Event-Triggered Fixed-Time Integral Sliding Mode Control for Nonlinear Multi-Agent Systems with Disturbances

In this paper, the leader-following consensus problem of first-order nonlinear multi-agent systems (FONMASs) with external disturbances is studied. Firstly, a novel distributed fixed-time sliding mode manifold is designed and a new static event-triggered protocol over general directed graph is proposed which can well suppress the external disturbances and make the FONMASs achieve leader-following consensus in fixed-time. Based on fixed-time stability theory and inequality technique, the conditions to be satisfied by the control parameters are obtained and the Zeno behavior can be avoided. In addition, we improve the proposed protocol and propose a new event-triggering strategy for the FONMASs with multiple leaders. The systems can reach the sliding mode surface and achieve containment control in fixed-time if the control parameters are designed carefully. Finally, several numerical simulations are given to show the effectiveness of the proposed protocols.


Introduction
In the past several years, more and more researchers are interested in cooperative control of multi-agent systems (MASs) because of its robustness, flexible deployment and high efficiency. Cooperative control is widely used in various research fields to solve engineering and non-engineering problems, such as formation of robots [1], sensor networks [2], attitude alignment [3] and so on. Among multitudinous cooperative control objectives, consensus is a basic problem in MASs. Its purpose is to design a controller which can ensure that all members agree on an interest signal according to local information. Therefore, the information exchange between agents on the shared network is regulated by the consensus algorithm or protocol.
Based on observation of nature, the emergence of leaders in animal groups led to the development of the leader-following problem in collective behavior of MASs. In the distributed consensus problem, the existing results of MASs can be roughly divided into three categories according to the number of leaders: leaderless consensus [4][5][6], leaderfollowing consensus [7][8][9] and containment control of multiple leaders [10,11]. In [4], the leaderless consensus of discrete-time MASs was studied by considering the connectivity of the network. In [5], the leaderless consensus of model-independent MASs was considered. In [6], the leaderless consensus of fractional-order MASs was investigated. In the case of single leader, the leader-following bipartite consensus problem was investigated for linear MASs in [7]. The leader-following consensus for MASs with Lipshitz-type node dynamics was considered in [8]. Furthermore, by using distributed impulsive control method, the authors studied the leader-following consensus of nonlinear MASs in [9]. In the case of multiple leaders, the reduplicative learning control problem for nonlinear heterogeneous MASs was investigated in [10]. In [11], a completely distributed control protocol was proposed to study the time-varying group formation tracking problem for linear MASs with multiple leaders. triggered finite-time consensus for multirobot systems with disturbances was considered via integral SMC strategy in [35]. The finite-time consensus for nonholonomic MASs with disturbances was studied by using event-triggered integral SMC method in [36]. However, in the existing research, the fixed-time event-triggered integral SMC method for single leader and multiple leaders of MASs with disturbances are rarely considered.
Inspired by the above considerations, this paper studies the fixed-time consensus problem for FONMASs with single leader and multiple leaders by using integral SMC and the theory of fixed-time stability. Firstly, to study the fixed-time leader-following consensus of FONMASs with external disturbances, a new event-triggered integral SMC protocol is devised for each agent. We show that both the systems can get to the sliding mode surface and all agents will achieve consensus in fixed-time under the proposed protocol. Moreover, the FONMAS with multiple leaders and external disturbances is considered. By generalizing our proposed event-triggered integral SMC protocol, it is proved that the containment control can also be achieved and the disturbance can be effectively suppressed in fixed-time. Compared with the existing works, the main contributions of the paper are at least the following three points: 1. In existing works [27,28], the disturbance rejection method was applied to study the fixed-time consensus of MASs with external disturbances. However, the integral SMC method combination with event-triggered control mechanism are introduced to design the distributed protocol in this paper, which can effectively suppress the disturbances and achieve consensus in fixed-time. 2. In [35,36], the finite-time event-triggered integral SMC protocols were proposed, in which the estimation of settling time was associated with the initial conditions. To overcome this disadvantage, the fixed-time event-triggered integral SMC protocols are proposed in this paper. According to the stability theory of fixed-time, we can prove that the consensus can be reached in fixed-time and the upper bound estimation of settling time is regardless of the initial conditions of MASs. 3. The containment control for FONMASs with multiple leaders is also considered, in which a generalized event-triggered integral SMC protocol is designed and the controller is updated only at some discrete instants. The sliding surface and the containment control can be reached in fixed-time. The Zeno phenomenon can be avoided.
The remainder of this paper is organized as follows. Section 2 introduces some preliminaries including graph theory, definitions, lemmas and problem formulation. In Section 3, the consensus protocols based on SMC technique are proposed, and some theorems are proved. In Section 4, the effectiveness of the proposed control protocols is verified by numerical simulations. Some conclusions are given in Section 5.

Algebraic Graph Theory
A graph that consists of N nodes is represented by G = (V, E , A) where V = {v 1 , · · · , v N } is the set of nodes, E denotes the edges set, in which (i, j) ∈ E if there exists an edge between v i and v j . The weighted adjacency matrix is denoted as A = [a ij ] ∈ R N×N , where a ij > 0 if (j, i) ∈ E , and a ij = 0, otherwise. The set of neighbors of agent i is denoted by N i = {j ∈ V : (j, i) ∈ E }. The graph G is called directed and strongly connected if there exists a directed path between each pair of nodes. The graph G contains a directed spanning tree if there exists at least one root. The Laplacian matrix L = [l ij ] N×N is defined by l ij = −a ij for i = j, and l ii = ∑ N j =i a ij .

Definitions and Lemmas
Consider the following differential equatioṅ where x(t) ∈ R n , and f : R n → R n is a nonlinear function with f (0) = 0. The following definitions and lemmas are given.

Definition 1 ([37]
). For any solution x(t, t 0 , x 0 ) of system (1), if there exists a positive number T(x 0 ) such that x(t, t 0 , x 0 ) = 0 for all t ≥ t 0 + T(x 0 ), then the solution x = 0 is said to be globally uniformly finite-time stable. T(x 0 ) is called the settling time. Moreover, x = 0 is said to be globally fixed-time stable if T(x 0 ) is independent of the initial value x 0 . (1), if there is a regular, positive definite and radially unbounded function W(x) : R n → R such that any solution of (1) satisfies the inequalitẏ

Problem Formulation
Consider a FONMAS consisting of N followers and a virtual leader indexed by i = 1, 2, · · · , N and 0, respectively. The dynamics is described bẏ where x i (t) ∈ R n , u i (t) ∈ R n and w i (t) ∈ R n are the state, the bounded control input and the external disturbance of the ith agent, respectively. f (x i (t)) is a nonlinear function, which represents the inherent dynamics. In addition, we assume that the external disturbance is bounded, which satisfies w i (t) ∞ ≤ D < ∞, for D > 0. x 0 (t) ∈ R n and u 0 (t) ∈ R n are the state and the bounded control input of the leader, respectively. f (x 0 (t)) is a nonlinear function, which also represents the inherent dynamics. The communication topology among followers is expressed as directed graphĜ, and the corresponding Laplacian matrix is described by the weighted matrixL. We use b i to represent the communication weight between the leader and the ith agent, in which b i > 0 if the ith agent can receive information from the leader, b i = 0 otherwise. In addition, we denote B = diag(b 0 , · · · , b N ).

Definition 2.
For the FONMAS (2), the fixed-time leader-following consensus is achieved for any initial conditions, if the following equations hold where T > 0 is called the settling time.

Assumption 1.
For the nonlinear function f (·), there exists a positive constant l 1 > 0 such that where z 1 (t), z 2 (t) ∈ R n .

Assumption 2.
The communication between the leader and all followers is represented by graph G which contains a directed spanning tree with the leader as the root. In addition, the communication topologyĜ is directed.

Fixed-Time Consensus with Single Leader
In this section, in order to achieve consensus between leader and followers, the integral SMC protocol will be designed for FONMAS described by (2). Before moving on, we define the following error variables Since the disturbances exist in the follower agent dynamics, the integral SMC technique is applied. Then, we define the following integral type sliding mode variable where η is the ratio of two positive odd numbers and η > 1. When the sliding mode surface is reached, σ i (t) = 0 andσ i (t) = 0. Hence, it has˙ In order to reduce the control cost and increase the rate of convergence, the eventtriggered consensus protocol is designed as follows where β > 0, K = K 1 + K 2 , K 1 , K 2 , K 3 , K 4 are constants to be determined. t i k is the triggering instant. Then, the novel measurement error is designed as In this paper, a distributed event-triggered sampling control is proposed. The trigger instant of each agent only depends on its trigger function. Based on the zero order hold, the control input is a constant in each trigger interval. In order to make FONMAS (2) achieve leader-following consensus under the proposed protocol (7), the following theorem is given. Theorem 1. Suppose that Assumptions 1 and 2 hold for the FONMAS (2). Under the protocol (7), the leader-following consensus can be achieved in fixed-time, if the following conditions are satisfied where ξ i > 0 for i = 1, 2, · · · , N. The triggering condition is defined as Proof. Firstly, we prove that the sliding mode surface σ i (t) =σ i (t) = 0 for i = 1, 2, · · · , N can be achieved in fixed-time. Consider the Lyapunov function as Based on Assumption 1, it has Based on conditions (9), we can geṫ According to triggering condition (10), we havė The closed-loop system will get to the sliding mode surface in fixed-time, which can be obtained according to Lemma 1. The settling time can be computed as Define T = max 1≤i≤N {T i }. Then, it is proved that the sliding mode surface σ i (t) = 0 can be reached for any t > T.
Secondly, we will prove that the leader-following consensus can be achieved in fixedtime. For convenience, χ i (t) for i = 1, 2, · · · , N can be rewritten in the following compact form χ(t) = −((L + B) ⊗ I n ) x(t) and sgn(χ(t)) ≤ √ Nn. LetL + B = H. Based on Assumption 2, the matrix H is invertible and all eigenvalues have positive real parts. Therefore, there exists a positive symmetric matrix P such that By Lemma 1, we can conclude that the closed-loop system will achieve consensus in fixed-time. The settling time can be computed as Remark 1. In this paper, the general directed network topology is considered, so the matrix H is asymmetric. We need to select the positive definite matrix P to make it symmetric. In particular, if the network topologyĜ is undirected, the matrix P corresponds to the identity matrix, and the construction of Lyapunov functionṼ(t) can be simplified. This reduces the computational burden.

Remark 2.
In [12][13][14], the finite-time consensus problem of MASs was studied. Compared with these literatures, we propose a fixed-time consensus protocol. Based on (17), we can find that the estimation of settling time is independent of initial values. In [15,16], the fixed-time consensus of MASs under ideal environment was studied. However, this paper considers a more complex environment in which agents of MASs are affected by external disturbances. We propose a new fixed-time consensus protocol based on integral sliding mode technique, which can suppress the disturbances better and improve the closed-loop performance of the system.

Remark 3.
There are generally three methods to deal with disturbances, namely internal made method, disturbances observation and sliding mode control. In [27,28], the disturbance rejection method was applied to eliminate the influence of disturbances in the protocols. However, in this paper, we adopt the integral sliding mode technique combined with event-triggered to suppress disturbances. Our research enriches the design method of control protocol and theoretical results. In [35,36], although the consensus of FONMASs with external disturbances was discussed by using integral sliding mode technique, only finite-time convergence was analyzed, and the estimation of settling time related to the initial conditions of system. To overcome this disadvantage, this paper proposes a new fixed-time event-triggered integral SMC protocol, in which the sliding mode surface can be reached and the consensus can be achieved in fixed-time. (2) with the event-triggered control protocol (7). If the triggering condition is defined by (10) and the conditions of Theorem 1 hold, then the Zeno behavior can be eliminated.

Theorem 2. Consider the FONMAS
Proof. The proof is divided into two parts, before and after reaching the sliding mode surface. On the one hand, we show the Zeno behavior does not exist before the systems achieve the sliding mode surface. Through the analysis of Theorem 1, the sliding mode surface will be reached when t > T. Therefore, we need to eliminate the Zeno behavior in the closed interval t ∈ [0, T]. Since χ i (t) is a continuous function, it must exist a maximum value.
Take the time derivative of e i (t) , it has where Using the triggering condition (10), the next trigger instant satisfies e i (t i k+1 ) = ξ i . Therefore, On the other hand, when the sliding mode surface is reached, σ i (t) = 0. Similar to the above proof, we can obtain Combination with e i (t i k ) = 0, one has Using the triggering condition (10), on can obtain where Based on the above analysis, the Zeno behavior can be avoided in control process.

Remark 4.
Since the trigger mechanism exists in the whole control process, the proof of Theorem 2 divided into two parts, i.e., before and after the system reaches the sliding mode surface. In this paper, a static distributed event-triggered strategy is developed. In order to reduce the number of triggers more effectively, we will consider the dynamic event-triggered control strategy in our future work.

Fixed-Time Containment Control with Multiple Leaders
In this section, we consider the MASs with multiple leaders. The main aim is to make MASs realize containment control in fixed-time by designing appropriate control protocol. That means all follower agents' states converge to the convex combination of leaders' states in fixed-time. In particular, if the MASs has only one leader, the containment control will degenerate into leader-following consensus.
For the sake of generality, we hypothesize that the FONMAS consisting of N followers and M leaders indexed by indexed by i = 1, · · · , N and j = N + 1, · · · , N + M, respectively. The dynamics of the FONMAS is described bẏ where x i (t) ∈ R n , u i (t) ∈ R n and w i (t) ∈ R n are the state, the bounded control input and the external disturbance of the ith agent, respectively. f (x i (t)) is a nonlinear function which represents the inherent dynamics. x j (t) ∈ R n , u j (t) ∈ R n and w j (t) ∈ R n are the state, the bounded control input and the internal disturbance of the jth leader, respectively. f (x j (t)) is a nonlinear function, which also represents the inherent dynamics. In addition, we assume that the disturbances are bounded, which satisfy w i (t) ∞ ≤ B < ∞, w j (t) ∞ ≤ F < ∞ for B > 0 and F > 0.

Assumption 3.
Suppose that the communication among the leaders and followers is represented by graph G. For each follower, there exists at least one leader that has a directed path to it.

Assumption 4.
Given scalars ρ 1 , ρ 2 , · · · , ρ M , satisfying ∑ M j=1 ρ j = 1 and ρ j ≥ 0. There exists a constant l 2 > 0 such that for Under Assumption 3, the Laplcain matrix of graph G is denoted by L, which can be decomposed into L = L 1 L 2 0 0 , where L 1 is a nonsingular matrix, L 2 ∈ R N×M has at least one positive entry and −L −1 Before moving on, we define the following error variables where Combination with Assumption 3 and the property of Laplacian matrix L, we can easily obtain that the containment control is achieve in fixed-time if and only if there exists a T > 0 such that lim t→T X(t) = 0 and X(t) ≡ 0 for t > T .
Considering the disturbances in the system, the consensus protocol can employ sliding mode approach. The integral type sliding variable is defined as follows whereχ i (t) = − X i (t), η is the ratio of two positive odd numbers and η > 1. The sliding mode manifold (26) is given by following comport form (χ η (s) + sgn(χ(s)))ds.
When the sliding mode surface is reached, σ(t) = 0 andσ(t) = 0. Hence, it haṡ X(t) =χ η (t) + sgn(χ(t)). (28) In order to reduce the control cost and increase the rate of convergence, the eventtriggered sample-data control protocol is presented as where β > 0, K = K 1 + K 2 , K 1 , K 2 , K 3 , K 4 are constants to be determined. t k is the triggering instant. Similarly, the controller (29) can be rewritten in the following comport form Then, the novel measurement error for the system (24) is designed as Theorem 3. Suppose that Assumptions 3 and 4 hold for the FONMAS (24). Under the protocol (30), the containment control can be achieved in fixed-time, if the following inequalities are satisfied: The triggering condition is defined as where ξ > 0.

Proof. Consider the Lyapunov function as
Define Based on (32), we can geṫ According to (33), we havė According to Lemma 1, the closed-loop system (24) will get to the sliding mode surface in fixed-time. The settling time can be estimated bȳ Then, it is proved that σ(t) = 0 is reached for t >T.
Then, we will prove that the containment control can be achieved in fixed-time. Define the Lyapunov function asV(t) =χ T (t)χ(t). Taking the time derivative ofV(t) for t >T yields˙V By Lemma 1, we can conclude that the closed-loop system will achieve containment control in fixed-time. The settling time can be computed aŝ The proof is finished.

Remark 5.
In [27], the fixed-time consensus problem of MASs with nonlinear dynamics and indeterminate disturbances was considered based on event-triggered method. Compared with [27], we introduce the integral sliding mode technique to deal with disturbances, and consider the containment control problem in the case of multiple leaders. In addition, the event-triggered strategy applied in this paper can greatly save computation and communication resources.

Theorem 4.
Consider the FONMAS (24) with the event-triggered control protocol (30). If the triggering condition is defined by (33) and all conditions of Theorem 3 are satisfied, then the Zeno behavior can be avoided.
Proof. Similar to the proof of Theorem 2, the proof is divided into two parts. First, we show that the Zeno behavior does not exist before the systems reach the sliding mode surface. Through the analysis of Theorem 3, we know that sliding mode surface will be reached when t >T. Therefore, we need to eliminate the Zeno behavior in the closed interval [0,T]. Since χ(t) is a continuous function, it must exist a maximum value.
Take the time derivative of e(t) , we have where ψ = ηγ Applying the triggering mechanism (33), it has e(t k+1 ) = ξ. Therefore, Denote , we can get ∆T k ≥ ξ π 1 > 0. Next, we prove that the Zeno behavior can be avoided when the sliding mode surface is reached. Similar to the above proof, we can obtain Combination with e(t k ) = 0, it yields When the event next event is triggered, it has e(t) = ξ. Therefore, Let π 2 = ηγV(0) η 2 + ηγ √ Nn and ∆T k = t k+1 − t k , we can get ∆T k ≥ ξ π 2 > 0. Based on above analysis, the lower bound of event-triggered interval is positive, then Zeno phenomenon is eliminated in the whole control process.
Then, Assumption 1 holds. The external disturbances are defined as w 11 (t) = w 12 (t) = 0.05 sin(t) + 0.1 cos(t), w 21 (t) = w 22 (t) = 0.05 sin(t) + 0.1 cos(t), w 31 (t) = w 32 (t) = 0.05 sin(t), and w 41 (t) = w 42 (t) = 0.1 cos(t). It follows that w i (t) ∞ ≤ 0.2, i = 1, 2, 3, 4. The control input of leader is u 01 (t) = u 02 (t) = 0.1 sin(t) + 0.1 cos(t). We choose the controller parameters K 1 = 0.2, K 2 = 1.7, K 3 = 1.5, K 4 = 2, η = 7 5 , β = 1.5, ξ i = 0.2 for i = 1, 2, 3, 4 and implement the proposed control protocol (7). Through the analysis, all conditions (9) of Theorem 1 are satisfied. The simulation results are presented in Figures 2-4. Specifically, Figure 2 depicts the states of all followers and the leader. It can be seen that all followers can track the leader's state in fixed-time under the proposed sliding mode control protocol (7) and the setting time isT ≤ 13.86. Based on analysis of Theorem 1, the sliding mode variable σ(t) converges to zero in fixed-time, and the setting time is T ≤ 2, which is verified in Figure 3. The event-triggered instants under the triggering mechanism (10) are shown in Figure 4. It is shown that the event-triggered instants of each agent are different. Therefore, the results of Theorem 1 are feasible and the proposed sliding mode control protocol (7) can effectively suppress the external disturbances and realize leader-following consensus in fixed-time.
It has w i (t) ∞ ≤ 0.2, i = 1, 2, 3, 4, w j (t) ∞ ≤ 0.2, j = 5, 6. The control input are u j1 (t) = u j2 (t) = 0.1 sin(t) + 0.1 cos(t), j = 5, 6. We choose the controller parameters K 1 = 1.7, K 2 = 2, K 3 = 1.5, K 4 = 24.5, η = 7 5 , β = 1.2, ξ = 1 and implement the proposed control protocol (30). Through simple calculation, we can verify that all conditions of Theorem 3 are satisfied. The simulation results are presented in Figures 5-7. Specifically, Figure 5 shows the states of four followers and two leaders. It can be seen that all followers' states gradually achieve consensus and fall into the convex hull of the leaders' states in fixed-time and the setting time isT ≤ 11.3. Figure 6 shows the evolution of sliding mode variable σ(t). The sliding mode surface can be reached in fixed-time, and the settling time isT ≤ 4.3. The triggering interval under the event-triggered mechanism (33) is presented in Figure 7. In order to show the event-triggered intervals more clearly, we only give the simulation result for a short period of time, from which we can see that the Zeno phenomenon can be excluded. Different from the distributed event triggering condition (10), we employ a centralized trigger function, which also can ensure the reachability of the consensus. In particular, if the FONMAS (24) with one leader, the containment control can be reduced into leader-following tracking problem.  Figure 5. The states of x i (t), i = 1, 2, · · · , 6. (a) The states of −x i1 (t) + 2z(x i1 (t)) − 1.2z(x i2 (t)); (b) The states of −x i2 (t) + 1.2z(x i1 (t)) + 2z(x i2 (t)).

Conclusions
In this paper, considering external disturbances, the leader-following consensus and containment control of FONMASs are studied. Two kinds of event-triggered integral SMC protocols are designed, which can well suppress the external disturbances and make the FONMASs achieve consensus in fixed-time. Based on fixed-time stability theory and inequality technique, some criteria are obtained and the Zeno behavior can be avoided. The effectiveness of the proposed protocols are verified by several numerical simulations.
In the future work, the consensus of higher-order MASs with dynamic event-triggered communication mechanism will be considered.