Fault Detection and Isolation of T–S Fuzzy Systems with Time-Delay Using Geometric Approach

Abstract

This paper investigates the fault detection and isolation problem for Takagi–Sugeno fuzzy time-delay systems under the framework of geometric method. First, the geometric concept of unobservability subspace is introduced and the algorithm for constructing the subspace is presented. Then, based on the above algorithm, the faults are divided into the different unobservability subspaces, and the reduced-order subsystems set corresponding to the fault set is constructed according to the geometrical properties of factor space and canonical projection. Finally, a set of structured residuals which are sensitive to a single fault and decoupled from other faults is established by designing an observer for the subsystems. In addition, $H_\infty$ performance is introduced to attenuate the effects of disturbances and error signals caused by a time-varying delay on the residual. Simulation example verifies the effectiveness of the proposed method.

Appendices

Appendix A: Proof of Lemma 3

Proof

First, for any ${\mathscr {Z}}\in {\mathscr {J}}$ that satisfies $(A_i+D_iC){\mathscr {Z}}\subseteq {\mathscr {Z}}$ and $(A_{\tau i}+D_{\tau i} C){\mathscr {Z}}\subseteq {\mathscr {Z}}$, we have

$$\begin{aligned} (A_i+D_iC){\mathscr {Z}}&\subseteq (A_i+D_iC)({\mathscr {S}}+\bigcap _{i=1}^s(A_i^{-1}{\mathscr {Z}}\cap \ker C))\\&\subseteq (A_i+D_iC)({\mathscr {S}}+A_i^{-1}{\mathscr {Z}}\cap \ker C)\\&\subseteq (A_i+D_iC){\mathscr {S}}+A_i(A_i^{-1}{\mathscr {Z}}\cap \ker C)\\&\subseteq {\mathscr {S}}+{\mathscr {Z}}\subseteq {\mathscr {Z}}. \end{aligned}$$

The same as above, we have $(A_{\tau i}+D_{\tau }C){\mathscr {Z}}\subseteq {\mathscr {Z}}$, where ${\mathscr {S}}\subseteq {\mathscr {Z}}$, hence ${\mathscr {Z}}\in {\mathfrak {W}}^{s}$. then we have ${\mathscr {Z}}^*\in {\mathfrak {W}}^{s}$. In addition, obviously if ${\mathscr {V}}\subseteq {\mathscr {S}}$, then ${\mathscr {V}}\subseteq {\mathscr {Z}}^*$ since ${\mathscr {S}}\subseteq {\mathscr {Z}}^*$. Consider the case of ${\mathscr {S}}\subseteq {\mathscr {V}}$, we have

$$\begin{aligned} A_i^{-1}{\mathscr {V}}\cap \ker C&=(A_i+D_iC)^{-1}{\mathscr {V}}\cap \ker C,\\ A_{\tau i}^{-1}{\mathscr {V}}\cap \ker C&=(A_{\tau i}+D_{\tau i}C)^{-1}{\mathscr {V}}\cap \ker C. \end{aligned}$$

We consider the facts that ${\mathscr {V}}\subseteq (A_i+D_i)^{-1}{\mathscr {V}}$ and if ${\mathscr {S}}\in {\mathscr {V}}$, then ${\mathscr {S}}+({\mathscr {V}}\cap \ker C)={\mathscr {V}}\cap ({\mathscr {S}}+\ker C)$. Clearly that ${\mathscr {V}}\subseteq {\mathscr {Z}}_0$. From the two facts, it can be concluded that if ${\mathscr {V}}\subseteq {\mathscr {Z}}_{k}$, then

$$\begin{aligned} {\mathscr {Z}}_{k+1}\supseteq&({\mathscr {S}}+\bigcap _{i=1}^s(A_i^{-1}{\mathscr {V}}\cap \ker C))\\&\bigcap \left( {\mathscr {S}}+\bigcap _{i=1}^s(A_{\tau i}^{-1}{\mathscr {V}}\cap \ker C)\right) \\ \supseteq&({\mathscr {S}}+\bigcap _{i=1}^s((A_i+D)^{-1}{\mathscr {V}})\cap \ker C))\\&\bigcap \left. \left( {\mathscr {S}}+\bigcap _{i=1}^s((A_{\tau i} +D_\tau )^{-1}{\mathscr {V}})\cap \ker C\right) \right) \\ \supseteq&{\mathscr {S}}+({\mathscr {V}}\cap \ker C)={\mathscr {V}}. \end{aligned}$$

Consequently, ${\mathscr {V}}\subseteq {\mathscr {Z}}_{k+1}$, and ${\mathscr {V}}\subseteq {\mathscr {Z}}^*$, then we have ${\mathscr {Z}}^*=\ll {\mathscr {S}}+\ker C|(A_i+D_iC),(A_{\tau i}+D_{\tau i} C)\gg _{i\in \{1\cdots s\}}$. $\square$

Appendix B: Proof of Lemma 4

Proof

First let us prove the “if” part. If ${\mathscr {S}}\in {\mathfrak {W}}^{s}$ and ${\mathscr {S}}={\mathscr {Z}}^*$ hold, then it can be got that ${\mathscr {S}}=\ll {\mathscr {S}}+\ker C|(A_i+D_iC),(A_{\tau i}+D_{\tau i}C)\gg _{i\in \{1\cdots s\}}$ according to Lemma 3. From Lemma 2. one gets ${\mathscr {S}}\in {\mathfrak {S}}^{s}$.

Now let us prove the “only if” part, if ${\mathscr {S}}$ is an unobservability subspace, we have ${\mathscr {S}}\in {\mathfrak {W}}^{s}$, and according to Lemma 3, one gets ${\mathscr {S}}={\mathscr {Z}}^*$. $\square$