Similar Fault Isolation of Discrete-Time Nonlinear Uncertain Systems: An Adaptive Threshold Based Approach

In this paper, a new concept of “similar fault” is introduced to the field of fault isolation (FI) of discrete-time nonlinear uncertain systems, which defines a new and important class of faults that have small mutual differences in fault magnitude and fault-induced system trajectories. Effective isolation of such similar faults is rather challenging as their small mutual differences could be easily concealed by other system uncertainties (e.g., modeling uncertainty/disturbances). To this end, a novel similar fault isolation (sFI) scheme is proposed based on an adaptive threshold mechanism. Specifically, an adaptive dynamics learning approach based on the deterministic learning theory is first introduced to locally accurately learn/identify the uncertain system dynamics under each faulty mode using radial basis function neural networks (RBF NNs). Based on this, a bank of sFI estimators are then developed using a novel mechanism of absolute measurement of fault dynamics differences. The resulting residual signals can be used to effectively capture the small mutual differences of similar faults and distinguish them from other system uncertainties. Finally, an adaptive threshold is designed for real-time sFI decision making. One important feature of the proposed sFI scheme is that: it is capable of not only isolating similar faults that belong to a pre-defined fault set (used in the training/learning process), but also identifying new faults that do not match any pre-defined faults. Rigorous analysis on isolatability conditions and isolation time is conducted to characterize the performance of the proposed sFI scheme. Simulation results on a practical application example of a single-link flexible joint robot arm are used to show the effectiveness and advantages of the proposed scheme over existing approaches.


I. INTRODUCTION
Fault isolation (FI) is an important and challenging problem that has been extensively investigated in the systems and control community [1]- [3]. It constitutes one of the critical steps in many complicated control system designs, e.g., faulttolerant control [4], [5] and condition-based maintenance [6], [7]. The past decades have witnessed considerable research The associate editor coordinating the review of this manuscript and approving it for publication was Haiquan Zhao .
interests in the field, leading to numerous interesting results on both analysis and design of various FI algorithms (see, e.g., [8]- [19]). In this paper, we will introduce a new concept of ''similar faults'' to the FI field, which defines an important class of faults that have small mutual differences in magnitude and associated system trajectories. The associated similar fault isolation (sFI) problem is of great importance for many modern engineering systems (e.g., [20]- [22]). A notable example is the high precision robotics [23], [24], which is often used for complex assembly or positioning VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ tasks to achieve high-precision motion. For such engineering applications, to meet stringent safety and precision requirements, a reliable sFI scheme is necessary. However, different from the control problem for which system faults could be effectively handled by advanced nonlinear control methods (e.g., [4], [5], [34]), the sFI problem is rather challenging, especially when the system is subject to various system uncertainties such as modeling uncertainty or disturbances. The technical difficulty lies in that for similar faults, their mutual differences are relatively small and could be easily concealed by other system uncertainties. This makes many existing FI schemes not applicable to dealing with the sFI problem. For example, in [15]- [17], since the effects of system uncertainties on measuring fault differences have not been properly dealt with, an isolatable fault is required to be sufficiently different from their mismatched faults, and generally the magnitude of fault differences is required to be larger than that of the system uncertainty, which may not be always feasible under the sFI context. In this paper, we focus on addressing the sFI problem for discrete-time nonlinear uncertain systems by proposing a novel adaptive threshold based sFI scheme. Specifically, we aim at achieving accurate isolation for various types of similar faults by distinguishing the occurring fault from system uncertainties. As shown in Fig. 1, the proposed sFI scheme consists of two phases, i.e., the fault dynamics identification phase, and fault isolation phase. The identification phase seeks to overcome the system uncertainty issue as discussed above. More specific, considering that the system operates under different faulty modes, the uncertain system dynamics will be accurately identified by using a deterministic learning (DL) based adaptive dynamics learning approach from [25]. The learned knowledge can be stored as a bank of constant radial basis function neural network (RBF NN) models. In the fault isolation phase, with the obtained RBF NN models, a bank of sFI estimators are designed through a novel mechanism of absolute measurement of fault dynamics difference. Their generated residuals are able to capture the fault differences (between the occurring fault and each trained fault), and distinguish such differences from other system uncertainties for sFI decision making. An adaptive threshold is designed based on these residuals, which can be used to pre-specify the required matching level between an isolatable fault and its matched fault in the training set. The realtime sFI decision making is based on an adaptive threshold mechanism, i.e., the occurring fault is identified similar to a unique trained fault when the related residual becomes smaller than the threshold. Through rigorous analysis, it is demonstrated that with the proposed sFI scheme, different types of similar faults that have been trained can be accurately isolated. In addition, for those faults that do not belong to the training fault set, the proposed sFI scheme is still capable of identifying them as new faults.
It should be noted that this research work is developed by extending our previous work of small fault detection (sFD) in [25]. Different from the sFD problem, the sFI problem considered in this paper is more challenging. The technical difficulty lies in how to effectively evaluate the mismatching/matching level between the occurring fault and each trained fault, especially when considering similar faults. Specifically, since the mutual differences among similar faults are relatively small, to accurately evaluate the fault matching level, it is necessary to study the functional structures of the fault differences in terms of not only the magnitude but also the sign. Most of the existing FI approaches only consider the magnitude factor. For example, in [14], [15], FI is achieved by monitoring the accumulated effect of fault differences. It requires that there is no change of sign and the magnitude of fault differences should be sufficiently large for a sufficiently long time. In the case when the fault differences have frequently-changing signs, their accumulated effect could be offset and approach zero, leading to possible FI misjudgment. In our scheme, this issue is addressed based on a novel mechanism of absolute measurement of fault dynamics differences. Furthermore, for sFI, a class of promising approaches [18], [19], [26] are developed based on smallest residual principle (SRP), which, however, cannot guarantee the desired fault matching level, leading to poor isolation reliability. For example, with the methods of [18], [19], [26], any kind of faults could be isolated (i.e., identified similar to a trained fault), even for those new faults that do not belong to the pre-defined set of trained faults. This deficiency has been overcome in this paper with our method by using an adaptive threshold mechanism instead of SRP.
The main contributions of this research work are summarized as follows: 1) The sFI problem of discrete-time nonlinear uncertain systems is addressed, where the considered similar faults are allowed to have relatively small mutual differences that could be easily concealed by other system uncertainties. 2) An adaptive threshold mechanism based sFI approach is proposed by developing a bank of sFI estimators through a novel mechanism of absolute measurement of fault dynamics differences, which are capable of capturing the small fault differences and distinguishing them from system uncertainties, and are also able to overcome the aforementioned sign issue suffered by existing FI approaches. 3) Rigorous performance analysis is conducted to derive the fault isolatability conditions and the fault isolation time under the proposed sFI scheme. The rest of this paper is organized as follows. In Section 2, the problem and some preliminary results are provided, including the adaptive dynamics learning approach for accurate identification of uncertain system dynamics. Section 3 presents the proposed sFI scheme and rigorous performance analysis. Simulation results are shown in Section 4 to verify the effectiveness and advantages of the proposed sFI scheme over existing approaches. Conclusions are made in Section 5.
Notation: Throughout this paper, R, R + and Z + denote, respectively, the set of real numbers, the set of positive real numbers and the set of positive integers; R m×n denotes the set of m × n real matrices; R n denotes the set of n × 1 real column vectors; I n denotes the n × n identity matrix; the open ball B r = {x ∈ R n : ||x|| < r} with r being an arbitrary positive constant; | · | is the absolute value of a real number; || · || is the 2-norm of a vector or a matrix, i.e. ||x|| = (x T x) 1 2 ; || · || 1 is the L 1 -norm of a vector or a matrix, i.e. ||x(k) ; a denotes the least integer greater than or equal to a real number a.

A. PROBLEM FORMULATION
Consider the following discrete-time nonlinear uncertain system: (1) where x ∈ R n , u ∈ R m are the system state and input, respectively, f : R n × R m → R n denotes the nominal healthy system dynamics, v : R n × R m → R n denotes the modeling system uncertainty, φ s : R n × R m → R n represents the functional structure of the deviation in system dynamics due to the fault s ∈ {1, · · · , N }, k 0 is an unknown fault occurrence time and β(k − k 0 ) characterizes a fault time profile, i.e., β(k − k 0 ) = 0 for k < k 0 , meaning that no fault occurs in system (1), and β(k − k 0 ) = 1 for k ≥ k 0 , meaning that the fault s occurs in (1).
Remark 1: In this paper, we assume that the system signals (x, u) in (1) are measurable in real-time for fault isolation. For those cases when full system states are not available (i.e., only partial states or output measurements are available), one can resort to state observer techniques (e.g., [35], [36]), which however is out of the scope of this paper and will be investigated in our future work.
Without losing any generality, for fault isolation purpose, we consider the case when system (1) operates in the s-th faulty mode, i.e., k ≥ k 0 . Note that the fault occurrence time k 0 does not need to be pre-known, that is, fault detection does not need to be achived in advance. Motivated by [18], [19], we characterize the so-called ''similar faults'' φ s (x, u) occurring in (1) as: 1) the difference between the fault s and each other faults (s ∈ {1, · · · , N }/{s}), denoted by φ s (x, u)− φ¯s(x, u), is small, which could be hidden within the modeling uncertainty, that is, the magnitude of each φ s (x, u)−φ¯s(x, u) could be smaller than that of the system uncertain dynamics v(x, u), and 2) the s-th faulty system trajectory (denoted by (x s , u s )) is close to eachs-th faulty system trajectory (denoted by (x¯s, u¯s)), i.e., dist (x s , u s ), (x¯s, u¯s) := max {min ||(x s , u s ), (x¯s, u¯s)||} < d ζ , (2) where d ζ is the constant number satisfying 0 < d ζ < d * ζ with d * ζ being the size of the NN approximation region to be specified later.

B. RBF NNs
Radial Basis Function Neural Network (RBF NN) is a neural network architecture that can solve any function approximation problem [27]. Specifically, the RBF networks can be described by ∈ R N n is the weight vector, N n is the NN node number, and S(Z ) = [s 1 (||Z − ς 1 ||), · · · , s N n (||Z − ς N n ||)] T , with s i (·) being a radial basis function, and ς i (i = 1, 2, · · · , N n ) being distinct points in state space. The Gaussian function ] is one of the most commonly used radial basis functions, where ς i = [ς i1 , ς i2 , · · · , ς iq ] T is the center of the receptive field. and η i is the width of the receptive field. The Gaussian function belongs to the class of localized RBFs in the sense that s i (||Z − ς i ||) → 0 as ||Z || → ∞. It is easily seen that S(Z ) is bounded and there exists a constant S M ∈ R + such that ||S(Z )|| S M [28]. VOLUME 8, 2020 It has been shown in [27], [29] that for any continuous function f (Z ) : Z → R where Z ⊂ R q is a compact set, and for the NN approximator, where the node number N n is sufficiently large, there exists an ideal constant weight vector W * , such that for any * > 0, f (Z ) = W * T S(Z ) + , ∀Z ∈ Z , where | | < * is the ideal approximation error. The ideal weight vector W * is an ''artificial'' quantity required for analysis, and is defined as the value of W that minimizes | | for Moreover, based on the localization property of RBF NNs [28], for any bounded trajectory Z (k) within the compact set Z , f (Z ) can be approximated by using a limited number of neurons located in a local region along the trajectory: It is shown in [28] that for a localized RBF network W T S(Z ) whose centers are placed on a regular lattice, almost any recurrent trajectory 1 Z (k) can lead to the satisfaction of the PE condition of the regressor subvector S ζ (Z ). This result is formally summarized in the following lemma.
Lemma 1 ( [30], [31]): Consider any recurrent trajectory Z (k): Z + → R q . Z (k) remains in a bounded compact set Z ⊂ R q , then for RBF network W T S(Z ) with centers placed on a regular lattice (large enough to cover compact set Z ), the regressor subvector S ζ (Z ) consisting of RBFs with centers located in a small neighborhood of Z (k) is persistently exciting (PE).

C. IDENTIFICATION OF FAULT DYNAMICS VIA DETERMINISTIC LEARNING
In our previous work [25], an adaptive dynamics learning approach has been proposed based on DL theory. It can be employed for the identification of the system uncertain dynamics under various faulty modes (i.e., the identification phase as shown in Fig. 1). Specifically, consider the following faulty dynamic systems: where s = 1, · · · , N is used to stand for the s-th faulty mode. Note that the modeling uncertainty v(x, u) and fault function φ s (x, u) in (3) cannot be decoupled from each other, we introduce a so-called general fault function η s (x, u) := v(x, u) + φ s (x, u), and rewrite the system (3) as: For this system, as typically adopted in the DL literature [25], [28], the following assumption is made.
Consider (4), it is known that there exists an ideal constant RBF NN weight W s * = [W s * 1 , · · · , W s * n ] ∈ R N n ×n (with N n denoting the number of NN nodes) such that where i = 1, · · · , n, s = 1, · · · , N , S(x, u) : R n × R m → R N n is a smooth RBF vector and s i,0 is the estimation error satisfying | s i,0 | < * with * being a positive constant that can be made arbitrarily small given sufficiently large number of neurons. Then, according to [25], we construct the following dynamical identifier: is a design parameter, andŴ s i ∈ R N n is the estimate of W s * i which is updated in real time using the following adaptive learning law: The rigorous analysis of the learning performance achieved by the identifier (6)- (7) has been provided in [25]. Specifically, from Lemma 1, the partial PE condition of the regression vector S(x s , u s ) is guaranteed by the given recurrent trajectory (x s , u s ) generated from the system (4). Based on this, from [25], the estimation errorx i will converge to a small neighborhood around the origin, and the con-structedŴ sT i S(x, u) will achieve locally-accurate approximation for η s i (x s , u s ). In addition, according to [25], the weightŝ W s i will converge to a small neighborhood of W s * i , and a constant NN weightW s i can be derived byW s i := with the estimation error s i is an arbitrarily small constant which can always be predesigned and obtained by constructing a sufficiently large number of neurons, as argued in [25].
Furthermore, according to [25], thanks to the generalization ability of constant RBF NN modelW sT i S(x s , u s ) in (8), its locally-accurate approximation for η s i (x, u) can be achieved in a local region along the trajectory (x s , u s ), with d * ζ > 0 characterizing the size of the NN approximation region. In such a region, a desirable accuracy level of the NN approximation (denoted by ξ * i ) will be guaranteed, i.e., Under the similar fault definition specified in (2), we have that each faulty trajectory (x¯s, u¯s) (generated from thē s-th faulty system,s ∈ {1, · · · , N }/{s}) satisfies dist((x s , u s ), (x¯s, u¯s)) < d * ζ , i.e., (x¯s, u¯s) ∈ s ζ . Thus, we obtain: In view of (8) and (9), it is demonstrated that no matter which faulty mode the system (1) operates in (i.e., ( with desired approximation accuracy. Consequently, the s-th faulty system (4) can be represented by: (1) operates ins-th faulty mode, i.e., (x, u) = (x¯s, u¯s).

III. MAIN RESULTS
In this section, based on the above learning results, a bank of novel sFI estimators will be designed and a sFI scheme will be developed based on adaptive threshold mechanism. Rigorous analysis of the performance of the proposed sFI scheme will also be provided.

A. DESIGN OF SIMILAR FAULT ISOLATION ESTIMATOR AND DECISION MAKING SCHEME
Assuming a fault l (similar to the trained fault l, l ∈ {1, · · · , N }) occurs, the monitored system is obtained as: (11) Recall that k 0 is the fault occurrence time. For this system, by utilizing the constant NN modelW sT i S(x, u) (i = 1, · · · , n, s = 1, · · · , N ) obtained from the learning phase, we propose to construct a bank of sFI estimators (i.e., residual systems) embedded with a novel mechanism of absolute measurement of faulty dynamics differences as follows: (12) where i = 1, · · · , n, s = 1, · · · , N ; e s i is the i-th state of the sth residual system with initial conditions set as e s i (0) = 0; b i is a design parameter satisfying 0 ≤ b i < 1; x i is the i-th state of the monitored system (11). Note that f i (x, u) +W sT i S(x, u) is able to represent the s-th faulty system (4), as verified in (10), (12) can thus be used to represent the absolute difference of the dynamics between the s-th faulty system (4) and the monitored system (11).
From Eqs. (10) and (11), the residual systems (12) can be rewritten as follows: Introducing a so-called fault mismatch function: s,l i (x, u) (13) is further equivalent to: where s i is the approximation error derived from (10) and can be made small.
Remark 2: The differences between the occurring fault l and the trained fault s can be represented by the fault mismatch function s,l i (x, u) (i = 1, · · · , n). The accumulated effects of such fault differences over the time [k 0 , k − 1] are characterized by the residual e s i in (14), which essentially manifests the matching level between the faults l and s. Remark 3: In the residual system (12), the constant model W sT i S(x, u) (i = 1, · · · , n) is employed to deal with the system uncertainty v i (x, u) in (11). This enables that the differences between the occurring fault l and each trained fault s can be captured and distinguished from the system uncertainty v i (x, u) by the residual e s i of (14). Furthermore, the residual system (12) is designed through a novel mechanism of absolute measurement of fault dynamics differences. This further enables that even when the fault difference s,l i (x, u) has frequently-changing signs, its accumulated effects would not be offset and the related residual will not approach zero.
Based on the residual systems (14), an adaptive threshold will be further developed for real-time sFI decision making. Such an adaptive threshold is a time-varying function used to bound the matched residual signal, i.e., e l i (k) (k ≥ k 0 ). For this purpose, we first consider an ideal scenario that the fault occurrence time is k 0 = 0 and the occurring fault l completely matches the trained fault l, i.e., l,l i (x, u) ≡ 0. Consider the l-th residual system in the form of (14). Noting that the system (11) operates in the l-th faulty mode for all k ≥ 0, i.e., (x, u) = (x l , u l ), and | l i | = | l i,1 | ≤ ε * i from (10), the residual e l i (k) in the form of (14) is obtained as: VOLUME 8, 2020 Based on (15), we further consider the general case when k 0 = 0 and l,l i (x, u) = 0, and design an adaptive threshold as:ē where q i > 1 is an auxiliary design parameter, b i is the parameter designed in residual systems (12), and ε * i represents the desired accuracy level of the NN approximation, as defined in (10), which can be pre-specified and obtained by setting a sufficiently large number of neurons in the identification phase (an example will be given in the simulation section for illustration).
Remark 4: In the adaptive threshold (16), the parameter q i is designed to pre-specify the required matching level between an isolatable fault (fault l ) and its matched fault (fault l). If the value of q i is selected small (close to 1), for successful isolation, the fault l is required to perfectly match the fault l (i.e., l,l i (x, u) ≈ 0), as verified in (15). Rigorous analysis on the effect of q i on the isolation performance (in terms of isolation accuracy and isolation time) will be conducted in the next section.
Based on the residual systems (12) and the threshold (16), an adaptive threshold based sFI scheme can be proposed, as illustrated in Fig. 2. Specifically, after a fault l occurs (i.e., k ≥ k 0 ), the matched residual e l i (k) will decrease and become very small such that e l i (k) <ē i (k) for k ≥ k l (with k l denoting absolute isolation time), whereas other mismatched residuals e¯l i (k) (l ∈ {1, · · · , N }/{l}) will remain large and satisfy e¯l i (k) ≥ē i (k) for all k ≥ k 0 . The idea is formalized as follows: sFI Decision Making Scheme: Compare the residual signals e s i (k) (i = 1, · · · , n, s = 1, · · · , N ) in (12) with the adaptive thresholdē i (k) in (16). For all i ∈ {1, · · · , n}, if there exists a unique l ∈ {1, · · · , N } and a finite time k l > 0 such that e l i (k l ) <ē i (k l ), then the occurring fault will be identified similar to fault l at time k l .
Remark 5: Existing FI methods (e.g., [18], [19]) are developed based the SRP, with which any possibly occurring faults, even a new fault that does not match any trained faults, will be identified similar to a trained fault, leading to a possible isolation misjudgment. Advanced over these schemes, our sFI scheme is based on an adaptive threshold mechanism, which can ensure desired isolation accuracy. More specific, only when the difference between the occurring fault l and the trained fault l is sufficiently small, the related residual e l i (i = 1, · · · , n) could become smaller than the thresholdē i . As for those occasions when no residual becomes smaller than the threshold, the occurring fault will be identified as a new fault.
Remark 6: With the proposed sFI scheme based on the adaptive threshold mechanism, similar fault isolation can be achieved without implementing fault detection in advance. This is because when no fault occurs, all residuals will remain larger than the adaptive threshold; when a trained fault recurs, the associated residual will then become smaller than the threshold, leading to successful fault isolation with no need of a priori fault detection.

B. FAULT ISOLATABILITY CONDITION AND ISOLATION TIME
In this section, we focus on studying the performance of the proposed sFI scheme in terms of isolatability condition and isolation time. For isolatability, we will examine under what conditions the occurring fault l could be identified similar to a unique trained fault l.
In view of (19) and (20), we have e l i (k f ) <ē i (k f ) and e¯l i (k f ) >ē i (k f ), thus, the occurring fault l will be identified similar to a unique trained fault l at time k f . This ends the proof.
Remark 7: Our approach is developed by considering a general case that the occurring fault l in system (11) does not necessarily have a perfect match to the trained fault l (i.e., l,l i (x, u) = 0). The isolatability condition (17) is thus proposed to restrict the matching level between the occurring fault l and the matched fault l. Particularly, if we consider a special case that the occurring fault l is completely identical to the trained fault l, i.e., l,l i (x, u) ≡ 0 (as typically adopted in the literature [15], [16]), the condition (17)  i (x, u)| is small. In the following, we denote the absolute fault isolation time as k l , and the fault isolation time K l as the time length between the occurrence time k 0 and the absolute fault isolation time k l , i.e., K l = k l − k 0 . To further investigate the performance of the proposed sFI scheme, we will derive the upper bound on the isolation time K l (denoted byK l ), i.e., sFI will be achieved at most inK l amount of time steps after the fault occurrence time k 0 .
Theorem 2 (Isolation Time): Consider the monitored system (11) and the fault isolation system consisting of (12) and (16). Under the proposed sFI scheme, if the following conditions hold: 1) for ∀i ∈ {1, · · · , n}, there exists a unique l ∈ {1, · · · , N } and some time steps with the total number of the elements in set u(k))|}. VOLUME 8, 2020 2) for each mismatched faultl (l ∈ {1, · · · , N }/{l}), condition (18) is satisfied; then, we have e l i (k l ) <ē i (k l ) and e¯l i (k l ) >ē i (k l ), i.e., the occurring fault l is isolated at time k l . The upper bound on the isolation time K l = k l − k 0 is given bȳ

Proof:
For the matched fault l, under condition (21), we have Recalling that for k > k 0 , (x, u) = (x l , u l ), and | l i | = | l i,1 | ≤ ε * i according to (10), and noting that E l i ≥ e l i (k 0 ), the state of the l-th residual system in (14) satisfies where From condition (21), we know that γ i > (q i − 1)ε * > µ i , and thus Therefore existence of T i can be guaranteed. Based on this, noting that 0 ≤ b i < 1, it can be verified that the term h=k i, which results in holds as long as under the condition (21). Furthermore, we have (21). Furthermore, we have e¯l i (k l ) >ē i (k l ) under the condition (18), as verified in Theorem 1. Thus, it can be deduced that the occurring fault l will be identified similar to the trained fault l in a finite time k l .
Using the constant modelsW sT i S(x, u) (i, s = 1, 2, 3, 4), from (12), we construct the following residual systems: To verify the superiority of our sFI scheme over existing approaches, we conduct simulation comparisons with existing approaches of the adaptive threshold mechanismbased FI scheme [15], [16] and the SRP-based FI schemes [18], [19]. All these approaches derive the residuals through a mechanism of direct measurements of system dynamics differences. Specifically, from [15], [16], the estimators for each trained faulty mode s (s = 1, 2, 3, 4) are constructed as: whereW sT i S(x, u) is the constant network obtained from the learning phase as discussed above, b i is set as 0.995. By comparing the dynamics of FI systems (31) with that of monitored system (29), the residuals used for FI are obtained as |x s i | = |x s i − x i |. In another class of FI schemes [18], [19], the estimators are constructed for not only the faulty modes s = 1, 2, 3, 4 but also the normal mode s = 0, which can be implemented following (31). The residual signals are generated in real time using an average L 1 norm, i.e., ||x s With the methods [15], [16] and [18], [19], for fault 3 (as defined above with F mf 1 = −0.0205), the FI is achieved in Figs. 8a and 8b, repsectively. It is seen that after fault occurrence, the mismatched residuals |x 4 1 | (of [15], [16]) and ||x 4 1 || 1 (of [18], [19]) will approach zero and finally be mixed up with the matched residuals, i.e., |x 3 1 | and ||x 3 1 || 1 . This implies that the fault difference between the faults 3 and 4 cannot be accurately captured by the residuals |x 4 1 |, ||x 4 1 || 1 . This is because the fault difference between the 3-rd fault type and 4-th fault type, i.e., 3,4 4 (x, u) shown in Fig. 5c, has frequently-changing signs. With the residual systems of [15], [16] and [18], [19], the accumulated effect of 3,4 4 (x, u) will be offset and cannot manifest itself in the residuals |x 4 1 |, ||x 4 1 || 1 . As a result, under the approaches [15], [16] and [18], [19], the fault 3 could be identified similar to fault 3 as well as fault 4. Similar observation can also be found when fault 4 (as defined above with F mf 2 = 0.302) occurs, as shown in Fig. 8c. These comparison results justify the superiority of our approach over the FI schemes of [15], [16], [18], [19].
In the following, we further investigate the relationship between the parameter q i of the adaptive threshold (16) and the isolation performance (in terms of isolation accuracy and isolation time). We consider a bank of tested faults that are similar to the fault 4 (with φ 4 4 (x, u) = − J m (F mf 2 cos(20 T s (k − 1)) + 1)x 4 with F mf 1 ≡ −0.02. The parameter F mf 2 are selected as F mf 2 = 0.302, 0.38, 0.48, 0.58 (or F mf 2 − F mf 2 = 0.002, 0.08, 0.18, 0.28), representing four tested faults with various fault differences. The fault occurrence time is assumed at k 0 = 1000. The simulation study is carried out by setting the parameter q i in different values, i.e., q i = 1.01, 2.5, 5, 7. The other system setups including the design of residual systems (30) and the parameters b i , ε * i VOLUME 8, 2020 (i = 1, 2, 3, 4) in (16) are kept the same as above. Due to the limitation of space, only the simulation results of i = 4 is presented (as shown in Fig. 9), the results of i = 1, 2, 3 are similar thus omitted here. We first consider the case of q 4 = 1.01. In Fig. 9, only the tested fault with F mf 2 − F mf 2 = 0.002 is isolatable (identified similar to trained fault 4), whereas those tested faults with F mf 2 − F mf 2 ≥ 0.08 will be identified as new faults. When increasing q 4 , as shown in Fig. 9, those tested faults with larger fault differences, i.e., the magnitude of F mf 2 − F mf 2 becomes larger, can be isolated, and the FI time k l is decreased. When the value q 4 becomes too large, e.g., q 4 = 5 and q 4 = 7, some tested faults (e.g., with F mf 2 − F mf 2 = 0.002, and F mf 2 − F mf 2 = 0.08) would be identified to match both the fault 4 and fault 3. In such cases, the 3-rd fault type and 4-th fault type will be considered as the same type of fault. These simulation results demonstrate the effect of the parameter q i on the isolation performance, which illustrate the discussions in Remark 10.

V. CONCLUSIONS
In this paper, we have proposed a novel adaptive threshold based sFI scheme for discrete-time nonlinear uncertain systems. The scheme consists of the fault dynamics identification phase, and the fault isolation phase. In identification phase, a DL-based adaptive dynamics learning approach has been employed to locally-accurately identify uncertain system dynamics. The learned knowledge can be obtained and stored in a bank of constant RBF NNs models. In fault isolation phase, with the obtained knowledge, a bank of sFI estimators (residual systems) have been developed through a novel mechanism of absolute measurement of fault dynamics differences. Their generated residuals are able to effectively capture the small fault differences and distinguish them from the system uncertainty. An adaptive threshold was then proposed based on the designed sFI estimators, such that the sFI decision making can be achieved by comparing the residuals with the designed threshold in real time. Rigorous analysis on isolatability condition and isolation time has been conducted to characterize the isolation capabilities of our scheme. The effectiveness of all the proposed results has been verified by extensive simulation studies on a practical application of a single-link robotic arm.
For the future work, we plan to extend the current results to (i) those cases when full system information is not available (e.g., only partial system states or measurement outputs are available) for fault isolation designs, promising approaches including the state observer techniques from [35], [36] will be explored; and (ii) applications of the proposed approach to fault isolation of real engineering systems, such as robotic manipulators or unmanned vehicles, etc.