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LMI-Based Sliding Mode Robust Control for a Class of Multi-Agent Linear Systems

The System
The state-space representation:


 * $$ \dot{x}(t)= Ax_i(t) + Bu_i(t)+ di,$$ i=1,⋯,N  .(1)

where $$x_i(t)$$∈$$R^{n}$$, $$u_i$$ ∈ $$R_n$$ are the state and control input of the $$i^{th}$$ agent, $$d_i$$∈$$R^{ n \times 1 }$$ is disturbance term, A and $$B$$∈ $$R^{n \times n}$$ are constant matrices and the initial state is defined by $$x_i(0)$$. The control targets are $$x_i$$→$$x_i^{y}$$, for i=1,...,N. $$x_i^{y}$$ 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
We define the tracking error as $$z_i(t)= x_i(t)-x_i^{y}(t)$$, then


 * $$\dot{z}_i(t)$$= $$\dot{x}_i(t) - \dot{x}_i^{y}(t)$$ = $$Ax_i(t)+Bu_i(t)+d_i - \dot{x}_i^{y}(t)$$  .(2)

Design the tracking error as a sliding mode function, we give the design of control law as


 * $$u_i(t)=F_i(t)x_i(t)+u_i^{y}(t)+u_i^{s}(t)$$   .(3)

Where $$F_i(t)$$ is a state feedback gain matrix which can be obtained by designing LMI.
 * Take forward-feedback control term:
 * $$u_i^{y}(t)=-F_i(t)x_i^{y}(t)-B^{-1}(t)A_i(t)x_i^{y}(t)+B^{-1}(t)\dot{x}_i^{y}(t)$$,


 * Sliding mode robust term:
 * $$u_i^{s}=-B^{-1}$$[$$\eta_isgn(z_i)], \eta_i$$∈$$R^{n x 1}$$,$$d_i^{j}-\eta_i^{j}<0$$ or$$ d_i{j}+\eta_i^{j}>0$$ ,
 * and $$\eta_isgn(zi)=[\eta_i^{1}sgnz_i^{1},...,\eta_i^{n}sgnz_i^{n}]^{T}$$.

The LMI
Theorem 1. Assume that
 * $$A^{T}P_i+M_i^{T}+P_iA+M_i<0$$ is true for any i=1,...,N,   .(4)
 * where $$F_i=(P_iB)^{-1}M_i$$,

then the closed-loop system consisting of (1) and (3) is asymptotic stability.

Implementation
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=$$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$$, $$A=\begin{bmatrix}-2.4 & 9.2 & 0 \\ 1 & -1 & 1 \\0 & -16.3 & 3\end{bmatrix}$$

the ideal matrix is [ sin(t)  cos(t)  sin(t) ], the interference matrix is
 * $$d_1=\begin{bmatrix}50 sin(t) \\ 50 cos(t) \\ 50sin(t) \end{bmatrix}$$, $$d_2=\begin{bmatrix}40 sin(t) \\ 40 cos(t) \\ 40sin(t) \end{bmatrix}$$,$$d_3=\begin{bmatrix}50 sin(t) \\ 50 cos(t) \\ 50sin(t) \end{bmatrix}$$,

corresponding to the ideal matrix. Solving LMI (4), let
 * $$P_1=\begin{bmatrix}10000 & 0 & 0 \\ 0 & 10000 & 0 \\0 & 0 & 10000\end{bmatrix}$$,
 * $$P_2=\begin{bmatrix}100000 & 0 & 0 \\ 0 & 100000 & 0 \\0 & 0 & 100000\end{bmatrix}$$,
 * $$P_3=\begin{bmatrix}5000000 & 0 & 0 \\ 0 & 5000000 & 0 \\0 & 0 & 5000000\end{bmatrix}$$,

respectively. Due to (3), let


 * $$\eta1=\begin{bmatrix}50 \\ 50  \\ 50) \end{bmatrix}$$,$$\eta2=\begin{bmatrix}40  \\ 40  \\ 40) \end{bmatrix}$$,$$\eta3=\begin{bmatrix}50  \\ 50  \\ 50) \end{bmatrix}$$

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
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.