# LMIs in Control/pages/HInf Optimal Filter

Optimal filtering is a means of adaptive extraction of a weak desired signal in the presence of noise and interfering signals. The goal of optimal filtering is to design a filter that acts on the output ${\displaystyle z}$ of the generalized plant and optimizes the transfer matrix from w to the filtered output.

## The System:

Consider the continuous-time generalized LTI plant with minimal states-space realization

{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+B_{1}w\\z&=C_{1}x+D_{11}w,\\y&=C_{2}x+D_{21}w,\end{aligned}}}

where it is assumed that ${\displaystyle A}$ is Hurwitz.

## The Data

The matrices needed as inputs are ${\displaystyle A,B_{1},C_{2},C_{1},,D_{11},D_{21}}$.

## The Optimization Problem:

An ${\displaystyle H\infty }$-optimal filter is designed to minimize the ${\displaystyle H_{\infty }}$ norm of ${\displaystyle {\tilde {P}}(s)}$ in following equation.

{\displaystyle {\begin{aligned}{\tilde {P}}(s)={\tilde {C}}_{1}(sI-{\tilde {A}})^{-}1{\tilde {B}}_{1}+{\tilde {D}}_{11},\\{\text{where}}\\{\tilde {A}}={\begin{bmatrix}A&&0\\B_{f}C_{2}&&A_{f}\end{bmatrix}}&<0\\{\tilde {B}}_{1}={\begin{bmatrix}B_{1}\\B_{f}D_{21}\end{bmatrix}}&<0\\{\tilde {C}}_{1}={\begin{bmatrix}C_{1}-D_{f}C_{2}-C_{f}\end{bmatrix}}&<0\\{\tilde {D}}_{11}=D_{11}-D_{f}D_{21}\\\end{aligned}}}

## The LMI: ${\displaystyle H_{\infty }}$- Optimal filter

Solve for ${\displaystyle A_{n}\in \mathbb {R} ^{n_{x}\times n_{x}},B_{n}\in \mathbb {R} ^{n_{x}\times n_{y}},C_{f}\in \mathbb {R} ^{n_{x}\times n_{x}}}$, ${\displaystyle X,Y\in \mathbb {S} ^{n_{x}}}$ and ${\displaystyle \nu \in \mathbb {R} _{>0}}$ that minimize ${\displaystyle \zeta (\nu )=\nu }$ subject to ${\displaystyle X>0,Y>0}$.

{\displaystyle {\begin{aligned}{\begin{bmatrix}YA+A^{T}Y+B_{n}C_{2}&&A_{n}+C_{2}^{T}B_{n}^{T}+A^{T}X&&YB_{1}+B_{n}D_{21}&&{C_{1}}^{T}-{C_{2}}^{T}{D_{f}}^{T}\\\star &&A_{n}+A_{n}^{T}&&XB_{1}+B_{n}D_{21}&&-{C_{f}}^{T}\\\star &&\star &&-\gamma I&&{D_{1}1}^{T}-{D_{2}1}^{T}{D_{f}}^{T}\\\star &&\star &&\star &&-\gamma I\end{bmatrix}}&<0\\\\Y-X>0\\\end{aligned}}}

## Conclusion:

The filter is recovered by ${\displaystyle A_{f}=X^{-1}A_{n}}$ and ${\displaystyle B_{f}=X^{-1}B_{n}}$.