Fault Detection Filtering for Uncertain Itô Stochastic Fuzzy Systems With Time-Varying Delays

Abstract

This paper considers the problem of robust $H_{\infty }$ fault detection for a class of Itô stochastic Takagi–Sugeno fuzzy systems with time-varying delays and parameter uncertainties. The purpose is to design fuzzy-rule-independent and fuzzy-rule-dependent fault detection filters, which guarantee the fault detection system is not only mean square asymptotically stable, but also satisfies a prescribed $H_{\infty }$-norm level for all admissible uncertainties. Via the application of Lyapunov stability theory and the linear matrix inequality technique, novel delay-dependent solvability conditions are obtained. Weighting fault signal approach is utilized to improve the performance of the fault detection system, and explicit expression of the desired filter parameters is characterized by congruence transformation, matrix decomposition, and convex optimization technique. A numerical example and a mass-spring-damper mechanical system are employed to illustrate the usefulness and effectiveness of the proposed method.

Appendix

Proof of Theorem 1

Define the following Lyapunov function candidate for the system $({{{\tilde{\Sigma } }_c}})$ in (12) as follows:

$$\begin{aligned} V({\xi (t),\;t})&= {\xi ^\mathrm{T}}(t)P\xi (t) + \int _{t - {\tau _1}(t)}^t {{\xi ^\mathrm{T}}(s){H^\mathrm{T}}QH\xi (s)\mathrm{d}s}\\&\quad +\,\int _{t - {\tau _2}(t)}^t {{\xi ^\mathrm{T}}(s ){H^\mathrm{T}}RH\xi (s)\mathrm{d}s}. \end{aligned}$$

Using Itô formula in Lemma 1, we obtain the stochastic differential as

$$\begin{aligned}&\mathrm{d}V({\xi (t),\;t}) = \mathcal {L}V({\xi (t),\;t})\mathrm{d}t + 2{\xi ^\mathrm{T}}(t)P{{\tilde{E}}_i}(t)H\xi ({t - {\tau _2}(t)})\mathrm{d}\varpi , \nonumber \\&\mathcal {L}V({\xi (t),\;t})= 2\sum \limits _{i = 1}^r {{h_i}({\theta (t)} )} {\xi ^\mathrm{T}}(t) P\left[ {{{\tilde{A}}_i}(t)\xi (t) \!+\! {{\tilde{A}}_{1i}}(t)H\xi ({t - {\tau _1}(t)}) \!+\! {{\tilde{B}}_i}(t)v(t)} \right] \nonumber \\&\quad +\,{\left[ {\sum \limits _{i = 1}^r {{h_i}({\theta (t)})} {{\tilde{E}}_i}(t)H\xi ({t - {\tau _2}(t)})} \right] ^\mathrm{T}} P\left[ {\sum \limits _{i = 1}^r {{h_i}({\theta (t)} )} {{\tilde{E}}_i}(t)H\xi ({t - {\tau _2}(t)})} \right] \nonumber \\&\quad +\,{\xi ^\mathrm{T}}(t){H^\mathrm{T}}({Q + R})H\xi (t) - ({1 - {{\dot{\tau }}_1}(t)}) {\xi ^\mathrm{T}}({t - {\tau _1}(t)} ){H^\mathrm{T}}QH\xi ({t - {\tau _1}(t)}) \nonumber \\&\quad -\,({1 - {{\dot{\tau } }_2}(t)}){\xi ^\mathrm{T}}({t - {\tau _2}(t)}){H^\mathrm{T}}RH\xi ({t - {\tau _2}(t)}). \end{aligned}$$

(38)

By Lemma 2, we have

$$\begin{aligned} \frac{{\mathcal {L}V({\xi (t),\;t})}}{{\sum \nolimits _{i = 1}^r {{h_i}({\theta (t)} )} }}&\leqslant {\xi ^\mathrm{T}}(t)\left[ {P{{\tilde{A}}_i}(t) + {{\tilde{A}}_i}^\mathrm{T}(t)P + {H^\mathrm{T}}({Q + R})H} \right] \xi (t)\nonumber \\&\quad +\,2{\xi ^\mathrm{T}}(t)P{{\tilde{A}}_{1i}}(t)x({t - {\tau _1}(t)})\nonumber \\&\quad +\, 2{\xi ^\mathrm{T}}(t)P{{\tilde{B}}_i}(t)v(t) + {x^\mathrm{T}}({t - {\tau _2}(t)} ){{\tilde{E}}^\mathrm{T}}_i(t)P{{\tilde{E}}_i}(t)x({t - {\tau _2}(t)}) \nonumber \\&\quad -\,({1 - {\mu _1}}){x^\mathrm{T}}({t - {\tau _1}(t)})Qx({t - {\tau _1}(t)}) \nonumber \\&\quad -\,({1 - {\mu _2}}){x^\mathrm{T}}({t - {\tau _2}(t)})Rx({t - {\tau _2}(t)}), \end{aligned}$$

(39)

where (5) and the relationship

$$\begin{aligned} x({t - {\tau _1}(t)}) = H\xi ({t - {\tau _1}(t)}),\quad x({t - {\tau _2}(t)}) = H\xi ({t - {\tau _2}(t)}) \end{aligned}$$

are used. From (21), it is easy to see

$$\begin{aligned} {P^{ - 1}} - \varepsilon _2^{-1}{{\tilde{M}}_{1i}}{{\tilde{M}}^\mathrm{T}}_{1i} > 0. \end{aligned}$$

(40)

Noting (7) and (40) and using Lemmas 3 and 4, we have

$$\begin{aligned}&2{\xi ^\mathrm{T}}(t)P\left[ {\Delta {{\tilde{A}}_i}(t)\xi (t) + \Delta {{\tilde{A}}_{1i}}(t)x({t - {\tau _1}(t)}) + \Delta {{\tilde{B}}_i}(t)v(t)} \right] \nonumber \\&\quad = 2{\xi ^\mathrm{T}}(t)P{{\tilde{M}}_{1i}}{G_i}(t)\left[ {{{\tilde{N}}_{1i}}\xi (t) + {N_{2i}}x({t - {\tau _1}(t)}) + {{\tilde{N}}_{3i}}v(t)} \right] \nonumber \\&\quad \leqslant {\xi ^\mathrm{T}}(t)\left[ {\varepsilon _1^{-1}P{{\tilde{M}}_{1i}}{{\tilde{M}}^\mathrm{T}}_{1i}P + {\varepsilon _1}{{\tilde{N}}_{1i}}^\mathrm{T}{{\tilde{N}}_{1i}}} \right] \xi (t)\nonumber \\&\quad \quad +\,{x^\mathrm{T}}({t - {\tau _1}(t)})\left( {{\varepsilon _1}{N_{2i}}^\mathrm{T}{N_{2i}}}\right) x({t - {\tau _1}(t)}) \nonumber \\&\quad \quad +\,{v^\mathrm{T}}(t)\left( {{\varepsilon _1}{{\tilde{N}}_{3i}}^\mathrm{T}{{\tilde{N}}_{3i}}}\right) v(t) + 2{\xi ^\mathrm{T}}(t)\left( {{\varepsilon _1}{{\tilde{N}}_{1i}}^\mathrm{T}{ N_{2i}}}\right) x({t - {\tau _1}(t)}) \nonumber \\&\quad \quad +\,2{\xi ^\mathrm{T}}(t)\left( {{\varepsilon _1}{{\tilde{N}}_{1i}}^\mathrm{T}{\tilde{N}_{3i}}}\right) v(t) + 2{x^\mathrm{T}}({t - {\tau _1}(t)})\left( {{\varepsilon _1}{N_{2i}}^\mathrm{T}{{\tilde{N}}_{3i}}}\right) v(t), \end{aligned}$$

(41)

and

$$\begin{aligned} {{\tilde{E}}^\mathrm{T}}_i(t)P{{\tilde{E}}_i}(t) \leqslant {{\tilde{E}}^\mathrm{T}}_i{\left( {{P^{ - 1}} - \varepsilon _2^{-1}{{\tilde{M}}_{1i}}{{{\tilde{M}}_{1i}}^\mathrm{T}}}\right) ^{ - 1}}{{\tilde{E}}_i} + {\varepsilon _2}{N_{5i}}^\mathrm{T}{N_{5i}}. \end{aligned}$$

(42)

Substituting (41) and (42) into (39) with $v(t) = 0$ results in

$$\begin{aligned} \mathcal {L}V({\xi (t),\;t}) \leqslant \sum \limits _{i = 1}^r {{h_i}({\theta (t)} )} \left[ {{\eta ^\mathrm{T}}(t)\Theta _i \eta (t)} \right] , \end{aligned}$$

(43)

where ${\eta ^\mathrm{T}}(t) = \left[ {{\xi ^\mathrm{T}}(t)\;\;{x^\mathrm{T}}({t - {\tau _1}(t)} )\;\;{x^\mathrm{T}}({t - {\tau _2}(t)})} \right]$ and

$$\begin{aligned} \Theta _i = \left[ {\begin{array}{c@{\quad }c@{\quad }c} {{\Upsilon _{11i}}} &{} {{\Upsilon _{12i}}} &{} 0 \\ * &{} {{\Upsilon _{22i}}} &{} 0 \\ * &{} * &{} {{\Upsilon _{33i}} + {{\tilde{E}}^\mathrm{T}}_i{{\left( {{P^{ - 1}} - \varepsilon _2^{-1}{{\tilde{M}}_{1i}}{{{\tilde{M}}_{1i}}^\mathrm{T}}}\right) }^{ - 1}}{{\tilde{E}}_i}} \\ \end{array} } \right] . \end{aligned}$$

By the Schur complement formula, it follows from (21) that $\Theta _i < 0$, which together with (43) implies

$$\begin{aligned} \mathcal {L}V({\xi (t),\;t}) < 0, \end{aligned}$$

(44)

for all

$$\begin{aligned} \eta (t) = {\left[ {{\xi ^\mathrm{T}}(t)\;\;{x^\mathrm{T}}({t - {\tau _1}(t)} )\;\;{x^\mathrm{T}}({t - {\tau _2}(t)})} \right] ^\mathrm{T}} \ne 0. \end{aligned}$$

Therefore, we have that the fuzzy stochastic FD system $({{{\tilde{\Sigma } }_c}})$ in (12) with $v(t) = 0$ is asymptotically mean square stable for all admissible uncertainties.

Now, we will establish the $H_{\infty }$ performance for the fuzzy stochastic FD system $({{{\tilde{\Sigma } }_c}})$ in (12). Assuming zero initial condition, we have

$$\begin{aligned} \mathcal { J }&= \mathcal {E}\left\{ {\int _0^\infty {\left( {{e_c}^\mathrm{T}(t){e_c}(t)- {\gamma ^2}{v^\mathrm{T}}(t)v(t)}\right) } \mathrm{d}t} \right\} \\&\leqslant \mathcal {E}\left\{ {\int _0^\infty {\left( {{e_c}^\mathrm{T}(t){e_c}(t) - {\gamma ^2}{v^\mathrm{T}}(t)v(t)}\right) } \mathrm{d}t}\right\} \\&\quad +\,\mathcal {E}\left\{ {V\left( {\xi (\infty ),\;\infty }\right) }\right\} - \mathcal {E}\{ {V({0,\;0})} \} \\&= \mathcal {E}\left\{ {\int _0^\infty {\left[ {{e_c}^\mathrm{T}(t){e_c}(t) - {\gamma ^2}{v^\mathrm{T}}(t)v(t) + \mathcal {L}V({\xi (t),\;t})} \right] } \mathrm{d}t}\right\} . \end{aligned}$$

According to Lemma 2, we can find

$$\begin{aligned} {e_c}^\mathrm{T}(t){e_c}(t) \leqslant \sum \limits _{i = 1}^r {{h_i} ({\theta (t)})} \left[ {{\xi ^\mathrm{T}}(t){{\tilde{C}}^\mathrm{T}}_i{{\tilde{C}}_i}\xi (t) + 2{\xi ^\mathrm{T}}(t){{\tilde{C}}^\mathrm{T}}_i{{\tilde{D}}_i}v(t) + {v^\mathrm{T}}(t){{\tilde{D}}^\mathrm{T}}_i{{\tilde{D}}_i}v(t)} \right] . \end{aligned}$$

It follows from (41), (42), that

$$\begin{aligned} {e_c}^\mathrm{T}(t){e_c}(t) - {\gamma ^2}{v^\mathrm{T}}(t)v(t) + \mathcal {L}V({\xi (t),\;t}) \leqslant \sum \limits _{i = 1}^r {{h_i}({\theta (t)})} {\psi ^\mathrm{T}}(t){\Omega _i}{\psi }(t), \end{aligned}$$

where ${\psi ^\mathrm{T}}(t) = \left[ {{\xi ^\mathrm{T}}(t)\;\;{x^\mathrm{T}}({t - {\tau _2}(t)} )\;\;{v^\mathrm{T}}(t)} \right]$ and

$$\begin{aligned} {\Omega _i} = \left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {{\Upsilon _{11i}} + {{\tilde{C}}^\mathrm{T}}_i{{\tilde{C}}_i}} &{} {{\Upsilon _{12i}}} &{} 0 &{} {{\Upsilon _{14i}}}+ {{\tilde{C}}_i}^\mathrm{T}{{\tilde{D}}_i}\\ * &{} {{\Upsilon _{22i}}} &{} 0 &{} {{\Upsilon _{24i}}} \\ * &{} * &{} {{\Upsilon _{33i}} - {{\tilde{E}}^\mathrm{T}}_i\Upsilon _{55i}^{ - 1}\tilde{E}_i} &{} 0 \\ * &{} * &{} * &{} {{\Upsilon _{44i}}}+ {{\tilde{D}}^\mathrm{T}}_i{{\tilde{D}}_i}\\ \end{array} } \right] . \end{aligned}$$

By Schur complement, (21) implies ${\Omega _i} < 0$, and thus

$$\begin{aligned} \mathcal {J} = \mathcal {E}\left\{ {\int _0^\infty {\left[ {{e_c}^\mathrm{T}(t){e_c}(t) - {\gamma ^2}{v^\mathrm{T}}(t)v(t)} \right] }\mathrm{d}t }\right\} < 0, \end{aligned}$$

which implies (18). The $H_{\infty }$ performance has been established and the proof is completed. $\square$

Proof of Theorem 2

By Schur complement, the matrix inequality condition (21) in Theorem 1 can be described as the following matrix inequality:

$$\begin{aligned} {\Phi _i} = \left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {{\Phi _{11i}}} &{} {{\Phi _{12i}}} &{} 0 &{} {{\Phi _{14i}}} &{} 0 &{} {{{\tilde{C}}_i}^\mathrm{T}} &{} 0 &{} 0 &{} {P{{\tilde{M}}_{1i}}} &{} {{{\tilde{N}}_{1i}}^\mathrm{T}} \\ * &{} {{\Phi _{22}}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {{N_{2i}}^\mathrm{T}} \\ * &{} * &{} {{\Phi _{33}}} &{} 0 &{} {{{\tilde{E}}^\mathrm{T}}_i} &{} 0 &{} {{N_{5i}}^\mathrm{T}} &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} { - {\gamma ^2}I} &{} 0 &{} {{{\tilde{D}}_i}^\mathrm{T}} &{} 0 &{} 0 &{} 0 &{} {{{\tilde{N}}_{3i}}^\mathrm{T}} \\ * &{} * &{} * &{} * &{} { - {P^{ - 1}}} &{} 0 &{} 0 &{} {{{\tilde{M}}_{1i}}} &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} { - I} &{} 0 &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} { - \varepsilon _2^{ - 1}I} &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - {\varepsilon _2}I} &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - {\varepsilon _1}I} &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - \varepsilon _1^{ - 1}I} \\ \end{array} } \right] < 0,\nonumber \\ \end{aligned}$$

(45)

where

$$\begin{aligned}&{\Phi _{11i}} = P{{\tilde{A}}_i} + {{\tilde{A}}_i}^\mathrm{T}P + {H^\mathrm{T}}({Q+R})H,\;\;{\Phi _{12i}} = P{{\tilde{A}}_{1i}} ,\ {\Phi _{14i}} = P{{\tilde{B}}_i} ,\\&{\Phi _{22}} = - ({1 - {\mu _1}})Q,\ {\Phi _{33}} = - ({1 - {\mu _2}})R. \end{aligned}$$

Performing a congruence transformation to (45) by diagonal matrix ${\text {diag}}(I,\;I,\;I,\;I,P,\;I,\;I,\;I,\;I,\;I )$, we obtain

$$\begin{aligned} \left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {{\Phi _{11i}}} &{} {{\Phi _{12i}}} &{} 0 &{} {{\Phi _{14i}}} &{} 0 &{} {{{\tilde{C}}_i}^\mathrm{T}} &{} 0 &{} 0 &{} {P{{\tilde{M}}_{1i}}} &{} {{{\tilde{N}}_{1i}}^\mathrm{T}} \\ * &{} {{\Phi _{22}}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {{N_{2i}}^\mathrm{T}} \\ * &{} * &{} {{\Phi _{33}}} &{} 0 &{} {{{\tilde{E}}^\mathrm{T}}_iP} &{} 0 &{} {{N_{5i}}^\mathrm{T}} &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} { - {\gamma ^2}I} &{} 0 &{}{{{\tilde{D}}_i}^\mathrm{T}} &{} 0 &{} 0 &{} 0 &{} {{{\tilde{N}}_{3i}}^\mathrm{T}} \\ * &{} * &{} * &{} * &{} { - P} &{} 0 &{} 0 &{} {P{{\tilde{M}}_{1i}}} &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} { - I} &{} 0 &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} { - \varepsilon _2^{ - 1}I} &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - {\varepsilon _2}I} &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - {\varepsilon _1}I} &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - \varepsilon _1^{ - 1}I} \\ \end{array} } \right] < 0.\nonumber \\ \end{aligned}$$

(46)

Let $P \triangleq \mathrm{diag}({U,\;V}) > 0$ in (46), where $U \in {\mathbb {R}^{2n \times 2n}}$ and $V \in {\mathbb {R}^{k \times k}}$; we get a new result. Specially, given a scalar $\gamma >0$, the fuzzy stochastic fault detection system $({{{\tilde{\Sigma } }_c}})$ in (12) is asymptotically mean square stable with an ${H_\infty }$ performance level $\gamma$ if there exist $U>0$ and $V>0$ such that the following LMI holds:

$$\begin{aligned} \left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {{\Psi _{11i}}} &{} 0 &{} {{\Psi _{13i}}} &{} 0 &{} {{\Psi _{15i}}} &{} 0 &{} {{{\hat{C}}_i}^\mathrm{T}} &{} 0 &{} 0 &{} {U{{\hat{M}}_{1i}}} &{} {{{\hat{N}}_{1i}}^\mathrm{T}} \\ * &{} {{\Psi _{22}}} &{} 0 &{} 0 &{} {{\Psi _{25i}}} &{} 0 &{} { - {C_w}^\mathrm{T}} &{} 0 &{} 0 &{} 0 &{} 0 \\ * &{} * &{} {{\Psi _{33}}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {{N_{2i}}^\mathrm{T}} \\ * &{} * &{} * &{} {{\Psi _{44}}} &{} 0 &{} {{{\hat{E}^{\mathrm{T}}}_i}U} &{} 0 &{} {{N_{5i}}^\mathrm{T}} &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} { - {\gamma ^2}I} &{} 0 &{} {{{\tilde{D}}_i}^\mathrm{T}} &{} 0 &{} 0 &{} 0 &{} {{{\tilde{N}}_{3i}}^\mathrm{T}} \\ * &{} * &{} * &{} * &{} * &{} { - U} &{} 0 &{} 0 &{} {U{{\hat{M}}_{1i}}} &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} { - I} &{} 0 &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - \varepsilon _2^{ - 1}I} &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - {\varepsilon _2}I} &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - {\varepsilon _1}I} &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - \varepsilon _1^{ - 1}I} \\ \end{array} } \right] < 0,\nonumber \\ \end{aligned}$$

(47)

where

$$\begin{aligned} {\Psi _{11i}}&= U{{\hat{A}}_i} + {{\hat{A}}^\mathrm{T}}_iU + W,\ {\Psi _{22}} = V{A_w} + A_w^\mathrm{T}V,\ {\Psi _{33}} = {\Phi _{22}}= - ({1 - {\mu _1}})Q,\\ {\Psi _{44}}&= {\Phi _{33}}= - ({1 - {\mu _2}})R,\ {\Psi _{13i}} = U{{\hat{A}}_{1i}} , {\Psi _{15i}} = U{{\hat{B}}_i} ,\ {\Psi _{25i}} = V{{\hat{B}}_w} , \\ {{\hat{A}}_i}&= \left[ {\begin{array}{c@{\quad }c} {{A_{i}}} &{} 0 \\ {{B_c}{C_i}} &{} {{A_c}} \\ \end{array} } \right] ,\quad {{\hat{A}}_{1i}} = \left[ {\begin{array}{c} {{A_{1i}}} \\ {{B_c}{C_{1i}}} \\ \end{array} } \right] ,\quad {{\hat{B}}_i} = \left[ {\begin{array}{c@{\quad }c@{\quad }c} {{B_{0i}}} &{} {{B_i}} &{} {{B_{1i}}} \\ {{B_c}{D_{0i}}} &{} {{B_c}{D_i}} &{} {{B_c}{D_{1i}}} \\ \end{array} } \right] , \\ {{\hat{B}}_w}&= \left[ {\begin{array}{c@{\quad }c@{\quad }c} 0 &{} 0 &{} {{B_w}} \\ \end{array} } \right] ,\quad {{\hat{E}}_i} = \left[ {\begin{array}{c} {{E_i}} \\ {{B_c}{F_i}} \\ \end{array} } \right] ,\quad W = \left[ {\begin{array}{c@{\quad }c} {Q + R} &{} 0 \\ 0 &{} 0 \\ \end{array} } \right] , \\ {{\hat{N}}_{1i}}&= \left[ {\begin{array}{c@{\quad }c} {{N_{1i}}} &{} 0 \\ \end{array} } \right] , \quad {{\hat{M}}^\mathrm{T}}_{1i} = \left[ {\begin{array}{c@{\quad }c} {{M_{1i}}^\mathrm{T}} &{} {{{({{B_c}{M_{2i}}})}^\mathrm{T}}} \\ \end{array} } \right] , \quad {{\hat{C}}_i} = \left[ {\begin{array}{c@{\quad }c} 0 &{} {{C_c}} \\ \end{array} } \right] . \end{aligned}$$

Now, partition $U$ as

$$\begin{aligned} U = \left[ {\begin{array}{c@{\quad }c} {{U_1}} &{} {{U_2}} \\ * &{} {{U_3}} \\ \end{array} } \right] > 0, \end{aligned}$$

(48)

where ${U_k} \in {\mathbb {R}^{n \times n}},\;k = 1,\;2,\;3.$

Without loss of generality, we assume $U_2$ is nonsingular; if not, $U_2$ may be perturbed by $\Delta {U_2}$ with sufficiently small norm such that $U_2+\Delta {U_2}$ is nonsingular and satisfying (47). Define the following matrices that are also nonsingular:

$$\begin{aligned} \aleph = \left[ {\begin{array}{c@{\quad }c} I &{} 0 \\ 0 &{} {{U_3}^{ - 1}{U_2}^\mathrm{T}} \\ \end{array} } \right] ,\quad \mathcal {U }= {U_1},\quad \mathcal {V }= {U_2}{U_3}^{ - 1}{U_2}^\mathrm{T}. \end{aligned}$$

(49)

and

$$\begin{aligned} \left[ {\begin{array}{c@{\quad }c} {{\mathcal {A}_c}} &{} {{\mathcal {B}_c}} \\ {{\mathcal {C}_c}} &{} 0 \\ \end{array} } \right] = \left[ {\begin{array}{c@{\quad }c} {{U_2}} &{} 0 \\ 0 &{} I \\ \end{array} } \right] \left[ {\begin{array}{c@{\quad }c} {{A_c}} &{} {{B_c}} \\ {{C_c}} &{} 0 \\ \end{array} } \right] \left[ {\begin{array}{c@{\quad }c} {{U_3}^{ - 1}{U_2}^\mathrm{T}} &{} 0 \\ 0 &{} I \\ \end{array} } \right] = \left[ {\begin{array}{c@{\quad }c} {{U_2}{A_c}{U_3}^{ - 1}{U_2}^\mathrm{T}} &{} {{U_2}{B_c}} \\ {{C_c}{U_3}^{ - 1}{U_2}^\mathrm{T}} &{} 0 \\ \end{array} } \right] .\nonumber \\ \end{aligned}$$

(50)

Performing a congruence transformation to (47) by diagonal matrix $diag(\aleph ,\;I,I,I,I,\aleph ,I,I,I,I,I)$, we get

$$\begin{aligned} \left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {{{\tilde{\Psi }}_{11i}}} &{} 0 &{} {{{\tilde{\Psi } }_{13i}}} &{} 0 &{} {{{\tilde{\Psi } }_{15i}}} &{} 0 &{} {{\aleph ^\mathrm{T}}{{\hat{C}}_i}^\mathrm{T}} &{} 0 &{} 0 &{} {{\aleph ^\mathrm{T}}U{{\hat{M}}_{1i}}} &{} {{\aleph ^\mathrm{T}}{{\hat{N}}_{1i}}^\mathrm{T}} \\ * &{} {{\Psi _{22}}} &{} 0 &{} 0 &{} {{\Psi _{25i}}} &{} 0 &{} { - {C_w}^\mathrm{T}} &{} 0 &{} 0 &{} 0 &{} 0 \\ * &{} * &{} {{\Psi _{33}}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {{N_{2i}}^\mathrm{T}} \\ * &{} * &{} * &{} {{\Psi _{44}}} &{} 0 &{} {{{\hat{E}^{\mathrm{T}}}_i}U\aleph } &{} 0 &{} {{N_{5i}}^\mathrm{T}} &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} { - {\gamma ^2}I} &{} 0 &{} {{{\tilde{D}}_i}^\mathrm{T}} &{} 0 &{} 0 &{} 0 &{} {{{\tilde{N}}_{3i}}^\mathrm{T}} \\ * &{} * &{} * &{} * &{} * &{} { - {\aleph ^\mathrm{T}}U\aleph } &{} 0 &{} 0 &{} {{\aleph ^\mathrm{T}}U{{\hat{M}}_{1i}}} &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} { - I} &{} 0 &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - \varepsilon _2^{ - 1}I} &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - {\varepsilon _2}I} &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - {\varepsilon _1}I} &{} 0 \\ * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} * &{} { - \varepsilon _1^{ - 1}I} \\ \end{array} } \right] < 0,\nonumber \\ \end{aligned}$$

(51)

where

$$\begin{aligned} {{\tilde{\Psi } }_{11i}}&= {\aleph ^\mathrm{T}}({U{{\hat{A}}_i} + {{\hat{A}}^\mathrm{T}}_iU + W}) \nonumber \\ \aleph&= \left[ {\begin{array}{c@{\quad }c@{\quad }c} {\mathcal {U}{A_i} \!+\! {A_i}^\mathrm{T}\mathcal {U} \!+\! {\mathcal {B}_c}{C_i} \!+\! {C_i}^\mathrm{T}{\mathcal {B}_c}^\mathrm{T} \!+\! Q \!+\! R} &{} {{\mathcal {A}_c} \!+\! {A_i}^\mathrm{T}\mathcal {V} + {C_i}^\mathrm{T}{\mathcal {B}_c}^\mathrm{T}} \\ * &{} {{\mathcal {A}_c} + {\mathcal {A}_c}^\mathrm{T}} \\ \end{array} } \right] , \nonumber \\ {{\tilde{\Psi } }_{13i}}&= {\aleph ^\mathrm{T}}({U{{\hat{A}}_{1i}} } ) = \left[ {\begin{array}{c} {\mathcal {U}{A_{1i}} + {\mathcal {B}_c}{C_{1i}} } \\ {\mathcal {V}{A_{1i}} + {\mathcal {B}_c}{C_{1i}}} \\ \end{array} } \right] , \ - {\aleph ^\mathrm{T}}U\aleph = \left[ {\begin{array}{c@{\quad }c} { - \mathcal {U}} &{} { - \mathcal {V}} \\ * &{} { - \mathcal {V}} \\ \end{array} } \right] ,\nonumber \\ {{\tilde{\Psi } }_{15i}}&= {\aleph ^\mathrm{T}}({U{{\hat{B}}_i} } ) = \left[ {\begin{array}{c@{\quad }c@{\quad }c} {\mathcal {U}{B_{0i}} + {\mathcal {B}_c}{D_{0i}} } &{} {\mathcal {U}{B_i} + {\mathcal {B}_c}{D_i}} &{} {\mathcal {U}{B_{1i}} + {\mathcal {B}_c}{D_{1i}} } \\ {\mathcal {V}{B_{0i}} + {\mathcal {B}_c}{D_{0i}}} &{} {\mathcal {V}{B_i} + {\mathcal {B}_c}{D_i}} &{} {\mathcal {V}{B_{1i}} + {\mathcal {B}_c}{D_{1i}} } \\ \end{array} } \right] , \nonumber \\ {\aleph ^\mathrm{T}}{{\hat{C}}_i}^\mathrm{T}&= \left[ {\begin{array}{c} 0 \\ {{\mathcal {C}_c}^\mathrm{T}} \\ \end{array} } \right] ,\nonumber \\ {\aleph ^\mathrm{T}}U{{\hat{M}}_{1i}}&= \left[ {\begin{array}{c} {\mathcal {U}{M_{1i}} + {\mathcal {B}_c}{M_{2i}}} \\ {\mathcal {V}{M_{1i}} + {\mathcal {B}_c}{M_{2i}}} \\ \end{array} } \right] ,\quad {\aleph ^\mathrm{T}}{{\hat{N}}_{1i}^\mathrm{T}} = \left[ {\begin{array}{c} {N_{1i}^\mathrm{T}} \\ 0 \\ \end{array} } \right] ,\nonumber \\ {{\hat{E}}^\mathrm{T}}_iU\aleph&= \left[ {\begin{array}{c@{\quad }c} {E_i^\mathrm{T}\mathcal {U} + F_i^\mathrm{T}{\mathcal {B}_c}^\mathrm{T}} &{} {E_i^\mathrm{T}\mathcal {V} + F_i^\mathrm{T}{\mathcal {B}_c}^\mathrm{T}} \\ \end{array} } \right] . \end{aligned}$$

(52)

Considering (52), we can get LMI (24) from (51). Moreover, note that (50) is equivalent to

$$\begin{aligned} \left[ {\begin{array}{c@{\quad }c} {{A_c}} &{} {{B_c}} \\ {{C_c}} &{} 0 \\ \end{array} } \right]&= \left[ {\begin{array}{c@{\quad }c} {U_2^{ - 1}} &{} 0 \\ 0 &{} I \\ \end{array} } \right] \left[ {\begin{array}{c@{\quad }c} {{\mathcal {A}_c}} &{} {{\mathcal {B}_c}} \\ {{\mathcal {C}_c}} &{} 0 \\ \end{array} } \right] \left[ {\begin{array}{c@{\quad }c} {{U_2}^{ - T}{U_3}} &{} 0 \\ 0 &{} I \\ \end{array} } \right] \nonumber \\&= \left[ {\begin{array}{c@{\quad }c} {{{({{U_2}^{ - T}{U_3}})}^{ - 1}}} &{} 0 \\ 0 &{} I \\ \end{array} } \right] \left[ {\begin{array}{c@{\quad }c} {{\mathcal {V}^{ - 1}}} &{} 0 \\ 0 &{} I \\ \end{array} } \right] \left[ {\begin{array}{c@{\quad }c} {{\mathcal {A}_c}} &{} {{\mathcal {B}_c}} \\ {{\mathcal {C}_c}} &{} 0 \\ \end{array} } \right] \left[ {\begin{array}{c@{\quad }c} {{U_2}^{ - T}{U_3}} &{} 0 \\ 0 &{} I \\ \end{array} } \right] \end{aligned}$$

(53)

Also note that the filter matrices $A_{c}, B_{c}$, and $C_{c}$ in (10) can be written as (53), which implies that ${{U_2}^{ - T}{U_3}}$ can be viewed as a similarity transformation on the state-space realization of the filter and, as such, has no effect on the filter mapping from $y$ to $\chi _{c}$. Without loss of generality, we set ${{U_2}^{ - T}{U_3}}=I$ and thus obtain (26). Therefore, the filter $({{{ \Sigma }_c}})$ in (10) can be constructed by (26). This completes the proof. $\square$