Data-Driven Adaptive Observer for Fault Diagnosis

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.


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 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 where 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 s Γ s 0, 2.5 for the so-called parity vector α s α s,0 α s,1 • • • α s,s ∈ R 1× s 1 m .The design of an observerbased residual generator is achieved by solving the so-called Luenberger equations TA − A z T LC, c z T gC, B z TB, 2.6 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 s y s,k − H s,u u s,k , 2.8 and the observer-based residual generation r k gy k − c z z k ∈ R.

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.

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 .
Step 1. Generate data sets Z and construct 1/N ZZ T .
Step 2. Compute the SVD of 1/N ZZ T

2.13
Step 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 ⊥ s H s,u and form them as

2.14
Step 2. Set A z , c z , L, g according to 2.11 -2.12 and Step 3. Construct the DO according to 2.9 -2.10 .

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 l 2 -norm.In addition, the persistent excitation condition for identification methods is assumed to be satisfied.

Adaptive Residual Generator-Based Fault Detection Scheme
According to Lemma 2.1, the system 2.1 -2.2 can be represented in following form: where z k Tx k , A z A z −L 0 c z , L 0 is a design parameter vector to ensure that the eigenvalues of A z lie in the unit circle and with col • denotes a column-wise reordering of a matrix; that is, In the following study, set L 0 0 for the purpose of simplicity.

Mathematical Problems in Engineering
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 where • denotes l 2 -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: it follows that the adaptive residual generator is stable and in the fault-free case the residual signal satisfies Moreover, if the persistent excitation condition is satisfied; that is, there exist positive constants β 1 , β 2 and integer Π such that for all k the adaptive residual generator is exponentially stable, and the parameter estimation θ k converges to the true value θ with an exponential convergence rate: 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 After a straightforward calculation, it follows that 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 where is a positive scalar such that k < and k 0 denotes the initial time sample; (ii) the residual signal r k converges exponentially to the set where is a positive scalar such that k < ; (iii) based on the assumption of the zero initial condition, that is, θ k 0 0, the normalized residual signal r k satisfies where r k r k / δ ϕ k ϕ T k .Furthermore, if the process corrupted by noise/disturbance, the residual signal can be formulated as where p k represents the influence of noise/disturbance on the residual signal.It follows that 23 Proof.According to 3.17 , for all k > k 0 , we have where S k,k 0 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 . Consequently, from 3.17 , we have

3.25
Since L 0 0, A z A z , and all the eigenvalues of A z are zero, it follows from 3.16 that The bound of residual signal can be straightforwardly obtained It is easier to prove that

3.28
Thus, set the normalized residual becomes 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,

3.31
The fault detection logic is given by |r k | > J th , alarm for fault.

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 .

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: 3.34 where 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 θ 1 , θ 2 ∈ Θ i f , it follows that

3.38
In case of the ith type of fault occurs, one has the following: (i) the ith fault isolation observer is stable and based on the assumption of the zero initial condition, the normalized residual signal satisfies 40 Proof.According to the property of the projection operator, it follows that θ i k 1 ∈ Θ i f .From 3.38 , we have

3.41
It is evident that for all k, and consequently,

3.43
Now, define a new parameter

3.44
Using 3.42 , we get

3.45
The right-hand side of 3.45 becomes

3.46
Note that, 3.35 can be reformulated as where

3.48
For the normalized residual signal r i k , it is known that

3.53
Since 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 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

3.56
Moreover, if the process corrupted by disturbance and/or noise, the normalized residual is 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

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 t a | > J i,iso th , then the occurrence of the ith type of fault is excluded; ii otherwise, if |J i t a | < 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.

3.62
Proof.For the rth fault isolation observer, we have

3.63
Straightforwardly, if for all r 1, . . ., S, r / i, ∃t a > t d such that the ith type of fault is isolable and directly 3.60 is proofed.In the case of s 1 and L 0 0, 3.62 is straightforward.

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.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 h 1 and h 2 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 h 1 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 h 1 35 sin 0.002t cm, h 2 25 cm, in which sin signal added to h 1 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 h 1 35 cm, h 2 25 cm with reduced order s 1 and L 0 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 2000 s seconds .The threshold J th 0.78 is determined according to 3.23 with the parameters μ 0.01, α 1 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.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 L 0 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.
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.
In the meanwhile, the fault isolation observers are activated, and the related fault isolation indices are shown in Figures 3 b -

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.
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.
• • • α s,s , with α s,i ∈ R 1×m , i 0, 1, . . ., s and process model 2.1 -2.2 , then 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 Theorem 3.8.

Table 1 :
The faults existed in TTS.