Second-Order Super-Twisting Sliding Mode Control for Finite-Time Leader-Follower Consensus with Uncertain Nonlinear Multiagent Systems

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 292437, 8 pages http://dx.doi.org/10.1155/2015/292437 Research Article Second-Order Super-Twisting Sliding Mode Control for Finite-Time Leader-Follower Consensus with Uncertain Nonlinear Multiagent Systems Nan Liu,1,2 Rui Ling,1,2 Qin Huang,1 and Zheren Zhu1 1 College of Automation, Chongqing University, Chongqing 400044, China Key Laboratory for Spacecraft TT&C and Communication under the Ministry of Education, Chongqing 400044, China 2 Correspondence should be addressed to Rui Ling; Received 18 September 2014; Revised 13 December 2014; Accepted 29 December 2014 Academic Editor: Peng Shi Copyright © 2015 Nan Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Consensus tracking problem of the leader-follower multiagent systems is resolved via second-order super-twisting sliding mode control approach. The followers’ states can keep consistent with the leader’s states on sliding surfaces. The proposed approach can ensure the finite-time consensus if the directed graph of the nonlinear system has a directed path under the condition that leader’s control input is unavailable to any followers. It is proved by using the finite-time Lyapunov stability theory. Simulation results verify availability of the proposed approach. 1. Introduction Recently, cooperative control of multiagent systems (MAS) received a lot of interest, such as consensus, containment control, formation control, coverage control, and flocking [1, 2]. It has attracted a lot of researchers due to having potential application in many fields. Compared with traditional control systems, agents in MAS need to work together cooperatively and achieve a common goal with shared information, such as position, speed, or other parameters, in spite of limited and unreliable communication. There has been many different control strategies, such as graph theory approach [3–5], decentralized control approach [6, 7], and virtual leader approach [8]. The application of this research involves unmanned air vehicles [9], cooperative robotic systems [4, 8], and so forth. Such control strategies have many advantages, such as easy implementation and low cost [10]. MAS based on positive systems was investigated [11]. Various results from formation control of MAS are addressed as well. The consensus tracking is an interesting problem in leader-follower MAS. Agents in MAS agree on a common value with cooperative control law or consensus protocol. A summary of approaches for consensus algorithms was introduced [12]. The consensus tracking algorithm was proposed and analyzed under variable undirected network topologies [13–15]. Some control approaches for MAS were presented [16, 17], provided that necessary and sufficiency conditions were fulfilled, to achieve consensus under directed communication topologies. A sliding mode control (SMC) with multisurface approach for leader-follower MAS was proposed to achieve the convergence in finite time [18]. Tracking errors of first-order or second-order agents in MAS can be forced to zero under directed fixed and switching network topologies. It is necessary for consensus algorithm to achieve the convergence in finite time even if there are external disturbances and system uncertainties in many applications. Compared with the infinite time property, finite-time leader-follower consensus can perform better, such as faster convergence rate. This paper aims to research finite-time consensus for MAS. As is well known, traditional sliding mode control (SMC) [19, 20] was robust against parameter uncertainty and external disturbances. It has been applied in many fields, such as aircrafts, electrical motors, and power systems [21]. But it is not common in the field of multiagent networks. And traditional SMC may cause high-frequency chattering in the 2 Mathematical Problems in Engineering vicinity of the sliding surfaces [22]. The chattering problem is main drawback for traditional SMC. Some approaches were proposed to attenuate chattering phenomenon, such as high-order sliding mode control (HOSMC). It features a continuous signal instead of switching signal [23]. This approach not only can maintain the merits of the traditional method but also can attenuate chattering. Second-order sliding mode control (SOSMC) is a class of HOSMC, such as super-twisting algorithm [24], suboptimal algorithm [25], improved suboptimal algorithm [25], and twisting algorithm [26]. Twisting algorithm needs the sign of 𝑠.̇ Suboptimal algorithm has memory characteristics through the most recent singular point. Super-twisting algorithm takes advantage of the fact that it steers the sliding variable to zero for the systems with relative degree of two without the time derivative of sliding variable. This paper adapts second-order super-twisting SMC to achieve the consensus tracking for leader-follower MAS in finite time and gives the condition of leader-follower consensus. This paper is organized as follows. Several concepts and theories are presented in Section 2. Section 3 is the problem statement. In Section 4, the proposed approach for leaderfollower consensus based on second-order super-twisting algorithm was presented, convergence analysis of the MAS was provided, and the condition of the finite-time consensus for leader-follower MAS is derived. Subsequently simulation results are presented in Section 5. Section 6 gives conclusion. 2. Preliminaries This section gives several preliminary concepts and theories in order to facilitate the subsequent analysis. 2.1. Concepts in Graph Theory for MAS. A directed graph G = (V, 𝜀, A) can be adapted to express the communication between the agents, where V = {0, 1, 2, . . . , 𝑚} is the set of nodes and 𝜀 is the set of edges. The edge in directed graph G is denoted by the sequence of edges (ℎ1 , ℎ2 ), (ℎ2 , ℎ3 ), . . . , (ℎ𝑚−1 , ℎ𝑚 ) if and only if the agents can exchange information with each other. A = [𝑎ℎ𝑗 ] is called the weighted adjacency matrix, if 𝜀ℎ𝑗 = (ℎ, 𝑗) ∈ 𝜀, and there exists edge between node 𝑗 and node ℎ; then 𝑎ℎ𝑗 > 0, and 𝑎ℎ𝑗 = 0 otherwise: 0 0 ⋅⋅⋅ 0 ] [𝑎 [ 10 𝑎11 ⋅ ⋅ ⋅ 𝑎1𝑚 ] ] [ A=[ . ∈ R(𝑚+1)×(𝑚+1) . .. ] ] [ . [ . d d . ] B = diag { 𝑏1, 𝑏2 , . . . , 𝑏𝑚 } . (3) Define G = {V, 𝜀, A} that is the subgraph of G, consisting of 𝑚 followers, where 𝑎11 ⋅ ⋅ ⋅ 𝑎1𝑚 ] [ ] [ A = [ ... d ... ] ∈ R𝑚×𝑚 . ] [ 𝑎 ⋅ ⋅ ⋅ 𝑎 𝑚𝑚 ] [ 𝑚1 (4) Moreover, let D = diag{𝑑1 , 𝑑2, . . . , 𝑑𝑚 }, and 𝑑ℎ = ∑𝑚 𝑗=1 𝑎ℎ𝑗 for ℎ = 1, 2, . . . , 𝑚. Defining the Laplacian of subgraph G, L = D − A ∈ R𝑚×𝑚 . (5) Theorem 1. If graph G = (V, 𝜀, A) has a directed path, then [L + B] is a nonsingular matrix [18, 28]. 2.2. Finite-Time Lyapunov Stability Theory. Consider a nonlinear system which satisfies 𝑥̇ = 𝑓 (𝑥) . (6) Assume that 𝑓(𝑥) is (...truncated)