# LMIs in Control/pages/LMI for Mixed H2 Hinf Output Feedback Controller

LMI for Mixed $H_{2}/H_{\infty }$ Output Feedback Controller
The mixed $H_{2}/H_{\infty }$ output feedback control has been known as an example of a multi-objective optimal control problem. In this problem, the control feedback should respond properly to several specifications. In the $H_{2}/H_{\infty }$ controller, the $H_{\infty }$ channel is used to improve the robustness of the design while the $H_{2}$ channel guarantees good performance of the system.

## The System

We consider the following state-space representation for a linear system:
${\begin{aligned}{\dot {x}}&=Ax+Bu\\y&=Cx+Du\end{aligned}}$
where $A$, $B$, $C$, and $D$ are the state matrix, input matrix, output matrix, and feedforward matrix, respectively.
These are the system (plant) matrices that can be shown as $P=(A,B,C,D)$.

## The Data

We assume that all the four matrices of the plant, $A,B,C,D$, are given.

## The Optimization Problem

In this problem, we use an LMI to formulate and solve the optimal output-feedback problem to minimize both the <> and <> norms. Giving equal weights to each of the norms, we will have the optimization problem in the following form:
${\begin{aligned}{\text{min}}\quad ||S(P,K)||_{H_{2}}^{2}+||S(P,K)||_{H_{\infty }}^{2}\end{aligned}}$

## The LMI: LMI for mixed $H_{2}$/$H_{\infty }$

Mathematical description of the LMI formulation for a mixed $H_{2}$/$H_{\infty }$ optimal output-feedback problem can be written as follows:
${\begin{aligned}&{\text{min}}\quad \gamma _{1}^{2}+\gamma _{2}^{2}\\&{\text{s.t.}}\\&{\begin{bmatrix}X_{1}&I\\I&Y_{1}\end{bmatrix}}>0\\&{\begin{bmatrix}AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}B_{2}^{\text{T}}&*^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\A^{\text{T}}+A_{n}+(B_{2}D_{n}C_{2})^{\text{T}}&X_{1}A+A^{\text{T}}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\(B_{1}+B_{2}D_{n}D_{21})^{\text{T}}&(X_{1}B_{1}+B_{n}D_{21})^{\text{T}}&-\gamma I&*^{\text{T}}\\C_{1}Y_{1}+D_{12}C_{n}&C_{1}+D_{12}D_{n}C_{2}&D_{11}+D_{12}D_{n}D_{21}&-\gamma I\\\end{bmatrix}}<0\\&{\begin{bmatrix}Y_{1}&I&(C_{1}Y_{1}+D_{12}C_{n})^{\text{T}}\\I&X_{1}&(C_{1}+D_{12}D_{n}C_{2})^{\text{T}}\\(C_{1}Y_{1}+D_{12}C_{n})&(C_{1}+D_{12}D_{n}D_{21}&Z\\C_{1}Y_{1}+D_{12}C_{n}&C_{1}+D_{12}D_{n}C_{2}&D_{11}+D_{12}D_{n}D_{21}&-\gamma I\\\end{bmatrix}}>0\\&{\begin{bmatrix}AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}{\text{T}}B_{2}{\text{T}}&*^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\(A^{\text{T}}+An+(B_{2}*D_{n}*C_{2})^{\text{T}})&X_{1}A+A^{\text{T}}X_{1}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\(B_{1}+B_{2}D_{n}D_{21})^{\text{T}}&(X_{1}B_{1}+B_{n}D_{21})^{\text{T}}&-\gamma _{2}^{2}I&*^{\text{T}}\\(C_{1}Y_{1}+D_{12}C_{n})&(C_{1}+D_{12}D_{n}C_{2})&(D_{11}+D_{12}D_{n}*D_{21})&-I\\\end{bmatrix}}<0\\&{\text{trace}}(Z)<\gamma _{1}^{2}\\&D_{11}+D_{12}D_{n}D_{21}=0\end{aligned}}$
where $\gamma _{1}^{2}$ and $\gamma _{1}^{2}$ are defined as the $H_{2}$ and $H_{\infty }$ norm of the system:
${\begin{aligned}&||S(P,K)||_{H_{2}}^{2}=\gamma _{1}^{2}\\&||S(P,K)||_{H_{\infty }}^{2}=\gamma _{2}^{2}\end{aligned}}$
Moreover, $X_{1}$, $Y_{1}$, $A_{n}$, $B_{n}$, $C_{n}$, and $D_{n}$ are variable matrices with appropriate dimensions that are found after solving the LMIs.

## Conclusion:

The calculated scalars $\gamma _{1}^{2}$ and $\gamma _{2}^{2}$ are the $H_{2}$ and $H_{\infty }$ norms of the system, respectively. Thus, the norm of mixed $H_{2}$/$H_{\infty }$ is defined as $\beta =\gamma _{1}^{2}+\gamma _{2}^{2}$. The results for each individual $H_{2}$ norm and $H_{\infty }$ norms of the system show that a bigger value of norms are found in comparison with the case they are solved separately.

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI_for_Mixed_H2_Hinf_Output_Feedback_Controller

## External Links

* - LMI in Control Systems Analysis, Design and Applications

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