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

=='''Robust <math>H_{2}</math> State Feedback Control'''==

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

:<math>

\begin{align}
||G_{zw}(s)||_{2} < \gamma
\end{align}</math> <br>

== '''The System''' ==

Consider System with following state-space representation.

:<math>

\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}</math> <br>

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

<math> \Delta{A} </math> and <math> \Delta{B_{1}} </math> are real valued matrix functions that represent the time varying parameter uncertainties and of the form
:<math>

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

\begin{align}
F^{T}F \leq I
\end{align}</math> <br>

For the perturbation, we obviously have

:<math>

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

\begin{bmatrix}
0 & 0
\end{bmatrix}
\end{align}</math>, for <math> F=0 </math> <br>
:<math>

\begin{align}
\begin{bmatrix}
\Delta{A} & \Delta{B_{1}}
\end{bmatrix} =
H
\begin{bmatrix}
E_{1} & E_{2}
\end{bmatrix}
\end{align}</math>, for <math> F=0 </math> <br>


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

== '''The LMI:''' == 
<math> \min \gamma^{2}
:: </math> <br>
:<math>
\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}</math> <br>
:<math>
\begin{align}
 \begin{bmatrix} 
-Z & CX + D_{1}W \\
(CX + D_{1}W)^{T} & -X
\end{bmatrix} < 0 
\end{align}</math> 
:<math>
\begin{align}
 trace(Z) < \gamma^2 
\end{align}</math> 
where <math> \langle M \rangle_{s} = (M+M^T) </math> is the definition that is need for the above LMI.



== '''Conclusion:''' ==

In this case, an <math> H_{2} </math> state feedback control law is given by <math> u(t)=WX^{-1}x(t) </math>.


== '''External Links ''' ==
* LMIs in Control Systems Analysis, Design and Applications - Duan and Yu
* A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

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