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Quadratic Polytopic Full State Feedback Optimal $H_{\infty }$ Control

For a system having polytopic uncertainties, Full State Feedback is a control technique that attempts to place the system's closed-loop system poles in specified locations based off of performance specifications given. $H_{\infty }$ methods formulate this task as an optimization problem and attempt to minimize the $H_{\infty }$ norm of the system.

The System

Consider System with following state-space representation.

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{1}q(t)+B_{2}w(t)\\p(t)&=C_{1}x(t)+D_{11}q(t)+D_{12}w(t)\\z(t)&=C_{2}x(t)+D_{21}q(t)+D_{22}w(t)\\\end{aligned}}$

where $x\in \mathbb {R} ^{m}$ , $q\in \mathbb {R} ^{n}$ , $w\in \mathbb {R} ^{g}$, $A\in \mathbb {R} ^{mxm}$, $B_{1}\in \mathbb {R} ^{mxn}$, $B_{2}\in \mathbb {R} ^{mxg}$, $p\in \mathbb {R} ^{p}$ , $C_{1}\in \mathbb {R} ^{pxm}$, $D_{11}\in \mathbb {R} ^{pxn}$, $D_{12}\in \mathbb {R} ^{pxg}$, $z\in \mathbb {R} ^{s}$, $C_{2}\in \mathbb {R} ^{sxm}$, $D_{21}\in \mathbb {R} ^{sxn}$ , $D_{22}\in \mathbb {R} ^{sxg}$ for any $t\in \mathbb {R}$.

Add uncertainty to system matrices

$A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12}$

New state-space representation

${\begin{aligned}{\dot {x}}(t)&=(A+A_{i})x(t)+(B_{1}+B_{i})q(t)+(B_{2}+B_{i})w(t)\\p(t)&=(C_{1}+C_{i})x(t)+(D_{11}+D_{i})q(t)+(D_{12}+D_{i})w(t)\\z(t)&=C_{2}x(t)+D_{21}q(t)+D_{22}w(t)\\\end{aligned}}$

The Optimization Problem:

Recall the closed-loop in state feedback is:
$S(P,K)=$

${\begin{aligned}{\begin{bmatrix}A+B_{2}F&&B_{1}\\C_{1}+D_{12}F&&D_{11}\end{bmatrix}}\\\end{aligned}}$

This problem can be formulated as $H\infty$ optimal state-feedback, where K is a controller gain matrix.

The LMI:

An LMI for Quadratic Polytopic $H\infty$ Optimal State-Feedback Control $||S(P(\Delta ),K(0,0,0,F))||_{H\infty }\leq \gamma$
$Y>0$

${\begin{aligned}{\begin{bmatrix}Y(A+A_{i})^{T}+(A+A_{i})Y+Z^{T}(B_{2}+B_{1,i})^{T}+(B_{2}+B_{1,i})Z&&*^{T}&&*^{T}\\(B_{1}+B_{1,i})^{T}&&-\gamma I&&*^{T}\\(C_{1}+C_{1,i})Y+(D_{12}+D_{12,i})Z&&(D_{11}+D_{11,i})&&-\gamma I\end{bmatrix}}<0\end{aligned}}$

Conclusion:

The $H\infty$ Optimal State-Feedback Controller is recovered by $F=ZY^{-1}$
Controller will determine the bound $\gamma$ on the $H_{\infty }$ norm of the system.

Implementation:

https://github.com/JalpeshBhadra/LMI/tree/master

Related LMIs

Full State Feedback Optimal $H_{\infty }$ Controller

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.