# LMIs in Control/pages/Robust stabilization of nonlinear systems

LMIs in Control/pages/Robust stabilization of nonlinear systems

Robust Stabilization of Nonlinear Systems

## The Optimization Problem

Consider a non linear system whos dynamics are given by

${\dot {x}}=Ax+Bu+h(t,x)$

where $x\in \mathbb {R} ^{n}$ ,$A\in \mathbb {R} ^{n\times n}$ , $B\in \mathbb {R} ^{n\times m}$  and $h:\mathbb {R} ^{n+1}\rightarrow \mathbb {R} ^{n}$ , $A$  is Hurwitz stable and $h(t,x)$  is piecewise continuous in both $t$  and $s$

We assume $(A,B)$  is stabilizable so

$u(x)=Kx$  , $K\in \mathbb {R} ^{m\times n}$

Assume that $h^{T}(t,x)h(t,x)\leq \alpha ^{2}x^{T}H^{T}Hx$

where $\alpha >0$  is the bounding parameter and $H\in \mathbb {R} ^{l\times n}$

## The Data

The matrices required to solve this problem are A, B, H

## The LMI: Nonlinear Systems Robust Stabilization

There exists scalars $\gamma$ , $\kappa _{Z}$ , and $\kappa _{Y}$ , along with the matrices $Y>0$  such that the following optimization problem is feasible.

{\begin{aligned}{\text{minimize}}\ &\gamma +\kappa _{Z}+\kappa _{Y}\\{\text{subject to}}\ &Y>0\\&{\begin{bmatrix}AY+YA^{T}+BZ+Z^{T}B^{T}&I&YH^{T}\\I&-I&0\\HY&0&-\gamma I\end{bmatrix}}&<0\\&\gamma -{\frac {1}{\alpha ^{2}}}<0\\&{\begin{bmatrix}-\kappa _{Z}I&Z^{T}\\Z&-I\end{bmatrix}}<0\\&{\begin{bmatrix}-Y&-I\\-I&-\kappa _{Y}I\end{bmatrix}}<0\end{aligned}}

## Conclusion:

The controller K can be recovered by the relation

$K=ZY^{-1}$