IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 1, JANUARY 2014

137

Adaptive Approximation for Multiple Sensor Fault Detection and Isolation of Nonlinear Uncertain Systems Vasso Reppa, Member, IEEE, Marios M. Polycarpou, Fellow, IEEE, and Christos G. Panayiotou, Senior Member, IEEE Abstract— This paper presents an adaptive approximationbased design methodology and analytical results for distributed detection and isolation of multiple sensor faults in a class of nonlinear uncertain systems. During the initial stage of the nonlinear system operation, adaptive approximation is used for online learning of the modeling uncertainty. Then, local sensor fault detection and isolation (SFDI) modules are designed using a dedicated nonlinear observer scheme. The multiple sensor fault isolation process is enhanced by deriving a combinatorial decision logic that integrates information from local SFDI modules. The performance of the proposed diagnostic scheme is analyzed in terms of conditions for ensuring fault detectability and isolability. A simulation example of a single-link robotic arm is used to illustrate the application of the adaptive approximation-based SFDI methodology and its effectiveness in detecting and isolating multiple sensor faults. Index Terms— Adaptive estimation, fault detection, fault diagnosis, learning systems.

I. I NTRODUCTION

D

URING the last few years, there has been a rapid increase in the use of sensing devices for monitoring and control applications such as in manufacturing, power systems, and environmental monitoring. Technological advances facilitated the wide deployment of spatially distributed sensors, which provide temporal and spatial information through wired and wireless links [1]. However, along with the key benefits associated with the wide use of distributed sensors, there are also some risks because of the severe consequences that may arise if their operation is subjected to temporary or permanent faults. Sensor faults can cause system instability and/or degradation of tracking performance, as well as affect the system supervision, leading to loss of information fidelity and wrong decisions that can jeopardize human safety. Therefore, the application of fault detection and isolation (FDI) mechanisms for capturing sensor faults and localizing the faulty sensor(s) as soon as possible becomes very critical.

Manuscript received May 29, 2012; accepted February 15, 2013. Date of publication March 29, 2013; date of current version December 13, 2013. The research leading to these results has received funding from the European Union Seventh Framework Programme ([FP7/2007-2013] under Grant Agreement n° 270428. The authors are with the Department of Electrical and Computer Engineering, KIOS Research Center for Intelligent Systems and Networks, University of Cyprus, Nicosia 1678, Cyprus (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2013.2250301

One approach to the problem of sensor fault detection and isolation (SFDI) is the physical (hardware) redundancy method [2], where multiple sensor devices (typically three sensors) are used to measure the same variable. However, in most applications, physical redundancy is not practical because of high cost of installation and maintenance, as well as because of space or weight restrictions. Therefore, analytical redundancy approaches for SFDI are widely used, relying on a nominal mathematical model describing the healthy system [3], [4]. Among them are the observer-based techniques, which are commonly used for detecting and isolating sensor faults in nonlinear systems. The observer-based SFDI techniques consist of two main stages: 1) the residual generation based on the mathematical model of the system; and 2) the decision-making process. In general however, the mathematical model is usually inaccurate because of modeling errors or disturbances. To handle the uncertainty, which is the main source of false alarms, and simultaneously enhance the sensitivity to faults, several robust FDI techniques are developed [5], [6]. The isolation of multiple sensor faults in nonlinear dynamic systems is a challenging problem. Several researchers, using nonlinear models of the monitored systems, developed diagnostic schemes based on a single nonlinear observer, capable of isolating multiple sensor faults with specific time profile or magnitude, or assuming a maximum number of their multiplicity [7]–[10]. On the other hand, there is also some research activity in addressing the problem of detecting and isolating multiple sensor faults using a bank of observers. The bank of observers is usually deployed in a generalized observer scheme (GOS) for isolating consecutive1 but not simultaneous2 sensor faults, in which each observer is driven by all sensor outputs except for one [11]; a differential geometry-based scheme for nonconcurrent faults [12], [13]; and a dedicated observer scheme (DOS) for isolating simultaneous faults, in which each observer of the bank is driven by one sensor output [14]–[16]. A powerful approach to robust FDI for nonlinear uncertain systems is based on the use of learning techniques [17]. The main concept behind the learning approach for FDI is the adaptive approximation of the unmodeled system behavior by using adaptive approximation models (e.g. sigmoidal neural networks, radial basis functions, and support vector machines) and nonlinear estimation schemes [18]–[21]. In [22], we developed an architecture for distributed SFDI 1 Consecutive faults: there is a time gap between the onset time of the faults. 2 Simultaneous faults: the onset time of the faults is the same.

2162-237X © 2013 IEEE

138

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 1, JANUARY 2014

applied to nonlinear uncertain systems. The key motivation of this paper is the enhancement of the observer-based SFDI methodology by exploiting the capability of the adaptive approximation to learn the modeling uncertainty during the initial stage of nonlinear system operation [23]. The nonlinear functional approximator of modeling uncertainty is used for optimizing the adaptive thresholds, thus enhancing the fault detectability and isolability of the proposed SFDI scheme for even simultaneous faults [24]. The proposed SFDI technique is developed for detecting and isolating multiple sensor faults in a distributed framework. Particularly, we design local SFDI modules, each of which is tailored to monitor the operation of a set of sensors (local sensing subsystem) and capture the occurrence of faults in it. This is realized by comparing the set of structured residuals to adaptive thresholds, which are both designed to be sensitive to the faults of the local sensing systems [6]. The multiple sensor fault isolation task aims to initially localize the set of sensors containing the faulty sensors through local SFDI modules and then to identify the individual faulty sensors by processing the information acquired from the local SFDI modules using a combinatorial decision logic. For analyzing the performance of the proposed distributed SFDI architecture, we establish conditions characterizing the sensor fault detectability and isolability. The main contribution of this paper is twofold. Initially, we derive a stable adaptive nonlinear estimation scheme, used for online learning of the modeling uncertainty. This scheme is based on the adaptive approximation of the modeling uncertainty, which may be function of unmeasured states, using measurements that are corrupted by noise. Next, we design distributed local SFDI modules that utilize the nonlinear functional approximator in the generation of observer-based residuals and adaptive thresholds. From the viewpoint of the design of the diagnostic scheme, we extend the classical DOS for linear systems [25], to obtain structured residuals and adaptive thresholds, sensitive to local sensing faults in a class of nonlinear uncertain systems. By modeling the effects of sensor faults and the effects of modeling uncertainty and noise on the residuals and adaptive thresholds, we show that the proposed learning approach can significantly improve the FDI capabilities of the SFDI scheme without introducing any false alarms. This paper is organized as follows. Problem formulation and design assumptions are presented in Section II. The details of the adaptive approximation methodology are presented in Section III. The design of the distributed SFDI methodology is presented in Section IV, while analytical results for the fault detectability and isolability are presented in Section V. The application of the proposed SFDI architecture to a single-link robotic arm is presented in Section VI, followed by concluding remarks in Section VII. II. P ROBLEM F ORMULATION Consider a class of nonlinear uncertain systems described by x(t) ˙ = Ax(t) + γ (x(t), u(t)) + η(x(t), u(t)) y(t) = C x(t) + d(t) + f (t)

(1) (2)

where x ∈ Rn is the state vector, u ∈ R is the input vector, y ∈ Rm is the sensor output vector, with m ≤ n, A ∈ Rn×n is a matrix representing the linearized part of the state equation, γ : Rn ×R → Rn is the nonlinear system dynamics, η: Rn × R → Rn is the system modeling uncertainty, d ∈ Rm is the measurement noise vector and f ∈ Rm is the sensor fault vector. The input vector u is generated by a feedback control algorithm based on the sensor output vector y and possibly some desired reference signal vector r . The fault vector is given by f = [ f 1 , . . . , f m ] , where fi , i ∈ {1, . . . , m} is the change in the output because of a fault in the i th sensor. It is assumed that A, γ , C are known, whereas the terms η, d, and f are unknown. Throughout this paper, the following assumptions are used: Assumption 1: The state vector x and input vector u generated by a feedback controller remain bounded before and after the occurrence of multiple sensor faults, i.e., there are compact regions of stability U ⊂ R and X ⊂ Rn for u(t) and x(t), respectively, such that (x(t), u(t)) ∈ X × U, for all t > 0. Assumption 2: The nonlinear vector field γ is locally Lipschitz in x ∈ X , for all u ∈ U and t > 0     γ (x(t), u(t)) − γ (x(t), ˆ  (3) ˆ u(t)) ≤ λγ x(t) − x(t) where λγ is the known Lipschitz constant. Assumption 3: The noise corrupting the measurements of each sensor is unknown but uniformly bounded |di (t)| ≤ d¯i ,

i ∈ {1, . . . , m}

(4)

where d¯i is known. Assumption 1 describes that the feedback controller, which generates the input u, can retain the boundedness of the state variables even in the presence of multiple sensor faults. This assumption is required because in this paper we deal exclusively with the FDI problem, not the fault accommodation issue. Assumption 2 describes the class of nonlinear systems, whose nonlinearity is differentiable with respect to the state vector, while its compact region of interest is known. This class of systems is encountered in many robotic and power systems applications, in which sinusoidal terms are used in their mathematical models. Assumption 3 describes a practical representation of the available knowledge for the sensor noise, which is provided in any given range of operation by the most manufacturers or introduced when a noise-free analog signal is converted into a digital one with a finite number of digits. The objective of this paper is to design and analyze a methodology for detecting and isolating multiple sensor faults in nonlinear uncertain systems described by (1) and (2), using the information provided by the application of an adaptive approximation technique for learning the unknown nonlinearilty η(x, u) [26]. The general architecture describing this methodology is shown in Fig. 1. The proposed adaptive approximation-based SFDI methodology is a two-step process. Particularly, we propose the use of an adaptive approximation methodology, which learns the modeling uncertainty η(x, u) for an initial time interval, assuming that no sensor is faulty. Then, a distributed SFDI methodology is applied aiming at detecting and isolating multiple sensor faults. The outcome of

REPPA et al.: ADAPTIVE APPROXIMATION FOR MULTIPLE SFDI

139

function between η and ηˆ for all x ∈ X and u ∈ U. Each subvector θi∗ ∈ Rqθi , i = 1, . . . , n corresponds to the weights associated with the  uncertainty ηi in the state x i , and qθ is defined by qθ = ni=1 qθi . The error term eη , defined as eη (x(t), u(t)) = η (x(t), u(t)) − η(x(t), ˆ u(t); θ ∗ )

(6)

is the (unknown) minimum functional approximation error (MFAE) [26], which is the minimum possible deviation between the unknown function η and the adaptive approximator ηˆ in the ∞-norm sense over the compact set X × U, with θ ∗ defined as      ∗   sup η (x, u) − ηˆ (x, u; θ) . (7) θ = arg min θ∈

Fig. 1. General architecture for distributed sensor fault detection and isolation with adaptive approximation.

the adaptive approximation leads to more accurate observerbased local SFDI modules, thus enhancing the fault detectability and isolability of the distributed SFDI methodology. The distributed SFDI is the procedure through which we detect the presence of multiple sensor faults by isolating the sets of sensors containing the faulty sensors. This is realized using local SFDI modules that are responsible for detecting faults in distributed sets of sensors. For the isolation of multiple faulty sensors, the information generated by the local SFDI modules is integrated and processed using a combinatorial decision logic. III. A DAPTIVE A PPROXIMATION The goal of the adaptive approximation is to learn for an initial time interval [0, TL ] the unknown functional nonlinearity η(x(t), u(t)), thus reducing the uncertainty in the system model and optimizing the thresholds in the distributed SFDI, which will be described in the next section. The following assumptions are required for the adaptive approximation of the unknown η(x, u). Assumption 4: During the time interval [0, TL ], there are no sensor faults, i.e., f (t) = 0 for t ∈ [0, TL ]. Assumption 5: The pair (A, C) is observable. A. Parametric Model The state equation (1) can be rewritten as x(t) ˙ = Ax(t) + γ (x(t), u(t)) + η(x(t), ˆ u(t); θ ∗ ) +eη (x(t), u(t))

(5)

→ Rn is an adaptive where η: ˆ Rn × R  × R q θ nonlinear approximation model (e.g., radial basis function, sigmoidal wavelet network, etc.) and  neural network,  

∈ Rqθ is the (unknown) optimal θ ∗ = θ1∗  , . . . , θn∗  parameter (weight) vector that minimizes a suitable cost

x∈X ,u∈U

The vector θ ∗ is constrained within a compact region  ⊂ Rqθ , which will be used by the projection operator later on. The set  is selected sufficiently large not to undermine the approximation power of ηˆ by increasing the MFAE error eη . In the special case of linearly parameterized approximators [26], ηˆ is of the form η(x(t), ˆ u(t), θ ∗ ) = G (x(t), u(t)) θ ∗ 

(8) 

G (x(t), u(t)) = (g1 (x(t), u(t))) ⊕ (g2 (x(t), u(t))) ⊕ . . . ⊕ (gn (x(t), u(t))) (9) where G: Rn × R → Rn×qθ is the regressor and gi : Rn × R → Rqθi , for i ∈ {1, . . . , n} is a bounded function with ⊕ denoting the direct sum [27]. For simplicity, in this paper we consider linearly parameterized approximators, thus replacing η(x, ˆ u; θ ∗ ) by G(x, u)θ ∗ hereafter. B. Adaptive Nonlinear Estimation Scheme The adaptive nonlinear estimation scheme consists of a nonlinear observer, a filter for the regressor G and the adaptive law for updating the adjustable parameter vector θˆ (t)  ˙ˆ x(t) = A x(t) ˆ + γ (x(t), ˆ u(t)) + L y(t) − C x(t) ˆ +G(x(t), ˆ u(t))θˆ (t) + (t)θˆ˙ (t) (10) ˙ (t) = (A − LC)(t) + G(x(t), ˆ u(t))

  ˙ ˆ θˆ (t) = P (C(t)) y(t) − C x(t)

(11) (12)

where xˆ ∈ Rn is the estimation of the state vector x with initial conditions set to x(0) ˆ = 0 following Assumption 4. The matrix L ∈ Rn×m is chosen such that the matrix A − LC is Hurwitz (all the eigenvalues are in the left half complex plane).     ˆ ˆ ∈ Rqθ contains The vector θ (t) = θ1 (t) , . . . , θˆn (t) the adjustable parameters of the adaptive approximator, with each subvector θˆi ∈ Rqθi , i = 1, . . . , n corresponding to the parameter estimate of θi∗ . The initial parameter vector θˆ (0) ˆ and matrix (0) are chosen as θ(0) = 0 and (0) = 0, respectively. In the adaptive law (12), ∈ Rqθ ×qθ is a symmetric, positive definite learning rate matrix, while the projection operator P restricts the adjustable parameter vector θˆ (t) to the predefined set  [26]. The reason for using the projection operator is to prevent parameter drift in the presence of nonzero MFAE error

140

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 1, JANUARY 2014

eη (x, u). The matrix  is the filtered version of G, necessary for ensuring the stability property of the adaptive nonlinear estimation scheme, while the term (t)θ˙ˆ (t) is added to compensate the estimation error because of θˆ . The idea of the linear filtering of G can also be found in adaptive nonlinear schemes used for fault diagnosis in both time invariant and time varying systems, which can be stochastic or deterministic [18], [28]–[30]. The adaptive nonlinear estimation scheme is applied for the time interval [0, TL ]. In general, the training period TL is a design constant, which should be sufficiently large to allow learning of the modeling uncertainty, while there should be no fault during the training period. Another approach for reducing the modeling error is offline identification of uncertainties. In general, the adaptive approximation scheme provides a more flexible methodology for learning the uncertainties in the sense that the training time, TL , can be adjusted online based on some criterion involving the estimation error. However, offline and online techniques can be combined for enhancing the learning performance [23]. Remark 1: In [28] and [31], a similar adaptive nonlinear estimation scheme was used for estimating the fault function for the special case in which the unknown function depends on the sensor outputs y and not the state x. Such schemes can be used when the unknown nonlinearity η is a function of the measured, noiseless states. When this is not the case, the design and analysis of the adaptive nonlinear estimation scheme become more challenging. C. Stability Analysis The stability of the adaptive nonlinear estimation scheme is analyzed in the following theorem. We define the state ˆ the output estimation estimation error as εx (t) = x(t) − x(t), ˆ and the parameter estimation error as ε y (t) = y(t) − C x(t) ˜ = θˆ (t) − θ ∗ . error as θ(t) Theorem 1: Under Assumptions 1–5, the adaptive nonlinear estimation scheme (10)–(12) guarantees that: 1) the output estimation error ε y (t), the state estimation ˜ are error εx (t), and the parameter estimation error θ(t) ˜ uniformly bounded; i.e., ε y (t), εx (t), θ (t) ∈ L ∞ ; 2) there exists a positive constant w1 and a bounded function v(t) (that depends on the MFAE and the uncertainty terms) such that for all finite t > 0, the output estimation error satisfies t t   ε y (τ )2 dτ ≤ w1 + |v(τ )|2 dτ ; (13) 0

0

3) in the absence of measurement noise (d(t) = 0), if the bounded function v is square integrable, it is implied that lim ε y (t) = 0 and lim θ˙ˆ (t) = lim θ˙˜ (t) = 0. t →∞ t →∞ t →∞ Proof: 1) Using (5) and (10), the state estimation error dynamics are described by ε˙ x (t) = A0 εx (t) + γ˜ (t) + G (x(t), u(t)) θ ∗ + eη (x(t), u(t)) ˆ − (t)θ˙ˆ (t) − Ld(t) − G(x(t), ˆ u(t))θ(t) (14)

where A0 = A − LC and γ˜ (t)  γ (x(t), u(t)) − γ (x(t), ˆ u(t)). (15)  ∗ By adding and subtracting G x(t), ˆ u(t) θ and using ˙ G x(t), ˆ u(t) = (t) − A0 (t), we obtain  ε˙ x (t) = A0 εx (t) + γ˜ (t) + G (x(t), u(t)) − G(x(t), ˆ u(t)) θ ∗ + eη (x(t), u(t)) − Ld(t)   ˙ − (t) − A0 (t) θˆ (t) − θ ∗ − (t)θ˙ˆ (t).

(16)

 = G(x(t), u(t)) − G(x(t), ˜ By letting G(t) ˆ u(t)) and θ(t) = ˆθ (t) − θ ∗ , we obtain

 ∗  ε˙ x (t) = A0 εx (t) + (t)θ˜ (t) + γ˜ (t) + G(t)θ + eη (x(t), u(t)) − Ld(t) −

d ((t)θ˜ (t)). dt

(17)

Let ε˜ x (t) = εx (t) + (t)θ˜ (t). Then (17) can be written as ∗  + eη (x(t), u(t)) − Ld(t). ε˙˜ x (t) = A0 ε˜ x (t) + γ˜ (t) + G(t)θ (18)

Therefore, given (0) = 0, we obtain t

˜ + e A0 t εx (0) + εx (t) = −(t)θ(t) e A0 (t −τ ) γ˜ (τ ) − Ld(τ ) 0 ∗  )θ + eη (x(τ ), u(τ )) dτ. (19) +G(τ   As A0 is Hurwitz, we can choose ρ0 , ξ0 such that e A0 t  ≤ ρ0 e−ξ0 t , for all t ≥ 0. Therefore, based on Assumptions 1–5, we obtain t −ξ0 t |εx (t)| ≤ E 0 (t) + ρ0 e λγ eξ0 τ |εx (τ )| dτ (20) 0

where

t     E 0 (t) = |(t)| θ˜ (t) + ρ0 e−ξ0 t |εx (0)| + ρ0 e−ξ0 (t −τ )(|L|d¯ 0     +  (21) G(τ ) |θ ∗ | + eη (x(τ ), u(τ )) dτ

and d¯ is a bound for d(t) such that |d(t)| ≤ d¯ for all t > 0. Applying the Bellman-Gronwall Lemma [32] results in t |εx (t)| ≤ E 0 (t) + ρ0 λγ E 0 (s)e−(ξ0 −ρ0 λγ )(t −s)ds. (22) 0

To guarantee that εx (t) remains bounded, it is required that the constants ξ0 , ρ0 are selected such that ξ0 > ρ0 λγ .

(23)

Considering that: 1) the MFAE eη and the regressor G are bounded functions in X × U; and 2) because of the use of parameter projection, θˆ (t) ∈ L ∞ and θ˜ (t) ∈ L ∞ , it yields that the state estimation error is bounded, i.e., εx (t) ∈ L ∞ . The latter implies that the output estimation error ε y (t) = y(t) − C x(t) ˆ is bounded, i.e., ε y (t) ∈ L ∞ as ε y (t) = Cεx (t) + d(t). 2) Equation (19) can be written as εx (t) = v 1 (t) + v 2 (t) − (t)θ˜ (t)

(24)

REPPA et al.: ADAPTIVE APPROXIMATION FOR MULTIPLE SFDI

141

where v 1 (t) and v 2 (t) are the solutions of

then according to Barbalat’s Lemma, lim ε y (t) = 0 and based t →∞ on (12) lim θ˙ˆ (t) = lim θ˙˜ (t) = 0.

∗  v˙1 (t) = A0 v 1 (t) + γ˜ (t) − Ld(t) + G(t)θ

+ eη (x(t), u(t)) v˙2 (t) = A0 v 2 (t)

(25) (26)

with initial conditions v 1 (0) = 0 and v 2 (0) = ε˜ x (0) = εx (0). Let the Lyapunov function candidate be V (t) =  1˜  −1 θ˜ (t) + ∞ |Cv (τ )|2 dτ . The time derivative of V θ (t) 2 t 2 considering (12) is given by ˜  −1 θ˙˜ (t) − |Cv 2 (t)|2 V˙ (t) = θ(t)

 ˜  −1 P (C(t)) ε y (t) − |Cv 2 (t)|2 . = θ(t)

(27)

The projection operator P restricts the parameter estimation to a predefined compact set . As θ ∗ ∈ , the projection operator cannot increase V˙ (as compared with the nonprojection case) [26]. Therefore V˙ (t) ≤ θ˜ (t) (C(t)) ε y (t) − |Cv 2 (t)|2

 ˜ = ε y (t) C (t)θ(t) − |Cv 2 (t)|2 .

(28)

Using (24) and by completing the squares, the time derivative of the Lyapunov function satisfies V˙ (t) ≤ ε y (t) C (−εx (t) + v 1 (t) + v 2 (t)) − |Cv 2 (t)|2    ε y (t)2   2 − ε y (t) |Cv 1 (t)| + |Cv 1 (t)| ≤− 4    ε y (t)2   2   ¯ ¯ − − ε y (t) d + d 4    ε y (t)2   2 − − ε y (t) |Cv 2 (t)| + |Cv 2 (t)| 4   ε y (t)2 − + |Cv 1 (t)|2 + d¯2 4   ε y (t)2 + |Cv 1 (t)|2 + d¯2 . ≤− (29) 4 The above inequality guarantees that V˙ (t) ≤ 0 for 2   1/2 2 2   ¯ |Cv 1 (t)| + d ≤ ε y (t) .  1/2 Let v(t)  2 |Cv 1 (t)|2 + d¯2 . Integrating (29) from 0 to t, we obtain t t   ε y (τ )2 dτ ≤ 4 [V (0) − V (t)] + |v(τ )|2 dτ 0 0 t |v(τ )|2 dτ = w1 + (30)

t →∞

t →∞

IV. D ISTRIBUTED S ENSOR FAULT D ETECTION AND I SOLATION This section provides the design details of the distributed sensor fault detection and isolation (SFDI) method. The distributed SFDI method is applied after the adaptive approximation procedure. A. Distributed SFDI Architecture In the proposed SFDI architecture, the m sensors used for monitoring and controlling the nonlinear dynamic system described by (1) and (2) constitute the sensing system S characterized by S: y(t) = C x(t) + d(t) + f (t)

(31)

while the j th sensor, j ∈ {1, . . . , m} is denoted as S{ j }, characterized by S{ j }: y j (t) = C j x(t) + d j (t) + f j (t)

(32)

where C j ∈ R1×n is the j th row of C. The sensing system S is decomposed into M local sensing subsystems S (I ) , I ∈ {1, . . . , M} and 1 < M ≤ m, whereas each local sensing subsystem contains m I sensors. Each subsystem S (I ) is defined by the local output vector y (I ) ∈ Rm I S (I ) : y (I ) (t) = C (I ) x(t) + d (I ) (t) + f (I ) (t)

(33)

where w1 = sup {4 [V (0) − V (t)]} is a positive constant

where d (I ) , f (I ) ∈ Rm I . The matrix C (I ) ∈ Rm I ×n is made up of m I rows of C, while y (I ) is a column vector made up of m I elements of y = [y1 , . . . , ym ] (correspondingly for d (I ) and f (I ) ). The sensing subsystems S (I ) may be disjoint or overlapping. In the latter case, some sensors may belong to more than one sensing subsystem. In any case, each sensor belongs to at least one local sensing subsystem. In some application domains, an overlapping sensing formulation is useful in handling largescale distributed systems, where some critical sensors are monitored by more than one local SFDI module. The idea of overlapping can also be found in [33] and [34], in which a distributed method for diagnosing process faults that affect shared states of interconnected systems is designed. Let’s define the set J = {1, . . . , m} ∈ Nm + as the index I set and J (I ) ∈ Nm as an extraction index set [34], which is + described by 

(34) J (I ) = j : S{ j } ∈ S (I )

(as V (t) is uniformly bounded). 3) As ε y (t), (t) ∈ L ∞ , then based on (12), θ˙ˆ (t) ∈ L ∞ . Given (14) and considering the boundedness of εx , G, d, eη , θ ∗ , θˆ , θ˙ˆ , it yields that ε˙ x ∈ L ∞ . In the absence of measurement noise, ε y (t) = Cεx (t), hence ε˙ y ∈ L ∞ . If the bounded function v is square integrable, i.e., v ∈ L 2 , the outcome of the second part of theorem can be valid for t → ∞, leading to ε y ∈ L 2 . As ε y , ε˙ y ∈ L ∞ , and ε y ∈ L 2 ,

(I ) = ∅ and and satisfies the following properties:  (I 1) J (I ) ) ⊂ J implying that m I  card J ∈ {1, . . . , m − 1} J (card(·) stands for the cardinality of a set); and 2) m I is the minimum number of sensors in the local sensing system S (I ) that satisfies Assumption 6 below. Assumption 6: The pair A, C (I ) is observable. The observability of the pair A, C (I ) in Assumption 6 is required for the design of nonlinear observers used in

0 t ≥0

142

Fig. 2.

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 1, JANUARY 2014

Architecture for distributed sensor fault detection and isolation.

the distributed SFDI scheme. An overview of the proposed distributed SFDI architecture is shown in Fig. 2. For each local sensing subsystem S (I ) , we design a local SFDI module (referred to as (I )-local SFDI module hereafter). The (I )-local SFDI module is responsible for monitoring the local sensing subsystem S (I ) and detecting the presence of faults in it. The latter is the outcome of the local decision process of each module, during which m I residuals are compared with m I adaptive thresholds. Particularly, the j th residual of the (I )) local SFDI module, j ∈ J (I ) denoted as ε(I y j , is computed using the output of a nonlinear observer that is the estimation of the state vector, denoted as xˆ (I ) ) (I ) (I ) ε(I y j (t) = y j (t) − C j xˆ (t), j ∈ J

(35)

where y j is the output of sensor S{ j } that belongs to S (I ) . The j th adaptive threshold of the (I )-local SFDI module, j ∈ J (I ) , ) (I ) denoted as ε¯ (I y j , is a function that varies with respect to xˆ and the system input u. The nonlinear functional approximator of the modeling uncertainty, derived during the learning period [0, TL ], is embedded in the nonlinear observer and also considered in the computation of the adaptive thresholds for enhancing fault detectability and isolability. The detection of faults in each local sensing subsystem is the first stage of multiple sensor fault isolation, as we can localize in a distributed manner the subsets of sensors from the global set that contains the faulty sensors. This capability stems from the fact that the observer-based residuals and adaptive thresholds are designed based on the local output vector and local estimated state vector, i.e. they are structured with respect to local sensor faults. The information generated by the (I )local SFDI module is a boolean function representing the decision about the presence of faults in S (I ) (zero, when there

is no sensor fault detection, and one when sensor faults are detected). At the second stage, the objective is to localize the individual faulty sensors in each local sensing subsystem by collecting and processing the information generated by M ∗ local SFDI modules in the combinatorial decision logic module shown in Fig. 2. These modules are designed to monitor M ∗ ≤ M neighboring and (possibly) overlapping local sensing subsystems. At this stage, a small group of neighboring local SFDI modules can be used for isolating faulty sensors in a local sensing subsystem that is detected as faulty. The need for the combinatorial process of the decisions of the local SFDI modules stems from the fact that the local sensing subsystems may contain more than one sensor and some sensors may belong to more than one local sensing subsystems. Remark 2: The idea of designing multiple process units (i.e. local SFDI modules) and an aggregation unit that fuses the information of some process units, can be found in the well-developed FDI methods based on interacting multiple models (IMM) [35] and multiple sensor fusion (MSF) for stochastic systems [2], [36]. However, in IMM-based techniques, the multiple models describe the system in healthy and various faulty system modes and are designed using the a priori knowledge of the possible system faults. Moreover, in IMM the fault isolation is realized when the measurements are consistent with the estimation provided by a filter that includes a fault model, with high probability. On the contrary, in the proposed scheme, the local SFDI modules do not use fault models, while the FDI is realized when the measurements are inconsistent with the estimation provided by the observer. Fault diagnosis using MSF-based techniques can be conducted in two different ways: 1) local filters, working in parallel, fuse multiple sensor data for generating

REPPA et al.: ADAPTIVE APPROXIMATION FOR MULTIPLE SFDI

143

state estimates, whereas a global filter combines the local state estimates to derive an improved global estimate that is further used in a fault diagnosis algorithm; and 2) each local processing unit using the measurements of a sensor executes hypothesis tests (hypothesizing a number of models describing the faults) and derive a local diagnosis decision, whereas the local diagnosis results are aggregated in a fusion center for obtaining a global decision. In contrast to the former MSF-based technique, the proposed SFDI method performs diagnosis both in the local process units (local SFDI modules), as well as in the aggregation (fusion) unit (combinatorial sensor fault decision logic). Moreover, for improving the state estimation generated in each module, we add the functional approximator of the modeling uncertainty, derived through the adaptive approximation methodology, instead of fusing local estimates generated by multiple sensors. In contrast to the latter MSF-based technique, the local SFDI modules are designed without using sensor fault models. B. Design of Local SFDI Modules The design details of the observer-based generation of the residuals and the computation of the adaptive thresholds of each local SFDI module are provided next. 1) Observer-Based Generation of Residuals: The (I )observer using the output of the local sensing subsystem S (I ) defined in (33), is structured as follows: xˆ˙ (I ) (t) = A xˆ (I ) (t) + γ (xˆ (I ) (t), u(t)) + ηˆ L (xˆ (I ) (t), u(t))

 +L (I ) y (I ) (t) − C (I ) xˆ (I ) (t) (36) where xˆ (I ) ∈ Rn is the estimation of the state vector x, with initial conditions xˆ (I ) (TL ) = x(T ˆ L ), y (I ) is defined in (I ) n×m I (33) and L ∈ R is the observer gain matrix. The vector field ηˆ L (xˆ (I ) (t), u(t))  G(xˆ (I ) (t), u(t))θˆ (TL ) is the nonlinear functional approximator with the parameter vector derived at the end of the learning period [0, TL ], which satisfies the following assumption. Assumption 7: The error between the modeling uncertainty η(x, u) and the functional approximator ηˆ L (x, u) is bounded for all x ∈ X , u ∈ U and t > TL by some known functional η   η(x(t), u(t)) − ηˆ L (x(t), u(t)) ≤ η(x(t), u(t)) (37) where η is locally Lipschitz in x ∈ X for all u ∈ U and t > 0. The time-varying bounding function η can be computed by applying a set-membership technique [37], [38] to dynamical nonlinear systems that satisfy Assumption 2 and can be transformed in or approximated by linear-in-the-parameters (or linear-in-the-weights) models [39]. (I ) Let us define εx (t)  x(t) − xˆ (I ) (t), the state estimation error for all t ≥ TL ; using (1) and (36), it yields ) (I ) (I ) (I ) ε˙ (I x (t) = A ε x (t) + γ (x(t), u(t)) − γ ( xˆ (t), u(t))

+ η(x(t), u(t)) − ηˆ L (xˆ (I ) (t), u(t))

 − L (I ) d (I ) (t) + f (I ) (t)

A(I ) = A − L (I ) C (I ) .

(38) (39)

The solution of (38) for all t ≥ TL satisfies the following equation: ) ε(I x (t)

=

(I ) ) e A (t −TL ) ε(I x (TL ) +



t

e

A(I ) (t −τ )

 γ (x(τ ), u(τ ))

TL

− γ (xˆ (I ) (τ ), u(τ )) + η(x(τ ), u(τ )) − ηˆ L (xˆ (I ) (τ ), u(τ ))

 −L (I ) d (I ) (τ ) + f (I ) (τ ) dτ.

(40)

The stability of the error dynamics given in (38) for all t ≥ TL is analyzed in Theorem 2, considering Assumption 7.  Theorem 2: For the observable pair A, C (I ) , if the observer gain L (I ) is chosen such that: 1) the matrix A(I ) (I ) (I ) in (39) is stable; and  2) there exist positive constants ρ , ξ  (I )  A(I ) t  such that e  ≤ ρ (I ) e−ξ t and ξ (I ) > ρ (I ) , where  = λγ + ληˆ + λη¯ (λγ , ληˆ , λη¯ are the Lipschitz constants of γ , ηˆ L , and η, ¯ respectively), the state estimation error given in (40) is uniformly bounded for all t ≥ TL if no sensor fault has occurred in S (I ) . In case of sensor faults in S (I ) , the state estimation error given in (40) is uniformly bounded if the sensor fault vector f (I ) is uniformly bounded. ) Proof: For t ≥ TL , the bound of ε(I x (t) given in (40), can be determined as        (I )   A(I ) (t −TL )   (I )  εx (TL ) εx (t) ≤ e t    A(I ) (t −τ )   + e  γ (x(τ ), u(τ )) TL   − γ (xˆ (I ) (τ ), u(τ )) + η(x(τ ), u(τ ))         − ηˆ L (xˆ (I ) (τ ), u(τ )) + L (I )  f (I ) (τ ) dτ t     A(I ) (t −τ ) (I )  (I )  L d (τ )dτ. + (41) e TL

(I )

Consider  that: 1) x¯ L is a bound for εx (TL ) such that   (I ) εx (TL ) ≤ x¯ L , for all x ∈ X ; and 2) d¯ (I ) is a bound for  (I )    ¯ (I ) that d (I ) (t) [defined in (33)] such   d (t) ≤ d . Choosing (I ) (I )   ρ (I ) , ξ (I ) > 0 such that e A t  ≤ ρ (I ) e−ξ t for t > 0 and   (I )  (I )  ρd(I ) , ξd(I ) > 0 such that e A t L (I )  ≤ ρd(I ) e−ξd t , for t > 0 we obtain (I )  

 (I ) ρ (I )  (I )  εx (t) ≤ ρ (I ) e−ξ (t −TL ) x¯ L + d(I ) d¯ (I ) 1 − e−ξd (t −TL ) ξ d  t    (I )  )   (I )  + ρ (I ) e−ξ (t −τ ) λγ ε(I x (τ ) + η˜ (τ ) TL     (I )   (I )  (42) + L   f (τ ) dτ

where η˜ (I ) (t) = η (x(t), u(t)) − ηˆ L (xˆ (I ) (t), u(t)).

(43)

144

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 1, JANUARY 2014

By adding and subtracting   ηˆ L (x(t), u(t)) in (43) and using (37), a bound for η˜ (I ) (t) can be computed as follows:    (I )  ¯ u(t)) η˜ (t) ≤ η(x(t),     (44) + ηˆ L (x(t), u(t)) − ηˆ L (xˆ (I ) (t), u(t)) . Assuming that the functional approximator ηˆ L is locally Lipschitz in x ∈ X , for all u ∈ U and t > 0 [40], i.e.,    ) ηˆ L (x(t), u(t)) − ηˆ L (xˆ (I ) (t), u(t)) ≤ ληˆ ε(I x (t), where ληˆ is the known Lipschitz constant, we have      )   (I )  ¯ u(t)) + ληˆ ε(I (45) η˜ (t) ≤ η(x(t), x (t) . By adding and subtracting η( ¯ xˆ (I ) (t), u(t)), inequality (45) can be rewritten as    (I )  ¯ xˆ (I ) (t), u(t)) η˜ (t) ≤ η(       )   ¯ u(t))−η( ¯ xˆ (I ) (t), u(t)) + ληˆ ε(I + η(x(t), (t) . (46) x

To guarantee that Z (I ) (t) is finite ∀ t, it is required to select the variables ρ (I ) , ξ (I ) such that ξ (I ) > ρ (I ) .

(53)

When there is no sensor fault in S (I ) , i.e., under healthy conditions, the state estimation error is defined as (I ) ) ε(I x H (t)  x(t) − xˆ H (t)

(54)

(I )

where xˆ H is the estimation of the state vector provided by the observer given in (36) under healthy conditions, i.e. y (I ) (t) = C (I ) x (I ) (t) + d (I ) (t) in (36). The solution of (38) with f (I ) (t) = 0 is t (I ) (I ) ) εx(IH) (t) = e A (t −TL ) ε(I (T ) + e A (t −τ ) (γ (x(τ ), u(τ )) L xH − −

TL (I ) γ (xˆ H (τ ), u(τ )) + η (x(τ ), u(τ ))  ) (I ) (I ) ηˆ L (xˆ (I H (τ ), u(τ )) − L d (τ ) dτ.

(55)

As stated in Assumption 7, η¯ is locally  Lipschitz, i.e.    (I )  η(x(t), ¯ u(t)) − η( ¯ xˆ (I ) (t), u(t)) ≤ λη¯ εx (t), where λη¯ is the known Lipschitz constant. Thus, it yields       (I )   ¯ xˆ (I ) (t), u(t)) + ληˆ + λη¯ εx(I ) (t) . (47) η˜ (t) ≤ η(

) (I ) Thus, under healthy conditions, ε(I x (t) = ε x H (t). Considering (32) with f j (t) = 0, (35) and (54), the j th residual, j ∈ J (I ) under healthy conditions is described by

Using (47) in (42), we obtain t     (I )  (I )   )  (I ) ρ (I ) e−ξ (t −τ ) ε(I εx (t) ≤ E (t) + x (τ ) dτ

Equation (56) represents the effects of modeling uncertainty and noise on the j th residual. After the first time instant of sensor fault occurrence in S (I ) , (I ) denoted as t◦ , the state estimation error is described by

(48)

TL

) (I ) (I ) ε(I . y j H (t) = C j ε x H (t) + d j (t), j ∈ J

where E (I ) is defined as

 (I ) L)x ¯ L + (I ) d¯(I ) 1 − e−ξd (t −TL ) E (I ) (t) = ρ (I ) e ξd t

(I ) ρ (I ) e−ξ (t −τ ) η( ¯ xˆ (I ) (τ ), u(τ )) + TL       (49) + L (I )   f (I ) (τ ) dτ. ρd(I )

−ξ (I ) (t −T

and  = λγ + λη¯ + ληˆ .

with ϒ(t, s) = exp

t 

eξ

(I ) τ

ρ (I ) e−ξ

(I ) τ

 dτ . Let Z (I ) (t) be

s

(I )

the bound of εx (t) such that    (I )  εx (t) ≤ Z (I ) (t).

(51)

Then, the time-varying bound Z (I ) is described by Z (I ) (t) = E (I ) (t) +ρ

(I )





where is the estimation of the state vector provided by the observer given in (36) under faulty conditions, i.e., y (I ) (t) (I ) in (36) is described by (33). The solution of (38) for t ≥ t◦ satisfies the following equation:



 (I ) A(I ) t −t◦ ) (I ) t◦ ε(I εx(I ) (t) = e x t (I ) + e A (t −τ ) (γ (x(τ ), u(τ )) (I )

t◦

E TL

(I )

(τ )e

 − ξ (I ) −ρ (I )  (t −τ )

dτ . (52)

(I )

−γ (xˆ F (τ ), u(τ )) + η (x(τ ), u(τ )) (I )

−ηˆ L (xˆ F (τ ), u(τ ))

 −L (I ) d (I ) (τ ) + f (I ) (τ ) dτ. (58)





 (I ) (I ) (I ) (I ) (I ) (I ) Given that εx t◦ = εx H t◦ , where εx H t◦ is defined through (55) for t = t◦(I ) , we obtain (I )

) A (t −TL ) (I ) ε(I εx H (TL ) x (t) = e t◦(I ) (I ) ) + e A (t −τ ) (γ (x(τ ), u(τ )) − γ (xˆ (I H (τ ), u(τ )) TL  ) (I ) (I ) + η (x(τ ), u(τ )) − ηˆ L (xˆ (I H (τ ), u(τ )) − L d (τ ) dτ t (I ) (I ) + e A (t −τ ) (γ (x(τ ), u(τ )) − γ (xˆ F (τ ), u(τ )) t◦(I )

t

(57)

xˆ F(I )

(50)

If no sensor fault has occurred in S (I ) , i.e. f (I ) (t) = 0, then E (I ) is uniformly bounded. In case of sensor faults in S (I ) , E (I ) is uniformly bounded if f (I ) is uniformly bounded. Applying the Bellman-Gronwall Lemma [32] results in t   (I ) (I )  (I )  E (I ) (s)eξ s ϒ(t, s)ds εx (t) ≤ E (I ) (t) + ρ (I ) e−ξ t  TL

(I ) ) (I ) ε(I x (t) = x(t) − xˆ F (t), t ≥ t◦

(56)

(I )

+ η (x(τ ), u(τ )) − ηˆ L (xˆ F (τ ), u(τ )) − L (I ) d (I ) (τ ) − L (I ) f (I ) (τ ))dτ.

(59)

REPPA et al.: ADAPTIVE APPROXIMATION FOR MULTIPLE SFDI

By adding and subtracting (59) becomes

t

(I )

t◦

eA

(I ) (t −τ )

145

) γ (xˆ (I H (τ ), u(τ ))dτ ,

) (I ) (I ) ε(I x (t) = ε x H (t) + ε x F (t)

where Y A(Ij) , Y B(Ij) are described by t (I ) (I ) (I ) −ζ (t −τ ) Y A j (t) = αj e j Z (I ) (τ )dτ

(60)

) (I ) −ζ (I j (t −TL )

(I )

Y B j (t) = α j e

(I )

=

t (I )

eA

(I ) (t −τ )

t◦

(I )

γ (xˆ H (τ ), u(τ ))

) −γ (xˆ F(I ) (τ ), u(τ ))+ηˆ L (xˆ (I )) H (τ ), u(τ  (I ) −ηˆ L (xˆ F (τ ), u(τ ))−L (I ) f (I ) (τ ) dτ.

(61)

) (I ) (I ) ε(I y j (t) = ε y j H (t) + ε y j F (t)

(62)

) ε(I y j F (t)

(63)

=

+ f j (t)

(I )

where ε y j F represents the effects of sensor faults on the j th residual. Based on (61)–(63), the j th residual, for j ∈ J (I ) , is sensitive to the subset of sensor faults f (I ) (t). When J (I ) is a singleton set, i.e., J (I ) = { j }, the j th residual is sensitive to the sensor fault f j (t). In the presence of modeling uncertainty and noise, a threshold is necessary for distinguishing between ) the effects of faults modeled by ε(I y j F and the effects of (I ) modeling uncertainty and noise represented by ε y j H on the j th residual, j ∈ J (I ) , as described in Section IV-B.2. 2) Computation of Adaptive Thresholds: The adaptive threshold is a time-varying function that bounds the residual at every time instant under healthy conditions and is necessary for ensuring robustness with respect to modeling uncertainty (I ) and noise. The adaptive threshold  of the  j th residual, j ∈ J  (I )  (I ) is computed so that it satisfies ε y j H (t) ≤ ε¯ y j (t), j ∈ J (I ) . The adaptive nature of the thresholds means that they are varying with respect to the values of measured or computable signals, reducing the conservativeness in the decision making in contrast to the utilization of fixed, constant thresholds. In the proposed methodology, the adaptive thresholds, implemented in the (I )-local SFDI module, are designed as functions of the estimation of the state vector provided by the (I )-observer, i.e. xˆ (I ) and the input vector u. Theorem 3: The j th adaptive threshold, j ∈ J (I ) , is designed to bound the effects of modeling uncertainties and  )  (I ) (I ) noise, i.e. ε(I y j H (t) ≤ ε¯ y j (t), for all TL ≤ t < t◦ , where ) ε¯ (I y j (t) is the adaptive threshold defined as

(I ) (I ) ) ε¯ (I y j (t) = Y A j (t) + Y B j (t)

(66)

with , Z (I ) defined in (50) and (52), respectively, and

) Hence, ε(I x F represents the effects of sensor faults on the state estimation error. Considering (56), (60) and (61), the j th residual, j ∈ J (I ) is defined for all t ≥ TL as

) C j ε(I x F (t)

x¯ L

  (I ) αd(Ij ) −ζd (t −TL ) (I ) j ¯ + (I ) d 1−e + d¯ j ζd j t (I ) ) −ζ j (t −τ ) + α (I η( ¯ xˆ (I ) (τ ), u(τ ))dτ j e TL

where εx H is given in (55) and ) ε(I x F (t)

(65)

TL

(64)

E (I ) (t) = ρ (I ) e−ξ

(I ) (t −T

L)

x¯ L (I )

 (I ) ρ + d(I ) d¯ (I ) 1 − e−ξd (t −TL ) ξ dt (I ) −ξ (I ) (t −τ ) (I ) ρ

+

η( ¯ xˆ

e

(τ ), u(τ ))dτ.

TL

(67) (I ) ε y j H (t)

Proof: Under healthy conditions, the bound of defined in (56) using (55) can be determined as      (I )  )  (I )    ε y j H (t) ≤ C j e A (t −TL )  ε(I x (TL ) t       (I )       + C j e A (t −τ ) γ˜ (I ) (τ )+ η˜ (I ) (τ ) dτ T Lt      (I )    + C j e A (t −τ ) L (I )  d (I ) (τ ) dτ + d j (t) (68) TL

 (I ) where γ˜ (I ) (t) = γ (x(t), u(t)) − γ x H (t), u(t) and η˜

(I ) is defined in (43), where xˆ (I ) (t) = xˆ H (t). Consider (I ) (I ) that: 1) α j , ζ j are positive constants chosen such that

|C j e A

(I ) t

(I )

(I ) −ζ j t

| ≤ αj e

(I )

(I )

for t > 0; and 2) αd j , ζd j are pos−ζ

(I )

itive constants chosen such that |C j e A t L (I ) | ≤ αd(Ij ) e d j , for t > 0. Then, based on Assumptions 1–3 and given (47), (68) is described by t (I ) (I ) −ζ (t −τ ) (I ) ) ) |ε(I (t)| ≤ αj e j |ε(I (69) yjH x (τ )|dτ + Y B j (t) (I )

t

TL

where

(I ) YB j

is given in (66). Given (51) and (52), and consid(I )

ering healthy conditions, i.e., xˆ (I ) (t) = xˆ H (t), the adaptive threshold is defined in (64). The j th adaptive threshold, for all j ∈ J (I ) , can be implemented using linear filtering techniques. Specifically, (I ) Y B j can be implemented using linear filters   (I ) Y B j (t) = H (s) Z (I ) (t) (70) H (s) =

) α (I j  (I )

s + ζj

.

(71)

Similarly, Z (I ) (t) can also be implemented using linear filters. After the first time instant of sensor fault occurrence in S (I ) , (I ) (I ) t◦ , xˆ (I ) corresponds to xˆ F that represents the estimation of the state vector provided by the observer given in (36) under faulty conditions, i.e., y (I ) (t) in (36) is described by (33).

146

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 1, JANUARY 2014

Thus, the adaptive threshold defined in (64)–(67) is affected by sensor faults. The adaptive threshold under healthy and faulty conditions is described by  ) ε¯ (I TL ≤ t < t◦(I ) y j H (t), (I ) ε¯ y j (t) = (72) (I ) (I ) (I ) ε¯ y j H (t) + ε¯ y j F (t), t ≥ t◦ (I )

where ε¯ y j H is defined through (64)–(67), where xˆ (I ) (t) in (66) and (67) equals to the healthy estimation of state vector ) (I ) xˆ (I H (t), whereas ε¯ y j F is the sensor fault effects on the j th adaptive threshold and is defined as t (I ) ) −ζ j (t −τ ) ) (I ) (I ) α (I Z (I ε¯ y j F (t) = j e F (τ )dτ + Y B j (t) (73) (I )

F

t◦

(I )

(I )

where Z F is defined through (52) by replacing E (I ) with E F (I ) and Y B j F defined as t (I ) (I ) E F (t) = ρ (I ) e−ξ (t −τ )δ η¯ (I ) (τ )dτ (74) (I ) t ◦t (I ) ) −ζ j (t −τ ) (I ) α (I δ η¯ (τ )dτ (75) Y B(Ij) (t) = j e F

(I )

t◦

(I )

(I )

¯ xˆ F (t), u(t)) − η( ¯ xˆ H (t), u(t)). where δ η¯ (I ) (t) = η(

) It is important to mention that ε¯ (I y j H is a time-varying threshold bounding the effects of modeling uncertainties and noise for all t ≥ TL    (I )  ) j ∈ J (I ) . (76) ε y j H (t) ≤ ε¯ (I y j H (t),

Condition (76) implies that the proposed distributed SFDI scheme is robust with respect to modeling uncertainties and noise, or else it does not suffer from false alarms. Remark 3: In the case that we do not use the nonlinear functional approximator, the computation of the adaptive thresholds can be realized, assuming that the modeling uncertainty is bounded for all x, u ∈ X × U, and t > 0 by a given known functional, i.e., |η(x(t), u(t))| ≤ η¯ ∗ (x(t), u(t)). The key point of using the nonlinear functional approximator in the proposed SFDI scheme is to obtain a functional η(x(t), u(t)) such that η(x(t), u(t)) < η¯ ∗ (x(t), u(t)), for all t, which will enhance the fault detectability and isolability, described in Section V. C. Multiple Sensor Fault Decision Logic This section describes the multiple sensor fault decision logic. The presence of faults in the local sensing subsystem S (I ) is detected by the (I )-local SFDI module according to the local decision logic, whereas the isolation of the faulty sensors is realized by processing the local information based on a combinatorial decision logic. 1) Local Decision Logic: The primary goal of the (I )-local SFDI module is to infer about the presence of faulty sensors in S (I ) based on the following theorem. Theorem 4: The presence of faults in the local sensing subsystem S (I ) , I ∈ {1, . . . , M}, is detected by the (I )-local SFDI module when    (I )  ) (77) ε y j (t) − ε¯ (I y j (t) > 0

for at least one j ∈ J (I ) , where J (I ) is defined in (34). Proof: Assume that no sensor fault has occurred. Then, (I ) (I ) (I ) (I ) ε y j (t) = ε y j H (t) and ε¯ y j (t) = ε¯ y j H (t) according to (56) and (72), respectively, implying that (76) is valid. This contradicts the validity of (77). Thus, sensor faults have occurred in S (I ) . The output of the (I )-local SFDI, i.e., the inference about the presence of sensor faults in S (I ) , is represented by a boolean decision function, defined as 

⎧ (I ) ⎪ ⎨ 0, if ∧ (I ) D j (t) = 0 j ∈J

 D(I ) (t) = (78) (I ) ⎪ D j (t) = 1 ⎩ 1, if ∨ j ∈J (I )

where

 (I ) D j (t) (I )

Tj

=

0,

(I )

if t < T j

(I )

if t ≥ T j  

  )  (I ) = min t > 0: ε(I (t) (t) . > ε ¯  yj yj 1,

(79) (80)

The symbols ∧ and ∨ are the logical conjunction and disjunction, respectively. The time instant of sensor fault detection using the (I )-local SFDI module is determined as  (I ) Tj . (81) TF(ID) = min j ∈J (I )

After the time instant TF(ID) , S (I ) is characterized as faulty by the (I )-local SFDI module, I ∈ {1, . . . , M}, and all m I sensors that belong to this local sensing subsystem are indicated as potentially faulty, until the isolation of the faulty sensors. On the other hand, before the time instant TF(ID) , we assume that S (I ) is working properly and is characterized as non-faulty, i.e. S (I ) is exonerated [41]. The indication of S (I ) as faulty guarantees the presence of the fault in the S (I ) local sensing subsystem, whereas the indication non-faulty corresponds to the case that either there is no fault or faults have occurred, but are not detected by the (I )-local SFDI module until that particular time instant. Remark 4: If S (I ) contains one sensor, i.e., S (I ) = S{ j }, then (I )-local SFDI module can isolate the fault in this sensor ) (I ) . If S (I ) contains more when D (I ) (t) = D (I j (t) = 1, j ∈ J than one sensor, then the (I )-local SFDI module is not capable of isolating the faulty sensor(s) in S (I ) and the combinatorial process of the local decisions is needed. 2) Combinatorial Decision Logic: When a local sensing system contains more than one sensor and some of the sensors belong also to some neighboring local sensing subsystems, then the combinatorial process of local decisions generated by M ∗ SFDI modules, M ∗ ≤ M, is conducted, where M ∗ is the number of a few neighboring and possibly overlapping local sensing subsystems. The combinatorial decision logic is designed considering the overlapping between a (possibly) small group of local sensing subsystems. Let us assume two overlapping local sensing subsystems, i.e. S (I ) ∩ S (Q) = S ∗ , I = Q, I, Q ∈ {1, . . . , M}, where S ∗ = ∅ contains the shared sensors. The local sensing subsystems S (I ) , S (Q) are monitored by the (I )- and (Q)-local SFDI module, respectively. The combinatorial decision logic

REPPA et al.: ADAPTIVE APPROXIMATION FOR MULTIPLE SFDI

147

that is applied for all I, Q ∈ {1, . . . , M} can be summarized as follows. 1) If D(I ) (t) = 1, while D(Q) (t) = 0 (S (I ) is faulty and S (Q) is non-faulty), it is inferred that S (I ) \ S ∗ is faulty (i.e. at least one of the nonshared sensors is faulty). 2) If D(Q) (t) = 1, while D(I ) (t) = 0, it is inferred that S (Q) \ S ∗ is faulty. 3) If D(I ) (t) = 1 and D(Q) (t) = 1, it is inferred that either S ∗ is faulty or combinations of S ∗ , S (I ) \ S ∗ and S (Q) \ S ∗ are faulty. Based on this decision logic, we can isolate smaller sensor sets in the local sensing subsystems, which contain faulty sensors. Let us now assume that the aforementioned local sensing subsystems S (I ) and S (Q) do not overlap, i.e., S ∗ = 0. In this case, if D(I ) (t) = 0, while D(Q) (t) = 1 i.e., S (I ) is non-faulty and S (Q) is faulty, the (I )-local SFDI module can be used for isolating faulty sensors in S (Q) . In order for the (I )-local SFDI module to do so, the (Q)-local SFDI module transmits of S (Q) , i.e., yi , for all i ∈  (Ithe  measurements (Q) ) (Q) = 0 to the (I )-local SFDI module. The J J J (I )-local SFDI module can generate m Q residuals using the estimated state vector xˆ (I ) , where each residual is defined ) (I ) (Q) . The residual as ε(I yi (t) = yi (t) − Ci xˆ (t), for i ∈ J (I ) (I ) (I ) ) ε yi (t) can be written as ε yi (t) = ε yi H (t) + ε(I yi F (t), where (I ) ε yi H is defined in (55), representing the effects of modeling ) uncertainty and noise in the residual, whereas ε(I yi F (t), i ∈ (Q) J , denotes the effect of sensor fault f i on the residual, when there is no sensor fault in S (I ) ) ε(I yi F (t) = f i (t).

(82)

Moreover, when there is no sensor fault in S (I ) , the adaptive threshold is not affected by the sensor fault f i ) ε¯ (I yi F (t)

= 0.

) (I ) (I ) where ε(I y j F , ε¯ y j F and ε¯ y j H are defined in (62), (63), and (72), respectively, the presence of sensor faults in S (I ) is guaranteed to be detected. Proof: The j th residual, j ∈ J (I ) , is described by (62) (I ) for all t ≥ TL . Hence, for any t ∗ > t◦

         (I ) ∗   (I ) ∗   (I ) ∗   (I ) ∗ ) ∗  ≥ − (t ) (t ) (t ) ε ε ε y j (t ) = ε y j H (t ) + ε(I   . yjF yjF yjH (85) Given (76), we obtain      (I ) ∗   (I ) ∗  ) ∗ ε y j (t ) ≥ ε y j F (t ) − ε¯ (I y j H (t ).

(86)

Combining (84) and (86), it yields      (I ) ∗   (I ) ∗  ) ∗ ε y j (t ) ≥ ε y j F (t ) − ε¯ (I y j H (t ) ) ∗ (I ) ∗ (I ) ∗ > 2¯ε(I y j H (t ) + ε¯ y j F (t ) − ε¯ y j H (t ) ) ∗ (I ) ∗ (I ) ∗ = ε¯ (I y j H (t ) + ε¯ y j F (t ) = ε¯ y j (t )

(87)

which implies that the presence of faults in S (I ) is detected at the time instant t ∗ . In the case that S (I ) contains a single sensor, implying that f (I ) (t) = f j (t), if condition (84) is satisfied, then the fault f j (t) affecting the j th sensor is guaranteed to be isolated. Given that there are no faults in the local sensing subsystem S (I ) , i.e., f (I ) (t) = 0, the (I )-local SFDI module is guaranteed to isolate a fault affecting the i th sensor that does ∗ not belong to S (I ) , if there exists a time  instant t such that the sensor fault fi (t), i ∈ J (Q), (J (Q) J (I ) = ∅), satisfies the condition    f i (t ∗ ) > 2¯ε(I ) (t ∗ ). yi H

(88)

(83)

Thus, the i th adaptive threshold, i ∈ J (Q) , is determined as ) (I ) (Q) , Q = I , is ε¯ (I yi (t) = ε¯ yi H (t). The faulty sensor i , i ∈ J (I ) (I ) isolated when |ε yi (t)| > ε¯ yi (t). The same procedure can be conducted if D(Q) (t) = 0, while D(I ) (t) = 1. However, if D(I ) (t) = 1 and D(Q) (t) = 1, no transmission of sensor measurements will be realized between the two local SFDI modules. However, other local SFDI modules monitoring neighboring local sensing subsystems can be used for isolating the faulty sensors in S (I ) and S (Q) . V. FAULT D ETECTABILITY AND I SOLABILITY The conditions, under which the occurrence of faults in a local sensing system is guaranteed to be detected and a faulty sensor is guaranteed to be isolated, are derived based on the following theorem. (I ) Theorem 5: If there exists a time instant t ∗ , where t ∗ > t◦ , (I ) such that at least one sensor fault f j , j ∈ J satisfies the condition    (I ) ∗  ) ∗ ) ∗ ε(I (84) ε y j F (t ) − ε¯ (I y j F (t ) > 2¯ y j H (t )

This can be proved by introducing (82) and (83) in (84). The condition in Theorem 5 can be regarded as a figure of merit, characterizing the ability of a local SFDI module to detect the presence of sensor faults in the underlying group of sensors. Hence, this condition can be taken into account during the design of the local SFDI modules, as it provides a relationship between the sensor faults f (I ) (t) and the selected design parameters used for the implementation of the nonlinear observer (e.g. L (I ) ) and the adaptive thresholds (ρ (I ) , ξ (I ) , and so on). However, it is important to note that the class of detectable sensor faults satisfying (84) are obtained under the worst-case detectability conditions. To illustrate the usefulness of the fault detectability conditions, let’s consider the special case of a single, abrupt bias (I ) sensor fault affecting the j th sensor after the time instant t◦ , (I ) i.e., f j (t) = φ j , j ∈ J (I ) , for all t ≥ t◦ and f (I ) = F j φ j , where F j is a vector with zeros except for the j th nonzero element, i.e., the j th element equals to 1. From the Appendix, if the magnitude of the fault φ j satisfies ) ∗   2¯ε(I (t ) φ j  > y j H Iφ (t ∗ )

(89)

148

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 1, JANUARY 2014

given that Iφ (t) > 0 for t > t◦(I ) , where Iφ is defined as   t   A(I ) (t −τ ) (I )  Iφ (t) = 1 − Cje L F j (τ )dτ  (I ) t◦   2

(I ) (I ) (I ) (I ) −α j μ λη¯ Ie (t) Ia (t) + ρ λη¯ Ic (t) + ρ

with μ

(I )

=

Ia (t) =

Ib (t) =

Ic (t) = Id (t) = Ie (t) =

 (I )  (I ) L Fj  ρ  ξ (I ) − ρ (I ) λγ + ληˆ t (I ) −ζ (t −τ ) e j (I ) t◦ 

 (I ) (I ) (I ) × 1 − e− ξ −ρ (λγ +ληˆ ) (τ −t◦ ) dτ t (I ) e−ξ (t −τ ) (I ) t◦ 

 (I ) (I ) (I ) × 1 − e− ξ −ρ (λγ +ληˆ ) (τ −t◦ ) dτ t (I ) Ib (τ )e−ζ j (t −τ ) dτ (I ) t ◦t (I ) (I ) Ib (τ )e−(ξ −ρ )(t −τ )dτ (I ) t ◦t (I ) −ζ (t −τ ) Id (τ )e j dτ (I )

m, the gravity constant g, the center of mass h, the amplifier gain kτ , and the sinusoidal input u are [42]: Jl = 4.5, Jm = 1, Fl = 0.5, Fm = 1, m = 4, h = 0.5, k = 2, g = 9.8, kτ = 1, and u = 2 sin(0.5 ∗ t). The parameter m is the inaccuracy in the mass, with |m| ≤ 15%m. By choosing x 1 = q˙m , x 2 = qm , x 3 = q˙l , and x 4 = ql , we obtain the nonlinear (90) uncertain model ⎤ ⎡ −Fm −k ⎤ ⎡ k ⎤⎡ x 1 (t) x˙1 (t) Jm Jm 0 Jm ⎢ x˙2 (t) ⎥ ⎢ 1 0 0 0 ⎥ ⎢ x 2 (t) ⎥ ⎥=⎢ ⎥ ⎥⎢ ⎢ l −k ⎦ ⎣ x (t) ⎦ (91) ⎣ x˙3 (t) ⎦ ⎣ 0 Jkl −F 3 Jl Jm x˙4 (t) x 4 (t) 0 0 1 0 ⎤ ⎡ ⎡ ⎤ kτ 0 Jm u(t) ⎥ ⎢ ⎢ ⎥ 0 0 ⎥ ⎢ ⎥ +⎢ ⎣ − mgh sin(x 4 (t)) ⎦ + ⎣ − mgh sin(x 4 (t)) ⎦. (92) Jl Jl 0 0 The state equation can be written in compact form as (93) (94) (95) (96)

t◦

then the (I )-local SFDI module is guaranteed to detect sensor faults in S (I ) . If during the design of the (I )-local SFDI module, we choose the design parameters according to Theorem 2 and to satisfy Iφ (t) > 0, i.e. choose the design parameters under worst-case conditions, then there does exist t ∗ given that the magnitude of the fault φ j satisfies (89). VI. S IMULATION E XAMPLE In this section, we apply the adaptive approximation methodology and the distributed SFDI (DSFDI) architecture in the example of a single-link robotic arm [42], aiming at detecting and isolating multiple faults affecting its sensors. The goal of this example is to present the application of the adaptive approximation-based DSFDI methodology and the improvement in detecting and isolating multiple sensor faults by utilizing the nonlinear functional approximator of the modeling uncertainty into the SFDI methodology. The latter is illustrated by implementing the DSFDI architecture with and without the nonlinear functional approximator of the modeling uncertainty. The motion dynamics of a single-link robotic arm with a revolute elastic joint are described by Jl q¨l + Fl q˙l + k(ql − qm ) + (m + m)gh sin ql = 0 Jm q¨m + Fm q˙m − k(ql − qm ) = kτ u where ql and qm are the angular positions of the link and the motor (the subscripts l and m are referred to the link and motor, respectively). The values of the link and the motor inertia Jl , Jm , the link viscous coefficient Fl , the motor viscous friction coefficients Fm , the elastic constant k, the link mass

x(t) ˙ = Ax(t) + γ (x 4 (t), u(t)) + H1η(x 4 (t)). where H1 = [0, 0, 1, 0] . The vector field γ (x 4 (t), u(t)) satisfies the Lipschitz condition in x 4 (t) ∈ R, ∀ u, t, with Lipschitz constant λγ = mgh/Jl . It is assumed that three sensors are used to measure the motor position, link position, and link velocity, whose outputs are described by ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎡ ⎤ x 1 (t) 0100 ⎢ f 1 (t) d1 (t) ⎥ x (t) 2 ⎥ ⎣ ⎦ ⎣ ⎦ y(t) = ⎣ 0 0 1 0 ⎦ ⎢ ⎣ x 3 (t) ⎦ + f 2 (t) + d2 (t) . f 3 (t) d3 (t) 0001 x 4 (t) The measurements of each sensor are corrupted by uniformly distributed noise, di , i = 1, 2, 3, with d¯i = 3%Yi °, where Yi ° is the amplitude of the noiseless measurements of the i th sensor. The objective of this simulation example is the detection and isolation of multiple abrupt, permanent sensor faults represented as f i (t) = B(t − Ti )φi (t − Ti ), i = 1, 2, 3, where Bi and Ti are the time profile and the time instant of occurrence of the i th fault, respectively, and φi is the fault signature. The time profile in abrupt and permanent sensor faults is described as B(t) = 0, if t < 0 and B(t) = 1, if t ≥ 0. In this simulation example, we will investigate the occurrence of offset faults, whose signature is described by φi (t) = φi° ∀t, i = 1, 2, 3, where φi° is a constant value. a) Adaptive approximation methodology: We apply the adaptive approximation methodology for obtaining a nonlinear functional approximator of the modeling uncertainty η(x 4 (t)) for the learning period [0, 500] s, implementing the adaptive nonlinear estimation scheme described in (10)–(12). The adaptive nonlinear approximation model ˆ 4 (t), θ ∗ ) = η(x ˆ 4 (t), θ ∗ ) is linearly parameterized, i.e., η(x ∗ ∗ 20 G η (x 4 (t))θ , where θ ∈ R is the unknown optimal parameter vector, and G η (x 4 (t)) = [g1 (x 4 (t)), . . . , g20 (x 4 (t))] ∈ R1×20 is the regressor. The function g j (x 4 (t)), j = 1, . . . , 20, is chosen to be a continuous radial basis function network with 20 fixed centers evenly distributed in the region of interest, i.e., x 4 ∈ [−0.2, 0.2], and the variance of each basis function is set to 0.03. The learning rate matrix of the adaptive algorithm is set to 0.05I, where I is the identity matrix. The gain L

REPPA et al.: ADAPTIVE APPROXIMATION FOR MULTIPLE SFDI

(a)

(b) Fig. 3. Online learning of modeling uncertainty using adaptive nonlinear estimation scheme. (a) Time histories of the modeling uncertainty and the adaptive nonlinear approximation model. (b) Error between the modeling uncertainty and the adaptive nonlinear approximation model.

is chosen via pole placement so that the eigenvalues of the matrix A0 are located at 1.5 ± 0.5i, 2 ± 0.5i . The learning capability of the adaptive nonlinear estimation scheme is shown in Fig. 3. Fig. 3(a) shows the actual function of the modeling uncertainty η(x 4 (t)) and the adaptive nonlinear approximation model η( ˆ xˆ4 (t); θˆ (t)) during the learning period [0, 500] s, while the error with respect to time is shown in Fig. 3(b). The online learning of the functional approximator through the proposed adaptive nonlinear estimation scheme is realized using noisy sensor measurements, leading to the error between the actual modeling uncertainty and its functional approximator. b) DSFDI architecture: The DSFDI architecture is applied as follows; the three sensors constitute the sensing system S. After checking Assumption 6, it yields that sensors 1 and 3 can be used for designing nonlinear observers in the SFDI modules. Hence, the sensing system S is initially decomposed into the local sensing subsystem S (1) containing sensor 1 and the local sensing subsystem S (3) containing sensor 3. Then, we design the (1)- and (3)-local  SFDI module (1) J (1) = {1} and driven by the local sensing subsystems S  S (3) J (3) = {3} , respectively. In contrast to sensors 1 and 3, sensor 2 cannot be used for designing a nonlinear observer, as Assumption 6 is not satisfied. For this reason, we create a third local sensing subsystem, denoted as S (2) , which consists  (2) 2 and sensor 3 J = {2, 3} such that the pair of sensor A, C (2) is observable.

149

To illustrate the benefit of using the nonlinear functional approximator of modeling uncertainty derived by the application of the adaptive approximation methodology, we implement two DSFDI schemes; the first one consists of the three local SFDI modules, in which the nonlinear functional approximator of modeling uncertainty is used [(A)-DSFDI scheme], whereas the second one consists of local SFDI modules without the nonlinear functional approximator of modeling uncertainty [(B)-DSFDI scheme]. The design of the (I )-local SFDI module, I = 1, 2, 3 of the (A)-DSFDI scheme is described in Section IV. The local SFDI modules of (B)-DSFDI scheme are also designed as described in Section IV, but now ηˆ L (xˆ4(I ) (t)) = 0 in (36). The design parameters are the same in both schemes. Particularly, the gains L (1) , L (2) , and L (3) are chosen via pole placement such that the eigenvalues of A(1), A(2) , and A(3), respectively, are: (1) (2) = {−5.5 ± = {−3 ± 4.5i, −3.5 ± 4.5i }, eig A eig A  0.5i ; −7.5 ± 0.5i } and eig A(3) = {−0.5 ± 1.5i ; −18.5 ±5i }. The functional approximator of the modeling uncertainty used in the local SFDI modules is described by η( ˆ xˆ4(I ) (t)) = (I ) G η (xˆ4 (t))θˆ (TL ), I = 1, 2, 3, where G η is the regressor used in the adaptive approximation methodology. In this simulation example, for the computation of the adaptive bounds in each local SFDI module, a worst-case constant bound η such that     (I ) ≤ (t)) − η ˆ ( x ˆ (t)) η, ∀ I , is used. The worst-case η(x 4  L 4 (I ) bound is defined as η¯ = max η¯ , I = 1, 2, 3 where η¯ (I ) I is estimated based

on the simulated time history of the error η(x 4 (t)) − ηˆ L xˆ4(I ) (t) , which is shown in Fig. 4 for I = 1, 2, 3. It is observed that the bound of the approximation error in the (1)- local SFDI module is greater than the corresponding bounds generated using (2)- and (3)-local SFDI modules. This is because of the noise level of the sensor 1, which is used as input to the (1)-nonlinear observer, is higher than the noise levels of the other two (d¯1 = 0.036, d¯2 = 0.0018, and d¯3 = 0.0036). This leads to a highly-noisy estimated signal xˆ4(1) and consequently to high approximation error. In (B)-DSFDI scheme, which is designed without the functional approximator of the modeling uncertainty, we use a constant bound for the modeling uncertainty, i.e., |η(x 4 (t))| ≤ mgh/Jl ∀ t. After the learning period [0, 500] s, we apply both DSFDI schemes for the time interval [500, 1000] s. It is assumed that the sensors 1 and 2 are affected by offset faults that occur at T1 = 800 s, T2 = 800.5 s and their signatures are φ1° = 20% Y1° and φ2° = 35% Y2° . The decision logic of the (A)- and (B)-DSFDI schemes for inferring the faulty or non-faulty status of the sensors simulating the aforementioned scenarios, is analyzed using the decision function defined in (78)–(80). The main goal of its local SFDI module is to infer about the presence of faults in its monitored local sensing subsystem. In this example, the (1)- and (3)-local SFDI modules are responsible for isolating the faults in sensors 1 and 3, respectively, whereas the (2)-local SFDI module is responsible for detecting the presence of faults in S (2) , i.e. detecting the presence of faults in sensors 2 or 3 or both. The local information of the (2)local SFDI module can be combined with the information

150

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 1, JANUARY 2014

(b)

(a)

(a)

(c) Fig. 4. Time history of error between actual modeling uncertainty and functional approximator used in (I )-local SFDI module. (a) (1)-local SFDI module. (b) (2)-local SFDI module. (c) (3)-local SFDI module.

generated by the third local SFDI module for isolating the fault in sensor 2. Figs. 5(a) and 5(b) shows the results of the multiple fault decision logic of the (A)- and (B)-DSFDI scheme, respectively. Each figure can be viewed as a matrix of nine subfigures; the subfigure in the I -row, I = 1, 2, 3, and j th column, j = 1, 2, 3 [denoted as e.g. Fig. 5(a)-{I, j}] shows the decision function D(Ij ) based on which the (I )-local SFDI module infers about the status of the j -th sensor. The time of detection of fault in the j th sensor by the (I )-local SFDI module is indicated in the horizontal axis along with the starting point 500 s. When no fault is detected, the end time of the simulated fault scenario Tend = 1000 s is labeled. Regarding the local decision logic, 1) the (1)-local SFDI module can infer about a fault in sensor 1 (1) (1) based on D(1) (t) = D1 (t), with D1 (t) shown in subfigure (1) {1, 1} (the decision functions D2 (t) and D3(1) (t) in subfigures {1, 2} and {1, 3}, respectively, are not used); 2) the (2)-local SFDI module infers about faults in sensor 2 or 3 or both (2) (2) based on D2 (t) and D3 (t) in subfigures {2, 2} and {2, 3}, respectively, (the decision function D1(2) (t) in subfigure {2, 1} is not used); and 3) the (3)-local SFDI module can infer about (3) (3) a fault in sensor 3 based on D(3) (t) = D3 (t), with D3 (t) (3) shown in subfigure {3, 3} (the decision functions D1 (t) and D2(3) (t) in subfigures {3, 1} and {3, 2}, respectively, are not used). According to Fig. 5(a)-{1, 1} and 5(b)-{1, 1} that show the decision function D1(1) , the fault in sensor 1 is immediately isolated by both DSFDI schemes at the same time instant. On the other hand, the status of sensor 3 is indicated as nonfaulty by both schemes [Fig. 5(a)-{3, 3} and 5(b)-{3, 3}], as the decision function D3(3) (t) is zero for t > TL . Moreover, the (A)-DSFDI scheme detects immediately the presence of faults (2) in the local sensing subsystem S (2) at the time instant T2 = (2) (2) (2) (2) 800.5 s, as D2 (t) ∨ D3 (t) = 1 for t > T2 . As D (t) = 1

(b) Fig. 5. Multiple sensor fault decision logic. (a) (A)-DSFDI scheme and (b) (B)-DSFDI scheme.

(2)

for t > T2 , whereas D(3) (t) = 0 (i.e. sensor 3 is indicated as non-faulty), the fault in sensor 2 is isolated. In contrast to the (A)-DSFDI scheme, the (B)-DSFDI scheme neither can detect the presence of faults in the local sensing subsystem S (2) (Fig. 5(b)-{2, 2} and 5(b)-{2, 3}) nor can isolate the fault in sensor 2. Both DSFDI schemes are robust, i.e., they do not suffer from false alarms. However, their performance can be investigated with respect to the number of detections/isolations and the mean value of isolation time, in multiple sensor faults. To do so, the aforementioned fault scenario is simulated using different values of the actual mass inaccuracy m within its bounds, i.e., m = {−0.15; −0.125; −0.1; −0.075; −0.05; −0.025; 0; 0.025; 0.05; 0.075; 0.1; 0.125; 0.15} ∗ m. For each value of m, the simulation is repeated 20 times. Fig. 6 shows the average number of isolations of the

REPPA et al.: ADAPTIVE APPROXIMATION FOR MULTIPLE SFDI

151

VII. C ONCLUSION

Fig. 6. Average number of isolations of sensor faults 1 and 2. Squared markers: using (A)-DSFDI scheme. Cross markers: using (B)-DSFDI scheme.

In this paper, we presented an adaptive approximation-based architecture for detecting and isolating multiple sensor faults in a class of nonlinear uncertain systems. During the initial stage of the nonlinear system operation and under normal conditions, a stable adaptive nonlinear estimation scheme was used for learning the modeling uncertainty. The backbone of the proposed SFDI methodology was the design of nonlinear observer-based local SFDI modules, using the nonlinear functional approximator of modeling uncertainty. Each module was dedicated to a subset of sensors, referred to as local sensing subsystem, and was responsible for detecting the presence of faults in the local sensing system. The multiple sensor fault isolation procedure was enhanced by processing the information of the local SDFI modules based on a combinatorial logic. Analytical results related fault detectability and isolability of the proposed SFDI method were derived. A single-link robotic arm was used for illustrating the application of the adaptive approximation-based distributed SFDI technique. Future research work will involve the integration of the proposed SFDI methodology with other techniques for actuator or plant faults, aiming at resolving the integrated isolation problem of multiple sensor/actuator/plant faults. From the viewpoint of application, we will elaborate on the validation of the proposed adaptive approximation-based SFDI methodology in real-world examples.

(a)

A PPENDIX For any t ∗ such that t ∗ > t◦(I ) , given (63) that describes the sensor fault effects on the j th residual and (61), we have      (I ) ∗   ) ∗ ∗  ε y j F (t ) = C j ε(I x F (t ) + f j (t )   t∗     ∗ A(I ) (t ∗ −τ ) (I ) (I ) ≥  f j (t ) − Cje L f (τ )dτ  (I )   t◦   ∗ − T1 (t ) (97) (b) Fig. 7. Mean value of isolation time of sensor faults 1 and 2. Squared markers: using (A)-DSFDI scheme. Cross markers: using (B)-DSFDI scheme. (a) Sensor fault 1. (b) Sensor fault 2.

faults in sensors 1 and 2 by the (A)-DSFDI scheme (squared markers) and (B)-DSFDI scheme (cross markers) for each value of the mass inaccuracy, showing the enhanced capability of the (A)-DSFDI scheme for isolating multiple sensor faults. Fig. 7(a) and 7(b) shows the mean value of isolation time of the faults in sensor 1 and 2, respectively, in the cases that the DSFDI schemes achieve to isolate the sensor faults. As observed, the performance of the two DSFDI schemes with respect to the isolation time of the sensor fault 1 is equivalent. On the other hand, the (A)-DSFDI scheme can isolate faster the fault in sensor 2. In all simulations, sensor 3 has been indicated correctly as non-faulty by both schemes.

where T1 (t) =

t t◦(I )

C jeA

(I ) (t −τ )

(I )

(I )

(γ (xˆ H (τ ), u(τ )) − γ (xˆ F (τ ), u(τ ))

(I )

(I )

+ηˆ L (xˆ H (τ ), u(τ )) − ηˆ L (xˆ F (τ ), u(τ )))dτ.

(98)

By bounding T1 and considering that γ and ηˆ L are Lipschitz, ) (I ) while α (I are positive constants chosen such that j , ζj |C j e A

(I ) t

|T1 (t)| ≤

(I )

(I ) −ζ j t

| ≤ αj e t

(I )

t◦

for t > 0, we obtain (I )

) −ζ j α (I j e

(t −τ ) 

) λγ + ληˆ |ε(I x F (τ )|dτ

(99)

) (I ) (I ) (I ) where ε(I x F (t) = xˆ H (t) − xˆ F (t), xˆ H denotes the estimation of the state vector provided by the observer given in (36) (I ) with y (I ) (t) = C (I ) x (I ) (t) + d (I ) (t) and xˆ F represents the estimation of the state vector provided by the observer given in (36) with y (I ) (t) described by (33). By bounding (61) and

152

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 1, JANUARY 2014

considering that f (I ) = F j φ j , j ∈ J (I ) , we obtain  (I ) ρ (I ) ) (I ) −ξ (I ) (t −t◦ ) |ε(I (t)| ≤ |φ ||L F | 1 − e j j xF ξ (I ) t (I ) ) + ρ (I ) e−ξ (t −τ ) (λγ + ληˆ )|ε(I x F (τ )|dτ. (100) (I )

t◦

Applying the Bellman-Gronwall Lemma results in 

) (I ) −ν (I ) (t −t◦(I )) |ε(I x F (t)| ≤ μ |φ j | 1 − e

(101)

where μ(I ) is defined in (91) and ν (I ) = ξ (I ) − ρ (I ) (λγ + ληˆ ). Using (99) and (101) yields ) (I ) |T1 (t)| ≤ |φ j |α (I j (λγ + ληˆ )μ Ia (t)

(102)

where Ia is defined in (92). Considering (73), we can compute a bound for the sensor fault effects on the j th adaptive threshold as follows: t (I ) (I ) (I ) ) (t)| ≤ α  |Z F (τ )||e−ζ j (t −τ ) |dτ |¯ε(I yjF j (I )

to +|Y B(Ij) (t)|. F

(103)

By bounding (74) and considering (101), we obtain   t      (I )   (I ) −ξ (I )(t −τ )   (I )  E F (t) ≤ ρ e λη¯ εx F (τ ) dτ (I )

to

≤ |φ j |ρ (I ) λη¯ μ(I ) Ib (t)

(104)

where Ib is defined in (93). Similarly, based on (75), we obtain (I )

(I )

|Y B j F (t)| ≤ |φ j |α j λη¯ μ(I ) Ia (t)

(105)

where Ia is defined in (92). Considering (52), (104), (105), (103), for t = t ∗ becomes

(I ) ) ∗ (I ) Ia (t) (t )| ≤ |φ |α λ μ |¯ε(I j η ¯ yjF j 2

 +ρ (I ) Ic (t) + ρ (I ) 2 Ie (t) (106) where Ia , Ic , Ie are defined in (92), (94) and (96). Considering ) ) that ε¯ (I ε(I y j F (t) ≤ |¯ y j F (t)| and given (97), (102) and (106), we obtain ) ∗ (I ) ∗ ∗ |ε(I y j F (t )| − ε¯ y j F (t ) ≥ |φ j |Iφ (t )

(107)

where Iφ is given in (90). If Iφ > 0 and (89) is valid, then sensor faults are detected in S (I ) . R EFERENCES [1] Y. Ding, E. Elsayed, S. Kumara, J. Lu, F. Niu, and J. Shi, “Distributed sensing for quality and productivity improvements,” IEEE Trans. Autom. Sci. Eng., vol. 3, no. 4, pp. 344–359, Oct. 2006. [2] S. Reece, S. Roberts, C. Claxton, and D. Nicholson, “Multi-sensor fault recovery in the presence of known and unknown fault types,” in Proc. 12th Int. Conf. Inf. Fusion, Jul. 2009, pp. 1695–1703. [3] R. Isermann, Fault-Diagnosis Systems: An Introduction from Fault Detection to Fault Tolerance. New York, USA: Springer-Verlag, 2006. [4] M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki, Diagnosis and Fault-Tolerant Control. New York, USA: Springer-Verlag, 2003. [5] J. Chen and R. Patton, Robust Model-Based Fault Diagnosis for Dynamic Systems. Norwell, MA, USA: Kluwer, 1999. [6] J. Gertler, Fault Detection and Diagnosis in Engineering Systems. Boca Raton, FL, USA: CRC Press, 1998. [7] R. Rajamani and A. Ganguli, “Sensor fault diagnostics for a class of nonlinear systems using linear matrix inequalities,” Int. J. Control, vol. 77, no. 10, pp. 920–930, Oct. 2004.

[8] S. Narasimhan, P. Vachhani, and R. Rengaswamy, “New nonlinear residual feedback observer for fault diagnosis in nonlinear systems,” Automatica, vol. 44, no. 9, pp. 2222–2229, Sep. 2008. [9] X. Yan and C. Edwards, “Sensor fault detection and isolation for nonlinear systems based on a sliding mode observer,” Int. J. Adapt. Control Signal Process., vol. 21, nos. 8–9, pp. 657–673, Oct.–Nov. 2007. [10] A. Vemuri, “Sensor bias fault diagnosis in a class of nonlinear systems,” IEEE Trans. Automa. Control, vol. 46, no. 6, pp. 949–954, Jun. 2001. [11] I. Samy, I. Postlethwaite, and D. Gu, “Survey and application of sensor fault detection and isolation schemes,” Control Eng. Pract., vol. 19, no. 7, pp. 658–674, Jul. 2011. [12] R. Mattone and A. De Luca, “Nonlinear fault detection and isolation in a three-tank heating system,” IEEE Trans. Control Syst. Technol., vol. 14, no. 6, pp. 1158–1166, Nov. 2006. [13] C. De Persis and A. Isidori, “A geometric approach to nonlinear fault detection and isolation,” IEEE Trans. Autom. Control, vol. 46, no. 6, pp. 853–865, Jan. 2001. [14] K. Adjallah, D. Maquin, and J. Ragot, “Non-linear observer-based fault detection,” in Proc. 3rd IEEE Conf. Control Appl., Aug. 1994, pp. 1115–1120. [15] S. Rajaraman, J. Hahn, and M. Mannan, “A methodology for fault detection, isolation, and identification for nonlinear processes with parametric uncertainties,” Ind. Eng. Chem. Res., vol. 43, no. 21, pp. 6774–6786, Sep. 2004. [16] S. Rajaraman, J. Hahn, and M. Mannan, “Sensor fault diagnosis for nonlinear processes with parametric uncertainties,” J. Hazardous Mater., vol. 130, nos. 1–2, pp. 1–8, Mar. 2006. [17] M. Polycarpou and A. Helmicki, “Automated fault detection and accommodation: A learning systems approach,” IEEE Trans. Syst., Man Cybern., vol. 25, no. 11, pp. 1447–1458, Nov. 1995. [18] A. Trunov and M. Polycarpou, “Automated fault diagnosis in nonlinear multivariable systems using a learning methodology,” IEEE Trans. Neural Netw., vol. 11, no. 1, pp. 91–101, Jan. 2000. [19] F. Caccavale, F. Pierri, and L. Villani, “Adaptive observer for fault diagnosis in nonlinear discrete-time systems,” J. Dyn. Syst., Meas., Control, vol. 130, no. 2, pp. 021005-1–021005-8, Feb. 2008. [20] H. Talebi, K. Khorasani, and S. Tafazoli, “A recurrent neural-networkbased sensor and actuator fault detection and isolation for nonlinear systems with application to the satellite’s attitude control subsystem,” IEEE Trans. Neural Netw., vol. 20, no. 1, pp. 45–60, Jan. 2009. [21] B. Thumati and S. Jagannathan, “A model-based fault-detection and prediction scheme for nonlinear multivariable discrete-time systems with asymptotic stability guarantees,” IEEE Trans. Neural Netw., vol. 21, no. 3, pp. 404–423, Mar. 2010. [22] V. Reppa, M. Polycarpou, and C. G. Panayiotou, “Distributed sensor fault detection and isolation for nonlinear uncertain systems,” in Proc. 8th IFAC Symp. Fault Detect., Supervis. Safety Tech. Process., 2012, pp. 1077–1082. [23] F. Caccavale, P. Cilibrizzi, F. Pierri, and L. Villani, “Actuators fault diagnosis for robot manipulators with uncertain model,” Control Eng. Pract., vol. 17, no. 1, pp. 146–157, Jan. 2009. [24] V. Reppa, M. Polycarpou, and C. G. Panayiotou, “A distributed architecture for sensor fault detection and isolation using adaptive approximation,” in Proc. Int. Joint Conf. Neural Netw., Jun. 2012, pp. 2340–2347. [25] R. Clark, “The dedicated observer approach to instrument failure detection,” in Proc. 18th IEEE Conf. Decision Control Includ. Symp. Adapt. Process., vol. 18. Dec. 1979, pp. 237–241. [26] J. Farrell and M. Polycarpou, Adaptive Approximation Based Control. New York, USA: Wiley, Mar. 2006. [27] S. Lipschutz and M. Lipson, Schaum’s Outline of Linear Algebra. New York, USA: McGraw-Hill, 2008. [28] X. Zhang, M. Polycarpou, and T. Parisini, “A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems,” IEEE Trans. Autom. Control, vol. 47, no. 4, pp. 576–593, Apr. 2002. [29] A. Xu and Q. Zhang, “Residual generation for fault diagnosis in linear time-varying systems,” IEEE Trans. Autom. Control, vol. 49, no. 5, pp. 767–772, May 2004. [30] A. Xu and Q. Zhang, “Nonlinear system fault diagnosis based on adaptive estimation,” Automatica, vol. 40, no. 7, pp. 1181–1193, Jul. 2004. [31] A. Vemuri and M. Polycarpou, “Robust nonlinear fault diagnosis in input-output systems,” Int. J. Control, vol. 68, no. 2, pp. 343–360, Feb. 1997. [32] P. A. Ioannou and J. Sun, Robust Adaptive Control. Englewood Cliffs, NJ, USA: Prentice-Hall, 1995. [33] R. Ferrari, T. Parisini, and M. Polycarpou, “Distributed fault diagnosis with overlapping decompositions: An adaptive approximation approach,” IEEE Trans. Autom. Control, vol. 54, no. 4, pp. 794–799, Apr. 2009.

REPPA et al.: ADAPTIVE APPROXIMATION FOR MULTIPLE SFDI

[34] R. Ferrari, T. Parisini, and M. Polycarpou, “Distributed fault diagnosis of large-scale discrete-time nonlinear systems: New results on the isolation problem,” in Proc. 49th IEEE Conf. Decision Control, Dec. 2010, pp. 1619–1626. [35] Y. Zhang and X. Li, “Detection and diagnosis of sensor and actuator failures using IMM estimator,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 4, pp. 1293–1313, Oct. 1998. [36] K. Salahshoor, M. Mosallaei, and M. Bayat, “Centralized and decentralized process and sensor fault monitoring using data fusion based on adaptive extended Kalman filter algorithm,” Measurement, vol. 41, no. 10, pp. 1059–1076, Oct. 2008. [37] M. Milanese and C. Novara, “Set membership identification of nonlinear systems,” Automatica, vol. 40, no. 6, pp. 957–975, Jun. 2004. [38] W. Yu and J. de Jesús Rubio, “Recurrent neural networks training with stable bounding ellipsoid algorithm,” IEEE Trans. Neural Netw., vol. 20, no. 6, pp. 983–991, Jun. 2009. [39] E. Kosmatopoulos, M. Polycarpou, M. Christodoulou, and P. Ioannou, “High-order neural network structures for identification of dynamical systems,” IEEE Trans. Neural Netw., vol. 6, no. 2, pp. 422–431, Mar. 1995. [40] Z. Uykan, C. Guzelis, M. Celebi, and H. Koivo, “Analysis of inputoutput clustering for determining centers of RBFN,” IEEE Trans. Neural Netw., vol. 11, no. 4, pp. 851–858, Jul. 2000. [41] M. Cordier, P. Dague, F. Lévy, J. Montmain, M. Staroswiecki, and L. Travé-Massuyès, “Conflicts versus analytical redundancy relations: A comparative analysis of the model based diagnosis approach from the artificial intelligence and automatic control perspectives,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34, no. 5, pp. 2163–2177, Oct. 2004. [42] X. Zhang, “Sensor bias fault detection and isolation in a class of nonlinear uncertain systems using adaptive estimation,” IEEE Trans. Autom. Control, vol. 56, no. 5, pp. 1220–1226, May 2011.

Vasso Reppa (S’06–M’12) received the Diploma in electrical and computer engineering and the Ph.D. degree in fault diagnosis from the University of Patras, Patras, Greece, in 2004 and 2010, respectively. She was an External Scientific Collaborator with Patras Scientific Park S.A., Patra, Greece, from 2006 to 2008. She was a Researcher in various Hellenic and European research and operational programmes. She was a Student Intern with the Storage Technologies Department, IBM Zurich Research Laboratory, Rüschlikon, Switzerland, in 2009. Since January 2011, she has been a PostDoctoral Researcher with the KIOS Research Center for Intelligent Systems and Networks, University of Cyprus, Nicosia, Cyprus. Her current research interests include fault diagnosis, adaptive learning, set-membership identification and control, with applications to microelectromechanical systems and large-scale engineering systems. Dr. Reppa is a reviewer for various conferences and journals.

153

Marios M. Polycarpou (M’92–SM’98–F’06) received the B.A. degree in computer science and the B.Sc. degree in electrical engineering from Rice University, Houston, TX, USA, in 1987, and the M.S. and Ph.D. degrees in electrical engineering from the University of Southern California, Los Angeles, CA, USA, in 1989 and 1992, respectively. He is a Professor of electrical and computer engineering and the Director of the KIOS Research Center for Intelligent Systems and Networks, University of Cyprus, Nicosia, Cyprus. Prior to joining the University of Cyprus as Founding Department Chair in 2001, he was Professor of electrical and computer engineering and computer science with the University of Cincinnati, Cincinnati, OH, USA. He has authored over 220 papers in his areas of expertise and is the holder of three patents. His teaching and research interests include intelligent systems and control, fault diagnosis, computational intelligence, adaptive and cooperative control systems, and large-scale systems. Prof. Polycarpou currently serves as the President of the IEEE Computational Intelligence Society. He served as the Editor-in-Chief of the IEEE T RANSACTIONS ON N EURAL N ETWORKS AND L EARNING S YSTEMS from 2004 until 2010. He participated in more than 60 research projects/grants, funded by several agencies and industry in Europe and the USA. He was the recipient of the prestigious European Research Council Advanced Grant Award in 2011.

Christos G. Panayiotou (M’94–SM’06) received the B.Sc. and Ph.D. degrees in electrical and computer engineering from the University of Massachusetts at Amherst, MA, USA, in 1994 and 1999, respectively, and the MBA degree from the Isenberg School of Management, University of Massachusetts at Amherst, MA, USA, in 1999. He was a Research Associate with the Center for Information and System Engineering (CISE) and the Manufacturing Engineering Department, Boston University, from 1999 to 2002. He was a Visiting Lecturer with the Electrical and Computer Engineering Department, University of Cyprus (UCY), Nicosia, Cyprus, from 2002 to 2003. He was an Assistant Professor from 2003 to 2008 and is currently an Associate Professor with the Electrical and Computer Engineering Department, UCY. His current research interests include wireless, ad hoc and sensor networks, computer communication networks, distributed control systems, fault diagnosis and fault tolerant systems, optimization and control of discrete-event systems, resource allocation, and simulation. Dr. Panayiotou is an Associate Editor for the Conference Editorial Board of the IEEE Control Systems Society, the Journal of Discrete-Event Dynamical Systems, and the European Journal of Control. He is also a reviewer for various conferences and journals, and has served on the organizing and program committees of various international conferences.