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:

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

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:

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

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:

$$ \begin{align} &\text{min} \quad \gamma_{1}^{2} + \gamma_{2}^{2} \\ &\text{s.t.} \\ &\begin{bmatrix} X_{1}     & I \\ I         & Y_{1} \end{bmatrix} > 0 \\ &\begin{bmatrix} AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}B_{2}^{\text{T}} & *^{\text{T}} & *^{\text{T}} & *^{\text{T}} \\ A^{\text{T}}+A_{n}+(B_{2}D_{n}C_{2})^{\text{T}} & X_{1}A+A^{\text{T}}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}} & *^{\text{T}} & *^{\text{T}} \\ (B_{1}+B_{2}D_{n}D_{21})^{\text{T}} & (X_{1}B_{1} + B_{n}D_{21})^{\text{T}} & -\gamma I & *^{\text{T}}\\ C_{1}Y_{1}+D_{12}C_{n} & C_{1}+D_{12}D_{n}C_{2} & D_{11}+D_{12}D_{n}D_{21} & -\gamma I\\ \end{bmatrix} < 0 \\ &\begin{bmatrix} Y_{1}  & I & (C_{1}Y_{1}+D_{12}C_{n})^{\text{T}} \\ I & X_{1} & (C_{1}+D_{12}D_{n}C_{2})^{\text{T}}\\ (C_{1}Y_{1}+D_{12}C_{n}) & (C_{1}+D_{12}D_{n}D_{21} & Z\\ C_{1}Y_{1}+D_{12}C_{n} & C_{1}+D_{12}D_{n}C_{2} & D_{11}+D_{12}D_{n}D_{21} & -\gamma I\\ \end{bmatrix} > 0 \\ &\begin{bmatrix} AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}\text{T}B_{2}\text{T} & *^{\text{T}} & *^{\text{T}} & *^{\text{T}} \\ (A^{\text{T}}+An+(B_{2}*D_{n}*C_{2})^{\text{T}})  & X_{1}A+A^{\text{T}}X_{1}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}}  & *^{\text{T}} & *^{\text{T}} \\ (B_{1}+B_{2}D_{n}D_{21})^{\text{T}}  & (X_{1}B_{1}+B_{n}D_{21})^{\text{T}} & -\gamma_{2}^{2}I  & *^{\text{T}} \\ (C_{1}Y_{1}+D_{12}C_{n}) &  (C_{1}+D_{12}D_{n}C_{2}) & (D_{11}+D_{12}D_{n}*D_{21}) & -I \\ \end{bmatrix} < 0 \\ & \text{trace}(Z) < \gamma_{1}^{2} \\ & D_{11} + D_{12}D_{n}D_{21} = 0 \end{align} $$

where $$ \gamma_{1}^{2} $$ and $$\gamma_{1}^{2}$$ are defined as the $$ H_{2}$$ and $$ H_{\infty}$$ norm of the system:

$$ \begin{align} &||S(P,K)||_{H_{2}}^{2} = \gamma_{1}^{2} \\ &||S(P,K)||_{H_{\infty}}^{2} = \gamma_{2}^{2} \end{align} $$

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

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