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

where $$\alpha > 0$$ and $$H \in \mathbb{R}^{lxn}$$

The Data
The data is dependent on the type the state-space representation of the 9-matrix plant; therefore the following must be known for this LMI to be calculated: $$ A \in \mathbb{R}^{m \times m} $$, $$ B_1 \in \mathbb{R}^{m \times n} $$, $$ B_2 \in \mathbb{R}^{m \times p} $$, $$ C_1 \in \mathbb{R}^{n \times m} $$, $$ D_{11} \in \mathbb{R}^{n \times n} $$, $$ D_{12} \in \mathbb{R}^{n \times p} $$, $$ C_2 = I $$, $$ D_{21} = 0 $$, and $$ D_{22} = 0 $$. What must also be considered is which matrix or matrices will be "switched" during optimization. This can be denoted as $$ A(\delta) $$.

The LMI: Switched Systems $$H_2$$ Optimization
There exists a scalar $$ \gamma $$, along with the matrices $$ X > 0 $$, $$ W $$, and $$ Z $$ where:



\begin{align}

||S(P,K(0,0,0,F))||^{2}_{H_2} \leq \gamma \\    \\     trace W < \gamma \\    \\     A X + X A^T + B_2 Z + Z' B_2'+B_1 B^{T}_{1} < 0 \\    \\     \begin{bmatrix} X & (C_1 X + D_{12} Z)^{T} \\ C_1 X + D_{12} Z & W \\ \end{bmatrix}&>0\\

\end{align}$$

Where $$ F = Z X^{-1} $$ is the controller matrix. This also assumes that the only switching matrix is $$ A $$; however, other matrices can be switched in states in order for more robustness in the controller.

Conclusion:
The results from this LMI gives a controller gain that is an optimization of $$ H_2 $$ for a switched system optimization.

Implementation

 * H2 switched state-feedback optimization example