LMIs in Control/Controller Synthesis/Continuous Time/LQ Regulation via H2 control

LQ Regulation via $H_2$ Control

The LQR design problem is to build an optimal state feedback controller $u=Kx$ for the system $\dot{x}=Ax+Bu,x(0)=x_0$ such that the following quadratic performance index.

$\begin{array}{c}J(x,u)=\underset{0}{\overset{\mathrm{\infty }}{\int }}(x^{T}Qx+u^{T}Ru)dt\end{array}$

is minimized, where

$\begin{array}{c}Q=Q^T\ge 0,R=R^T>0\end{array}$

The following assumptions should hold for a traditional solution.

$𝑨1.(A,B)$ is stabilizable.
$𝑨2.(A,L)$ is observable, with $L=Q^{1/2}$.

Relation to $H_2$ performance

For the system given above an auxiliary system is constructed

$\begin{array}{c}\dot{x}=Ax+Bu+x_0\omega ,y=Cx+Du\end{array}$

where

$\begin{array}{c}C=\left[\begin{array}{c}{Q}^{1/2}\\ 0\end{array}\right],D=\left[\begin{array}{c}0\\ R^{1/2}\end{array}\right]\end{array}$

Where $\omega$ represents an impulse disturbance. Then with state feedback controller $u=Kx$ the closed loop transfer function from disturbance $\omega$ to output $y$ is

$\begin{array}{c}{G}_{y\omega }(s)=(C+DK)[sI-(A+BK){]}^{-1}{x}_0\end{array}$

Then the LQ problem and the $H_2$ norm of $G_{y\omega }$ are related as

$\begin{array}{c}J(x,u)=||G_{y\omega }(s)|{|}_2^2\end{array}$

Then $H_2$ norm minimization leads minimization of $J$.

Data

The state-representation of the system is given and matrices $Q,R$ are chosen for the optimal LQ problem.

The Problem Formulation:

Let assumptions $A1$ and $A2$ hold, then the state feedback control of the form $u=Kx$ exists such that $J(x,u)<\gamma$ if and only if there exist $X\in {𝕊}^n$, $Y\in {𝕊}^r$ and $W\in {ℝ}^{rxn}$. Then $K$ can be obtained by the following LMI.

The LMI:

$\min \gamma ::$

$\begin{array}{c}(AX+BW)+(AX+BW{)}^T+x_0{x}_0^T<0\end{array}$

$\begin{array}{c}trace(Q^{1/2}X(Q^{1/2}))+trace(Y)<\gamma \end{array}$

$\begin{array}{c}\left[\begin{array}{cc}-Y& R^{1/2}W\\ (R^{1/2}W{)}^T& -X\end{array}\right]<0\end{array}$

Conclusion:

In this case, a feedback control law is given as $K=W{X}^{-1}$.

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.