7.4 Sparse Structured *H*_{∞} Filter Design

In this section, the non-fragile *H*_{∞} filtering problem with *sparse structure* is studied.

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

$\begin{array}{c}\xi \left(k+1\right)={A}_{F}\xi \left(k\right)+{B}_{Fy}\left(k\right),\\ {z}_{F}\left(k\right)={C}_{F}\xi \left(k\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$ |
(7.46) |

where *ξ*(*k*) ∈ *R*^{n} 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

${G}_{0{z}_{e}\omega}={C}_{e0}{\left(zI-{A}_{e0}\right)}^{-1}{B}_{e0},$ |
(7.47) |

where

$\begin{array}{c}{A}_{e0}=\left[\begin{array}{cc}A& 0\\ {B}_{F}{C}_{2}& {A}_{F}\end{array}\right],\\ {B}_{e0}=\left[\begin{array}{c}{B}_{1}\\ {B}_{F}{D}_{21}\end{array}\right],{C}_{e0}=\left[\begin{array}{ccc}{C}_{1}& -{C}_{F}& -\text{\Delta}{C}_{F}\end{array}\right].\end{array}$

Let filter gain matrices *A*_{F},

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