Classification of parameter changes in a dynamic system with the use of wavelet analysis and neural networks

Abstract

In this paper a neural detector of internal parameter changes in a stationary, non-linear SISO dynamic system is considered. A dynamic system is usually described by an input–output relation or by a set of state equations. Each change of parameter values creates a new non-nominal model of a dynamic system (sometimes with different values of parameters, sometimes with different structure and different values of parameters). Thus the detection of parameter changes can be formulated as a multi-model classification. The LVQ (Learning Vector Quantisation) neural network has been proposed as a classifier. Selected aggregated properties of discrete wavelet decomposition coefficients of the system output have been chosen as the inputs of the LVQ classifier. The output of the classifier points out the current model. The proposed approach to classification can be adopted as a fault detection method where faults are represented by changes of values of internal parameters of a system. The algorithm has been evaluated on the example of a non-linear fluid system with a non-ideal pipe which internal state is characterised by one value of a parameter, chosen from the known set.

Introduction

In this study a neural detector of internal parameter changes in a stationary, non-linear SISO dynamic system is considered. The continuous systems described by differential equations should be transformed to the discrete form because a lot of control, classification or signal processing algorithms are performed in software environment and implemented in discrete devices. The discrete system is represented by the difference equation y(n) = f{y(n − 1), … , y(n − p), u(n), … , u(n − q), Γ}, where f is a non-linear function, y(n), … , y(n − p) are the output samples, u(n), … , u(n − q) are the input samples respectively. The set of characteristic internal parameters in the input–output relation is represented by the vector Γ.

$$y(k)=f({\mathrm{\Delta }}^{p}y(k)\text{,}\dots \text{,}y(k)\text{,}{\mathrm{\Delta }}^{q}u(k)\text{,}\dots \text{,}u(k)\text{,}\mathit{\Gamma })$$

State equations are another way of the system description with the set of internal parameters Θ and only this description is considered in this paper. It is also assumed that the values of the elements of the vector Θ can be changed in random moments of time, but these values belong to the finite set, so that detection of parameter changes can be considered as classification of signals acquired for different values of changeable parameters.

$$\begin{array}{cc}\hfill & x(k+1)=\Lambda (x(k)\text{,}u(k)\text{,}\mathit{\Theta })\hfill \\ \hfill & y(k)=\Omega (x(k)\text{,}u(k)\text{,}\mathit{\Theta })\hfill \end{array}$$

Such a formulation of the problem can be suitable in industrial applications where the change of parameters can model selected faults (or changes of an operating point) of an industrial dynamic system. It is very important because the system monitoring and diagnostics is performed in programming environment. The mathematical model allows to simulate system behaviour and to develop advanced signal processing algorithms for efficient decision making.

On the other hand the presented algorithm does not require a mathematical model and can be implemented directly on the industrial plant having large database of input–output relations. However effectiveness of the algorithm is strongly dependent on the possibility of acquisition of real output signals carrying information about known, faulty behaviour of a plant as a reaction on an exciting signal. Such a set of signals should be large enough in order to constitute the training and the testing data for a classifier. Sometimes real acquisition of input–output relations may be very expensive, complex and time consuming for creating rich enough database, so the usage of a model of a real plant and applying simulation environment is fully justified.

The literature on classification is vast and scattered in several disciplines (e.g. applied statistics, machine learning, neural networks, and signal and image processing). Most of classification algorithms require probabilistic information: P(ω_i) – class prior probability, p(y|ω_i) – class-conditional density and p(ω_i|y) – posterior probability, which is rarely given a priori [1]. The stochastic classification rules use the most frequently the following approaches: discriminant functions, optimal Nearest Neighbour classifiers, Logistic classifier, Parzen classifier, Binary Decision Tree, Support Vector classifier [2]. Each of these rules has some advantages and disadvantages. The other approaches utilise expert systems or other artificial intelligence techniques, such as neural networks and fuzzy logic [3].

To decrease dimensionality of classified data, extraction of specific distinctive aspects or characteristics is strongly needed. The combination of d extracted features is represented as a d-dimensional vector Φ, which defines the d-dimensional feature space. Among different linear or non-linear extraction strategies two techniques are commonly used: Principal Component Analysis (PCA) and Fisher’s Linear Discriminant Analysis (LDA). In this paper the feature space is created by the aggregation of properties of the discrete wavelet decomposition coefficients. Different subsets of coefficient properties at a different decomposition level and its ability to discriminate examples from different classes are extensively examined.

The goal of a classifier is to partition the feature space into class-labelled decision regions. Majority of classification algorithms is based on a statistical approach, where the optimal Bayes decision rule assigns a pattern to the class with the maximum posterior probability. However, the class-conditional densities are usually not known in practice and must be obtained from the available training patterns. The simplest and the most intuitive approach to classifier design is based on the concept of similarity: patterns that are similar should be assigned to the same class. Patterns can be classified by template matching or the minimum distance classifier using a few prototypes per class. Advanced techniques for computing prototypes are: vector quantisation and learning vector quantisation.

The LVQ (Learning Vector Quantisation) neural network classifier has been chosen because of its ability of learning data classification, where the similar input vectors are grouped into a region represented by a prototype called a coded vector (CV). Selected aggregated properties of wavelet decomposition coefficients have been chosen as the inputs of the LVQ classifier.

A lot of papers address discrete wavelet decomposition (DWT) as a discrete decomposition with multi-scale wavelet transforming of signals for feature extraction. Unlikely to continuous wavelet algorithms, discrete algorithms are represented by a collection of a finite number of decomposition coefficients, what is a compressed form for a signal representation [4]. A vector Φ could be for example: mean of the absolute values of the wavelet decomposition coefficients, maximum of the absolute values of the wavelet coefficients, average power of the wavelet coefficients, standard deviation of the wavelet, the absolute sum of the wavelet coefficients at each resolution level, ratio of the absolute mean values of adjacent sub-bands, distribution distortion of the coefficients [5], [6], [7], [8]. By the use of signal processing methods we could reduce the signal (i.e. the original waveform) to a lower dimension represented by a vector Φ. Next a vector Φ has to be entered to an input of a neural LVQ classifier, performing an appropriate classification and pointing out the most probable values of a vector of parameters, Θ.

The algorithm of the classification has been performed and evaluated on the example of the model of a simple fluid dynamic system with two tanks connected by a non-ideal pipe. This model seems like a good idea for testing the new algorithm, especially that non-ideal pipe with flaws and corrosion is one of the major problems in industrial and civil plants, as water, oil, steam, and gas pipelines. In industrial conditions the non-destructive testing of inaccessible pipe without dismantling and interrupting the service, favouring the least uneasiness and economic loss is strongly required. The presented algorithm can be though as a form of such a non-destructive approach. The condition (in this case the blockage condition) of a pipe is represented by one parameter in the considered model. The trained classifier is capable to classify the current value of that parameter, processing only the output response to an exciting signal.