Wenchao Huang, Hailin Liu, and Jie Huang,

Abstract—In this paper, we consider the robust output containment problem of linear heterogeneous multi-agent systems under fixed directed networks. A distributed dynamic observer based on the leaders’ measurable output was designed to estimate a convex combination of the leaders’ states. First, for the case of followers with identical state dimensions, distributed dynamic state and output feedback control laws were designed based on the state-coupled item and the internal model compensator to drive the uncertain followers into the leaders’ convex hull within the output regulation framework. Subsequently, we extended theoretical results to the case where followers have nonidentical state dimensions. By establishing virtual errors between the dynamic observer and followers, a new distributed dynamic output feedback control law was constructed using only the states of the compensator to solve the robust output containment problem. Finally, two numerical simulations verified the effectiveness of the designed schemes.

I. INTRODUCTION

RECENTLY, multi-agent techniques have received increasing attention due to their potential applications in industrial, agricultural and military fields [1]–[4]. The cooperative problem is one of the basic research directions of multi-agent systems, and considerable research on this topic have been presented in the literature [5]–[11]. The main focus of cooperative control is the design of distributed strategies under which the agents reach control objectives accessing only local information from itself and its neighbors rather than global information. Distributed cooperative control has been successfully applied in large-scale complex system control problems that are difficult or even impossible to solve via centralized or decentralized strategies.

Containment control, as a typical distributed cooperative control problem, has also attracted substantial attention. The control purpose is to drive followers to asymptotically converge to the convex hull spanned by the leaders [8],[10]–[14]. The containment control problem originates from natural phenomena and has broad practical applications, for example, when a team of autonomous vehicles or robots has to move from one area to another but only some individuals have the ability to detect dangerous areas. The method adopted is to treat autonomous vehicles or robots with detection capability as leaders and the rest as followers to drive the followers to enter the convex hull spanned by the leaders and move together, so as to ensure group security in the transfer process [12], [13].

Many theoretical studies on the containment problem have been conducted, and the integrator system has played an important role. Among them, containment problems with single-integrator and double-integrator agents were investigated in [8], and the necessary and sufficient conditions were provided with continuous-time and sampled-data based protocols under a fixed directed topology. Moreover, the final output of the follower systems were found to be completely dependent on topology structure and the initial states of the leaders. Subsequently, the containment problem of integrator agents with switching topology was considered in [15]–[17],and sufficient conditions for containment control were given utilizing different control approaches. Moreover, the containment problem of the integrator system subject to complex situations has begun to receive attention, with many meaningful results, including nonlinear system containment[18], [19], event-triggered containment [20], [21], finite-time containment [22]–[24], communication-delay containment[25]–[28], and input-saturation containment [29]–[31]. Due to the simplicity of the integrator system structure, it can give full play to the mathematical operation function based on the topology network, and we can obtain many meaningful results. However, it is difficult to directly extend this to problem solving of high-order systems.

Compared with the integrator system, the high-order system has complex system dynamics, so it is more difficult to solve the containment problem, but it is more in line with the needs of practical applications. The forementioned achievements have been reconsidered for high-order systems. For example,the distributed output and state feedback approaches were adopted for state containment [12], [32], and the distributed adaptive approach with agent output was proposed for output containment [33]. The containment problem for communication delay was investigated in [34], [35]. The containment problems for switching topology and performance optimization were considered in [11] simultaneously. To explore the impact of parameter uncertainty of system plants, the robust containment problem was discussed utilizing diverse control ideas [36]–[39]. In [37], a composite feedback containment controller was synthesized to solve the containment problem of unknown nonlinear dynamics based on neural network approximation theory and the Lyapunov theory. In [38], the adaptive factor was embedded in the robust containment protocol to address uncertain leaders utilizing an adaptive internal model and a recursive stabilization approach. In [39],the robust containment problem of fractional-order system dynamics was investigated, and the sufficient condition to state containment was provided according to the linear matrix inequalities technique. The containment problem with input saturation was considered in [40]. The bipartite containment problem was studied in [41] and [42] for linear multi-agent systems and descriptor multi-agent systems, respectively.

Notably, a common drawback associated with most of the forementioned results is that each agent must have the relative values of the state with respect to its neighbors. One major reason is that many containment methods involve the differential term of containment error, and containment error itself is an expression of variable coupling [15], [32],[39]–[42]. Therefore, these methods can only be applied in the case where the agents have identical dynamics or state dimensions and absolutely rules out the possibility of agents with nonidentical state dimensions. However, multi-agent systems with nonidentical state dimensions are common in practical applications. For example, different types of robots must work cooperatively to complete complex production tasks in industry.

In addition, the method in this paper is based upon output regulation theory, which can realize asymptotic tracking of the reference input [43] and has been broadly used in the cooperative control problem of multi-agent systems with leaders [5], [12], [13], [38], [44]–[46]. Under the framework of output regulation, the design of the compensator is an important issue. For example, a cooperative output regulation protocol was introduced to solve the containment problem of linear heterogeneous multi-agent systems using a dynamic compensator based on the leaders’ state information [12](Details can be found in Remark 3). Output regulation protocols with similar compensators can be found in [5], [13],[44], [47]. However, the dynamic compensators are associated with the leaders’ state information: However, the state information is not always available due to various practical factors, such as high acquisition cost and difficult acquisition.

Recently, an interesting design idea was adopted in [11] to realize output containment by driving all agents to track embedded compensator systems. Inspired by [11], we have solved the robust output containment control problem of linear multi-agent systems with a directed network. The main differences are that the problem we considered involves robust containment, and the control strategies do not take measures against the leaders. The main contributions of this paper include the following three points. First, we modify the conventional state observer to produce an estimate of the convex combination of the leaders’ states by applying a directed network, which lends itself to the design of the distributed protocols. Moreover, the distributed observer can also be viewed as an extension of the compensators associated with the leaders’ states in [5], [12], [13], [44], [47] as these compensators can be regarded as a special case of our observer when the output matrix of the leader systems is of full column rank.

Second, based on the internal model principle and the compensator technique, distributed dynamic state and output feedback control laws were introduced to drive the containment errors to converge to the origin asymptotically,such that uncertain followers with identical nominal dynamics entered the convex hull spanned by the leaders under the output regulation framework. In addition, for the closed-loop stability analysis, the closed-loop system was divided into multiple subsystems via nonsingular transformation, and a Lyapunov inequality method was used to make multiple subsystems reach a stable state simultaneously, thereby stabilizing the closed-loop system.

Finally, we extended the theoretical results to a more general case where the followers have nonidentical state dimensions, where the robust containment problem was converted into a new tracking problem between the distributed observer systems and the follower systems by constructing a virtual error vector. A distributed dynamic output feedback control law was further devised by modifying the statecoupled item in the previous method to drive virtual error to converge to the origin asymptotically, such that multi-agent systems achieved output containment control. In this way, we have avoided the dependence of most existing containment protocols on the relative states of followers, such that the distributed control law is capable of solving the robust output containment problem for linear heterogeneous multi-agent systems with nonidentical state dimensions.

The remainder of this paper is organized as follows. Section II presents the preliminary knowledge and the problem formulation. Section III provides the main result, where the robust output containment control problem was solved for multi-agent systems with followers with identical state dimensions. Section IV presents further results about heterogeneous followers with nonidentical state dimensions.Section V discusses two numerical simulation examples to verify the effectiveness of the designed schemes. Finally,Section VI concludes this paper.

II. PRELIMINARIES AND PROBLEM FORMULATION

A. Algebraic Graph Theory

B. System Model

In this paper, we consider the robust output containment problem of linear heterogeneous multi-agent systems with uncertain followers. Assume theNfollower subsystems have the following dynamics:

Additionally, virtual errors (5) can be regarded as a special containment error with a single leader. When the digraphG has a directed spanning tree, the leader-following consensus of [46] can be achieved if and only if the virtual errors converge to the origin asymptotically. However, this result is not valid when it is extended to a containment problem with more than one leader. Here, if the digraph G has a united spanning tree, the containment errors (3) converge to the origin asymptotically is only a sufficient condition for containment control.

C. Problem Formulation

It is well-known that the internal model principle can eliminate the steady-state error of the uncertain system by embedding a model of the exogenous signals [50]–[52]. It is also a key part of the solution to the cooperative robust output regulation problem.

The robust output regulation for a single plant has been provided in detail in [43] via the internal model principle, and it is theoretically proven that pure state feedback protocol cannot solve this problem. Subsequently, the robust leaderfollowing consensus and containment problem of multi-agent systems were further considered in [6], [46], [47], [53] via the internal model principle and the compensator technique. In this paper, we continue to consider the robust output containment problem of linear multi-agent systems within the output regulation framework.

In practice, on the one hand, partial followers can not directly access information from the leader due to the lack of existence of an edge from the leaders to the followers in the communication network; on the other hand, the leaders’ state variate may not be available due to measurement difficulties,no actual physical meaning, etc. To address these problems,we design a distributed dynamic compensator for the followers based on the leaders’ measurable output:

III. SOLUTION TO REGULATOR PROBLEM

In this section, two main results are presented under control laws (15) and (16).

A. Distributed Dynamic State Feedback Control Law

B. Distributed Dynamic Output Feedback Control Law

Based on the above analysis, we have achieved robust output containment for linear heterogeneous multi-agent systems (1) and (2) using output regulation theory. Control laws (15) and (16) ensure that an agent accesses only local information from itself and its neighbors instead of global information, so they follow full information distributed strategies. For the stability analysis of the closed-loop system,the Lyapunov inequality method provided in [49] can make multiple subsystems reach a stable state simultaneously.However, a shortcoming remains; that is, the control protocols are associated with the relative state information between followers, so they can only cope with the case where followers have identical state dimensions. Moreover, our research framework is similar to that in [46]; the main differences include the following: 1) Different problems: The problem we consider is containment control with multiple leaders, while the problem in [46] is leader-following consensus with a single leader. 2) Different stability analysis methods for closed-loop system: Our method is based on a Lyapunov inequality, while [46] is based on an algebraic Riccati equation.

IV. ROBUST CONTAINMENT CONTROL OF HETEROGENEOUS MULTI-AGENT SYSTEMS

In this section, we further generalize the theoretical results to a more general case, where the followers have nonidentical state dimensions and the above strategies cannot be applied directly. Here, with the aid of the compensator (11), we introduce a new type of distributed dynamic output feedback control law by replacing the state-coupled item with the state variate of the compensator in the previous control law that lends itself to the avoidance of dependence on relative states,such that the new control protocol can cope with the case where the followers have nonidentical dynamic dimensions.

ConsiderNheterogeneous follower subsystems with the following dynamics:

V. EXAMPLE SIMULATION

In this section, two numerical simulation examples are adopted to illustrate the validity of the results of Theorems 2 and 3.

A. Example 1

Fig. 1. The topology network.

By solving the Riccati equation and Lyapunov inequality:

Fig. 2. The output trajectories of the followers and the leaders for the closed loop multi-agent system in Example 1.

Fig. 3. The output trajectories of the followers and the leaders over time for the closed loop multi-agent system in Example 1.

Fig. 4. The containment error of the followers over time for the closed loop multi-agent system in Example 1.

B. Example 2

Consider a team of agents consisting of three heterogeneous

Fig. 5. The state trajectories of the compensators and leaders over time.

Fig. 6. The output trajectories of the followers and the leaders for the closed loop multi-agent system in Example 2.

Fig. 7. The output trajectories of the followers and the leaders over time for the closed loop multi-agent system in Example 2.

VI. CONCLUSIONS

Fig. 8. The containment error of the followers over time.

Fig. 9. The virtual error of the followers over time.

In this paper, we have solved the robust output containment control problem for linear multi-agent systems with a directed fixed network G. A novel distributed dynamic compensator with measurable leaders’ output was designed to estimate the convex combination of the leaders’ states in combination with the communication network. First, based on the state-coupled item, the internal model compensator and the follower dynamic compensator, the distributed dynamic state and output feedback laws were adopted to drive uncertain followers with identical state dimensions into the convex hull spanned by the leaders. Subsequently, based on the leader compensator, the robust containment problem was converted into an equivalent tracking problem. A new distributed dynamic output feedback control law was designed to drive uncertain followers into the leaders’ convex hull under the output regulation framework. Unlike most existing control approaches, our method addressed the dependence on the relative states; therefore, it is capable of coping with containment problems where the followers have nonidentical state dimensions.

APPENDIX

A. The Proof of Lemma 1

B. The Proof of Lemma 7