Second-Order Containment Control of Multiagent Systems in the Presence of Uncertain Topologies with Time-Varying Delays

This paper considers the containment control problem of second-order multiagent systems in the presence of time-varying delays and uncertainties with dynamically switching communication topologies. Moreover, the control algorithm is proposed for containment control, and the stability of the proposed containment control algorithm is studiedwith the aid of Lyapunov-Krasovskii function when the communication topology is jointly connected. Some sufficient conditions in terms of linear matrix inequalities (LMIs) are provided for second-order containment control with multiple stationary leaders. Finally, simulations are given to verify the effectiveness of the obtained theoretical results.


Introduction
Recently, there has been a growing interest towards development of the distributed cooperative control of multiagent systems (MASs).To our knowledge, plenty of theoretical results about consensus [1][2][3][4][5] and containment control [6][7][8][9] in distributed cooperative control of MASs have been obtained.However, in practical applications, due to communication delays and uncertain topologies that always emerge, analysis and synthesis of distributed cooperative problem have become more complex and difficult.Meanwhile, it is practically significant to investigate the containment control of MASs with delays and uncertainties.
For consensus problems of MASs with uncertain topologies, the average consensus with time-varying communication delays is investigated in [10], and several sufficient conditions for average consensus are derived in terms of linear matrix inequalities (LMIs).In [11], the robust discretetime consensus problem of MASs with uncertain topologies is addressed, and a necessary and sufficient condition for robust discrete-time consensus is obtained by the Lyapunov stability theory.Considering both fixed and switching directed topologies, the average consensus problem in MASs with uncertain topologies and multiple time-varying delays is studied in [12].For linear time-invariant MASs over Markovian switching networks, stochastic consensus problems with time-varying delays and uncertain topologies are analyzed in [13].In [14], the consensus problem of MASs in the presence of uncertain topologies with time-varying delays is analyzed by a new approach, and a condition in terms of linear matrix inequalities is presented for consensus for MASs with switching topologies.
As a kind of extended consensus problem, containment control has been paid much attention, which aims to design appropriate control protocols to drive the followers to a target area (convex hull formed by the leaders) asymptotically.For linear MASs, the cooperative containment control problem is discussed in [15], and several necessary and sufficient containment conditions are presented by using spectral analysis and matrix theory.Considering unconnected topologies of MASs, the containment control for both first-order and second-order MASs with jointly connected topologies is studied in [16].Considering communication delays, containment control for second-order MASs with time-varying delays is studied in [17]; both the case with multiple dynamic leaders and the case with multiple stationary leaders are discussed.In [18], the containment control problem for secondorder MASs with time-varying delays and jointly connected topologies is investigated, and the stability of the proposed control algorithm is analyzed by Lyapunov-Krasovskii method.For uncertain linear MASs, the containment control problem with the dynamic agents described by fractionalorder differential equations is investigated in [19], and some sufficient conditions are presented by the stability theory of fractional-order systems and matrix theory.The abovementioned works implicitly assume the link weights or interaction strengths can be exactly measured; however, environmental uncertainties and measurement error cannot be ignored in real systems.Motivated by these considerations, the containment control of first-order MASs with uncertain topologies and communication time delays is studied in [20], and the sufficient condition for containment control of MASs under jointly connected topologies is derived with the aid of Lyapunov-Krasovskii function.
Two important factors always emerge in some real situations: one is that some uncertainties may exist in the MASs due to measurement error and environmental uncertainties cannot be avoided in real systems and another is that time delays are usually inevitable because of the possible slow process of interactions among the agents in communication networks.Motivated by these two factors, lots of distributed consensus protocols have been developed by some researchers for MASs with uncertainties and delayed communications.However, different from most of those current literatures, for the MASs with uncertainties and time-varying delays, we aim to analyze and investigate the distributed containment control problem with multiple leaders and jointly connected topologies.The algorithms proposed here can ensure that the uncertain MASs achieve containment control in the presence of switching topologies with multiple stationary leaders.By applying linear matrix inequality method, the convergence of the algorithm for the MASs with dynamically switching topologies is analyzed by Lyapunov-Krasovskii method.Finally, numerical simulations are provided to illustrate the effectiveness of the conclusion in this paper.
The paper is organized as follows.In Section 2, some basic concepts in algebraic graph theory and some related lemmas are presented.In Section 3, the main results for second-order MASs with uncertain topologies and time-varying delays are obtained.Then the containment control protocol is presented for MASs with multiple stationary leaders and dynamically switching topologies.Finally, numerical simulations and conclusion are given in Sections 4 and 5, respectively.
A path between two distinct notes  and  is a finite ordered sequence of distinct edges of  with the form ( 1 ,  2 ), ( 2 ,  3 ), . . ., ( −1 ,   ), where  1 =  and   = .The undirected graph  is said to be connected if there is a path between any distinct pair of nodes.The union of a collection of graphs  1 ,  2 , . . .,   with the same note set  is defined as the graph  1− with the note set  and edge set equal to the union of the edge sets of all of the graphs in the collection.Moreover,  1 ,  2 , . . .,   is jointly connected if its union graph  1− is connected [21].
In MASs, an agent is called a leader if it has no neighbor or a follower if it has at least one neighbor [22].Considering a MAS with  followers and  leaders, since the leaders have no neighbors, the Laplacian matrix  associated with the communication graph  can be partitioned as where   ∈ R × and  R ∈ R × .The following lemma about   and  R is brought in, which is useful for deriving the main results.
Lemma 1 (see [23]).If there exists at least a path to a leader for each follower, then   in ( 1) is positive definite, − −1   R is a nonnegative matrix, and the sum of the entries in each row equals 1.
denotes the convex hull of the set .

Main Results
Consider second-order MASs consisting of  followers and  stationary leaders.The followers' set and the leaders' set are denoted by  = {1, 2, . . ., } and R = { + 1,  + 2, . . .,  + }, respectively.The dynamics of each agent can be described by the following equation: where   () ∈ R  ,   () ∈ R  , and   () ∈ R  are the position vector, the velocity vector, and the control input vector of the th agent, respectively.For simplicity, we assume that  = 1 in this paper and the case of  > 1 can be obtained with the Kronecker product.
In a network with uncertain topology, we consider the following containment control algorithm for (5): where  1 > 0 is the feedback gain, Δ  () and Δ  () = 0 are uncertainties, and () is the time-varying communication delay.Here the uncertainties Δ  () exist in the control algorithm, since measurement error and environmental uncertainties cannot be avoided in real systems.Assumption 6.The time-varying communication delay () in ( 6) is bounded; that is, 0 ≤ () < ℎ, where ℎ > 0,  ≥ 0. In this paper, we assume that the norm bounded parameter uncertainty satisfies where  1 ,  2 ,  1 , and  2 are constant matrixes with appropriate dimensions, and the diagonal matrix () satisfies It follows that the uncertainties of MASs satisfy where  > 0 is an appropriate constant.
Considering the second-order MASs in the presence of dynamically switching and uncertain topologies with timevarying delays, dynamics (13) can be written as where In each subinterval [ , ,  ,+1 ), dynamics (17) can be transformed into the following subsystems: where x  () =    () +  Theorem 11.Consider a second-order dynamic system with dynamics (5) of  followers and  stationary leaders with the switching topologies, and suppose Assumptions 6, 7, and 10 hold.Control protocol (6) can solve the containment control of second-order MASs in the presence of uncertain topologies with time-varying delays, for each subinterval [ , ,  ,+1 ), if there exists a constant  > 0 and an appropriate constant  > 0, such that where Proof.For MASs (17), we choose a Lyapunov-Krasovskii function candidate as Evidently, () is essentially a distributed Lyapunov function.By (16), () can be rewritten as Calculating the derivative of () along the trajectories of ( 18) yields By Assumption 6 and Lemma 3, we can get where Denote

Conclusion
In this paper, the distributed containment control for secondorder MASs with multiple stationary leaders and uncertain topologies is investigated.The control algorithm of MASs with time-varying delays and jointly connected topologies is proposed.By applying modern control theory and algebraic graph theory, the convergence of MASs for the proposed containment control algorithm is analyzed via Lyapunov-Krasovskii method.Some sufficient conditions in terms of linear matrix inequalities (LMIs) are given for containment control.The correctness and effectiveness of our theoretical results for uncertain MASs with time-varying delays are demonstrated by some simulation examples.The containment control problem for nonlinear systems will be our future work.

Figure 1 :
Figure 1: Three possible graphs of the communication topology.

Figure 2 :
Figure 2: State trajectories of MASs with uncertain topologies.