LMIs in Control/Click here to continue/Optimal control systems/Discrete time H2 optimal filter

Introduction[edit | edit source]

The goal of optimal filtering is to design a filter that acts on the output z of the generalized plant and optimizes the transfer matrix from w to the filtered output. An H2-optimal filter is designed to minimize the $H_{2}$ norm of ${\overline {P}}(s)$ (will be defined below).

System Dynamics[edit | edit source]

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

${\begin{aligned}&\quad \quad {\begin{cases}X_{k+1}=A_{d}X_{k}+B_{d1}w_{k},\\Z_{k}=C_{d1}X_{k}+D_{d11}w_{k},\\Y_{k}=C_{d2}X_{k}+D_{d21}w_{k},\\\end{cases}}\end{aligned}}$

where it is assumed that A_{d} is Schur. A discrete-time dynamics LTI filter with state-space realization

${\begin{aligned}&\quad \quad {\begin{cases}X_{f,k+1}=A_{f}X_{f,k}+B_{f}Y_{k},\\{\hat {Z}}_{k}=C_{f}X_{f,k}+D_{f}y_{k},\\\end{cases}}\end{aligned}}$

is to be designed to optimize the transfer function from w{k} to ${\overline {z}}_{k}=z_{k}-{\hat {z}}_{k}$ , given by ${\overline {P}}(z)={\overline {C}}(d1)(zI-{\overline {A}}_{d})^{-1}{\overline {B}}_{d1}+{\overline {D}}_{d11}$ ,

where

${\overline {A}}_{d}={\begin{bmatrix}A_{d}&0\\B_{f}C_{d2}&A_{f}\end{bmatrix}},{\overline {B}}_{d1}={\begin{bmatrix}B_{d1}\\B_{f}D_{d21}\end{bmatrix}},{\overline {C}}_{d1}={\begin{bmatrix}C_{d1}-D_{f}C_{d2}-C_{f}\\\end{bmatrix}},{\overline {d}}_{d11}=D_{d11}-D_{f}D_{d21}.$

The Optimization Problem[edit | edit source]

Solve for $A_{n}\in \mathbb {R} ^{n_{x}Xn_{x}},B_{n}\in \mathbb {R} ^{n_{x}Xn_{y}},C_{f}\in \mathbb {R} ^{n_{z}Xn_{x}},D_{f}\in \mathbb {R} ^{n_{z}Xn_{y}},X,Y\in \mathbb {S} ^{n_{z}},and$ $v\in \mathbb {R} >0$ that minimize $J(v)=v$ subject to $X>0,Y>0,Z>0.$

LMI[edit | edit source]

${\begin{bmatrix}YA+A^{T}Y+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&A_{n}+C_{2}^{T}B_{n}^{T}+A^{T}X&YB_{1}+B_{n}D_{21}\\*&A_{n}+A_{n}^{T}&XB_{1}+B_{n}D_{21}\\*&*&-I\end{bmatrix}}$ < 0 ,