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

LMI for Mixed ${\displaystyle H_{2}/H_{\infty }}$ Output Feedback Controller

The mixed ${\displaystyle H_{2}/H_{\infty }}$ 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 ${\displaystyle H_{2}/H_{\infty }}$ controller, the ${\displaystyle H_{\infty }}$ channel is used to improve the robustness of the design while the ${\displaystyle H_{2}}$ channel guarantees good performance of the system.

The System

We consider the following state-space representation for a linear system:

{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+Bu\\y&=Cx+Du\end{aligned}}}

where ${\displaystyle A}$ , ${\displaystyle B}$ , ${\displaystyle C}$ , and ${\displaystyle D}$  are the state matrix, input matrix, output matrix, and feedforward matrix, respectively.

These are the system (plant) matrices that can be shown as ${\displaystyle P=(A,B,C,D)}$ .

The Data

We assume that all the four matrices of the plant, ${\displaystyle A,B,C,D}$ , 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:

{\displaystyle {\begin{aligned}{\text{min}}\quad ||S(P,K)||_{H_{2}}^{2}+||S(P,K)||_{H_{\infty }}^{2}\end{aligned}}}

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

Mathematical description of the LMI formulation for a mixed ${\displaystyle H_{2}}$ /${\displaystyle H_{\infty }}$  optimal output-feedback problem can be written as follows:

{\displaystyle {\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 ${\displaystyle \gamma _{1}^{2}}$  and ${\displaystyle \gamma _{1}^{2}}$  are defined as the ${\displaystyle H_{2}}$  and ${\displaystyle H_{\infty }}$  norm of the system:

{\displaystyle {\begin{aligned}&||S(P,K)||_{H_{2}}^{2}=\gamma _{1}^{2}\\&||S(P,K)||_{H_{\infty }}^{2}=\gamma _{2}^{2}\end{aligned}}}

Moreover, ${\displaystyle X_{1}}$ , ${\displaystyle Y_{1}}$ , ${\displaystyle A_{n}}$ , ${\displaystyle B_{n}}$ , ${\displaystyle C_{n}}$ , and ${\displaystyle D_{n}}$  are variable matrices with appropriate dimensions that are found after solving the LMIs.

Conclusion:

The calculated scalars ${\displaystyle \gamma _{1}^{2}}$  and ${\displaystyle \gamma _{2}^{2}}$  are the ${\displaystyle H_{2}}$  and ${\displaystyle H_{\infty }}$  norms of the system, respectively. Thus, the norm of mixed ${\displaystyle H_{2}}$ /${\displaystyle H_{\infty }}$  is defined as ${\displaystyle \beta =\gamma _{1}^{2}+\gamma _{2}^{2}}$ . The results for each individual ${\displaystyle H_{2}}$  norm and ${\displaystyle H_{\infty }}$  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: