Multi-agent system finite-time consensus control in the presence of disturbance and input saturation by using of adaptive terminal sliding mode method

Abstract The paper develops finite-time consensus control for multi-agent systems by considering disturbances and input saturation. A new adaptive-terminal sliding mode control is suggested to solve consensus control within a finite time. Two cases are solved in the paper. In the first case, it is assumed that disturbances are with known upper. To achieve the consensus purpose within the finite time, in this case, the control inputs are designed based on terminal sliding mode technique by considering the input signal saturation. Also, the control inputs are modified to reduce the high dependency of reaching times to initial speeds. In the second case, the agents are subjected to disturbances with unknown upper bounds. To handle the problem, the control signals are acquired by combining the adaptive and terminal sliding mode methods. By considering saturation boundary and disturbances with unknown upper band, a new adaptive-terminal sliding mode method is designed to control the multi-agent system in reduced settling and reaching times. The proposed techniques efficiency is confirmed by numerical simulations.

On the other hand, in the consensus problem, two important practical issues including agent disturbances (and uncertainties) and each agent actuator saturation should be considered. If these two issues are not considered in multi-agent system consensus problems, some seriousundesired problems, e.g. convergence rate and tracking precision decrease and even divergence or instability, will appear. In Hu, Yu, Chen, & Xie (2013) and Zhu et al. (2016), the asymptotic consensus for multi-agent systems in the presence of agents' disturbance and saturation is guaranteed. The finite-time consensus for a typical multi-agent system with disturbance free and actuator saturation agents is investigated in (Lyu et al., 2016;. The finitetime consensus problem of disturbed multi-agent systems with agents without saturation actuators is considered (Li et al., 2013;Zhao & Hua, 2014;Zhou et al., 2015).
Due to the importance of the three reviewed problems, including finite-time consensus, agent disturbances and actuator saturation of each agent, a novel robust approach is proposed and generalized in this paper to guarantee the consensus control goal by.
Here, the finite-time consensus control problem is discussed and studied for a typical multiagent system possessing double-integrator agents and a fixed speed leader. Each system agent is subjected to control input disturbances (or uncertainty) and saturation. It is assumed that parameters related to agents' control inputs saturations are known. Agent disturbance is assumed to be bounded, while their upper bounds can be known (Case 1) or unknown (Case 2). As Case 1 is considered a developed TSMC method (or a generalized Fast TSMC method) is used to fulfill the finite-time consensus for the described multi-agent system. For Case 2, a novel adaptive TSMC (ATSMC) method is suggested to both estimate these upper bounds in finite time and also to solve the multi-agent system finite time consensus problem. It is worth noting that for the two aforementioned cases two inequalities are solved to determine the two finite times for achieving the finite-time consensus objective. In addition, the global dynamic finite-time stability of tracking errors (between agents and leader dynamics) is proven in several theorems in this paper.
Further, basic definitions and mathematical preliminaries are presented in Section 2. Section 3 is devoted to finite-time consensus tracking for Case 1. Section 4 investigates on the fast finite-time consensus tracking problem. In Section 5 adaptive terminal sliding mode is generalized to solve the Case 2. Finally, numerical examples and conclusions are presented in Sections 6 and 7, respectively.

Graph theory
and an adjacency matrix A. Each edge e k is defined by a pair of vertices v i ; v j À Á .
Matrix A ¼ a ij Â Ã 2 R NÂN shows the connections between vertices, so that a ij ¼ 1 if v j ; v i À Á 2 E and a ij ¼ 0. Else, if matrix A is symmetric, the graph G is known as undirected. A path is a sequence of edges from vertex i to vertex j. G is called connected if there exist at least one path between any two arbitrary separate vertices.

Finite-time stability
In this section, the main finite-time stability definition and two useful lemmas are presented. These are later used throughout the paper.
Definition 1 (Bhat & Bernstein, 1998). Assume a nonlinear time-invariant system as: where f : U 0 ! R n is a continuous vector function on an open neighborhood U 0 of the origin x ¼ 0. The equilibrium point x ¼ 0 of system (1) is called locally finite-time stable if the following conditions hold.
(i) It should be finite-time convergent inÛ 0 , namely, there is a convergence time T x 0 ð Þ : (ii) It should be Lyapunov stable in an open neighborhoodÛ 0 such thatÛ 0 U 0 .
Lemma 1 (Bhat & Bernstein, 1998). Consider the nonlinear system (1). Assume that there exist a C 1 Then, the equilibrium point x ¼ 0 of system (1) is locally finite-time stable. Furthermore, the convergence time T x 0 ð Þ satisfies the following inequality.
Lemma 2 (Hong, Huang, & Xu, 2001). Consider the nonlinear system (1). Suppose there exist a C 1 positive function V x ð Þ : U 0 ! R and real numbers c 1 ; c 2 > 0 and 0 Then, the convergence time T x 0 ð Þ is given by the following inequality.

Finite-time consensus tracking
The dynamic models of N agents are assumed to be: where x i and v i are the i th agent position and velocity, respectively. u i and d i denote the control input and bounded disturbance satisfying the inequality d i It is assumed that l i is a known constant and the control input of each agent is subjected to saturation such that u i j j < Υ s . It is worth noting that the saturation bound Υ s is known.
The leader dynamic is defined as: Based on finite-time consensus tracking, positions and velocities of all agents should converge to the position and velocity of the leader in a specific adjustable finite time. This goal can be defined mathematically as: where T is the required finite time for achieving the defined goal. Tracking errorsx i andṽ i are defined as, Assumption 1. In the multi-agent system of (4), it is assumed that each agent is connected to the leader independently or through other agents. To clarify this assumption mathematically, matrix B Remark 1. As the upper disturbance bound is known in sections 3 and 4, the powerful robust finitetime stabilization method TSMC control will be adopted for Case 1. But in section 5, for Case 2 ATSMC will be used where several finite-time adaptation laws are proposed for the unknown upper bound estimation.
3. Finite-time consensus with known bounded disturbance and saturation To satisfy the described consensus problem, a TSMC is designed. The terminal sliding surfaces s i , i ¼ 1; Á Á Á ; N are proposed as: in which ϕ i is defined as: The optional parameter α 1 is chosen as α 1 2 0; 1 ð Þ and the parameter α 2 is determined as α 2 ¼ 2α 1 1þα1 .
Theorem 1. Considering the agents, leader, tracking errors, and sliding surfaces described by (4), (5), (6), and (8), respectively, the sliding mode dynamics (sliding motions) N are globally finite-time stable. This means that tracking errorsx i andṽ i on sliding motion s i ¼ _ s i ¼ 0 will exactly converge to zero in the finite settling time, T s .
Proof. Assume that the sliding mode dynamic s i ¼ _ s i ¼ 0 has been achieved for the i th agent (input control for the i th agent will be designed later to guarantee sliding motion existence s i ¼ _ s i ¼ 0). Based on (7) and (8), sliding mode dynamic According to the definition of ϕ i and by referring to Theorem 1 (Guan et al., 2012), it can be demonstrated that there exist a T s such thatx i andṽ i in (10) become zero for times larger than T s . Consequently, sliding motions The control inputs are designed to assure the existence of The control law for the i th agent is proposed as: where k i , i ¼ 1; Á Á Á ; N are optional constants satisfying inequalities ∑ N j¼1 2 a ij þ 2b i þ k i þ l i Υ s . It is worth noting that T r is dependent on these optional constants (demonstrated later). In Theorem 2, it will be shown that (11) can ensure the existence of sliding motions in finite time.
Theorem 2. Relation (11) ensure the existence of s i ¼ _ s i ¼ 0, i ¼ 1; Á Á Á ; N for all agents at times larger than T r described by where k m is defined as k m ¼ min Proof. Consider the Lyapunov function to be i from (7) and u i from Replacing (13) in _ V ¼ ∑ N i¼1 s i _ s i and by considering the definition k m ¼ min i k i ð Þ, the following inequality is obtained.
Since d i j j l i , ∑ N i¼1 s i j j d i j j À l i ð Þis always less or equal to zero. Thus, (14) could be simplified as Finally, by setting c ¼ ffiffiffi 2 p k m and α ¼ 0:5, and applying Lemma 1, it is seen that the sliding mode dynamics s i ¼ _ s i ¼ 0, i ¼ 1; Á Á Á ; N are always fulfilled for t ! T r , where T r can be estimated by (12). This ends the proof. □ Remark 1. The defined consensus tracking object will be fulfilled for t ! T t , where T t ¼ T s þ T r .
Remark 2. Since the k i parameters, i ¼ 1; Á Á Á ; N are selected to satisfy the inequalities

Fast finite-time consensus with known bounded disturbance
The inequality T r k m ð Þ À1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ N i¼1 s 2 i 0 ð Þ q is strongly dependent on initial conditions. To reduce this high dependency, (16) is defined by modifying (11) by fast terminal sliding mode control method.
where k i ; ω i , i ¼ 1; Á Á Á ; N and 0<γ<1 are positive arbitrary constants and are tuned to The sliding surfaces s i , i ¼ 1; Á Á Á ; N are identical to (8). The finite-time stability proof for s i ¼ _ s i ¼ 0, i ¼ 1; Á Á Á ; N is similar to Theorem 1. Therefore, it can be claimed that there exists a T s such that allx i andṽ i , described by (10), will converge to zero for t ! T s . In Theorem 3, it is demonstrated that (16) is able to fulfill s i ¼ _ s i ¼ 0, i ¼ 1; Á Á Á ; N within a finite time.
Theorem 3. Consider the multi-agent system (4) with bounded disturbances. By applying (16), where ω m and k m are defined as ω m ¼ min

Proof. Consider the Lyapunov function to be
By substituting (18) into _ V ¼ ∑ N i¼1 s i _ s i and considering the definitions k m ¼ min i k i ð Þ and ω m ¼ min i ω i ð Þ, the following inequality is obtained.
Since d i , bounded as d i j j l i , ∑ N i¼1 s i j j d i j j À l i ð Þ , always is none positive, and the inequality is correct for all real values y i and 0<γ<1, (19) is simplified to: By considering the definition of V, (20) can be written as: Þ, and applying Lemma 2, it can be proven that N are always fulfilled for t ! T r where T r is calculated by (17). This ends the proof. □ Remark 4. Since k i ; ω i , i ¼ 1; Á Á Á ; N and 0<γ<1 are chosen to satisfy inequalities j Υ s , it could be proven that the maximum values of (16) are always less than the saturation bounds and, consequently, actuator saturation does not occur.

Finite-time consensus with unknown-bounded disturbance
Here, it is assumed that the upper disturbance bounds l i , i ¼ 1; Á Á Á ; N are constant but unknown. By this assumption, (11) and (16) can be expressed as: where k i , i ¼ 1; Á Á Á ; N are optional positive constants (introduced later).l i is the unknown upper bound estimations l i .
N are arbitrary parameters that satisfy λ i >1. By considering Lemma 1 in (Plestan, Shtessel, Brégeault, & Poznyak, 2010), it can be shown that 0 l i l Ã i , in which the constant l Ã i is not necessarily equal to the nominal value of l i . Therefore, l Ã i can be assumed to be l Ã i ¼ l i þ η i in which η i >0 is an arbitrary number. Notice that optional positive constants k Nis similar to that in Theorem 1. In Theorem 4, the existence of s i ¼ _ s i ¼ 0, i ¼ 1; Á Á Á ; N for t ! T r will be shown by applying (22) and (23).
By adopting the well-known inequality is converted to _ V À ffiffiffi 2 p θV 1 2 . Finally, by setting c ¼ ffiffiffi 2 p θ, a ¼ 0:5, and applying Lemma 1, it is proven that s i ¼ _ s i ¼ 0, i ¼ 1; Á Á Á ; N are always fulfilled for t ! T r where T r is estimated by (24). This ends the proof. □ Remark 5. As arbitrary constants k i , i ¼ 1; Á Á Á ; N are selected such that 2 ∑ N j¼1 a ij þ 2b i þ k i þ l Ã i Υ s is satisfied, it is concluded that the maximum values of the proposed control inputs (22) are always less than the saturation bounds.

Numerical simulations
In this section, a multi-agent system consisting of five agents and one leader is simulated and the results are discussed. In all simulations, matrices A and B are considered to be as presented in (29).  (30), is assumed to be time variant (Yu & Long, 2015).
Scenario 3. Unlike the two previous scenarios, the considered disturbance upper bounds are assumed to be unknown and should be estimated. In this scenario, the control inputs are based on (22). The tuning parameters are selected as k i ¼ 20 and λ i = 1.01 for i ¼ 1; Á Á Á ; 5.

Conclusion
In this work, finite-time consensus problem for multi-agent systems with leader in the presence of bounded disturbances and saturation constraints on control inputs have been discussed. To solve the problem, control inputs were designed by considering different assumptions on the upper disturbance bounds. First control laws were proposed, based on a new TSMC method, to tackle the finite-time consensus for disturbed multi-agent systems while the upper disturbance bounds were known.
It has been demonstrated that the proposed control inputs were bounded while their maximum amplitudes could be adjustable by proper tuning parameters selection. Then, by applying a new fast TSMC approach, the control inputs were modified to reduce the high dependency of the finitereaching time on agent initial conditions. In the second scenario, the same problem was solved for the case where the disturbance upper bounds were unknown. In this case, for fulfilling the finitetime consensus goal, the control inputs and the finite-time estimation laws were designed by applying the adaptive TSMC method. Mathematical analysis of the paper was demonstrated that all suggested control inputs are able to satisfy the finite-time consensus aim within the total adjustable finite-time. Finally, three computer based numerical simulations were illustrated to validate the theoretical results presented in the paper.