Actuator fault diagnosis for uncertain T–S fuzzy systems with local nonlinear models

Abstract

This article investigates the problem of fault diagnosis (FD) for a class of nonlinear state-feedback control systems subject to parameter uncertainties. The considered nonlinear systems are described by T–S fuzzy models with local nonlinear parts and uncertain grades of membership. First, a general actuator fault model is proposed, which considers bias faults and gain faults. Then, a switching technique is introduced to address the unknown membership functions, external disturbances, faults, and their coupling. Furthermore, an adaptive FD observer design method combined with the switching technique is proposed to estimate the occurred actuator fault. It is noted that the obtained fault errors converge exponentially to zero. Finally, a numerical example of NSV reentry dynamic model is given to confirm the effectiveness of the new results.

Appendix

In this section, a simple state-feedback controller design method is proposed in the form of linear matrix inequalities, and the controller is described as

$$\begin{aligned} u(t) =K_{a}\xi (t)+K_{b}\phi (x(t)) \end{aligned}$$

(50)

Define the performance output

$$\begin{aligned} z(t)&= \sum _{i=1}^{N}\alpha _{i}(\theta (t))(C_{zi}\xi (t)+D_{i1}u^{f}(t)+D_{i2}w(t)\\&+N_{i2}\times \phi (x(t))), \end{aligned}$$

where $C_{zi}, D_{i1}, D_{i2}, N_{i2} (i=1,\ldots , N)$ are known matrices.

The considered $H_{\infty }$ state-feedback controller (50) is designed such that,

• (1) For a prescribed $H_{\infty }$ performance bound $\gamma >0$, the closed-loop system (11) in fault-free case is stable with

  $$\begin{aligned} \int \limits _{0}^{\infty }z^{T}(t)z(t)dt\le \gamma ^{2}\int \limits _{0}^{\infty }\overline{w}^{T}(t)\overline{w}(t)dt \end{aligned}$$

  (51)

• (2) For a prescribed $H_{\infty }$ performance bound $\gamma _{1}>0$, the closed-loop system (11) in faulty case is stable with

$$\begin{aligned} \int \limits _{0}^{\infty }z^{T}(t)z(t)dt\le \gamma _{1}^{2}\int \limits _{0}^{\infty }\overline{w}^{T}(t)\overline{w}(t)dt \end{aligned}$$

(52)

with $\overline{w}(t)=[ w^{T}(t) y_{r}^{T}(t)]^{T}$.

Lemma 1

For the prescribed $\gamma >0$, $\gamma _{1}>0$, the closed-loop augmented system (11) is stable, and the performances (51) and (52) are satisfied if there exist matrices $Q=Q^{T}>0, L_{a}, L_{b}$, and $\lambda$ such that

$$\begin{aligned} \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} He(\overline{A}_{i}Q+\overline{B}_{i}L_{a}) &{} \lambda \overline{N}_{i}+\overline{B}_{i}L_{b}+Q\overline{E}^{T} &{} \overline{G}_{i} &{} B_{r} \\ * &{} -2\lambda &{} 0 &{} 0 \\ * &{} * &{} -\gamma ^{2}I &{} 0 \\ * &{} * &{} * &{} -\gamma ^{2}I \\ * &{} * &{} * &{} *\\ \end{array} \right. \end{aligned}$$

$$\begin{aligned} \left. \begin{array}{c} (C_{zi}Q+D_{i1}L_{a})^{T} \\ (\lambda N_{i2}+D_{i1}L_{b})^{T} \\ D_{i2}^{T} \\ 0 \\ -I \\ \end{array} \right] <0 \end{aligned}$$

(53)

$$\begin{aligned} \left[ \begin{array}{c@{\quad }c} He(\overline{A}_{i}Q+\overline{B}_{i}(I-\varrho _{j})L_{a}) &{} \lambda \overline{N}_{i}+\overline{B}_{i}(I-\varrho _{j})L_{b}+Q\overline{E}^{T}\\ * &{} -2\lambda \\ * &{} * \\ * &{} * \\ * &{} *\\ \end{array} \right. \end{aligned}$$

$$\begin{aligned} \left. \begin{array}{c@{\quad }c@{\quad }c} \overline{G}_{i} &{} B_{r} &{} (C_{zi}Q+D_{i1}(I-\varrho _{j})L_{a})^{T} \\ 0 &{} 0 &{} (\lambda N_{i2}+D_{i1}(I-\varrho _{j})L_{b})^{T} \\ -\gamma _{1}^{2}I &{} 0 &{} D_{i2}^{T} \\ * &{} -\gamma _{1}^{2}I &{} 0 \\ * &{} * &{} -I \\ \end{array} \right] <0\nonumber \\ \end{aligned}$$

(54)

for $i=1,\ldots , N, j=1,\ldots , m$ with $\overline{E}=[0 E ] , \varrho _{j}=diag\{0,\ldots \rho _{j},\ldots 0\}, \rho _{j}\in \{\underline{\rho }_{j},\ \overline{\rho }_{j}\}$, and the controller gains are given as $K_{a}=L_{a}Q^{-1},\ K_{b}=L_{b}\lambda ^{-1}.$

Proof

The proof is easily obtained from the techniques in [29] and omitted.