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

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{align} \text{minimize}\ & \gamma + \kappa_Z + \kappa_Y \\ \text{subject to}\ &Y > 0\\ &\begin{bmatrix} AY+YA^T + BZ +Z^TB^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{align}$$

Conclusion:
The controller K can be recovered by the relation

$$K = ZY^{-1}$$