LMIs in Control/Controller Synthesis/Continuous Time/Robust H2 State Feedback Control

Robust $$H_{2}$$ State Feedback Control
For the uncertain linear system given below, and a scalar $$ \gamma > 0 $$. The goal is to design a state feedback control $$ u(t) $$ in the form of $$ u(t) =Kx(t) $$ such that the closed-loop system is asymptotically stable and satisfies.



\begin{align} \end{align}$$
 * G_{zw}(s)||_{2} < \gamma

The System
Consider System with following state-space representation.



\begin{align} \dot x(t)&=(A + \Delta{A})x(t)+ (B_{1} + \Delta{B_{1}}) u(t) + B_2 w(t)\\ z(t)& = C x(t) + D_{1} u(t) + D_{2} w(t)\\ \end{align}$$

where $$x\in\mathbb{R}^{n}$$, $$u\in\mathbb{R}^{r}$$ , $$w\in\mathbb{R}^{p}$$, $$z\in\mathbb{R}^{m}$$. For $$ H_{2} $$ state feedback control $$ D_{2}=0 $$

$$ \Delta{A} $$ and $$ \Delta{B_{1}} $$ are real valued matrix functions that represent the time varying parameter uncertainties and of the form

\begin{align} \begin{bmatrix} \Delta{A} & \Delta{B_{1}} \end{bmatrix} = HF \begin{bmatrix} E_{1} & E_{2} \end{bmatrix} \end{align}$$ where matrices $$ E_{1},E_{2}$$ and $$ H $$ are some known matrices of appropriate dimensions, while $$ F $$ is a matrix which contains the uncertain parameters and satisfies.

\begin{align} F^{T}F \leq I \end{align}$$

For the perturbation, we obviously have



\begin{align} \begin{bmatrix} \Delta{A} & \Delta{B_{1}} \end{bmatrix} =

\begin{bmatrix} 0 & 0 \end{bmatrix} \end{align}$$, for $$ F=0 $$

\begin{align} \begin{bmatrix} \Delta{A} & \Delta{B_{1}} \end{bmatrix} = H \begin{bmatrix} E_{1} & E_{2} \end{bmatrix} \end{align}$$, for $$ F=0 $$

The Problem Formulation:
The $$ H_{2} $$ state feedback control problem has a solution if and only if there exist a scalar $$ \beta $$, a matrix $$ W $$, two symmetric matrices $$ Z $$ and $$ X $$ satisfying the following LMI's problem.

The LMI:
$$ \min \gamma^{2}

\begin{align} \begin{bmatrix} \langle AX+B_{1}W \rangle_{s} + B_{2}B_{2}^{T} + \beta{H}H^(T) & (E_{1}X + E_{2}W)^{T} \\ E_{1}X + E_{2}W & -\beta{I} \end{bmatrix} < 0 \end{align}$$

\begin{align} \begin{bmatrix} -Z & CX + D_{1}W \\ (CX + D_{1}W)^{T} & -X \end{bmatrix} < 0 \end{align}$$

\begin{align} trace(Z) < \gamma^2 \end{align}$$ where $$ \langle M \rangle_{s} = (M+M^T) $$ is the definition that is need for the above LMI.

Conclusion:
In this case, an $$ H_{2} $$ state feedback control law is given by $$ u(t)=WX^{-1}x(t) $$.