Fixed-Time Connectivity Preserving Tracking Consensus of Multiagent Systems with Disturbances

Key Lab of Intelligent Analysis and Decision on Complex Systems, School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany Institute of Physics, Humboldt University of Berlin, Berlin 12489, Germany Key Lab of Intelligent Air-Ground Cooperative Control for Universities in Chongqing, College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, China


Introduction
Consensus of multiagent systems (MASs) is a very active research topic in control field [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. e key to consensus problem is through designing an appropriate control protocol, such that all the agents can achieve a certain value. e convergence velocity is an important performance index of a control protocol, and the fast convergence velocity is very important for improving the performance and robustness of a system [4,[7][8][9][10][11][12][13][14][15]. Olfati-Saber and Murray [9] proved that the algebraic connectivity of the interaction graph determines the convergence velocity. e larger the second small eigenvalue of the Laplace matrix, the faster the convergence velocity. Zhou and Wang [4] characterized the convergence velocity for the distributed discrete-time consensus algorithm over a variety of random networks with arbitrary weights. ey proposed the asymptotic and per-step convergence factors as measures of the convergence velocity and derived the exact value for the per-step convergence fact. Draief and Vojnović [11] derived an exact relation between the expected convergence time and the voting margin for some of these graphs, which revealed how the expected convergence time tends to infinity as the voting margin approaches zero. By maximizing the algebraic connectivity of the interaction graph, one can increase the convergence velocity with respect to the linear protocol, but the consensus can never be reached in finite time [4,[9][10][11][12][13][14][15]. at is, the convergence time is infinite or inestimable. However, in practice, it is often required the consensus to be reached in finite time. Comparing with the asymptotic consensus, the finite-time consensus has strong antiinterference, antiuncertainty, robustness, and fast convergence velocity [16][17][18][19]. ough the finite-time consensus can ensure the system converge in finite time, the convergence time depends on the initial states. However, due to the complexity of environment, the initial states are usually not available in advance [20][21][22][23][24][25][26][27][28]. In this case, the convergence time cannot be estimated. Correspondingly, the practical application of the multiagents systems will be restricted. To overcome this shortcoming, the fixed-time stability is proposed and analyzed in [20][21][22][23]. Based on the fixed-time stability concept, Parsegov et al. [22] first proposed the fixed-time consensus (FdTC), which means that the convergence time is limited and the upper bound of convergence time is independent of the initial states. Zuo and Tie [24] studied the FdTC of firstorder MASs over undirected topology. It is shown that the convergence time of the proposed general framework is upper bounded for any initial states. Fu and Wang [25] studied the fixed-time tracking problem for second-order systems with bounded input uncertainties. Based on a fixedtime distributed observer and sliding mode control method, the fixed-time tracking consensus (FdTTC) for high-order MASs with unknown bounded external disturbances is studied in [26]. A new cascade control structure is developed to achieve the FdTTC. Wang et al. [27] studied the FdTC of the second-order nonlinear MASs with time delay over undirected topology. Shang and Ye studied the leader-follower fixed-time group consensus control of MASs over directed topology [28]. A display estimation of convergence time is given. ough there are many results on FdTC, the preserving connectivity between agents communicating with each other is seldom taken into account. However, in practice, the communication ability of multiple agents is usually limited, so it is very important to ensure that the agents communicate with each other within a limited communication radius [29][30][31][32]. In addition, for some complex systems, the Lipschitz condition is hard to satisfy. Hence, it is very important to study the system with the nonlinear term regardless of the Lipschitz conditions [29,30]. By the nonsmooth analysis method, Cao et al. [29] studied the finite-time consensus of MASs with unknown Lipschitz terms and ensured that the communication between agents are always within a limited communication radius. Sun et al. [30] studied the finite-time connectivitypreserving consensus of second-order MASs with nonlinear terms which did not satisfy the Lipschitz conditions. In [30], by using the energy function idea and the technique of arc method, it is proved that the designed control protocol can ensure that the neighbor agents are always in a limited communication radius. Hong et al. [31] studied the finitetime consensus problem for second-order nonlinear MASs under communication constraints, and the interaction patterns can be preserved for the system with the objection of disturbance rejection, but the display estimation of convergence time is not given. Hong et al. [32] studied fixedtime connectivity-preserving distributed average tracking for first-order MASs without considering the interference and the nonlinear terms. Zuo et al. [33] studied the FdTC for MASs over directed and switching interaction topology. ey addressed that the explicit bounds of the convergence time for both protocols are independent of the initial condition. However, as far as we know, there are few results related to the FdTTC of MASs with the disturbance and general nonlinearity under the limited communication radius between neighbor agents. Hence, inspired by [27][28][29][30][31][32][33][34], this work studies the FdTTC of the first-order leader following MASs with disturbances and general nonlinearities. Based on the integral sliding surface, a new control protocol is designed. e main contributions of this paper are as follows: (a) the FdTTC of MASs with disturbances and nonlinear term is studied; (b) the agents communicating with each other are always maintaining in a limited communication radius; (c) the upper and lower bounds of convergence time that are independent of initial conditions are given; (d) the unknown disturbances and nonlinear terms are not required to satisfy the Lipschitz condition. e remainder of this paper is as follows. Notations and preliminaries are described in Section 2. Problem statement and some necessary lemmas are given in Section 3. Main results are given in Section 4. A simulation example is presented in Section 5 to illustrate the correctness of the obtained theoretical results. And the conclusion is concluded in Section 6.

Notations and Preliminaries
In this paper, R is the real set, R n is the n-dimensional real space, E ⊗ F is the Kronecker product of matrices E and F, ‖•‖ is the 2-norm of a matrix, ‖·‖ 1 is the 1-norm of a matrix, 1 n � (1, . . . , n) T is the n-dimensional column vector with all elements being 1, and (·) T is the transposition of a matrix or a vector. Let sign( where a ij � 0 if node φ j does not belong to N i or the relative distance between nodes φ i and φ j is larger than r, and r > 0 is the communication radius of the agent; otherwise a ij > 0. In the context, assume that it is called there is a path from agent i 1 to agent i s . If there is a path between any two different nodes of G, graph G is a connected graph. e degree matrix of G is

Problem Formulation
is section studies the FdTTC for MASs with N agents. e dynamics of the i-th follower is e dynamics of the leader, that is, agent 0, is described as where p 0 (t) ∈ R is the position state and f 0 (p 0 (t), t) ∈ R is the nonlinear item.

Complexity
Lemma 1 (see [21]). If there is a continuous unbounded positive definite function V(x), such that for some en, the origin of systems (1) and (2) is globally fixedtime stable, and the convergence time T satisfies
en, according to (iii) there is Lemma 3 (see [18]). Assume that function φ: Lemma 4 (see [9]). Laplace matrix L of the undirected connected graph G satisfies the following: (ii) e eigenvalues of matrix L are recorded as Assumption 1. Graph G is connected, and there is at least one agent that can access the leader's information, and at the initial time, the distance between any follower and its neighbors is less than r.
ere is a constant c > 0, such that Definition 1. For any initial state, there is where T is the settling time and the bound of T is independent of the initial states; then, systems (1) and (2) are said to achieve the FdTTC.

Definition 2.
If for any t > 0, graph G is connected, and the distance between any agent and its neighbor agents is less than the communication radius r and the states of systems (1) and (2) satisfy Definition 1; then, systems (1) and (2) are said to achieve the fixed-time connectivity-preserving tracking consensus. For systems (1) and (2), design the following sliding mode control protocol: with and u i2 (t) � − k 5 sign s i − k 6 sig α 1 s i , where k 1 , k 2 ,k 5 , and k 6 are the positive constants; α 1 > 1, 0 < α 2 < 1, α 3 > 1, and the sliding surface s i (t) ∈ R, i � 1, . . . , N, is designed as Complexity 3 where p i � p i (t) − p i (0) with p i (0) being the initial position state of agent i, i � 1, . . . , N. roughout this paper, we denote en, (16) can be rewritten as

Theorem 1. Suppose Assumptions 1 and 2 hold, and for any t ∈ [0, t 1 ), the distance between followers and their neighbors is less than r. en, under the control protocol (13), each agent of systems (1) and (2) can maintain its dynamic behavior on the sliding surface in a limited time. at is, s i (t) � 0 and
t ∈ [0, t 1 ), ∀ i � 1, . . . , N, provided that the control gain α 1 > 1 and k 5 > c and the convergence time (17) is From Lemma 1, one can get the convergence time which satisfies which implies that (18) holds. e proof is completed. □ Theorem 2. For systems (1) and (2) under the control protocol (13)
□ From eorems 1-3, one can find that systems (1) and (2) can achieve the fixed-time connectivity-preserving tracking consensus and the convergence time T � T 1 + T 2 satisfies

Examples
Example 1. Consider MASs (1) and (2) consisting of four followers i � 1, 2, 3, and 4 and one leader i � 5. Under the consensus protocol (13) over the communication network as shown in Figure 1, choose α 1 � 2, α 2 � 0.3, α 3 � 3, r � 20, , t) � 2 sin(0.5it + (iπ/6)), f 0 (p 0 (t), t) � 2 sin t. Let By calculation, one can get T max � 28.465, T min � 4.785, and T � 7.690, where T is the convergence time and T min andT max are the lower and upper bound of T. Hence, the convergence time is limited. Trajectories of the position error between followers and the leader are given in Figure 2, which shows that the position error between followers and the leader tends to zero. e distances between neighbor agents are shown in Figure 3. e maximum distance (d max ) and minimum distance (d min ) are given in Table 1. Figure 3 and Table 1 show that the distance between followers and their neighbors is always within a given limited range. e simulations illustrate that under the consensus protocol (13), systems (1) and (2) can achieve the FdTTC, which verifies the correctness of the main results in the work.

Conclusion
e FdTTC of first-order MASs with general nonlinearity and disturbances is studied in this paper. Firstly, the integral sliding surface is designed to solve the nonlinear term and disturbance in the system; then, the Lyapunov function theory and principle of fixed-time stability are adopted to prove that the integral sliding surface can switch to the surface in a finite time, and the upper and lower bounds independent of the initial conditions of the time to the sliding surface are obtained. Secondly, it is proved that the agents communicating with each other are always keeping within a limited communication radius by using the idea of arc method and designing the energy function. irdly, it is proved that the system can achieve FdTTC and the upper and lower bounds of convergence time that are independent of the initial conditions are given. Finally, a simulation example is presented, which shows that all the states of the followers can converge to that of the leader, and the convergence time is between the given lower and upper bounds. Table 1 gives that the agents communicating with each other are always within the given limited communication radius. Hence, the presented example verifies the correctness of theoretical results and the validity of the adopting methods.
e FdTTC for the nonlinear MASs with time delay or noise over the switching network is the future work to be done.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.