Admissible consensus for descriptor multi-agent systems via distributed observer-based protocols

Abstract

In this paper, the admissible consensus problem of descriptor multi-agent systems via distributed observer-based protocols is considered. The dynamics of agents is modeled by the general form of continuous-time linear descriptor systems, and the interaction topology among the agents is modeled by a directed graph. It is assumed that the output information but not the state information of the agents is available. To achieve control objective, two types of state observers are adopted for the agents to estimate its states and the related state disagreements respectively, by which two different architectures of connecting observers and controllers are proposed. Based on the generalized Riccati equation and Lyapunov admissible theory, a sufficient admissible consensus condition is established for the descriptor multi-agent systems. The full-order and reduced-order observer-based consensus protocols can be unified in our proposed framework. Based on the special solutions of the established condition, some full-order and reduced-order observer-based consensus protocols can be obtained. Finally, a simple simulation example is provided to illustrate our established result.

Introduction

In the past few decades, a huge and rapidly growing literature on multi-agent systems due to its numerous potential applications [1]. Although the practical multi-agent systems have different backgrounds, the fundamental idea in coordination control of multi-agent systems is that while each agent can only access to the information of its neighbor and itself, a goal of the whole system can be achieved without centralized controller. It is well-known that consensus problem is one of the most important and fundamental issues of coordination control, whose main task is to design the distributed protocols based on the local relative information so that the states of a group of agents can reach an agreement [2]. Till now, numerous interesting results have been established for the multi-agent consensus problems with different system dynamics, including first-order system [2], [3], second-order system [4], [5], general linear system [6], [7], [8], descriptor system [9], [10], discrete-time system [12], [11], and non-linear system [13], [14], [15].

Most of the existing approaches to solve consensus problem are based the state information. But, in many practical systems, some state variables can not be obtained directly due to the technical constraints or economic cost. To achieve control aim in this case, the observer-based consensus protocols are proposed by adopting the state observers to estimate those unmeasurable state variables. The observer-based consensus problem with first-order dynamics was first addressed by [16], whose protocols based on a velocity observer to estimate the velocity of second-order leader. Furthermore, to track the leader with general linear dynamics, a distributed observer-based consensus protocol was proposed in [17], and its communication delay case discussed by [19]. In [18], the $H^{\infty }$ consensus problem was addressed based on estimation of uncertain systems via dissipativity theory. To track the leader with acceleration motion, a distributed acceleration observer based consensus protocol was provided by [4]. The second consensus problem via sampled-data and observer-based control was investigated by [11]. For multi-agent systems with general linear dynamics, [6] introduced a unified framework to design the distributed observer based consensus protocols, and three kinds of consensus protocols based on different connected architectures of controllers and observers were proposed by [7]. Then, the reduced order and functional observer based consensus protocols were investigated by [8], [20], [21] and [22], [23] respectively. In [12], two kinds of distributed observer-based consensus protocols were proposed to solve the consensus problem with discrete-time general linear dynamics.

The descriptor system is also referred to as singular state-space system, generalized system, or implicit system [24], which is more general than normal physical systems and especially suitable for modeling many engineering systems, such as robotic system, mechanical system, power system, and so on. Actually, the descriptor system is characterized by the differential-algebraic equations, which may be more complicated and challenging to deal with than normal systems. Till now, the related control problem, such as stability, admission, observer-based control, robust control, for descriptor systems is still attracting, and a great number of useful results have been obtained [25], [26], [27], [28], [29]. The applications of descriptor multi-agent systems can be found in practical, such as the multi-agent supporting systems are wildly adopted in earthquake damage prevention in buildings, large-diameter parabolic antennae or telescopes and water-floating plants. It is worth to noted that the multi-agent supporting systems can be described by descriptor systems when they consist of many independent blocks and each block is supported by several pillars, more details can be found in [30]. Usually, each subsystem (i.e., agent) has three types of modes which are finite-dynamic modes, impulse modes and non-dynamic modes. Recently, the coordination control problems for descriptor multi-agent systems have been drawing increasing attention. In [31], the network-based control problem was addressed for a class of discrete-time descriptor systems with random delays. The necessary and sufficient conditions of consensusability with respect to a set of admissible consensus protocols were investigated in [9]. The descriptor multi-agent consensus problem was solved by the distributed dynamic output feedback approach in [10]. In [32], the observer-based design approach of [7] was generalized to solve the descriptor consensus problem. The admissible consensus analysis and consensualizing controller design problems for high-order linear time-invariant descriptor swarm systems were investigated in [33], and time delay case was discussed in [34]. The containment problem for descriptor swarm systems solved by state and output feedback laws were discussed in [35] and [36] respectively.

Motivated by the above works, we investigate the multi-agent consensus problem with the agent׳s dynamics modeled by descriptor linear systems under directed fixed interaction topology. The main contribution of this paper is that two kinds of novel observer-based consensus protocols are proposed for multi-agent systems with descriptor dynamics. The separation principle of observer design is shown to be valid in our control structure. Then, in virtue of the generalized Riccati equation and Lyapunov admissible theory, a sufficient admissible consensus condition is established for the descriptor multi-agent systems. In comparison with the existing references, our design approach at least has several advantages as follows: (1) The proposed protocols are based on output information. For descriptor multi-agent systems, most of the existed references adopted the state feedback laws. Here, by assumption that the state information can not be directly available, the local and cooperative state observers are adopted to estimate agent׳s states and the related state disagreements respectively. Then, two different architectures of connecting observers and controllers are provided to design the consensus protocols. (2) The observer involved in this paper is the normal state observer. In comparison with full-order descriptor observer used in [10], [32], the full-order descriptor observer-based consensus protocols were constructed. In comparison with descriptor observer, the normal observer proposed by this paper may be easier to realize and have lower dimension. (3) A unified framework of observer-based protocols is proposed, which includes the full-order and reduce-order observer-based consensus protocols. As a special case, the widely studied observer-based consensus problem with general linear dynamics can be unified in our proposed framework, and the related full-order local observer- based protocols of [6], [7] and reduced-order local-observer-based protocols of [8], [20], [21] can be also obtained by our proposed approach. (4) The established consensus condition is mainly based on the well-known generalized Riccati equation and generalized Sylvester equation, whose solvability is also discussed. Furthermore, some algorithms are provided to compute the explicit solution of the established condition, by which several special full-order and reduced-order observer-based protocols can be obtained.

The rest of the paper is organized as follows. In Section 2, some based concepts of the descriptor systems and the formulation of the considered problem are introduced. Two kinds of distributed observer-based consensus protocols are proposed in Section 3. Then in Section 4, the full-order and reduced-order observer gain matrices are obtained by the explicit solution of the established condition. Following that, a simulation example is given in Section 5, and finally, Section 6 provides some concluding remarks.

All the notations of this paper are standard. $R^{m\times n}$ and $C^{m\times n}$ are the set of $m\times n$ real matrices and complex matrices, respectively. For any $s\in C$, Re(s) represents its real part. I represents unit matrix. $\sigma (E,A)$ represents the generalized eigenvalues of pair $(E,A)$. The transpose and conjugate transpose of matrix A are represented by A^T and A^H respectively. For symmetric matrices A and B, $A>(\ge )B$ means that $A-B$ is positive (semi-) definite. Denote $E^{\perp }$ as a full row rank matrix satisfying $E^{\perp }E=0$ and $E^{\perp }{E}^{\perp T}>0$. Rank(A) represents the rank of matrix A. ⊗denotes the Kronecker product of two matrices A and B, which satisfies: (i) $(A\otimes B)(C\otimes D)=\left(\mathit{AC}\right)\otimes \left(\mathit{BD}\right)$; (ii) If $A\ge 0$ and $B\ge 0$, then $A\otimes B\ge 0$.