LMIs in Control/pages/Full-State Feedback Optimal Control H2 LMI

Full State Feedback Optimal $$H_2$$ Control
Full State Feedback in general has the goal of positioning a system's closed loop poles in a desired location. This allows us to specify the performance of the system such as requiring stability or bounding the overshoot of the output. By minimizing the $$H_2$$ norm of this system we are minimizing the effect noise has on the system as part of the performance specifications, particularly when there is information about the distribution of the noise.

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 $$K$$ such that $$||S(K,P)||_{H_{2}}<\gamma$$

2) There exists $$X>0$$, $$Z$$ and $$W$$ such that


 * $$\begin{bmatrix} A & B_2 \end{bmatrix} \begin{bmatrix}

X \\ Z \end{bmatrix}+ \begin{bmatrix} X & Z^T \end{bmatrix} \begin{bmatrix} A^T \\ B_2^T \end{bmatrix}+ B_1B_1^T <0 $$
 * $$\begin{bmatrix}

X & *^T \\ C_1X+D_{12}Z & W \end{bmatrix} > 0 $$
 * $$trace(W)<\gamma^2$$

where $$K=ZX^{-1}$$

Conclusion:
This LMI solves the $$H_2$$ optimal full state feedback problem and finds the upper bound of the $$H_2$$ norm of the system, $$\gamma$$. In addition to this the controller $$K$$ is also found in the process.

Implementation
This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/FSF_H2.m

Related LMIs
Full State Feedback Optimal $H_{\infty}$ LMI