- Email: [email protected]
Output consensus for heterogeneous multi-agent systems with linear dynamics
Applied Mathematics and Computation 271 (2015) 548–555
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Output consensus for heterogeneous multi-agent systems with linear dynamics Qian Ma a,∗, Guoying Miao b a b
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, PR China School of Information and Control, Nanjing University of Information Science and Technology, Nanjing 210044, PR China
a r t i c l e
i n f o
Keywords: Multi-agent systems Output consensus Heterogenous networks Regulation equation
a b s t r a c t This paper deals with output consensus problem of heterogeneous multi-agent systems. The cases of leaderless and leader-following are considered. For each case, a dynamic consensus protocol is proposed, in which the parameters are dependent on the solution of a regulation equation and a homogeneous system. The homogeneous system can be analyzed by fulldeveloped technique in homogenous multi-agent systems. Furthermore, dynamic regulators based on the state observers also are presented which is suitable for the case that the system states cannot be obtained. Simulation examples are provided finally to demonstrate the effectiveness of the proposed design methods. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The consensus problem as one of the most fundamental research topics in the field of coordination control of multi-agent systems has attracted considerable attention over the past few years due to its extensive applications in cooperative control of mobile autonomous robots, the design of distributed sensor networks, spacecraft formation flying, and other areas. Much of the attention has been devoted to achieving state consensus in homogeneous networks, i.e., networks where the agent models are identical and with the same state dimensions. The consensus problems of homogeneous multi-agent systems, such as consensus of systems with second-order dynamics [1–4] and high-order dynamics [5,6], distributed containment control [7–9], consensus of agents with time-delay [10,11], and consensus with switching topology [12–14], just to mention a few, have been intensively studied recently. In practical applications, it is often impossible to require all agents are identical. Therefore, the difference between the agents which can be called the heterogeneity cannot be neglected in the design of consensus protocol. In this case, the individual agents are not identical and in particular the state dimensions may be different, then state consensus among all agents cannot be achieved. Therefore, the objective is to realize output consensus. This problem is challenging, and there are few available results on output consensus of heterogeneous system can be found, see [15–22] for instance. In [15], leader-following consensus of heterogeneous agents with nonlinear intrinsic dynamics was addressed with a fuzzy disturbance observer. In [16,17], it has been shown that the agents have to possess an internal model of the consensus trajectory. Also based on the internal model principle, a constructive method for designing local dynamic controllers with relative information was proposed in [18]. In [19,20], by using of the notion of system inclusion and system intersection, the agents can be synchronized by an appropriate networked controller. By means of state transformation, the new agents in [21] were almost identical except for different exponentially ∗
Corresponding author. Tel.: +86 13951769356. E-mail address: [email protected] (Q. Ma).
http://dx.doi.org/10.1016/j.amc.2015.08.117 0096-3003/© 2015 Elsevier Inc. All rights reserved.
Q. Ma, G. Miao / Applied Mathematics and Computation 271 (2015) 548–555
549
decaying signals and then a decentralized controller was designed. A similar state transformation approach was taken in [22], in which the output synchronization problem for a heterogeneous network of non-introspective agents was considered. In this paper, we address output consensus problem of heterogeneous multi-agent systems. The cases of leaderless and leaderfollowing are considered. For each case, a dynamic consensus protocol is proposed, in which the parameters are dependent on the solution of a regulation equation and a homogeneous system. The homogeneous system can be analyzed by full-developed technique in homogenous multi-agent systems. And it can be seen that there is a close link with the parameters in homogeneous system and the eigenvalue of topology matrix. Furthermore, dynamic regulators based on the state observers also are presented which is suitable for the case that the system states are not accessible. Notations. Throughout this paper, for real symmetric matrices X and Y, the notation X ≥ Y (respectively, X > Y) means that the matrix X − Y is positive semi-definite (respectively, positive definite). I denotes an identity matrix of appropriate dimension. 1N ∈ RN be the vector with all entries being 1. The notation ‘∗’ is used as an ellipsis for terms that are induced by symmetry. (X) denotes the set of the eigenvalues of X and + (X ) the set of the eigenvalues with positive real part of X. The Kronecker product of matrices X and Y is denoted as X⊗Y. (x) represent the real parts of a complex number x. 2. Preliminaries Let G = {V, E, A} be a weighted directed graph with the set of nodes V = {1, 2, . . . , N}, the set of directed edges E ⊆ V × V, and a weighted adjacency matrix A = (ai j )N×N . A directed edge eij in network G is denoted by the ordered pair of nodes (i, j), meaning that node j can receive information from node i. The elements of the adjacency matrix A are defined as aij > 0 if and only if there is a directed edge (j, i) in G; otherwise, ai j = 0. A directed path is a sequence of nodes 1, 2, … , r such that (i, i + 1) ∈ E, i ∈ {1, 2, . . . , r − 1}. A directed graph is strongly connected if there is a directed path for any two distinct nodes j and i. A directed graph has a directed spanning tree if there exists at least one node called root node which has a directed path to all the other nodes. The Laplacian matrix L = (li j )N×N of graph G is defined as N
lii = −
li j ,
j=1, j=i
li j = −ai j ,
i = j; i, j = 1, 2, . . . , N,
which ensures the diffusion property that Nj=1 li j = 0. Consider a heterogeneous multi-agent system given by:
x˙ i = Ai xi + Bi ui , yi = Ci xi , i = 1, 2, . . . , N, R ni ,
Rq ,
(1)
R mi
where xi ∈ yi ∈ and ui ∈ are the state vector, output vector and control input, respectively. Since the every agents are not identical and in particular the state dimensions may be different, state consensus cannot be achieved. Therefore, our objective is to achieve output consensus among all agents. Definition 1. Output consensus in heterogeneous multi-agent system (1) is achieved if for any initial conditions,
lim yi − y j = 0,
t→∞
∀i, j = 1, 2, . . . , N.
3. Output consensus without a leader From [16], if the heterogeneous agents (1) achieve output consensus, then the all individual systems are able to track the same virtual exosystem defined by the dynamics matrix S and the output matrix D, which can be regarded as an internal model of the virtual exosystem. To reach output consensus of the heterogeneous agents, the following protocol is proposed:
ui = Ki (xi − i ηi ) + i ηi ,
η˙ i = Sηi + QF
N
ai j (η j − ηi ), i = 1, . . . , N,
(2)
j=1
where Ki ∈ Rmi ×ni , F ∈ Rk × r and Q ∈ Rr × k are needed to be designed. S ∈ Rr × r represents state of the virtual exosystem. i ∈ Rni ×r and i ∈ Rmi ×r can be obtained by the following regulation equation:
i S = Ai i + Bi i , 0 = Ci i − D.
(3)
Lemma 1. [8,23] Consider formula (2). Suppose that the pair (S, Q) is stabilizable and network G has a directed spanning tree. Let F = max{1, min 1 (λ) }Q T P, where P = P T is the solution to λ∈+ (L)
T
S P + PS + Ir − PQQ T P = 0.
(4)
550
Q. Ma, G. Miao / Applied Mathematics and Computation 271 (2015) 548–555
Then, one has
lim ηi = lim η j = η0 ,
t→∞
t→∞
∀i, j = 1, 2, . . . , N.
(5)
where η0 is a real valued function. Theorem 1. Suppose that (Ai , Bi ) is stabilizable and network G has a directed spanning tree. Then, output consensus in heterogeneous multi-agent system (1) can be achieved under protocol (2), if Ki be such that Ai + Bi Ki is Hurwitz, i and i are a solution of regulation Eq. (3), and F is designed in Lemma 1 with stabilizable pair (S, Q). Proof. Let
εi = xi − i ηi , ε = col (ε1 , ε2 , . . . , εN ), η = col (η1 , η2 , . . . , ηN ). We have
ε˙ i = x˙ i − i η˙ i
= Ai xi + Bi ui − i Sηi + QF
N
ai j (η j − ηi )
j=1
= Ai xi + Bi Ki εi + Bi i ηi − Ai i ηi − Bi i ηi − i QF = (Ai + Bi Ki )εi − i QF
N
N
ai j (η j − ηi )
j=1
ai j (η j − ηi ).
(6)
j=1
Then, one has
ε˙ = (A + BK )ε − blockdiag(i )(L ⊗ QF )η.
(7)
From Lemma 1, it follows that
lim η = 1N ⊗ η0 ,
(8)
t→∞
which implies that
lim blockdiag(i )(L ⊗ QF )η = 0,
t→∞
(9)
Thus, it can be concluded that limt→∞ ε = 0, which then implies that limt→∞ yi = limt→∞ Ci i ηi = Dη0 . This completes the proof. Remark 1. In Theorem 1, a dynamic controller (2) is proposed to guarantee output consensus of the heterogeneous multi-agent system (1). The key point lies in making the state of each heterogenous system approach to a function which involves the state of virtual exosystem and the solution of regulation equation, namely, xi → i ηi . It is worth noting that η˙ i = Sηi + QF Nj=1 ai j (η j − ηi ) is a homogeneous system which can be analyzed by some full-developed techniques in homogenous multi-agent systems. By Lemma 1, it can be concluded that ηi → ηj under the certain conditions. In reality, the states of systems often are not accessible. In this case, it is efficient to provide a dynamic regulator based on the state observers as follows:
ui = Ki (xˆi − i ηi ) + i ηi ,
η˙ i = Sηi + QF
N
ai j (η j − ηi ),
j=1
xˆ˙ i = Ai xˆi + Bi ui − Hi (Ci xi − Ci xˆi ), i = 1, . . . , N,
(10)
where xˆi represents the estimate of the state xi . Theorem 2. Suppose that (Ai , Bi ) is stabilizable, (Ai , Ci ) is detectable, and network G has a directed spanning tree. Then, output consensus in heterogeneous multi-agent system (1) can be achieved under protocol (10), if Ki and Hi be such that Ai + Bi Ki , Ai + HiCi are Hurwitz, i and i are a solution of regulation Eq. (3), and F is designed in Lemma 1 with stabilizable pair (S, Q). Proof. Let
εi = xi − i ηi , ei = xi − xˆi ,
ε = col (ε1 , ε2 , . . . , εN ),
Q. Ma, G. Miao / Applied Mathematics and Computation 271 (2015) 548–555
551
η = col (η1 , η2 , . . . , ηN ), e = col (e1 , . . . , eN ). We have
ε˙ i = x˙ i − i η˙ i
= Ai xi + Bi ui − i Sηi + QF
N
ai j (η j − ηi )
j=1
= Ai xi + Bi Ki (xˆ − i ηi ) + Bi i ηi − Ai i ηi − Bi i ηi − i QF
N
ai j (η j − ηi )
j=1
= Ai xi − Ai i ηi + Bi Ki (xi − i ηi ) + Bi Ki (xˆi − xi ) − i QF
N
ai j (η j − ηi )
j=1
= (Ai + Bi Ki )εi − Bi Ki ei − i QF
N
ai j (η j − ηi ).
(11)
j=1
One has
ε˙ = (A + BK )ε − blockdiag(Bi Ki ei ) − blockdiag(i )(L ⊗ QF )η.
(12)
Note that
e˙ i = Ai xi + Bi ui − Ai xˆi − Bi ui + Hi (yi − yˆi ) = (Ai + HiCi )ei .
(13)
Then, it follows that limt→∞ ε = 0, which then implies that limt→∞ yi = limt→∞ Ci i ηi = Dη0 . This completes the proof. 4. Output consensus with a leader In this section, we consider output consensus problem of heterogeneous agents with a leader. The dynamics of the leader is given by
v˙ = Sv, w = Dv.
(14)
If agent i can access the leader, a virtual edge (i, 0) is said to exist with weighting gain ai0 = 1; otherwise, ai0 = 0. Denote the weighting matrix as G = diag{ai0 }N×N . We provide the following output consensus protocol:
ui = Ki (xi − i ρi ) + i ηi ,
ρ˙ i = Sρi + μM
N
ai j (ρ j − ρi ) + ai0 (v − ρi ) , i = 1, . . . , N,
(15)
j=1
where Ki ∈ Rmi ×ni , M ∈ Rr × r and μ ∈ R are needed to be designed. S ∈ Rr × r represents state of the virtual exosystem. i ∈ Rni ×r and i ∈ Rmi ×r can be obtained by the regulation Eq. (3). Lemma 2. [24] Suppose that network G contains a directed spanning tree and a root node r can obtain the information of the leader, i.e., ar0 = 1. Let matrices X and Y be positive definite. Design control gain M as M = Y −1 P, where P is the unique positive definite solution of the control algebraic Riccati equation:
ST P + PS + X − PY −1 P = 0.
(16)
Then it follows that
lim ρi − v = 0, i = 1, 2, . . . , N,
t→∞
(17)
if
μ≥
1 . 2minλ∈(L+G) (λ)
(18)
Theorem 3. Suppose that (Ai , Bi ) is stabilizable and network G contains a directed spanning tree and a root node r can obtain the information of the leader. Then, output consensus in heterogeneous multi-agent system (1) with a leader (14) can be achieved under protocol (15), if Ki be such that Ai + Bi Ki is Hurwitz, i and i are a solution of regulation Eq. (3), μ and M are designed in Lemma 2.
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Q. Ma, G. Miao / Applied Mathematics and Computation 271 (2015) 548–555
Fig. 1. Topology structure of network in Example 1.
Proof. Let
εi ε ρ¯ i ρ¯
= xi − i ρi , = col (ε1 , ε2 , . . . , εN ), =
ρi − v,
= col (ρ¯ 1 , ρ¯ 2 , . . . , ρ¯ N ).
One has
ε˙ i = x˙ i − i ρ˙ i
= Ai xi + Bi ui − i Sρi + μM
N
ai j (ρ j − ρi ) + ai0 (v − ρi )
j=1
= Ai xi + Bi Ki εi + Bi i ρi − Ai i ρi − Bi i ρi
N −i μM ai j (ρ j − ρi ) + ai0 (v − ρi ) .
(19)
j=1
Namely,
ε˙ = (A + BK )ε − μblockdiag(i )(L + G) ⊗ F ρ¯ .
(20)
From Lemma 2, we have
lim ρ¯ = 0.
(21)
t→∞
Then, it can be easily seen that limt→∞ ε = 0. Finally, one has limt→∞ yi = limt→∞ Ci i ρi = Dv = w. This completes the proof. Remark 2. Similar to Theorem 1, the parameters in dynamic output consensus protocol (15) are dependent on the solution of regulation equation and a homogeneous system. The parameters in homogeneous system ρ˙ i = Sρi + μM( Nj=1 ai j (ρ j − ρi ) + ai0 (v − ρi )) can be designed by Lemma 2. It can be seen that ρi → v under the certain conditions. For the case of that the states of systems are not accessible, a dynamic regulator based on the state observers can be designed as follows:
ui = Ki (xˆi − i ρi ) + i ρi ,
ρ˙ i = Sρi + μM
N
ai j (ρ j − ρi ) + ai0 (v − ρi ) ,
j=1
xˆ˙ i = Ai xˆi + Bi ui − Hi (Ci xi − Ci xˆi ), i = 1, . . . , N,
(22)
where xˆi represents the estimate of the state xi . Theorem 4. Suppose that (Ai , Bi ) is stabilizable, (Ai , Ci ) is detectable, and network G contains a directed spanning tree and a root node r can obtain the information of the leader. Then, output consensus in heterogeneous multi-agent system (1) with a leader (14) can be achieved under protocol (22), if Ki and Hi be such that Ai + Bi Ki , Ai + HiCi are Hurwitz„ i and i are a solution of regulation Eq. (3), μ and M are designed in Lemma 2. Proof. Through the same line as the proof in Theorems 2 and 3, the results can be obtained easily which is omitted here. 5. Numerical examples In this section, we provide two examples to demonstrate the effectiveness of the proposed methods. Example 1. Consider the heterogeneous multi-agent system (1) with the topology shown in Fig. 1 with the parameters given by:
Q. Ma, G. Miao / Applied Mathematics and Computation 271 (2015) 548–555
553
Fig. 2. The trajectories of εi1 , i = 1, 2, 3.
Fig. 3. The trajectories of εi2 , i = 1, 2, 3.
A1 =
−1 1
C1 = 1
1 , C2 = 1
1 −2 , A3 = 0.5 1
1 1 , A2 = 1 −1
1 , C3 = 2
1 0 1 1 , B1 = , B2 = , B3 = , 1 1 0 2
1 , D= 1
1 .
Suppose that the matrix pair (S, Q) read
S=
0 −1
1 0 , Q= 0 1
.
Then i , i , i = 1, 2, 3, for the linear regulator Eq. (3) can be obtained as:
1 =
0.6 0.4
1 = −1.8
0.2 0.85 , 2 = 0.8 0.15
0.23 0.29 , 3 = 0.77 1.42
−0.6 , 2 = −1.23
0.26 , 0.48
−0.15 , 3 = −1.09
0.34 .
From Theorem 1, output consensus can be achieved. The trajectories of εi , i = 1, 2, 3 are shown in Figs. 2 and 3. Example 2. Consider the heterogeneous multi-agent system (1) with the topology shown in Fig. 4. Assume that the agent 1 can obtain the leader’s information directly. With the same parameters in Example 1 and applying Theorem 3, the trajectories of εi , i = 1, 2, 3 are shown in Figs. 5 and 6. 6. Conclusions In this paper, output consensus problem of heterogeneous multi-agent systems is studied. The cases of leaderless and leaderfollowing are considered. For each case, a dynamic consensus protocol is proposed, in which the parameters are dependent on
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Q. Ma, G. Miao / Applied Mathematics and Computation 271 (2015) 548–555
Fig. 4. Topology structure of network in Example 2.
Fig. 5. The trajectories of εi1 , i = 1, 2, 3.
Fig. 6. The trajectories of εi2 , i = 1, 2, 3.
the solution of a regulation equation and a homogeneous system. The homogeneous system can be analyzed by full-developed technique in homogenous multi-agent systems. Furthermore, dynamic regulators based on the state observers also are presented which is suitable for the case that the system states cannot be accessible. Finally, simulation examples are provided to demonstrate the effectiveness of the proposed design methods. Acknowledgments This work was supported by the National Natural Science Foundation of China under grant nos. 61403199, 61403178, 61503189, 61374153, the Natural Science Foundation of Jiangsu Province under grant nos. BK20140770 and BK20150926. References [1] W. Ren, E. Atkins, Distributed multi-vehicle coordinated control via local information exchange, Int. J. Robust Nonlinear Control 17 (10) (2007) 1002–1033. [2] W. Yu, G. Chen, M. Cao, Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems, Automatica 46 (6) (2010) 1089–1095. [3] W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control 58 (6) (2008) 1503–1509.
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[4] Q. Ma, S. Xu, F. Lewis, Second-order consensus for directed multi-agent systems with sampled data, Int. J. Robust Nonlinear Control 24 (16) (2014) 2560– 2573. [5] Y. Liu, Y. Jia, Consensus problem of high-order multi-agent systems with external disturbances, Int. J. Robust Nonlinear Control 20 (14) (2010) 1579–1593. [6] Z. Li, W. Rei, X. Liu, L. Xie, Distributed consensus of linear multi-agent systems with adaptive dynamic protocols, Automatica 49 (7) (2013) 1986–1995. [7] J.Z. Li, W. Ren, S.Y. Xu, Distributed containment control with multiple dynamic leaders for double-integrator dynamics using only position measurements, IEEE Trans. Autom. Control 57 (6) (2012) 1553–1559. [8] H. Liu, G. Xie, L. Wang, Containment of linear multi-agent systems under general interaction topologies, Syst. Control Lett. 61 (4) (2012) 528–534. [9] Z. Meng, W. Ren, Y. Zheng, Distributed finite-time attitude containment control for multiple rigid bodies, Automatica 46 (12) (2010) 2092–2099. [10] L. Li, D.W.C. Ho, J. Lu, A unified approach to practical consensus with quantized data and time delay, IEEE Trans. Circuits Syst. I Regul. Pap. 60 (10) (2013) 2668–2678. [11] Y. Sun, L. Wang, G. Xie, Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays, Syst. Control Lett. 57 (2008) 175–183. [12] H. Zhao, W. Ren, D. Yuan, J. Chen, Distributed discrete-time coordinated tracking with Markovian switching topologies, Syst. Control Lett. 61 (7) (2012) 766–772. [13] J. Shen, J. Cao, Consensus of multi-agent systems on time scales, IMA J. Math. Control Inf. 29 (4) (2012) 507–517. [14] J. Qin, H. Gao, W.X. Zheng, Second-order consensus for multi-agent systems with switching topology and communication delay, Syst. Control Lett. 60 (6) (2011) 390–397. [15] T.H. Lee, J. Park, D.H. Ji, H.Y. Jung, Leader-following consensus problem of heterogeneous multi-agent systems with nonlinear dynamics using fuzzy disturbance observer, Complexity 19 (4) (2014) 20–31. [16] P. Wieland, R. Sepulchre, F. Allgöwer, An internal model principle is necessary and sufficient for linear output synchronization, Automatica 47 (5) (2011) 1068–1074. [17] H. Kim, H. Shim, J. Seo, Output consensus of heterogeneous uncertain linear multi-agent systems, IEEE Trans. Autom. Control 56 (1) (2011) 200–206. [18] J. Wu, F. Allgöwer, A constructive approach to synchronization using relative information, in: Proceedings of 51st IEEE Conference on Decision and Control, 2012, pp. 5960–5965. [19] J. Lunze, An internal-model principle for the synchronisation of autonomous agents with individual dynamics, in: Proceedings of 50th IEEE Conference on Decision and Control, 2011, pp. 2106–2111. [20] J. Lunze, Synchronization of heterogeneous agents, IEEE Trans. Autom. Control 57 (11) (2012) 2885–2890. [21] T. Yang, A. Saberi, A. Stoorvogel, H. Grip, Output synchronization for heterogeneous networks of introspective right-invertible agents, Int. J. Robust Nonlinear Control 24 (13) (2014) 1821–1844. [22] H. Grip, T. Yang, A. Saberi, A. Stoorvogel, Output synchronization for heterogeneous networks of non-introspective, right-invertible agents, Automatica 48 (10) (2012) 2444–2453. [23] S. Tuna, LQR-based coupling gain for synchronization of linear systems (2008).[Online] http://arxiv.org/abs/0801.3390 [24] H. Zhang, F. Lewis, A. Das, Optimal design for synchronization of cooperative systems: state feedback, observer and output feedback, IEEE Trans. Autom. Control 56 (8) (2011) 1948–1952.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Output consensus for heterogeneous multi-agent systems with linear dynamics Qian Ma a,∗, Guoying Miao b a b
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, PR China School of Information and Control, Nanjing University of Information Science and Technology, Nanjing 210044, PR China
a r t i c l e
i n f o
Keywords: Multi-agent systems Output consensus Heterogenous networks Regulation equation
a b s t r a c t This paper deals with output consensus problem of heterogeneous multi-agent systems. The cases of leaderless and leader-following are considered. For each case, a dynamic consensus protocol is proposed, in which the parameters are dependent on the solution of a regulation equation and a homogeneous system. The homogeneous system can be analyzed by fulldeveloped technique in homogenous multi-agent systems. Furthermore, dynamic regulators based on the state observers also are presented which is suitable for the case that the system states cannot be obtained. Simulation examples are provided finally to demonstrate the effectiveness of the proposed design methods. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The consensus problem as one of the most fundamental research topics in the field of coordination control of multi-agent systems has attracted considerable attention over the past few years due to its extensive applications in cooperative control of mobile autonomous robots, the design of distributed sensor networks, spacecraft formation flying, and other areas. Much of the attention has been devoted to achieving state consensus in homogeneous networks, i.e., networks where the agent models are identical and with the same state dimensions. The consensus problems of homogeneous multi-agent systems, such as consensus of systems with second-order dynamics [1–4] and high-order dynamics [5,6], distributed containment control [7–9], consensus of agents with time-delay [10,11], and consensus with switching topology [12–14], just to mention a few, have been intensively studied recently. In practical applications, it is often impossible to require all agents are identical. Therefore, the difference between the agents which can be called the heterogeneity cannot be neglected in the design of consensus protocol. In this case, the individual agents are not identical and in particular the state dimensions may be different, then state consensus among all agents cannot be achieved. Therefore, the objective is to realize output consensus. This problem is challenging, and there are few available results on output consensus of heterogeneous system can be found, see [15–22] for instance. In [15], leader-following consensus of heterogeneous agents with nonlinear intrinsic dynamics was addressed with a fuzzy disturbance observer. In [16,17], it has been shown that the agents have to possess an internal model of the consensus trajectory. Also based on the internal model principle, a constructive method for designing local dynamic controllers with relative information was proposed in [18]. In [19,20], by using of the notion of system inclusion and system intersection, the agents can be synchronized by an appropriate networked controller. By means of state transformation, the new agents in [21] were almost identical except for different exponentially ∗
Corresponding author. Tel.: +86 13951769356. E-mail address: [email protected] (Q. Ma).
http://dx.doi.org/10.1016/j.amc.2015.08.117 0096-3003/© 2015 Elsevier Inc. All rights reserved.
Q. Ma, G. Miao / Applied Mathematics and Computation 271 (2015) 548–555
549
decaying signals and then a decentralized controller was designed. A similar state transformation approach was taken in [22], in which the output synchronization problem for a heterogeneous network of non-introspective agents was considered. In this paper, we address output consensus problem of heterogeneous multi-agent systems. The cases of leaderless and leaderfollowing are considered. For each case, a dynamic consensus protocol is proposed, in which the parameters are dependent on the solution of a regulation equation and a homogeneous system. The homogeneous system can be analyzed by full-developed technique in homogenous multi-agent systems. And it can be seen that there is a close link with the parameters in homogeneous system and the eigenvalue of topology matrix. Furthermore, dynamic regulators based on the state observers also are presented which is suitable for the case that the system states are not accessible. Notations. Throughout this paper, for real symmetric matrices X and Y, the notation X ≥ Y (respectively, X > Y) means that the matrix X − Y is positive semi-definite (respectively, positive definite). I denotes an identity matrix of appropriate dimension. 1N ∈ RN be the vector with all entries being 1. The notation ‘∗’ is used as an ellipsis for terms that are induced by symmetry. (X) denotes the set of the eigenvalues of X and + (X ) the set of the eigenvalues with positive real part of X. The Kronecker product of matrices X and Y is denoted as X⊗Y. (x) represent the real parts of a complex number x. 2. Preliminaries Let G = {V, E, A} be a weighted directed graph with the set of nodes V = {1, 2, . . . , N}, the set of directed edges E ⊆ V × V, and a weighted adjacency matrix A = (ai j )N×N . A directed edge eij in network G is denoted by the ordered pair of nodes (i, j), meaning that node j can receive information from node i. The elements of the adjacency matrix A are defined as aij > 0 if and only if there is a directed edge (j, i) in G; otherwise, ai j = 0. A directed path is a sequence of nodes 1, 2, … , r such that (i, i + 1) ∈ E, i ∈ {1, 2, . . . , r − 1}. A directed graph is strongly connected if there is a directed path for any two distinct nodes j and i. A directed graph has a directed spanning tree if there exists at least one node called root node which has a directed path to all the other nodes. The Laplacian matrix L = (li j )N×N of graph G is defined as N
lii = −
li j ,
j=1, j=i
li j = −ai j ,
i = j; i, j = 1, 2, . . . , N,
which ensures the diffusion property that Nj=1 li j = 0. Consider a heterogeneous multi-agent system given by:
x˙ i = Ai xi + Bi ui , yi = Ci xi , i = 1, 2, . . . , N, R ni ,
Rq ,
(1)
R mi
where xi ∈ yi ∈ and ui ∈ are the state vector, output vector and control input, respectively. Since the every agents are not identical and in particular the state dimensions may be different, state consensus cannot be achieved. Therefore, our objective is to achieve output consensus among all agents. Definition 1. Output consensus in heterogeneous multi-agent system (1) is achieved if for any initial conditions,
lim yi − y j = 0,
t→∞
∀i, j = 1, 2, . . . , N.
3. Output consensus without a leader From [16], if the heterogeneous agents (1) achieve output consensus, then the all individual systems are able to track the same virtual exosystem defined by the dynamics matrix S and the output matrix D, which can be regarded as an internal model of the virtual exosystem. To reach output consensus of the heterogeneous agents, the following protocol is proposed:
ui = Ki (xi − i ηi ) + i ηi ,
η˙ i = Sηi + QF
N
ai j (η j − ηi ), i = 1, . . . , N,
(2)
j=1
where Ki ∈ Rmi ×ni , F ∈ Rk × r and Q ∈ Rr × k are needed to be designed. S ∈ Rr × r represents state of the virtual exosystem. i ∈ Rni ×r and i ∈ Rmi ×r can be obtained by the following regulation equation:
i S = Ai i + Bi i , 0 = Ci i − D.
(3)
Lemma 1. [8,23] Consider formula (2). Suppose that the pair (S, Q) is stabilizable and network G has a directed spanning tree. Let F = max{1, min 1 (λ) }Q T P, where P = P T is the solution to λ∈+ (L)
T
S P + PS + Ir − PQQ T P = 0.
(4)
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Then, one has
lim ηi = lim η j = η0 ,
t→∞
t→∞
∀i, j = 1, 2, . . . , N.
(5)
where η0 is a real valued function. Theorem 1. Suppose that (Ai , Bi ) is stabilizable and network G has a directed spanning tree. Then, output consensus in heterogeneous multi-agent system (1) can be achieved under protocol (2), if Ki be such that Ai + Bi Ki is Hurwitz, i and i are a solution of regulation Eq. (3), and F is designed in Lemma 1 with stabilizable pair (S, Q). Proof. Let
εi = xi − i ηi , ε = col (ε1 , ε2 , . . . , εN ), η = col (η1 , η2 , . . . , ηN ). We have
ε˙ i = x˙ i − i η˙ i
= Ai xi + Bi ui − i Sηi + QF
N
ai j (η j − ηi )
j=1
= Ai xi + Bi Ki εi + Bi i ηi − Ai i ηi − Bi i ηi − i QF = (Ai + Bi Ki )εi − i QF
N
N
ai j (η j − ηi )
j=1
ai j (η j − ηi ).
(6)
j=1
Then, one has
ε˙ = (A + BK )ε − blockdiag(i )(L ⊗ QF )η.
(7)
From Lemma 1, it follows that
lim η = 1N ⊗ η0 ,
(8)
t→∞
which implies that
lim blockdiag(i )(L ⊗ QF )η = 0,
t→∞
(9)
Thus, it can be concluded that limt→∞ ε = 0, which then implies that limt→∞ yi = limt→∞ Ci i ηi = Dη0 . This completes the proof. Remark 1. In Theorem 1, a dynamic controller (2) is proposed to guarantee output consensus of the heterogeneous multi-agent system (1). The key point lies in making the state of each heterogenous system approach to a function which involves the state of virtual exosystem and the solution of regulation equation, namely, xi → i ηi . It is worth noting that η˙ i = Sηi + QF Nj=1 ai j (η j − ηi ) is a homogeneous system which can be analyzed by some full-developed techniques in homogenous multi-agent systems. By Lemma 1, it can be concluded that ηi → ηj under the certain conditions. In reality, the states of systems often are not accessible. In this case, it is efficient to provide a dynamic regulator based on the state observers as follows:
ui = Ki (xˆi − i ηi ) + i ηi ,
η˙ i = Sηi + QF
N
ai j (η j − ηi ),
j=1
xˆ˙ i = Ai xˆi + Bi ui − Hi (Ci xi − Ci xˆi ), i = 1, . . . , N,
(10)
where xˆi represents the estimate of the state xi . Theorem 2. Suppose that (Ai , Bi ) is stabilizable, (Ai , Ci ) is detectable, and network G has a directed spanning tree. Then, output consensus in heterogeneous multi-agent system (1) can be achieved under protocol (10), if Ki and Hi be such that Ai + Bi Ki , Ai + HiCi are Hurwitz, i and i are a solution of regulation Eq. (3), and F is designed in Lemma 1 with stabilizable pair (S, Q). Proof. Let
εi = xi − i ηi , ei = xi − xˆi ,
ε = col (ε1 , ε2 , . . . , εN ),
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551
η = col (η1 , η2 , . . . , ηN ), e = col (e1 , . . . , eN ). We have
ε˙ i = x˙ i − i η˙ i
= Ai xi + Bi ui − i Sηi + QF
N
ai j (η j − ηi )
j=1
= Ai xi + Bi Ki (xˆ − i ηi ) + Bi i ηi − Ai i ηi − Bi i ηi − i QF
N
ai j (η j − ηi )
j=1
= Ai xi − Ai i ηi + Bi Ki (xi − i ηi ) + Bi Ki (xˆi − xi ) − i QF
N
ai j (η j − ηi )
j=1
= (Ai + Bi Ki )εi − Bi Ki ei − i QF
N
ai j (η j − ηi ).
(11)
j=1
One has
ε˙ = (A + BK )ε − blockdiag(Bi Ki ei ) − blockdiag(i )(L ⊗ QF )η.
(12)
Note that
e˙ i = Ai xi + Bi ui − Ai xˆi − Bi ui + Hi (yi − yˆi ) = (Ai + HiCi )ei .
(13)
Then, it follows that limt→∞ ε = 0, which then implies that limt→∞ yi = limt→∞ Ci i ηi = Dη0 . This completes the proof. 4. Output consensus with a leader In this section, we consider output consensus problem of heterogeneous agents with a leader. The dynamics of the leader is given by
v˙ = Sv, w = Dv.
(14)
If agent i can access the leader, a virtual edge (i, 0) is said to exist with weighting gain ai0 = 1; otherwise, ai0 = 0. Denote the weighting matrix as G = diag{ai0 }N×N . We provide the following output consensus protocol:
ui = Ki (xi − i ρi ) + i ηi ,
ρ˙ i = Sρi + μM
N
ai j (ρ j − ρi ) + ai0 (v − ρi ) , i = 1, . . . , N,
(15)
j=1
where Ki ∈ Rmi ×ni , M ∈ Rr × r and μ ∈ R are needed to be designed. S ∈ Rr × r represents state of the virtual exosystem. i ∈ Rni ×r and i ∈ Rmi ×r can be obtained by the regulation Eq. (3). Lemma 2. [24] Suppose that network G contains a directed spanning tree and a root node r can obtain the information of the leader, i.e., ar0 = 1. Let matrices X and Y be positive definite. Design control gain M as M = Y −1 P, where P is the unique positive definite solution of the control algebraic Riccati equation:
ST P + PS + X − PY −1 P = 0.
(16)
Then it follows that
lim ρi − v = 0, i = 1, 2, . . . , N,
t→∞
(17)
if
μ≥
1 . 2minλ∈(L+G) (λ)
(18)
Theorem 3. Suppose that (Ai , Bi ) is stabilizable and network G contains a directed spanning tree and a root node r can obtain the information of the leader. Then, output consensus in heterogeneous multi-agent system (1) with a leader (14) can be achieved under protocol (15), if Ki be such that Ai + Bi Ki is Hurwitz, i and i are a solution of regulation Eq. (3), μ and M are designed in Lemma 2.
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Fig. 1. Topology structure of network in Example 1.
Proof. Let
εi ε ρ¯ i ρ¯
= xi − i ρi , = col (ε1 , ε2 , . . . , εN ), =
ρi − v,
= col (ρ¯ 1 , ρ¯ 2 , . . . , ρ¯ N ).
One has
ε˙ i = x˙ i − i ρ˙ i
= Ai xi + Bi ui − i Sρi + μM
N
ai j (ρ j − ρi ) + ai0 (v − ρi )
j=1
= Ai xi + Bi Ki εi + Bi i ρi − Ai i ρi − Bi i ρi
N −i μM ai j (ρ j − ρi ) + ai0 (v − ρi ) .
(19)
j=1
Namely,
ε˙ = (A + BK )ε − μblockdiag(i )(L + G) ⊗ F ρ¯ .
(20)
From Lemma 2, we have
lim ρ¯ = 0.
(21)
t→∞
Then, it can be easily seen that limt→∞ ε = 0. Finally, one has limt→∞ yi = limt→∞ Ci i ρi = Dv = w. This completes the proof. Remark 2. Similar to Theorem 1, the parameters in dynamic output consensus protocol (15) are dependent on the solution of regulation equation and a homogeneous system. The parameters in homogeneous system ρ˙ i = Sρi + μM( Nj=1 ai j (ρ j − ρi ) + ai0 (v − ρi )) can be designed by Lemma 2. It can be seen that ρi → v under the certain conditions. For the case of that the states of systems are not accessible, a dynamic regulator based on the state observers can be designed as follows:
ui = Ki (xˆi − i ρi ) + i ρi ,
ρ˙ i = Sρi + μM
N
ai j (ρ j − ρi ) + ai0 (v − ρi ) ,
j=1
xˆ˙ i = Ai xˆi + Bi ui − Hi (Ci xi − Ci xˆi ), i = 1, . . . , N,
(22)
where xˆi represents the estimate of the state xi . Theorem 4. Suppose that (Ai , Bi ) is stabilizable, (Ai , Ci ) is detectable, and network G contains a directed spanning tree and a root node r can obtain the information of the leader. Then, output consensus in heterogeneous multi-agent system (1) with a leader (14) can be achieved under protocol (22), if Ki and Hi be such that Ai + Bi Ki , Ai + HiCi are Hurwitz„ i and i are a solution of regulation Eq. (3), μ and M are designed in Lemma 2. Proof. Through the same line as the proof in Theorems 2 and 3, the results can be obtained easily which is omitted here. 5. Numerical examples In this section, we provide two examples to demonstrate the effectiveness of the proposed methods. Example 1. Consider the heterogeneous multi-agent system (1) with the topology shown in Fig. 1 with the parameters given by:
Q. Ma, G. Miao / Applied Mathematics and Computation 271 (2015) 548–555
553
Fig. 2. The trajectories of εi1 , i = 1, 2, 3.
Fig. 3. The trajectories of εi2 , i = 1, 2, 3.
A1 =
−1 1
C1 = 1
1 , C2 = 1
1 −2 , A3 = 0.5 1
1 1 , A2 = 1 −1
1 , C3 = 2
1 0 1 1 , B1 = , B2 = , B3 = , 1 1 0 2
1 , D= 1
1 .
Suppose that the matrix pair (S, Q) read
S=
0 −1
1 0 , Q= 0 1
.
Then i , i , i = 1, 2, 3, for the linear regulator Eq. (3) can be obtained as:
1 =
0.6 0.4
1 = −1.8
0.2 0.85 , 2 = 0.8 0.15
0.23 0.29 , 3 = 0.77 1.42
−0.6 , 2 = −1.23
0.26 , 0.48
−0.15 , 3 = −1.09
0.34 .
From Theorem 1, output consensus can be achieved. The trajectories of εi , i = 1, 2, 3 are shown in Figs. 2 and 3. Example 2. Consider the heterogeneous multi-agent system (1) with the topology shown in Fig. 4. Assume that the agent 1 can obtain the leader’s information directly. With the same parameters in Example 1 and applying Theorem 3, the trajectories of εi , i = 1, 2, 3 are shown in Figs. 5 and 6. 6. Conclusions In this paper, output consensus problem of heterogeneous multi-agent systems is studied. The cases of leaderless and leaderfollowing are considered. For each case, a dynamic consensus protocol is proposed, in which the parameters are dependent on
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Fig. 4. Topology structure of network in Example 2.
Fig. 5. The trajectories of εi1 , i = 1, 2, 3.
Fig. 6. The trajectories of εi2 , i = 1, 2, 3.
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