LMIs in Control/pages/Full-State Feedback Optimal Control Hinf LMI

Full State Feedback Optimal $H_{\mathrm{\infty }}$ Control

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. $H_{\mathrm{\infty }}$ methods formulate this task as an optimization problem and attempt to minimize the $H_{\mathrm{\infty }}$ norm of the system. In a single-input single-output (SISO) system this norm represents the maximum gain on a magnitude Bode plot. In the case of multi-input multi-output (MIMO) systems it can be interpreted as maximum response to a perturbation introduced to the system. In either, by minimizing the $H_{\mathrm{\infty }}$ we are minimizing the worst case effect of a disturbance to the system, whether it is noise or another perturbation.

The System

The system is represented using the 9-matrix notation shown below.

$\left[\begin{array}{c}\dot{x}\\ z\\ y\end{array}\right]=\left[\begin{array}{ccc}A& B_1& B_2\\ C_1& D_{11}& D_{12}\\ C_2& D_{21}& D_{22}\end{array}\right]\left[\begin{array}{c}x\\ w\\ u\end{array}\right]$

where $x(t)\in {ℝ}^n$ is the state, $z(t)\in {ℝ}^p$ is the regulated output, $y(t)\in {ℝ}^q$ is the sensed output, $w(t)\in {ℝ}^r$ is the exogenous input, and $u(t)\in {ℝ}^m$ is the actuator input, at any $t\in ℝ$.

The lower linear fractional transformation (LFT) is used to implement a controller $K$ into the system. The lower LFT is denoted as $\underset{\_}{S}(P,K)$ and is formed by $\underset{\_}{S}(P,K)=P_{11}+P_{12}(I-K{P}_{22}{)}^{-1}K{P}_{21}$ with $\left[\begin{array}{c}z\\ y\end{array}\right]=\left[\begin{array}{cc}{P}_{11}& P_{12}\\ P_{21}& P_{22}\end{array}\right]\left[\begin{array}{c}w\\ u\end{array}\right]$. For full-state feedback we consider a controller of the form $u(t)=Fx(t)$. This is a special case where $y(t)=x(t)$ and results in a controller of the form $K=\left[\begin{array}{cc}0& 0\\ 0& F\end{array}\right]$.

The Data

$A$, $B_1$, $B_2$, $C_1$, $C_2$, $D_{11}$, $D_{12}$, $D_{21}$, $D_{22}$ are known.

The LMI:Full State Feedback Optimal $H_{\mathrm{\infty }}$ Control LMI

The following are equivalent.

1) There exists a $F$ such that $||\underset{\_}{S}(P,K(0,0,0,F)|{|}_{H_{\mathrm{\infty }}}\le \gamma$

2) There exists $Y>0$ and $Z$ such that

$\left[\begin{array}{ccc}Y{A}^T+AY+Z^T{B}_2^T+B_{2}Z& B_1& Y{C}_1^T+Z^T{D}_{12}^T\\ B{1}_1^T& -\gamma I& D_{11}^T\\ C_{1}Y+D_{12}Z& D_{11}& -\gamma I\end{array}\right]<0$.

Then $F=Z{Y}^{-1}$.

Conclusion:

The above LMI, if feasible, will determine the bound $\gamma$ on the $H_{\mathrm{\infty }}$ norm of the system. In addition to this $F$ is also determined allowing the closed loop system to be determined using the controller $\hat{K}(0,0,0,F)$ found during the optimization.

Implementation

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/FSF_Hinf.m

Related LMIs

Full State Feedback Optimal H2 LMI

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.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.
• A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Chapter 7.