LMIs in Control/pages/Robust stabilization of nonlinear systems

LMIs in Control/pages/Robust stabilization of nonlinear systems

Robust Stabilization of Nonlinear Systems

The Optimization Problem

Consider a non linear system whos dynamics are given by

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

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

We assume $(A,B)$ is stabilizable so

$u(x)=Kx$ , $K\in \mathbb {R} ^{m\times n}$

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 required to solve this problem are A, B, H

The LMI: Nonlinear Systems Robust Stabilization

There exists scalars $\gamma$, $\kappa _{Z}$, and $\kappa _{Y}$, along with the matrices $Y>0$ such that the following optimization problem is feasible.

${\begin{aligned}{\text{minimize}}\ &\gamma +\kappa _{Z}+\kappa _{Y}\\{\text{subject to}}\ &Y>0\\&{\begin{bmatrix}AY+YA^{T}+BZ+Z^{T}B^{T}&I&YH^{T}\\I&-I&0\\HY&0&-\gamma I\end{bmatrix}}&<0\\&\gamma -{\frac {1}{\alpha ^{2}}}<0\\&{\begin{bmatrix}-\kappa _{Z}I&Z^{T}\\Z&-I\end{bmatrix}}<0\\&{\begin{bmatrix}-Y&-I\\-I&-\kappa _{Y}I\end{bmatrix}}<0\end{aligned}}$

Conclusion:

The controller K can be recovered by the relation

$K=ZY^{-1}$

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