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

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

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

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

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

## The Data

${\displaystyle A}$, ${\displaystyle B_{1}}$, ${\displaystyle B_{2}}$, ${\displaystyle C_{1}}$, ${\displaystyle C_{2}}$, ${\displaystyle D_{11}}$, ${\displaystyle D_{12}}$, ${\displaystyle D_{21}}$, ${\displaystyle D_{22}}$ are known.

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

The following are equivalent.

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

2) There exists ${\displaystyle Y>0}$ and ${\displaystyle Z}$ such that

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

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

## Conclusion:

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

## Implementation

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