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LMI-Based Sliding Mode Robust Control for a Class of Multi-Agent Linear Systems
Introduction[edit | edit source]
The System[edit | edit source]
The state-space representation:
- i=1,⋯,N .(1)
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.
The Design of Controllers[edit | edit source]
We define the tracking error as , then
- = = .(2)
Design the tracking error as a sliding mode function, we give the design of control law as
- .(3)
Where is a state feedback gain matrix which can be obtained by designing LMI.
- Take forward-feedback control term:
- ,
- Take forward-feedback control term:
- Sliding mode robust term:
- [∈, or ,
- and .
- Sliding mode robust term:
The LMI[edit | edit source]
Theorem 1. Assume that
- is true for any i=1,...,N, .(4)
- where ,
then the closed-loop system consisting of (1) and (3) is asymptotic stability.
Implementation[edit | edit source]
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.
Conclusion[edit | edit source]
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.
References[edit | edit source]
- 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