where ∈, ∈ are the state and control input of the agent, ∈ is disturbance term, A and ∈ are constant matrices and the initial state is defined by . The control targets are → , for i=1,...,N . is an ideal instruction.
Assumption. This study deals with the information exchange among agents is modeled by an undirected graph. We assume that the communication topology is connected.
We focus on the multi-agent linear system, without loss of generality, we assume that the system has three agents and B is the unit matrix.
According to (1),
B=,
the ideal matrix is [ sin(t) cos(t) sin(t) ] , the interference matrix is
, ,,
corresponding to the ideal matrix. Solving LMI (4), let
,
,
,
respectively. Due to (3), let
,,
replacing switching function with saturation function and choosing the boundary layer as Δ=0.05 . We give simulations are in the following (Figures 1-3).
It is clear that from three figures the closed-loop system with disturbance is asymptotic stability, hence, the proposed method is effective.
The multi-agent linear system was studied in this paper. Based on linear matrix inequality technology and sliding mode control, the forward-feedback control term was given. Sufficient conditions for the closed-loop system were established by Lyapunov stability theory. Simulations show that the proposed method was effective.
Open Access Library Journal Vol.9 No.1, January 2022: "LMI-Based Sliding Mode Robust Control for a Class of Multi-Agent Linear Systems" by Tongxing Li, Wenyi Wang, Yongfeng Zhang, Xiaoyu Tan, School of Mathematics and Statistics, Taishan University, Taian, China. DOI: 10.4236/oalib.1108342