LMIs in Control/pages/Hinf Output Optimal Control

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

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

The System

$\begin{array}{cc}\dot{x}& =Ax+B_{1}v+B_{2}u,x(0)=x_0,\\ z& =C_{1}x+D_{11}v+D_{12}u,\\ y& =C_{2}x+D_{21}v,\end{array}$

where $x\in {ℝ}^n$ is the state, $v\in {ℝ}^r$ is the exogenous input, $u\in {ℝ}^m$ is the control input, $y\in {ℝ}^p$ is the measured output and $z\in {ℝ}^s$ 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 $v\in L_2[0,\mathrm{\infty })$. $N_1,N_2$ 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 $H_{\mathrm{\infty }}$ condition needs to be fulfilled:

${\gamma }_w=su{p}_{\Vert v{\Vert }_{\mathrm{\infty }}^2+x_0^{\mathrm{\top }}R{x}_0\ne 0}\frac{\Vert z{\Vert }_{\mathrm{\infty }}}{(\Vert v{\Vert }_{\mathrm{\infty }}^2+x_0^{\mathrm{\top }}R{x}_0{)}^{1/2}}<{\gamma }_w,$

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

$\begin{array}{cc}& {\text{min}}_{\gamma ,X_{11},Y_{11}}:\gamma \\ & \text{subj. to: }{X}_{11}>0,Y_{11}>0,\\ & {\left[\begin{array}{cc}{N}_1& 0\\ 0& I\end{array}\right]}^{\mathrm{\top }}\left[\begin{array}{ccc}{A}^{\mathrm{\top }}{X}_{11}+X_{11}A& X_{11}{B}_1& C_1^{\mathrm{\top }}\\ *& -{\gamma }^{2}I& D_{11}^{\mathrm{\top }}\\ *& *& -I\end{array}\right]\left[\begin{array}{cc}{N}_1& 0\\ 0& I\end{array}\right]<0,\\ & {\left[\begin{array}{cc}{N}_2& 0\\ 0& I\end{array}\right]}^{\mathrm{\top }}\left[\begin{array}{ccc}A{Y}_{11}+Y_{11}{A}^{\mathrm{\top }}& Y_{11}{C}_1^{\mathrm{\top }}& B_1\\ *& -I& D_{11}\\ *& *& -{\gamma }^{2}I\end{array}\right]\left[\begin{array}{cc}{N}_2& 0\\ 0& I\end{array}\right]<0,\\ & \left[\begin{array}{cc}{X}_{11}& I\\ I& Y_{11}\end{array}\right]\ge 0,X_{11}<{\gamma }^{2}R,\end{array}$

Conclusion:

Solution of the above LMI gives a check to see if an $H_{\mathrm{\infty }}$ 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 $H_{\mathrm{\infty }}$-optimal control with transients Link to the original article.