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

## Robust ${\displaystyle H_{2}}$ State Feedback Control

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

{\displaystyle {\begin{aligned}||G_{zw}(s)||_{2}<\gamma \end{aligned}}}

## The System

Consider System with following state-space representation.

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

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

{\displaystyle {\begin{aligned}{\begin{bmatrix}\Delta {A}&\Delta {B_{1}}\end{bmatrix}}=HF{\begin{bmatrix}E_{1}&E_{2}\end{bmatrix}}\end{aligned}}}

where matrices ${\displaystyle E_{1},E_{2}}$ and ${\displaystyle H}$ are some known matrices of appropriate dimensions, while ${\displaystyle F}$ is a matrix which contains the uncertain parameters and satisfies.

{\displaystyle {\begin{aligned}F^{T}F\leq I\end{aligned}}}

For the perturbation, we obviously have

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

## The Problem Formulation:

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

## The LMI:

${\displaystyle \min \gamma ^{2}::}$

{\displaystyle {\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}}}
{\displaystyle {\begin{aligned}{\begin{bmatrix}-Z&CX+D_{1}W\\(CX+D_{1}W)^{T}&-X\end{bmatrix}}<0\end{aligned}}}
{\displaystyle {\begin{aligned}trace(Z)<\gamma ^{2}\end{aligned}}}

where ${\displaystyle \langle M\rangle _{s}=(M+M^{T})}$ is the definition that is need for the above LMI.

## Conclusion:

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