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

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

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

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

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

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:

$\begin{array}{cc}\left[\begin{array}{ccc}AY+Y{A}^T& I& Y{H}^T\\ I& -I& 0\\ HY& 0& -\gamma I\end{array}\right]& <0\\ \end{array}$

Conclusion:

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

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• Robust stabilization of nonlinear systems: The LMI approach