LMIs in Control/pages/quadratic polytopic h2 optimal state feedback control

LMIs in Control/pages/quadratic polytopic h2 optimal state feedback control

Quadratic Polytopic Full State Feedback Optimal $H_2$ 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 on performance specifications given, such as requiring stability or bounding the overshoot of the output. By minimizing the $H_2$ norm of this system we are minimizing the effect noise has on the system as part of the performance specifications.

The System

Consider System with following state-space representation.

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

where $x\in {ℝ}^m$ , $q\in {ℝ}^n$ , $w\in {ℝ}^g$, $A\in {ℝ}^{mxm}$, $B_1\in {ℝ}^{mxn}$, $B_2\in {ℝ}^{mxg}$, $p\in {ℝ}^p$ , $C_1\in {ℝ}^{pxm}$, $D_{11}\in {ℝ}^{pxn}$, $D_{12}\in {ℝ}^{pxg}$, $z\in {ℝ}^s$, $C_2\in {ℝ}^{sxm}$, $D_{21}\in {ℝ}^{sxn}$ , $D_{22}\in {ℝ}^{sxg}$ for any $t\in ℝ$.

Add uncertainty to system matrices

$A,B_1,B_2,C_1,C_2,D_{11},D_{12}$

New state-space representation

$\begin{array}{cc}\dot{x}(t)& =(A+A_i)x(t)+(B_1+B_i)q(t)+(B_2+B_i)w(t)\\ p(t)& =(C_1+C_i)x(t)+(D_{11}+D_i)q(t)+(D_{12}+D_i)w(t)\\ z(t)& =C_{2}x(t)+D_{21}q(t)+D_{22}w(t)\\ \end{array}$

The Data

The matrices necessary for this LMI are

The Optimization Problem:

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

$\begin{array}{c}\left[\begin{array}{ccc}A+B_{22}F& & B_1\\ C_1+D_{12}F& & D_{11}\end{array}\right]\\ \end{array}$

This problem can be formulated as $H_2$ optimal state-feedback, where K is a controller gain matrix.

The LMI: An LMI for Quadratic Polytopic $H_2$ Optimal

State-Feedback Control
$||S(P(\mathrm{\Delta }),K(0,0,0,F))|{|}_{H_2}\le \gamma$
$X>0$

$\begin{array}{c}\left[\begin{array}{ccc}AX+B_{2}Z+X{A}^T+Z^T{B}_2^T& & B_1\\ B_1^T& & -I\end{array}\right]+\left[\begin{array}{ccc}{A}_{i}X+B_{2,i}Z+X{A}_i^T+Z^T{B}_{2,I}^T& & B_{1,i}\\ B_{1,i}^T& & 0\end{array}\right]<0i=1,......,k\end{array}$

$\begin{array}{c}\left[\begin{array}{ccc}X& & (C_{1}X+D_{12}Z{)}^T\\ C_{1}X+D_{12}Z& & W\end{array}\right]+\left[\begin{array}{ccc}0& & (C_{1,i}X+D_{12,i}Z{)}^T\\ C_{1,i}X+D_{12,i}Z& & 0\end{array}\right]>0i=1,......,k\end{array}$

$\begin{array}{c}\\ TraceW<\gamma \end{array}$

Conclusion:

The $H_2$ Optimal State-Feedback Controller is recovered by $F=Z{X}^{-1}$

Implementation:

https://github.com/JalpeshBhadra/LMI/blob/master/H2_optimal_statefeedback_controller.m

Related LMIs

$H_2$ Optimal State-Feedback 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.