# LMIs in Control/pages/Hinf Output Optimal Control

$H_{\infty }$ Optimal Output Controllability for Systems With Transients

This LMI provides an $H_{\infty }$ optimal output controllability problem to check if such controllers for systems with unknown exogenous disturbances and initial conditions can exist or not.

## The System

{\begin{aligned}{\dot {x}}&=Ax+B_{1}v+B_{2}u,x(0)=x_{0},\\z&=C_{1}x+D_{11}v+D_{12}u,\\y&=C_{2}x+D_{21}v,\end{aligned}}

where $x\in \mathbb {R} ^{n}$  is the state, $v\in \mathbb {R} ^{r}$  is the exogenous input, $u\in \mathbb {R} ^{m}$  is the control input, $y\in \mathbb {R} ^{p}$  is the measured output and $z\in \mathbb {R} ^{s}$  is the regulated output.

## The Data

System matrices $(A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12},D_{21},D_{22})$  need to be known. It is assumed that $v\in L_{2}[0,\infty )$ . $N_{1},N_{2}$  are matrices with their columns forming the bais of kernels of $C_{2}D_{21}$  and $C_{2}D_{12}$  respectively.

## The Optimization Problem

For a given $\gamma$ , the following $H_{\infty }$  condition needs to be fulfilled:

$\gamma _{w}=sup_{\|v\|_{\infty }^{2}+x_{0}^{\top }Rx_{0}\neq 0}{\frac {\|z\|_{\infty }}{(\|v\|_{\infty }^{2}+x_{0}^{\top }Rx_{0})^{1/2}}}<\gamma _{w},$

## The LMI: $H_{\infty }$ Output Feedback Controller for Systems With Transients

{\begin{aligned}&{\text{min}}_{\gamma ,X_{11},Y_{11}}:\gamma \\&{\text{subj. to: }}X_{11}>0,Y_{11}>0,\\&\quad {\begin{bmatrix}N_{1}&0\\0&I\end{bmatrix}}^{\top }{\begin{bmatrix}A^{\top }X_{11}+X_{11}A&X_{11}B_{1}&C_{1}^{\top }\\*&-\gamma ^{2}I&D_{11}^{\top }\\*&*&-I\end{bmatrix}}{\begin{bmatrix}N_{1}&0\\0&I\end{bmatrix}}<0,\\&\quad {\begin{bmatrix}N_{2}&0\\0&I\end{bmatrix}}^{\top }{\begin{bmatrix}AY_{11}+Y_{11}A^{\top }&Y_{11}C_{1}^{\top }&B_{1}\\*&-I&D_{11}\\*&*&-\gamma ^{2}I\end{bmatrix}}{\begin{bmatrix}N_{2}&0\\0&I\end{bmatrix}}<0,\\&\quad {\begin{bmatrix}X_{11}&I\\I&Y_{11}\end{bmatrix}}\geq 0,X_{11}<\gamma ^{2}R,\end{aligned}}

## Conclusion:

Solution of the above LMI gives a check to see if an $H_{\infty }$  optimal output controller for systems with transients can exist or not.

## Implementation

A link to CodeOcean or other online implementation of the LMI