# LMIs in Control/pages/Full-State Feedback Optimal Control H2 LMI

## Full State Feedback Optimal ${\displaystyle H_{2}}$ Control

Full State Feedback in general has the goal of positioning a system's closed loop poles in a desired location. This allows us to specify the performance of the system such as requiring stability or bounding the overshoot of the output. By minimizing the ${\displaystyle H_{2}}$ norm of this system we are minimizing the effect noise has on the system as part of the performance specifications, particularly when there is information about the distribution of the noise.

## 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 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: Optimal Output Feedback ${\displaystyle H_{\infty }}$ Control LMI

The following are equivalent.

1) There exists a ${\displaystyle K}$ such that ${\displaystyle ||S(K,P)||_{H_{2}}<\gamma }$

2) There exists ${\displaystyle X>0}$, ${\displaystyle Z}$ and ${\displaystyle W}$ such that

${\displaystyle {\begin{bmatrix}A&B_{2}\end{bmatrix}}{\begin{bmatrix}X\\Z\end{bmatrix}}+{\begin{bmatrix}X&Z^{T}\end{bmatrix}}{\begin{bmatrix}A^{T}\\B_{2}^{T}\end{bmatrix}}+B_{1}B_{1}^{T}<0}$
${\displaystyle {\begin{bmatrix}X&*^{T}\\C_{1}X+D_{12}Z&W\end{bmatrix}}>0}$
${\displaystyle trace(W)<\gamma ^{2}}$

where ${\displaystyle K=ZX^{-1}}$

## Conclusion:

This LMI solves the ${\displaystyle H_{2}}$ optimal full state feedback problem and finds the upper bound of the ${\displaystyle H_{2}}$ norm of the system, ${\displaystyle \gamma }$. In addition to this the controller ${\displaystyle K}$ is also found in the process.

## Implementation

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/FSF_H2.m