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{aligned}||G_{zw}(s)||_{2}<\gamma \end{aligned}} The System

Consider System with following state-space representation.

{\begin{aligned}{\dot {x}}(t)&=(A+\Delta {A})x(t)+(B_{1}+\Delta {B_{1}})u(t)+B_{2}w(t)\\z(t)&=Cx(t)+D_{1}u(t)+D_{2}w(t)\\\end{aligned}} 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{aligned}{\begin{bmatrix}\Delta {A}&\Delta {B_{1}}\end{bmatrix}}=HF{\begin{bmatrix}E_{1}&E_{2}\end{bmatrix}}\end{aligned}} 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{aligned}F^{T}F\leq I\end{aligned}} For the perturbation, we obviously have

{\begin{aligned}{\begin{bmatrix}\Delta {A}&\Delta {B_{1}}\end{bmatrix}}={\begin{bmatrix}0&0\end{bmatrix}}\end{aligned}} , for $F=0$ {\begin{aligned}{\begin{bmatrix}\Delta {A}&\Delta {B_{1}}\end{bmatrix}}=H{\begin{bmatrix}E_{1}&E_{2}\end{bmatrix}}\end{aligned}} , 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{aligned}{\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{aligned}} {\begin{aligned}{\begin{bmatrix}-Z&CX+D_{1}W\\(CX+D_{1}W)^{T}&-X\end{bmatrix}}<0\end{aligned}} {\begin{aligned}trace(Z)<\gamma ^{2}\end{aligned}} 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 $u(t)=WX^{-1}x(t)$ .