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 $x\in \mathbb {R} ^{n}$,$A\in \mathbb {R} ^{n\times n}$ 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 $h^{T}(t,x)h(t,x)\leq \alpha ^{2}x^{T}H^{T}Hx$

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:

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

Conclusion:

The system is robustly stable to degree $\alpha$ 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