LMIs in Control/pages/Switched systems Hinf Optimization

Switched Systems $$H_\infty$$ Optimization

The Optimization Problem
This Optimization problem involves the use of the State-feedback plant design, with the difference of optimizing the system with a system matrix which "switches" in properties during optimization. This is similar to considering the system with variable uncertainty; an example of this would be polytopic uncertainty in the matrix $$ A $$. which ever matrix is switching states, there must be an optimization for both cases using the same variables for both.

This is first done by defining the 9-matrix plant as such: $$ A \in \mathbb{R}^{m \times m} $$, $$ B_1 \in \mathbb{R}^{m \times n} $$, $$ B_2 \in \mathbb{R}^{m \times p} $$, $$ C_1 \in \mathbb{R}^{n \times m} $$, $$ D_{11} \in \mathbb{R}^{n \times n} $$, $$ D_{12} \in \mathbb{R}^{n \times p} $$, $$ C_2 = I $$, $$ D_{21} = 0 $$, and $$ D_{22} = 0 $$. Using this type of optimization allows for stacking different LMI matrix states in order to achieve the controller synthesis for $$H_\infty $$.

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 $$H_\infty$$ Optimization
There exists a scalar $$ \gamma $$, along with the matrices $$ X > 0 $$ and $$ Z $$ where:



\begin{align}

||S(P,K(0,0,0,F))||_{H_\infty} \leq \gamma \\    \\     \begin{bmatrix} X A(\delta)^T + Z^T B^{T}_2 + A(\delta) X + B_2 Z & B_1 & X C^{T}_1 + Z^{T} D^{T}_{12} \\ B1^{T}_1 & -\gamma I & D^{T}_{11} \\ C_1 X + D_{12} Z & D_{11} & -\gamma I    \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 $$ H_\infty $$ for a switched system optimization.

Implementation

 * Hinf switched state-feedback optimization example