Data-Driven Adaptive Observer for Fault Diagnosis

Abstract

This paper presents an approach for data-driven design of fault diagnosis system. The proposed fault diagnosis scheme consists of an adaptive residual generator and a bank of isolation observers, whose parameters are directly identified from the process data without identification of complete process model. To deal with normal variations in the process, the parameters of residual generator are online updated by standard adaptive technique to achieve reliable fault detection performance. After a fault is successfully detected, the isolation scheme will be activated, in which each isolation observer serves as an indicator corresponding to occurrence of a particular type of fault in the process. The thresholds can be determined analytically or through estimating the probability density function of related variables. To illustrate the performance of proposed fault diagnosis approach, a laboratory-scale three-tank system is finally utilized. It shows that the proposed data-driven scheme is efficient to deal with applications, whose analytical process models are unavailable. Especially, for the large-scale plants, whose physical models are generally difficult to be established, the proposed approach may offer an effective alternative solution for process monitoring.

1. Introduction

During the last two decades, diagnostic observers and parity space-based fault detection and isolation (FDI) schemes for linear time invariant (LTI) systems are intensively studied [1–6]. The core of the parity space FDI technique is, based on state space representation of the system, construction of residual generator by means of the so-called parity vector, which is the null space of the observability matrix. As pointed out in Ding [6], the design of an observer-based residual generator can be equivalently formulated as a similar problem.

Since the majority of observer and parity space-based FDI schemes involve rigorous development of process models based on the first principles, later identification techniques that extracts transfer function [7] or state space model become a necessary step prior to the design. For this purpose, subspace identification methods (SIM) that identify the complete state space matrices have been successfully implemented see Overschee and Moor [8], Favoreel et al. [9], and Qin [10]. Provided the process model is known a priori, observer and parity space-based FDI systems can be designed with a large number of applications [11–13]. The approaches of filtering and control for such complex systems have been well studied in the literature; see Shen et al. [14], Shen et al. [15], and Dong et al. [16], Dong et al. [17]. Recently, an alternative data-driven approach has been proposed that does not require the identification of complete set of process model but only the so-called primary form of the residual generator from the process data; see Ding et al. [18]. Based on it, the advanced observer-based FDI system can be designed in an efficient way [19–23]. Thanks to its simple forms and less requirements on the design and engineering efforts, the data-driven FDI approach becomes more efficient in many industry sectors, especially for large-scale industry applications [24]. Recent survey given by Ding et al. [22, 23] provided the reader with a comprehensive overview on the basic and advanced data-driven FDI schemes.

Our study is motivated by the aforementioned data-driven FDI approach, in that we also recognize the wide existence of systems with uncertain or normal variation parameters in practice, which have not been paid enough attention in research study. Extension of the data-driven FDI scheme to such processes will improve the safety and reliability of these applications and further reduce the complexity to perform FDI especially on the large-scale systems. For this purpose, a data-driven fault diagnosis approach was proposed in this paper, in which the issues of fault isolation and threshold setting were studied to complete the earlier work given by Ding et al. [19, 20]. The structure of the fault diagnosis scheme consists of an adaptive residual generator and a bank of isolation observers, whose parameters are directly updated from the plant data with standard adaptive technique to cope with normal variations in the process. The threshold for fault detection can be determined either analytically or by probability density function estimation technique. When a fault is detected, the fault isolation scheme is activated, in which each isolation observer indicates the occurrence of a particular type of fault in the process. Fault isolation is successfully achieved when all the isolation indices, except the one responsible for the fault, exceed the thresholds. For the realization of the isolation scheme, the standard projection algorithm is implemented [25, 26]. The sufficient condition of fault isolability is also analyzed in this work.

The rest of the paper is organized as follows. In Section 2, the mathematical preliminaries and problem formulation are presented. Section 3 addresses the theoretical core of the proposed fault diagnosis scheme, in which both fault detection and isolation issues will be analyzed in detail. In Section 4, the simulation of laboratory-scale three-tank system is used to illustrate the performance of the proposed scheme. The paper ends with concluding remarks in the last section.

2. Preliminaries and Problem Formulation

2.1. Preliminaries of Model-Based Residual Generator

Consider a discrete-time LTI system which is described by 𝑥_𝑘+1=𝐴𝑥_𝑘+𝐵𝑢_𝑘,𝑦_𝑘=𝐶𝑥_𝑘, (2.1)(2.2) where 𝑥_𝑘∈ℛ^𝑛, 𝑢_𝑘∈ℛ^𝑙, and 𝑦_𝑘∈ℛ^𝑚 represent the vector of state variables and process input and output, respectively. 𝐴, 𝐵, and 𝐶 are system matrices with appropriate dimensions. Reformulate (2.1)-(2.2) into 𝑌𝑈=Γ𝑍=^𝑠𝐻_𝑠,𝑢𝑋0𝐼𝑈^𝑖∈ℛ^{(𝑠+1)(𝑚+𝑙)×𝑁}, (2.3) where 𝑋_𝑖=[𝑥_𝑖𝑥_𝑖+1⋯𝑥_{𝑖+𝑁−1}]∈ℛ^𝑛×𝑁, 𝑈=[𝑢_𝑠,𝑘⋯𝑢_{𝑠,𝑘+𝑁−1}]∈ℛ^{(𝑠+1)𝑙×𝑁}, 𝑌=[𝑦_𝑠,𝑘⋯𝑦_{𝑠,𝑘+𝑁−1}]∈ℛ^{(𝑠+1)𝑚×𝑁}, and 𝑦_𝑠,𝑘=𝑦_𝑘−𝑠⋮𝑦_𝑘,𝑢_𝑠,𝑘=𝑢_𝑘−𝑠⋮𝑢_𝑘,Γ_𝑠=𝐶⋮𝐶𝐴𝐶𝐴^𝑠,𝐻_𝑠,𝑢=00⋯0𝐶𝐵0⋯0⋮⋱⋱⋮𝐶𝐴^𝑠−2𝐵⋯𝐶𝐵0, (2.4) and 𝑠(≥𝑛) and 𝑁(≫𝑠) are integers. On the assumption of known 𝐴, 𝐵, and 𝐶, the design of a parity space-based residual generator consists in solving 𝛼_𝑠Γ_𝑠=0, (2.5) for the so-called parity vector 𝛼_𝑠[𝛼_𝑠,0𝛼_𝑠,1⋯𝛼_𝑠,𝑠]∈ℛ^{1×(𝑠+1)𝑚}. The design of an observer-based residual generator is achieved by solving the so-called Luenberger equations 𝑇𝐴−𝐴_𝑧𝑇=𝐿𝐶,𝑐_𝑧𝑇=𝑔𝐶,𝐵_𝑧𝐴=𝑇𝐵,_𝑧∈ℛ^𝑠×𝑠,𝑇∈ℛ^𝑠×𝑛,𝑐_𝑧∈ℛ^1×𝑠,𝑔∈ℛ^1×𝑚 (2.6)(2.7) for 𝐴_𝑧 (should be stable), 𝐵_𝑧, 𝑐_𝑧, 𝑔, 𝐿 together with a transformation matrix 𝑇. It follows then the construction of the parity space-based residual generator 𝑟_𝑘=𝛼_𝑠𝑦_𝑠,𝑘−𝐻_𝑠,𝑢𝑢_𝑠,𝑘, (2.8) and the observer-based residual generation 𝑧_𝑘+1=𝐴_𝑧𝑧_𝑘+𝐵_𝑧𝑢_𝑘+𝐿𝑦_𝑘∈ℛ^𝑠,𝑟_𝑘=𝑔𝑦_𝑘−𝑐_𝑧𝑧_𝑘∈ℛ. (2.9)(2.10) In the above equations, 𝑟_𝑘 is called residual signal and 𝑠 the order of the parity space or the observer-based residual generator. The following lemma given by Ding [6] describes the one-to-one mapping between the parity vector and the solutions of Luenberger equations.

Lemma 2.1 (see Ding [6]). Given any parity vector 𝛼_𝑠=[𝛼_𝑠,0𝛼_𝑠,1⋯𝛼_𝑠,𝑠], with 𝛼_𝑠,𝑖∈ℛ^1×𝑚, 𝑖=0,1,…,𝑠 and process model (2.1)-(2.2), then 𝐴_𝑧=00⋯010⋯0⋮⋱⋱⋮0⋯10∈ℛ^𝑠×𝑠𝛼,𝐿=−^𝑠,0𝛼_𝑠,1⋮𝛼_{𝑠,𝑠−1}𝛼,𝑇=^𝑠,1𝛼_𝑠,2⋯𝛼_𝑠,𝑠𝛼_𝑠,2…𝛼_𝑠,𝑠0𝛼⋮⋰⋰⋮_s,𝑠𝐶⋮0⋯0𝐶𝐴𝐶𝐴^𝑠−1𝑐_𝑧,=0⋯01∈ℛ^1×𝑠,𝑔=𝛼_𝑠,𝑠∈ℛ^1×𝑚, (2.11)(2.12) solve the Luenberger equations (2.6).

2.2. Preliminaries of Data-Driven Residual Generator Design

It is assumed that system matrices 𝐴, 𝐵, and 𝐶 and system order 𝑛 are unknown a priori; Ding et al. [19, 20] proposed an approach for data-driven design of observer-based residual generator, which briefly consists of two algorithms, that is, (i)Algorithm D2PS (from data to parity subspace),(ii)Algorithm PS2DO (from parity vector to diagnostic observer (DO)).

Algorithm 2.2. D2PS (from data to parity subspace).

Step 1. Generate data sets 𝑍 and construct (1/𝑁)𝑍𝑍^𝑇.

Step 2. Compute the SVD of (1/𝑁)𝑍𝑍^𝑇1𝑁𝑍𝑍^𝑇=𝑈_𝑧Σ_𝑧,100Σ_𝑧,2𝑈^𝑇_𝑧,𝑈_𝑧=𝑈_𝑧,11𝑈_𝑧,12𝑈_𝑧,21𝑈_𝑧,22,Σ_𝑧,2=0∈ℛ^{((𝑠−1)𝑚−𝑛)×((𝑠−1)𝑚−𝑛)},𝑈_𝑧,11∈ℛ^{(𝑠+1)𝑚×((𝑠+1)𝑙+𝑛)},𝑈^𝑇_𝑧,12∈ℛ^{((𝑠+1)𝑚−𝑛)×(𝑠+1)𝑚}. (2.13)

Step 3. Set Γ^⟂_𝑠=𝑈^𝑇_𝑧,12,Γ^⟂_𝑠𝐻_𝑠,𝑢=−𝑈^𝑇_𝑧,22.

Note that any row of matrix Γ^⟂_𝑠 is a parity vector. For a system with multiple output (𝑚>1), (𝑠+1)𝑚 may be significantly larger than 𝑛. In order to reduce the online computation, an order reduction algorithm is given by Ding et al. [19, 20] to achieve a reduced order 𝑠≤𝑛. For multiple output systems, 𝑠 may be significantly smaller than 𝑛.

Algorithm 2.3. PS2DO (from parity vector to DO).

Step 1. Select 𝛼_𝑠∈Γ^⟂_𝑠 and corresponding row 𝛽_𝑠∈Γ^⟂_𝑠𝐻_𝑠,𝑢 and form them as 𝛼_𝑠=𝛼_𝑠,0𝛼_𝑠,1⋯𝛼_𝑠,𝑠,𝛼_𝑠,𝑖∈ℛ^1×𝑚,𝛽_𝑠=𝛽_𝑠,0𝛽_𝑠,1⋯𝛽_𝑠,𝑠,𝛽_𝑠,𝑖∈ℛ^1×𝑙. (2.14)

Step 2. Set 𝐴_𝑧, 𝑐_𝑧, 𝐿, 𝑔 according to (2.11)-(2.12) and 𝐵^𝑇_𝑧=[𝛽^𝑇_𝑠,0⋯𝛽^𝑇_{𝑠,𝑠−1}].

Step 3. Construct the DO according to (2.9)-(2.10).

2.3. Problem Formulation

So far in our study, the data-driven fault detection scheme has been developed for LTI systems. However, The wide existence of systems with uncertain or normal variation parameters has not been considered enough in the literatures. In order to develop an efficient data-driven fault diagnosis scheme for such systems, it is necessary to(i)propose an efficient residual generator to deal with normal parameter variations in the process,(ii)determine proper threshold for fault detection purpose,(iii) develop related fault isolation strategy to complete the diagnosis task.

Without loss of generality, in the remaining part of this paper, the parameter variation rate is assumed bound in term of 𝑙²-norm. In addition, the persistent excitation condition for identification methods is assumed to be satisfied.

3. Data-Driven Design of Fault Diagnosis Scheme

3.1. Adaptive Residual Generator-Based Fault Detection Scheme

According to Lemma 2.1, the system (2.1)-(2.2) can be represented in following form: 𝑧_𝑘+1=𝐴_𝑧𝑧_𝑘+𝑄𝑢_𝑘,𝑦_𝑘𝜃, (3.1) where 𝑧_𝑘=𝑇𝑥_𝑘, 𝐴_𝑧=𝐴_𝑧−𝐿₀𝑐_𝑧, 𝐿₀ is a design parameter vector to ensure that the eigenvalues of 𝐴_𝑧 lie in the unit circle and 𝑄𝑢_𝑘,𝑦_𝑘=𝑄𝑢_𝑘,𝑦_𝑘𝐿₀𝑦^𝑇_𝑘∈ℛ^{𝑠×[𝑠(𝑚+𝑙)+𝑚]},𝑄𝑢_𝑘,𝑦_𝑘=𝒰_𝑘𝒴_𝑘∈ℛ^{𝑠×𝑠(𝑚+𝑙)},𝒰_𝑘=𝑢_1,𝑘×𝐼_𝑠×𝑠⋯𝑢_𝑙,𝑘×𝐼_𝑠×𝑠,𝒴_𝑘=𝑦_1,𝑘×𝐼_𝑠×𝑠⋯𝑦_𝑚,𝑘×𝐼_𝑠×𝑠,𝜃=𝜃𝑔^𝑇∈ℛ^{𝑠(𝑚+𝑙)+𝑚},𝐵𝜃=col_𝑧col(𝐿)∈ℛ^{𝑠(𝑚+𝑙)}, (3.2) with col(•) denotes a column-wise reordering of a matrix; that is, 𝑃=𝑝₁⋯𝑝_𝛼∈ℛ^𝛽×𝛼(𝑃)=𝑝,col⋮¹𝑝_𝛼∈ℛ^𝛽𝛼×1. (3.3) In the following study, set 𝐿₀=0 for the purpose of simplicity.

Note that in (3.1) the system matrices 𝐴, 𝐵, and 𝐶 are integrated into vector 𝜃, and the input and output signals are included in 𝑄(𝑢_𝑘,𝑦_𝑘). Any parameter variation in the original system can be reflected through the parameter variation rate defined as Δ_𝑘=𝜃_𝑘+1−𝜃_𝑘, which is bounded by Δ_𝑘≤𝑣, (3.4) where ‖•‖ denotes 𝑙²-norm. Let us firstly consider the basic case, that is, a constant parameter 𝜃; the adaptive residual generator is stated in the following theorem.

Theorem 3.1. Given the following adaptive residual generator which consists of three subsystems.
(i) Residual generator: 𝑧_𝑘+1=𝐴_𝑧𝑧_𝑘+𝑄𝑢_𝑘,𝑦_𝑘𝜃_𝑘+𝑉_𝑘+1𝜃_𝑘+1−𝜃_𝑘,𝑟_𝑘=𝑔_𝑘𝑦_𝑘−𝑐_𝑧𝑧_𝑘. (3.5)(3.6)
(ii) Auxiliary filter 𝑉_𝑘+1=𝐴_𝑧𝑉_𝑘+𝑄𝑢_𝑘,𝑦_𝑘∈ℛ^{𝑠×[𝑠(𝑚+𝑙)+𝑚]},𝜑_𝑘=𝑐_𝑧𝑉_𝑘−0⋯0𝑦^𝑇_𝑘∈ℛ^{𝑠(𝑚+𝑙)+𝑚}. (3.7)(3.8)
(iii) Parameter estimator 𝜃_𝑘+1=𝛾_𝑘𝜑^𝑇_𝑘𝑟_𝑘+𝜃_𝑘∈ℛ^{𝑠(𝑚+𝑙)+𝑚},𝛾_𝑘=𝜇𝛿+𝜑_𝑘𝜑^𝑇_𝑘𝜃,𝛿>0,0<𝜇<2,_𝑘=𝜃_𝑘𝑔_𝑘^𝑇,𝜃_𝑘∈ℛ^{𝑠(𝑚+𝑙)},𝑔_𝑘∈ℛ^1×𝑚. (3.9)(3.10)(3.11) it follows that the adaptive residual generator is stable and in the fault-free case the residual signal satisfies lim_𝑘→∞𝑟_𝑘=0. (3.12) Moreover, if the persistent excitation condition is satisfied; that is, there exist positive constants 𝛽₁, 𝛽₂ and integer Π such that for all 𝑘0<𝛽₁𝐼≤^𝑘+Π−1_𝑖=𝑘𝜑^T_𝑖𝜑_𝑖≤𝛽₂𝐼<∞, (3.13) the adaptive residual generator is exponentially stable, and the parameter estimation 𝜃_𝑘 converges to the true value 𝜃 with an exponential convergence rate: lim_𝑘→∞𝜃_𝑘=𝜃. (3.14)

Proof. The proof can be found in the earlier study by Ding et al. [19, 20].

Until now, the unknown parameter 𝜃 has been assumed constant. We would like to further consider the behavior of the adaptive residual generator (3.5)–(3.10) in case 𝜃 is a time-varying parameter and bounds by (3.4). To simplify the notations, define 𝜂_𝑘=𝑧_𝑘−𝑉_𝑘𝜃_𝑘,𝑧_𝑘=𝑧_𝑘−𝑧_𝑘,𝜃_𝑘=𝜃_𝑘−𝜃_𝑘. (3.15) After a straightforward calculation, it follows that 𝜂_𝑘+1=𝐴_𝑧𝜂_𝑘+𝜖_𝑘,𝜃_𝑘+1=𝐼−𝛾_𝑘𝜑^𝑇_𝑘𝜑_𝑘𝜃_𝑘+𝜖_𝑘,𝑟_𝑘=𝑐_𝑧𝜂_𝑘+𝜑_𝑘𝜃_𝑘, (3.16)(3.17)(3.18) with Θ_𝑘=−𝛾_𝑘𝜑^𝑇_𝑘𝑐_𝑧, Δ_𝑘=𝜃_𝑘+1−𝜃_𝑘, 𝜖_𝑘=Θ_𝑘𝜂_𝑘+Δ_𝑘, and 𝜖_𝑘=−𝑉_𝑘+1Δ_𝑘. According to (3.18), the residual 𝑟_𝑘 has a nonzero value since Δ_𝑘≠0. Assume that persistent excitation condition (3.13) is satisfied, the properties of adaptive residual generator (3.5)–(3.10) can be generalized in the following theorem.

Theorem 3.2. In case of Δ_𝑘≠0 and bounded by (3.4), the adaptive residual generator (3.5)–(3.10) ensures the following:
(i) the estimation error 𝜃_𝑘 converges exponentially to the set ℬ=𝜃_𝑘∣𝜃_𝑘≤𝛼^𝑘−𝑘0₁𝜃_𝑘0+1−𝛼^𝑘−𝑘0₁1−𝛼₁𝜖,0<𝛼₁<1, (3.19) where 𝜖 is a positive scalar such that ‖𝜖_𝑘‖<𝜖 and 𝑘₀ denotes the initial time sample;
(ii) the residual signal 𝑟_𝑘 converges exponentially to the set ℛ=𝑟_𝑘∣𝑟_𝑘≤𝜖+𝜑_𝑘𝛼^𝑘−𝑘0₁𝜃_𝑘0+𝜑_𝑘1−𝛼^𝑘−𝑘0₁1−𝛼₁𝜖, (3.20) where 𝜖 is a positive scalar such that ‖𝜖_𝑘‖<𝜖;
(iii) based on the assumption of the zero initial condition, that is, 𝜃_𝑘0=0, the normalized residual signal 𝑟_𝑘 satisfies 𝑟_𝑘≤√√𝑠𝑣+1+𝑠1−𝛼₁𝑣, (3.21) where 𝑟_𝑘=𝑟_𝑘/𝛿+𝜑_𝑘𝜑^𝑇_𝑘. Furthermore, if the process corrupted by noise/disturbance, the residual signal can be formulated as 𝑟_𝑘=𝑐_𝑧𝜂_𝑘+𝜑_𝑘𝜃_𝑘+𝑝_𝑘, (3.22) where 𝑝_𝑘 represents the influence of noise/disturbance on the residual signal. It follows that 𝑟_𝑘≤√√𝑠𝑣+1+𝑠1−𝛼₁𝑣+𝑝, (3.23) with 𝑝=sup_∀𝑘(𝑝_𝑘/𝛿+𝜑_𝑘𝜑^𝑇_k).

Proof. According to (3.17), for all 𝑘>𝑘₀, we have 𝜃_𝑘=𝑆_𝑘,𝑘0𝜃_𝑘0+^𝑘−1_𝑖=𝑘0𝑆_𝑘,𝑖+1𝜖_𝑘, (3.24) where 𝑆_𝑘,𝑘0 is the transition matrix of the linear time-varying system (3.17). Since (3.13) is satisfied, the system (3.17) is exponentially stable; and Astrom and Wittenmark [27]. Therefore, there exists a positive constant 0<𝛼₁<1 such that ‖𝑆_𝑘,𝑘0‖≤𝛼^𝑘−𝑘0₁. Consequently, from (3.17), we have 𝜃_𝑘≤𝛼^𝑘−𝑘0₁𝜃_𝑘0+^𝑘−1_𝑖=𝑘0𝑆_{𝑘⋅𝑖+1}𝜖_𝑘≤𝛼^𝑘−𝑘0₁𝜃_𝑘0+1−𝛼^𝑘−𝑘0₁1−𝛼₁𝜖. (3.25) Since 𝐿₀=0, 𝐴_𝑧=𝐴_𝑧, and all the eigenvalues of 𝐴_𝑧 are zero, it follows from (3.16) that 𝜂_𝑘≤𝜖. (3.26) The bound of residual signal can be straightforwardly obtained 𝑟_𝑘≤𝑐_𝑧𝜂_𝑘+𝜑_𝑘𝜃_𝑘≤𝜖+𝜑_𝑘𝛼^𝑘−𝑘0₁𝜃_𝑘0+𝜑_𝑘1−𝛼^𝑘−𝑘0₁1−𝛼₁𝜖. (3.27) It is easier to prove that 𝜀_𝑘𝛿+𝜑_𝑘𝜑^𝑇_𝑘≤√𝑠Δ_𝑘≤√𝑠𝑣,𝜀_𝑘≤1+√𝑠Δ_𝑘≤1+√𝑠𝑣. (3.28) Thus, set 𝜀𝛿+𝜑_𝑘𝜑^𝑇_𝑘=√𝑠𝑣,𝜖=1+√𝑠𝑣, (3.29) the normalized residual becomes 𝑟_𝑘≤√√𝑠𝑣+1+𝑠1−𝛼₁𝑣. (3.30) Equations (3.22)-(3.23) can be easily proved, and thus they are omitted here.

According to Theorem 3.2, in case 𝜃 is a time-varying parameter and bounds by (3.4) the residual 𝑟_𝑘 is bounded and the threshold 𝐽_th can be set as the right-hand side presented by (3.23); that is, 𝐽_th=√√𝑠𝑣+1+𝑠1−𝛼₁𝑣+𝑝. (3.31) The fault detection logic is given by 𝑟_𝑘≤𝐽_th,faultfree,𝑟_𝑘>𝐽_th,alarmforfault. (3.32)

Remark 3.3. It is of great interest to detect the faults that cause abnormal changes on physical parameters of the process. Although the identified parameter 𝜃 is physically meaningless, any abnormal physical parameters variations can influence 𝜃 and should be finally discovered by the residual signal. In practice, the bound of normal variation rate of 𝜃 given by (3.4) can be determined through the offline test data. The related threshold of residual signal is designed for the detection of abnormal parameter change, which is supposed to be faster than the normal parameter variation.

Remark 3.4. The order of residual generator 𝑠 could be significantly smaller than system order 𝑛 in multiple output systems (𝑚>1): 𝑠=(𝑛+1)/𝑚−1. An algorithm is proposed in Ding et al. [19, 20] for constructing the reduced order residual generator. Thus, if persistent excitation condition is satisfied, the estimation error 𝜃_𝑘 converges exponentially to a set determined by Δ_𝑘. For industrial process, the excitation mainly comes from the variation of process variables and measurement noise.

Remark 3.5. Another efficient way to determine the threshold for residual signal is based on statistical methods. Without special assumption on process data, the so-called kernel density estimation (KDE) is widely used in practice for estimating the probability density function of residual signals. Based on a large number of offline test data, a proper threshold can be chosen under given confidence level. More detailed description on KDE can be found in Silverman [28] and Martin and Morris [29].

3.2. Fault Isolation Scheme Design

In this subsection, the fault isolation scheme will be further introduced in the framework of adaptive residual generator.

Suppose that there exist 𝑆 classes of faults in the process including all potential abnormalities in sensors and actuators. Under the influence of 𝑖th class of fault, 𝑖=1,…,𝑆, the unknown parameter becomes 𝜃^𝑖_𝑓, which is assumed to belong to a known compact and convex set Θ^𝑖_𝑓∈ℛ^{𝑠(𝑚+𝑙)+𝑚}. Note that, the set Θ^𝑖_𝑓 could be offline identified by using the faulty data within the framework of adaptive scheme (3.5)–(3.10).

The proposed fault isolation strategy can be developed by integrating the fault information, that is, Θ^𝑖_𝑓. Based on it, 𝑆 fault isolation observers are constructed, in which the 𝑖th observer is only responsible for the 𝑖th set of fault. According to the fault detection scheme discussed in the last subsection, after a fault is detected at time 𝑡_𝑑, the fault isolation scheme is activated, such that the 𝑖th isolator is insensitive to the 𝑖th type of fault, but sensitive to other faults; see Zhang et al. [30]. In order to realize these requirements, the parameter projection method is utilized and the 𝑖th fault isolation observer has the following form: 𝑧^𝑖_𝑘+1=𝐴_𝑧𝑧^𝑖_𝑘+𝑄𝑢_𝑘,𝑦_𝑘𝜃^𝑖_𝑘+𝑉^𝑖_𝑘+1𝜃^𝑖_𝑘+1−𝜃^𝑖_𝑘,𝑟^𝑖_𝑘=𝑔^𝑖_𝑘𝑦_𝑘−𝑐_𝑧𝑧^𝑖_𝑘,𝑉^𝑖_𝑘+1=𝐴_𝑧𝑉^𝑖_𝑘+𝑄𝑢_𝑘,𝑦_𝑘∈ℛ^{𝑠×[𝑠(𝑚+𝑙)+𝑚]},𝜑^𝑖_𝑘=𝑐_𝑧𝑉^𝑖_𝑘−0⋯0𝑦^𝑇_𝑘∈ℛ^{𝑠(𝑚+𝑙)+𝑚},𝜃^𝑖∗_𝑘+1=𝜃^𝑖_𝑘+𝜑^𝑖_𝑘^𝑇𝑟^𝑖_𝑘𝛿+𝜑^𝑖_𝑘𝜑^𝑖_𝑘^𝑇∈ℛ^{𝑠(𝑚+𝑙)+𝑚},𝜃^𝑖_𝑘+1=𝒫_Θ^𝑖_𝑓𝜃^𝑖∗_𝑘+1, (3.33)(3.34)(3.35)(3.36) where 𝜃^𝑖_𝑘=𝜃^𝑖_𝑘𝑔^𝑖_𝑘^𝑇,𝜃^𝑖_𝑘∈ℛ^{𝑠(𝑚+𝑙)},𝑔^𝑖_𝑘∈ℛ^𝑚, (3.37) the 𝛿>0, 𝒫_Θ^𝑖_𝑓 denotes a projection operator that ensures 𝜃^𝑖_𝑘+1 lies in a known bounded convex subset Θ^𝑖_𝑓∈ℛ^{𝑠(𝑚+𝑙)+𝑚}. Details on the projection operator can be founded in Tao [26]. The following theorem states the properties of the 𝑖th isolation observer in case of the 𝑖th type of fault occurred.

Theorem 3.6. Given the 𝑖𝑡ℎ fault isolation observer in the form (3.33)–(3.36), suppose that there is a positive constant 𝑑_𝑖, such that for all 𝜃₁,𝜃₂∈Θ^𝑖_𝑓, it follows that 𝑑_𝑖=sup_{𝜃1,𝜃2∈Θ𝑖𝑓}𝜃₁−𝜃₂. (3.38) In case of the 𝑖𝑡ℎ type of fault occurs, one has the following:
(i) the 𝑖𝑡ℎ fault isolation observer is stable and 𝜃^𝑖_𝑘+1≤𝑑_𝑖,𝜃^𝑖_𝑘+1−𝜃^𝑖_𝑘≤𝑟^𝑖_𝑘, (3.39) where 𝑟^𝑖_𝑘=𝑟^𝑖_𝑘/𝛿+𝜑^𝑖_𝑘(𝜑^𝑖_𝑘)^𝑇, 𝜃^𝑖_𝑘+1=𝜃^𝑖_𝑘+1−𝜃^𝑖_𝑘+1,
(ii) based on the assumption of the zero initial condition, the normalized residual signal satisfies 𝑟^𝑖_𝑘²≤𝑑²_𝑖+2𝑠+2√𝑠+1𝑣²_𝑖+2+2√𝑠𝑣_𝑖𝑑_𝑖, (3.40) where 𝑣_i is a positive scalar such that ‖Δ^𝑖_𝑘‖≤𝑣_𝑖 with Δ^𝑖_𝑘=𝜃^𝑖_𝑘+1−𝜃^𝑖_𝑘.

Proof. According to the property of the projection operator, it follows that 𝜃^𝑖_𝑘+1∈Θ^𝑖_𝑓. From (3.38), we have 𝜃^𝑖_𝑘+1=𝜃^𝑖_𝑘+1−𝜃^𝑖_𝑘+1≤𝑑_𝑖. (3.41) It is evident that for all 𝑘, 𝜃^𝑖_𝑘+1−𝜃^𝑖_𝑘+1≤𝜃^𝑖∗_𝑘+1−𝜃^𝑖_𝑘+1, (3.42) and consequently, 𝜃^𝑖_𝑘+1−𝜃^𝑖_𝑘≤𝜃^𝑖∗_𝑘+1−𝜃^𝑖_𝑘≤𝜑^𝑖_𝑘^𝑇𝑟^𝑖_𝑘𝛿+𝜑^𝑖_𝑘𝜑^𝑖_𝑘^𝑇≤𝑟^𝑖_𝑘. (3.43) Now, define a new parameter 𝜃^𝑖∗_𝑘+1=𝜃^𝑖∗_𝑘+1−𝜃^𝑖_𝑘+1. (3.44) Using (3.42), we get 𝜃^𝑖_𝑘+1²−𝜃^𝑖_𝑘²≤𝜃^𝑖∗_𝑘+1²−𝜃^𝑖_𝑘². (3.45) The right-hand side of (3.45) becomes 𝜃^𝑖∗_𝑘+1²−𝜃^𝑖_𝑘²=𝜃^𝑖_𝑘+1−𝜃^𝑖∗_𝑘+1−𝜃^𝑖_𝑘+𝜃^𝑖_𝑘^𝑇𝜃^𝑖_𝑘+1−𝜃^𝑖∗_𝑘+1+𝜃^𝑖_𝑘−𝜃^𝑖_𝑘. (3.46) Note that, (3.35) can be reformulated as 𝜃^𝑖∗_𝑘+1=𝜃^𝑖_𝑘+𝜑^𝑖_𝑘^𝑇𝑟^𝑖_𝑘, (3.47) where 𝜑^𝑖_𝑘^𝑇=𝜑^𝑖_𝑘^𝑇𝛿+𝜑^𝑖_𝑘𝜑^𝑖_𝑘^𝑇. (3.48) For the normalized residual signal 𝑟^𝑖_𝑘, it is known that 𝑟^𝑖_𝑘=𝜑^𝑖_𝑘𝜃^𝑖_𝑘+𝜂^𝑖_𝑘, (3.49) with 𝜂^𝑖_𝑘=𝑐_𝑧𝜂^𝑖_𝑘𝛿+𝜑^𝑖_𝑘𝜑^𝑖_𝑘^𝑇,𝜂^𝑖_𝑘=𝐴_𝑧𝜂^𝑖_𝑘−1−𝑉^𝑖_𝑘Δ^𝑖_𝑘−1,Δ^𝑖_𝑘−1=𝜃^𝑖_𝑘−𝜃^𝑖_𝑘−1,0≤𝜑_𝑘𝜑^𝑇_𝑘≤1. (3.50) Combining (3.46), (3.47), and (3.49), we have 𝜃^𝑖_𝑘+1²−𝜃^𝑖_𝑘²≤−𝑟^𝑖_𝑘²+2𝑟^𝑖_𝑘𝜂^𝑖_𝑘+−2𝜑^𝑖_𝑘𝑟^𝑖_𝑘+2𝜃^𝑖_𝑘^𝑇+𝜃^𝑖_𝑘+1^𝑇−𝜃^𝑖_𝑘^𝑇Δ^𝑖_𝑘. (3.51) Equation (3.51) can be reformulated as 𝑟^𝑖_𝑘²≤−𝜃^𝑖_𝑘+1²+𝜃^𝑖_𝑘²+2𝜂^𝑖_𝑘²+2𝜂^𝑖_𝑘𝜃^𝑖_𝑘+𝜃^𝑖_𝑘+1−𝜃^𝑖_𝑘2𝜃^𝑖_𝑘+2𝜂^𝑖_𝑘+𝜃^𝑖_𝑘+1−𝜃^𝑖_𝑘. (3.52)(3.53) Since 𝜂^𝑖_𝑘≤𝜂^𝑖_𝑘𝛿+𝜑^𝑖_𝑘𝜑^𝑖_𝑘^𝑇≤√𝑠𝑣_𝑖, (3.54) according to (3.52), the result represented by (3.40) can be easily proved.

For the 𝑖th isolation observer, define the fault isolation index 𝐽^𝑖_𝑘=𝑟^𝑖_𝑘, (3.55) and the related threshold 𝐽^𝑖,iso_th. Based on Theorem 3.6, we have the following corollary.

Corollary 3.7. For the 𝑖𝑡ℎ isolation observer, the fault isolation threshold 𝐽^{𝑖,𝑖𝑠𝑜}_𝑡ℎ can be determined by 𝐽^{𝑖,𝑖𝑠𝑜}_𝑡ℎ=𝑑²_𝑖+2𝑠+2√𝑠+1𝑣²_𝑖+2+2√𝑠𝑣_𝑖𝑑_𝑖. (3.56) Moreover, if the process corrupted by disturbance and/or noise, the normalized residual is 𝑟^𝑖_𝑘=𝜑^𝑖_𝑘𝜃^𝑖_𝑘+𝜂^𝑖_𝑘+𝑝ⁱ_𝑘, (3.57) where 𝑝^𝑖_𝑘 represents the influence of noise/disturbance on the normalized residual signal. In this case, the threshold for fault isolation purpose is given by 𝐽^{𝑖,𝑖𝑠𝑜}_𝑡ℎ=𝑑²_𝑖+2𝑠+2√𝑠+1𝑣²_𝑖+𝑐𝑣_𝑖𝑑_𝑖+𝑐𝑝^𝑖𝑣_𝑖, (3.58) with 𝑝^𝑖=sup_∀𝑘𝑝^𝑖_𝑘,𝑐=2+2√𝑠. (3.59)

Proof. The proof is straightforward based on Theorem 3.6 and omitted here.

The fault isolation logic can be described as the following:(i)for the 𝑖th isolation observer, if ∃𝑡_𝑎>𝑡_𝑑 such that |𝐽^𝑖_𝑡𝑎|>𝐽^𝑖,iso_th, then the occurrence of the 𝑖th type of fault is excluded; (ii) otherwise, if |𝐽^𝑖_𝑡𝑎|<𝐽^𝑖,iso_th for ∀𝑡_𝑎>𝑡_𝑑, the 𝑖𝑡ℎ type of fault is occurred.

The sufficient condition of fault isolability is given by the following theorem.

Theorem 3.8. Based on the fault isolation observer (3.33)–(3.36), the 𝑖th type of fault, which detected at time 𝑡_𝑑, is isolable, if for the other 𝑆−1 fault isolation observers, ∃𝑡_𝑎>𝑡_𝑑 such that the following inequality is satisfied 𝜑^𝑟_𝑡𝑎𝜃^𝑟_𝑡𝑎>√𝑠𝑣_𝑟+𝑠𝑣²_𝑟+𝐽^{𝑟,𝑖𝑠𝑜}_𝑡ℎ², (3.60) where 𝐽^{𝑟,𝑖𝑠𝑜}_th=𝑑²_𝑟+2𝑠+2√𝑠+1𝑣²_𝑟+2+2√𝑠𝑣_𝑟𝑑_𝑟,𝑟=1,…,𝑆,𝑟≠𝑖. (3.61) Furthermore, if 𝑠=1, 𝐿₀=0, it follows that 𝜑^𝑟_𝑡𝑎𝜃^𝑟_𝑡𝑎>𝑣_𝑟+𝑑²_𝑟+6𝑣²_𝑟+4𝑣_𝑟𝑑_𝑟. (3.62)

Proof. For the 𝑟th fault isolation observer, we have 𝐽^𝑟_𝑡𝑎²=𝜑^𝑟_𝑡𝑎𝜃^𝑟_𝑡𝑎²+𝜂^𝑟_𝑡𝑎²+2𝜑^𝑟_𝑡𝑎𝜃^𝑟_𝑡𝑎𝜂^𝑟_𝑡𝑎≥𝜑^𝑟_𝑡𝑎𝜃^𝑟_𝑡𝑎²−2√𝑠𝑣_𝑟𝜑^𝑟_𝑡𝑎𝜃^𝑟_𝑡𝑎. (3.63) Straightforwardly, if for all 𝑟=1,…,𝑆,𝑟≠𝑖, ∃𝑡_𝑎>𝑡_𝑑 such that 𝜑^𝑟_𝑡𝑎𝜃^𝑟_𝑡𝑎²−2√𝑠𝑣_𝑟𝜑^𝑟_𝑡𝑎𝜃^𝑟_𝑡𝑎>𝐽^𝑟,iso_th², (3.64) the 𝑖th type of fault is isolable and directly (3.60) is proofed. In the case of 𝑠=1 and 𝐿₀=0, (3.62) is straightforward.

4. Application to Three-Tank System

The three-tank system (TTS) considered in our study is a laboratory setup located in the laboratory of Institute for Automatic Control and Complex Systems, University of Duisburg-Essen. The sketch is shown in Figure 1, which has typical characteristics of tanks, pipelines, and pumps used in the chemical industry and thus often serves as benchmark process for many control and monitoring relevant studies.

Figure 1 

Structure of TTS.

The plant consists of three cylindrical tanks which are serially interconnected with each other by cylindrical pipes with the cross-section of 𝑆_𝑛. The outflowing water is collected in a reservoir, which supplies pumps 1 and 2. 𝐻_max denotes the maximal height of tanks. The flow rates and water levels of tanks, represented by ℎ_𝑖, 𝑖=1,2,3, are measured throughout the process. By integrating a nonlinear controller, water levels ℎ₁ and ℎ₂ can be controlled. The detailed description of TTS can be found in Ding [6].

It is well known that the system matrices of TTS, which are achieved from linearization at different operation points, are different. In our experiment, the operation point of water level ℎ₁ is periodically changed in order to simulate the normal parameter variations in the process. An experiment including the following steps has been performed.(i)Place TTS at the operating point ℎ₁=35+sin(0.002𝑡)cm,ℎ₂=25cm, in which sin signal added to ℎ₁ leads to normal parameter variations. (ii)Use the adaptive scheme (3.5)–(3.10) to identify 𝜃 through the data collected at the operating point ℎ₁=35cm, ℎ₂=25cm with reduced order 𝑠=1 and 𝐿₀=0. Note that the system order 𝑛=3 can be determined by Algorithm D2PS, and based on it the reduced order 𝑠 is calculated according to the relationship 𝑠=(𝑛+1)/𝑚−1 with two system outputs; that is, 𝑚=2. (iii)Construct two residual generators: (a) an adaptive residual generator (3.5)–(3.10) (b) a standard one without adaptive scheme. (iv)Both the residual generators run for 2000s (seconds). The threshold 𝐽_th=0.78 is determined according to (3.23) with the parameters 𝜇=0.01, 𝛼₁=0.9997, 𝑣=7.3668×10⁻⁵, and 𝑝=0.55, which are chosen according to the offline test data.

Figure 2 shows the residual signals with and without the adaptive scheme. It is clear that the standard process monitoring method is unsuitable to monitor TTS with normal parameters variations that is apparent by the numerous false alarms.

(a)

(b)

Figure 2 

Residual signal.

The faults occurred in TTS can be classified as process fault, sensor fault, and actuator fault, which are shown in Table 1. To verify the performance of the proposed fault diagnosis scheme, the following experiment is carried out. (i)Offline: apply adaptive scheme (3.5)–(3.10) to identify Θ^𝑖_𝑓 through the 𝑖th type of faulty data with 𝑠=1 and 𝐿₀=0. (ii)Online: use adaptive residual generator (3.5)–(3.10) for fault detection purpose. If there exists time 𝑡_𝑑 such that |𝑟_𝑘|>𝐽_𝑡ℎ, the alarm is released. Simultaneously, the 𝑆−1 fault isolation observers (3.33)–(3.36) are activated, and the threshold 𝐽^𝑖_iso=0.8822 is determined according to (3.58) with the parameters 𝑣_𝑖=1×10⁻², 𝑑_𝑖=0.85, and 𝑝_𝑖=0.55.

Table 1 

The faults existed in TTS.

The fault diagnosis results of faults 1, 4, and 9, which represent the process, actuator, and sensor fault, are mainly presented in the following study. All these faults occur at the 500th second. The sensor fault 9 has 30% offset compared to the normal value and the actuator fault 4 represents 100% offset to the desired value. Figures 3(a), 4(a), and 5(a) show the residual signal from adaptive residual generator for fault detection purpose. It can be seen that the faults are successfully detected at the 505 s, the 507 s, and the 501 s, respectively.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 3 

Fault diagnosis for fault 1.

(a)

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Figure 4 

Fault diagnosis for fault 4.

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Figure 5 

Fault diagnosis for fault 9.

In the meanwhile, the fault isolation observers are activated, and the related fault isolation indices are shown in Figures 3(b)–3(j), 4(b)–4(j), and 5(b)–5(j) for faults 1, 4, 9, respectively. It is evident that the fault isolation indices from the 1st isolation observer (Figure 3(b)), the 4th isolation observer (Figure 4(e)), and the 9th isolation observer (Figure 5(j)) are consistently maintained under respective thresholds which indicate occurrence of these faults. On the other hand, the other subfigures show the isolation indices associated to other isolation observers. It is obvious that all of them exceed the related thresholds, which indicate the absence of these faults in the process.

5. Conclusion

In this paper, we have proposed an approach for data-driven design of fault diagnosis system, which consists of an adaptive residual generator and a bank of observers for fault detection and isolation purposes. Analytical results regarding the issues of adaptive observers, threshold calculation, and fault isolation strategy are discussed. The proposed design scheme is demonstrated on the simulation of laboratory-scale three-tank system, which shows satisfactory fault diagnosis performance.