Simultaneous Sensor and Actuator Fault Reconstruction by Using a Sliding Mode Observer, Fuzzy Stability Analysis, and a Nonlinear Optimization Tool

Over the past few decades, the reliability and safety of industrial systems has attracted considerable attention. As a consequence, fault-tolerant control (FTC) has received considerable attention in different fields [1,2]. There are different classifications for FTCs. In general, FTCs are classified into passive and active classifications. Active fault-tolerant controllers compensate for the effects of the occurred faults by using early information obtained from fault detection and isolation (FDI) schemes, which leads to a more flexible dynamic [3]. Consequently, FDI is becoming an attractive topic in different research fields. Observer-based methods are one of the most popular model-based FDIs. The main idea of observer-based FDIs is to construct a residual based on the measured output of the systems or to reconstruct the fault directly. Sliding mode observer (SMO) works based on the second approach, which detects the faults while determining the dynamic behavior [4,5]. SMOs are more insensitive to the unknown uncertainties occurring in the system compared to other observers like unknown input observers (UIOs) [6].

First, SMO observers were developed for linear dynamic systems; however, most actual physical systems are often nonlinear. Currently, lots of SMO-based fault reconstruction methods have been developed for uncertain nonlinear systems. In ref. [7], by considering a filter of the measured output vector, the original system with sensor and actuator faults is transformed into an augmented system with just the actuator fault and unknown inputs. Nevertheless, the classes of nonlinear systems considered in most of the papers are limited and cannot represent a general model for real systems [8,9].

Takagi–Sugeno (TS) fuzzy models can represent the behavior of nonlinear systems while keeping the simplicity of the linear models. A TS fuzzy representation is a convex nonlinear aggregation of several linear systems. Because the parameters of a TS fuzzy representation satisfy the convex sum, it is interesting to investigate the properties of the TS system based on its local linear vertices. With the advent of TS fuzzy systems, TS-based FDI techniques emerged to tackle a broader range of nonlinear systems [10]. By changing a nonlinear system to a TS system, some local linear systems are created, representing the behavior of the nonlinear system in a specific operating area. These local linear systems can be aggregated by using an interpolation mechanism. Thus, TS fuzzy models can represent the actual nonlinear behavior while maintaining the simplicity of linear models. Thus, an efficient FDI can be obtained by combining the SMO, which is robust to the uncertainties, and the TS fuzzy model, which causes simplicity in the design process. Recently, several researchers have utilized TS-based SMOs for fault detection and isolation in continuous-time and discrete-time systems [11,12]. However, in the methods developed in these articles, it is assumed that the premise variables are measurable, which reduces the applicability of these approaches. To deal with this problem, an FDI approach for stability analysis of the TS fuzzy systems with immeasurable premise variables was proposed in [13,14].

In [15], simultaneous actuator and sensor faults in a nonlinear system represented by a TS fuzzy model are reconstructed by using an SMO and considering $H_{\infty }$ performance criteria to reduce the effect of disturbance, whereas [16] does the same procedure for the fault reconstructions and both of the exogenous disturbance and the system faults are reconstructed. However, in refs. [15,16] quadratic Lyapunov functions (QLFs) are used to design the observers. By using the QLF for TS fuzzy systems with a large number of fuzzy rules can cause undesired performance or unfeasible solutions. Consequently, refs. [17,18] offered to use a non-quadratic Lyapunov function (NQLF) to design the TS-based SMO for the FDI purposes. In all these papers, a linear optimization approach based on linear matrix inequalities (LMIs) is utilized, making the stability analysis more complex and using some approximations and lemmas to prove the stability conditions.

In this paper, a TS fuzzy-based SMO with immeasurable premise variables is designed to reconstruct simultaneous actuator and sensor faults in a nonlinear system exposed to an unknown disturbance. Then, the states and faults are estimated. The stability of the proposed observer is guaranteed by using the NQLF and fmincon as a nonlinear optimization tool in MATLAB. In addition, $H_{\infty }$ performance criteria are considered to minimize the effect of disturbances and uncertainties on the estimation error and the fault estimations. By using the NQLF, a generalized eigenvalue problem is proposed, which maximizes the admissible Lipschitz constant and minimizes the disturbance effects on the estimation error through a nonlinear optimization problem.

The main advantages of the proposed approach over the existing methods can be summarized as follows:

This paper is organized as follows. Section 2 presents a TS fuzzy model with simultaneous actuator and sensor faults and disturbance and how to construct a fictitious system with just an actuator fault. In Section 3, the main results of this paper, including the sliding mode observer design and the sufficient conditions of stability of the estimation errors, are proposed and guarantee the $H_{\infty }$ performance simultaneously. Section 4 discusses the procedure of the actuator and sensor fault reconstructions. In Section 5, simulation results are given, and comparisons are discussed. Finally, in Section 6, the concluding remarks are given.

Assume that a continuous-time nonlinear system affected by actuator and sensor faults and disturbance is given as where $x\left(t\right)\in R^n$, $u\left(t\right)\in R^m$, $y\left(t\right)\in R^p$, $f_a\left(t\right)\in R^q$, $f_s\left(t\right)\in R^h$ and $d\left(x\left(t\right),u\left(t\right),t\right)\in R^l$ are the state, input, output, unknown actuator, and sensor faults, and the system uncertainty vectors, respectively. $f$ and $g$ are nonlinear smooth functions. By using sector nonlinearity transformation, the nonlinear model (1) can be replaced by the following TS fuzzy model

Where $C$ and $N$ are known full rank matrices with appropriate dimensions. $A_i$, $B_i$, $M_i$, and $D_i$ are real known matrices, $r$ represents the number of fuzzy rules and ${\mu }_i\left(\xi \left(t\right)\right)$ are the fuzzy membership functions depending on the unmeasurable variable vector $\xi \left(t\right)$ and satisfy the following so-called convex sum property

In the rest of the paper, $\left(t\right)$ is dropped from the equations, $d$, ${\mu }_i$ and ${\widehat{\mu }}_i$ denote $d\left(x,u,t\right)$, ${\mu }_i\left(\xi \left(t\right)\right)$, and ${\mu }_i\left(\widehat{\xi }\left(t\right)\right)$ and the mark $(*)$ denotes the transposed element in a symmetric matrix.

To build a system with just an actuator fault and then use the actuator fault reconstruction concepts, the output is passed through an orthogonal matrix $T_r\in R^{p\times p}$ and an augmented TS system of order $n+h$ can be obtained as where $X={\left[x^T{z}^T\right]}^T\in R^{n+h}$, $Y={\left[y_1^T{z}^T\right]}^T\in R^p$, and

$-A_f\in R^{h\times h}$ is an arbitrary stable matrix, $z\in R^h$ and $N_2\in R^{h\times h}$. $T_r$ can be obtained by QR reduction of the matrix $N$.

By defining where $\widehat{x}$ is the estimation of the $x$, the TS system (4) can be derived as

Moreover, the nonlinear term $\varphi$ is assumed to satisfy the Lipschitz condition as

To design a sliding mode observer, some assumptions and lemmas are needed as follows.

The proposed TS sliding mode observer for the nonlinear system (2) in the new coordinate (10) is as follows: where ${\mathcal{G}}_{n.i}$ and ${\mathcal{G}}_{l.i}$ are design matrices of the observer that will be derived through Theorem 1. $e_Y:=Y-\widehat{Y}$ represents the output error estimation, ${\nu }_{a.i}$ and ${\nu }_s$ are the equivalent output error injections that are used to compensate the errors due to the actuator fault and sensor fault, respectively, and have the following structure: where ${\eta }_{a.i}$ and ${\eta }_s$ are two positive scalars such that

$w_{a.i}$ and $w_s$ are two arbitrary positive constants.

The observer (16) guarantees that the state estimation error converges to a pre-designed sliding surface in finite time and then, asymptotically to zero. Define state estimation error as $e:=X-\widehat{X}$. By subtracting the observer dynamics from the system dynamic (7) in the new coordinate (12), the state estimation error dynamic can be given as

By partitioning $\varphi$ as $\varphi ={\left[{\varphi }_1^T {\varphi }_2^T\right]}^T$ and applying a further change of coordinates where ${\overline{L}}_i\in R^{\left(n+h-p\right)\times \left(p-q-h\right)}$ is a stabilizing gain matrix, it is straightforward to see that where ${\mathcal{A}}_{s.i}$ are arbitrary stable design matrices. Through the new coordinate, the error dynamic (19) can be re-written as where

The goal is to design the matrices $L_i$ such that the asymptotic stability of (22) is assured while the following specified $H_{\infty }$ performance is guaranteed:

The following theorem provides sufficient conditions to ensure asymptotic stability of the state estimation error (22) with maximized admissible Lipschitz constant $\gamma$ in (8) and minimized $H_{\infty }$ performance gain $\vartheta$ in (24).

In Section 3, an $H_{\infty }$ sliding mode observer is designed in which two discontinuous terms (19) are considered to reconstruct simultaneous faults in the presence of an unknown disturbance based on the measured signals $u$ and $y$. Along the sliding surface $e_Y=\dot{e_Y}=0$. Consequently, (22) on the sliding surface changes to where ${\nu }_{eqa,i}$ and ${\nu }_{eqs}$ are approximations of the equivalent output error injection terms (17) required to maintain the sliding motion and can be defined as where ${\delta }_f$ and ${\delta }_d$ are small positive constants. Consequently, (40) leads to

On the other hand, using (8) and (24) can show that the term $\begin{array}{c}{\mathcal{A}}_{21.\mathrm{i}}{e}_1+{\varphi }_2\end{array}+{\mathcal{D}}_{2.i}d$ is bounded as where $ϵ=\mu \left(\parallel {\mathcal{A}}_{21,\mathrm{i}}\parallel +\gamma \parallel T_L{}^{-1}\parallel \right)+\parallel {\mathcal{D}}_{2,i}\parallel$. Therefore, for small values of $ϵ\parallel d\parallel$, the actuator and sensor faults can be estimated as where $†$ shows the pseudo-inverse of a matrix.

In this section, a three-state variable continuous stirred tank reactor (CSTR) system is utilized to show the effectiveness of the proposed sliding mode observer in both actuator and sensor faults reconstruction in the presence of an unknown disturbance. To show the performance improvement of the proposed approach, the obtained results are compared to the LMI approach presented in ref. [17].

Consider a well-mixed variable CSTR in which a multi-component chemical reaction A ⇌ B → C is being carried out. The nonlinear dynamics of the CSTR is given by the following model [21], where $x={\left[x_1 x_2 x_3\right]}^T,$ and the states represent the concentrations of the species $A$, $B,$ and $C$, respectively. To check the advantage of the proposed method, two faults and a disturbance are added to the dynamic (45) as

It is supposed that the concentration of $B$ is dimensionless, which means that $x_2\in \left[-1 1\right]$. Consequently, by using TS rules, two membership functions can be defined as

Therefore, the local linear TS matrices can be determined as