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

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

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

Assume that $$h^T(t,x)h(t,x) \leq \alpha^2x^TH^THx$$

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

The Data
The matrices necessary for this LMI are A and H.

The LMI: Switched Systems $$H_2$$ Optimization
There exists a scalar $$ \gamma $$, along with the matrices $$ Y > 0 $$ such that:



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

\end{align}$$

Conclusion:
The system is robustly stable to degree $$\alpha$$ is the LMI is feasible.