LMIs in Control/Controller Synthesis/Continuous Time/LQ Regulation via H2 control

LQ Regulation via $$H_{2}$$ Control
The LQR design problem is to build an optimal state feedback controller $$ u=Kx $$ for the system $$ \dot x = Ax + Bu, x(0)=x_0 $$ such that the following quadratic performance index.



\begin{align} J(x,u) = \int_{0}^{\infty}(x^TQx+u^TRu)dt \end{align}$$ is minimized, where

\begin{align} Q = Q^T \geq 0, R = R^T > 0 \end{align}$$ The following assumptions should hold for a traditional solution.

$$ \boldsymbol{A1}. (A,B) $$ is stabilizable. $$ \boldsymbol{A2}. (A,L) $$ is observable, with $$ L = Q^{1/2} $$.

Relation to $$ H_{2} $$ performance
For the system given above an auxiliary system is constructed

\begin{align} \dot x =Ax + Bu + x_0\omega, y= Cx +Du \end{align}$$ where



\begin{align} C = \begin{bmatrix} Q^{1/2}\\ 0 \end{bmatrix}, D = \begin{bmatrix} 0\\ R^{1/2} \end{bmatrix} \end{align}$$ Where $$ \omega $$ represents an impulse disturbance. Then with state feedback controller $$ u=Kx $$ the closed loop transfer function from disturbance $$ \omega $$ to output $$ y $$ is



\begin{align} G_{y\omega}(s)=(C+DK)[sI - (A+BK)]^{-1}x_0 \end{align}$$ Then the LQ problem and the $$ H_{2} $$ norm of $$ G_{y\omega} $$ are related as

\begin{align} J(x,u) = ||G_{y\omega}(s)||_{2}^2 \end{align}$$ Then $$ H_{2} $$ norm minimization leads minimization of $$ J $$.

Data
The state-representation of the system is given and matrices $$ Q,R $$ are chosen for the optimal LQ problem.

The Problem Formulation:
Let assumptions $$ A1 $$ and $$ A2 $$ hold, then the state feedback control of the form $$ u=Kx $$ exists such that $$ J(x,u) < \gamma $$ if and only if there exist $$ X \in \mathbb{S}^n $$, $$ Y \in \mathbb{S}^{r} $$ and $$ W \in \mathbb{R}^{rxn} $$. Then $$ K $$ can be obtained by the following LMI.

The LMI:
$$ \min \gamma

\begin{align} (AX + BW) + (AX + BW)^T +x_0x_0^T < 0 \end{align}$$

\begin{align} trace(Q^{1/2}X(Q^{1/2})) + trace(Y) < \gamma \end{align}$$

\begin{align} \begin{bmatrix} -Y & R^{1/2}W \\ (R^{1/2}W)^T & -X \end{bmatrix} < 0 \end{align}$$

Conclusion:
In this case, a feedback control law is given as $$ K = WX^{-1} $$.