Abstract
This paper considers the containment control problem for uncertain nonlinear multiagent systems under directed graphs. The followers are governed by nonlinear systems with unknown dynamics while the multiple leaders are neighbors of a subset of the followers. Fuzzy logic systems (FLSs) are used to identify the unknown dynamics and a distributed state feedback containment control protocol is proposed. This result is extended to the output feedback case, where observers are designed to estimate the unmeasurable states. Then, an output feedback containment control scheme is presented. The developed state feedback and output feedback containment controllers guarantee that the states of all followers converge to the convex hull spanned by the dynamic leaders. Based on Lyapunov stability theory, it is proved that the containment control errors are uniformly ultimately bounded (UUB). An example is provided to show the effectiveness of the proposed control method.
1. Introduction
Since the scales of practice control system became larger and larger, much attention has been paid to complex systems, such as interconnected systems and multiagent systems. An interconnected system means a system that consists of interacting subsystems. The main control objective of interconnected systems is to find some decentralized feedback laws for adapting the interconnections from the other subsystems, where no state information is transferred [1–4]. Multiagent systems consist of some intelligent agents, which have the capability of reacting to the variety of environments automatically, such as robots, automatic vehicles, and sensors. The main control objective of multiagent systems is to establish distributed control laws based on the information of the agent and its neighbors to realize collective behavior [5–8]. In the past decades, cooperative control problem of multiagent systems has attracted significant research interests, which mainly focuses on consensus [9–14], formation control [15, 16], and containment control [17].
Containment control aims at guiding the states or outputs of the followers to converge to a convex hull formed by the multiple leaders using a distributed control protocol. The problem has many applications, for example, securing a group of followers in the area spanned by the leaders so that they can be away from dangerous sources outside the area. Recently, distributed containment control problem has been investigated and numerous research results have been obtained [18–27]. Containment control strategies were proposed for multiagent systems with single-integrator [17–19], double-integrator [20–22], or general linear dynamics [23]. However, the reported methods can only deal with the containment control problem of linear multiagent systems. By now, there have been some results on containment control for nonlinear multiagent systems in [24–27]. It should be noted that the proposed containment controllers required each agent satisfying Lagrangian dynamics with known nonlinearities [24, 25], linearly parameterized nonlinearities [26], or unknown nonlinearities [27]. Therefore, containment control problem for uncertain nonlinear multiagent systems needs to be further investigated.
Motivated by the above observations, in this paper, containment control problem for multiagent systems with more general nonlinear dynamics is studied. The nonlinear dynamics of each follower can be totally unknown. Using FLSs to identify the unknown nonlinear dynamics, distributed state feedback and output feedback containment control schemes are proposed to drive the states of all followers into the convex hull spanned by the leaders. It is proved that the containment control errors converge to a residual set. The rest of the paper is organized as follows. Section 2 formulates problem formulation. Section 3 provides the design of distributed state feedback containment controllers. Section 4 provides the design of distributed state feedback containment controllers. Section 5 gives an illustrative example to show the effectiveness of the proposed approaches. Section 6 concludes the paper.
Compared with the existing results on nonlinear multiagent systems, the main advantages of the proposed containment control scheme in this paper are listed as follows.(1)In [14], the consensus scheme was proposed for general nonlinear multiagent systems, which drive all followers to track the states of the leader. In this paper, we develop a containment control method for the nonlinear multiagent systems to drive all followers to converge to a convex hull formed by the multiple leaders. It should be noted that consensus and containment control are two different problems in the cooperative control of multiagent systems.(2)In [17–23], containment control methods were proposed for single-integrator, double-integrator, or general linear multiagent systems. From a piratical perspective, we consider the containment control problem for nonlinear multiagent systems in this paper.(3)In [24–27], containment control scheme was developed for nonlinear multiagent systems with Lagrangian dynamics, where the nonlinearities were assumed to be known, linearly parameterized, or unknown. In this paper, we design a state feedback containment control scheme for more general nonlinear multiagent systems with unknown dynamics. Besides, considering that some states in the systems are unmeasurable in practice, an output feedback containment control scheme is proposed.
Notations. Throughout this paper, is a set of positive real numbers. is a set of real matrices. is an identity matrix with the dimension of . is the Euclidean norm of a vector. is the Frobenius norm of a matrix. is the trace of a matrix. and are the maximum and minimum singular values of a matrix, respectively. is a diagonal matrix with being the th diagonal element. is the Kronecker product.
2. Problem Formulation
Consider a class of nonlinear multiagent systems consisting of followers and leaders. The dynamics of follower are described by where , , and are the state vectors, inputs, and outputs of the systems. are unknown nonlinear functions. , , and are known matrices.
The dynamics of leader are given by where are the state vectors. are unknown bounded inputs.
The information flow among the agents can be described by a directed graph which consists of a vertex set , an edge set , and an adjacency matrix . represents agent node . means that there is a directed information flow from agent to agent . The neighbor set of node is denoted by . Each element of is defined as , if , and , if . Throughout this paper, it is assumed that . If , for all , the graph is undirected; otherwise the graph is directed. A directed graph has a spanning tree if there is a root node, such that there is a directed path from the root node to every other node in the graph. The Laplacian matrix is defined as Then, the Laplacian matrix , where is the degree matrix with .
An agent is called a follower if the agent has at least one neighbor. An agent is called a leader if the agent has no neighbor. Without loss of generality, we assume that the agents indexed by are followers, whereas the agents indexed by are leaders. Then, the Laplacian matrix can be partitioned as where and .
Assumption 1. For each follower, there exists at least one leader that has a directed path to that follower.
Lemma 2 (see [24]). Under Assumption 1, all the eigenvalues of have positive real parts, each entry of is nonnegative, and each row of has a sum equal to 1.
Definition 3. The set is said to be convex if, for any and any , the point is in . The convex hull for a set of points is the minimal convex set containing all points in and is defined as .
The control objective is to design containment controllers , such that the states of all followers converge to the convex hull formed by the leaders , that is, , for all , where , .
Let and . From Lemma 2, we can obtain with . Therefore, the control objective can be transformed as
where . The containment control errors are defined as , .
In this paper, we adopt the singleton fuzzifier, product inference, and the center-defuzzifier to deduce the following fuzzy rules [28–30].
: IF is , and and is , THEN is , where and are the input and output of the fuzzy system, respectively. and are fuzzy sets in . The fuzzy inference engine performs a mapping from fuzzy sets in to a fuzzy set in based on the IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The fuzzifier maps a crisp point into a fuzzy set in . The defuzzifier maps a fuzzy set in to a crisp point in . Since the strategy of singleton fuzzification, center-average defuzzification, and product inference is used, the output of the fuzzy system can be formulated as
where is the point at which fuzzy membership function achieves its maximum value. It is assumed that . Let , , and . Then the fuzzy logic system (6) can be rewritten as
It has been proved in [31] that if Gaussian functions are used as membership functions, the following lemma holds.
Lemma 4. Let be a continuous function defined on a compact set . Then, for any constant , there exists an FLS such as where is a compact region for . is an adjustable vector. is a fuzzy basis function vector. Optimal parameter vector is defined as where is the compact set of . Then where is the minimum fuzzy approximation error with an unknown bound.
3. The Design of Distributed State Feedback Containment Controllers
3.1. State Feedback Containment Controller Design
Distributed containment controllers are proposed as where and are designed as follows: where is a coupling gain. is a controller gain with , and is positive definite satisfying the following Riccati inequality: where is positive definite. By Lemma 4, the multiple-input multiple-output unknown dynamics can be approximated by FLSs as [24, 25] Then, are designed as where are the estimations of .
Let , . Then, one has where , , , .
Let and . Then Substituting (16) into the derivative of (17), we have
3.2. Stability Analysis
Theorem 5. Consider the multiagent systems given by (1), (2). Under Assumption 1, the communication graph is directed and has a spanning tree. Select the containment controllers (11), (12), and (15) with the coupling gain satisfying where are the eigenvalues of . are updated by where , . Then, all the signals in the closed-loop multiagent systems are UUB, and the containment control errors satisfy where , .
Proof. Consider the Lyapunov function candidate where . Substituting and (18) into the derivative of (22), we have It follows from (20) that By Assumption 1 and Lemma 2, all the eigenvalues of have positive real parts. Thus, there exists a unitary matrix such that , . Let , where . Then, it follows from (24) that Substituting (13) and (19) into (25), one has By Lemma 4, are bounded and use the fact that are bounded. Then, there exist positive constants , and , such that , , and . It follows from (26) that Rewrite (27) in the following matrix form: where Noting the fact that and , it follows that is positive definite. Then Let Then From (30) and (32), we have where , . Then Since , we obtain that all signals in the closed-loop multiagent systems are UUB. Then Then, it follows from (17) that Then, we get (21) with . It means that the states of the followers converge to the convex hull formed by those of the leaders with the containment errors being UUB. The containment control problem is solved.
Remark 6. In [24–27], the distributed containment control approaches were proposed for nonlinear Lagrangian systems. However, the previous approaches cannot be applied to the nonlinear multiagent systems (1), (2). Therefore, it is significant to investigate the distributed containment control problem for more general nonlinear multiagent systems in the presence of unknown dynamics.
4. The Design of Distributed Output Feedback Containment Controllers
The method proposed in Section 3 required the states of the followers being measurable. However, in practice, some states in the systems are unmeasurable. In this section, the output feedback containment controllers will be designed. We assume here that the states of the leaders are measurable and .
4.1. Output Feedback Containment Controller Design
Design distributed observers to estimate the unmeasurable states. Let be the estimations of . Similar to [32–34], the observers are designed in the following form: where is a coupling gain. . is an observer gain with , and is positive definite satisfying the following linear matrix inequality (LMI): where is an adjustable parameter to guarantee the existence of . and is positive definite.
Based on the developed observers, the output feedback containment controllers are designed in (11) with Let . Then, one has where with being the state estimation errors. The state estimation error equation is described by Let and . Then Then
4.2. Stability Analysis
Theorem 7. Consider the multiagent systems given by (1), (2). Under Assumption 1, the communication graph is directed and has a spanning tree. Select the output feedback containment controllers (11), (39) with the coupling gains and satisfying (19) and (44): are updated by where , , and . Then, all the signals in the closed-loop systems are UUB, and the containment control errors satisfy where .
Proof. Consider the Lyapunov function candidate
Substituting and (41) into the derivative of (47), we have
By (45) and , we have
Using Young’s inequality,
Then
It follows from (38) and (44) that
By Lemma 4, it follows that and are bounded. Then, there exist positive constants and , such that , . Then, (52) can be rewritten as
Let
Rewrite (53) in the following form:
Using a similar analysis process to Section 3.2, it follows that and are UUB and the bound of is given by
where with , .
Consider another Lyapunov function candidate
Substituting and (43) into the derivative of (57), we have
Then
Noting the fact that
indicates . Considering (42) and (56), it follows that
Note the fact that . Then, (46) is obtained with
Then, the containment control problem is solved.
Remark 8. In [14], consensus scheme was developed for nonlinear multiagent systems (1); that is, the proposed method can guarantee all states of the followers synchronize to that of a single leader. In this paper, containment control approach is designed to guarantee all states of the followers stay in a dynamic convex hull formed by multiple leaders.
5. Simulation
In this section, a simulation example is provided to show the effectiveness of the proposed distributed output feedback containment controllers.
Consider a network of harmonic oscillators described by (1), with [35] Choose fuzzy membership functions as where .
The communication graph is described in Figure 1. Let and solve (13) and (38). Then In simulation, , , , and . The containment results and containment errors using the output feedback containment controllers of this paper are shown in Figures 2 and 3. It can be observed that the proposed containment scheme can realize that the states of followers converge to the convex hull formed by those of the leaders; that is, the states of all followers stay in the area formed by the leaders. Figure 4 shows the states of developed observer, from which we can see that the designed observer can estimate unmeasurable states with the estimation errors in a small neighborhood of the origin. The profiles of the designed distributed output feedback containment controllers are shown in Figure 5. It can be observed that the designed containment controllers guarantee both the stability and good containment performance of the closed-loop multiagent systems with unknown dynamics. Figure 6 shows the profiles of , from which it can be observed that the unknown dynamics can be compensated by .
6. Conclusions
In this paper, the containment control problems were considered for uncertain nonlinear multiagent systems with measurable and unmeasurable states under directed graphs. Based on FLSs identifying the unknown dynamics of the followers, distributed state feedback containment controllers were designed first. Then, adaptive fuzzy observers were designed to estimate the unmeasurable states. Based on the developed observers, distributed output feedback containment controllers were designed. Both of the developed containment controllers ensure that the states of the followers converge to the convex hull formed by those of the leaders with the containment control errors in a small residual set. Future research efforts will be devoted to the containment control problem of uncertain nonlinear multiagent systems with time-delay.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is partially supported by the Scientific Research Fund of Liaoning Provincial Education Department (L2013244) and in part by Research Foundation of Liaoning University of Technology for Young Teachers (X201313).