LMIs in Control/pages/Optimal Output Feedback H2 LMI

Optimal Output Feedback $$H_2$$ LMI
Similar to state feedback, output feedback is necessary when information about the output is not known. Often techniques such as Kalman filtering are implemented to tackle this problem. The method below, however, does not use a filtering technique and instead uses a combination of LMI constraints to perform the output feedback as well as find the minimal bound on the $$H_2$$ norm of the system. $$H_2$$ is often done using more classical tools such as Riccati equations. More recently LMI techniques have been created to solve problems such as full state feedback or output feedback as seen below.

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 $$H_2$$ 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 $$||S(K,P)||_{H_2}<\gamma$$

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



\begin{bmatrix} AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}B_{2}^{\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}}\\ (B_{1}+B_{2}D_{n}D_{21})^{\text{T}} & (X_{1}B_{1} + B_{n}D_{21})^{\text{T}} & -\gamma I \\ \end{bmatrix} < 0 $$

\begin{bmatrix} Y_{1}  & *^T & *^T \\ I & X_{1} & *^T\\ C_{1}Y_{1}+D_{12}C_{n} & C_{1}+D_{12}D_{n}C_{2} & Z \end{bmatrix} > 0 $$

D_{11} + D_{12}D_{n}D_{21} = 0 $$

\text{trace}(Z) < \gamma^{2} $$

Conclusion:
The above LMI determines the the upper bound $$\gamma$$ on the H2 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_H2.m

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Optimal Output Feedback Hinf