Designing Unknown Input Observers for Fault Reconstruction in Disturbed Takagi-Sugeno Fuzzy Systems ^†

Abstract

Fault occurrence in practical systems, if not addressed, can cause diminished performance or even system breakdown. Therefore, fault detection has emerged as a crucial challenge in ensuring system safety and reliability. This paper presents a novel fuzzy observer aimed at reconstructing actuator and sensor faults in nonlinear systems, even when subjected to external disturbances. The approach we propose utilizes the Takagi-Sugeno fuzzy model and Lyapunov function. Initially, by filtering the system output, we construct a system where actuator faults correspond to the original actuator and sensor faults. Subsequently, the impact of disturbance on state estimations is minimized by employing the H-infinity performance criteria. We demonstrate that, for non-disturbed systems, these estimations gradually converge to their true values. In designing the observer gains, transformation matrices are derived by solving linear matrix inequalities. Our approach boasts some advantages over existing methods. By assuming that the premise variables are immeasurable, we enhance the usability of our approach. As a proof of concept, we evaluate two practical systems. The simulation results underline the benefits of our proposed method in terms of rapid and accurate fault detection performance.

1. Introduction

Takagi-Sugeno Fuzzy (TSF) systems, particularly effective in engineering for observer and fault detection, use observers like the Proportional Integral Observer (PIO) and its enhanced version, the Proportional Multi-Integral Observer (PMIO) [1], for complex nonlinear systems such as the Continuous Stirred Tank Reactor (CSTR) in chemical engineering. The CSTR, a multi-input multi-output (MIMO) system, presents control and estimation challenges due to its nonlinear dynamics [2]. Luenberger’s observers, initially for linear systems, have been adapted for nonlinear systems, offering cost-effective state estimation solutions [3].

The TS multi-model approach simplifies state estimation in CSTRs by interpolating between linear models for different behaviors [4]. This is crucial in scenarios with simultaneous unknown input actuator and sensor faults, necessitating integrated control and diagnostic systems. For TS models, several state and unknown input estimation methods have been developed. These include PI observers for decoupled Multiple Models [5], UI-PI observers with measurable premise variables [6,7], and a TS multi-model based PI observer for simultaneous state and input estimation [8]. However, PI observers have limitations with time-varying inputs, leading to the development of PMI observers, like the Thau-Luenberger observer [9], capable of assessing all unknown input derivatives. This research aims to refine PMI-based unknown input observers for TS-model systems, focusing on convergence conditions as linear matrix inequalities.

The paper’s structure is as follows: Section 2 introduces the TS fuzzy model and constructs a fictitious system with a fault, with the design of a PMI observer, ensuring estimation error stability and $H\infty$ performance. In Section 3, we validate the approach with simulation results. Conclusions are drawn in Section 4.

2. Problem Statement

Consider the Takagi-Sugeno model described by the system of equations:

$$\left\{\begin{array}{c}\dot{x}\left(t\right)={\sum }_{i=1}^Q{\mu }_i\left(x\left(t\right)\right)(A_{i}x\left(t\right)+B_{i}u\left(t\right)+E_i{f}_a\left(t\right))+Mw\left(t\right)\hfill \\ y\left(t\right)=Cx\left(t\right)+F{f}_a+Hw\left(t\right)\hfill \end{array}\right.$$

(1)

Here, $x\left(t\right)\in {\mathbb{R}}^n$ represents the state vector, $u\left(t\right)\in {\mathbb{R}}^{n_u}$ is the control input vector, $f_a\left(t\right)\in {\mathbb{R}}^{n_{f_a}}$ signifies the unknown input vector including actuator and sensors faults, $w\left(t\right)$ is the disturbance, and $y\left(t\right)\in {\mathbb{R}}^{n_y}$ corresponds to the output vector.

Consider the matrices $A_i$, $B_i$, $E_i$, M, C, F, and H, which are constants with appropriate dimensions. The activation functions ${\mu }_i\left(x\left(t\right)\right)$, dependent on the system’s state, adhere to the following convexity properties [10,11]:

$$\left\{\begin{array}{c}{\sum }_{i=1}^Q{\mu }_i\left(x\left(t\right)\right)=1\hfill \\ \forall i\in \{1,\dots ,Q\}\hfill \end{array}\right.$$

(2)

The scalar Q designates the number of local models.

Hypothesis 1. 

Unknown input $f_a\left(t\right)$ satisfies:

$$f_a^{\left(q\right)}\left(t\right)=0$$

(3)

Generally, the sequences ${f_a}^{\left(1\right)}\left(t\right)$, ${f_a}^{\left(2\right)}\left(t\right)$, ..., ${f_a}^{(q-1)}\left(t\right)$ denote the continuous derivatives of $f_a\left(t\right)$, expressed as:

$$\left[\begin{array}{c}\dot{f_a}\left(\mathrm{t}\right)\\ {\dot{f_a}}_1\left(t\right)\\ ⋮\\ {\dot{f_a}}_{q-1}\left(t\right)\end{array}\right]=\left[\begin{array}{c}{f_a}_1\left(t\right)\\ {f_a}_2\left(t\right)\\ ⋮\\ {f_a}_q\left(t\right)\end{array}\right]$$

(4)

System (1) can be expressed as a perturbed system with weighting functions ${\mu }_i$ based on the estimated state, where the signals $u\left(t\right)$, $f_a\left(t\right)$, and $w\left(t\right)$ are bounded.

$$\left\{\begin{array}{c}\dot{x}\left(t\right)={\sum }_{i=1}^Q{\mu }_i\left(\widehat{x}\right)\left(A_{i}x+B_{i}u+E_i{f}_a+Mw\left(t\right)+\delta \left(t\right)\right)\hfill \\ y\left(t\right)=Cx\left(t\right)+F{f}_a+Hw\left(t\right)\hfill \end{array}\right.$$

(5)

where:

$$\delta \left(t\right)=\sum _{i=1}^Q\left({\mu }_i\left(x\right)-{\mu }_i\left(\widehat{x}\right)\right)\left(A_{i}x+B_{i}u+E_i{f}_a+Mw\left(t\right)\right)$$

Then, system (1) can be articulated as:

$$\left\{\begin{array}{c}{\dot{x}}_a\left(t\right)={\sum }_{i=1}^Q{\mu }_i\left(\widehat{x}\left(t\right)\right)\left({\overline{A}}_i{x}_a\left(t\right)+{\overline{B}}_{i}u\left(t\right)+{\overline{\sigma }}_i\overline{\Omega }\left(t\right)\right)\hfill \\ y\left(t\right)=\overline{C}{x}_a\left(t\right)+\overline{F}\overline{\Omega }\left(t\right)\hfill \end{array}\right.$$

(6)

$$x_a=\left[\begin{array}{c}x\\ f_a\\ {f_a}_1\\ ⋮\\ {f_a}_{q-1}\end{array}\right],{\overline{A}}_i=\left[\begin{array}{cccccc}{A}_i& E_i& 0& \dots & 0& 0\\ 0& 0& I_{n_{f_a}}& \dots & 0& 0\\ 0& 0& 0& \ddots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & 0& I_{n_{f_a}}\\ 0& 0& 0& \dots & 0& 0\end{array}\right],{\overline{B}}_i=\left[\begin{array}{c}{B}_i\\ 0\\ ⋮\\ 0\\ 0\end{array}\right],{\overline{\sigma }}_i=\left[\begin{array}{c}{\sigma }_i^T\\ 0\\ ⋮\\ 0\end{array}\right],$$

(7)

$$\overline{\Omega }\left(t\right)=\left[\begin{array}{c}\sigma \left(t\right)\\ w\left(t\right)\end{array}\right], and\overline{C}=\left[\begin{array}{cccccc}C\hfill & F\hfill & 0\hfill & \dots \hfill & 0\hfill & 0\hfill \end{array}\right]$$

(8)

where: ${\sigma }_i=\left[\begin{array}{cc}{I}_n\hfill & M\hfill \end{array}\right],and\overline{F}=\left[\begin{array}{cc}0\hfill & H\hfill \end{array}\right],$

Hypothesis 2. 

The pairs $\left(\overline{A_i},\overline{C}\right)$ are observable for all i.

The Proportional Multiple Integral (PMI) observer can be described by the Equation (9), and the principle of this observer is presented in Figure 1:

$$\left\{\begin{array}{l}\dot{\widehat{x}}\left(t\right)={\sum }_{i=1}^Q{\mu }_i\left(\widehat{x}\left(t\right)\right)(A_i\widehat{x}\left(t\right)+B_{i}u\left(t\right)+E_i\widehat{f_a}\left(t\right)+L_{Pi}(y\left(t\right)-\widehat{y}\left(t\right)))\\ {{\dot{\widehat{f}}}_a}_j\left(t\right)={\sum }_{i=1}^Q{\mu }_i\left(\widehat{x}\left(t\right)\right)({\widehat{f_a}}_{j+1}+L_{Ii}^j(y\left(t\right)-\widehat{y}\left(t\right))),j=1,\dots ,q-2\\ {{\dot{\widehat{f}}}_a}_j\left(t\right)={\sum }_{i=1}^Q{\mu }_i\left(\widehat{x}\left(t\right)\right)\left(L_{Ii}(y\left(t\right)-\widehat{y}\left(t\right))\right)j=q-1\\ \widehat{y}\left(t\right)=C\widehat{x}\left(t\right)+F{\widehat{f}}_a\end{array}\right.$$

(9)

With the above, the augmented PMI observer that incorporates an output error into its activation functions is articulated.

$$\left\{\begin{array}{c}{\widehat{x}}_a\left(t\right)={\sum }_{i=1}^Q{\mu }_i\left(\widehat{x}\left(t\right)\right)\left({\overline{A}}_i{\widehat{x}}_a\left(t\right)+{\overline{B}}_{i}u\left(t\right)+{\overline{L}}_i(y\left(t\right)-\widehat{y}\left(t\right))\right)\hfill \\ \widehat{y}\left(t\right)=\overline{C}{\widehat{x}}_a\left(t\right)\hfill \end{array}\right.$$

(10)

where ${\overline{L}}_i={\left[\begin{array}{cccccc}{L}_{Pi}^T\hfill & L_{Ii}^T\hfill & L_{Ii}^{1^T}\hfill & \dots \hfill & L_{Ii}^{q-2^T}\hfill & L_{Ii}^{q-1^T}\hfill \end{array}\right]}^T$.

The dynamic of the state estimation error is presented in the following augmented format:

$$\dot{\overline{e}}=\sum _{i=1}^Q{\mu }_i\left(\widehat{x}\right)\left(\left({\overline{A}}_i-{\overline{L}}_i\overline{C}\right)\overline{e}+\left({\overline{\sigma }}_i-{\overline{L}}_i\overline{F}\right)\overline{\Omega }\right)$$

(11)

Theorem 1. 

The $PMI$ Observer, as defined by (9), designed to Simultaneously estimate the state and the unknown inputs of the fuzzy system represented in (1). This is achieved while minimizing the ${\mathcal{L}}_2$-gain $\overline{\gamma }$ from the unknown inputs to the augmented state estimation error $\overline{e}$. This can be obtained by determining a positive definite matrix P, matrices $M_i$, and a positive scalar $\overline{\gamma }$ that satisfy the following LMI constraints for $i=1,\dots ,r$:

$$\left[\begin{array}{cc}{\overline{A}}_i^{T}P+P{\overline{A}}_i-M_i\overline{C}-{\overline{C}}^T{M}_i^T+I& P{\overline{\sigma }}_i-M_i\overline{F}\\ {\overline{\sigma }}_i^{T}P-{\overline{F}}^T{M}_i^T& -\overline{\gamma }I\end{array}\right]<0$$

(12)

where $\gamma =\sqrt{\overline{\gamma }}$.

The observer gains: ${\overline{L}}_i=P^{-1}{M}_i$

Proof. 

Where System (1) stands as a paragon of stability and all preceding signals remain bounded, a transformative revelation unfolds. By invoking Lemma of Perturbation attenuation and satisfying the condition ${∥\overline{e}\left(t\right)∥}_2<\gamma {\parallel \tilde{\omega }\left(t\right)\parallel }_2$ yields an enigmatic Linear Matrix Inequality (LMI):

$$\left[\begin{array}{cc}{\overline{A}}_i^{T}P+P{\overline{A}}_i-P{\overline{L}}_i\overline{C}-{\overline{C}}^T{\overline{L}}_i^{T}P+I& P{\overline{\sigma }}_i-P{\overline{L}}_i\overline{F}\\ {\overline{\sigma }}_i^{T}P-{\overline{F}}^T{\overline{L}}_i^{T}P& -{\gamma }^{2}I\end{array}\right]<0$$

(13)

The essence of Theorem 1 obtain through variable transformations:

$$M_i=P{\overline{K}}_i,\overline{\gamma }={\gamma }^2$$

□

3. Practical Example

As a demonstration of the methodology, we examine a nonlinear CSTR system depicted by a multi-model with unmeasurable premise variables. This multi-model comprises two local models, each with three states [3].

Consider a thoroughly mixed CSTR where the multi-component chemical reaction $\mathrm{A}\rightleftharpoons \mathrm{B}\to \mathrm{C}$ takes place. The Schematic of the Continuous Stirred Tank Reactor (CSTR) is shown in Figure 2, illustrating the concentrations of species $A,B,$ and C as $C_A,C_B,$ and $C_C$ respectively. The reactor temperature is denoted by T, with $C_{Af}$ and $T_f$ representing the feed’s actual concentration and temperature. The flow rate and temperature of the cooling water are indicated by $q_c$ and $T_c$. The system’s nonlinear dynamics can be represented as:

$$\dot{x}=\left[\begin{array}{ccc}-4& 0.8796& 0\\ 3& -3.6388& 0\\ 0& 1.7592& -1\end{array}\right]x+\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]u+\left[\begin{array}{c}0.5x_2^2\\ -1.5x_2^2\\ x_2^2\end{array}\right],$$

(14)

where $x={\left[\begin{array}{ccc}{x}_1& x_2& x_3\end{array}\right]}^T$ denotes the concentrations of species $A,B$, and C respectively.

To evaluate the effectiveness of our proposed approach, we introduce a fault and disturbance to the dynamics, leading to:

$$\left\{\begin{array}{c}\dot{x}=\left[\begin{array}{ccc}-4& 0.8796+0.5x_2& 0\\ 3& -3.6388-1.5x_2& 0\\ 0& 1.7592+x_2& -1\end{array}\right]x+\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]u+\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]f_a+\left[\begin{array}{c}0.6\\ 0.05\\ 0.03\end{array}\right]w\left(t\right)\hfill \\ y=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]x+\left[\begin{array}{c}0.01\\ 0.001\end{array}\right]f_a+\left[\begin{array}{c}0.01\\ 0.03\end{array}\right]w\left(t\right)\hfill \end{array}\right.$$

(15)

Assuming the concentration of species B is dimensionless, it is given by $x_2\in [-1,1]$. This allows the definition of two membership functions using Takagi-Sugeno (TS) rules as:

$$h_1=\frac{1-x_2}{2}\mathrm{and}{h}_2=\frac{1+x_2}{2}.$$

(16)

With this, the corresponding local linear TS matrices can be established as:

$$A_1=\left[\begin{array}{ccc}-4& 0.8796-0.5& 0\\ 3& -3.6388+1.5& 0\\ 0& 1.7592-1& -1\end{array}\right],A_2=\left[\begin{array}{ccc}-4& 0.8796+0.5& 0\\ 3& -3.6388-1.5& 0\\ 0& 1.7592+1& -1\end{array}\right],B_1=B_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],$$

$$E_1=E_2=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\mathrm{and}F=\left[\begin{array}{c}0.01\\ 0.001\end{array}\right]$$

(17)

Given that these TS fuzzy system matrices comply with all the preconditions, the TS fuzzy observer (9) is constructible.

For the simulation setup, parameters and input signals were chosen as follows: $u=sin\left(t\right)$, with initial conditions $x_0={\left[\begin{array}{ccc}0.15& 0.2& 0.1\end{array}\right]}^T$ and ${\widehat{x}}_0={\left[\begin{array}{ccc}0.15& 1& 3\end{array}\right]}^T$. The disturbance profile depicted in Figure 3.

The Linear Matrix Inequality (LMI) outlined in the provided theorem is addressed using the Matlab Yalmip toolbox, leading to the determination of the observer’s gains. The results of state and fault estimation are visually depicted in Figure 4 and Figure 5.

In Figure 4, it is evident that the estimated states converge rapidly to their true signals right from the beginning. Figure 5 displays a robust estimation of the fault. These simulation outcomes distinctly demonstrate that the proposed observer not only guarantees accurate state estimation but also effectively handles concurrent fault detection, even in the presence of unknown disturbances.

4. Conclusions

In this study, we focus on estimating states and unknown inputs in Takagi-Sugeno systems that have unmeasurable premise variables, utilizing a Proportional Multiple Integral (PMI) observer. We define the stability conditions using Linear Matrix Inequalities to ensure robust performance. To validate the PMI observer’s efficacy, we conduct a simulation using the Continuous Stirred Tank Reactor (CSTR) system as a practical example.