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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.

${\begin{aligned}J(x,u)=\int _{0}^{\infty }(x^{T}Qx+u^{T}Ru)dt\end{aligned}}$

is minimized, where

${\begin{aligned}Q=Q^{T}\geq 0,R=R^{T}>0\end{aligned}}$

The following assumptions should hold for a traditional solution.

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

Relation to $H_{2}$ performance

For the system given above an auxiliary system is constructed

${\begin{aligned}{\dot {x}}=Ax+Bu+x_{0}\omega ,y=Cx+Du\end{aligned}}$

where

${\begin{aligned}C={\begin{bmatrix}Q^{1/2}\\0\end{bmatrix}},D={\begin{bmatrix}0\\R^{1/2}\end{bmatrix}}\end{aligned}}$

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

${\begin{aligned}G_{y\omega }(s)=(C+DK)[sI-(A+BK)]^{-1}x_{0}\end{aligned}}$

Then the LQ problem and the $H_{2}$ norm of $G_{y\omega }$ are related as

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

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 $X\in \mathbb {S} ^{n}$, $Y\in \mathbb {S} ^{r}$ and $W\in \mathbb {R} ^{rxn}$. Then $K$ can be obtained by the following LMI.

The LMI:

$\min \gamma ::$

${\begin{aligned}(AX+BW)+(AX+BW)^{T}+x_{0}x_{0}^{T}<0\end{aligned}}$

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

${\begin{aligned}{\begin{bmatrix}-Y&R^{1/2}W\\(R^{1/2}W)^{T}&-X\end{bmatrix}}<0\end{aligned}}$

Conclusion:

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

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.