Admissible Consensus for Descriptor Multi-Agent Systems with Exogenous Disturbances

In this paper, we study the admissible consensus for descriptor multi-agent systems (MASs) with exogenous disturbances that are generated by some linear systems. The topology among agents is represented by a directed graph. For solving the admissible consensus problem, the exogenous disturbance observer and distributed control protocol are proposed. With the help of the graph theory and the generalized Riccati equation, some conditions for admissible consensus of descriptor MASs with exogenous disturbances are obtained. Finally, we provide a numerical simulation to effectively illustrate the results we have reached before.

multi-agent systems with external stochastic inputs was studied in [29]. In addition, [30] also studied the consensus problem of linear MASs with exogenous disturbances generated from heterogeneous exosystems. However, there is little research on the admissible consensus problem for descriptor MASs with exogenous disturbances. For example, [31] only studied cooperative output regulation of singular heterogeneous MASs. This situation also appears in other literature, and there are few relevant results on this issue. Therefore, in this paper, we further study the admissible consensus problem of descriptor MASs with exogenous disturbances. The main contributions of this paper are (1) Consider a descriptor MASs with exogenous disturbances, the consensus problem of general linear system is extended to the descriptor MASs; (2) The disturbance observer for descriptor MASs is designed by state feedback; (3) The disturbance observer is used for disturbance attenuation.
The rest of the paper is organized as follows: In Section 2, we introduce some basic concepts and related theorems to describe descriptor MASs. Then, the formulation of the considered problem is introduced. In Section 3, the disturbance observer is proposed, the corresponding distributed control protocol is designed and sufficient conditions that ensure the states of a group of agents can reach an agreement are obtained. Section 4 gives a numerical example. Finally, conclusions are given in Section 5.

Preliminaries
In this subsection, some notations and preliminaries involved in this paper are introduced. I n denotes the n × n identity matrix. For a matrix A (or a vector x), A T (or x T ) represents the transpose of A (or x). Let σ(A) be the set of all eigenvalues of the square matrix A, and let σ(E, A) be {λ|λ ∈ C, det(λE − A) = 0}. C − represents the open left-half complex plane. Let ⊗ denotes the Kronecker product of matrix A ∈ R m×n and B ∈ R p×q , which is defined as In this paper, the information interaction topology is modeled by a weighted digraph is a weighted adjacency matrix. An edge of G that is from node v i to node v j is denoted by (v i , v j ), which represents node v j can get information from node v i , but not necessarily vice versa. The set of neighbors of node v i is Then, the Laplacian matrix of G is defined as L = D − A, which has at least one zero eigenvalue with 1 = [1, 1, · · · , 1] T as its corresponding right eigenvector. In addition, L has exactly one zero eigenvalue if and only if the directed graph G contains a directed spanning tree. Definition 1. [21] Let E, A ∈ R n×n . (i) The pair (E, A) is said to be regular if det(sE − A) is not identically zero for some s ∈ C; (ii) The pair (E, A) is said to be impulse free if (E, A) is regular and deg(det(sE − A)) = rankE for ∀s ∈ C; (iii) The pair (E, A) is said to be stable if σ(E, A) ⊆ C − ; (iv) The pair (E, A) is said to be admissible if (E, A) is impulse free and stable.

Problem Formulation
In this subsection, we will establish the model of the descriptor MASs with exogenous disturbances and propose the corresponding consensus problem for this system. In addition, the corresponding model of the exogenous disturbances system is also given.
Consider a descriptor MASs composing of n identical agents. The dynamic of agent i is modeled by the following descriptor linear system where , C, D are real constant matrices with appropriate dimensions, and rankE = r < m.
Since E is a singular matrix, the initial state cannot be arbitrarily chosen, which needs to satisfy some algebraic conditions.
The descriptor MASs (1) is said to achieve consensus, if the states of all agents satisfy for any initial states x i (0), i = 0, 1, · · · , n. If the closed-loop system is admissible and achieves consensus, we say the protocol u i (i = 1, 2, · · · , n) can solve the admissible consensus problem.
In this paper, we suppose that the disturbance ω i (t) is generated by the following linear exogenous systemξ where ξ i (t) ∈ R l is the state of the exogenous system, G ∈ R l×l and F ∈ R p×l are the matrices of the disturbance system. Lemma 1. [21] Assume that (E, A) is impulse free and (E, A, B) is stable. Then, the generalized Riccati equation A T P + P T A − P T BB T P + I n = 0 (5) has at least one admissible solution P, that is, (E, A − BB T P) is admissible. Furthermore, the admissible solution P is unique in the sense of E T P.

Lemma 2.
[9] Assume (E, A, C) is detectable, and the matrix P is the admissible solution of the following Riccati equation Then, σ(E, A − (a + bi)BB T P) ⊂ C − , when a ≥ 1 2 , b ∈ R, where i is imaginary units.

Main Results
In this section, we will design a disturbance observer to solve the problem caused by exogenous disturbances. Moreover, in order to resolve admissible consensus problem, we propose the distributed consensus protocol.
The disturbance observer is designed as followṡ where η i ∈ R l is the internal state variable of the observer,ξ i andω i are the estimated values of ξ i and ω i , respectively. H ∈ R l×m is the gain matrix of the observer. Let e i = ξ i −ξ i , we can geṫ That isė Denoting e = (e T 1 , e T 2 , · · · , e T n ) T , then, one has (I n ⊗ I l )ė = [I n ⊗ (G + HBF)]e.
We can get that the tracking errors e i , i = 1, 2, . . . , n, converge to 0, when the matrix G + HBF is Hurwitz.
In order to resolve admissible consensus problem, we consider the following distributed consensus protocol where K is the gain matrix to be designed.

Theorem 1.
For the descriptor MASs (1) whose interaction topology G contains a directed spanning tree, suppose that the pair (E, A) is regular and impulse free, (E, A, B) is stable, and (E, A, C) is detectable. Then, the protocol (10) can solve the consensus problem if (i) The matrix G + HBF is Hurwitz; (ii) K = αB T P, where matrix P is the unique admissible solution of (5) and α ≥ 1 2min λ i (L) =0 {Re(λ i )} , i = 1, 2, · · · , n.
Proof. By substituting the control protocol (10) into the system (1), one has Next, we write formula (11) in the following form Denote x = (x T 1 , x T 2 , · · · , x T n ) T , e = (e T 1 , e T 2 , · · · , e T n ) T . Then, it follows that Then, after manipulations with combining (9) and (13), the closed-loop system can be expressed as Since the topology G contains a directed spanning tree, L has exactly one zero eigenvalue. Let r T = (r 1 , r 2 , · · · , r n ) be the left zero eigenvector of L with r T 1 = 1 Then, (14) can be divided into the following two subsystems and wherex = [x 0T ,x 1T ] T ,ē = [ē 0T ,ē 1T ] T , andx 0T is the first m components ofx,ē 0T is the first l components ofē. At the same time, we have Obviously, if the system (16) is admissible, we can getx 1 (t) → 0, as t → ∞. Thus, x(t) − 1 ⊗ x 0 (t) → 0, as t → ∞, which means that lim t→∞ (x j − x i ) = 0, i, j = 1, 2, . . . , n.
Next, we will prove that the descriptor system (16) is admissible. Due to the matrix G + HBF is Hurwitz, the admissibility of the system (16) is equivalent to the admissibility of all pairs (E, A − λ i BK). According to Lemma 1, P is the unique admissible solution of (5), therefore, (E, A − BB T P) is admissible.

Simulations
In this section, a simulation result is presented to illustrate the previous theoretical results. The network includes five agents. The topology can be described in Figure 1. The following matrix is the weighted adjacency matrix A of the topology: Solving the Equation (5), we obtain that Then, we have For the simulation, let initial states of the agents be the following:   Figure 4 as an example, although its initial state and direction of movement are very different from those of other agents, they can eventually reach the consensus agreement by the distributed control protocol (10).

Conclusions
In this paper, we study the admissible consensus for descriptor MASs with exogenous disturbances. A design method of disturbance observer is proposed, so that the states of all agents reach an agreement. Through the use of graph theory and the generalized Riccati equation, some conditions were obtained for admissible consensus of descriptor MASs with exogenous disturbances. The future work is to consider the admissible consensus of the descriptor MASs with leaders. In addition, since this paper does not consider the time delay, the perspective of considering the time delay will also be an interesting topic.