# LMIs in Control/Click here to continue/Applications of Linear systems/LMI-Based Sliding Mode Robust Control for a Class of Multi-Agent Linear Systems

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:
             $u_{i}^{s}=-B^{-1}$[$\eta _{i}sgn(z_{i})],\eta _{i}$∈$R^{nx1}$,$d_{i}^{j}-\eta _{i}^{j}<0$ or$d_{i}{j}+\eta _{i}^{j}>0$ ,
       and $\eta _{i}sgn(zi)=[\eta _{i}^{1}sgnz_{i}^{1},...,\eta _{i}^{n}sgnz_{i}^{n}]^{T}$ .

## The LMI

Theorem 1. Assume that
       $A^{T}P_{i}+M_{i}^{T}+P_{i}A+M_{i}<0$ is true for any i=1,...,N, .(4)
       where $F_{i}=(P_{i}B)^{-1}M_{i}$,
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
       $d_{1}={\begin{bmatrix}50sin(t)\\50cos(t)\\50sin(t)\end{bmatrix}}$, $d_{2}={\begin{bmatrix}40sin(t)\\40cos(t)\\40sin(t)\end{bmatrix}}$,$d_{3}={\begin{bmatrix}50sin(t)\\50cos(t)\\50sin(t)\end{bmatrix}}$,
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
       $\eta 1={\begin{bmatrix}50\\50\\50)\end{bmatrix}}$,$\eta 2={\begin{bmatrix}40\\40\\40)\end{bmatrix}}$,$\eta 3={\begin{bmatrix}50\\50\\50)\end{bmatrix}}$
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.

## References

* 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