Diagnostic observer-based fault detection for nonlinear parabolic PDE systems via dual sampling approaches

Abstract

This paper investigates the fault detection problem for nonlinear parabolic PDE systems. In contrast to the existing works, the designed fault detection observer utilizes less state information in both time domain and space domain, the details of which are illustrated as follows. First, based on Takagi–Sugeno fuzzy theory, a novel fuzzy state observer is developed via sampling in time and space approach. Then, by using Lyapunov direct method, some sufficient conditions that guarantee both the disturbance/control-inputs attenuation performance and the faulty sensitivity performance of the fault detection systems are established in the form of linear matrix inequality, which provides an effective approach to optimize the systems’ performance. Finally, two simulation examples are given to illustrate the effectiveness of the designed fault detection systems.

Introduction

Recent years, considerable attention has been paid to distributed parameter systems. From the perspective of modeling approach, distributed parameter systems can be classified to parabolic partial differential equation (PDE) systems, hyperbolic PDE systems, and elliptic PDE systems, etc. Compared with ordinary differential equation (ODE) systems, the analysis and synthesis of PDE systems are more difficult because of their infinite dimensional feature. More recently, fruitful results have been achieved for parabolic PDE systems due to their wide applications in biology and chemistry fields [1]. For parabolic PDE systems, we can achieve our goals by reducing the PDE to a large number of ODE systems and then design the controller or state observer (see [2], [3], and [4]). However, it is noteworthy that the infinite dimensional feature of distributed parameter systems was neglected in this design method. Thus, to deal with this problem, some scholars propose PDE-based design approaches; furthermore, some meaningful results have been reported for nonlinear PDE systems. For example, [5] investigated fuzzy boundary control design, [6] studied the synchronization and consensus, [7] considered fuzzy output feedback control and [8] and [9] investigated the reliable H_∞ control and filtering problem, respectively.

It is worth pointing out that the above mentioned results were all achieved based on an assumption that the systems’ states were measurable in the whole time and space domains. However, the condition is high cost and even impossible to hold in real applications. How to achieve our aims by utilizing incomplete measurements becomes a hot research direction in PDE systems. Sampled-data control [10] that contains many advantages may provide a good choice to solve this problem. Accordingly, [11] designed sampled-data distributed fuzzy controller, [12] considered sampled-data boundary feedback control problem, and [13] investigated sampled-data control problem for nonlinear parabolic PDE systems with control inputs missing. It is easy to observe that these sampling approaches are considered in time domain. In fact, there also exists literature that considered space sampling. For example, Wu et al. designed a novel state estimator using pointwise measurements [14], Wang et al. achieved pointwise exponential stabilization via using non-collocated pointwise observation [15], and so on [16], [17], [18].

From the discussion above, we can easily find that various issues such as control synthesis, observer design, and state synchronization, have all been considered in the existing works. However, in real applications, various faults (including sensor fault and process fault) are also a significant obstacle to obtain better performance, higher reliability, and superior safety. Many unexpected phenomena including stochastic variation of signals, sudden changes of working conditions, and stochastic occurring of external disturbance, will destroy systems’ performance. Therefore, it is of great importance to detect them in time. Whereas, limited related work has been carried out for PDE systems. To the authors’ best knowledge, literature [19] considers fault detection problem for hyperbolic PDE systems, in which distributed sensors were required. Thus, it is natural to consider sampled-data fault detection problem for PDE systems. This is the main motivation of the present work.

Note that in [19], the authors assumed that the premise variables of plant and observer keep the same. While the premise variables might be unknown and unmeasurable in engineering environment. Furthermore, the digital communication channel in networked systems will also bring much difficulty to keep the premise variables same (refer to [20]). In fact, many works that are related to asynchronous fuzzy fault detection, have been carried out for ODE systems. To mention a few, [21] considered fuzzy fault detection observer design for switched fuzzy systems under unmeasurable premise variables, [22] considered asynchronous diagnostic observer design for Takagi–Sugeno (T–S) fuzzy systems, and [23] investigated the fault detection problem for T–S fuzzy systems with partly unmeasurable premise variables. Therefore, asynchronous fuzzy fault detection is more worth exploring for PDE systems. Furthermore, it it not difficult to observe that many nonlinear phenomena often exist in various systems [21], [22], [23], [24], [25], [26]; thus, the research of nonlinear PDE systems is of great importance. This is another motivation of this study.

In addition, disturbance attenuation performance plays a key role in fault detection systems. As the basic research, ${\mathcal{H}}_{\infty }$ performance draws considerable attention. For instance, in ODE systems, [27] considered event-triggered ${\mathcal{H}}_{\infty }$ fault detection for Markovian jump systems, and [28] was concerned with ${\mathcal{H}}_{\infty }$ fault detection filter design for discrete-time nonlinear systems with past output measurements. These literature provides many valuable inspirations for PDE systems’ diagnostic observer design. However, it is not enough to only guarantee the fault detection systems’ robustness to disturbance; sometimes, the control input is also an important factor that will influence the fault detection performance. On another hand, it should be noted that most of literature forgot the essential aim of fault detection; namely, the systems’ sensitivity to faults was ignored.

After the discussion above, by taking into account the robustness to disturbance/control-inputs and sensitivity to faults, this work aims to investigate the fault detection problem for nonlinear parabolic PDE systems based on sampling in time and space approach. The main works are listed as

• The nonlinear PDE systems are remodelled by several fuzzy rules by using sector nonlinearity approach.

• A new asynchronous fuzzy diagnostic observer is designed via utilizing incomplete measurements.

• The sufficient conditions to ensure the robustness and sensitivity are established in terms of linear matrix inequality (LMI), and then the observer design approach is proposed. Furthermore, two simulation studies are shown to verify the effectiveness of the developed result.

Finally, the novelties and contributions of this paper are summarized as follows:

• In the existing work [19], the designed fault detection system requires a lot of sensors, which certainly brings much capital cost. Therefore, this paper takes a bold and meaningful attempt to investigate the fault detection problem for parabolic PDE systems by only utilizing less sensors. The main advantages of this design method is reflected in the easy implementation, and low cost.

• In many literatures that are related to fuzzy fault detection observer design (including ODE and PDE systems), the premise variables of the plant and observer are assumed to be the same (for example, [19] and [29], [30], [31], [32]). However, the unmeasurable states and strict communication environment will bring much difficulty to the condition’s holding. Therefore, this study determines to consider asynchronous fuzzy fault detection system design.

• For fault detection systems, it is necessary to take into account the robustness to disturbance/control-inputs and the sensitivity to faults. However, few scholars considered them simultaneously (refer to [22] and [33]). In most obtained results, only ${\mathcal{H}}_{\infty }$ disturbance attenuation performance was satisfied. As one novelty of the present paper, spatial ${\mathcal{L}}^{\infty }$ norm based disturbance/control-inputs attenuation performance and fault sensitivity performance are both studied.

Notations. The sets of real numbers and n-dimensional vectors are denoted by $\mathbb{R}$ and ${\mathbb{R}}^n,$ respectively. $\mathcal{N}$ denotes the set of non-negative integers. col[ · ] denotes a column vector, diag{ · } denotes a diagonal matrix, and $Sym\left[A\right]\genfrac{}{}{0pt}{}{\Delta }{=}A+A^T$. For a symmetric matrix P, the notations P > 0 and P ≥ 0 means that matrix P is positive definite and positive semi-definite, respectively. λ_max(P) and λ_min(P) represent the maximum and minimum eigenvalues of P, respectively. ${\mathcal{H}}_n={\mathcal{L}}_2([l_1,l_2];{\mathbb{R}}^n)$ is a Hilbert space of square integrable vector functions, $\vartheta \left(x\right):[l_1,l_2]\to {\mathbb{R}}^n,$ with the norm ${\parallel \vartheta \left(x\right)\parallel }_2=\sqrt{{\int }_{l_1}^{l_2}{\vartheta }^T\left(x\right)\vartheta \left(x\right)dx}$. ${\mathcal{H}}_n^k\left([l_1,l_2]\right)={\mathcal{W}}^{k,2}([l_1,l_2];{\mathbb{R}}^n)$ is a real Sobolev space of absolutely continuous vector functions $\vartheta \left(x\right):[l_1,l_2]\to {\mathbb{R}}^n$ with square integrable derivatives d^kϑ(x)/dx^k of the order k and with the norm ${\parallel \vartheta \left(x\right)\parallel }_{{\mathcal{H}}_n^k\left([l_1,l_2]\right)}=\sqrt{{\int }_{l_1}^{l_2}{\sum }_{i=0}^k\frac{d^i{\vartheta }^T\left(x\right)}{d{x}^i}\frac{d^i\vartheta \left(x\right)}{d{x}^i}dx}$. For any $s(x,t)\in {\mathcal{H}}_n^k\left([l_1,l_2]\right),$ the spatial ${\mathcal{L}}^{\infty }$ norm is defined as ${\parallel s(x,t)\parallel }_{\infty }=\underset{x\in [l_1,l_2]}{max}\parallel s(x,t)\parallel$. Furthermore, to simplify the description, let define $s_{xx}(x,t)=\frac{{\partial }^{2}s(x,t)}{\partial x^2},$ $s_x(x,t)=\frac{\partial s(x,t)}{\partial x},$ and $s_t(x,t)=\frac{\partial s(x,t)}{\partial t}$.

The outline of this paper is as follows. The problem description is given in Section 2. The ${\mathcal{L}}^{\infty }/{\mathcal{L}}^2$ robustness and the $\mathcal{L}{}_{-}$ fault sensitivity conditions are established in Sections 3 and 4, respectively. In addition, Section 5 gives the fault detection system design approach, and Section 6 presents two simulation studies. Finally, the conclusions of this study are listed in 7.