# LMIs in Control/pages/Stability of nonlinear systems

LMIs in Control/pages/Stability of nonlinear systems

Robust Stability of Nonlinear Systems

## The Optimization Problem

Consider a non linear system whos dynamics are given by

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

where ${\displaystyle x\in \mathbb {R} ^{n}}$ ,${\displaystyle A\in \mathbb {R} ^{n\times n}}$  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 ${\displaystyle s}$

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 necessary for this LMI are A and H.

## The LMI: Switched Systems ${\displaystyle H_{2}}$ Optimization

There exists a scalar ${\displaystyle \gamma }$ , along with the matrices ${\displaystyle Y>0}$  such that:

{\displaystyle {\begin{aligned}{\begin{bmatrix}AY+YA^{T}&I&YH^{T}\\I&-I&0\\HY&0&-\gamma I\end{bmatrix}}&<0\\\end{aligned}}}

## Conclusion:

The system is robustly stable to degree ${\displaystyle \alpha }$  is the LMI is feasible.