Based Fault-Tolerance Consensus of Second-Order Heterogeneous System under Input Saturation with Dynamics and Static Leader

. We focus on fault-tolerant consensus for heterogeneous dynamics systems with static and dynamic leaders under input saturation in this article. We apply theory of fnite-time stability to multiagent system cooperative control. Also, we use integral sliding mode to overcome disturbance. Te primary goal is a set of second-order linear and second-order nonlinear agents moving along the leader’s trajectory. We use topology graph to describe communication between multiple agents. By using integral sliding mode control way, corresponding controller is introduced to make system stable. Finally, correctness of experiment was confrmed by MATLAB numerical simulation.


Introduction
Recently, we have witnessed great progress in the development of control systems.Due to characteristics of cooperative control of multiagent systems [1][2][3], such as its increasingly wide application and high execution efciency, it is receiving increased attention from researchers.Multiagent systems have a wide range of applications, such as formation of unmanned aerial vehicles, encirclement of unmanned boats at sea, and containment control consisting of leaders who can detect obstacles and followers who cannot detect danger.Consensus [4][5][6][7][8] is the most fundamental research question of cooperative control, which represents that states of all agents reach the same state under the action of the consensus control protocol, i.e., their state errors converge to zero.Consensus issues can be categorized into leaderless consensus [9][10][11] and leader-following consensus [12][13][14] depending on the criterion of the number of leaders.
Depending on whether the leader will cause movement, it can be divided into static leader [15] and dynamic leader [16].Te simple understanding is that the dynamic leader has a certain speed, and the state of leader will cause certain changes.Also, the static leader has no speed and will not move.In [15], the authors designed new control protocols and introduced a nonlinear feedback control to solve fnitetime containment control with dynamic and static leaders.Multiagent system with model uncertainties was introduced in [16] to address chattering reduction containment control problem.
Convergence time is a key research point in the study of consensus, and based on convergence time of consensus of multiagent systems, consensus can be categorized into asymptotic time consensus [17,18], fnite-time consensus [19,20], and fxed-time consensus [21,22].As the name suggests, asymptotic time consensus is the consensus that converges at an exponential rate over an infnite amount of time.Regarding asymptotic time consensus, fnite-time consensus is proposed because convergence time of asymptotic time consensus is not well calculated.In [17], highorder multiagent system with input quantization, actuator faults, unknown nonlinear functions, and directed communication topology were studied, and asymptotic consensus was considered.Leader-following bipartite consensus of Euler-Lagrange systems was investigated in [18] under system uncertain and deception attacks.Finite-time consensus can calculate its corresponding convergence time compared to asymptotic time consensus.Convergence time of fnite-time consensus is associated with primary value of state.Controller was designed by considering relative position, and velocity measurements were investigated to deal with fnite-time input saturation consensus in work of [19].Authors considered fnite-time output consensus in [20] of dynamics system with directed network and disturbance.According to this limitation, fxed-time consensus is proposed and convergence time of fxed-time consensus is independent on primary value.Because of the advantage of fxed-time consensus, the study of fxed-time consensus is more interested.Fixed-time consensus of heterogeneous dynamics systems consisting of frst-and second-order systems was regarded in [21].Fixed-time consensus of uncertain system was focused on with state constraints and input delay in [22].
In research of dynamics system, a healthy actuator and controller are generally studied.However, in real life, due to the needs of industrial engineering and the age of the actuator, some damage will inevitably occur.Terefore, it is necessary to study the situation of how to maintain stability of system when actuator fails [23][24][25].In [23], under the condition of considering actuator fails, authors investigated fuzzy fxed-time consensus of nonlinear dynamics system with adding-a-power-integrator method.In [24], fnite-time consensus control of nonlinear discretetime system with Markov jump parameters and actuator faults was focused on.In [25], leader-following consensus of nonlinear dynamic system under actuator faults was taken into account, and communication graph is directed and connected.Teoretically, any system can be stabilized if the control is only large enough, but this is not realistic.Tis is because we study the input saturation of the system [26][27][28].In [26], the authors demonstrate in detail that bipartite consensus of linear dynamics system and input control was regarded as saturation.Time-varying formation control of linear dynamics system could be achieved by authors in [27], and input saturation was considered.In the work of [28], consensus control of mixed second-order linear and nonlinear system was studied.For a general system, each agent has the same model and application environment.With the wide application of dynamics systems, in many cases, the agents will have diferent models or have diferent application environments.Terefore, it is important to study heterogeneous systems [29][30][31][32].In [29], the authors explored bipartite output formation containment of heterogeneous linear dynamics system.In [30], bipartite output consensus of heterogeneous linear system was introduced, and fniteand fxed-time could be reached.Finite-time heterogeneous consensus was studied in [31] with integral sliding mode control and pinning control methods.In [32], the author studied tracking consensus for heterogeneous group system based on switching topology and input time delay.
Tus, based on some of the above articles, we study fxedtime consensus control of heterogeneous nonlinear dynamics systems according to input saturation and actuator fault.Main contributions are as follows: (1) Compared to each agent having the same dynamic, we are studying heterogeneous multiagent systems, which mean that each agent has its own state equation, but they can still satisfy consensus through controller.Compared to homogeneous multiagent systems, heterogeneous systems have more research signifcance in practical applications.(2) Compared to general linear or terminal sliding modes, we use integral sliding mode.Te integral sliding mode has better robustness and avoids the drawbacks of the conventional sliding mode approach stage.Te use of integral sliding mode can not only avoid singular phenomena but also achieve better robustness performance.(3) Compared to a normal controller, we are studying a faulty system.When the actuator of an agent faults, how to maintain system stability is the focus of our research.When the input of the controller is too large, the method of input saturation can be used to solve this problem.
Te remaining parts of this work are structured as follows.Section 2 introduces preamble and formulation of problem.In Section 3, main results of analysis are provided.Simulations results are provided in Section 4. Section 5 summarizes work done in this paper.

Graph Teory.
What is discussed in this section is a topology with n followers and one leader.Tat topology is denoted by G � (V G , E G , A), and graph G is a directed graph, where is the weighted adjacency matrix of graph G.
Node indexes belong to a nonempty fnite index set Γ � 0, 1, . . ., n { }, and followers' nodes belong to If there is a path between any two distinct vertices, then directed graph G is called strongly connected.
Connection weight between any of followers and a leader is displayed by b 2 Discrete Dynamics in Nature and Society

Some Useful Lemmas and Defnitions
Defnition 1 (see [33]).Connected graph with leader is connected if there exists one or more agents in G that can connect to the leader via an edge.
Defnition 2 (see [34,35]).System _ y s � h(y s ) is nonlinear, with h(0) � 0, y s ∈ R n , where h(•): R n ⟶ R n is a continuous function.If system's equilibrium point is zero and is Lyapunov stable in a fnite time, it is stable for a fnite time.Finite-time attractive means that there is a function T h (y s0 ) such that lim t⟶T h (y s0 ) y s (t, y s0 ) � 0 that x(t, y s0 ) � 0, ∀t ≥ T h (y s0 ), where y s (t, y s0 ) is the solution of system starting from y s (0) � y s0 .If the system is fnite-time stable and fxed-time attractive, then it is called fxed-time stable.Fixed-time attractive requires that there is a constant T h such that pervious fnite convergent time T h (y s0 ) satisfes T h (y s0 ) ≤ T for all y s0 ∈ R n .Defnition 3 (see [36]).A vector fled f(y s ) � (f 1 (y s ), . . ., f n (y s )) T is said to be homogeneous in the 0-limit or ∞-limit with associated triple (r p , k p , f p ) where r p � [r p1 , . . ., r pn ] ∈ R n is weight, k p is degree, and f p is approximating vector feld, if k p + r pj > 0, and then, function f i (y s ) is homogeneous in 0-limit or ∞-limit with associated triple (r p , k p + r p , f p ) for each i.
Lemma 4 (see [36]).For _ y s � f(t, y s ), y s (0) � y s0 , suppose that f(y s ) is homogeneous in the 0-limit or ∞-limit with associated triples (r 0 , k 0 , f 0 ) and (r ∞ , k ∞ , f ∞ ).If the origins of system _ y s � f(y s ), _ y s0 � f 0 (y s ), and _ y s∞ � f ∞ (y s ) are global asymptotically stable, then the origin of _ y s � f(t, y s ), y s (0) � y s0 is fxed-time stable when condition k ∞ > 0 > k 0 holds.Lemma 5 (see [35]).Nonlinear dynamics system is then the origin is fxed-time stable equilibrium of system and settling time satisfes Assumption 6. Communication topology among followers is directed.For any follower, there is at least a directed path from leader to follower.Assumption 7. Nonlinear dynamic continuous function f i (x i , v i , t) is assumed to be bounded and satisfed where c 1 and c 2 are any nonnegative constants. where ) is nonlinear dynamics, and d i is external disturbance.In this paper, loss of efectiveness was concerned about actuator faults.Assume that the leader is not subject to actuator faults.Also, u a i is the actual control input, which is expressed by u a i � c i u i , where u i is ideal input and c i is unknown faults, where 0 Dynamics of leader is where x 0 ∈ R n is position, v 0 ∈ R n is velocity, and u 0 ∈ R n is control input, correspondingly.To organize the above systems ( 3) and ( 4), it can obtain (5) becomes a matrix of where D � (c − I)u + d describes so-called lumped faults, in which external disturbances and actuator faults are included.
Also, defne We can get the following assumption for nonlinear dynamics of this agent.Assumption 9. Following nonlinear continuous function f i (x i , v i , t) is supposed to be bounded and satisfes where α 1 and α 2 are any constants other than negative numbers.System error is defned as Matrix form of the error ( 9) is where Te consensus error can be obtained by (10): According to error, integral sliding mode is designed as where According to (12), _ s i can be obtained that Te saturated protocol is written as where κ 1 > 1, 0 < κ 2 < 1.

Main Result
Theorem 10.According to Assumptions 6 and 9 hold.Introducing (12) as sliding mode, achievement of s � 0 makes e x and e v converge to 0 in fxed time.
Proof.When s � 0, one can get Consider candidate Lyapunov function Terefore, e vi converges to 0 asymptotically.Moreover, based on equation (15), it has e xi converged to 0 asymptotically. Considering error system (9) in 0-limit is written as follows: Also, error system (9) in ∞-limit is written as follows: , and its derivative is Terefore, it is obtained that both the 0-limit and ∞-limit systems ( 18) and ( 20) are globally asymptotically stable.
For 0-limit, and according to Defnition 3, one can obtain  Theorem 11.Suppose Assumptions 6 and 9 hold.For systems ( 4) and ( 3), controller is designed as (14), and sliding mode is shown as (8), and then, system will reach sliding mode surface s � 0 in fxed time.
Te proof is completed.□ 3.2.Static Leader.In the above, the main study is on dynamic leaders.Within this, there is a special case of a static leader, where the speed of the leader is zero.For a static leader, the velocity of the leader is zero, and then the leader's dynamic is Ten, one can get state error Matrix form of error is Consensus error is obtained by (28): Te same sliding mode control is used as (12), and the consensus control topology can be designed as where κ 3 > 1, 0 < κ 4 < 1.
Theorem 1 .Suppose Assumptions 6 and 9 hold.For systems (26) and ( 3), controller is designed as (30), and sliding mode is shown as 8, and then, system will reach sliding mode surface s � 0 in fxed time.
Te proof is completed.

Simulations
To verify validity of the proposed controller, we give two examples to verify its validity.In this section, we give a topology graph consisting of six followers and a leader whose topology is satisfying the assumed conditions.Topology is designed by Figure 1, where agents labeled as 1, 2, 3, 4, 5, and 6 are followers and labeled as 0 is the leader.
After having these data above, we simulate it by performing simulation on it.We can get several pictures as shown below.
Figure 2 shows the error graph of position and velocity, and it can be seen that state error converges to zero in picture.According to defnition, when state errors converge to zero, then consensus is satisfed.According to the trajectory of state and the trajectory of state error, we can know that it can be seen our controller is efcient and correct.
Figure 3 shows the position and velocity error change.We can see that the position and velocity error can converge to 0. When state errors converge to zero, then it is to meet consensus.According to the position and velocity change of agents and error, we can see that the consensus can be achieved.Our controller is efective and correct.
Example 2. In this section, we consider the case of a static leader.Figures 4 and 5 are obtained through simulation.
As shown in Figure 4, it is known that position and velocity of the followers converge to leader's position and speed, consensus with leader's dynamics.Again, because it is a static leader, position and velocity of the followers converge to 0. Actual is consensus with the theory.
As shown in Figure 5, error between followers' position and velocity and leader's position and velocity converges to 0. According to defnition of consensus, the same conclusion as in Figure 5 can be obtained.Terefore, it can be obtained from Figures 4 and 5 that the system will realize to fxed-time consensus in the case of a static leader and our controller is efcient and correct.

Conclusion
In this section, we focus on a summary of the work done in the whole article, as well as an outlook on what we would like to do in the future.In this paper, we focus on fxed-time consensus problem for nonlinear heterogeneous systems under input saturation and actuator faults.In this article, we mainly use a heterogeneous second-order system, which mainly contains second-order linear system and secondorder nonlinear system.Fixed-time consensus with actuator faults is addressed by using a sliding mode control approach.Finally, simulations are used to verify that the proposed controller is efective.In future work, one would like to study heterogeneous systems that are in diferent locations on land, sea, and air and study how these agents perform tasks such as consensus and containment.

□ 3 . 1 .
Dynamic Leader.In this part, the main research is about dynamics leader.

Figure 5 :
Figure 5: Position and velocity error change of multiagent system with static leader.