LMIs in Control/pages/Optimal Output Feedback Hinf LMI

Optimal Output Feedback $$H_{\infty}$$ LMI
Optimal output feedback control is a problem which arises from not knowing all information about the output of the system. It correlates to the state feedback situation where the part of the state is unknown. This issue can arise in decentralized control problems, for example, and requires the use of an "observer-like" solution. One such method is the use of a Kalman Filter, a more classical technique. However, other methods exist that do not implement a Kalman Filter such as the one below which uses an LMI to preform the output feeback. The $$H_{\infty}$$ control methods form an optimization problem which attempts to minimize the $$H_{\infty}$$ norm of the system.

The System
The system is represented using the 9-matrix notation shown below.
 * $$\begin{bmatrix} \dot{x}\\z\\y \end{bmatrix}=\begin{bmatrix} A & B_{1} & B_{2}\\ C_{1} & D_{11} & D_{12}\\ C_{2} & D_{21} & D_{22} \end{bmatrix} \begin{bmatrix} x\\w\\u \end{bmatrix}

$$ where $$x(t)\in \R^n$$ is the state, $$z(t)\in \R^p$$ is the regulated output, $$y(t)\in \R^q$$ is the sensed output, $$w(t)\in \R^r$$ is the exogenous input, and $$u(t)\in \R^m$$ is the actuator input, at any $$t\in \R$$.

The Data
$$A$$, $$B_{1}$$, $$B_{2}$$, $$C_{1}$$, $$C_{2}$$, $$D_{11}$$, $$D_{12}$$, $$D_{21}$$, $$D_{22}$$ are known.

The LMI: Optimal Output Feedback $$H_{\infty}$$ Control LMI
The following are equivalent.

1) There exists a $$\hat{K}=\begin{bmatrix} A_K & B_K \\ C_K & D_K \end{bmatrix}$$ such that $$||S(K,P)||_{H_{\infty}}<\gamma$$

2) There exists $$X_1$$, $$Y_1$$, $$Z$$, $$A_n$$, $$B_n$$, $$C_n$$, $$D_n$$ such that


 * $$\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 $$

Conclusion:
The above LMI determines the the upper bound $$\gamma$$ on the $$H_{\infty}$$ norm. In addition to this the controller $$\hat{K}(A_K,B_K,C_K,D_K)$$ can also be recovered.
 * $$ D_K=(I+D_{K2}D_{22})^{-1}D_{K2}$$
 * $$ B_K=B_{K2}(I+D_{22}D_K)$$
 * $$ C_K=(I+D_{K}D_{22})C_{K2}$$
 * $$ A_K=A_{K2}-B_K(I+D_{22}D_K)^{-1}D_{22}C_K$$

where,
 * $$\begin{bmatrix} A_{K2} & B_{K2} \\ C_{K2} & D_{K2} \end{bmatrix}= {\begin{bmatrix} X_{2} & X_{1}B_2 \\ 0 & I \end{bmatrix}}^{-1} \left[\begin{bmatrix} A_n & B_n \\ C_n & D_n \end{bmatrix}-\begin{bmatrix} X_1AY_1 & 0 \\ 0 & 0 \end{bmatrix}\right] {\begin{bmatrix} Y_2^T & 0 \\ C_2Y_1 & I \end{bmatrix}}^{-1}

$$ for any full-rank $$X_2$$ and $$Y_2$$ such that
 * $$\begin{bmatrix} X_1 & X_2 \\ X_2^T & X_3 \end{bmatrix}= {\begin{bmatrix} Y_1 & Y_2 B_2 \\ Y_2^T & Y_3\end{bmatrix}}^{-1}

$$.

Implementation
This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/OF_Hinf.m

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