LMIs in Control/pages/Mixed H2 Hinf optimal state feedback control

Mixed $$H_\infty/H_2$$ Optimal State Feedback Control

The Optimization Problem
This Optimization problem involves the same process used on the full-feedback control design; however, instead of optimizing the full output-feedback design of the Optimal output-feedback control design. This is 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 of optimization LMIs in order to achieve the controller synthesis for both $$H_\infty $$ and $$ H_2$$.

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 $$ D_{22} = 0 $$.

The LMI: Mixed $$H_\infty/H_2$$ Optimal State Feedback Control
There exists the scalars $$ \gamma_1 $$, $$ \gamma_2 $$, along with the matrices $$ X > 0 $$, $$ W $$ and $$ Z $$ where:



\begin{align}

||S(P,K(0,0,0,F))||^{2}_{H_\infty} + ||S(P,K(0,0,0,F))||^{2}_{H_2} \leq \gamma_1 + \gamma_2 \\    \\     \begin{bmatrix} X A^T + Z^T B^{T}_2 + A X + B_2 Z & B_1 & X C^{T}_1 + Z^{T} D^{T}_{12} \\ B1^{T}_1 & -I & D^{T}_{11} \\ C_1 X + D_{12} Z & D_{11} & -\gamma_1 I    \end{bmatrix}&<0\\ \\    \\     trace W < \gamma_2 \\    \\     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.

Conclusion:
The results from this LMI give a controller that is a mixed optimization of both an $$ H_\infty $$, and $$ H_2 $$ optimization.

Implementation

 * Hinf/H2 Mixed state-feedback optimization example