Finite-time and ﬁxed-time sliding mode control for second-order nonlinear multiagent systems with external disturbances *

. In this paper, the leader-following consensus of second-order nonlinear multiagent systems (SONMASs) with external disturbances is studied. Firstly, based on terminal sliding model control method, a distributed control protocol is proposed over undirected networks, which can not only suppress the external disturbances, but also make the SONMASs achieve consensus in ﬁnite time. Secondly, to make the settling time independent of the initial values of systems, we improve the protocol and ensure that the SONMASs can reach the sliding surface and achieve consensus in ﬁxed time if the control parameters satisfy some conditions. Moreover, for general directed networks, we design a new ﬁxed-time control protocol and prove that both the sliding mode surface and consensus for SONMASs can be reached in ﬁxed time. Finally, several numerical simulations are given to show the effectiveness of the proposed protocols.


Introduction
In recent years, distributed cooperative control of multiagent systems (MASs) has attracted great attentions because its wide practical applications in engineering, social science, biological science, and other fields. A particularly challenging problem in these fields is commonly referred to consensus. The main task on consensus of MASs is to design some feasible controllers such that all agents can achieve an agreement by sharing information among their neighbors. The consensus of MASs has been studied by many researchers since it can be used in many potential fields, such as the coordination control of UAV [10,20], formation control of vehicles [8,18,26], and so on.
Depending on the number of leaders, the existing consensus results on MASs can be roughly divided into three categories: leaderless consensus [1], single leader-following consensus [11], and containment control of multiple leaders [9]. These results in [1,9,11] did not consider the influence of disturbances in the systems. However, in many practical applications, agents of MASs may face various disturbance signals. Therefore, the consensus of MASs with disturbances has become a hot research topic in distributed cooperative control. There are many techniques to eliminate disturbances in MASs, such as disturbance observation [6], output regulation [5], and internal model principle [13,24]. Since the sliding mode technology can achieve fast convergence rate, sliding mode control method is widely used to the control systems with disturbances. With the development of distributed cooperative control for MASs, many sliding mode control protocols [17,21,30] were proposed to solve the consensus problems of MASs.
As we all know, the convergence rate is one of the key elements in consensus of MASs. In [17,21,27,30], the asymptotical consensus was considered, that means the consensus can be achieved only when the time tended to infinity. However, in practical applications, all agents should reach consensus in a finite-time interval. Hence, the finite-time L 2 leader-follower consensus was investigated for networked Euler-Lagrange systems in [12]. In [31], the authors studied the consensus problem of MASs by using terminal sliding mode. In [15], the finite-time containment control for SONMASs with undirected networks was considered. However, the estimation of settling time depends on initial conditions of MASs in finite time consensus. To overcome this shortcoming, some fixedtime consensus and synchronization problems were considered in [7,16,22,25,32].
The above works [7,15,22,25,31,32] show that the finite-time and fixed-time consensus of SONMASs with external disturbances were rarely considered by using sliding mode control. Thus, it is very meaningful to develop some sliding mode control protocols to study the finite-time and fixed-time consensus of SONMASs with external disturbances. Inspired by this, we propose three terminal sliding mode control protocols to study the finite-time and fixed-time consensus of SONMASs with external disturbances on undirected and directed networks, respectively. Compared with the previous works, the main contributions of the paper are at least the following three points: [17,21,30], a finite-time terminal sliding mode control protocol is designed, which can not only suppress the external disturbances, but also make the SONMASs achieve consensus in finite time. (ii) In [15], although the sliding mode control protocols were proposed, the estimation of settling time depends on initial values of MASs. In this paper, we overcome this disadvantage and propose a fixed-time distributed protocol, which can suppress the external disturbances and make the SONMASs achieve consensus in fixed time.
https://www.journals.vu.lt/nonlinear-analysis (iii) For SONMASs with external disturbances, a novel sliding mode control protocol is carefully designed over general directed networks. Based on fixed-time stability theory, we also rigorously prove that the external disturbances can be suppressed, and the leader-following consensus can be reached in fixed time.
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, three consensus control protocols based on sliding mode technique are proposed, and relevant theorems are given. In Section 4, the effectiveness of the proposed control protocols is verified by numerical simulations. Finally, some conclusions are given in Section 5.
Notations. In this paper, R n denotes the n-dimensional Euclidean space. I n denotes n dimensional unit matrix. For q = [q 1 , q 2 , . . . , q N ] T , q 1 , q , and q ∞ represent the 1-norm, 2-norm, and ∞-norm of vector q, respectively.
where α > 0. For a matrix A ∈ R N ×N , let A T represents its transpose, A −1 represents its inverse, λ max (A) and λ min (A) represent the maximum eigenvalue and minimum eigenvalue of A, respectively. 1 N is the column vector with all elements of 1. The symbol ⊗ denotes the Kronecker product of matrices. sgn(·) represents the sign function. diag(·) represents the diagonal matrix.

Algebraic graph theory
A MAS consisting of N agents is represented by a graph G = (V, E, A) where V = {v 1 , . . . , v N } is a set of nodes, E denotes a set of edges in which (i, j) ∈ E if there is 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}. G is an undirected graph if a ij = a ji . The graph G is called directed and strongly connected if there is a directed path between each pair of nodes. The graph G contains a directed spanning tree if there is at least one root. A node is called globally reachable if it is a 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 . For a leader-following MAS, b i > 0 if the ith agent can receive information from the leader, b i = 0 otherwise. Let B = diag(b 1 , . . . , b N ). The matrix L + B is invertible if the network topology G has a directed spanning tree.

Definitions and lemmas
Consider the following differential equation: where x(t) ∈ R n denotes the state variable, and f (·) : R n → R n is a nonlinear function. Assume that the origin is the equilibrium point of system (1).
The following definitions and lemmas are given.
Definition 1. (See [19].) The origin of system (1) is said to be globally finite-time stable if for any solution x(t) with x 0 ∈ R n , there is a positive number T (x 0 ) such that x(t) = 0 for all t T (x 0 ). The positive number T (x 0 ) is called the settling time. Moreover, if the positive number T (x 0 ) independent of initial value x 0 , the origin is said to be globally fixed-time stable.

Problem formulation
Consider a SONMAS consisting of N followers and a leader. The dynamics of the ith (i = 1, 2, . . . , N ) follower is described bẏ where x i (t) ∈ R n and v i (t) ∈ R n denote the position and velocity, respectively. w i (t) ∈ R n is the external disturbance, u i (t) ∈ R n is the control input, and f (x i (t), v i (t)) : R n × R n × R → R n is a nonlinear function, which represents the inherent dynamics. In addition, we assume that the external disturbance is bounded, which satisfies The dynamics of the leader is described bẏ where x 0 (t) ∈ R n and v 0 (t) ∈ R n are the position and velocity of the leader, respec- is a nonlinear function, which represents the inherent dynamics. For convenience, we write f (x 0 (t), v 0 (t)) as f 0 (t). In addition, we also assume that the external disturbance is bounded, which satisfies w 0 (t) ∞ c < ∞ for c > 0.
Definition 2. For the SONMASs (2)-(3), the fixed-time leader-following consensus is achieved for any initial conditions if there exists a positive constant T such that Before moving on, the following assumptions are presented.

Assumption 1.
For the nonlinear function f (·), there are two nonnegative constants l 1 and l 2 such that where x(t), y(t), z(t), ς(t) ∈ R n . Remark 1. Assumption 1 is very mild and usually used in SONMASs [17,21,28], which is so-called QUAD condition on vector fields [3]. This condition can be satisfied for all linear and piecewise-linear continuous functions. In addition, the condition is also satisfied if ∂f i /∂x j (i, j = 1, 2, . . . , n) are uniformly bounded, which includes many well-known systems.
Assumption 2. The leader is globally reachable and the communication topology of followers is undirected.
The leader is globally reachable and the communication topology of followers is directed.

Main results
In this section, three different control protocols are designed for SONMASs described by (2) and (3). To achieve consensus between leader and followers, we need to design the following error variables.
Define the consensus errors as follows: Let where

Finite-time consensus with undirected networks
We choose the sliding mode manifold as follows: where 0 < α < 1 and β > 1. The sliding mode manifold (8) can be rewritten in the following comport form: In order to guarantee the closed-loop system reaching the sliding mode manifold in finite time, the following control protocol is proposed for the ith agent: where k 1 , k 2 , k 3 , k 4 , and k 5 are positive constants to be determined. Similarly, controller (9) can be rewritten in the following comport form: According to (7) and (10), we obtaiṅ Theorem 1. Suppose that Assumptions 1 and 2 hold. The finite-time leader-following consensus can be achieved for SONMASs (2)-(3) under the protocol (10) if the following inequalities are satisfied: Proof. Consider the following Lyapunov function candidate: The time derivative of V (t) along system (11) can be calculated aṡ where Combining (4) and (6), it is easy to verify that Furthermore, we have Based on (12), the following inequalities hold: Therefore, equality (13) can be written aṡ By Lemma 1, it can be concluded that the closed-loop system will reach the sliding mode surface σ(t) = 0 in finite time. The settling time can be estimated as where V (0) is the initial value of V (t).
Define the Lyapunov function V 1 (t) = e x T (t)e x (t)/2. Then taking the time derivative of V 1 (t), we obtaiṅ By using Lemma 2, it is shown that V 1 (t) converges to 0 in fixed time, which means e x (t) → 0 and e v (t) → 0 in fixed time. The estimation of settling time is Remark 2. In existing works [17,21,30], although the sliding mode control technique was applied in the consensus of MASs, the main results focused on asymptotical convergence. In addition, similar to [14], the traditional variable structure control adopts linear sliding mode, and the tracking error will converge in infinite time. In order to overcome this shortcoming, a new terminal sliding mode control protocol, which both can suppress the external disturbances and make all agents achieve consensus in finite time, is proposed in this paper.

Fixed-time consensus with undirected networks
In this section, in order to solve the drawback that the settling time depends on initial value of the system, we improve the proposed protocol (10) and prove that both the sliding surface and the consensus of SONMASs can be achieved in fixed time.
The following control protocol is proposed: where 0 < α < 1, β > 1, h > 1, θ, k 1 , k 2 , k 3 , k 4 , and k 5 are positive constants to be determined. According to Eqs. (7) and (14), we havė  (14) if the following inequalities are satisfied: Proof. Consider the following Lyapunov function candidate: The time derivative of V (t) along system (15) can be calculated aṡ Based on Lemma 3, we have . Similar to analysis of Theorem 1, combining with (16), inequality (17) can be written asV Using Lemma 2, we can conclude that the closed-loop system will reach the sliding mode surface σ(t) = 0 in fixed time. Moreover, the settling time can be estimated as Therefore, for t T 2 , we have e v (t) = − sig α (e x (t)) − sig β (e x (t)). Sinceė x (t) = e v (t), then we can obtain thatė x (t) = − sig α (e x (t)) − sig β (e x (t)).
Define the Lyapunov function V 1 (t) = e T x (t)e x (t)/2. Then taking the time derivative of V 1 (t) yieldṡ According to Lemma 2, we can conclude that V 1 (t) converges to 0 in fixed time, which means that e x (t) → 0 and e v (t) → 0 in fixed time. The estimation of settling time is Remark 3. In control protocol (9), the sign function is applied to make the systems reach the sliding mode surface in finite time. However, in protocol (14), we make an improvement by adding a term −θsig(σ(t)) h to improve the convergence time. In theoretical analysis, we also prove the fixed-time reachability of sliding mode surface. Moreover, the improved protocol (14) can make all agents reach consensus in fixed time.

Fixed-time consensus with directed networks
The consensus of SONMASs with undirected networks is concerned in previous sections. In some practical, the communication among agents may be directed. Hence, the fixedtime consensus of SONMASs with directed networks will be discussed in this section. We choose the sliding mode manifold as follows: The following control protocol is proposed: where η, k 1 , k 2 , k 3 , k 4 , and k 5 are positive constants to be determined, T . According to Eq. (7) and (18), we obtaiṅ Theorem 3. Suppose that Assumptions 1 and 3 hold. The fixed-time leader-following consensus can be achieved for SONMASs (2)-(3) under protocol (18) if the following inequalities are satisfied: Proof. Consider the following Lyapunov function candidate: The time derivative of V 2 (t) along system (19) can be calculated aṡ Similar to analysis of Theorem 2, based on (20), it yieldṡ Using Lemma 2, we obtain that the sliding mode surfaceσ(t) = 0 can be reached in fixed time. Furthermore, the settling time can be estimated as Consequently, for t T 3 , the sliding mode surfaceσ(t) = 0 is reached. Then we haveė x (t) = −µ sig α (e x (t)) − µ sig β (e x (t)). Similar to analysis of Theorem 2, we also can prove that e x (t) → 0 and e v (t) → 0 in fixed time. The estimation of settling time is Remark 4. In Theorems 1 and 2, some finite-time and fixed-time consensus conditions are obtained for SONMASs with undirected networks, respectively. Since network topology is undirected, the convergence analysis mainly utilizes the symmetry of the Lapalcian matrix. However, the Lapalcian matrix is asymmetry in directed networks. Therefore, the design of control protocol (18) and the selection of Lyapunov (21) are different form previous ones.
Remark 5. Since the sign function is employed in protocol (10), (14), and (18), there exists chattering in the control process. How to reduce or eliminate chattering is an interesting issue. At present, some methods including the fuzzy control [4] and the neural network control [23] are used to solve the chattering problem. In our future work, we will further improve the control protocol and use some special functions such as saturation function to reduce the chattering phenomenon.
Remark 6. In protocols (10), (14), and (18), we may find that when the states of the followers move to e x (t) = 0 and e v (t) = 0, the denominator control law equation will be zero. That means e x (t) → 0 will lead to a singular problem, thus rendering a big peak torque value. Whereas, in practice it is difficult to implement a large value of torque. How to design a nonsingular terminal sliding mode control protocol will be further considered in our future work.

Numerical example
In this section, we give three numerical examples to demonstrate the effectiveness of the theoretical results.
Example 1. Consider the SONMAS (2)-(3) with one leader and four followers. The interaction topology among the leader and followers is shown in Fig. 1(a). The nonlinear function are defined as follows: x 03 (t) x 13 (t) x 23 (t) x 33 (t) x 43 (t) (c) Figure 2. The states of x i (t) with control protocol (10).     (t),i=0,1,2,3,4 x 03 (t) x 13 (t) x 23 (t) x 33 (t) x 43 (t) (c) Figure 6. The states of x i (t) with control protocol (14).    Figure 7 describes the velocity states of four followers and one leader. Figure 8 describes the evolution of sliding mode variables, and the estimation of the setting time is T 19.56. Figure 9 describes the evolution of the control protocol (14).
Example 3. Consider the SONMAS (2)-(3) with one leader and four followers. The directed interaction topology among the leader and followers is shown in Fig. 1(b). The nonlinear function and external disturbances are same as the ones in Example 1. In control protocol (18), we choose k 1 = 5.4, k 2 = 4.8, k 3 = 8, k 4 = 3, k 5 = 2.5, α = 0.9, β = 1.5, µ = 2, h = 1.2, and η = 1. Then all conditions of Theorem 3 are satisfied. The simulation results are presented in Figs. 10-13. Figure 10 describes the position states of four followers and the leader. Figure 11 describes the velocity states of four followers and the leader. Figure 12 describes the evolution of sliding mode variables, and the estimation of the setting time is T 17.10. Figure 13 describes the evolution of the control protocol (18). (t),i=0,1,2,3,4 x 03 (t) x 13 (t) x 23 (t) x 33 (t) x 43 (t) (c) Figure 10. The states of x i (t) with control protocol (18).

Conclusion
In this paper, the leader-following consensus of SONMASs with external disturbances was studied. Three control protocols were developed to guarantee the consensus performance of SONMASs over undirected and directed networks, respectively. Firstly, a fixedtime sliding mode manifold was designed, and some control protocols over undirected networks were proposed, which can make all agents achieve consensus in finite time and fixed time, respectively. Moreover, we improved the proposed protocol and studied the fixed-time consensus of SONMASs with directed networks. In addition, a fixed-time control protocol was designed for directed networks. The settling time was estimated by using an improved lemma. Finally, some numerical simulations were given to show the effectiveness of the proposed control protocols. In the future work, the consensus of higher-order MASs with dynamic event-triggered communication mechanism will be considered. https://www.journals.vu.lt/nonlinear-analysis