# 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

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

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

We assume ${\displaystyle (A,B)}$  is stabilizable so

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

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

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

## The Data

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

## The LMI: Nonlinear Systems Robust Stabilization

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

{\displaystyle {\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

${\displaystyle K=ZY^{-1}}$