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

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

$u(x)=Kx$ , $K\in {ℝ}^{m\times n}$

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 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.

$\begin{array}{cc}\text{minimize}& \gamma +{\kappa }_Z+{\kappa }_Y\\ \text{subject to}& Y>0\\ & \left[\begin{array}{ccc}AY+Y{A}^T+BZ+Z^T{B}^T& I& Y{H}^T\\ I& -I& 0\\ HY& 0& -\gamma I\end{array}\right]& <0\\ & \gamma -\frac{1}{{\alpha }^2}<0\\ & \left[\begin{array}{cc}-{\kappa }_{Z}I& Z^T\\ Z& -I\end{array}\right]<0\\ & \left[\begin{array}{cc}-Y& -I\\ -I& -{\kappa }_{Y}I\end{array}\right]<0\end{array}$

Conclusion:

The controller K can be recovered by the relation

$K=Z{Y}^{-1}$

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