Distributed containment control of singular heterogeneous multi-agent systems

Distributed containment control of singular heterogeneous multi-agent systems

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Distributed containment control of singular heterogeneous multi-agent systems Xuxi Zhang, Xianping Liu, Zhiguang Feng PII: DOI: Reference:

S0016-0032(19)30758-6 https://doi.org/10.1016/j.jfranklin.2019.10.025 FI 4224

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Journal of the Franklin Institute

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Please cite this article as: Xuxi Zhang, Xianping Liu, Zhiguang Feng, Distributed containment control of singular heterogeneous multi-agent systems, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.10.025

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Distributed containment control of singular heterogeneous multi-agent systems Xuxi Zhanga,∗, Xianping Liua , Zhiguang Fengb a

College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, P. R. China b College of Automation, Harbin Engineering University, Harbin 150001, P. R. China

Abstract This paper investigates the containment control problem of singular heterogeneous multi-agent systems under directed interaction topology. Firstly, a distributed observer, in which the eigenvalue information of the leaders is not needed, is proposed to estimate the convex hull spanned by the states of the leaders. Secondly, state feedback control and reduced-order normal observer based output feedback control are presented, respectively. Finally, some numerical simulation examples are performed to demonstrate the efficiency of the proposed theoretical results. Keywords: Containment control, heterogeneous, multi-agent systems, output feedback, reduced-order normal observer. 1. Introduction In recent years, the problem of cooperative control of multi-agent systems has attracted significant attention owing to its broad applications, such as, spacecraft formation flying, sensor networks, cooperative surveillance, voltage and frequency regulation of micro-grid systems, cooperation of robotic systems, and so on [1–6]. In the area of cooperative control, one of the fundamental problems is consensus, which means that a flock of agents achieves an agreement on a physical quantity of interest by collaborating with their local neighbors. The consensus problem can usually be categorized into three ∗

Corresponding author. Email address: [email protected] (Xuxi Zhang)

Preprint submitted to Journal of the Franklin Institute

November 8, 2019

cases, i.e., leaderless consensus [7–9], consensus with one leader [10–14], and consensus with multiple leaders [15–27]. In the case of multiple leaders, the main objective is to drive the follower agents into the convex hull spanned by the leader agents via some appropriate distributed control laws, and such a problem is also called the containment control problem. In fact, the research of containment control is motivated by some interesting natural phenomena, and has many potential applications in practice. For instance, the containment control algorithm can guarantee that a group of robots reach their destination safely when only a few robots are equipped with sensors to detect the hazardous obstacle [28]. In [16], under undirected interacting topology, the problem of attitude containment control was investigated for multiple rigid bodies by applying distributed finite time control method. In [17], under switching communication topologies, containment control problem was discussed for first-order multi-agent systems. In [18] and [19], under directed networks, distributed containment control problem for single-integrator dynamics and double-integrator dynamics with stationary or dynamic multiple leaders were discussed, respectively. In [20], some necessary and sufficient criteria have been proposed for achieving containment control for first-order and second-order multi-agent systems, respectively. In [21], under random switching topologies, and by using some techniques from convex analysis and stochastic process, the containment control problem of second-order multiagent systems was considered. In [26] and [27], under asynchronous setting, the containment control problem were investigated for discrete-time secondorder and linear multi-agent systems, respectively. On the other hand, the singular system, which is also called the descriptor system [29], is an important part in the field of control systems, and has also received significant attention in the last decades. Singular systems appear in many areas, such as, power systems, electrical networks, aerospace engineering, network analysis, biological systems, and so on [30]. Recently, the cooperative control problems for singular multi-agent systems have been attracting increasing attention [31–36]. In [31], by introducing a consensus subspace and a complement consensus subspace, a necessary and sufficient condition was given for admissible consensus problem of high-order linear time-invariant singular multi-agent systems. In [32], the consensus problem for singular linear multi-agent systems was investigated by using dynamic output feedback method. In [33], the admissible consensus problem of singular multi-agent systems was considered via proposing a distributed observer-based protocol. In [34], the containment problems of high-order lin2

ear time-invariant singular swarm systems was investigated on directed graph with time delays. Note that, the multi-agent systems considered in [31–34] are homogeneous, where all agents have identical dynamics. However, in practical applications, the dynamics of the agents are usually different due to various restrictions. Therefore, the cooperative control problem of heterogeneous multi-agent systems, where the agent models are non-identical, has also earned increasing attention recently. The leader-following consensus problem and containment control problem of heterogeneous multi-agent systems have been recently addressed in [35] and [36], respectively. In [35], the leaderfollowing consensus problem of singular heterogeneous multi-agent systems was investigated by using the techniques from output regulation. In [36], based on distributed observer design for estimating the convex hull of the leaders, the containment control problem of singular heterogeneous multiagent systems was investigated by using state feedback and dynamic output feedback control, respectively. However, different from the distributed observer proposed in [35], where the coupling gain of the distributed observer only depends on the spectrum of the communication graph, the coupling gain of the distributed observer presented in [36] depends on both the spectrum of the communication graph and the spectrum of the coefficient matrices of the leaders. On the other hand, in [35, 36], the full-order singular observers were designed to estimate the states of singular systems for the output-feedback control. However, in fact, some part of the state information of the system are usually already reflected in the output of the system. Therefore, the full-order observer has a certain degree of redundancy because it is used to estimate all state information of the system. Then, some works have been devoted to the reduced-order observer design, which has lower dimension [37]. Nevertheless, there is still no result on the containment control problem of singular heterogeneous multi-agent systems using reduced-order normal observer based output feedback control laws. Motivated by the above analysis, in this paper, we further investigate the containment control problem of a class of singular heterogeneous multiagent systems. The main contributions of this paper are stated as follows. Firstly, a new distributed observer, in which the coupling gain only depends on the spectrum of the communication graph, is proposed for the containment control problem of singular heterogeneous multi-agent systems and can be used for estimating the convex hull spanned by the states of the leaders. 3

Secondly, reduced-order normal observer based output feedback control laws are presented, which is more easier to implement physically than the output feedback control protocols based on full-order singular observer design. The remainder of this paper is organized as follows. In Section 2, some necessary preliminaries, the problem formulation and some useful lemmas are presented. In Section 3, distributed observers are proposed for estimating the convex hull of the leader agents. The state feedback controllers and reduced-order normal observer based output feedback controllers are designed in Section 4 and Section 5, respectively. In Section 6, some examples are given to illustrate the validity of the proposed control strategy. Conclusions are drawn in Section 7. 2. Preliminaries and Problem Formulation In this section, some notations, basic preliminaries, and the problem formulation are provided. 2.1. Notations Throughout this paper, the following notation will be used. R and C denote the sets of all real numbers and all complex numbers, respectively; Rn×m denotes the set of n × m real matrices; In stands for an identity matrix with dimension n and 1n represents a column vector with all entries equal to 1. For a matrix A, A > 0(A ≥ 0) means that the matrix A is positive definite (positive semi-definite). The Kronecker product of matrices A and B is represented as A ⊗ B. k · k is the Euclidean norm of a vector; λmin (A) is the minimum eigenvalue of the matrix A. diag(a1 , a2 , . . . , an ) denotes a diagonal matrix with complex numbers ai , i = 1, 2, . . . , n, on the diagonal; diag(A1 , . . . , An ) stands for a block-diagonal matrix with matrices Ai , i = 1, . . . , n, being its diagonal elements. Re(λ) is the real part of a complex number λ. For matrices E, A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n , the pair (E, A) is called to be stable when σ(E, A) = {λ| det(λE − A) = 0} ⊂ C− , where C− = {λ|Re(λ) < 0}; the triplet (E, A, B) is stabilizable if there is a matrix K such that the pair (E, A+BK) is stable; the triplet (E, A, C) is detectable if its dual system (E T , AT , C T ) is stabilizable; the pair (E, A) is called regular if there exists a constant γ ∈ C such that det(γE − A) 6= 0; (E, A) is standard if deg det(λE − A) = rank(E), where deg(·) denotes the degree of a polynomial. The pair (E, A) is called impulse free if it is both regular

4

and standard. The distance from x ∈ Rn to a set C ⊂ Rn is denoted as dist(x, C) = inf y∈C kx − yk. 2.2. Basic graph theory In the framework of multi-agent systems, graphs are often exploited to characterize the communication topology among agents. For a group of N agents, let G = (V, E, A) be a weighted digraph, where the set of vertices V = {1, 2, · · · , N }, the set of edges E ⊆ V × V, and a weighted adjacency matrix A = [aij ] ∈ RN ×N with nonnegative adjacency elements aij satisfying that aij > 0 ⇔ (j, i) ∈ E and aij = 0, otherwise. Also, it is assumed that there are no self-loops, i.e., aii = 0, ∀i ∈ V. The set of neighbors of every agent i is Ni = {j : (j, i) ∈ E}. The in-degree matrix P D is defined as a diagonal matrix D = diag{degin (i)} with degin (i) = j∈Ni aij . Then, the Laplacian matrix corresponding to the weighted graph G can be defined as L = D−A. A directed path from agent i1 to agent in is a sequence of ordered edges of the form (ik , ik+1 ), k = 1, · · · , n − 1. 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. An agent is called a leader if the agent has no neighbor, and a follower if the agent has at least one neighbor. 2.3. Problem formulation In this paper, we consider a singular heterogeneous linear multi-agent system composed of M (< N ) follower agents and N − M leader agents. Here, the sets of followers and leaders are denoted by F = {1, 2, . . . , M } and L = {M + 1, M + 2, . . . , N }, respectively. Note that, the Laplacian matrix L associated with G corresponding to the N agents can then be partitioned as   L1 L2 L= 0(N −M )×M 0(N −M )×(N −M ) where L1 ∈ RM ×M and L2 ∈ RM ×(N −M ) , since the last N − M agents are the leaders, who have no neighbors. The dynamics of the ith follower agent is given as the following singular linear system Ei x˙ i = Ai xi + Bi ui , yi = Ci xi , i ∈ F, 5

(1)

where xi ∈ Rn , ui ∈ Rm and yi ∈ Rp are the state, control input and output of the ith follower, respectively. Ei ∈ Rn×n , Ai ∈ Rn×n , Bi ∈ Rn×m , Ci ∈ Rp×n are constant matrices, with 0 < rank(Ei ) ≤ n, and there exists at least one follower agent i ∈ F such that 0 < rank(Ei ) < n. Note that the dynamics of the agents in (1) are non-identical because the coefficient matrices Ei , Ai , Bi and Ci , i ∈ F, are different. Therefore, the multi-agent system (1) is heterogeneous. The leaders of the concerned multi-agent system are described by w˙ i = Swi , i ∈ L,

(2)

where wi ∈ Rn is the state of the ith leader, and S ∈ Rn×n is a constant matrix. Definition 1. [38] A set C ⊂ Rn is said to be convex if (1 − γ)x + γy ∈ C whenever x ∈ C, y ∈ C and γ ∈ [0, 1]. The convex hull of a finite set of points X = {x1 , · · · , xn } in Rn is the minimal convex set containing Pn all points xi , i = 1, · ·P · , n, denoted by Co{X}. In particular, Co{X} = { i=1 γi xi |xi ∈ X, γi ≥ 0, ni=1 γi = 1}. Now, the containment control problem is defined as follows.

Definition 2. The multi-agent system (1)–(2) is said to achieve containment if some distributed control laws ui , i = 1, . . . , M , can be designed for the follower agents such that the states of all followers will converge to the convex hull spanned by states of the leader agents, that is lim dist(xi , Co{wk , k ∈ L}) = 0, ∀i ∈ F.

t→∞

(3)

The following assumptions are made in this paper. Assumption 1. The communication topology among the follower agents is directed. And, for each follower, there exists at least one leader that has a directed path to it. Assumption 2. (Ei , Ai ) is regular, and (Ei , Ai , Bi ) is stabilizable. Assumption 3. (Ei , Ai ) is regular, and (Ei , Ai , Ci ) is detectable.

6

Assumption 4. The following linear matrix equation Ei S = Ai + Bi Ui ,

(4)

has a solution Ui for i ∈ F. Before proceeding further, we recall the following lemmas, which will be used in deriving the main theoretical results. Lemma 1. [16] If the interaction topology satisfies the Assumption 1, then all the eigenvalues of L1 have positive real parts, each entry of −L−1 1 L2 is nonnegative, and each row of −L−1 L has a sum equal to one. 2 1 Remark 1. Lemma 1 reveals that, for the overall graph Laplacian matrix L, the zero eigenvalues can only appear in the part corresponding to the submatrix 0(N −M )×(N −M ) , and the submatrix L1 is nonsingular because all its eigenvalues have positive real parts. The submatrix L2 has no eigenvalue when N − M 6= M . Moreover, the submatrix L2 has no influence on the eigenvalues of L even when N − M = M . Lemma 2. [30] Under the Assumption 2, then, for any Qi > 0 and Ri > 0, there exists a matrix Pi > 0 satisfying the following generalized Riccati equation ATi Pi Ei + EiT Pi Ai + EiT Qi Ei − EiT Pi Bi Ri−1 BiT Pi Ei = 0.

(5)

3. Distributed observer design for leaders’ convex hull In this section, a distributed observer is presented to estimate the convex hull spanned by the states of the leader agents. Consider the distributed observer as following "M # N X X η˙ i = Sηi + µF aij (ηj − ηi ) + aij (wj − ηi ) , (6) j=1

j=M +1

where ηi ∈ Rn is the state of the observer (6), µ > 0 and F ∈ Rn×n are coupling gain and feedback gain matrix which will be determined later, respectively.

7

Lemma 3. Consider the leaders’ dynamics (2) and the distributed observer (6), under Assumption 1, let the gain matrix F be designed as F = W −1 N,

(7)

where N is the unique positive definite matrix solution to the following algebraic Riccati equation ¯ − N W −1 N = 0, ST N + N S + Q

(8)

¯ and W ; then, the state of observer (6) for some positive definite matrices Q will converge to the convex hull spanned by states of the leader agents, that is lim dist (ηi , Co{wk , k ∈ L}) = 0, ∀i ∈ F,

t→∞

if the gain µ satisfies µ>

1 , 2 mini Re(λi (L1 ))

where λi (L1 ), (i = 1, 2, . . . , M ), is the ith eigenvalue of matrix L1 . Proof: Firstly, let  T  T T T w = wM , +1 wM +2 . . . wN   T T T T . η = η1 η2 . . . ηM

(9)

w˙ = (IN −M ⊗ S)w,

(10)

η˙ = [IM ⊗ S − µ(L1 ⊗ F )]η − µ(L2 ⊗ F )w,

(11)

Then, from (2) and (6), the global form of the leaders’ dynamics and the distributed observer can be written as

and

respectively. Define the global estimation error as η˜ = η + (L−1 1 L2 ⊗ In )w, 8

(12)

then, from (10) and (11), the dynamics of the global estimation error can be given as η˜˙ = η˙ + (L−1 ˙ 1 L2 ⊗ In )w

= [IM ⊗ S − µ(L1 ⊗ F )]η − µ(L2 ⊗ F )w + (L−1 1 L2 ⊗ In )(IN −M ⊗ S)w = [IM ⊗ S − µ(L1 ⊗ F )]˜ η. (13)

According to the Lemma 1, one knows that all the eigenvalues of L1 have positive real parts; thus, there exists a nonsingular matrix T such that   λ1 ∗   ... T −1 L1 T =   , Λ, 0 λM where λi , i = 1, 2, . . . , M , are the eigenvalues of the matrix L1 . Meanwhile, by (7) and (8), one has the following Lyapunov equation ¯ − (2µRe(λi ) − 1)N W −1 N, N (S − µλi F ) + (S − µλi F )H N = −Q

which means that S − µλi F is Hurwitz, so is IM ⊗ S − µ(L1 ⊗ F ), since 1 ¯ > 0 and µ > N > 0, Q . 2 mini Re(λi (L1 )) Next, from the fact that IM ⊗ S − µ(L1 ⊗ F ) is Hurwitz and dynamics of the global estimation error (13), it follows that lim η˜ = 0.

t→∞

(14)

Finally, by (12), (14) and Lemma 1, we have lim dist (ηi , Co{wk , k ∈ L}) = 0, ∀i ∈ F,

t→∞

−1 since each entry of −L−1 1 L2 is nonnegative, and each row of −L1 L2 has a sum equal to one, which completes the proof. 2

Remark 2. Unlike the distributed observer proposed in [36], in which it is max Re(λ(S)) required that the coupling gain µ satisfies µ > min , an extra gain mai Re(λi (L1 )) trix F has been introduced into the distributed observer (6), which also results 1 in that the coupling gain µ of (6) only needs to satisfy µ > 2 mini Re(λ as i (L1 )) indicated by Lemma 3. It should be pointed out that the introduction of the extra gain matrix F is motivated by the results in [35], and the distributed observer (6) contains the existing ones proposed in [35, 36] as special cases, because (6) is reduced to the distributed observer given in [35] when only one leader is involved, and is reduced to the ones presented in [36] when the gain matrix F is chosen as identity matrix In . 9

4. State feedback control design In this section, we will propose a state feedback controller for the containment problem of the multi-agent system (1) and (2). To this end, define the local relative state information of the ith follower as ei =

M X j=1

aij (xi − xj ) +

N X

j=M +1

aij (xi − wj ), i ∈ F,

(15)

and let  T x = xT1 xT2 . . . xTM , T  e = eT1 eT2 . . . eTM .

Then, the global form of (15) can be given as   e = (L1 ⊗ In ) x + (L−1 1 L2 ⊗ In )w .

(16)

From (16) and the results in Lemma 1, it is easy to see that the containment problem is solved if and only if e converges to zero as t → ∞. Hence, we only need to show that e converges to zero as t → ∞. To this end, based on the disturbance observer presented by (6), we propose the following state feedback control law for the followers: ui = Ki xi + Gi ηi ,

(17)

where Ki and Gi are feedback gain matrices, which will be specified later. Theorem 1. Suppose that the Assumptions 1,2 and 4 are satisfied. Then, the containment problem of multi-agent system (1) and (2) is solved by the distributed observer (6) and state feedback control (17) if the gains of the 1 , σ(Ei , Ai + observer and controller are chosen such that µ > 2 mini Re(λ i (L1 )) − Bi Ki ) ⊂ C , and Gi = Ui − Ki with Ui being the solution to equation (4). Proof: By the Assumptions 2 and 4, we know that there exists a matrix Ki such that σ(Ei , Ai + Bi Ki ) ⊂ C− , and then Gi = Ui − Ki , where Ui being the solution to (4), is also well-defined. By substituting the state feedback control law (17) into (1), one has the closed-loop system of the ith follower as follows Ei x˙ i = (Ai + Bi Ki )xi + Bi Gi ηi . 10

(18)

Define E = diag(E1 , E2 , . . . , EM ), A = diag(A1 , A2 , . . . , AM ), B = diag(B1 , B2 , . . . , BM ), G = diag(G1 , G2 , . . . , GM ), K = diag(K1 , K2 , . . . , KM ), then, the global form of the closed-loop system (18) can be given as E x˙ = (A + BK)x + BG˜ η − BG(L−1 1 L2 ⊗ In )w,

(19)

where we have used the transformation (12) to get (19). Denote the global containment error by x˜ = x + (L−1 1 L2 ⊗ In )w

(20)

and by combining equations (10), (13), (16) and (19), we have E x˜˙ = (A + BK)˜ x + BG˜ η, η˜˙ = [IM ⊗ S − µ(L1 ⊗ F )]˜ η, e = (L1 ⊗ In )˜ x,

(21)

where the Eq. (4) in Assumption 4 has been used to get the first equation in (21). For the system (21), by using the Lemma 3 under the Assumption 1, 1 one gets that limt→∞ η˜ = 0 if the gain µ satisfies µ > 2 mini Re(λ ; on the i (L1 )) − other hand, it is easy to see that σ(E, A + BK) ⊂ C , since the matrix − gain SM Ki is chosen such that σ(Ei , Ai + Bi Ki ) ⊂ C and σ(E, A + BK) = ˜ will converge to 0 as t → ∞, so i=1 σ(Ei , Ai + Bi Ki ), which implies that x is e, that is, limt→∞ e = 0, which completes the proof. 2 5. Output feedback control based on reduced-order normal observer design In this section, we propose a reduced-order normal observer based output feedback control, which has lower dimension and is more easier to realize physically than the singular ones. To this end, some transformations based on matrix analysis techniques are firstly presented, in order to divide the states of the agents into the measurable part and the unmeasurable part. 11



 Ei Assumption 5. rank(Ci ) = p, and rank = n. Ci Remark 3. Assumption 5 is standard, and is commonly used in [30, 39, 40] Under the Assumption 5, without loss of generality, we assume that   Ci = Ci1 Ci2 ,

where Ci1 ∈ Rp×p with rank(Ci1 ) = p. Now, define the following matrices:  −1 Ci1 Ξi = 0  E¯ Ei Ξi = ¯i11 Ei21

 −1 −Ci1 Ci2 , In−p  E¯i12 , E¯i22

where E¯i11 ∈ Rp×p . Then, it follows that the matrix Ξi is nonsingular and   Ci Ξi = Ip 0 ,     E¯i11 E¯i12 E n = rank i = rank E¯i21 E¯i22  . Ci Ip 0 Therefore, we have

 E¯i12 = n − p. rank ¯ Ei22   E˜i11 Selecting a matrix ˜ ∈ Rn×p be such that Ei21   E˜i11 E¯i12 rank ˜ = n, Ei21 E¯i22

and defining

then, we have



 −1 E˜i11 E¯i12 Πi = ˜ , Ei21 E¯i22 

   E¯i12 0 Πi ¯ = . Ei22 In−p 12

(22)

Next, denote 

 Ai11 Ai12 Πi Ai Ξi = , Ai21 Ai22   Ei11 0 Πi Ei Ξi = , Ei21 In−p   Bi1 Πi Bi = . Bi2

(23)

Lemma 4. [30] Under the Assumption 3, the matrix pair (Ai22 , Ai12 ) is detectable. Now, we design the reduced-order normal observer as following: ξ˙i = (Ai22 − Li Ai12 )ξi + (Bi2 − Li Bi1 )ui + [Ai21 − Li Ai11 − (Ai22 − Li Ai12 )(Ei21 − Li Ei11 )]yi ,      0 Ip xˆi = Ξi y , i ∈ F, ξ + −(Ei21 − Li Ei11 ) i In−p i

(24)

where ξi ∈ Rn−p is the state of the observer (24), Li is the observer gain to be specified later, and xˆi is the estimate of xi . Lemma 5. Under the Assumption 3, using the observer (24), it holds that lim (ˆ xi − xi ) = 0,

t→∞

(25)

if the observer gain Li is chosen such that the matrix Ai22 −Li Ai12 is Hurwitz. Proof: Firstly, from the Lemma 4, we know that there exists a matrix Li such that Ai22 − Li Ai12 is Hurwitz. Denote   x −1 Ξi xi = i1 , xi2

then, from (1), (22) and (23), we have

Ei11 x˙ i1 = Ai11 xi1 + Ai12 xi2 + Bi1 ui , Ei21 x˙ i1 + x˙ i2 = Ai22 xi2 + Ai21 xi1 + Bi2 ui , yi = xi1 . 13

(26)

Next, to show (25), from (24) and (26), it suffices to show that lim [ξi − (Ei21 − Li Ei11 )yi − xi2 ] = 0.

t→∞

(27)

To this end, let exi2 = ξi − (Ei21 − Li Ei11 )yi − xi2 . Then, from (24) and (26), we can obtain that e˙ xi2 = ξ˙i − (Ei21 − Li Ei11 )y˙ i − x˙ i2 = (Ai22 − Li Ai12 )ξi + (Bi2 − Li Bi1 )ui + [Ai21 − Li Ai11 − (Ai22 − Li Ai12 )(Ei21 − Li Ei11 )]yi − (Ei21 − Li Ei11 )y˙ i − [Ai22 xi2 + Ai21 xi1 + Bi2 u − Ei21 x˙ i1 ] = (Ai22 − Li Ai12 )exi2 , which implies that (27) is hold since the matrix Ai22 − Li Ai12 is Hurwitz. 2 Remark 4. It should be noted that, compared with the full-order singular observer used in [36], the reduced-order normal observer (24) has lower dimension, and only the unmeasurable parts of the system are estimated. Thus, the reduced-order observer can exclude the redundancy caused by using fullorder observer. Now, we are ready to present the main result of this section. Theorem 2. Under the Assumptions 1-5, let the output feedback control protocol be ui = Ki xˆi + Gi ηi , (28) where Ki is a matrix gain such that σ(Ei , Ai +Bi Ki ) ⊂ C− , and Gi = Ui −Ki with Ui being the solution to equation (4). Then, the containment control of multi-agent system (1) and (2) can be achieved under the output feedback control protocol (28) based on the distributed observer (6) and the reduced order normal observer (24), if the coupling gain µ of (6) satisfies µ>

1 , 2 mini Re(λi (L1 ))

where λi (L1 ), (i = 1, 2, . . . , M ), is the ith eigenvalue of matrix L1 . 14

Proof: Let exi = xˆi − xi , and substituting the output feedback control protocol (28) into (1), we have Ei x˙ i = (Ai + Bi Ki )xi + Bi Ki exi + Bi Gi ηi .

(29)

Define  ex = eTx1 eTx2 · · ·

eTxM

T

,

then, by (9), (10), (16), and (29), we obtain the global form of the closed-loop system as follows E x˙ = (A + BK)x + BKex + BGη, η˙ = [IM ⊗ S − µ(L1 ⊗ F )]η − µ(L2 ⊗ F )w, w˙ = (IN −M ⊗ S)w,   e = (L1 ⊗ In ) x + (L−1 1 L2 ⊗ In )w .

(30)

Furthermore, let

x˜ = x + (L−1 1 L2 ⊗ In )w, −1 η˜ = η + (L1 L2 ⊗ In )w,

(31)

then, by (30) and (31), it is obtained that E x˜˙ = (A + BK)˜ x + BG˜ η + BKex , η˜˙ = [IM ⊗ S − µ(L1 ⊗ F )]˜ η, e = (L1 ⊗ In )˜ x,

(32)

where we have used (4) in the Assumptioin 4 to get the first equation in (32). From Lemma 5, one obtains that limt→∞ ex = 0. Moreover, by using Lemma 3 under the Assumption 1, one gets that limt→∞ η˜ = 0 if the gain 1 µ satisfies µ > 2 mini Re(λ . Therefore, similar to the proof of Theorem 1, i (L1 )) and by (32), we can see that lim x˜ = 0.

t→∞

This implies that limt→∞ e = 0, which completes the proof. 15

2

Remark 5. Compared with the output feedback control presented in [36], where full-order singular observer was designed to estimate the state of the singular system, a reduced-order normal observer based output feedback control consisting of (24) and (28) is proposed in the current paper, which can reduce costs since it has lower dimension, and is more easier to realize physically than the singular ones. Remark 6. For the feedback control (17) and (28), from the Lemma 2, the control gain matrix Ki can be given by Ki = Ri−1 BiT Pi Ei , where Pi > 0 is the solution of the Riccati equation (5). Then, by solving the equation (4) to get Ui , we have the gain matrix Gi = Ui − Ki . For the reduced-order observer (24), the observer gain Li can be given by normal eigenvalue assignment method. Remark 7. In this paper, we mainly focus on the containment control of singular heterogeneous linear multi-agent systems under fixed topology and synchronous updates scheme, where all agents can collaborate with their neighbors or update their states at any time. As pointed out in [11] and [26], switching topologies are more general, and the agents in the system usually interact with each other under asynchronous setting, which means that each of the agents updates its information only by own clock that is independent of other agents’ update times. Therefore, how to combine the results proposed in this paper and asynchronous scheme under switching topologies is an interesting topic for future work. 6. Illustrative Examples In this section, two numerical examples are provided to demonstrate the effectiveness of proposed containment algorithms in the previous sections. Consider a singular heterogeneous multi-agent system with five follower agents (labeled from 1 to 5) and three leader agents (labeled from 6 to 8), whose dynamics are taken from [36] with some adjustments:     βi 0 0 0 0 βi 0.3βi 0.1βi  0 αi 0 0    , Ai =  −αi 0 0.2αi −0.1αi  , Ei =   0 0 0 0  0 4αi 10βi  0 0 0 0 0 5γi 0 0 7αi 16

Figure 1: The communication graph

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The parameters of each agent are selected as α1 = 6.4378, β1 = 8.8764, γ1 = 5.5948, α2 = 2.0881, β2 = 1.3795, γ2 = 3.3420, α3 = 4.0188, β3 = 3.9537, γ3 = 7.2034, α4 = 2.3193, β4 = 1.9218, γ4 = 2.8404, α5 = 3.7843, β5 = 4.6487, γ5 = 2.4859. Assume that the communication topology among the agents are shown in Fig.1. Then, it is easy to verify that the Assumptions 1-5 are satisfied. According to Lemma 3, the coupling gain µ is taken as 3, since the smallest real part of eigenvalues of matrix L1 is 0.2. For the case with state feedback control, let     7.626 4.9425 0 0 15.1086 9.1197 0 0 K1 = , K2 = , 7.9897 −7.7917 0 0 1.9831 −1.8081 0 0     12.174 7.9612 0 0 11.9562 7.4158 0 0 K3 = , K4 = , 3.2824 −3.2987 0 0 3.1947 −2.9739 0 0   7.1258 4.5295 0 0 K5 = , 7.6979 −7.2867 0 0 17

then one obtains that σ(Ei , Ai + Bi Ki ) ⊂ C− , i = 1, 2, · · · , 5. Furthermore, to obtain Ui , by solving the Eq. (4), gives   0 −4.6027 −15.8654 0 U1 = , −3.1515 0 0 −5.0769   0 −2.4992 −4.1278 0 U2 = , −12.1131 0 0 −10.5957   0 −2.2316 −5.4887 0 U3 = , −9.1097 0 0 −7.1153   0 −3.2662 −6.7659 0 U4 = , −7.3899 0 0 −8.4479   0 −6.0892 −18.7003 0 U5 = , −2.6738 0 0 −5.6984 The initial states of xi and wk , i = 1, · · · , 5; k = 6, 7, 8, are chosen randomly from [−5, 5], while the initial values of the remaining states are all selected as 0. Simulation results are shown in Figures 2-4. Figure 2 depicts trajectories of the leader and follower agents under the state feedback control (17) based on the distributed observer (6), from which it can be observed that, all followers are asymptotically driven into the convex hull formed by the leaders. The containment errors, which are defined by Eq. (20), are shown in the Figures 3 and 4. From Figures 3 and 4, we see that the containment errors go to zero asymptotically. For the case with output feedback control based on reduced-order observer, the gain matrices Li of the reduced-order observer (24) are selected as     0.0388 0.0357 0.1197 0.0598 L1 = , L2 = , 0.0777 −0.0357 0.2395 −0.0598     0.0622 0.0278 0.1078 0.0704 L3 = , L4 = , 0.1244 −0.0278 0.2156 −0.0704   0.0661 0.0805 L5 = , 0.1321 −0.0805 and use the same Ki and Ui as the case of state feedback control. In the simulations, the initial states of the leaders and followers are also chosen randomly from [−5, 5], and all the other initial conditions are set to zero. The simulation results are shown in Figures 5-9. Figure 5 indicates that the 18

followers move into the convex hull formed by the leaders under the output feedback control (28) based on the distributed observer (6) and reducedorder observer (24). Figures 6-7 show that the containment errors converge to zero asymptotically. The estimate errors between the states of system (1) and observer (24) are described in Figures 8-9, where only xi1 and xi2 are estimated because xi3 and xi4 are already reflected in the output of the system. From Figures 8-9, it can be observed that all the estimate errors converge to zero. For the purpose of comparison, under the same initial conditions and pole assignment, the trajectories of the multi-agent system controlled by output feedback control based on full order singular observer proposed in [36] are shown in Figures 10-14. Figure 10 gives the state trajectories of all the agents, and Figures 11-12 present the containment errors. The estimation errors between the states of the system and the full-order singular observer are depicted in Figures 13-14. The advantage of our proposed output feedback control based on reduced-order normal observer can be illustrated by comparing the Figures 6-9 and Figures 11-14, and it can observed that the approach proposed in the present paper can achieve good tracking performances. 7. Conclusion This paper has studied the containment control problem of singular heterogeneous multi-agent systems with directed interaction topology. A new distributed observer is designed to estimate asymptotically the convex hull spanned by the states of the leaders. Both a state feedback control and an output feedback control are presented to solve the containment control problem. Moreover, the proposed reduced-order observer based output feedback control has lower dimension and is more easier to realize physically than the singular ones. Finally, some numerical examples are provided to verify the effectiveness of the proposed results. Acknowledgment This work was supported in part by the National Natural Science Foundation of China under Grants 61503089 and 11871017, Postdoctoral Research Startup Foundation of Heilongjiang under Grant LBH-Q16068, and the Fundamental Research Funds for the Central Universities under Grant 3072019CF2404. 19

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The authors declare that there is no conflict of interests.

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