Adaptive distributed observer design for containment control of heterogeneous discrete-time swarm systems

Shimin WANG, Zhi ZHAN, Rnxin ZHONG,c,*, Yuanqing WU,Zhouhua PENG

a Department of Electrical and Computer Engineering, University of Alberta, Alberta T6G 2R3, Canada

b School of Intelligent Systems Engineering, Sun Yat-Sen University, Guangzhou 510006, China

c Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, China

d School of Automation, Guangdong University of Technology, Guangzhou 510006, China

e School of Marine Engineering, Dalian Maritime University, Dalian 116026, China

KEYWORDS

Abstract This paper develops both adaptive distributed dynamic state feedback control law and adaptive distributed measurement output feedback control law for heterogeneous discrete-time swarm systems with multiple leaders.The convex hull formed by the leaders and the system matrix of leaders is estimated via an adaptive distributed containment observer.Such estimations will feed the followers so that every follower can update the system matrix of the corresponding adaptive distributed containment observer and the system state of their neighbors.The followers cooperate with each other to achieve leader–follower consensus and thus solve the containment control problem over the network. Numerical results demonstrate the effectiveness and computational feasibility of the proposed control laws.

1. Introduction

Cooperative control of swarm systems (also known as multiagent systems)has been a recent research hotspot for its extensive applications in engineering.1Generally speaking,cooperative control problems of multi-agent systems can be mainly classified into the consensus problems2–5and the containment control problems.6–8The consensus problem is to design the adaptive distributed control strategy regarding local information to fulfill the condition that the state errors between any two agents in the network tend to zero. On the other hand,the containment control problem is to design control laws so that all the followers converge to the convex hull formed by the leaders. In this paper, we investigate the containment control problem for heterogeneous discrete-time swarm systems with multiple leaders.

Extensive research effort has been dedicated to containment control problems for continuous-time and discrete-time swarm systems of first-order integrators and second-order integrators with stationary or dynamic leaders. Ref.9studied the distributed containment control of a group of mobile autonomous agents with multiple stationary or dynamic leaders under both fixed and switching network topologies using continuoustime formulation.Ref.10studied the containment control problem for multiple continuous-time and discrete-time single integrator systems. Control protocols were devised by exploiting the control input information of neighbors so that the leaders will converge to the desired convex formation while the followers converge to the convex hull of the leaders. Ref.11considered the containment control problems for both continuoustime and discrete-time multi-agent systems with general linear dynamics under general directed communication topologies.Distributed dynamic containment controllers based on the relative outputs of neighboring agents were constructed. Ref.12solved containment control problems for networked single integrator systems with multiple stationary or dynamic leaders over directed graphs.Ref.13proposed the containment control problem for a group of agents with heterogeneous dynamics modeled by both first-order integrators and second-order integrators. Due to the packet drop and node failure phenomena during the information transmission, the containment control of multi-agent systems over switching communication topologies is of importance. Ref.14considered the distributed containment control for second-order multi-agent systems guided by multiple leaders with random switching topologies.Ref.15investigated the containment control of linear multiagent systems with input saturation on switching topologies.Both state feedback and output feedback containment control protocols were proposed. The containment control problems are achieved via local relative position and velocity measurements with constraints when the velocity and acceleration are difficult or impossible to measure in certain scenarios.Ref.16investigated the distributed containment control problem for a group of autonomous vehicles modeled by doubleintegrator dynamics with multiple dynamic leaders over a fixed network that the velocities and the accelerations of both leaders and followers are not available. Ref.17investigated the robust global containment control problem for continuoustime second-order multi-agent systems subject to input saturation wherein only local velocity measurements, relative positions, and velocity measurements are involved in the controller design. Distributed observers were constructed for followers to estimate the states of the leaders since they are not available for the followers. Ref.18addressed the design of distributed observers for agents with identical linear discretetime dynamics over a directed graph interaction topology.Ref.19addressed the cooperative output regulation problem for discrete-time linear multi-agent systems with a new type of adaptive distributed observer for the leader systems.Ref.20studied the cooperative output regulation problem for the discrete-time linear time-delay multi-agent systems by a distributed observer approach. Refs.21,22considered the containment problem of heterogeneous linear multi-agent systems via the dynamic compensator technique wherein the containment control problem was converted into a cooperative output regulation problem.

A common assumption adopted in the literature addressing the containment control problem for discrete-time swarm systems is that every follower needs to know the system matrix of its leader. However, this assumption seems rather vulnerable.To tackle this challenge, we revisit the containment control problem for discrete-time swarm systems using an adaptive distributed containment observer inspired by the adaptive distributed observer design in Refs.23,24. In this paper, the discrete-time swarm systems under consideration consist of multiple leaders. The containment control problem is addressed by a new design of distributed control laws with the aid of an adaptive distributed containment observer for the followers to infer the system matrices of their leaders and the convex hull formed by the leaders. The proposed control laws circumvent the restrictive assumption that every follower needs to know the system matrix and signal of its leader as proposed in Refs.20,21,22.

The rest of this paper is organized as follows:Section 2 formulates the problem. Section 3 presents some existing results from Refs.23,24and devises a new lemma for the design of an adaptive distributed containment observer. The main results are outlined in Section 4.A numerical example is conducted in Section 5.Companion materials are provided in the appendix.

2. Problem formulation and assumptions

Fig.4 shows the tracking errors under the state feedback control law(21).As shown in Fig.4,it is clear that the tracking errors converge to zero as time tends to infinity.The state trajectories of the followers converge to the convex hull formed by the leaders as demonstrated in Fig.6. The tracking errors and the state trajectories under the dynamic measurement output feedback control law Eq. (22a), Eq. (22b) are shown in Figs. 5 and 7, respectively. Similar to the results of state feedback control law (21), the tracking errors converge to zero asymptotically under the dynamic measurement output feedback control law Eq. (22a), Eq. (22b) while the state trajectories of the followers converge to the convex hull formed by the leaders. Fig.8 shows the convergence of the estimation of the leader’s system matrix S by the adaptive distributed observer.Satisfactory tracking performance confirms the analytical results of Theorems 1–2.

Fig.3 Network topology G－.

Fig.4 Tracking errors under state feedback control law (21).

Fig.5 Tracking errors under dynamic measurement output feedback control law (22).

Fig.6 State trajectories under state feedback control law (21).

Fig.7 State trajectories under dynamic measurement output feedback control law (22).

Fig.8 Estimation of the leader’s system matrix by adaptive distributed observer.

6. Conclusions

In this paper, the containment control problem for discretetime swam system has been investigated. To circumvent a restrictive assumption in the literature that every follower needs to know the system matrix and signal of its leader, we extended the design of an adaptive distributed observer in Ref.24to estimate the leader’s system matrix and signal. Both analytical and numerical results confirm that the followers cooperate with each other to achieve consensus and convergence to the convex hull spanned by the leaders under both the distributed adaptive dynamic state feedback and distributed adaptive measurement output feedback control laws.

Three issues deserve further research effort.First,the eigenvalues of S are assumed to have modulus smaller than or equal to 1 in Assumption 2. Removing this restricted assumption would yield more general and practical results. Second, it is interesting to look into the containment control problem of singular discrete-time linear heterogeneous swarm systems.Third,the topology of the communication network is confined to be static in the containment control problem.Extending the results for containment control problems, wherein the topology of the communication network is time-varying, is of importance.

Acknowledgements

This study was co-supported by the National Key R&D Program of China (No. 2018YFB1600500).

Appendix A. This appendix presents a brief introduction of

graph theory in line with Ref.25.