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.

$\begin{array}{c}||G_{zw}(s)|{|}_2<\gamma \end{array}$

The System

Consider System with following state-space representation.

$\begin{array}{cc}\dot{x}(t)& =(A+\mathrm{\Delta }A)x(t)+(B_1+\mathrm{\Delta }{B}_1)u(t)+B_{2}w(t)\\ z(t)& =Cx(t)+D_{1}u(t)+D_{2}w(t)\\ \end{array}$

where $x\in {ℝ}^n$ , $u\in {ℝ}^r$ , $w\in {ℝ}^p$, $z\in {ℝ}^m$. For $H_2$ state feedback control $D_2=0$

$\mathrm{\Delta }A$ and $\mathrm{\Delta }{B}_1$ are real valued matrix functions that represent the time varying parameter uncertainties and of the form

$\begin{array}{c}\left[\begin{array}{cc}\mathrm{\Delta }A& \mathrm{\Delta }{B}_1\end{array}\right]=HF\left[\begin{array}{cc}{E}_1& E_2\end{array}\right]\end{array}$

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.

$\begin{array}{c}{F}^{T}F\le I\end{array}$

For the perturbation, we obviously have

$\begin{array}{c}\left[\begin{array}{cc}\mathrm{\Delta }A& \mathrm{\Delta }{B}_1\end{array}\right]=\left[\begin{array}{cc}0& 0\end{array}\right]\end{array}$, for $F=0$

$\begin{array}{c}\left[\begin{array}{cc}\mathrm{\Delta }A& \mathrm{\Delta }{B}_1\end{array}\right]=H\left[\begin{array}{cc}{E}_1& E_2\end{array}\right]\end{array}$, 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:

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

$\begin{array}{c}\left[\begin{array}{cc}⟨AX+B_{1}W{⟩}_s+B_2{B}_2^T+\beta H{H}^{(}T)& (E_{1}X+E_{2}W{)}^T\\ E_{1}X+E_{2}W& -\beta I\end{array}\right]<0\end{array}$

$\begin{array}{c}\left[\begin{array}{cc}-Z& CX+D_{1}W\\ (CX+D_{1}W{)}^T& -X\end{array}\right]<0\end{array}$

$\begin{array}{c}trace(Z)<{\gamma }^2\end{array}$

where $⟨M{⟩}_s=(M+M^T)$ is the definition that is need for the above LMI.

Conclusion:

In this case, an $H_2$ state feedback control law is given by $u(t)=W{X}^{-1}x(t)$.

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.