Aggregation-based fault diagnosis algorithms⁎

10th IFAC Symposium on Fault Detection, 10th IFAC Symposium on Fault Detection, Supervision and Safetyon for Technical Processes 10th IFAC Symposium Detection, Available online at www.sciencedirect.com Supervision and Safety forFault Technical Processes 10th IFACPoland, Symposium on29-31, Fault Detection, Warsaw, August 2018 Supervision and Safety Technical Warsaw, Poland, Augustfor 29-31, 2018 Processes Supervision and Safety Technical Warsaw, Poland, Augustfor 29-31, 2018 Processes Warsaw, Poland, August 29-31, 2018

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IFAC PapersOnLine 51-24 (2018) 488–493

Aggregation-based Aggregation-based Aggregation-based  fault diagnosis algorithms Aggregation-based fault fault diagnosis diagnosis algorithms algorithms  fault diagnosis algorithms ∗ ∗

Przemysław Przemysław Przemysław Przemysław

∗ Śliwiński Śliwiński ∗∗∗ Paweł Paweł Wachel Wachel∗∗∗∗ Zygmunt Zygmunt Hasiewicz Hasiewicz ∗∗∗ Śliwiński Szymon Paweł Wachel Zygmunt Hasiewicz Łagosz ∗ ∗ Łagosz ∗∗∗ Zygmunt Hasiewicz ∗ Śliwiński Szymon Paweł Wachel Szymon Łagosz Szymon Łagosz ∗ ∗ ∗ Department of Control Systems and Mechatronics, ∗ Department of Control Systems and Mechatronics, ∗ Department Control and Systems and Mechatronics, Wrocław University of Science Technology, Wrocław, Poland ∗ University of Science and Technology, Wrocław, Poland Wrocław of Control Systems and Mechatronics, WrocławDepartment University of Science and Technology, Wrocław, Poland (e-mail: [email protected]). (e-mail: [email protected]). Wrocław University of Science and Technology, Wrocław, Poland (e-mail: [email protected]). (e-mail: [email protected]). Abstract: We propose and discuss aa fault fault diagnosis diagnosis algorithm algorithm in in which which the the aggregative aggregative modeling modeling Abstract: We propose and discuss Abstract: We propose and discuss a fault diagnosis algorithm in whichworking the aggregative modeling approach is explored. We assume that a user has a model of a properly system, a number approach is We explored. We assume that a user has a model of a properly working system, amodeling number Abstract: propose and discuss a and fault diagnosis algorithm in which theon aggregative approach is explored. We assume that a user has a model of a properly working system,expansion a number of models of the known system faults an auxiliary generic model based Volterra of models is of the knownWe system faults an generic model based on approach assume thatand a user has a model of a two properly working system,expansion a number of ofexplored. the system faults and an auxiliary auxiliary generic model based on Volterra Volterra expansion to models represent the known unknown malfunctions. The algorithm has phases: to represent the unknown malfunctions. The algorithm has two phases: of models of the known system faults and an auxiliary generic model based onthe Volterra expansion to represent the unknown malfunctions. The algorithm has two phases: (1) The aggregation algorithm is used to evaluate the Volterra model from set of the (1) The aggregation algorithm is used toThe evaluate the Volterra model from the set of the system system to the unknown malfunctions. algorithm has twomodel phases: (1)represent The aggregation algorithm is{(x used to evaluate the Volterra from the set of the system input-output measurements , y )} , n = 1, . . . , N . n , yn )} , n = 1, . . . , N . input-output measurements {(x n, y n )} (1) The aggregation algorithmwith is{(x used model fromand the the set of thewith system input-output measurements ,set n =of1,the . . .Volterra , N .measurements, n to n evaluate (2) All models are compared another system one the n n (2) All models are compared with another set of system measurements, and the one with the measurements {(x , y )} , n = 1, . . . , N . n n (2) input-output All models are compared with another set of system measurements, and the one with the smallest error indicates the type of the system fault (or lack thereof). smallest error indicates the type of the system fault (or lack thereof). (2) illustrative All models areindicates comparedthe with another set ofWiener-Hammerstein system measurements, and the one with of the smallest error type of the system fault (or lack thereof). The experiments are performed on a system in presence The illustrative experiments are type performed a Wiener-Hammerstein system in presence of aa smallest error indicates the of the on system fault (or lack thereof). The illustrative experiments are performed on a Wiener-Hammerstein system in presence of a heavy noise. heavy noise. The experiments are performed on a Wiener-Hammerstein system in presence of a heavyillustrative noise. © 2018,noise. IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. heavy Keywords: Model Keywords: Model selection, selection, aggregative aggregative algorithms, algorithms, Volterra Volterra expansion, expansion, model/signal-based model/signal-based Keywords: Model selection, aggregative algorithms, Volterra expansion, model/signal-based fault detection fault detection Keywords: Model selection, aggregative algorithms, Volterra expansion, model/signal-based fault detection fault detection essential part of the algorithm is the mentioned Volterra 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION essential part of algorithm is mentioned Volterra 1. essential part task of the the algorithm is the the mentioned Volterra model whose is to effectively model such a variety in model whose task is to effectively model such a variety in essential part task of the algorithm is the mentioned Volterra 1. INTRODUCTION model whose is to effectively model such a variety in order to distinguish the unknown system failure from the In the paper we examine an applicability of the aggregationorder to distinguish the unknown model systemsuch failure from the In the paper we examine an applicability of the aggregationmodel whose task is to effectively a variety in order to distinguish the unknown system failure from the known ones (and from the properly working system itself). In the paper we examine an applicability of the aggregationbased system system modeling modeling algorithms algorithms to to the the fault fault diagnosis diagnosis known ones (and from the properly working system itself). based order to distinguish the unknown system failure from the In the paper wemodeling examine an of the aggregationones (andapplication from the properly system itself). Thanks to the of the working aggregative modeling, based system algorithms the fault diagnosis known problems. The following useapplicability case is to considered: Thanks to application of aggregative modeling, problems. The following use considered: known ones (andrequires from thea properly system itself). to the the application of the the working aggregative modeling, based system the fault diagnosis Thanks the algorithm relatively (w.r.t. the number problems. Themodeling following algorithms use case case is is to considered: the algorithm requires a relatively (w.r.t. themodeling, number Thanks to the of application of the aggregative • The nonlinear and dynamic system is of an unknown problems. The following use case is considered: the algorithm requires a relatively (w.r.t. the number of parameters the Volterra model) small number of •• The nonlinear of parameters the Volterra model) smallthe number of The nonlinear and and dynamic dynamic system system is is of of an an unknown unknown the algorithm –of requires aa relatively (w.r.t. number type. of parameters of the Volterra model) small number of measurements enabling quasi-on-line working régime. type. aa quasi-on-line working régime. • type. The nonlinear and dynamic system is ofare an available. unknown measurements of parameters ––ofenabling the Volterra model) small number of • Models of select known system’s faults measurements enabling quasi-on-line working régime. •• Models of select known system’s faults are available. type.unmodeled The proposed algorithm model-based fault detection Models of select system known failures system’sare faults are available. measurements – enablingfits a the quasi-on-line working régime. The admitted. The proposed algorithm fits the model-based fault detection •• The unmodeled admitted. Models of select system known failures system’sare faults are available. The proposed algorithm fits the model-based fault detection scheme; cf. Ding [2013], Gao et al. [2015] but can also be The unmodeled system failures are admitted. cf. [2013], Gao et al. [2015] but can also be The proposed algorithm fault In particular, in the proposed diagnosis algorithm we we asas- scheme; scheme; cf. Ding Ding [2013], fits Gaothe etamodel-based al.pattern [2015] recognition but candetection alsowith be •particular, The unmodeled system failures are admitted. seen as an implementation of In in the proposed diagnosis algorithm seen as an implementation of a pattern recognition with In particular, in the proposed diagnosis algorithm we asscheme; cf. Ding [2013], Gao et al. [2015] but can also be sume that a user has an access to the (noisy) input-output seen as an implementation of a pattern recognition with rejection approach (see [1970], Homenda al. sume that has an access to the (noisy) input-output rejection approach (see e.g. e.g. Chow Chow [1970], recognition Homenda et etwith al. In particular, inofthe proposed diagnosis algorithm we assume that aa user user has an access {(x to the (noisy) input-output seen as an implementation of a pattern measurements the system , y )} , n = 1, 2 . . ., to n n rejection approach (see e.g. Chow [1970], Homenda et al. [2016]). )} , n = 1, 2 . . ., to [2016]). measurements of the {(x n , yn sume that a user has accessof tothe the (noisy) input-output measurements of theansystem system {(x approach (see e.g. Chow [1970], Homenda et al. = 1,properly 2 . . ., to rejection n n the measurements model the n, y n )} , nof the (noise-less) (noise-less) of measurements of the model of the measurements the system {(x , y )} , n = 1,properly 2 . . .,systo [2016]). n model n [2016]). the (noise-less) measurements of the of the properly working system and of several, F say, models of known 2. working system and of several, F say, models of known sys2. PROBLEM PROBLEM STATEMENT STATEMENT the (noise-less) measurements of model of the properly working system and ofn )} several, Fnthe say, models of known system faults, {x , m (x and {x , m (x )} , f = 1, . . . , F , 2. PROBLEM STATEMENT n f n tem faults, {x ,and m (xofn )} and {x , mfmodels (xn )} , of f= 1, . . .sys,F, nsay, working system several, F known 1n 2. PROBLEM STATEMENT tem faults, {x , m (x )} and {x , m (x )} , f = 1, . . . , F , nIn order n to handlen nthef funknown n respectively. system failn n 2.1 Assumptions 1n respectively. order toand handle the unknown system fail2.1 Assumptions 1 tem faults, {x , m (x )} {x , m (x )} , f = 1, . . . , F , 1 nIn n n f n 2.1 Assumptions respectively. In order to handle the unknown system failures, the expansion-based model, m ˆ is 1 ures, the Volterra Volterra expansion-based m ˆ θθˆˆ(x), (x), is used. used. respectively. In(that order to the handle themodel, unknown system fail- 2.1 Assumptions ures, the Volterra expansion-based model, m ˆ θθˆˆ(x), is used. In the paper Its parameters is, Volterra kernel coefficients) In the paper we we assume assume that that the the following following properties properties of of Its parameters (that is, the Volterra kernel coefficients) ures, the Volterra expansion-based model, m ˆ (x), is used. In the paper we assume that the following properties of θˆ coefficients) Its parameters (that is, the Volterra kernel the system and measurement signals hold: are evaluated using an aggregative approach; see Wachel the system and measurement signals hold: are evaluated using an aggregative approach; see Wachel In paper wemeasurement assume that signals the following Its parameters (thatan is,aggregative the [2016], Volterra kernel coefficients) thethe system and hold: properties of are evaluated using approach; see Wachel and Śliwiński Wachel Śliwiński al. [2017], and Śliwiński [2015], [2015], Wachel [2016], approach; Śliwiński et etsee al.Wachel [2017], (A1) The nonlinear dynamic system m (x) the system and measurement signals hold: are evaluated using an aggregative (A1) The nonlinear dynamic system m (x) and and its its faulty faulty and Śliwiński [2015], Wachel [2016], Śliwiński et al. [2017], Wachel [2017]. The that matches best the Wachel [2017].[2015], The model model that matches best et theal.working working (A1) versions The nonlinear dynamic system m (x) and its faulty (x) , . . . , m (x), are continuous maps m 1 F and Śliwiński Wachel [2016], Śliwiński [2017], (x) , . . . , m (x), are continuous maps versions m Wachel [2017]. The model that matches best the working system measurements (has the smallest residual error ) 1 F The nonlinear m (x) and its faulty , . .longer . , mF (x), are continuous maps versions m11 (x)dynamic system measurements (hasthat thematches smallestbest residual error ) (A1) with Fsystem a memory no than some τ . Wachel [2017]. The model the working with a memory no longer than some τ . system measurements (has the smallest residual error ) determines the system state: no fault, a known fault event ,{x . .longer . ,}m are continuous maps versions msignal 1 (x)no Fa(x), determines the system state: no fault, a known fault event with a memory than some τ . (A2) The input is white stationary process n } is a white stationary process of system measurements (has the smallest residual determines the system state: no fault, a known faulterror event) (A2) with The input signalno {xlonger of or an unknown one. n a memory than some τ . or an unknown one. (A2) The input signal {x } is a white stationary process of n an unknown yet bounded distribution. The additive n determines the system unknown yet {x bounded distribution. Theprocess additive or an unknown one. state: no fault, a known fault event (A2) an The input signal } is a white stationary of n an unknown yet bounded distribution. The additive output noise {z } is a stationary, white, zero-mean Clearly, the class of nonlinear dynamic systems is ample n or an unknown one. output noise {z is aa stationary, white, zero-mean Clearly, the class of nonlinear dynamic systems is ample n} 2 an unknown yet bounded distribution. The additive output noise {z } is stationary, white, zero-mean Clearly, the class of nonlinear dynamic systems is ample n process with variance σ {x and so the set failures. Hence, the n process noise with aa{zfinite finite variance σ 222 .. Processes Processes {xnn } } and and and so is is setofof of their their possible possible failures. Hence, the output a stationary, white, zero-mean Clearly, thethe class dynamic systems is ample n } isvariance process with a finite σ . Processes {x } and and so is the set ofnonlinear their possible failures. Hence, the n {z } are independent. n n 2 {z } are independent. n process with a finite variance σ . Processes {x } and so is the set of their possible failures. Hence, the  The work is supported by the Wrocław University of Science and n and {znn } are independent.  The work is supported by the Wrocław University of Science and Let us shortly discuss the above assumptions: according {z } are independent.  n Let us shortly discuss the above assumptions: according Technology S0401/0155/17. The work Grant is supported by the Wrocław University of Science and Technology Grant S0401/0155/17.  Let us shortly discuss the above assumptions: according to Assumption A1, aa structure of system 1 The work is supported bythe thesoftware Wrocławones University Science and models can e.g. be that areofoften created to Assumption A1, neither neither structure of the the system Technology Grant S0401/0155/17. 1 These Let us shortly discuss the above assumptions: according These models can e.g. be the software ones that are often created 1 to Assumption A1, neither a structure of the system nor the characteristics of its building blocks are known. 1 Technology Grant S0401/0155/17. byThese system designers for fast prototyping. Theythat can are alsooften be remotely models can e.g. be the software ones created nor the characteristics of its building blocks are known. by system designers for fast prototyping. They can also be remotely to Assumption A1, neither a structure of the system 1 nor the characteristics of its building blocks are known. These models can for e.g. be the software ones that are created This implies that neither parametric nor nonparametric accessed todesigners protect thefast intellectual property assets ofoften theremotely system by system prototyping. They can also be This implies that neither parametric nor nonparametric accessed to protect the intellectual property assets of the system nor the characteristics of its building blocks are known. This implies that neither parametric nor nonparametric by system designers for fast prototyping. They can also be remotely accessed to protect the intellectual property assets of the system designer/manufacturer. algorithm dedicated to system structure designer/manufacturer. algorithm dedicated to the the particular particular structure (be (be This implies that neither parametricsystem nor nonparametric accessed to protect the intellectual property assets of the system designer/manufacturer. algorithm dedicated to the particular system structure (be designer/manufacturer. algorithm dedicated to the particular system structure (be 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Copyright © 2018 IFAC 488 Copyright 2018 IFAC 488 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 488 10.1016/j.ifacol.2018.09.621 Copyright © 2018 IFAC 488

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it a Wiener or Hammerstein one) can be used to identify and model the system (and its failures). For this reason the approach based on Volterra expansion is employed, as the continuity assumption, together with the finite memory restriction, allow the Volterra expansion to model the system (admitting infinite memory is possible, but it raises some implementation issues and results in a slightly less efficient aggregation; cf. Wachel [2016, 2017]). 2 In turn, Assumption A2 is a standard one and allows the modeling problem to be solved with the help of statistics methods. 3 3. THE ALGORITHM First, we shortly recollect the aggregation algorithm and its properties under assumptions A1-A2; cf. Śliwiński et al. [2017]. 3.1 Aggregation modeling algorithm

V (xn ) = h0 + H1L (xn ) + · · · + HP L (xn ) = h0 +

(1)

HpL (xn ) ,

p=1

where xn = [xn , xn−1 , . . . , xn−L+1 ] is a vector of inputs, h0 is a constant and HpL (xn ) are truncated Volterra operators (see e.g. Alper [1965], Boyd et al. [1984], Pearson and Ogunnaike [2002]) HpL (xn ) =

L−1 

k1 =0

···

L−1 

kp =0

hk1 ...kp ·

p 

xn−ki ,

i=1

and where hk1 ...kp are the pth order Volterra kernels; P denotes the degree of the expansion and L ≤ τ is the model memory length. The number ofmodel  parameters, denoted further by D is thus equal to L+P and is large even for L small values of P and L; cf. Kekatos and Giannakis [2011].

Because the structure of the Volterra expansion is not relevant in the algorithm, we will use an equivalent yet simpler representation of the model (1), in which all kernels hk1 ...kp , p = 1, . . . , P , kj = 0, . . . , L − 1 and j = 0, . . . , p, are gathered into a single parameter vector T θ = [θ1 , θ2 , . . . , θD ] . The corresponding Volterra terms will be denoted as vi (x) , i = 1, . . . , D. The aggregative model to be used in the fault diagnosis algorithm can be now expressed as m ˆ θˆ(x) =

D 

θˆi vi (x) ,

from the measurement set {(xn , yn )} , n = 1, . . . , N , by minimizing the empirical quadratic criterion N  2 ˆ (θ) = 1 [m ˆ θ (xi ) − yi ] , (3) Q N − τ i=τ +1

with the following norm constraint (4) θ1 ≤ 1. Remark 1. The constraint in (4) means that the sequence of Volterra kernel coefficients θ are absolutely summable. This assumption is satisfied for all Volterra models with finite P and L (and hence with finite D) up to some (unknown) multiplicative constant R > 0; see Section 3.2. The performance of the empirical model (2) will be, for a given D, compared to the best possible one D  θi∗ vi (x) , (5) m ˆ θ∗ (x) = i=1

The model of unknown faults is based on a standard double-truncated Volterra series

P 

489

(2)

i=1

where θˆ = [θˆ1 , θˆ2 , . . . , θˆD ]T is now the vector of empirical coefficients (estimates of unknown elements of θ) obtained Actually, neither the system m (x) nor their known faulty versions m1 (x) , . . . , mF (x) need to have a finite memory; we only require the unmodeled fault to have this property. 3 A2 can be relaxed so that both the input signal, {x } , and the n noise, {zk } , can be the correlated processes; see Wachel and Śliwiński [2015]. 2

489

where the vector θ∗ minimizes the following least squares error (cf. (3) and (4)): 2

Q (θ) = E {m ˆ θ (xn ) − yn } ,

(6)

under the constraint θ1 ≤ 1.

(7)

The mean accuracy of m ˆ θˆ(x) with respect to the accuracy of mθ∗ (x) can be upper bounded (for Assumptions A1– A2 and any D > 2), by the following inequality √ N  ˆ − Q (θ∗ ) ≤ C · (τ + 1) ln D, (8) E{Q(θ)} N −τ

with the constant √ (9) C = 32 e(M σ + 2M 2 ), = sup {m (x)} where M = max{Mm , Mv } with Mm x  and Mv = maxx supi=1,...,D {vi (x)} ; see Wachel and Śliwiński [2015], Śliwiński et al. [2017].

The bound implies a good behavior of the model (2), especially in case when D is large with respect to N. In particular, for D > N , the bound in (8) is lower than the order D/N usually obtained for least squares algorithms; cf. e.g. Ljung [2010]). Note finally that the error grows logarithmically with D, which is of special importance when combined with Volterra models, since it compensates (to some extent) the exponential growth of D w.r.t. both memory, L, and degree, P , parameters in (1). 3.2 Tuning the algorithm We do not know a priori if the constraint (4) holds for the system of interest. However, by multiplying all dictionary terms {vi (x)} by a sufficiently large factor R > 0, we can make any system compliant with (4). From the analysis in Śliwiński et al. [2017], we known that the aggregation error bound (8) constant C depends on R as follows; cf. (9): √ 2 C = 32 e(RM σ + 2 (RM ) ). A user can be interested in small R, however, too small R reduces the class of effectively modeled systems. In the mentioned paper, a simple heuristics was proposed to select R in case of lack of information about the system available to the user.

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zn

4. FAULT DIAGNOSIS ALGORITHM The proposed fault diagnosis algorithm has two phases: (1) Evaluation of the Volterra model m ˆ θˆ (x) using the measurements of the system, {(xn , yn )} , n = 1, . . . , N . (2) Comparing the outputs of all models with a separated set of system measurements, {(xw , yw )} , w = 1, . . . , W . The system state is indicated by the model with the smallest residue error, here in the form of the empirical mean squared error W  1 2 errori = [mi (xw ) − yw ] , (10) W − τ w=τ +1   where mi ∈ m, m1 , . . . , mF , m ˆ θˆ .

The algorithm is based on the following: Observation 1. Let Assumptions A1-A2 hold. For any fault not represented by the available fault models, m1 (x) , . . . mF (x), and for sufficiently large size D of the Volterra model m ˆ θˆ (x), there is a number of measurements   N such that the criterion in (3), that is Q θˆ of m ˆ ˆ (x) θ

is smaller than errors of the fault models in (10).

The observation is trivial, however, together with Assumptions A1-A2, it offers both a theoretical framework and a practical justification for the algorithm. Note that the model size, D, needs to be sufficiently large in order to effectively model the faulty system so that the modelling error (the difference between the faulty system and its model) is negligible. Usually, we are interested in a fast fault detection. Application of the aggregative algorithm allows for a relatively small (w.r.t. to D) number of measurements N needed to evaluate the Volterra model – which translates to relatively short detection time – and thus reconciles these two opposite requirements. 5. EXPERIMENTAL ILLUSTRATION A system with an LNL structure is taken as an example. This system seems to be simple, however, to the best of Authors’ knowledge, there is no single model (other than the Volterra expansion) 4 , which is capable of representing its structure and its various faults (which can change a system structure) simultaneously. 5 The nonlinear part (N) of the system was characterized by a function µ(x) = arctan(x), whereas input (L1 ) and output (L2 ) linear subsystems had finite, oscillating −i impulse responses, αi = 2−i and βi = (−2) , i = 0, 1, . . . , 19, respectively. The system was excited by a random i.i.d. sequence xn ∼ U [−5, 5] and disturbed by an additive output noise zn ∼ U [−0.5, 0.5]. The learning and validation sets had equal cardinality M = W = 500. Or its orthogonalized version, the Wiener expansion; see e.g. Wiener [1966], Nelles [2001], Westwick and Kearney [2003], Marmarelis [2004], Giri and Bai [2010]. 5 For simpler systems, like the linear dynamic, nonlinear static, Hammerstein or Wiener ones, the dedicated identification algorithms exist (see e.g. Hasiewicz and Śliwiński [2002], Pawlak et al. [2007], Greblicki and Pawlak [2008], Śliwiński [2013], Śliwiński et al. [2018]) and can be used to represent their unknown faults as well 4

490

N

L1

xn

L2

yn

(a) Wiener-Hammerstein system L1

L2

N

s/c

L1

s/c

Wiener system

L1

s/c

N

N

L2 Hammerstein system

L2

NF

L1

Cascade linear system

L2

Wiener-Hammerstein system

(b) Modeled and not modeled system faults

Fig. 1. A Wiener-Hammerstein test system and its faulty versions The main objective of the experiment was to test two different aggregation-based diagnosis algorithms in case of various system’s fault scenarios. To this end, we simulated two types of possible damage done to the system, that is: • Type I (unmodeled) fault: the nonlinear characteristic of the nonlinear subsystem is altered in an unknown way (in the example we applied function µ ˜(x) = arctan(x) · I[x≥0] (x), where I[·] (x) stands for an indicator function), and • Type II (structural) fault: one of the subsystems breaks down and the system reduces to either L1 N, NL2 or L1 L2 structure. The first algorithm (minimum error, ME ) was an implementation of the one described in Section 4. The other one (maximum weight, MW ) was its slightly modification, where the aggregative algorithm operates on all models and requires only a single measurement set {(xn , yn )} , n = 1, . . . , N . 5.1 Minimum error algorithm Five different models were at our disposal: L1 NL2 , L1 N, NL2 , L1 L2 and the Volterra one of order P = 3 and memory length L = 10 or 20. The results of the experiment are presented on Fig. 2-3. 5.2 Maximum weight algorithm The dictionary for this method was composed of a model of the true system, together with its faulty counterparts, i.e. the L1 N, NL2 and L1 L2 structures. Additionally, it contained unique terms {vi (x)} of a truncated Volterra model with P = 3 and L = 10 or 20. The purpose of the Volterra part was to capture type I faults, which were unmodeled by other dictionary’s elements. In the first step, the dictionary elements were aggregated only with the help of a learning set {(xn , yn )} of N measurements, obtained from either unbroken or faulty system. Then, by comparing weights assigned to models during the estimation process, the highest one was picked up as the indicator of the system’s structure, which directly translated into the type of an undergone fault (or it’s lack); see Figs. 4-5. 5.3 Results The experiments demonstrate that:

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101

101 Model

Model

L1 NL2 L1 N NL2 L1 L2 Volterra

L1 NL2 L1 N NL2 L1 L2 Volterra

error

100

error

100

491

10!1

10!1

10!2

10!2 none

L1 N

NL2

L1 L2

unmodeled

none

L1 N

fault type

NL2

L1 L2

unmodeled

L1 L2

unmodeled

fault type

(a) R = 1, L = 10

(a) R = 0.001, L = 20

101

101 Model

error

100

error

100

L1 NL2 L1 N NL2 L1 L2 Volterra

10!1

10!1

10!2

10!2 none

L1 N

NL2

L1 L2

unmodeled

none

fault type

L1 N

NL2

fault type

(b) R = 1, L = 20

(b) R = 100, L = 20

Fig. 2. Outcomes of ME algorithm: R tuned properly for system with short (a) and longer (b) memory

Fig. 3. Outcomes of ME algorithm: the value of R was either too low (a) or too high (b)

• if R is tuned properly, both methods adapt to given circumstances and accurately detect the type of the examined system’s fault (Fig. 2-5). However, • too large R can be a reason behind false outputs of proposed algorithms. The problem is especially seen, when the number of a dictionary’s elements exceeds the size of the learning set, which is the case when we use a Volterra model with memory of length L = 20. In such situations, the advantage of aggregation’s 1 constraint is heavily diminished, as the estimator is effectively reduced to the standard least squares method, what, in turn, entails higher variance of estimates. With regards to MW algorithm, this leads to inability of differentiation unmodeled faults from properly working system or structural faults (Fig. 5b), making the method completely ineffective. The ME algorithm seems to be more robust against an improper choose of a scaling factor R (Fig. 3b). It can be explained by the fact, that in this approach only Volterra kernels are aggregated and therefore it cannot influence detecting faults of the second type.

• If R is too small, then both algorithms may have trouble with detecting unmodeled faults (Fig. 5a, 3a). This is due to the fact, that too tight constraint (4) makes the optimal (in terms of given P and L) Volterra model infeasible. Another explanation is that for a finite dictionary and R → 0, the output of the model goes to zero as well. As a result, it may happen that one of the remaining models yields a better approximation of the working system, leading to incorrect estimate of the current state of the system. Remark 2. In accordance with Reviewer’s suggestion, a simple experiment has been conducted in order to compare the aggregation algorithm with a standard Principal Component Regression (PCR) method in the task of nonlinear systems modeling. A plain L1 N system, with a short memory and third order polynomial nonlinearity has been taken under consideration. For the record, the dictionary was composed of unique terms of truncated Volterra model with P = 3 and L = 20 (i.e. D = 1771); the system was excited by a random i.i.d. sequence xn ∼ U [−3, 3] and disturbed by an additive output noise zn ∼ U [−0.1, 0.1].

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Fig. 5. Outcomes of MW algorithm: the value of R was either too low (a) or too high (b)

The results, presented in Fig. 6, reveal that when the size of measurements set is small (relatively to the number of dictionary’s elements), the aggregation algorithm is much more effective and thus should be regarded as the first choice method when the user is interested in fast fault detection.

ACKNOWLEDGEMENTS

6. CONCLUSION In the paper we have proposed a fault detection algorithm based on aggregative modeling. The algorithm is designed to work with dynamic nonlinear systems which can be effectively modeled by the Volterra expansion. Application of aggregative modeling makes Volterra models viable in practice due to the fact that for all √N ≤ D, the model m ˆ θˆ (x) has an error of order order O( N −1 ln D) which is smaller than the order O (D/N ), typical for models based on the least squares method. Remark 3. One can examine off-line the structure of Volterra kernels in m ˆ θˆ (x) to create the new signatures of unknown faults and add them to the set of known fault models. 492

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