# LMIs in Control/pages/Hinf Output Optimal Control

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

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

## The System

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

## The Data

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

## The Optimization Problem

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

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

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

{\displaystyle {\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 ${\displaystyle H_{\infty }}$  optimal output controller for systems with transients can exist or not.

## Implementation

A link to CodeOcean or other online implementation of the LMI