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

Related LMIs

Links to other closely-related LMIs

External Links

• LMI-based $H_{\infty }$-optimal control with transients Link to the original article.