Robust Fault Detection Filter Design for Discrete-Time Fuzzy Models

Abstract

This New fault detection observer design conditions for discrete-time fuzzy systems with unmeasurable premise variables are proposed. In this study, the considered Takagi-Sugeno (T-S) fuzzy system is subject to sensor faults and unknown bounded disturbances. The T-S observer is used to estimate jointly states and faults by means of a mixed $H_{ - } /H_{\infty }$ performance index. Using the technique of descriptor system representation new conditions are proposed in terms of a Linear Matrix Inequality (LMI) by considering the sensor fault as an auxiliary state variable. Simulation results are presented to demonstrate the effectiveness of the approach.

Appendix A

The following Lemmas are required in the development of the Theorem 1 proof.

Lemma A.1.

If $\Phi > 0$, then

$$ \Xi _{\text{i}}^{\text{T}}\Phi \Xi _{\text{j}} +\Xi _{\text{j}}^{\text{T}} {\Phi} {\Xi }_{\text{i}} \le\Xi _{\text{i}}^{\text{T}} {\Phi} {\Xi }_{\text{i}} +\Xi _{\text{j}}^{\text{T}} {\Phi} {\Xi }_{\text{j}} $$

(A.1)

Lemma A.2.

if ${\text{Z}} > 0$, then

$$ {\text{SZ}}^{ - 1} {\text{S}}^{\text{T}} \ge {\text{S}}^{\text{T}} + {\text{S}} - {\text{Z}} $$

(A.2)

Lemma A.3 [13].

Consider two real matrices X and Y with appropriate dimensions, for any positive scalar $\Omega$ the following inequality is verified:

$$ {\text{X}}^{\text{T}} {\text{Y}} + {\text{Y}}^{\text{T}} {\text{X}} \le {\text{X}}^{\text{T}}\Omega {\text{X}} + {\text{Y}}^{\text{T}}\Omega ^{ - 1} {\text{Y}}\quad\Omega > 0 $$

(A.3)

Lemma B [14].

Consider the system (B.1). If there exist symmetric matrices ${\text{X}}_{\text{i}}$ and any matrices ${\text{L}}$, such that the following stability conditions are satisfied for ${\text{i}},{\text{j}},{\text{l}} = 1, \cdots ,{\text{r}}$

$$ \Pi _{\text{l}} < 0, {\text{X}}_{\text{i}} > 0, \quad (i,l \in {\mathcal{R}}) $$

(B.1)

where

$$ \Pi _{\text{l}} = \left[ {\begin{array}{*{20}c} {{\mathcal{Q}}_{{11{\text{l}}}} } & * & \cdots & * \\ {{\mathcal{Q}}_{{21{\text{l}}}} } & {{\mathcal{Q}}_{{22{\text{l}}}} } & \cdots & * \\ \vdots & \vdots & \ddots & \vdots \\ {{\mathcal{Q}}_{{{\text{r}}1{\text{l}}}} } & {{\mathcal{Q}}_{{{\text{r}}2{\text{l}}}} } & \cdots & {{\mathcal{Q}}_{\text{rrl}} } \\ \end{array} } \right] $$

(B.2)

$$ {\mathcal{Q}}_{\text{ijl}} = \left[ {\begin{array}{*{20}c} {\frac{1}{2}\left( {\Gamma _{\text{i}} +\Gamma _{\text{j}} } \right)} & * \\ {\frac{1}{2}\left( {{\text{A}}_{\text{i}} + {\text{A}}_{\text{j}} } \right){\text{L}}} & { - {\text{X}}_{\text{l}} } \\ \end{array} } \right] $$

(B.3)

$$ \Gamma _{\text{i}} = {\text{X}}_{\text{i}} - {\text{L}}^{\text{T}} - {\text{L}} $$

(B.4)

Then the closed loop fuzzy model is globally asymptotically stable.