Linear Systems

7.4    Sparse Structured H_∞ Filter Design

In this section, the non-fragile H_∞ filtering problem with sparse structure is studied.

7.4.1    Problem Statement

First, a sparse structure is defined. Consider system (7.1) and a full-order filter given by

$\begin{array}{c}\xi \left(k+1\right)=A_F\xi \left(k\right)+B_{Fy}\left(k\right),\\ z_F\left(k\right)=C_F\xi \left(k\right),\end{array}$

(7.46)

where ξ(k) ∈ Rⁿ is the filter state, z_F(k) is the estimation of z(k), and the constant matrices A_F, B_F, and C_F are filter matrices to be designed.

First, a method of finding a class of feasible sparse structures for the designed filter to meet a prescribed H_∞ performance requirement is presented as follows.

Denote

$G_{0z_e\omega }=C_{e0}{\left(zI-A_{e0}\right)}^{-1}{B}_{e0},$

(7.47)

where

$\begin{array}{c}{A}_{e0}=\left[\begin{array}{cc}A& 0\\ B_F{C}_2& A_F\end{array}\right],\\ B_{e0}=\left[\begin{array}{c}{B}_1\\ B_F{D}_{21}\end{array}\right],C_{e0}=\left[\begin{array}{ccc}{C}_1& -C_F& -\text{Δ}{C}_F\end{array}\right].\end{array}$

Let filter gain matrices A_F,