B T P B B T P A, it follows that A + sbk P A + sbk P = A T P A + s K T B T P A + sa T P BK + ss K T B T P BK P = A T P A + s s + ss A T P BB T P B B T P A P = A T P A A T P BB T P B B T P A P + s s + ss A T P BB T P B B T P A = Q + s 2 A T P BB T P B B T P A <. 4 From Lemma 2, there exists symmetric matrix ˇQ that Q = ˇQ ˇQ. 4 is rewritten as I n + s 2 ˇQ A T P BB T P B B T P A ˇQ <, therefore, the stability region is given as Ψ = s C : s 2 < ρ}, 5 with ρ = λ max[ ˇQ A T P BB T P B B T P A ˇQ. ] Sufficiency. When s lies in the region Ψ, and if λ ri, i =, 2,, n are the eigenvalues of ˇQ A T P BB T P B B T P A ˇQ, there has > s 2 ρ λ ri, i.e., λ ri s 2 < T. 6 Under Lemma, λ ri s 2 are the eigenvalues of I n + s 2 ˇQ A T P BB T P B B T P A ˇQ and 6 implies that I n + s 2 ˇQ A T P BB T P B B T P A ˇQ <. 7 Pre- and post-multiplying 7 by ˇQ, yields Q + s 2 A T P BB T P B B T P A A + sbk P A + sbk P <. Then A + sbk is stable. Theorem. Under Assumptions 4, if φ i = a i+ d i, K = K K 2 = B T P B B T P A and σφh Ψ, then the consensus robust output regulation problem can be solved by the distributed consensus protocol 9. INn Proof. Let T = ΦH I. The nominal psm closed-loop system matrix A c can be transformed into Λ = T A c T = IN A + ΦH BK ΦH BK 2 = I N G 2 C I N G I N A + ΦH BK, 8 with K = K K 2. Denote the eigenvalues of ΦH by λ i, i =,, N. There exists a unitary matrix U such that U ΦHU = T = [t ij ] is upper triangular, with diagonal entries t ii = λ i, i =,, N. Let T 2 = U In U I. Then psm Λ 2 = T2 Λ T 2 = IN A + T BK T BK 2 = I N G 2 C I N G I N A + T BK 9 Because the elements of the transformed system matrix Λ 2 are either block diagonal or block upper-triangular, Λ 2 is Schur if and only if A + λ i BK, i =,, N is Schur. By the Geršgorin disc theorem, all the eigenvalues λ i, i =,, N of ΦH are located in the union of N discs N z C : z φi d i + a i } φi di, 2 i= If φ i = a i+ d i, then 2 is rewritten as N } d i z C : z a i + d, 2 i i= which represents that the eigenvalues of ΦH span in the neighbouring of. By Lemmas 5 and 7, the stability region of A + sbk is } Ψ = s C : s 2 < ρ, 22 If all the eigenvalues λ i, i =,, N are located in Ψ, i.e., σφh Ψ, then A + λ i BK, i =,, N is Schur. Therefore, the nominal closed-loop system matrix A c is Schur. For each W, where W is an open neighborhood of = such that Āc is Schur and under Assumption 2, λ i S+ λ j Āc, i =,, s, j =,, Nn + ps m. Then there exists a unique solution Π R Nn+psm Nn+psm of Sylvester equation ΠS = ĀcΠ + Ēc N I s. 23 Π Let Π =, with Π Π R Nn Nn, Π 2 2 R Npsm Npsm. Equation 23 has the following form: Π S = I N Ā + ΦH BK Π + I N BK 2 Π 2 + Ē N I s, Π 2 S = ΦH G 2 CΠ + I N G Π 2 ΦH G 2 F N I s = I N G 2 ΦH I q I N CΠ N F + I N G Π Since I N G, I N G 2 incorporates a pn-copy internal model of S, under Lemma 6 and by the invertibility of ΦH I q, one gets I N CΠ N F =. 25 Leting ˆζ = ζ Πω and calculating the difference of ˆζ yields ˆζ = Ācζ + Ēc N I s ω ΠSω = Āc ˆζ 26 Let e = e T, e T 2,, e T N T. The tracking errors 6 have the following form: e = I N C x N F ω = IN C ζ N F ω = IN C ˆζ + Πω N F ω = IN C ˆζ + IN CΠ N F ω. From 25, one gets e = I N C ˆζ.

5 28 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL., NO. 2, APRIL 24 Since Āc is Schur, ˆζ t. Thus, we get lim e = lim IN C ˆζ =. k k That is, the consensus robust output regulation problem is solved. Remark 2. For the system dynamics 3, if =, robust output regulation problem for uncertain discrete-time multiagent systems becomes output regulation problem for certain systems. Remark 3. The robust output regulation for linear multiagent systems has been deeply studied in [27] and a common gain matrix K could be found to ensure N nominal closedloop systems stability. We can determine the gain matrix only based on continuous-time algebraic Riccati equation for any system matrix and any information graph. But for discrete-time multi-agent systems, it is hard to find a gain K to make the closed-loop system matrix in the form of A + H BK stable without considering the relationship between system matrix A and Laplacian matrix H of the information graph. Theorem 3.2 in [34] gave the relationship between them that can make the closed-loop system matrix stable. In our paper, parameters φ i are used to regulate all the eigenvalues of H to lie in the neighbor of, which can reduce the conservatism. Φ = block diag,,, /4, /2,, /4}. By some simple calculations, the stability region is shown as Ψ = s C : s 2 <.298}. The control gain K =.454, and K 2 =.5486 which can make Āc stable in an open neighborhood of =. Thus, we obtain the simulation results as shown in Fig. 2. The regulated errors converge to the origin asymptotically. V. CONCLUSIONS This paper has investigated the design of distributed consensus protocol for the consensus robust output regulation problem for discrete-time multi-agent systems. A directed graph has been used to describe the information exchange among agents. We assume that the digraph should contain a spanning tree so that all the nodes could communicate with their neighbors and there are no isolated nodes. The internal model method is also used to solve the robust regulation problem. We have also provided an example to illustrate the result. IV. AN EXAMPLE In this section, an example is given to validate the effectiveness of the theoretical result. Consider a network of discretetime multi-agent systems described by 3 with x i = x i, x i2 T, i =,, 7, A =, B =, C =,. A =, B =, C =.2,.. E i =, i =, 2, 3, i E i =, i = 4, 5, 6, 7.i E i =, i =,, 7. Fig.. The communication topology. The exosystem is in the form of 5 with ω = ω, ω 2, ω 3 T and S =, F =. By the exosystem matrix S, we could find a pair of matrices G =, G 2 =, which incorporate a -copy internal model of S. It is easy to check that Assumptions 3 hold. The system topology is described by the digraph showed in Fig. with V = v, v,, v 7 and the adjacent weighted values are also showed in Fig.. Obviously, digraph Ḡ contains a spanning tree and node as its root. The parametric matrix Fig. 2. Regulated errors of seven agents. REFERENCES [] Jadbabaie A, Lin J, Morse A. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 23, 486: 988

6 LIANG et al.: CONSENSUS ROBUST OUTPUT REGULATION OF DISCRETE-TIME LINEAR MULTI-AGENT SYSTEMS 29 [2] Hong Y G, Gao L X, Cheng D Z, Hu J P. Lyapunov-based approach to multiagent systems with switching jointly connected interconnection. IEEE Transactions on Automatic Control, 27, 525: [3] Fax J A, Murray M R. Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, 24, 499: [4] Ren W, Beard R W. Decentralized scheme for spacecraft formation flying via the virtual structure approach. Journal of Guidance, Control, Dynamics, 24, 27: [5] Olfati-Saber R, Murray R M. Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 24, 499: [6] Olfati-Saber R, Fax J A, Murray R M. Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 27, 95: [7] Hengster-Movric K, You K Y, Lewis F L, Xie L H. Synchronization of discrete-time multi-agent systems on graphs using Riccati design. Automatica, 23, 492: [8] Yang H, Jiang B, Cocquempot V, Zhang H G. Stabilization of switched nonlinear systems with all unstable modes: applications to multiagent systems. IEEE Transactions on Automatic Control, 2, 569: [9] Yang H, Jiang B, Zhang H G. Stabilization of non-minimum phase switched nonlinear systems with application to multi-agent systems. Systems and Control Letters, 22, 6: 23 3 [] Hong Y G, Wang X L. Multi-agent tracking of a high-dimensional active leader with switching topology. Journal of Systems Science and Complexity, 29, 224: [] Huang Q Z. Consensus analysis of multi-agent discrete-time systems. Acta Automatica Sinica, 22, 387: [2] Yan J, Guan X P, Luo X Y, Yang X. Consensus and trajectory planning with input constraints for multi-agent systems. Acta Automatica Sinica, 22, 387: [24] Wang X L, Hong Y G, Huang J, Jiang Z P. A distributed control approach to a robust output regulation problem for multi-agent linear systems. IEEE Transactions on Automatic Control, 2, 552: [25] Wang X L, Han F. Robust coordination control of switching multiagent systems via output regulation approach. Kybernetika, 2, 475: [26] Wieland P, Sepulchre R, Allgöwer F. An internal model principle is necessary and sufficient for linear output synchronization. Automatica, 2, 475: [27] Su Y F, Hong Y G, Huang J. A general result on the robust cooperative output regulation for linear uncertain multi-agent systems. IEEE Transactions on Automatic Control, 23, 585: [28] Su Y F, Huang J. Cooperative output regulation of linear multiagent systems. IEEE Transactions on Automatic Control, 22, 574: [29] Su Y F, Huang J. Cooperative output regulation with application to multi-agent consensus under switching network. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 22, 423: [3] Horn R A, Johnson C R. Matrix Analysis. Cambridge, UK: Cambridge University Press, 987 [3] Godsil C, Royle G. Algebraic Graph Theory. New York: Springer-Verlag, 2 [32] Li Z K, Duan Z S, Chen G R, Huang L. Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint. IEEE Transactions on Circuits and Systems I: Regular Papers, 2, 57: [33] Huang J. Nonlinear Output Regulation: Theory and Applications. Phildelphia, PA: SIAM, 24 [34] You K Y, Xie L H. Network topology and communication data rate for consensusability of discrete-time multi-agent systems. IEEE Transactions on Automatic Control, 2, 56: [3] Wang Y C, Zhang H G, Wang X Y, Yang D S. Networked synchronization control of coupled dynamic networks with time-varying delay. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 2, 46: [4] Fu J, Zhang H G, Ma T D, Zhang Q L. On passivity analysis for stochastic neural networks with interval time-varying delay. Neurocomputing, 2, 734 6: [5] Zhang H G, Liu D R, Luo Y H, Wang D. Adaptive Dynamic Programming for Control-Algorithms and Stability. London: Springer-Verlag, 23 [6] Hong Y G, Hu J P, Gao L X. Tracking control for multi-agent consensus with an active leader and variable topology. Automatica, 26, 427: [7] Wang X L, Ni W, Wang X S. Leader-following formation of switching multirobot systems via internal model. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 22, 423: Hongjing Liang Ph. D. candidate at the School of Information Science and Engineering, Northeastern University. His research interest covers multi-agent systems, complex systems, and output regulation. Huaguang Zhang Professor in the Department of Automatic Control, Northeastern University. His research interest covers neural-network-based control, fuzzy control, and their industrial applications. Corresponding author of this paper. [8] Hong Y G, Wang X L, Jiang Z P. Distributed output regulation of leader-follower multi-agent systems. International Journal of Robust and Nonlinear Control, 23: [9] Liu S C, Tan D L, Liu G J. Robust leader-follower formation control of mobile robots based on a second order kinematics model. Acta Automatica Sinica, 27, 339: [2] Francis B A. The linear multivariable regulator problem. SIAM Journal on Control and Optimization, 977, 53: Zhanshan Wang Professor in the Department of Automatic Control, Northeastern University. His research interest covers stability analysis of recurrent neural networks, fault diagnosis, fault-tolerant controls, and nonlinear controls. [2] Isidori A, Byrnes C I. Output regulation of nonlinear systems. IEEE Transactions on Automatic Control, 99, 352: 3 4 [22] Li R R, Khalil H. Nonlinear output regulation with adaptive conditional servocompensator. Automatica, 22, 48: [23] Xiang J, Wei W, Li Y J. Synchronized output regulation of networked linear systems. IEEE Transactions on Automatic Control, 29, 546: Junyi Wang Ph. D. candidate at the School of Information Science and Engineering, Northeastern University. His research interest covers complex networks, neural networks, and network control.