LMIs in Control/pages/H2Optimalfilter

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Optimal filtering is a means of adaptive extraction of a weak desired signal in the presence of noise and interfering signals. Optimal filters normally are free from stability problems. There are simple operational checks on an optimal filter when it is being used that indicate whether it is operating correctly. Optimal filters are probably easier to make adaptive to parameter changes than suboptimal filters.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 ${\displaystyle A,B_{1},C_{2},D_{21}}$.

The Optimization Problem:

An ${\displaystyle H_{2}}$-optimal filter is designed to minimize the ${\displaystyle H_{2}}$ 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}}}

To ensure that ${\displaystyle {\tilde {P}}(s)}$ has a finite ${\displaystyle H_{2}}$ norm, it is required that ${\displaystyle D_{f}=D_{11}}$, which results in ${\displaystyle {\tilde {D}}_{11}=D_{11}-D_{f}=0}$

The LMI: ${\displaystyle H_{2}}$- Optimal filter

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

{\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}\\\star &&A_{n}+A_{n}^{T}&&XB_{1}+B_{n}D_{21}\\\star &&\star &&-1\end{bmatrix}}&<0\\{\begin{bmatrix}-Z&&C_{1}-D_{f}C_{2}&&-C_{f}\\\star &&-Y&&-X\\\star &&\star &&-X\end{bmatrix}}&<0\\Y-X>0\\trZ

Conclusion:

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