Output containment control for swarm systems with general linear ...
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Output containment control for swarm systems with general linear dynamics: A dynamic output feedback approach✩ Xiwang Dong, Fanlin Meng, Zongying Shi, Geng Lu, Yisheng Zhong ∗ Department of Automation, TNlist, Tsinghua University, Beijing, 100084, PR China
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Article history: Received 4 January 2014 Received in revised form 5 May 2014 Accepted 13 June 2014 Available online xxxx Keywords: Output containment High-order Swarm system Dynamic output feedback
abstract Output containment control problems for high-order linear time-invariant swarm systems under directed interaction topologies are investigated using a dynamic output feedback approach. Firstly, to propel the outputs of followers to converge to the convex hull formed by the outputs of leaders, a dynamic output containment protocol is presented. Then necessary and sufficient conditions for swarm systems to achieve output containment are proposed. To ensure the scalability of the criteria, a sufficient condition which only includes two linear matrix inequality constraints independent of the number of agents is further presented. Moreover, an approach independent of the number of agents is proposed to determine the gain matrices in the dynamic output containment protocols. Finally, numerical simulations are presented to demonstrate theoretical results. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Recently, distributed cooperative control of swarm systems has gained much attention due to its broad applications including flocking [1,2], synchronization [3,4], and formation control [5,6]. Many existing works in distributed cooperative control of swarm systems focus on the consensus problem, where all agents reach an agreement on certain variables of interest in the case that there is no leader in the swarm system, such as [7–16] and references therein. In some practical applications, the existence of leaders can broaden the applications of consensus due to that a group objective can be brought by the leaders. Consensus with one leader is often known as consensus tracking or coordinated tracking. Consensus tracking problems for first-order continuous and discrete swarm systems with directed interaction topologies were studied in [17,18]. Hong et al. presented sufficient conditions for first-order and second-order linear time-invariant (LTI) swarm systems with fixed and switching interaction topologies to achieve consensus tracking in [19,20]. For second-order LTI swarm systems with switching interaction topologies and time-delays, Peng and Yang [21] derived sufficient
✩ This work was supported by the National Natural Science Foundation of China under Grants 61374034 and 61174067. ∗ Corresponding author. Tel.: +86 10 62772573. E-mail address: [email protected] (Y. Zhong).
http://dx.doi.org/10.1016/j.sysconle.2014.06.007 0167-6911/© 2014 Elsevier B.V. All rights reserved.
conditions to achieve consensus tracking with uniformly and ultimately bounded errors. Ni and Cheng [22] addressed consensus tracking problems for high-order LTI swarm systems with both fixed and switching undirected interaction topologies. Zhang et al. [23] investigated the distributed finite-time consensus tracking problems for nonlinear swarm systems with external disturbances. In the case of multiple leaders, containment problems arise, which require that the states/outputs of followers converge to the convex hull formed by the states/outputs of leaders. Ji et al. [24] proposed a hybrid stop–go control strategy for first-order swarm systems to achieve containment. Meng et al. [25] presented nonlinear control protocols for rigid body swarm systems to achieve containment in a finite time. Notarstefano et al. [26] discussed containment problems for first-order swarm systems with switching undirected interaction topologies. Cao et al. dealt with containment problems for first-order and second-order swarm systems with both stationary and dynamic leaders in [27,28]. Liu et al. [29] proposed necessary and sufficient conditions for firstorder and second-order swarm systems to achieve containment. Containment problems for second-order swarm systems with random switching interaction topologies were studied in [30]. However, in many practical cases, the dynamics of each agent is of high-order, as shown in [8]. Containment problems for high-order swarm systems are more challenging since each agent contains more complex dynamics. Moreover, they have generality and it can be shown that containment problems for first-order and secondorder swarm systems (e.g., [26–30]) can be regarded as special
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cases of high-order swarm systems. Therefore, containment problems for high-order swarm systems make more sense. Containment problems for both continuous and discrete highorder LTI swarm systems were investigated in [31]. Dong et al. [32] presented sufficient conditions for high-order LTI singular swarm systems with time-delays to achieve containment. By classifying the agents into boundary agents and internal agents, Liu et al. [33] proposed a criterion for the states of internal agents to converge to a convex combination of the states of the boundary agents. It should be pointed out that in [24–30,32,33], the relative states of neighboring agents are required to construct the containment protocols, and in [24–33], the states of followers are required to converge to the convex hull formed by the states of leaders. However, in some practical applications, only the relative outputs of neighboring agents are available, and only outputs of followers are required to converge to the convex hull formed by the outputs of leaders. Because outputs are linear combinations of states and may not be related with all states, both the analysis and design for output containment problems of high-order LTI swarm systems become complicated and challenging. Dong et al. [34] dealt with output containment problems using a static output containment protocol. However, they only proposed sufficient conditions to achieve output containment and the existence of the controller cannot be ensured. To the best of our knowledge, necessary and sufficient conditions for high-order swarm system to achieve output containment have not been obtained before. In the current paper, output containment analysis and design problems for high-order LTI swarm systems with dynamic protocols are discussed. Firstly, to partly improve the output behavior of leaders and propel the outputs of followers to converge to the convex hull formed by the outputs of leaders, dynamic output containment protocols are presented for leaders and followers respectively. Then necessary and sufficient conditions for swarm systems to achieve output containment are presented. An approach is introduced to decrease the calculation complexity of the criteria. Finally, an approach to determine the gain matrices in the dynamic output containment protocols is proposed. Compared with the existing works on consensus and containment, the novel features of the current paper are threefold. Firstly, both output containment analysis and design problems are investigated for high-order LTI swarm systems. In [15], output consensus problems were studied, where there do not exist multiple leaders. In [24–33], only state containment problems were discussed. Secondly, necessary and sufficient criteria for swarm systems to achieve output containment are proposed. The criteria in [34] are only sufficient. Thirdly, the approach to determine the gain matrices in the dynamic protocols has less calculation complexity, which only needs to solve an algebraic Riccati equation. In [34], it is difficult to design the output containment protocol since the output containment problems were transformed into static output feedback stabilization problems, and the existence of the controller cannot be ensured. Moreover, the results in this paper can be applied to solve the output consensus tracking problems for high-order LTI swarm systems. The rest of this paper is organized as follows. In Section 2, basic concepts on graph theory are introduced and the problem description is given. In Section 3, necessary and sufficient conditions for swarm systems to achieve dynamic output containment are proposed. In Section 4, an approach to determine the gain matrices in the dynamic protocols is presented. Numerical simulations are provided in Section 5. Finally, Section 6 concludes the whole work. Throughout this paper, for simplicity of notation, 0 is used to denote zero matrices of appropriate size with zero vectors and zero number as special cases, and superscript H represents the Hermitian transpose of matrices. I stands for an identity matrix with appropriate dimension and ⊗ refers to the Kronecker product.
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2. Preliminaries and problem description In this section, basic concepts on graph theory are introduced and the problem description is presented. 2.1. Basic concepts on graph theory Let G = {V , E , W } be a directed graph of order N with V = {v1 , v2 , . . . , vN } the set of nodes, E ⊆ (vi , vj ) : vi , vj ∈ V , i ̸= j the set of edges and W = [wij ] ∈ RN ×N the weighted adjacency matrix. An edge of G is denoted by eij = (vi , vj ). The weighted matrix elements are positive, i.e., wji > 0 if and only if eij ∈ E. Moreover, wii = 0 for all i ∈ {1, 2, . . . , N }. The set of neighbors of node vi is denoted by Ni = vj ∈ V : (vj , vi ) ∈ E . The in-degree of N node vi is defined as degin (vi ) = j=1 wij . The degree matrix of G is denoted by D = diag {degin (vi ), i = 1, 2, . . . , N }. The Laplacian matrix of G is defined as L = D − W . A directed path from node vi1 to vil is a sequence of ordered edges with the form of (vik , vik+1 ), where vik ∈ V (k = 1, 2, . . . , l − 1). More details on graph theory can be found in [35]. 2.2. Problem description For a swarm system with N agents, the interaction topology of the swarm system can be described by directed graph G, each agent can be regarded as a node in G. For i, j ∈ {1, 2, . . . , N }, the available interaction channel from agent i to agent j can be denoted by the edge and the interaction strength can be denoted by eij . Consider the following high-order LTI dynamics for each agent
x˙ i (t ) = Axi (t ) + Bui (t ), yi (t ) = Cxi (t ),
(1)
where i ∈ {1, 2, . . . , N }, xi (t ) ∈ Rn is the state, yi (t ) ∈ Rq is the output, and ui (t ) ∈ Rm is the control input. Assumption 1. C is of full row rank and (A, B) is stabilizable. Remark 1. In Assumption 1, C is assumed to be of full row rank, which means that the elements in the output y(t ) are independent of each other. If C is not of full row rank, then some of the elements in y(t ) can be expressed as linear combination of the others. In this case, those elements in y(t ) which are linearly dependent on others can be ignored due to that they involves no new information. Definition 1. An agent is called a leader if it has no neighbors. An agent is called a follower if it has at least one neighbor. Assume that in swarm system (1) there are M (M < N ) followers with states xk (t ) (k = 1, 2, . . . , M ) and N − M leaders with states xi (t ) (i = M + 1, M + 2, . . . , N ). Let F = {1, 2, . . . , M } and J = {M + 1, M + 2, . . . , N } be the follower subscript set and leader subscript set respectively. Under Definition 1, the Laplacian matrix corresponding to G has the following form
L=
L1 0
L2 , 0
where L1 ∈ RM ×M and L2 ∈ RM ×(N −M ) . Assumption 2. For each follower, there exists at least one leader that has a directed path to it. Under Assumption 2, the following lemma holds. Lemma 1 ([25]). If the interaction topology satisfies Assumption 2, then all the eigenvalues of L1 have positive real parts, each entry of 1 −1 −L − 1 L2 is nonnegative, and each row of −L1 L2 has a sum equal to one.
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Definition 2. Swarm system (2) is said to achieve output containment if for any k ∈ F , there exist nonnegative constants αk,j (j ∈ E )
3. Output containment analysis
satisfying
In this section, firstly swarm systems (5) and (6) are transformed into systems which include subsystems with states yF (t ) and yJ (t ) respectively. Then necessary and sufficient conditions for swarm system (1) under dynamic protocols (3) and (4) to achieve output containment are presented. A sufficient condition with less calculation complexity is further proposed. By Assumption 1, there exists C¯ ∈ R(n−q)×n such that T = T ¯T T [C , C ] is nonsingular. Let y¯ i (t ) = C¯ xi (t ) (i = 1, 2, . . . , N ), y¯ F (t ) = [¯yT1 (t ), y¯ T2 (t ), . . . , y¯ TM (t )]T , y¯ J (t ) = [¯yTM +1 (t ), y¯ TM +2 (t ), . . . , y¯ TN (t )]T , and
lim
t →∞
αk,j = 1 such that N yk (t ) − αk,j yj (t ) = 0. N
j=M +1
(2)
j=M +1
Consider the following dynamic output containment protocols for followers and leaders respectively
z˙i (t ) = K1 zi (t ) + K2 wij zi (t ) − zj (t ) j∈Ni + K3 w yi (t ) − yj (t ) , i ∈ F , ij j∈Ni ui (t ) = K4 zi (t ), i ∈ F , z˙i (t ) = K1 zi (t ), i ∈ J , ui (t ) = K4 zi (t ), i ∈ J ,
(3) TAT
(4)
where zi (t ) ∈ R (i = 1, 2, . . . , N ) are the states of the protocols, and K1 , K2 , K3 and K4 are constant gain matrices with appropriate dimensions. Let p
T zF (t ) = [z1T (t ), z2T (t ), . . . , zM (t )]T ,
xF (t ) = [xT1 (t ), xT2 (t ), . . . , xTM (t )]T , yF (t ) = [yT1 (t ), yT2 (t ), . . . , yTM (t )]T , T T T T zJ (t ) = [zM +1 (t ), zM +2 (t ), . . . , zN (t )] ,
xJ (t ) = [xTM +1 (t ), xTM +2 (t ), . . . , xTN (t )]T , yJ (t ) = [yTM +1 (t ), yTM +2 (t ), . . . , yTN (t )]T . Under protocols (3) and (4), the dynamics of swarm system (1) can be written in a compact form as
z˙ (t ) = (IM ⊗ K1 + L1 ⊗ K2 )zF (t ) + (L2 ⊗ K2 )zJ (t ) F + (L1 ⊗ K3 )yF (t ) + (L2 ⊗ K3 )yJ (t ), x˙ F (t ) = (IM ⊗ A)xF (t ) + (IM ⊗ BK4 )zF (t ), yF (t ) = (IM ⊗ C )xF (t ), z˙J (t ) = (IN −M ⊗ K1 )zJ (t ), x˙ J (t ) = (IN −M ⊗ A)xJ (t ) + (IN −M ⊗ BK4 )zJ (t ), yJ (t ) = (IN −M ⊗ C )xJ (t ).
(5)
Remark 2. A variety of observer-like dynamic protocols were proposed for swarm systems to achieve consensus or state containment, such as protocol (13) in [36] and protocol (2) in [31]. It can be verified that protocol (13) in [36] can be regarded as a special case of protocol (3) by choosing K1 = A + BK , K2 = −I, K3 = I, K4 = K . The dimension of zi (t ) must be the same with that of xi (t ) in protocol (13) of [36] and protocol (2) of [31]. However, in protocol (3) of this paper, the dimension of zi (t ) can be less than that of xi (t ). Moreover, protocol (3) has more degrees of freedom for swarm system (1) to achieve output containment. From (6), one sees that K1 , K4 and zi (0) (i ∈ J ) can be used to partly change the behavior of leaders, which is useful for the scenarios that the behavior of leaders needs to be improved. It should be pointed out that by choosing zi (0) = 0 (i ∈ J ), the output containment problems in this paper become the ones in which leaders have no control input. In the current paper, the following two problems for swarm system (1) under dynamic protocols (3) and (4) are mainly addressed: (i) under what conditions output containment can be achieved; and (ii) how to determine the gain matrices in dynamic protocols (3) and (4) to achieve output containment.
A¯ = ¯ 11 A21
B¯ TB = ¯ 1 . B2
A¯ 12 , A¯ 22
Then swarm systems (5) and (6) can be converted into
z˙F (t ) = (IM ⊗ K1 + L1 ⊗ K2 ) zF (t ) + (L2 ⊗ K2 ) zJ (t ) + (L1 ⊗ K3 ) yF (t ) +(L2 ⊗ K3 ) yJ (t ), y˙ F (t ) = IM ⊗ A¯ 11 yF (t ) + IM ⊗ A¯ 12 y¯ F (t ) + IM ⊗ B¯ 1 K4 zF (t ), y˙¯ F (t ) = IM ⊗ A¯ 21 yF (t ) + IM ⊗ A¯ 22 y¯ F (t ) + IM ⊗ B¯ 2 K4 zF (t ), z˙ (t ) = (IN −M ⊗ K1 ) zJ (t ), J y ˙ J (t ) = IN−M ⊗ A¯ 11 yJ (t ) + IN −M ⊗ A¯ 12 y¯ J (t ) ¯ + IN −M ⊗ B1 K4 zJ (t ), ˙ ¯ y¯ (t ) = IN −M ⊗ A21 yJ (t ) + IN −M ⊗ A¯ 22 y¯ J (t ) J + IN −M ⊗ B¯ 2 K4 zJ (t ).
(7)
(8)
In (7) and (8), the subsystems with states yF (t ) and yJ (t ) can be used to analyse the output containment problems. It can be found that only the observable components of (A¯ 22 , A¯ 12 ) influence the subsystems with states yF (t ) and yJ (t ). Therefore, the observability decomposition of (A¯ 22 , A¯ 12 ) is presented firstly. Let T˜ be a nonsingular matrix such that
(6)
−1
T˜ −1 A¯ 22 T˜ , A¯ 12 T˜
˜1 D ˜2 D
=
0 ˜ ˜D3 , E1
0
,
˜ 1 ∈ Rg and (D˜ 1 , E˜ 1 ) is completely observable. Let yˆ i (t ) = where D − 1 T˜ y¯ i (t ) = [ˆyTio (t ), yˆ Tio¯ (t )]T (i = 1, 2, . . . , N ), T˜ −1 A¯ 21 = [F˜1T , F˜2T ]T , T˜ −1 B¯ 2 = [B˜ T1 , B˜ T2 ]T , and
yˆ Fo (t ) = [ˆyT1o (t ), yˆ T2o (t ), . . . , yˆ TMo (t )]T , yˆ F o¯ (t ) = [ˆyT1o¯ (t ), yˆ T2o¯ (t ), . . . , yˆ TM o¯ (t )]T , yˆ Jo (t ) = [ˆyT(M +1)o (t ), yˆ T(M +2)o (t ), . . . , yˆ TNo (t )]T , yˆ J o¯ (t ) = [ˆyT(M +1)¯o (t ), yˆ T(M +2)¯o (t ), . . . , yˆ TN o¯ (t )]T . Then systems (7) and (8) can be further converted into
z˙F (t ) = (IM ⊗ K1 + L1 ⊗ K2 ) zF (t ) + (L2 ⊗ K2 ) zJ (t ) + (L1 ⊗ K3 ) yF (t ) +(L2 ⊗ K3) yJ (t ), ˙ y ( t ) = IM ⊗ A¯ 11 yF (t ) + IM ⊗ E˜ 1 yˆ Fo (t ) F + IM ⊗ B¯1 K4 zF (t ), ˜ 1 yˆ Fo (t ) y˙ˆ Fo (t ) = IM ⊗ F˜1 yF (t ) + IM ⊗ D + IM ⊗ B˜ 1 K4 zF (t ), y˙ˆ F o¯ (t ) = IM ⊗ F˜2 yF (t ) + IM ⊗ D˜ 2 yˆ Fo (t ) + IM ⊗ D˜ 3 yˆ F o¯ (t ) + IM ⊗ B˜ 2 K4 zF (t ),
(9)
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˙ zJ (t ) = (IN −M ⊗ K1 ) zJ (t ), y˙ J (t ) = IN −M ⊗ A¯ 11 yJ (t ) + IN −M ⊗ E˜ 1 yˆ Jo (t ) + IN −M ⊗ B¯1 K4 zJ (t ), ˙ˆ ˜ 1 yˆ Jo (t ) yJo (t ) = IN −M ⊗ F˜1 yJ (t ) + IN −M ⊗ D ˜ 1 K4 zJ (t ), + I ⊗ B N − M ˙ˆ (t ) = IN −M ⊗ F˜2 yJ (t ) + IN −M ⊗ D˜ 2 yˆ Jo (t ) y ¯ J o + IN −M ⊗ D˜ 3 yˆ J o¯ (t ) + IN −M ⊗ B˜ 2 K4 zJ (t ).
λ i K3 A¯ 11 F˜1
(10)
0 E˜ 1 . ˜1 D
Proof. Sufficiency: Let φF (t ) = [zFT (t ), yTF (t ), yˆ TFo (t )]T , φJ (t ) = [zJT (t ), yTJ (t ), yˆ TJo (t )]T , and φ(t ) = [φFT (t ), φJT (t )]T . From (9) and (10), one has
˙ t ) = Φ1 φ( 0
Φ2 φ(t ), Φ3
(11)
where
IM ⊗ K1 + L 1 ⊗ K2 IM ⊗ B¯ 1 K4 Φ1 = IM ⊗ B˜ 1 K4 L2 ⊗ K2 0 0
L 2 ⊗ K3 0 0
Φ2 =
IN − M ⊗ K1 Φ3 = IN −M ⊗ B¯ 1 K4 IN −M ⊗ B˜ 1 K4
L 1 ⊗ K3 IM ⊗ A¯ 11 IM ⊗ F˜1
0
IM ⊗ E˜ 1 , ˜1 IM ⊗ D
eΦ1 t φ(t ) = 0
Θ1
eΦ3 t
0 IN −M ⊗ E˜ 1 . ˜1 IN − M ⊗ D
Θ1 = L−1 (sI − Φ1 )−1 Φ2 (sI − Φ3 )−1 , with L−1 the inverse Laplace transform and s the Laplace operator. It can be shown that
(sI − Φ1 )−1 Φ2 (sI − Φ3 )−1 = (sI − Φ1 )−1 Θ2 − Θ2 (sI − Φ3 )−1 1 L− 1 L2 ⊗ I Θ2 = 0 0
It is well known that one typical feature of swarm systems is of large scale. For swarm systems with a very large M, it may be time costly to check the conditions in Theorem 1. By encapsulating λi (i = 1, 2, . . . , M ) into a convex set, a sufficient condition with less calculation complexity is obtained in the following theorem. Let 0 A¯ 11 F˜1
0 . 0 1 L− L ⊗ I 2 1
K2 0 0
ψ2 =
K3 0 0
0 0 . 0
Re(λ)I Im(λ)I
−Im(λ)I . Re(λ)I
Lemma 2 ([37]). For given real symmetric matrices Ω0 , Ω1 and Ω2 ¯ i (i = 1, 2, 3, 4), if Ω0 + independent of λi (i = 1, 2, . . . , M ) and λ ¯ i Ω1 + Im λ¯ i Ω2 < 0 (i = 1, 2, 3, 4), then Ω0 + Re(λi )Ω1 + Re λ Im(λi )Ω2 < 0 (i = 1, 2, . . . , M ). Theorem 2. If the interaction topology satisfies Assumption 2, swarm system (1) under dynamic protocols (3) and (4) achieves output containment if there exists a matrix R¯ = R¯ T > 0 such that
T Λψ1 + Υλ¯ i Λψ2 ΛR¯ + ΛTR¯ Λψ1 + Υλ¯ i Λψ2 < 0 (i = 1, 3). (16)
T
Proof. Let Γi = Λψ1 + Υλi Λψ2 ΛR¯ + ΛTR¯ Λψ1 + Υλi Λψ2 1, 2, . . . , M ), then one can obtain
0 1 L− L 2 ⊗I 1 0
0 E˜ 1 , ˜1 D
¯ 1,2 = Re(λ1 ) ± jµF and λ¯ 3,4 = Re(λM ) ± jµF where µF = Set λ max{Im(λi ), i = 1, 2, . . . , M }. Then the following lemma holds.
where
Remark 3. From the structure of ΨFi (i = 1, 2, . . . , M ), one sees that the output containment of swarm systems depends on the dynamics of each agent, the observable components of (A¯ 22 , A¯ 12 ), the interaction topology and Ki (i = 1, 2, 3, 4).
the imaginary part of λ respectively, and Υλ =
(12)
where
From (15) and Lemma 1, one knows that swarm system (1) under dynamic protocols (3) and (4) achieves output containment. Necessity: The necessity is proven by contradiction. Choose φJ (0) = 0 or φJ (0) in the unobservable subspace of (Φ3 , [0 I 0]). In this case, yJ (t ) ≡ 0. If swarm system (1) under dynamic protocols (3) and (4) achieves output containment but subsystem ΨFi , 0 I 0 has at least one observable mode which is not stable for some i ∈ {1, 2, . . . , M }, then subsystem asymptotically Φ1 , 0 I 0 has atleast one observable mode that is not asymptotically stable. Hence, one can find nonzero φF (0) such that limt →∞ yF (t ) ̸= 0; that is, swarm system (1) under dynamic protocols (3) and (4) does not achieve output containment. This results a contradiction. the condition that the Therefore, observable modes of ΨFi , 0 I 0 are asymptotically stable is required. The proof for Theorem 1 is completed.
Then ΨFi = ψ1 + λi ψ2 . For any real matrix R and any λ ∈ C, let ΛR = diag{R, R}, Re(λ) and Im(λ) denote the real part and
φ(0),
0
K1 ψ1 = B¯ 1 K4 B˜ 1 K4
The solution to Eq. (11) can be written as
t →∞
0 0 , 0
0 IN −M ⊗ A¯ 11 IN −M ⊗ F˜1
1 I 0 eΦ1 t φF (0) + L− 1 L2 ⊗ I φJ (0) −1 + −L1 L2 ⊗ I yJ (t ), (13) Φt 3 yJ (t ) = 0 I 0 e φJ (0). (14) If the observable modes of ΨFi , 0 I 0 (i = 1, 2, . . . , M ) ¯ are asymptotically stable, by the structure of U F and ΛF , it can be obtained that limt →∞ 0 I 0 eΦ1 t = 0. Therefore, 1 lim yF (t ) − −L− (15) 1 L2 ⊗ I yJ (t ) = 0. yF (t ) =
Theorem 1. If the interaction topology satisfies Assumption 2, swarm system (1) under dynamic protocols (3) and (4) with any bounded initial states achieves output containment if and onlyif for ∀i ∈ {1, 2, . . . , M } the observable modes of ΨFi , 0 I 0 are asymptotically stable, where K1 + λi K2 ΨFi = B¯ 1 K4 B˜ 1 K4
–
From (12), one has
Let UF ∈ CM ×M be a nonsingular matrix such that UF−1 L1 UF = ¯ F , where Λ ¯ F is an upper-triangular matrix with λi (i = Λ 1, 2, . . . , M ) as its diagonal entries and Re(λ1 ) ≤ Re(λ2 ) ≤ · · · ≤ Re(λM ). The following theorem presents necessary and sufficient conditions for swarm system (1) under dynamic protocols (3) and (4) to achieve output containment.
)
Γi = Ω0 + Re(λi )Ω1 + Im(λi )Ω2 ,
(i =
X. Dong et al. / Systems & Control Letters (
Ω1 =
T ψ1 R¯ + R¯ T ψ1 0
T ψ2 R¯ + R¯ T ψ2 0
0
0
ψ ¯ + R¯ ψ1 T 1R
T
5
0
K + λ i K2 ΨFi = 1 Bo K4
0 T¯ = I
Ao
(i = 1, 2, . . . , M ).
(18)
, ψ2T R¯ + R¯ T ψ2 ψ2T R¯ − R¯ T ψ2
Let
λi K3 Co
,
. −ψ2T R¯ + R¯ T ψ2 0 T Similarly, let Ξi = Λψ1 + Υλ¯ i Λψ2 ΛR¯ + ΛTR¯ Λψ1 + Υλ¯ i Λψ2 (i = 1, 2, 3, 4), then it follows Ξi = Ω0 + Re(λ¯ i )Ω1 + Im(λ¯ i )Ω2 . Ω2 =
–
then ΨFi (i = 1, 2, . . . , M ) can be rewritten as
where
Ω0 =
)
Theorem 3. If the interaction topology satisfies Assumptions 1 and 2, then for any bounded initial states, swarm system (1) achieves output containment by dynamic protocols (3) and (4) with K4 satisfying that 1 T Ao − Bo K4 is Hurwitz, K1 = Ao − Bo K4 , K3 = −[Re(λ1 )]−1 (R− o Co Po ) , T and K2 = K3 Co , where Po = Po > 0 is the solution to the algebraic Riccati equation 1 Po ATo + Ao Po − Po CoT R− o Co Po + Qo = 0,
for Ro =
I . 0
RTo
> 0 and Qo =
(19)
≥ 0 with ( ,
DTo Do
Do ATo
) detectable.
Proof. Let K1 = Ao − Bo K4 and K2 = K3 Co , then one can obtain
¯ 1 = −Im λ¯ 2 and Im λ¯ 3 = −Im λ¯ 4 , one has Since Im λ Ξ1 = T¯ Ξ2 T¯ −1 and Ξ3 = T¯ Ξ4 T¯ −1 . Therefore, if Ξ1 < 0 and Ξ3 < 0, then Ξ2 < 0 and Ξ4 < 0 respectively. By Lemma 2, one knows that if condition (16) holds, then Γi < 0 (i = 1, 2, . . . , M ). Consider the stability of the following subsystems
ξ˙i (t ) = Λψ1 + Υλi Λψ2 ξi (t ) (i = 1, 2, . . . , M ).
(17)
(i = 1, 2, . . . , M ),
λi K3 Co
Ao
(i = 1, 2, . . . , M ).
It can be verified that ΨFi (i = 1, 2, . . . , M ) are similar to
A + λi K3 Co Ψ¯ Fi = o
0 Ao − Bo K4
Bo K4
Choose the Lyapunov functional candidates as follows Vi (t ) = ξiH (t )ΛR¯ ξi (t )
A − Bo K4 + λi K3 Co ΨFi = o Bo K4
(i = 1, 2, . . . , M ).
Since T and T˜ are nonsingular, if (A, B) is stabilizable, then by the PBH criterion for stabilizability, one has
where ξiH stand for the Hermitian transpose of ξi . Taking the time derivative of Vi (t ) along the trajectory of (17), one has
sI − A¯ 11 rank −F˜1 −F˜2
T V˙ i (t ) = ξiH (t ) Λψ1 + Υλi Λψ2 ΛR¯ + ΛR¯ Λψ1 + Υλi Λψ2 ξi (t ).
¯ + = {s|s ∈ C, Re(s) ≥ 0}. It also can be shown that where s ∈ C
Since Γi < 0 (i = 1, 2, . . . , M ), V˙ i (t ) < 0; that is, Λψ1 + Υλi Λψ2 (i = 1, 2, . . . , M ) are Hurwitz. By the decomposition of real and imaginary parts, it can be verified that ΨFi (i = 1, 2, . . . , M ) are Hurwitz due to that Λψ1 + Υλi Λψ2 (i = 1, 2, . . . , M ) are Hurwitz. Therefore, the observable modes of ΨFi , 0 I 0 are asymptotically stable. From Theorem 1, one knows that swarm system (1) under dynamic protocols (3) and (4) achieves output containment. The proof of Theorem 2 is completed. Remark 4. The condition in Theorem 1 is necessary and sufficient and Theorem 2 only includes two linear matrix inequality (LMI) constraints independent of the number of agents. It should be pointed out that the condition in Theorem 1 of [34] is only sufficient and Theorem 2 in [34] includes four LMI constraints. As a result, the criteria in this paper are more general than those in [34] and the calculation efficiency of this paper is better than that in [34]. Although condition (16) is independent of the number of agents, it needs the global information of the interaction topology due to that the eigenvalues of the Laplacian matrix depend on the global information of the interaction topology. The approaches in [36,38] may be used to estimate the eigenvalues of the Laplacian matrix without using the global information of the interaction topology. 4. Output containment protocol design In this section, an approach to determine Ki (i = 1, 2, 3, 4) in dynamic protocols (3) and (4) for swarm system (1) to achieve output containment is proposed. For simplicity of expression, let A¯ 11 F˜1
Ao =
E˜ 1 ˜1 , D
B¯ 1 Bo = , B˜ 1
Co = I
0 ,
sI − A¯ 11 −F˜1
rank
−E˜ 1 ˜1 sI − D −D˜ 2
−E˜ 1 ˜1 sI − D
˜3 sI − D
B¯ 1 B˜ 1
B¯ 1 B˜ 1 = n, B˜ 2
0 0
= q + g (∀s ∈ C¯ + ),
¯ +. which means that [sI − Ao , Bo ] is of full row rank for any s ∈ C Therefore, (Ao , Bo ) is stabilizable and there exists K4 such that Ao − Bo K4 is Hurwitz. ˜ 1 , E˜ 1 ) is completely observable, one knows that for Note that (D any s ∈ C,
sI − A¯ 11 −F˜1 I
−E˜ 1 ˜ 1 sI − D 0
is of full column rank, which means that (Ao , Co ) is completely observable. As a result, (ATo , CoT ) is completely controllable. Therefore, for any given Ro = RTo > 0 and Qo = DTo Do > 0 with (Do , ATo ) detectable, the algebraic Riccati equation (19) has a unique solution Po = PoT > 0. By Lemma 1, one knows that Re(λ1 ) > 0. Consider the stability of the following subsystems
ζ˙i (t ) = ATo + λHi CoT K3T ζi (t ) (i = 1, 2, . . . , M ).
(20)
Choose Lyapunov functional candidates as follows V¯ i (t ) = ζiH (t )Po ζi (t )
(i = 1, 2, . . . , M ).
(21)
1 T Let K3 = −[Re(λ1 )]−1 (R− o Co Po ) . Taking the time derivative of ¯Vi (t ) along the trajectory of (20), it holds that
V˙¯ i (t ) = −ζiH (t )Qo ζi (t ) + ζiH (t ) 1 − 2Re(λi )[Re(λ1 )]−1
1 × Po CoT R− o Co Po ζi (t ) ≤ 0.
Note that (Do , ATo ) is detectable and Ro = RTo > 0, one knows T T that ATo + λH i Co K3 (i = 1, 2, . . . , M ) are Hurwitz, which means
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X. Dong et al. / Systems & Control Letters (
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It can be shown that (A¯ 22 , A¯ 12 ) is not completely observable, then choose the following nonsingular matrix T˜ for observability decomposition T˜ =
that Ao + λi K3 Co (i = 1, 2, . . . , M ) are Hurwitz. Therefore, Ψ . . . , M ) are Hurwitz and the observable modes of Fi (i = 1, 2, ΨFi , 0 I 0 are asymptotically stable. From Theorem 1, the conclusion of Theorem 3 can be obtained. Remark 5. From Theorem 3, one sees that the output behavior of leaders can be partly improved by choosing K4 satisfying that Ao − Bo K4 is Hurwitz and zi (0) (i ∈ J ). If (A, B) is stabilizable, then the gain matrices in the dynamic output containment protocol can be determined by solving an algebraic Riccati equation. Although output containment problems have been investigated in [34], only sufficient conditions were obtained, and both a cone complementarity linearization algorithm and an interactive linear matrix inequality algorithm were required to determine the gain matrices in the protocol. The existence of the controller cannot be ensured using the results in [34]. Moreover, in the case that M = N–1, the results in the current paper can be applied to solve the output consensus tracking problems. 5. Numerical simulations In this section, a numerical example is given to illustrate the effectiveness of theoretical results obtained in the previous sections. The interaction topology of the swarm system is shown in Fig. 1. For simplicity, it is assumed that the interaction topology has 0 − 1 weights. Consider a fifth-order swarm system with three leaders and eight followers. The dynamics of each agent is described by (1) with xi (t ) = [xi1 (t ), xi2 (t ), . . . , xi5 (t )]T , yi (t ) = [yi1 (t ), yi2 (t ), yi3 (t )]T (i = 1, 2, . . . , 11), and
−9 −23 A = −2 −7 1 1 1 0
C =
0 0 1
6 10 0 −2 −2 0 1 0
7 5 −2 1 1 0 0 −1
−2 −6 0
−2 2
9 23 −6 , −9 −5
1 3 B = 0 , 1 −1
1 −1 . 0
Choose C¯ =
1 0
0 −1
(a) t = 0 s.
−1 1
0 0
−1
1 −1
2
.
Let K4 = [−2.5395, 6.9380, 1.9890, −7.1270], then Ao − Bo K4 is Hurwitz. Since (A, B) is stabilizable, by Theorem 3, one can obtain that
Fig. 1. Directed interaction topology G.
−1 . −1
(b) t = 5 s.
−4 4 0 −4 1.079 −13.876 0.022 10.254 K1 = , 9.079 −17.876 0.022 6.254 13.079 −9.876 0.022 10.254 −0.1088 −0.0750 −0.3467 0 −0.0750 −0.3460 −0.6300 0 , K2 = −0.3467 −0.6300 −1.8779 0 0.0488 −0.5132 −0.6358 0 −0.1088 −0.0750 −0.3467 −0.0750 −0.3460 −0.6300 . K3 = −0.3467 −0.6300 −1.8779 0.0488 −0.5132 −0.6358 Let zi (t ) = [zi1 (t ), zi2 (t ), zi3 (t ), zi4 (t )]T (i = 1, 2, . . . , 11). For simplicity let the initial states of each agent be of description, ¯ − 0.5 (i = 1, 2, . . . , 11; j = 1, 2, . . . , 5) and xij (0) = 4 Θ
¯ − 0.5 (i = 1, 2, . . . , 11; j = 1, 2, 3, 4) where Θ ¯ is zij (0) = 4 Θ a pseudorandom value with a uniform distribution on the interval (0, 1). Figs. 2 and 3 show the output snapshots of the eleven agents at different time, where the outputs of leaders and followers are denoted by pentagrams and asterisks respectively, and the convex hull formed by the output of leaders is marked by solid lines. From Figs. 2 and 3, one sees that the outputs of followers converge to the convex hull formed by those of leaders, which means that the swarm system achieves output containment.
6. Conclusions Output containment analysis and design problems for highorder linear time-invariant swarm systems with directed interaction topologies were investigated. Dynamic output containment protocols were presented for leaders and followers respectively. Necessary and sufficient conditions for swarm systems to achieve output containment were presented. An approach to decrease the calculation complexity of the criteria was given. An approach to determine the gain matrices in the dynamic output containment protocol was presented. As an interesting and important research topic, one can aim at designing the output containment protocol without using the global information of the interaction topology.
(c) t = 50 s.
Fig. 2. Output snapshots of eleven agents at different time.
X. Dong et al. / Systems & Control Letters (
(a) View of yi1 (t ) − yi2 (t ).
(b) View of yi1 (t ) − yi3 (t ).
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(c) View of yi2 (t ) − yi3 (t ).
Fig. 3. Different views of output snapshots at t = 50 s.
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[19] Y.G. Hong, J.P. Hu, L.X. Gao, Tracking control for multi-agent consensus with an active leader and variable topology, Automatica 42 (7) (2006) 1177–1182. [20] Y.G. Hong, G.R. Chen, L. Bushnell, Distributed observers design for leaderfollowing control of multi-agent networks, Automatica 44 (3) (2008) 846–850. [21] K. Peng, Y. Yang, Leader-following consensus problem with a varying-velocity leader and time-varying delays, Physica A 388 (2–3) (2009) 193–208. [22] W. Ni, D.Z. Cheng, Leader-following consensus of multi-agent systems under fixed and switching topologies, Systems Control Lett. 59 (3–4) (2010) 209–217. [23] Y.J. Zhang, Y. Yang, Y. Zhao, G.H. Wen, Distributed finite-time tracking control for nonlinear multi-agent systems subject to external disturbances, Internat. J. Control 86 (1) (2013) 29–40. [24] M. Ji, G. Ferrari-Trecate, M. Egerstedt, A. Buffa, Containment control in mobile networks, IEEE Trans. Automat. Control 53 (8) (2008) 1972–1975. [25] Z.Y. Meng, W. Ren, Z. You, Distributed finite-time attitude containment control for multiple rigid bodies, Automatica 46 (12) (2010) 2092–2099. [26] G. Notarstefano, M. Egerstedt, M. Haque, Containment in leader-follower networks with switching communication topologies, Automatica 47 (5) (2011) 1035–1040. [27] Y.C. Cao, W. Ren, M. Egerstedt, Distributed containment control with multiple stationary or dynamic leaders in fixed and switching directed networks, Automatica 48 (8) (2012) 1586–1597. [28] Y.C. Cao, D. Stuart, W. Ren, Z.Y. Meng, Distributed containment control for multiple autonomous vehicles with double-integrator dynamics: algorithms and experiments, IEEE Trans. Control Syst. Technol. 19 (4) (2011) 929–938. [29] H.Y Liu, G.M. Xie, L. Wang, Necessary and sufficient conditions for containment control of networked multi-agent systems, Automatica 48 (7) (2012) 1415–1422. [30] Y.C. Lou, Y.G. Hong, Target containment control of multi-agent systems with random switching interconnection topologies, Automatica 48 (5) (2012) 879–885. [31] Z.K. Li, W. Ren, X.D. Liu, M.Y. Fu, Distributed containment control of multiagent systems with general linear dynamics in the presence of multiple leaders, Int. J. Robust Nonlinear Control 23 (5) (2013) 534–547. [32] X.W. Dong, J.X. Xi, Z.Y. Shi, Y.S. Zhong, Containment analysis and design for high-order linear time-invariant singular swarm systems with time delays, Int. J. Robust Nonlinear Control 24 (7) (2014) 1189–1204. [33] H.Y. Liu, G.M. Xie, L. Wang, Containment of linear multi-agent systems under general interaction topologies, Systems Control Lett. 61 (4) (2012) 528–534. [34] X.W. Dong, Z.Y. Shi, G. Lu, Y.S. Zhong, Output containment analysis and design for high-order linear time-invariant swarm systems, Int. J. Robust Nonlinear Control (2013) http://dx.doi.org/10.1002/rnc.3118. Published online in Wiley online library. [35] C. Godsil, G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001. [36] L. Scardovi, R. Sepulchre, Synchronization in networks of identical linear systems, Automatica 45 (11) (2009) 2557–2562. [37] J.X. Xi, N. Cai, Y.S. Zhong, Consensus analysis and design for high-order linear swarm systems with time-varying delays, Physica A 390 (23) (2011) 4114–4123. [38] Z.K. Li, W. Ren, X.D. Liu, L.H. Xie, Distributed consensus of linear multiagent systems with adaptive dynamic protocols, Automatica 49 (7) (2013) 1986–1995.
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Contents lists available at ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Output containment control for swarm systems with general linear dynamics: A dynamic output feedback approach✩ Xiwang Dong, Fanlin Meng, Zongying Shi, Geng Lu, Yisheng Zhong ∗ Department of Automation, TNlist, Tsinghua University, Beijing, 100084, PR China
article
info
Article history: Received 4 January 2014 Received in revised form 5 May 2014 Accepted 13 June 2014 Available online xxxx Keywords: Output containment High-order Swarm system Dynamic output feedback
abstract Output containment control problems for high-order linear time-invariant swarm systems under directed interaction topologies are investigated using a dynamic output feedback approach. Firstly, to propel the outputs of followers to converge to the convex hull formed by the outputs of leaders, a dynamic output containment protocol is presented. Then necessary and sufficient conditions for swarm systems to achieve output containment are proposed. To ensure the scalability of the criteria, a sufficient condition which only includes two linear matrix inequality constraints independent of the number of agents is further presented. Moreover, an approach independent of the number of agents is proposed to determine the gain matrices in the dynamic output containment protocols. Finally, numerical simulations are presented to demonstrate theoretical results. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Recently, distributed cooperative control of swarm systems has gained much attention due to its broad applications including flocking [1,2], synchronization [3,4], and formation control [5,6]. Many existing works in distributed cooperative control of swarm systems focus on the consensus problem, where all agents reach an agreement on certain variables of interest in the case that there is no leader in the swarm system, such as [7–16] and references therein. In some practical applications, the existence of leaders can broaden the applications of consensus due to that a group objective can be brought by the leaders. Consensus with one leader is often known as consensus tracking or coordinated tracking. Consensus tracking problems for first-order continuous and discrete swarm systems with directed interaction topologies were studied in [17,18]. Hong et al. presented sufficient conditions for first-order and second-order linear time-invariant (LTI) swarm systems with fixed and switching interaction topologies to achieve consensus tracking in [19,20]. For second-order LTI swarm systems with switching interaction topologies and time-delays, Peng and Yang [21] derived sufficient
✩ This work was supported by the National Natural Science Foundation of China under Grants 61374034 and 61174067. ∗ Corresponding author. Tel.: +86 10 62772573. E-mail address: [email protected] (Y. Zhong).
http://dx.doi.org/10.1016/j.sysconle.2014.06.007 0167-6911/© 2014 Elsevier B.V. All rights reserved.
conditions to achieve consensus tracking with uniformly and ultimately bounded errors. Ni and Cheng [22] addressed consensus tracking problems for high-order LTI swarm systems with both fixed and switching undirected interaction topologies. Zhang et al. [23] investigated the distributed finite-time consensus tracking problems for nonlinear swarm systems with external disturbances. In the case of multiple leaders, containment problems arise, which require that the states/outputs of followers converge to the convex hull formed by the states/outputs of leaders. Ji et al. [24] proposed a hybrid stop–go control strategy for first-order swarm systems to achieve containment. Meng et al. [25] presented nonlinear control protocols for rigid body swarm systems to achieve containment in a finite time. Notarstefano et al. [26] discussed containment problems for first-order swarm systems with switching undirected interaction topologies. Cao et al. dealt with containment problems for first-order and second-order swarm systems with both stationary and dynamic leaders in [27,28]. Liu et al. [29] proposed necessary and sufficient conditions for firstorder and second-order swarm systems to achieve containment. Containment problems for second-order swarm systems with random switching interaction topologies were studied in [30]. However, in many practical cases, the dynamics of each agent is of high-order, as shown in [8]. Containment problems for high-order swarm systems are more challenging since each agent contains more complex dynamics. Moreover, they have generality and it can be shown that containment problems for first-order and secondorder swarm systems (e.g., [26–30]) can be regarded as special
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cases of high-order swarm systems. Therefore, containment problems for high-order swarm systems make more sense. Containment problems for both continuous and discrete highorder LTI swarm systems were investigated in [31]. Dong et al. [32] presented sufficient conditions for high-order LTI singular swarm systems with time-delays to achieve containment. By classifying the agents into boundary agents and internal agents, Liu et al. [33] proposed a criterion for the states of internal agents to converge to a convex combination of the states of the boundary agents. It should be pointed out that in [24–30,32,33], the relative states of neighboring agents are required to construct the containment protocols, and in [24–33], the states of followers are required to converge to the convex hull formed by the states of leaders. However, in some practical applications, only the relative outputs of neighboring agents are available, and only outputs of followers are required to converge to the convex hull formed by the outputs of leaders. Because outputs are linear combinations of states and may not be related with all states, both the analysis and design for output containment problems of high-order LTI swarm systems become complicated and challenging. Dong et al. [34] dealt with output containment problems using a static output containment protocol. However, they only proposed sufficient conditions to achieve output containment and the existence of the controller cannot be ensured. To the best of our knowledge, necessary and sufficient conditions for high-order swarm system to achieve output containment have not been obtained before. In the current paper, output containment analysis and design problems for high-order LTI swarm systems with dynamic protocols are discussed. Firstly, to partly improve the output behavior of leaders and propel the outputs of followers to converge to the convex hull formed by the outputs of leaders, dynamic output containment protocols are presented for leaders and followers respectively. Then necessary and sufficient conditions for swarm systems to achieve output containment are presented. An approach is introduced to decrease the calculation complexity of the criteria. Finally, an approach to determine the gain matrices in the dynamic output containment protocols is proposed. Compared with the existing works on consensus and containment, the novel features of the current paper are threefold. Firstly, both output containment analysis and design problems are investigated for high-order LTI swarm systems. In [15], output consensus problems were studied, where there do not exist multiple leaders. In [24–33], only state containment problems were discussed. Secondly, necessary and sufficient criteria for swarm systems to achieve output containment are proposed. The criteria in [34] are only sufficient. Thirdly, the approach to determine the gain matrices in the dynamic protocols has less calculation complexity, which only needs to solve an algebraic Riccati equation. In [34], it is difficult to design the output containment protocol since the output containment problems were transformed into static output feedback stabilization problems, and the existence of the controller cannot be ensured. Moreover, the results in this paper can be applied to solve the output consensus tracking problems for high-order LTI swarm systems. The rest of this paper is organized as follows. In Section 2, basic concepts on graph theory are introduced and the problem description is given. In Section 3, necessary and sufficient conditions for swarm systems to achieve dynamic output containment are proposed. In Section 4, an approach to determine the gain matrices in the dynamic protocols is presented. Numerical simulations are provided in Section 5. Finally, Section 6 concludes the whole work. Throughout this paper, for simplicity of notation, 0 is used to denote zero matrices of appropriate size with zero vectors and zero number as special cases, and superscript H represents the Hermitian transpose of matrices. I stands for an identity matrix with appropriate dimension and ⊗ refers to the Kronecker product.
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2. Preliminaries and problem description In this section, basic concepts on graph theory are introduced and the problem description is presented. 2.1. Basic concepts on graph theory Let G = {V , E , W } be a directed graph of order N with V = {v1 , v2 , . . . , vN } the set of nodes, E ⊆ (vi , vj ) : vi , vj ∈ V , i ̸= j the set of edges and W = [wij ] ∈ RN ×N the weighted adjacency matrix. An edge of G is denoted by eij = (vi , vj ). The weighted matrix elements are positive, i.e., wji > 0 if and only if eij ∈ E. Moreover, wii = 0 for all i ∈ {1, 2, . . . , N }. The set of neighbors of node vi is denoted by Ni = vj ∈ V : (vj , vi ) ∈ E . The in-degree of N node vi is defined as degin (vi ) = j=1 wij . The degree matrix of G is denoted by D = diag {degin (vi ), i = 1, 2, . . . , N }. The Laplacian matrix of G is defined as L = D − W . A directed path from node vi1 to vil is a sequence of ordered edges with the form of (vik , vik+1 ), where vik ∈ V (k = 1, 2, . . . , l − 1). More details on graph theory can be found in [35]. 2.2. Problem description For a swarm system with N agents, the interaction topology of the swarm system can be described by directed graph G, each agent can be regarded as a node in G. For i, j ∈ {1, 2, . . . , N }, the available interaction channel from agent i to agent j can be denoted by the edge and the interaction strength can be denoted by eij . Consider the following high-order LTI dynamics for each agent
x˙ i (t ) = Axi (t ) + Bui (t ), yi (t ) = Cxi (t ),
(1)
where i ∈ {1, 2, . . . , N }, xi (t ) ∈ Rn is the state, yi (t ) ∈ Rq is the output, and ui (t ) ∈ Rm is the control input. Assumption 1. C is of full row rank and (A, B) is stabilizable. Remark 1. In Assumption 1, C is assumed to be of full row rank, which means that the elements in the output y(t ) are independent of each other. If C is not of full row rank, then some of the elements in y(t ) can be expressed as linear combination of the others. In this case, those elements in y(t ) which are linearly dependent on others can be ignored due to that they involves no new information. Definition 1. An agent is called a leader if it has no neighbors. An agent is called a follower if it has at least one neighbor. Assume that in swarm system (1) there are M (M < N ) followers with states xk (t ) (k = 1, 2, . . . , M ) and N − M leaders with states xi (t ) (i = M + 1, M + 2, . . . , N ). Let F = {1, 2, . . . , M } and J = {M + 1, M + 2, . . . , N } be the follower subscript set and leader subscript set respectively. Under Definition 1, the Laplacian matrix corresponding to G has the following form
L=
L1 0
L2 , 0
where L1 ∈ RM ×M and L2 ∈ RM ×(N −M ) . Assumption 2. For each follower, there exists at least one leader that has a directed path to it. Under Assumption 2, the following lemma holds. Lemma 1 ([25]). If the interaction topology satisfies Assumption 2, then all the eigenvalues of L1 have positive real parts, each entry of 1 −1 −L − 1 L2 is nonnegative, and each row of −L1 L2 has a sum equal to one.
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Definition 2. Swarm system (2) is said to achieve output containment if for any k ∈ F , there exist nonnegative constants αk,j (j ∈ E )
3. Output containment analysis
satisfying
In this section, firstly swarm systems (5) and (6) are transformed into systems which include subsystems with states yF (t ) and yJ (t ) respectively. Then necessary and sufficient conditions for swarm system (1) under dynamic protocols (3) and (4) to achieve output containment are presented. A sufficient condition with less calculation complexity is further proposed. By Assumption 1, there exists C¯ ∈ R(n−q)×n such that T = T ¯T T [C , C ] is nonsingular. Let y¯ i (t ) = C¯ xi (t ) (i = 1, 2, . . . , N ), y¯ F (t ) = [¯yT1 (t ), y¯ T2 (t ), . . . , y¯ TM (t )]T , y¯ J (t ) = [¯yTM +1 (t ), y¯ TM +2 (t ), . . . , y¯ TN (t )]T , and
lim
t →∞
αk,j = 1 such that N yk (t ) − αk,j yj (t ) = 0. N
j=M +1
(2)
j=M +1
Consider the following dynamic output containment protocols for followers and leaders respectively
z˙i (t ) = K1 zi (t ) + K2 wij zi (t ) − zj (t ) j∈Ni + K3 w yi (t ) − yj (t ) , i ∈ F , ij j∈Ni ui (t ) = K4 zi (t ), i ∈ F , z˙i (t ) = K1 zi (t ), i ∈ J , ui (t ) = K4 zi (t ), i ∈ J ,
(3) TAT
(4)
where zi (t ) ∈ R (i = 1, 2, . . . , N ) are the states of the protocols, and K1 , K2 , K3 and K4 are constant gain matrices with appropriate dimensions. Let p
T zF (t ) = [z1T (t ), z2T (t ), . . . , zM (t )]T ,
xF (t ) = [xT1 (t ), xT2 (t ), . . . , xTM (t )]T , yF (t ) = [yT1 (t ), yT2 (t ), . . . , yTM (t )]T , T T T T zJ (t ) = [zM +1 (t ), zM +2 (t ), . . . , zN (t )] ,
xJ (t ) = [xTM +1 (t ), xTM +2 (t ), . . . , xTN (t )]T , yJ (t ) = [yTM +1 (t ), yTM +2 (t ), . . . , yTN (t )]T . Under protocols (3) and (4), the dynamics of swarm system (1) can be written in a compact form as
z˙ (t ) = (IM ⊗ K1 + L1 ⊗ K2 )zF (t ) + (L2 ⊗ K2 )zJ (t ) F + (L1 ⊗ K3 )yF (t ) + (L2 ⊗ K3 )yJ (t ), x˙ F (t ) = (IM ⊗ A)xF (t ) + (IM ⊗ BK4 )zF (t ), yF (t ) = (IM ⊗ C )xF (t ), z˙J (t ) = (IN −M ⊗ K1 )zJ (t ), x˙ J (t ) = (IN −M ⊗ A)xJ (t ) + (IN −M ⊗ BK4 )zJ (t ), yJ (t ) = (IN −M ⊗ C )xJ (t ).
(5)
Remark 2. A variety of observer-like dynamic protocols were proposed for swarm systems to achieve consensus or state containment, such as protocol (13) in [36] and protocol (2) in [31]. It can be verified that protocol (13) in [36] can be regarded as a special case of protocol (3) by choosing K1 = A + BK , K2 = −I, K3 = I, K4 = K . The dimension of zi (t ) must be the same with that of xi (t ) in protocol (13) of [36] and protocol (2) of [31]. However, in protocol (3) of this paper, the dimension of zi (t ) can be less than that of xi (t ). Moreover, protocol (3) has more degrees of freedom for swarm system (1) to achieve output containment. From (6), one sees that K1 , K4 and zi (0) (i ∈ J ) can be used to partly change the behavior of leaders, which is useful for the scenarios that the behavior of leaders needs to be improved. It should be pointed out that by choosing zi (0) = 0 (i ∈ J ), the output containment problems in this paper become the ones in which leaders have no control input. In the current paper, the following two problems for swarm system (1) under dynamic protocols (3) and (4) are mainly addressed: (i) under what conditions output containment can be achieved; and (ii) how to determine the gain matrices in dynamic protocols (3) and (4) to achieve output containment.
A¯ = ¯ 11 A21
B¯ TB = ¯ 1 . B2
A¯ 12 , A¯ 22
Then swarm systems (5) and (6) can be converted into
z˙F (t ) = (IM ⊗ K1 + L1 ⊗ K2 ) zF (t ) + (L2 ⊗ K2 ) zJ (t ) + (L1 ⊗ K3 ) yF (t ) +(L2 ⊗ K3 ) yJ (t ), y˙ F (t ) = IM ⊗ A¯ 11 yF (t ) + IM ⊗ A¯ 12 y¯ F (t ) + IM ⊗ B¯ 1 K4 zF (t ), y˙¯ F (t ) = IM ⊗ A¯ 21 yF (t ) + IM ⊗ A¯ 22 y¯ F (t ) + IM ⊗ B¯ 2 K4 zF (t ), z˙ (t ) = (IN −M ⊗ K1 ) zJ (t ), J y ˙ J (t ) = IN−M ⊗ A¯ 11 yJ (t ) + IN −M ⊗ A¯ 12 y¯ J (t ) ¯ + IN −M ⊗ B1 K4 zJ (t ), ˙ ¯ y¯ (t ) = IN −M ⊗ A21 yJ (t ) + IN −M ⊗ A¯ 22 y¯ J (t ) J + IN −M ⊗ B¯ 2 K4 zJ (t ).
(7)
(8)
In (7) and (8), the subsystems with states yF (t ) and yJ (t ) can be used to analyse the output containment problems. It can be found that only the observable components of (A¯ 22 , A¯ 12 ) influence the subsystems with states yF (t ) and yJ (t ). Therefore, the observability decomposition of (A¯ 22 , A¯ 12 ) is presented firstly. Let T˜ be a nonsingular matrix such that
(6)
−1
T˜ −1 A¯ 22 T˜ , A¯ 12 T˜
˜1 D ˜2 D
=
0 ˜ ˜D3 , E1
0
,
˜ 1 ∈ Rg and (D˜ 1 , E˜ 1 ) is completely observable. Let yˆ i (t ) = where D − 1 T˜ y¯ i (t ) = [ˆyTio (t ), yˆ Tio¯ (t )]T (i = 1, 2, . . . , N ), T˜ −1 A¯ 21 = [F˜1T , F˜2T ]T , T˜ −1 B¯ 2 = [B˜ T1 , B˜ T2 ]T , and
yˆ Fo (t ) = [ˆyT1o (t ), yˆ T2o (t ), . . . , yˆ TMo (t )]T , yˆ F o¯ (t ) = [ˆyT1o¯ (t ), yˆ T2o¯ (t ), . . . , yˆ TM o¯ (t )]T , yˆ Jo (t ) = [ˆyT(M +1)o (t ), yˆ T(M +2)o (t ), . . . , yˆ TNo (t )]T , yˆ J o¯ (t ) = [ˆyT(M +1)¯o (t ), yˆ T(M +2)¯o (t ), . . . , yˆ TN o¯ (t )]T . Then systems (7) and (8) can be further converted into
z˙F (t ) = (IM ⊗ K1 + L1 ⊗ K2 ) zF (t ) + (L2 ⊗ K2 ) zJ (t ) + (L1 ⊗ K3 ) yF (t ) +(L2 ⊗ K3) yJ (t ), ˙ y ( t ) = IM ⊗ A¯ 11 yF (t ) + IM ⊗ E˜ 1 yˆ Fo (t ) F + IM ⊗ B¯1 K4 zF (t ), ˜ 1 yˆ Fo (t ) y˙ˆ Fo (t ) = IM ⊗ F˜1 yF (t ) + IM ⊗ D + IM ⊗ B˜ 1 K4 zF (t ), y˙ˆ F o¯ (t ) = IM ⊗ F˜2 yF (t ) + IM ⊗ D˜ 2 yˆ Fo (t ) + IM ⊗ D˜ 3 yˆ F o¯ (t ) + IM ⊗ B˜ 2 K4 zF (t ),
(9)
4
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˙ zJ (t ) = (IN −M ⊗ K1 ) zJ (t ), y˙ J (t ) = IN −M ⊗ A¯ 11 yJ (t ) + IN −M ⊗ E˜ 1 yˆ Jo (t ) + IN −M ⊗ B¯1 K4 zJ (t ), ˙ˆ ˜ 1 yˆ Jo (t ) yJo (t ) = IN −M ⊗ F˜1 yJ (t ) + IN −M ⊗ D ˜ 1 K4 zJ (t ), + I ⊗ B N − M ˙ˆ (t ) = IN −M ⊗ F˜2 yJ (t ) + IN −M ⊗ D˜ 2 yˆ Jo (t ) y ¯ J o + IN −M ⊗ D˜ 3 yˆ J o¯ (t ) + IN −M ⊗ B˜ 2 K4 zJ (t ).
λ i K3 A¯ 11 F˜1
(10)
0 E˜ 1 . ˜1 D
Proof. Sufficiency: Let φF (t ) = [zFT (t ), yTF (t ), yˆ TFo (t )]T , φJ (t ) = [zJT (t ), yTJ (t ), yˆ TJo (t )]T , and φ(t ) = [φFT (t ), φJT (t )]T . From (9) and (10), one has
˙ t ) = Φ1 φ( 0
Φ2 φ(t ), Φ3
(11)
where
IM ⊗ K1 + L 1 ⊗ K2 IM ⊗ B¯ 1 K4 Φ1 = IM ⊗ B˜ 1 K4 L2 ⊗ K2 0 0
L 2 ⊗ K3 0 0
Φ2 =
IN − M ⊗ K1 Φ3 = IN −M ⊗ B¯ 1 K4 IN −M ⊗ B˜ 1 K4
L 1 ⊗ K3 IM ⊗ A¯ 11 IM ⊗ F˜1
0
IM ⊗ E˜ 1 , ˜1 IM ⊗ D
eΦ1 t φ(t ) = 0
Θ1
eΦ3 t
0 IN −M ⊗ E˜ 1 . ˜1 IN − M ⊗ D
Θ1 = L−1 (sI − Φ1 )−1 Φ2 (sI − Φ3 )−1 , with L−1 the inverse Laplace transform and s the Laplace operator. It can be shown that
(sI − Φ1 )−1 Φ2 (sI − Φ3 )−1 = (sI − Φ1 )−1 Θ2 − Θ2 (sI − Φ3 )−1 1 L− 1 L2 ⊗ I Θ2 = 0 0
It is well known that one typical feature of swarm systems is of large scale. For swarm systems with a very large M, it may be time costly to check the conditions in Theorem 1. By encapsulating λi (i = 1, 2, . . . , M ) into a convex set, a sufficient condition with less calculation complexity is obtained in the following theorem. Let 0 A¯ 11 F˜1
0 . 0 1 L− L ⊗ I 2 1
K2 0 0
ψ2 =
K3 0 0
0 0 . 0
Re(λ)I Im(λ)I
−Im(λ)I . Re(λ)I
Lemma 2 ([37]). For given real symmetric matrices Ω0 , Ω1 and Ω2 ¯ i (i = 1, 2, 3, 4), if Ω0 + independent of λi (i = 1, 2, . . . , M ) and λ ¯ i Ω1 + Im λ¯ i Ω2 < 0 (i = 1, 2, 3, 4), then Ω0 + Re(λi )Ω1 + Re λ Im(λi )Ω2 < 0 (i = 1, 2, . . . , M ). Theorem 2. If the interaction topology satisfies Assumption 2, swarm system (1) under dynamic protocols (3) and (4) achieves output containment if there exists a matrix R¯ = R¯ T > 0 such that
T Λψ1 + Υλ¯ i Λψ2 ΛR¯ + ΛTR¯ Λψ1 + Υλ¯ i Λψ2 < 0 (i = 1, 3). (16)
T
Proof. Let Γi = Λψ1 + Υλi Λψ2 ΛR¯ + ΛTR¯ Λψ1 + Υλi Λψ2 1, 2, . . . , M ), then one can obtain
0 1 L− L 2 ⊗I 1 0
0 E˜ 1 , ˜1 D
¯ 1,2 = Re(λ1 ) ± jµF and λ¯ 3,4 = Re(λM ) ± jµF where µF = Set λ max{Im(λi ), i = 1, 2, . . . , M }. Then the following lemma holds.
where
Remark 3. From the structure of ΨFi (i = 1, 2, . . . , M ), one sees that the output containment of swarm systems depends on the dynamics of each agent, the observable components of (A¯ 22 , A¯ 12 ), the interaction topology and Ki (i = 1, 2, 3, 4).
the imaginary part of λ respectively, and Υλ =
(12)
where
From (15) and Lemma 1, one knows that swarm system (1) under dynamic protocols (3) and (4) achieves output containment. Necessity: The necessity is proven by contradiction. Choose φJ (0) = 0 or φJ (0) in the unobservable subspace of (Φ3 , [0 I 0]). In this case, yJ (t ) ≡ 0. If swarm system (1) under dynamic protocols (3) and (4) achieves output containment but subsystem ΨFi , 0 I 0 has at least one observable mode which is not stable for some i ∈ {1, 2, . . . , M }, then subsystem asymptotically Φ1 , 0 I 0 has atleast one observable mode that is not asymptotically stable. Hence, one can find nonzero φF (0) such that limt →∞ yF (t ) ̸= 0; that is, swarm system (1) under dynamic protocols (3) and (4) does not achieve output containment. This results a contradiction. the condition that the Therefore, observable modes of ΨFi , 0 I 0 are asymptotically stable is required. The proof for Theorem 1 is completed.
Then ΨFi = ψ1 + λi ψ2 . For any real matrix R and any λ ∈ C, let ΛR = diag{R, R}, Re(λ) and Im(λ) denote the real part and
φ(0),
0
K1 ψ1 = B¯ 1 K4 B˜ 1 K4
The solution to Eq. (11) can be written as
t →∞
0 0 , 0
0 IN −M ⊗ A¯ 11 IN −M ⊗ F˜1
1 I 0 eΦ1 t φF (0) + L− 1 L2 ⊗ I φJ (0) −1 + −L1 L2 ⊗ I yJ (t ), (13) Φt 3 yJ (t ) = 0 I 0 e φJ (0). (14) If the observable modes of ΨFi , 0 I 0 (i = 1, 2, . . . , M ) ¯ are asymptotically stable, by the structure of U F and ΛF , it can be obtained that limt →∞ 0 I 0 eΦ1 t = 0. Therefore, 1 lim yF (t ) − −L− (15) 1 L2 ⊗ I yJ (t ) = 0. yF (t ) =
Theorem 1. If the interaction topology satisfies Assumption 2, swarm system (1) under dynamic protocols (3) and (4) with any bounded initial states achieves output containment if and onlyif for ∀i ∈ {1, 2, . . . , M } the observable modes of ΨFi , 0 I 0 are asymptotically stable, where K1 + λi K2 ΨFi = B¯ 1 K4 B˜ 1 K4
–
From (12), one has
Let UF ∈ CM ×M be a nonsingular matrix such that UF−1 L1 UF = ¯ F , where Λ ¯ F is an upper-triangular matrix with λi (i = Λ 1, 2, . . . , M ) as its diagonal entries and Re(λ1 ) ≤ Re(λ2 ) ≤ · · · ≤ Re(λM ). The following theorem presents necessary and sufficient conditions for swarm system (1) under dynamic protocols (3) and (4) to achieve output containment.
)
Γi = Ω0 + Re(λi )Ω1 + Im(λi )Ω2 ,
(i =
X. Dong et al. / Systems & Control Letters (
Ω1 =
T ψ1 R¯ + R¯ T ψ1 0
T ψ2 R¯ + R¯ T ψ2 0
0
0
ψ ¯ + R¯ ψ1 T 1R
T
5
0
K + λ i K2 ΨFi = 1 Bo K4
0 T¯ = I
Ao
(i = 1, 2, . . . , M ).
(18)
, ψ2T R¯ + R¯ T ψ2 ψ2T R¯ − R¯ T ψ2
Let
λi K3 Co
,
. −ψ2T R¯ + R¯ T ψ2 0 T Similarly, let Ξi = Λψ1 + Υλ¯ i Λψ2 ΛR¯ + ΛTR¯ Λψ1 + Υλ¯ i Λψ2 (i = 1, 2, 3, 4), then it follows Ξi = Ω0 + Re(λ¯ i )Ω1 + Im(λ¯ i )Ω2 . Ω2 =
–
then ΨFi (i = 1, 2, . . . , M ) can be rewritten as
where
Ω0 =
)
Theorem 3. If the interaction topology satisfies Assumptions 1 and 2, then for any bounded initial states, swarm system (1) achieves output containment by dynamic protocols (3) and (4) with K4 satisfying that 1 T Ao − Bo K4 is Hurwitz, K1 = Ao − Bo K4 , K3 = −[Re(λ1 )]−1 (R− o Co Po ) , T and K2 = K3 Co , where Po = Po > 0 is the solution to the algebraic Riccati equation 1 Po ATo + Ao Po − Po CoT R− o Co Po + Qo = 0,
for Ro =
I . 0
RTo
> 0 and Qo =
(19)
≥ 0 with ( ,
DTo Do
Do ATo
) detectable.
Proof. Let K1 = Ao − Bo K4 and K2 = K3 Co , then one can obtain
¯ 1 = −Im λ¯ 2 and Im λ¯ 3 = −Im λ¯ 4 , one has Since Im λ Ξ1 = T¯ Ξ2 T¯ −1 and Ξ3 = T¯ Ξ4 T¯ −1 . Therefore, if Ξ1 < 0 and Ξ3 < 0, then Ξ2 < 0 and Ξ4 < 0 respectively. By Lemma 2, one knows that if condition (16) holds, then Γi < 0 (i = 1, 2, . . . , M ). Consider the stability of the following subsystems
ξ˙i (t ) = Λψ1 + Υλi Λψ2 ξi (t ) (i = 1, 2, . . . , M ).
(17)
(i = 1, 2, . . . , M ),
λi K3 Co
Ao
(i = 1, 2, . . . , M ).
It can be verified that ΨFi (i = 1, 2, . . . , M ) are similar to
A + λi K3 Co Ψ¯ Fi = o
0 Ao − Bo K4
Bo K4
Choose the Lyapunov functional candidates as follows Vi (t ) = ξiH (t )ΛR¯ ξi (t )
A − Bo K4 + λi K3 Co ΨFi = o Bo K4
(i = 1, 2, . . . , M ).
Since T and T˜ are nonsingular, if (A, B) is stabilizable, then by the PBH criterion for stabilizability, one has
where ξiH stand for the Hermitian transpose of ξi . Taking the time derivative of Vi (t ) along the trajectory of (17), one has
sI − A¯ 11 rank −F˜1 −F˜2
T V˙ i (t ) = ξiH (t ) Λψ1 + Υλi Λψ2 ΛR¯ + ΛR¯ Λψ1 + Υλi Λψ2 ξi (t ).
¯ + = {s|s ∈ C, Re(s) ≥ 0}. It also can be shown that where s ∈ C
Since Γi < 0 (i = 1, 2, . . . , M ), V˙ i (t ) < 0; that is, Λψ1 + Υλi Λψ2 (i = 1, 2, . . . , M ) are Hurwitz. By the decomposition of real and imaginary parts, it can be verified that ΨFi (i = 1, 2, . . . , M ) are Hurwitz due to that Λψ1 + Υλi Λψ2 (i = 1, 2, . . . , M ) are Hurwitz. Therefore, the observable modes of ΨFi , 0 I 0 are asymptotically stable. From Theorem 1, one knows that swarm system (1) under dynamic protocols (3) and (4) achieves output containment. The proof of Theorem 2 is completed. Remark 4. The condition in Theorem 1 is necessary and sufficient and Theorem 2 only includes two linear matrix inequality (LMI) constraints independent of the number of agents. It should be pointed out that the condition in Theorem 1 of [34] is only sufficient and Theorem 2 in [34] includes four LMI constraints. As a result, the criteria in this paper are more general than those in [34] and the calculation efficiency of this paper is better than that in [34]. Although condition (16) is independent of the number of agents, it needs the global information of the interaction topology due to that the eigenvalues of the Laplacian matrix depend on the global information of the interaction topology. The approaches in [36,38] may be used to estimate the eigenvalues of the Laplacian matrix without using the global information of the interaction topology. 4. Output containment protocol design In this section, an approach to determine Ki (i = 1, 2, 3, 4) in dynamic protocols (3) and (4) for swarm system (1) to achieve output containment is proposed. For simplicity of expression, let A¯ 11 F˜1
Ao =
E˜ 1 ˜1 , D
B¯ 1 Bo = , B˜ 1
Co = I
0 ,
sI − A¯ 11 −F˜1
rank
−E˜ 1 ˜1 sI − D −D˜ 2
−E˜ 1 ˜1 sI − D
˜3 sI − D
B¯ 1 B˜ 1
B¯ 1 B˜ 1 = n, B˜ 2
0 0
= q + g (∀s ∈ C¯ + ),
¯ +. which means that [sI − Ao , Bo ] is of full row rank for any s ∈ C Therefore, (Ao , Bo ) is stabilizable and there exists K4 such that Ao − Bo K4 is Hurwitz. ˜ 1 , E˜ 1 ) is completely observable, one knows that for Note that (D any s ∈ C,
sI − A¯ 11 −F˜1 I
−E˜ 1 ˜ 1 sI − D 0
is of full column rank, which means that (Ao , Co ) is completely observable. As a result, (ATo , CoT ) is completely controllable. Therefore, for any given Ro = RTo > 0 and Qo = DTo Do > 0 with (Do , ATo ) detectable, the algebraic Riccati equation (19) has a unique solution Po = PoT > 0. By Lemma 1, one knows that Re(λ1 ) > 0. Consider the stability of the following subsystems
ζ˙i (t ) = ATo + λHi CoT K3T ζi (t ) (i = 1, 2, . . . , M ).
(20)
Choose Lyapunov functional candidates as follows V¯ i (t ) = ζiH (t )Po ζi (t )
(i = 1, 2, . . . , M ).
(21)
1 T Let K3 = −[Re(λ1 )]−1 (R− o Co Po ) . Taking the time derivative of ¯Vi (t ) along the trajectory of (20), it holds that
V˙¯ i (t ) = −ζiH (t )Qo ζi (t ) + ζiH (t ) 1 − 2Re(λi )[Re(λ1 )]−1
1 × Po CoT R− o Co Po ζi (t ) ≤ 0.
Note that (Do , ATo ) is detectable and Ro = RTo > 0, one knows T T that ATo + λH i Co K3 (i = 1, 2, . . . , M ) are Hurwitz, which means
6
X. Dong et al. / Systems & Control Letters (
)
–
It can be shown that (A¯ 22 , A¯ 12 ) is not completely observable, then choose the following nonsingular matrix T˜ for observability decomposition T˜ =
that Ao + λi K3 Co (i = 1, 2, . . . , M ) are Hurwitz. Therefore, Ψ . . . , M ) are Hurwitz and the observable modes of Fi (i = 1, 2, ΨFi , 0 I 0 are asymptotically stable. From Theorem 1, the conclusion of Theorem 3 can be obtained. Remark 5. From Theorem 3, one sees that the output behavior of leaders can be partly improved by choosing K4 satisfying that Ao − Bo K4 is Hurwitz and zi (0) (i ∈ J ). If (A, B) is stabilizable, then the gain matrices in the dynamic output containment protocol can be determined by solving an algebraic Riccati equation. Although output containment problems have been investigated in [34], only sufficient conditions were obtained, and both a cone complementarity linearization algorithm and an interactive linear matrix inequality algorithm were required to determine the gain matrices in the protocol. The existence of the controller cannot be ensured using the results in [34]. Moreover, in the case that M = N–1, the results in the current paper can be applied to solve the output consensus tracking problems. 5. Numerical simulations In this section, a numerical example is given to illustrate the effectiveness of theoretical results obtained in the previous sections. The interaction topology of the swarm system is shown in Fig. 1. For simplicity, it is assumed that the interaction topology has 0 − 1 weights. Consider a fifth-order swarm system with three leaders and eight followers. The dynamics of each agent is described by (1) with xi (t ) = [xi1 (t ), xi2 (t ), . . . , xi5 (t )]T , yi (t ) = [yi1 (t ), yi2 (t ), yi3 (t )]T (i = 1, 2, . . . , 11), and
−9 −23 A = −2 −7 1 1 1 0
C =
0 0 1
6 10 0 −2 −2 0 1 0
7 5 −2 1 1 0 0 −1
−2 −6 0
−2 2
9 23 −6 , −9 −5
1 3 B = 0 , 1 −1
1 −1 . 0
Choose C¯ =
1 0
0 −1
(a) t = 0 s.
−1 1
0 0
−1
1 −1
2
.
Let K4 = [−2.5395, 6.9380, 1.9890, −7.1270], then Ao − Bo K4 is Hurwitz. Since (A, B) is stabilizable, by Theorem 3, one can obtain that
Fig. 1. Directed interaction topology G.
−1 . −1
(b) t = 5 s.
−4 4 0 −4 1.079 −13.876 0.022 10.254 K1 = , 9.079 −17.876 0.022 6.254 13.079 −9.876 0.022 10.254 −0.1088 −0.0750 −0.3467 0 −0.0750 −0.3460 −0.6300 0 , K2 = −0.3467 −0.6300 −1.8779 0 0.0488 −0.5132 −0.6358 0 −0.1088 −0.0750 −0.3467 −0.0750 −0.3460 −0.6300 . K3 = −0.3467 −0.6300 −1.8779 0.0488 −0.5132 −0.6358 Let zi (t ) = [zi1 (t ), zi2 (t ), zi3 (t ), zi4 (t )]T (i = 1, 2, . . . , 11). For simplicity let the initial states of each agent be of description, ¯ − 0.5 (i = 1, 2, . . . , 11; j = 1, 2, . . . , 5) and xij (0) = 4 Θ
¯ − 0.5 (i = 1, 2, . . . , 11; j = 1, 2, 3, 4) where Θ ¯ is zij (0) = 4 Θ a pseudorandom value with a uniform distribution on the interval (0, 1). Figs. 2 and 3 show the output snapshots of the eleven agents at different time, where the outputs of leaders and followers are denoted by pentagrams and asterisks respectively, and the convex hull formed by the output of leaders is marked by solid lines. From Figs. 2 and 3, one sees that the outputs of followers converge to the convex hull formed by those of leaders, which means that the swarm system achieves output containment.
6. Conclusions Output containment analysis and design problems for highorder linear time-invariant swarm systems with directed interaction topologies were investigated. Dynamic output containment protocols were presented for leaders and followers respectively. Necessary and sufficient conditions for swarm systems to achieve output containment were presented. An approach to decrease the calculation complexity of the criteria was given. An approach to determine the gain matrices in the dynamic output containment protocol was presented. As an interesting and important research topic, one can aim at designing the output containment protocol without using the global information of the interaction topology.
(c) t = 50 s.
Fig. 2. Output snapshots of eleven agents at different time.
X. Dong et al. / Systems & Control Letters (
(a) View of yi1 (t ) − yi2 (t ).
(b) View of yi1 (t ) − yi3 (t ).
)
–
7
(c) View of yi2 (t ) − yi3 (t ).
Fig. 3. Different views of output snapshots at t = 50 s.
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