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

## LQ Regulation via ${\displaystyle H_{2}}$ Control

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

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

is minimized, where

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

The following assumptions should hold for a traditional solution.

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

## Relation to ${\displaystyle H_{2}}$ performance

For the system given above an auxiliary system is constructed

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

where

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

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

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

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

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

Then ${\displaystyle H_{2}}$ norm minimization leads minimization of ${\displaystyle J}$.

## Data

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

## The Problem Formulation:

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

## The LMI:

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

{\displaystyle {\begin{aligned}(AX+BW)+(AX+BW)^{T}+x_{0}x_{0}^{T}<0\end{aligned}}}
{\displaystyle {\begin{aligned}trace(Q^{1/2}X(Q^{1/2}))+trace(Y)<\gamma \end{aligned}}}
{\displaystyle {\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 ${\displaystyle K=WX^{-1}}$.