12th International Conference on DEVELOPMENT AND APPLICATION SYSTEMS, Suceava, Romania, May 15-17, 2014

978-1-4799-5094-2/14/$31.00 ©2014 IEEE

On Quick-Change Detection based on Process Adaptive Modelling and Identification

Dorel Aiordachioaie “Dunarea de Jos” University of Galati

Electronics and Telecommunications Department Galati, Romania

E-mail: Dorel.Aiordachioaie@ugal.ro

Abstract—This work introduces a simple method and

implementation algorithm for quick-change detection based on process measurements, adaptive modelling and identification, which will be called model-based quick-change detection (M-QCD). The idea of the method is to select a proper model for process identification purposes, and to continuously estimate the parameters of this model. A change in process implies a change in the set of parameters, which is detected using a criterion based on gradient and energy of the previous values of the criterion. The results are compared to those given by an off-line classical algorithm used in change detection area, based on variance. Computer simulations show better results for the proposed method and confirm that - some times - simple methods and algorithms, free of constraints, can work at the same performance level with change detection complex algorithms, which use advanced processing and optimization.

Keywords—process; identification; modelling; signal; change detection; segmentation; parameter estimation.

I. INTRODUCTION In the large field of fault detection and diagnosis, the

advanced processing techniques of data, provided mainly by measurements, are the key element of the automatic monitoring and conditional maintenance systems. A refined analysis of the data collected by the sensors allows for a more precise estimation of the equipment state. It also allows initiating inspections or maintenance procedures only if they are necessary. In this respect, change detection of the process behavior is an important task.

The general topic of fault detection and diagnosis is well covered by scientific and application oriented literature, e.g. [1] and [2], fault detection based on model in [3], fault detection in [4] and [5], and change detection problem and adequate solution in [6] and [7].

Many applications on this subject make use of theories based on statistics [8], which provide the theoretical instruments of solving the problem of early detection. Such a theory is referred to as local approach [9] and is based on the transformation of the general detection problem into a classical problem of monitoring the mean of a Gaussian vector variable. This theory assumes that a mathematical model of the monitored system is available. The hypothesis is reasonable,

because many industrial processes are based on known physical principles, with available equations as analytical models. When the models are very complicated or unknown, the local approach can still use semi-physical models or black-box models.

The detection and diagnosis methods aim at solving the following problems: (1) detection of the change (the alarm); (2) diagnosis of the change (source isolation) and (3) evaluation of the change (estimation). These represent different complexity levels of the change detection and diagnosis (CDD) problem. The alarm marks the event of a change and is a simple binary decision, whether the change appeared or not. The purpose of the isolation is to identify the cause (source) of the change and its spatial origin. The estimation evaluates the extent and the characteristics of the change. The weights of these objectives depend on application requirements and on the specific conditions of the equipment.

A detection/diagnosis algorithm usually implies two stages: residuals generation and decision. Residuals are the difference vectors between the observed (measured) behavior and the expected behavior of the system. The classical instruments for residuals generation are the filters and the estimators, e.g. [10]. Within the CDD problem, already in its mature phase, there is a gap between theory and practice. At present, the research effort of this domain is targeted to developing applications that use robust CDD methods, reduced order models and appropriate distances (as signs of the change) [11]. The research in the field investigates the cases where reducing the ambitious requests of the CDD problem may be beneficial to the practical application. High quality models, complex and precise, can bring good performance in the CDD solutions, but – in other cases – reduced order parametric models, with drifted values of the parameters, can be successfully used. The drifted values can reduce but not wipe the performance of the CDD procedures.

Concerning the existing challenges of the investigated domain, there is still a “gap” between the theoretical results and the applications. This is mainly caused by the requirements of some “strong” hypotheses in the existing algorithms, which are not easily verified in practice. These hypotheses are connected to the non-unique model, the nonlinear and variant properties of the process and high-level noise with unknown properties. If

25

these hypotheses are not satisfied, the CDD algorithms will not be able to separate the external from internal changes, leading to possibly incorrect decisions. Another challenge is the time restriction, in the case of real time applications, when low computation effort is possible and reduced order models have to be used, but without reducing the performance of the detection and diagnosis system.

In CDD research area, two main categories of methods could be recognized, as it is presented in Fig. 1. In the first category of methods, the measured signals are directly processed mainly based on statistical averages up to order 4. Examples of such solutions are presented in [8] or [12].

The second category of method uses transformations, from signal space to other data spaces, with or without mandatory meaning. The time–frequency transformation, [13] or [14], and Renyi entropy, as in [15], are just two examples.

The work introduces a quick-change detection (QCD) method, based on on-line measurements and not on batch sets of data. The results are compared to those given by an off-line classical algorithm used in the change detection area. One conclusion is that, some-times, simple methods and algorithms, free of constraints, can work at the same performance level with advanced processing and optimization CDD algorithms. Section II presents the proposed solution and section III presents the results obtained by computer simulation. The results of two methods are compared and discussed, i.e. a classical method based on variance and the one developed in this work.

Fig. 1. Basic structures of the CDD methods

II. DESCRIPTION OF THE SOLUTION

A. The reference signal In order to check the validity of the proposed techniques, a

synthetic signal is considered. It is composed of seven multicomponent frames or epochs (Ne=7) with the duration of five seconds. The sampling frequency is Fs = 100 Hz. The noise-free signal is built by

( ) Ne,...,,i,)tfcosA)t(u ijiji 212 =⋅⋅⋅=∑ π (1)

A Gaussian signal noise is added to the previous signal:

)()()( tntuts += (2)

with zero mean and 0.1 variance. The values of the amplitudes and frequencies are presented in Eq. 3. Fig. 2 shows the global

evolution of the analyzed signal. The moments of changes are represented with spikes, and the vector of changes is done by Eq. (4). Such a signal is common in biomedical engineering, industrial vibrational processes and seismic processes.

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

0.00.30.18.00.00.05.25.00.00.05.25.05.38.05.15.00.00.05.40.10.00.00.45.10.00.45.15.0

A

,⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

045.25.10045.100838315.0005.35.000710425.0

F

(3)

[ ]30252015105=MC (4)

B. The method The objective of this work is to find a method to indicate

the exact time changes, with minimum computational resources, and as quickly as possible. The proposed method is built around a model behavior, and its name is model-based quick-change detection (M-QCD). Fig. 3 presents the structure of the method. The signals coming from process are inputs to a structure with modelling and parameter estimation tasks. Depending on input data, simple models as AR (Auto Regressive), MA (Moving Average) or ARMA are used. Estimation is made in adaptive fashion and on-line framework, i.e. after each value of inputs, an update of the parameters is made. Depending on available resources, RLS (Recursive Least Squares) or LMS (Least Mean Square) or Kalman adaptive estimators could be used, as described in [16] or [17]. The parameters are inputs to a change detection block, which indicated the moments of changes based on the changes of the parameters’ values. It uses a set of three algorithms (parameter estimation, computation of detection variables and testing), as described in the M_QCD procedure.

Fig. 2. The reference signal

Fig. 3. The structure of the proposed method (M-QCD)

26

M_QCD Procedure: Loop #1: Read signals of the process; #2: Estimate parameters of the model; #3: Compute the detection variables; #4: Test the change detection condition. Until STOP; End M_QCD.

C. Estimation of the model parameters Referring to the structure of the method, in Fig. 3, the data

analysis block indicates an AR model described by

,...,,n),k(v)in(ua)n(uM

ii 210

1=+−⋅−= ∑

=

(5)

where ai refers AR parameters and v(k) means white noise with zero mean and finite variance. The model is a common choice for modelling signals, which have harmonic components.

The estimation of the parameters is made under adaptive framework and using RLS algorithm. The requirement is to estimate the unknown parameter vector of a multiple regression model that relates the desired response d(n) to the input vector u(n). The structure of the algorithm is presented below and details are presented e.g. in [16]. The global error criterion J is defined by the instantaneous squared error.

RLS Algorithm: #1. Initializations:

0w =)( 0 ; % weights δ =200; % regularization

IP ⋅= −1)0( δ ; % inv. correlation 990.=λ ; % forgetting

n = 0; % index #2. LOOP n:

n = n+1; [ ]T)Mn(u),...,n(u),n(u)n( −−−= 21u

#2.1. Compute: )()1()( nnn uPπ ⋅−=

#2.2. Compute gain: ( ))n()n(/)n()n( T πuπk ⋅+= λ

#2.3. Estimate error: )n()n(ˆ)n(d)n( T uw ⋅−−= 1ξ

#2.4. Adaptation: )()()1(ˆ)(ˆ * nnnn ξ⋅+−= kww

#2.5. Compute: )n()n()n()n()n( T 11 11 −⋅−−= −− PukPP λλ

UNTIL minJ)n(J < END.

D. Computation and testing of detection variables If w of size Mx1 is the vector of the estimated parameters, dw of size Mxm is a matrix with the gradient of w. The basic equations, which describe detection, use a variable of change detection, cdw, which is a sliding window of length m:

)(diff MxmMxm wdw = (6)

m,i,)j,i(dw)i(cdwM

jj 1

1=⋅= ∑

=α (7)

where: diff is an operator making consecutive differences; M is the number of parameters; m is the memory order (the number of past values); α is a weight vector satisfying the constraint

M... ααα ≥≥≥ 21 (8)

which considers the sensitivity of model’s parameters. The last value of the change detection variable, cdw, is tested with a level γ defined on the energy of the last m samples:

∑=

=m

i)i(cdw

1

2γ (9)

The global change detection variable, CD, is switched to TRUE value if cdw is greater than γ:

TRUETHENIF => )i(CD,)i(cdw γ (10)

III. RESULTS Some simulation results are presented in this section. Fig. 4

shows the evolution of the parameters during the estimation stage (M=4 parameters). During each segment, the estimation is asymptotically converging to some values. At the end of each segment, there are high variations of the estimated values. At the bottom of the figure, the evolution of the error criterion based on squared errors is presented.

Fig. 5 presents the evolution of the detection criterion, as defined by (7). Fig. 6 shows the results of the change detection. On the upper side, the results are obtained with an algorithm based on variance, which is described in [18] and [19]. It was developed by Lavielle for detection of multiple changes in a sequence of dependent variables. The Matlab implementation function “wvarchg” meant to find variance change points has the following parameters: the minimum delay between two change points D = 200; the maximum number of change points is K = 10. It should be mentioned here this is an off-line algorithm (i.e. batch data processing) and it requires zero mean signals.

The segmentation made by the reference function introduces two false results: around 5 s (no change) and around 32 s (change). The results from the bottom of Fig. 6 are for M > 4. For model order M < 4 the results are not appropriate.

27

Fig. 4. Parameter estimation and error criterion

Fig. 5. Change detection criterion

Fig. 6. Reference and obtained results for change detection

CONCLUSION The objective of the paper was to evaluate a quick method

for change detection. The proposed method is based on the gradient of the parameter estimated values. The global behavior of the estimation is defined by the weighted sum of the gradient of the changes, which is the change detection variable. At each instants of time, the variable is compared to its energy computed over twenty past values. The simulation results are compared to the results given by a classical algorithm based on change of the variance. Despite the fact that the reference needs more time than the proposed one, the results of the proposed methods (named M-QCD) are better.

Comparing with other off-line methods based on complex transformations (time-frequency and entropy), e.g. [15], or statistic analysis, e.g. [12], the proposed method is faster. The minimum time interval between two consecutive changes is conditioned by the a priori knowledge of the process model (identification time). The proposed method uses only linear models. Nonlinear models could also be used, but the detection criterion must be changed, as structure and parameters.

Further simulations and experiments with real data and scenarios need to be considered, in order to check and to validate the performances of the proposed method in correlation with various dynamic processes. Each class of processes (i.e. with high/small output ranges and fast/slow dynamic) needs particular values for the parameters of the detection criterion, which depends also on the imposed performances, but the structure of the detection method remains unchanged.

REFERENCES [1] Patton, R.J., Frank, P. and Clark, R., Eds., Fault Diagnosis in Dynamic

Systems – Theory and Application. Prentice Hall, 1989. [2] Gertler, J., Fault Detection and Diagnosis in Engineering Systems.

Marcel Dekker, 1998. [3] Chen, J. and Patton, R.J., Robust Model-Based Fault Diagnosis for

Dynamic Systems, Kluwer Academic Publishers, 1999. [4] Mangoubi, R.S., Robust Est. and Failure Detection, Springer, 1998. [5] Isermann, R., “Supervision, fault-detection and fault-diagnosis methods

– An introduction”, Control Engineering Practice, vol. 5, no. 5, 1997. [6] Basseville, M. and Nikiforov, I., Detection of Abrupt Changes – Theory

and Applications. Prentice Hall, N.J., 1993. [7] Gustafsson, F., Adaptive Filtering and Change Detection, Wiley, 2001. [8] Basseville, M., “Statistical approaches to industrial monitoring problems

– Fault detection and isolation”, Proc. of the 11th IFAC/IFORS Symposium on Identification and System Parameter Estimation – SYSID’97, Kitakyushu, Japan, July 8-11, 1997.

[9] Basseville, M., On-board component fault detection and isolation using the statistical local approach, Publication Interne no. 1122, IRISA,1997.

[10] Popescu, Th.D., “Detection and diagnosis of model parameter and noise variance changes with application in seismic signal processing”, Mechanical Syst. and SP, vol. 25, no. 5, 2011, pp. 1598-1616.

[11] Ninness B.M. and Goodwin G.C., Robust fault detection based on low order models, Proc. of the Int. Symp. on Fault Detection, Supervision and Safety of Technical Proc. (SAFEPROCESS), Baden-Baden, 1991.

[12] Aiordachioaie, D., “Signal Segmentation Based on Direct Use of Statistical Moments and Renyi Entropy”, 10th IEEE Int. Conf. on Electronics, Computer & Comput.(ICECCO’13), 2013, pp. 359-362.

[13] Popescu. Th.D., “Change detection in systems. Time and frequency approaches”, Neural Network World, vol.10, no 1-2, 2000, pp. 81–88.

[14] Gabarda, S. and Cristobal G, “Detection of events in seismic time series by time-frequency methods”, IET SP, vol. 4, no. 4, pp. 413-420, 2010.

[15] Popescu T. and Aiordachioaie D., “Signal Segmentation in Time-Frequency Plane using Renyi Entropy - Application in Seismic Signal Processing”, 2nd Int. Conf. on Control and Fault-Tolerant Systems, SysTol-2013, Oct. 9-11, 2013, Nice, France, pp. 312-317.

[16] Haykin S., Adaptive Filter Theory, 4th Edition, Prentice Hall, 2002. [17] Bozic S.M., Digital and Kalman Filtering. An Introduction to discrete

time-filtering and optimum linear estimation, Halsted Pr., 2nd Ed., 1996. [18] Bleakley K., Lavielle M., “Effective strategies for segmenting data into

coherent subsets”, HAL, http://hal.archives-ouvertes.fr/hal-00642621, 2012.

[19] Lavielle, M., “Detection of multiple changes in a sequence of dependent variables”, Stochastic Processes and their Applications, Elsevier, vol. 83, 1999, pp. 79-102.

28