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

Full State Feedback Optimal $H_{\infty }$ 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. $H_{\infty }$ methods formulate this task as an optimization problem and attempt to minimize the $H_{\infty }$ 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 $H_{\infty }$ 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.

${\begin{bmatrix}{\dot {x}}\\z\\y\end{bmatrix}}={\begin{bmatrix}A&B_{1}&B_{2}\\C_{1}&D_{11}&D_{12}\\C_{2}&D_{21}&D_{22}\end{bmatrix}}{\begin{bmatrix}x\\w\\u\end{bmatrix}}$

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

The lower linear fractional transformation (LFT) is used to implement a controller $K$ into the system. The lower LFT is denoted as ${\underline {S}}(P,K)$ and is formed by ${\underline {S}}(P,K)=P_{11}+P_{12}(I-KP_{22})^{-1}KP_{21}$ with ${\begin{bmatrix}z\\y\end{bmatrix}}={\begin{bmatrix}P_{11}&P_{12}\\P_{21}&P_{22}\end{bmatrix}}{\begin{bmatrix}w\\u\end{bmatrix}}$. 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 $K={\begin{bmatrix}0&0\\0&F\end{bmatrix}}$.

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 $H_{\infty }$ Control LMI

The following are equivalent.

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

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

${\begin{bmatrix}YA^{T}+AY+Z^{T}B_{2}^{T}+B_{2}Z&B_{1}&YC_{1}^{T}+Z^{T}D_{12}^{T}\\B1_{1}^{T}&-\gamma I&D_{11}^{T}\\C_{1}Y+D_{12}Z&D_{11}&-\gamma I\end{bmatrix}}<0$.

Then $F=ZY^{-1}$.

Conclusion:

The above LMI, if feasible, will determine the bound $\gamma$ on the $H_{\infty }$ 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.