H CONTROL OF NETWORKED MULTI-AGENT SYSTEMS

Transcription

1 Jrl Syst Sci & Complexity (2009) 22: H CONTROL OF NETWORKED MULTI-AGENT SYSTEMS Zhongkui LI Zhisheng DUAN Lin HUANG Received: 17 April 2008 c 2009 Springer Science Business Media, LLC Abstract This paper concerns the disturbance rejection problem arising in the coordination control of a group of autonomous agents subject to external disturbances. The agent network is said to possess a desired level of disturbance rejection, if the H norm of its transfer function matrix from the disturbance to the controlled output is satisfactorily small. Undirected graph is used to represent the information flow topology among agents. It is shown that the disturbance rejection problem of an agent network can be solved by analyzing the H control problem of a set of independent systems whose dimensions are equal to that of a single node. An interesting result is that the disturbance rejection ability of the whole agent network coupled via feedback of merely relative measurements between agents will never be better than that of an isolated agent. To improve this, local feedback injections are applied to a small fraction of the agents in the network. Some criteria for possible performance improvement are derived in terms of linear matrix inequalities. Finally, extensions to the case when communication time delays exist are also discussed. Key words Coordination control, graph theory, H control, linear matrix inequality (LMI), pinning control. 1 Introduction Coordination control of a group of agents has received compelling attentions from scientific community. An agent can represent a spaceship, a ground/underwater vehicle, an internet router, a cellular phone, or even a smart sensor with microprocessors. Typical applications of coordinated multi-agent systems include satellite clusters [1, automated highway systems [2, air traffic control [3, sensor networks [4, and so forth. One unique feather of coordination control is that the agents are physically decoupled, whereas are coupled via exchanging information with each other when trying to accomplish certain assigned tasks. Besides, only limited information about the other agents is available to each agent, thereby causing centralized controllers unrealistic. There is a rich body of literature on the coordination control. Fax and Murray [5 6 studied the formation control problem of autonomous vehicles by combining methods from the graph theory and the control theory. A Nyquist type criterion is derived, which determines the effect Zhongkui LI Zhisheng DUAN Lin HUANG State Key Lab for Turbulence and Complex Systems and Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing , China. zhongkli@pku.edu.cn; duanzs@pku.edu.cn; hl35hj75@pku.edu.cn. This research is supported by the Natural Science Foundation of China under Grants Nos and

2 36 ZHONGKUI LI ZHISHENG DUAN LIN HUANG of the communication topology on the formation stability by analyzing the eigenvalues of the graph Laplacian matrix. Lafferriere et al. [7 further investigated the graph theoretic method. Gupta et al. [8 considered the synthesis of LQG control law of a network of dynamical agents when the control additionally constrained to lie in a particular space. Vicsek et al. [9 addressed the alignment problem of a group of self-driven particles all moving in the plane with the same speed but different heading. Jadbabaie et al. [10 provided a theoretical explanations for the behavior observed by Vicsek et al. in [9. It is shown that the headings of all agents will converge to a steady-state value if the union of a collection of graphs is connected frequently enough. Olfati-Saber [11 proposed a theoretical framework for analysis and design of distributed flocking algorithms. Olfati-Saber and Murray [12 examined a general framework of consensus problem for networks of dynamic agents with fixed/switching topology and communication time-delays. They showed that average consensus was achieved if the communication topology was strongly connected and balanced. Ren and Beard [13 relaxed the conditions given by Olfati- Saber and Murray in [12 by showing that consensus can be achieved asymptotically if and only if the communication graph has spanning tree. This work addresses the disturbance rejection problem arising in the coordination control of a group of autonomous agents subject to external disturbances. The agent network is said to possess a desired level of disturbance rejection, if the H norm of its transfer function matrix from the disturbance to the controlled output is satisfactorily small. Undirected graph is used to represent the information flow topology among agents. It is shown that the disturbance rejection problem of an agent network can be solved by analyzing the H control problem of a set of independent systems whose dimensions are equal to that of a single node. An interesting result is that the disturbance rejection ability of the whole agent network coupled via feedback of merely relative measurements between agents will never be better than that of an isolated agent. To improve this, pinning control technique, i.e., applying local control injections only to a small fraction of nodes, is employed. The advantage of pinning control strategy over the decentralized control method used by Fax and Murray [5 6 is that the structure of the former is simpler, with less variables to be designed, thus is more attractive especially when the number of agents is large. The rest of this paper is organized as follows. The disturbance rejection problem of a group of agents is formulated in Section 2. In Section 3 the disturbance rejection level of the agent network coupled via feedback of relative output measurements between agents is analyzed. Pinning control strategy is applied to improve the disturbance rejection ability in Section 4. Extensions to the case when communication time delays exist are discussed in Section 5. In Section 6 numerical simulation examples are provided to verify the effectiveness of the obtained results. Finally, the paper is concluded in Section 7. Throughout this paper, for real symmetric matrices X and Y, the notation X > Y means that matrix X Y is positive definite. R n n are the set of n n real matrices. The superscript T means the transpose for real matrices and H the conjugate transpose for complex matrices. diag(a 1, A 2,, A n ) denotes a block-diagonal matrix with matrices A i, i 1, 2,, n, on its diagonal. X Y denotes the Kronecker product of matrices X and Y. I N means the identity matrix of dimension N, and I the identity matrix of appropriate dimension. Matrices, if not explicitly stated, are assumed to have compatible dimensions. 2 Problem Formulation Consider a group of N identical agents, with each agent being a linear dynamical system subject to external disturbances:

3 H CONTROL OF NETWORKED MULTI-AGENT SYSTEMS 37 x i Ax i B 1 ω i B 2 u i, z i C 1 x i D 11 ω i, y i C 2 x i D 21 ω i, i 1, 2,, N, where x i R n is the state, u i R m 1 is the control input, w i R m 2 is the external disturbance, y i R p is the measured output, and z i R q is the controlled output of agent i. It is assumed that (A, B 2 ) is stabilizable, (A, C 2 ) is detectable, and without loss of generality, B 2 is of full column rank. The topology of information exchange between the agents in the network is represented by an undirected graph G (V, E), where V {1, 2,, N} is the set of nodes (i.e., agents), and E V V is the set of unordered pairs of nodes, called edges (i.e., communication links among agents). The graph is assumed here to be simple, i.e., it has neither self-loops nor repeated edges. A graph with either self-loops or repeated edges is called a multigraph. If there exists an edge between two nodes of graph G, then the two nodes are called adjacent, or neighboring. The set of neighbors of node α i is defined as N i {α j V : (α i, α j ) E}, and its degree as N i. A path from vertex α 0 to α l is a sequence of ordered edges of the form (α i, α i1 ), i 0, 1,, l 1. The information graph is called connected if there exists a path between every pair of distinct nodes, otherwise disconnected. A dynamical coordinating law based on the relative output measurements between agents is proposed as follows: v i A c v i B c (y i y j ), (2) u i C c v i D c (y i y j ), where v i R n c, n c is a preassigned dimension of the coordinating law. If n c 0, (2) is reduced to be a static output coordinating law. Define the system matrix K of (2) as [ Ac B K c. (3) C c D c Let x [x 1, x 2,, x N T, v [v 1, v 2,, v N T, ω [ω1 T, ω2 T,, ωn T T, and z [z 1, z 2,, z N. Then, by stacking up the individual dynamics in (1), the closed-loop network dynamics can be written as: [ẋ [ [ [ IN A L B 2 D c C 2 I N B 2 C c x IN B 1 L B 2 D c D 21 ω, v L B c C 2 I N A c v L B c D 21 z [ I N C 1 0 [ x [ I v N D 11 ω, (1) (4) where L (L ij ) N N is the Laplacian matrix of the communication graph G, defined by 1, i j, j N i, L ij 0, i j, j N i, N i, i j. (5) Some results concerning the spectrum of L are stated as follows [14 15 : i) Zero is an eigenvalue of L with eigenvector [1, 1,, 1 T. Further, zero is a simple eigenvalue, if the graph is

4 38 ZHONGKUI LI ZHISHENG DUAN LIN HUANG connected; ii) All the eigenvalues of L are real, and lie within the interval [0, 2β on the real coordinate, where β denotes the largest node degree of G. Assume, hereafter, that graph G is connected. Then, zero is a simple eigenvalue of L, and all the other eigenvalues are strictly positive. The intention of this paper is to investigate the disturbance rejection ability of network (4). The H norm will be used as the measure to quantify the disturbance rejection level of the network. A system is said to possess a desired level of disturbance rejection, if the H norm of its transfer function matrix from the disturbance to the controlled output is satisfactorily small. Denote by T wz the transfer function matrix from ω to z of system (4). Then, the robustness to disturbance problem of system (4) can be stated as follows: for a given γ > 0, find appropriate coordinating law in the form of (2), such that i) network (4) is asymptotically stable; ii) T wz < γ, where T wz denotes the H norm of T wz. 3 H Performance Analysis Let 0 λ 1 < λ 2 λ N be the eigenvalues of the Laplacian matrix L. Since L is symmetric, there exists a unitary matrix U such that U 1 LU Λ diag(λ 1, λ 2,, λ N ). Let x (U I n )ξ, and v (U I nc )ϑ, with ξ [ξ 1, ξ 2,, ξ N T, ϑ [ϑ 1, ϑ 2,, ϑ N T. Then, network (4) can be rewritten as: [ [ [ [ ξ IN A Λ B 2 D c C 2 I N B 2 C c ξ U 1 B 1 U 1 L B 2 D c D 21 ϑ Λ B c C 2 I N A c ϑ U 1 ω, L B c D 21 z [ U C 1 0 [ (6) ξ [ I ϑ N D 11 ω. Further, reformulate the disturbance variable w and the performance variable z via w (U I m1 ) w, z (U I m2 ) z, where w [ w 1, w 2,, w N, z [ z 1, z 2,, z N. Then, substituting (7) into (6) gives [ [ [ [ ξ IN A Λ B 2 D c C 2 I N B 2 C c ξ IN B 1 Λ B 2 D c D 21 ω, ϑ Λ B c C 2 I N A c ϑ Λ B c D 21 z [ I N C 1 0 [ (8) ξ [ I ϑ N D 11 ω. (7) It is worth noting that (8) is composed of the following N isolated systems: [ [ [ [ ξi A λi B 2 D c C 2 B 2 C c ξi B1 λ i B 2 D c D 21 ω i, ϑ i λ i B c C 2 A c ϑ i λ i B c D 21 z i [ C 1 0 [ ξ i D ϑ 11 ω i, i 1, 2,, N. i (9) Denote by T w z and T wi z i the transfer function matrices of systems (8) and (9), respectively. Then, it follows from (6) (9) that T ω z diag(t ω1 z 1, T ω2 z 2,, T ωn z N ) (U 1 I m2 )T ωz (U I m1 ). (10)

5 H CONTROL OF NETWORKED MULTI-AGENT SYSTEMS 39 Recall that the H norm of a transfer function matrix F (s) is defined by [16 F (s) sup σ(f (jw)), w R where σ(f ) represents the maximal singular value of F. By observing that both U I m1 and U I m2 are unitary matrices, one obtains the relationships between the H norms of T wz, T w z, T wi z i from (10) as follows: T ω z T ωz max i1,2,,n T ω i z i. (11) The above discussions are summarized in the following theorem. Theorem 1 For a given scalar γ > 0, network (4) is asymptotically stable and T wz < γ, if and only if there exists coordinating law in the form of (2) such that all the systems in (9) are simultaneously asymptotically stable, and satisfy T wi z i < γ, for i 1, 2,, N. Remark 1 When the number N of agents is large, the stability and disturbance rejection level of the agent network is generally very hard to determine. The above theorem relaxes this difficulty significantly by converting the disturbance rejection problem of the network into those of a set of independent systems whose dimensions are the same as that of an isolated agent. The key tools leading to this result rely heavily on the input and output transformations, which meanwhile preserve the H norm of network (4). The number of systems to be verified is equal to the number of distinct eigenvalues of the Laplacian matrix L, sometimes much less than N. To name one, for the star-shaped graph, in which a node is adjacent to the N 1 others, only three systems need to be tested, since in this case L has only three distinct eigenvalues: N, 1, and 0. Remark 2 It deserves noticing that the system in (9) corresponding to λ 1 0 is [ [ [ [ ξ1 A B2 C c ξ1 B1 ω 1, ϑ 1 0 A c ϑ 1 0 z 1 [ C 1 0 [ (12) ξ 1 D ϑ 11 ω 1, 1 whose transfer function T w1 z 1 is the same as that of an isolated agent. Therefore, Theorem 1 yields an interesting result that the H norm of the whole agent network (4) is bigger than or equal to that of an isolated agent, implying that the disturbance rejection level of the network coupled via (2) does not become any better, as compared to that of an isolated agent. Remark 3 In the above, the H performance of agent network (4) with respect to external disturbances is concerned. Here, consider another case when the H 2 norm is taken as the performance specification for the agent network. It is known that the H 2 norm can be taken as a measure of the expected root mean square value of the output in response to white noise excitation. In this case, D 11 in (1) needs to be zero in order to guarantee the existence of H 2 norms. The H 2 norm of transfer function F (s) is defined as [16 Then, it follows from (10) that F (s) π T ωz (s) 2 2 T ω z 2 2 tr(f (jω) H F (jω))dω. N T ωi z i 2 2, (13) i1

6 40 ZHONGKUI LI ZHISHENG DUAN LIN HUANG which states that the square of the H 2 norm of network (4) is equal to the sum of those of all the N systems in (9). As the system in (9) corresponding to λ 1 0 is the same as a single agent, one obtains that the H 2 performance of the agent network coupled via relative measurements between agents is always worse than that of an isolated agent. Comparing to what obtained in Remark 2 yields that the couplings in the form of (2) is more deleterious in the sense of H 2 norm than in the sense of H norm. The reason leading to this lies essentially in the definitions of H 2 and H norms. To be specific, roughly speaking, the H norm depicts the peak of the largest singular value, whereas the H 2 norm is the average of all singular values over all frequencies. 4 H Pinning Control In the above section, it was shown that the disturbance rejection level of the agent network (4) under the coordinating law (2) is not any better than that of an isolated agent. This is improved in this section by applying self feedbacks to a limited fraction of the agents, i.e., inserting terms representing the absolute measured outputs of agents into the coordinating laws corresponding to the controlled agents. Such a control action is essentially the pinning control strategy, which has wide applications in chaos control, synchronization, and control of complex dynamical networks [ Without loss of generality, we assume that the first τ agents are to be pinned. Then, the new coordinating law is given by v i A c v i B c (y i y j ) d i B c y i, (14) u i C c v i D c (y i y j ) d i D c y i, where d i is the feedback gains to be determined, satisfying d i > 0, i {1, 2,, τ}; d i 0, i {τ 1, τ 2,, N}. (15) Let D diag(d 1, d 2,, d N ), x [x T 1, x T 2,, x T N T, z [z1 T, z2 T,, zn T T, and w [w1 T, w2 T,, wn T T. Then, the closed-loop system in this case can be rewritten in the following form as did in the last section: [ẋ [ IN A L [ [ B 2 D c C 2 I N B 2 C c x IN B 1 L B 2 D c D 21 ω, v L B c C 2 I N A c v L B c D 21 z [ I N C 1 0 [ (16) x [ I v N D 11 ω, where L is a matrix defined by L L D. Lemma 1 [19 20 Let M (M ij ) N N be a symmetric matrix satisfying M ij 0, i j, and N j1 M ij 0, for i 1, 2,, N, and R diag(r 1, r 2,, r N ) be a nonzero matrix with r i 0, i 1, 2,, N. Then, all the eigenvalues of matrix M R are negative. From Lemma 1 and the definitions of matrices L and D, it is obvious that L L D is positive definite. Let 0 < λ 1 λ 2 λ N be the eigenvalues of L. Denote by T wz the transfer function matrix from w to z of network (16). By following the same steps in deriving Theorem 1, one can obtain the following result.

7 H CONTROL OF NETWORKED MULTI-AGENT SYSTEMS 41 Theorem 2 For a given scalar γ > 0, network (16) is asymptotically stable and T wz < γ, if and only if there exists some coordinating law in the form of (14) such that the following N systems are simultaneously asymptotically stable and the H norms of their transfer function matrices are all less than γ: x i Ãi x i B i ω i, z i C x i D 11 ω i, i 1, 2,, N, (17) where x i R nnc, Ã i A B 2 KH i B i B 1 B 2 KD i [ A [ 0 B2 K I nc 0 [ 0 Inc λ i C 2 0, [ [ B1 0 [ B 0 2 K, C C C1 0. λ i D 21 (18) Remark 4 The pinning control technique applied here can be reinterpreted by means of graph theory. Given a group of agents, whose information flow topology is denoted by graph G. If agent i is pinned with feedback gain d i > 0, then a self-loop weighted by d i is formed around node i of G. The resulting graph G is no longer simple, instead turns to a special multigraph containing self-loops. Correspondingly, the Laplacian matrix of G can be obtained by adding d i onto the (i, i) element of the Laplacian matrix of G, as the degree of node i in this case becomes N i d i, which is possibly not integer, depending on d i. In light of Lemma 1, the Laplician matrix of G is positive definite, thus possibly improving the disturbance rejection level of network (16) for certain coordinating law in the form of (14). It is easy to see that one may even achieve this by pinning only one single agent in the network. For the disturbance rejection problem of multiple agents considered by this paper, an interesting observation is that the multigraph with self-loops is advantageous over the simple graph, implying possible further applications to other problems arising in the multi-agent systems. Remark 5 The coordinating law proposed for the stabilization problem by Fax and Murray [6 is as follows: v i K A v i K B1 y i K B2 (y i y j ), u i K C v i K D1 y i K D2 (y i y j ). One can see that (19) is more complex, with more variables to be designed, as apposed to (14). The reason leading to this is that (19) acts self feedbacks to all the agents in the network, thereby containing too much information, inadequate in fully exploiting the favorable effects of the couplings among agents. In what follows, some sufficient conditions in terms of LMIs are derived for the existence of the coordinating law (14) in Theorem 2. Before moving further, the next lemma is introduced, which gives another representation of the Bounded Real Lemma (BRL) [16. Lemma 2 [16,21 22 Consider system (1) with u i 0, whose transfer function matrix from ω i to z i is denoted by T ωi z i. Then, T ωi z i < γ, where γ > 0 is a prescribed value, if and only if there exist matrices P P T > 0, F, and G such that G T G P F T GA GB 1 0 P F A T G T A T F T F A F B 1 C T 1 B1 T G T B1 T F T γi D11 T < 0. (20) 0 C 1 D 11 γi (19)

8 42 ZHONGKUI LI ZHISHENG DUAN LIN HUANG By introducing two slack variables, F and G, the Lyapunov matrix P in LMI (20) is separated from the dynamical matrices of system (1). The introduced matrices F and G are even not restricted to be symmetric, thus providing extra degrees of freedom for the design. Theorem 3 For a given scalar γ > 0, network (16) is asymptotically stable and satisfies T wz < γ, if for certain nonzero scalars ε i, κ i, there exist matrices P i > 0, F i, G i, V, with [ [ [ εi F 11 F12 i κi F 11 G i 12 V1 F i 0 F22 i, G i 0 G i, V, (21) 22 0 satisfying the following LMIs for i 1, N: where G i T T T G T i Ψ 1 G i T B 1 κ i V D i 0 Ψ1 T Ψ 2 F i T B 1 ε i V D i C T B1 T T T G T i κ i Di TV T B T 1 F T i ε i Di TV T γi D11 T < 0, (22) 0 C D 11 γi Ψ 1 P i T T F T i G i T A κ i V H i, Ψ 2 T T A T F T i ε i H T i V T F i T A ε i V H i, and T is a nonsingular matrix such that T B 2 [I 0 T. If (22) holds for i 1, N, then a feasible coordinating law is given by K F11 1 V 1. (23) Proof By the definitions of V and (23), one can obtain [ [ [ [ V1 F11 K F11 I V K, (24) where denotes items that don t matter. Then, it follows from the definitions of F i, G i, and T that F i T B 2 K ε i V, G i T B 2 K κ i V. (25) From Theorem 2 and Lemma 2, system (16) is asymptotically stable and satisfies T wz < γ, if and only if there exist matrices P i > 0, F i, and G i such that G T i G i P i Fi T G i à i G i Bi 0 P i F i ÃT i GT i à T i F T i F i à i F i Bi CT B i TGT i B i TF i T γi D11 T < 0, i 1, 2,, N. (26) 0 C D11 γi By letting F i F i T, G i G i T, substituting Ãi A B 2 KH i, Bi B 1 B 2 KD i, C C into (26), and invoking (25), one obtains after some algebraic calculations that (22) hold for i 1, 2,, N. The above result can be further refined. It is known that each eigenvalue of L can be represented by a convex combination of the largest eigenvalue λ N and the smallest eigenvalue λ 1, and (22) are linear with respect to variables P i, V, F i, G i, and γ, for any given ε i and κ i. Similarly, the N 2 LMIs in (22) for i 2, 3,, N 1, can be written as convex combinations

9 H CONTROL OF NETWORKED MULTI-AGENT SYSTEMS 43 of the two LMIs in (22) corresponding to λ N and λ 1. Therefore, the feasibility of N LMIs in (22) for i 1, 2,, N, can be reduced to the feasibility of the two LMIs in (22) for i 1, N. Furthermore, if (22) are feasible for i 1, N, then G i T T T G T i > 0, implying that G i, and implicitly F 11 are nonsingular, as T is invertible. Therefore, one obtains a feasible dynamical coordinating law as (23). Remark 6 The key technique leading to Theorem 3 is replacing general matrices F i and G i by F i T, G i T, respectively, with F i and G i having the special forms in (21). On the other hand, such a replacing in turn results in the main conservatism of the condition in Theorem 3. Scalars ε i and κ i are introduced so as to provide extra searching dimensions in the solution space, thereby can reduce the conservatism for certain degree. Obviously, ε 1 can be set to be 1 without involving any conservatism. For a given B 2, the transformation matrix T satisfying T B 2 [I 0 T is not unique. However, it can be shown that the feasibility of the condition in Theorem 3 is independent of the choices of T [23. A special T is given by T, where B 2 denotes the matrix with maximal row rank such that B 2 B Extension to Communication with Time Delay [ (B T 2 B 2) 1 B T 2 B 2 In practice, the agents are spatially located. Thus, the information exchanges among them have to be transmitted through certain communication media, whose limited transmission speed and bandwidth may introduce time delays into the information flow among agents. One extends above sections by considering the case when there exist time delays in the coordinating protocols. For illustration, assume that the time delays are uniform, i.e., the same on every communication link. In this case, the coordinating law with time delays is proposed as v i A c v i B c (y i (t h) y j (t h)), (27) u i C c v i D c (y i (t h) y j (t h)), where v i R nc, n c is a preassigned dimension of the coordinating law, and h is the time delay satisfying 0 < h h. As did in Section 3, by letting x [x 1, x 2,, x N T, v [v 1, v 2,, v N T, ω [ω1 T, ω2 T,, ωn T T, and z [z 1, z 2,, z N, one obtains the closed-loop dynamics of the agent network as follows: [ẋ [ [ [ [ IN A I N B 2 C c x L B2 D c C 2 0 x(t h) v 0 I N A c v L B c C 2 0 v(t h) [ IN B 1 L B 2 D c D 21 ω, (28) L B c D 21 z [ I N C 1 0 [ x v [ I N D 11 ω, where L is the Laplacian matrix as defined by (5). Denote by T wz the transfer function matrix from w to z of network (28). The following Theorem presents a time-delay version of Theorem 1. Theorem 4 For a given scalar γ > 0, time-delay network (28) is asymptotically stable and T wz < γ, if and only if there exist coordinating law in the form of (27) such that

10 44 ZHONGKUI LI ZHISHENG DUAN LIN HUANG the following N systems are simultaneously asymptotically stable and the H norms of their transfer function matrices are all less than γ: [ [ [ [ [ [ ξi A B2 C c ξi λi B 2 D c C 2 0 ξi (t h) B1 λ i B 2 D c D 21 ω i, ϑ i 0 A c ϑ i λ i B c C 2 0 ϑ i (t h) λ i B c D 21 z i [ C 1 0 [ (29) ξ i D ϑ 11 ω i, i 1, 2,, N, i where ξ i R n, ϑ i R nc, and λ i, i 1,, N, are the eigenvalues of L. Remark 7 Note that the time delay existing in (27) does not affect the system in (29) corresponding to λ 1 0, whose transfer function is also the same as that of an isolated agent. Thus, one obtains a similar observation as in Section 3, that is, the H norm of the time-delay agent network (28) is equal to or even bigger than that of an isolated agent. On the other hand, the communication time delay may degrade the disturbance rejection level of network (28), by rendering some of H norms of the other N 1 systems larger than that of an isolated agent. In what follows, one will possibly improve the disturbance rejection level of network (28) by including terms representing the absolute measured outputs of agents into the coordinating law, similar to what did in Section 4. What is unlike Section 4 is that the pinning control strategy will not work out here due to the communication time delay. One proposes the following new coordinating law: v i K A v i K B1 y i K B2 (y i (t h) y j (t h)), (30) u i K C v i K D1 y i K D2 (y i (t h) y j (t h)). Then, the controlled network under (30) can be written as [ẋ [ [ [ [ IN A I N B 2 K D1 C 2 I N B 2 K C x L B2 K D2 C 2 0 x(t h) v I N K B1 C 2 I N K A v L K B2 C 2 0 v(t h) [ IN B 1 I N B 2 K D1 D 21 L B 2 K D2 D 21 ω, I N K B1 D 21 L K B2 D 21 z [ I N C 1 0 [ x [ I v N D 11 ω. (31) Denote by T wz the transfer function matrix from w to z of network (28). By following the steps in deriving Theorem 1, one can obtain the following result. Theorem 5 For a given scalar γ > 0, network (31) is asymptotically stable and T wz < γ, if and only if there exist some coordinating law in the form of (30) such that the following N systems are simultaneously asymptotically stable and the H norms of their transfer function matrices are all less than γ: where ζ i R nn c, ζ i Ăζ i Ăhζ i (t h) B i ω i, z i Cζ i D 11 ω i, i 1, 2,, N, (32)

11 H CONTROL OF NETWORKED MULTI-AGENT SYSTEMS 45 [ A 0 Ă A B 2 KR 0 0 [ [ 0 B2 KA K B1 K B2 I nc 0 K C K D1 K D2 Ă h B 2 KSi B 2 K , C C1 [ C 1 0, λ i C 2 0 [ 0 B1 B i B 1 B 2 KEi B 0 2 K D 21. λ i D 21 0 I n c C , (33) Before proceeding further, the next lemma is introduced, which gives a delay-dependent BRL for system (32). Lemma 3 Consider system (32), whose transfer function matrix from w i to z i is denoted by T wi z i. Then, for any h satisfying 0 < h h, system (32) is asymptotically stable and satisfies T wiz i < γ, if there exist matrices P i > 0, Q i > 0, W i > 0, F i, G i, H i, X i, Y i, Z i such that where Ω 1 Ω 2 Ω 3 F i Bi hx i CT Ω 2 T Ω 4 G i Z T i ĂT h HT i G i Bi hy i 0 Ω3 T G T i Z i H i Ă h hw i H i Hi T H i Bi hz i 0 B i T F i T B i TGT i B < 0, (34) 1 T Hi T γ 2 I 0 D11 T hxi T hyi T hzi T 0 hw i 0 C D 11 I Ω 1 Q i X i X T i F i Ă ĂT F T i, Ω 2 X i Y T i F i Ă h ĂT G T i, Ω 3 P i F i Z T i ĂT H T i, Ω 4 Q i Y i Y T i G i Ă h ĂT h G T i. Proof It can be shown easily on the basis of the result given by Gao et al. [24, and is omitted for brevity. Theorem 6 For a given scalar γ > 0, network (31) is asymptotically stable and T wz < γ, if there exist scalars κ i 1, κ i 2, κ3, i matrices P i > 0, Q i > 0, W i > 0, X i, Y i, Z i, Fi, Ğ i, Hi, V, with [ [ [ [ κ i F i 1 F 11 F12 i κ i 0 F22 i, Ğ i 2 F 11 G i 12 κ i 0 G i, Hi 3 F 11 H12 i V H22 i, V, 0 satisfying the following LMIs for i 1, N: Θ 11 Θ 12 Θ 13 Θ 14 hx i C1 T Θ T 12 Θ 22 Θ 23 Θ 24 hy i 0 Θ13 T Θ23 T Θ 33 Θ 34 hz i 0 Θ14 T Θ24 T Θ34 T γ 2 < 0, (35) I 0 DT i hxi T hyi T hzi T 0 hw i 0 C Di I

12 46 ZHONGKUI LI ZHISHENG DUAN LIN HUANG where Θ 11 Q i X i X T i F i T A A T T T F T i κ i 1V R κ i 1R T V T, Θ 12 X i Y T i κ i 1V S i A T T T F T i κ i 2R T V T, Θ 13 P i F i T Z T i A T T T HT i κ i 3R T V T, Θ 14 F i T B 1 κ i 1V E i, Θ 22 Q i Y i Y T i κ i 1V S i κ i 2S T i V T, Θ 23 ĞiT Z T i κ i 3S T i V T, Θ 24 ĞiT B 1 κ i 2V E i, Θ 33 hw i H i T T T HT i, Θ 34 H i T B 1 κ i 3V E i, and T is a nonsingular matrix such that T B 2 [I 0 T. If (35) holds for i 1, N, then the coordinating law is given by K F11 1 V 1. Proof It can be shown by following the steps in deriving Theorem 3, and is omitted here. 6 Simulation Examples In this section, some simulation examples will be provided to demonstrate the effectiveness of the conditions derived in above sections. Example 1 Consider a group of 8 agents, described by (1) with [ 0 5 A, B [ 4 2, B 2 [ , C 1 [ 2 0, C 2 [ 1 0.2, D , D (36) The H norm of an isolated agent with u i 0 is 6.8. The information graph G and the corresponding Laplician matrix L are defined in Fig L Figure 1 The information graph G and Laplacian matrix L The objective is to stabilize this agent network and meanwhile render the disturbance rejection level γ to be less than 2 by designing a coordinating law in the form of (14). For simplicity, (14) is assumed to be static, i.e., n c 0, with K as the feedback gain. Consider here two different pinning control schemes: to pin the node with the biggest degree, i.e., node 1, and to pin the node with the smallest degree, i.e., node 2. To pin node 1, choose d 1 1. As explained in Remark 4, this is equivalent to adding the dashed self-loop weighted by 1 around node 1 of graph G. The resulting multigraph is Ĝ, whose Laplacian matrix is defined as L Ldiag(1, 0,, 0). The smallest and largest eigenvalues of L are 0.11 and , respectively. The transformation matrix T is taken as T [ such that T B 2 [1 0 T. Let γ 2, ε 1 ε N κ 1 κ N 1 in Theorem 3. By employing the

13 H CONTROL OF NETWORKED MULTI-AGENT SYSTEMS 47 LMI Toolbox of MATLAB [25, one can find a feasible feedback gain as K , along with the following matrices: [ [ F 1, F N, [ [ G 1, G N The disturbance rejection level of the network with K is , which is less than γ 2. To pin node 2, choose d 2 1. Similarly, this is equivalent to adding the dotted self-loop weighted by 1 around node 2 of graph G. The Laplacian matrix of the resulting multi-graph is L L diag(0, 1, 0,, 0), whose smallest and largest eigenvalues are and , respectively. Let the feedback gain be as before, i.e., K Then, the disturbance rejection level of the network is in this case, which is also less than γ 2 but larger than the one obtained above. Comparing the above two pinning schemes, one can see that the agent network yields a better level of disturbance rejection by applying pinning control strategy to the nodes with higher degrees than to those with lower degrees. This is in accordance with the intuition that the big nodes with higher degrees play a more important role in stabilizing the network and in attenuating external disturbances. Example 2 In this example, communication time delays will be taken into consideration. The agent dynamics are the same as in Example 1, and the communication graph is defined in Fig. 1. The largest eigenvalue of the corresponding Laplacian matrix is It is desired that by designing a static coordinating law of the form (30), the agent network will be stabilized and the disturbance rejection level γ less than 2. Assume that the time delay in (30) satisfies 0 < h 0.2. Let γ 2, κ 1 1 κ N 1 κ 1 2 κ N 2 κ 1 3 κ N 3 1 in Theorem 6. By using the LMI Toolbox, one can find a feasible feedback gain of the self feedbacks as K and the feedback gain of delayed couplings as K d , along with the following matrices: [ [ [ F 1, F N, G 1, G N 7 Conclusion [ , H , H N [ [ In this paper, the disturbance rejection problem of a group of autonomous agents perturbed by external disturbances has been addressed. It is shown that the disturbance rejection problem of a high-dimensional agent network can be solved by converting it into the H control problem of a set of independent systems with dimensions equal to that of a single agent. A surprising result is that the disturbance rejection level of the whole network coupled via merely feedbacks of relative measurements between each other will never be better than that of an isolated agent. To improve this, pinning control strategy is applied to a small fraction of agents in the network. The pinning control strategy is further reinterpreted in the terms of graph theory. It is then shown, via a simulation example, that the network will yield a better level of disturbance rejection by pinning the nodes with higher degrees than pinning those with lower degrees. It implies that the big nodes with higher degrees play a more important role in stabilizing

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