# 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.