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

LMI for Mixed $H_2/H_{\mathrm{\infty }}$ Output Feedback Controller

The mixed $H_2/H_{\mathrm{\infty }}$ 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 $H_2/H_{\mathrm{\infty }}$ controller, the $H_{\mathrm{\infty }}$ 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:

$\begin{array}{cc}\dot{x}& =Ax+Bu\\ y& =Cx+Du\end{array}$

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:

$\begin{array}{c}\text{min}||S(P,K)|{|}_{H_2}^2+||S(P,K)|{|}_{H_{\mathrm{\infty }}}^2\end{array}$

The LMI: LMI for mixed $H_2$/$H_{\mathrm{\infty }}$

Mathematical description of the LMI formulation for a mixed $H_2$/$H_{\mathrm{\infty }}$ optimal output-feedback problem can be written as follows:

$\begin{array}{cc}& \text{min}{\gamma }_1^2+{\gamma }_2^2\\ & \text{s.t.}\\ & \left[\begin{array}{cc}{X}_1& I\\ I& Y_1\end{array}\right]>0\\ & \left[\begin{array}{cccc}A{Y}_1+Y_1{A}^{\text{T}}+B_2{C}_n+C_n{B}_2^{\text{T}}& {*}^{\text{T}}& {*}^{\text{T}}& {*}^{\text{T}}\\ A^{\text{T}}+A_n+(B_2{D}_n{C}_2{)}^{\text{T}}& X_{1}A+A^{\text{T}}+B_n{C}_2+C_2^{\text{T}}{B}_n^{\text{T}}& {*}^{\text{T}}& {*}^{\text{T}}\\ (B_1+B_2{D}_n{D}_{21}{)}^{\text{T}}& (X_1{B}_1+B_n{D}_{21}{)}^{\text{T}}& -\gamma I& {*}^{\text{T}}\\ C_1{Y}_1+D_{12}{C}_n& C_1+D_{12}{D}_n{C}_2& D_{11}+D_{12}{D}_n{D}_{21}& -\gamma I\end{array}\right]<0\\ & \left[\begin{array}{ccc}{Y}_1& I& (C_1{Y}_1+D_{12}{C}_n{)}^{\text{T}}\\ I& X_1& (C_1+D_{12}{D}_n{C}_2{)}^{\text{T}}\\ (C_1{Y}_1+D_{12}{C}_n)& (C_1+D_{12}{D}_n{D}_{21}& Z\\ C_1{Y}_1+D_{12}{C}_n& C_1+D_{12}{D}_n{C}_2& D_{11}+D_{12}{D}_n{D}_{21}& -\gamma I\end{array}\right]>0\\ & \left[\begin{array}{cccc}A{Y}_1+Y_1{A}^{\text{T}}+B_2{C}_n+C_n\text{T}{B}_2\text{T}& {*}^{\text{T}}& {*}^{\text{T}}& {*}^{\text{T}}\\ (A^{\text{T}}+An+(B_2*D_n*C_2{)}^{\text{T}})& X_{1}A+A^{\text{T}}{X}_1+B_n{C}_2+C_2^{\text{T}}{B}_n^{\text{T}}& {*}^{\text{T}}& {*}^{\text{T}}\\ (B_1+B_2{D}_n{D}_{21}{)}^{\text{T}}& (X_1{B}_1+B_n{D}_{21}{)}^{\text{T}}& -{\gamma }_2^{2}I& {*}^{\text{T}}\\ (C_1{Y}_1+D_{12}{C}_n)& (C_1+D_{12}{D}_n{C}_2)& (D_{11}+D_{12}{D}_n*D_{21})& -I\end{array}\right]<0\\ & \text{trace}(Z)<{\gamma }_1^2\\ & D_{11}+D_{12}{D}_n{D}_{21}=0\end{array}$

where ${\gamma }_1^2$ and ${\gamma }_1^2$ are defined as the $H_2$ and $H_{\mathrm{\infty }}$ norm of the system:

$\begin{array}{cc}& ||S(P,K)|{|}_{H_2}^2={\gamma }_1^2\\ & ||S(P,K)|{|}_{H_{\mathrm{\infty }}}^2={\gamma }_2^2\end{array}$

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 $H_{\mathrm{\infty }}$ norms of the system, respectively. Thus, the norm of mixed $H_2$/$H_{\mathrm{\infty }}$ is defined as $\beta ={\gamma }_1^2+{\gamma }_2^2$. The results for each individual $H_2$ norm and $H_{\mathrm{\infty }}$ 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

• [1] - LMI in Control Systems Analysis, Design and Applications

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