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

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

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

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

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

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

$$\begin{bmatrix} -Z& C_{1}-D_{f}C_{2}&-C_{f}\\* &-Y& -X\\*&*&-X \end{bmatrix}$$ < 0 ,

$$ D_{11}-D_{f}D_{21}=0, $$

$$

Y-X>0, $$

$$

tr(Z)<v. $$

Conclusion
The filter is recovered by the state-space matrices $$ A_{f} = X^{-1}A_{n}, Bf = X^{-1}B_{n}, C_{f}, and D_{f}.$$