Recognising patterns in electromechanical systems

Ž .Pattern Recognition Letters 18 1997 1335–1346

Recognising patterns in electromechanical systems

P.J. Costa Branco a,), J.F. Martins a,1,b, A.J. Pires b, J.A. Dente a

a Instituto Superior Tecnico, DEECrSMEEP – CAUTL-Lab. de Mecatronica, AÕ. RoÕisco Pais 1, 1096 Lisboa Codex, Portugal´ ´b Escola Superior de Tecnologia, Instituto Politecnico de Setubal, Rua do Vale de ChaÕes, Estefanilha 2910 Setubal, Portugal´ ´ ´

Abstract

Pattern recognition using fuzzy logic and neural network techniques is usually related with image and speech processing.In a different approach, this paper shows results concerning the use of these techniques to extract models of electromechani-cal system’s. This approach is important because such systems always contain internal non-linear relations that are difficultto model and, in most cases, there are unknown internal phenomena dominating their dynamics. So, these techniques enableto recognise system patterns and to include them in a more complete system’s model. q 1997 Elsevier Science B.V.

Keywords: Fuzzy modelling; Neural modelling; Pattern recognition; Systems modelling

1. Introduction

An electromechanical system can be described asa functional relationship between certain variablesexpressing the system’s condition and an observedoutput variable. So, pattern recognition in this caseconsists of identifying the relationships between thestate conditions and the output value.

Fuzzy logic expresses such relations as IF–THENrules. In neural networks, such relations are implicitin the weights that divide the system’s operatingdomain by hyperplanes. Both approaches partition ina different way the domain in clusters, each onecorresponding to an extracted relation.

The application of pattern recognition techniquesin electromechanical systems enables a better sys-

Žtems modelling Costa Branco et al., 1996, Costa

) Corresponding author. E-mail: [email protected] E-mail: [email protected].

.Branco and Dente, 1997a,b and so better controllersŽ .Costa Branco and Dente, 1997a,b . The conven-tional modelling approaches using differential equa-tions yield insufficient representations of complexelectromechanical systems. Such models cannotcompletely represent the internal dynamics of thephysical system because of non-linear relations suchas hysteresis, temperature effects, and saturation, thatare often time-variant or even unknown.

This paper describes the application of fuzzy logicand neural network techniques in modelling twoexperimental systems. It emphasises the followingthree practical modelling aspects common to bothtechniques:1. What are the most significant system variables

and what is their influence on the model perfor-mance?

2. How to obtain a good training set for the learningprocess?

3. What are the generalisation capabilities of fuzzylogic and neural network approaches?

0167-8655r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved.Ž .PII S0167-8655 97 00136-0

( )P.J. Costa Branco et al.rPattern Recognition Letters 18 1997 1335–13461336

2. The learning algorithms

This section gives an overview of the fuzzy andneural network learning algorithms used throughoutthe paper.

2.1. The fuzzy algorithm

We illustrate the algorithm by an example of afuzzy system shown in Fig. 1 with two inputs, x1

and x , and one output variable, y. A data set2

composed of system examples was acquired during atraining test. Each data sample is symbolised in Fig.Ž . Ž X X . X1 a by a relation like x , x ™y . The membership1 2

functions attributed to each variable establish a struc-ture grouping the examples in clusters where eachone is related with a model rule RŽ l . as shown in Fig.Ž .1 a .

The learning mechanism for extraction of the ruleŽ . Ž .expressed in 1 , for example, is shown in Fig. 1 a .

The rule conclusion v Ž l . is derived based on valuesin the region covered by membership functions Posi-

Ž . Ž .tive-Medium PM and Negative-Medium NM asrepresented in the figure by dark elements.

IF x is PM and x is NMŽ . Ž .1 2

Ž l .THEN y is v . 1Ž .

In the next learning step, to each example in thecluster a grade is attributed indicating the degree towhich it contributes to the conclusion extraction.

Ž X X . XSuppose that x , x ™y is an example in the rule1 2Ž X.region. Its contribution degree m y is computed as

Ž .indicated in 2 by the product of each membershipdegree in the antecedent fuzzy sets PM and NM.

m yX sm xXm xX . 2Ž . Ž . Ž . Ž .PM 1 NM 2

The obtained contribution degree weighs the cor-responding output value yX. Since this procedure isperformed for all n examples inside the rule region,it induces a membership function defined for corre-

X Ž .sponding output values y as Fig. 1 b shows for ruleŽ . Ž l . Ž .1 . The conclusion value v of rule l is thencomputed from the induced membership function

Ž . Ž .Fig. 1. a Examples related to the extraction of rule with antecedent linguistic terms PM and NM. b Induced membership function and itsgravity-center v Ž l . as the conclusion value for rule RŽ l ..

( )P.J. Costa Branco et al.rPattern Recognition Letters 18 1997 1335–1346 1337

Fig. 2. Picture of the electro-hydraulic actuator.

through the calculation of its centroid using expres-Ž .sion 3 .

nX X

m y k y kŽ . Ž .Ž .Ýks1Ž l .v s . 3Ž .n

Xm y kŽ .Ž .Ý

ks1

2.2. The neural network algorithm

Error-backpropagation is a well-known trainingalgorithm for neural networks. We used a singlehidden layer neural network with a sigmoid transfer

Ž .function 4 for the hidden layer with 3 neurons, andŽ .a linear transfer function 5 for the output layer with

one neuron. In both equations, h is the neuronthreshold and b its bias.

2ft h ,b s y1. 4Ž . Ž .s y2Žhqb.1qe

ft h ,b shqb. 5Ž . Ž .l

The weights and biases are updated using the gener-alised delta rule with an adaptive learning rate and

Ž .momentum Hertz et al., 1991 . The weights areŽ .updated using 6 , where E is the quadratic cost

function, h and g the adaptive learning rate andmomentum, respectively, and w the weights.

E EDw kq1 syh k qg k Dw k . 6Ž . Ž . Ž . Ž . Ž .

E w


3. The experimental systems

3.1. Electro-hydraulic actuator

This system is illustrated in Fig. 2. It is composedof an electrical drive that controls the speed of amotor connected to a pump that sends fluid to ahydraulic circuit moving a linear piston. The system

Ž .is described by the piston position y as a functionŽ . Ž .of the reference position y , the pump speed v ,ref

Ž . Ž .and the piston speed signal n : ys f y ,v,n .ref

The fuzzy algorithm extracts the patterns amongthese variables and constructs a rule-based modelexpressing its behaviour.

3.2. Electrical driÕe system

The second experimental system is illustrated inFig. 3. It is composed of an induction motor drivenby a power converter. These systems are modelledusing elements of the electromechanical power con-

Ž .version theory Fitzgerald et al., 1985 . With somesimplifying assumptions, the electrical drive systemis usually represented by a 12 equation state modelŽ .Martins et al., 1997a . This large number of vari-ables obstructs any attempt to perform on-line learn-ing and the neural network learning algorithm woulddemand huge computational costs. So, it is needed tofurther simplify the electrical drive model.

For command purposes, the system has an inter-Žnal control loop for the input variables Martins et

.al., 1997b . The electrical drive model can be re-Ž .duced from a 12th to a 3rd order system 7 , where

C and C denote the internal magnetic fluxes, vdr qr

the motor speed, i and i the reference statordsref qsref

currents, t and t the electrical time constants, Ms r

and L the mutual and self-inductances, B and J ther

friction and inertia coefficients. This model still pre-sents some fundamental problems such as accuratemeasurement of some system parameters, the self-in-ductances and the internal fluxes C and C .dr qr

dc 1 Mdrsy c q v yv c q i ,Ž .dr r qr dsrefd t t tr r

dc 1 Mqrsy v yv c y c q i ,Ž .r dr qr qsrefd t t tr r

dv M B 1s yi c q i c y v T .Ž .dsref qr qsref dr extd t JL J Jr

7Ž .

4. Results and discussion

Although using two different approaches in twodifferent systems, both situations present some gen-eral problems: the training data acquisition, choosingthe best variable set characterising the system, andthe generalisation capability of each model. Theseproblems are analysed using two modelling situa-tions:Ø Situation 1: fuzzy static modelling of the electro-

hydraulic system;Ø Situation 2: neural network dynamic modelling of

the electrical drive system.

4.1. The training data set acquisition

4.1.1. Situation 1This situation consisted of applying a command

signal to conduct the piston to the operating regions

Fig. 3. Schematic diagram of the electrical drive system.


where we want to acquire system examples. To dothis, and considering the electro-hydraulic systemwith a coarse closed-loop position regulation madeby a proportional controller, a sinusoidal referenceposition y with different amplitudes and frequen-ref

cies was imposed in order to get a large number ofdifferent working patterns from the electro-hydraulicsystem. The obtained training set is displayed in Fig.4.

4.1.2. Situation 2As described, the electrical drive system can be

Ž .modelled using a 3rd order model 7 , where themeasurement of the motor internal magnetic fluxes isa difficult task. To overcome this drawback, aninput-output modelling approach using a dynamicmodel is considered. This avoids the measurement of

Ž .the internal fluxes C and C , and still achievesdr qr

an accurate representation of the electrical drive’sbehaviour. Using fixed sampling intervals, it is pos-sible to express the dynamic behaviour of the electri-cal drive as a discrete model where the actual speedcan be computed exclusively from the previous in-

put-output variables as stated by its dynamic relationŽ .8 .

The neural network models the system using onlyeasily obtainable input-output system’s information.Its input signals are the previous reference currentsŽ .i and i and the previous motor speed signalqsref dsref

Ž .values v . The output signal of the neural networkis the motor speed prediction.

v sg v ,v ,v ,i ,i ,Žkq1 k ky1 ky2 qsref qsrefk ky1

i ,i ,i ,i . 8Ž ..qsref dsref dsref dsrefky 2 k ky1 ky2

To obtain a training set, we imposed a sinusoidalreference signal with a combination of amplitudesand frequencies, as for the electro-hydraulic system,on both input system variables yi and i . Theqsref dsref

training and test data sets are presented in Fig. 5,where the test data is displayed in bold.

4.2. Denoting the most significant system Õariables

In real circumstances, the number of system vari-ables is very high. If all variables are considered, the

Ž . Ž . Ž .Fig. 4. Training data set. a Reference and position signal. b Motor speed. c Piston speed.


Fig. 5. Training and test data sets for the electrical drive system.

model is of high complexity with high computationalcosts as occurred with the electrical drive. The statis-tical technique of Principal Component AnalysisŽ .PCA can be used to select the main variable set.PCA is a technique to describe the multivariate

Ž .structure of data as described in Jackson, 1982 . Inorder to examine the relationships among a set of pcorrelated variables, PCA transforms the original set

Ž .of variables xs x , x , . . . , x by an orthogonal1 2 p

rotation in p-dimensional space to a new set ofuncorrelated variables called principal components

Ž .zs z , z , . . . , z .1 2 p

The principal components are described by theeigenvectors u of a covariance matrix S computedi

from the experimental data. Every ith principal com-Ž .ponent calculated using 9 is a linear combination of

the original variables. Each principal component has

a variance l associated with it that is given by thei

ith eigenvalue of the matrix S.

z suTx . 9Ž .i i

If some of the original variables are highly corre-lated, they are effectively ‘‘saying the same thing’’and one of these variables is redundant.

4.2.1. Situation 1The analysis of the electro-hydraulics’ training set

using PCA permits to investigate the relative impor-tance of each pre-selected variable y , v, n and y.ref

The eigenvectors and eigenvalues computed from thecovariance matrix are displayed in Table 1. Theeigenvalues l measure the variance of each princi-i

Ž .pal component z , the highest one l s8.3648i 1

being attributed to z , accounting for almost 88% of1

total data variance. The components of each individ-ual eigenvector are linearly related with the originalvariables, as shown for the principal component z1

Ž .in 10 .

z s q0.7694 y q 0.0460 nq 0.0170 PlŽ . Ž . Ž .1 ref

q 0.0809 vq 0.6317 y. 10Ž . Ž . Ž .Taking the absolute values of each weight, the

signal y is the most significant variable, followedref

by y, v, n and piston Pl.To show how the chosen variable set influences

the model performance, we consider a case whereinitially only the variables y , n and y, are consid-ref

ered to characterise the system, with the informationof v not considered, instead of n signal, in themanner to better emphasise the modelling effects.

Ž .The system is then represented by relation 11 .

ys f y ,n . 11Ž . Ž .ref

Fig. 6 displays the corresponding training set.Using this data, the fuzzy algorithm is applied to

Table 1

Ž .Eigenvalues l l 8.3648 l 0.9057 l 0.1023 l 0.0034 l 0.0010i 1 2 3 4 5

Eigenvectors u u u u u1 2 3 4 5

y 0.7694 y0.3174 0.2726 y0.4769 y0.0739ref

n 0.0460 y0.5978 y0.7963 0.0045 0.0806Pl 0.0170 y0.0746 0.1567 0.0140 0.9846v 0.0809 y0.5345 0.3984 0.7319 y0.1157y 0.6317 0.5006 y0.3292 0.4865 0.0725


Fig. 6. Training set composed by signals y , n and y.ref

Ž . Ž .extract relation 11 . Fig. 7 a shows the domain ofŽ .the training set for this relation. Fig. 7 b shows the

domain partitioning by the rule set of the fuzzymodel for the electro-hydraulic system. In this fig-ure, each domain region presents a certain grey levelrelated with the piston position value extracted asconclusion for each rule. The 13 grey levels quantizethe piston displacement interval from 0 m to itsmaximum value of 0.2 m. The domain regions with-out learning examples cause the appearing of ruleswith null conclusion, if we consider the worst situa-tion without any type of information to complete

Ž .them. These rules are represented in Fig. 7 b bywhite regions.

Ž .Fig. 8 a shows the obtained fuzzy modellingresults with y) the inferred piston position and y

Ž .the corresponding measured position. Fig. 8 b showsŽ ) .the error signal y yy located between "10%

with large oscillations in time. These oscillationsindicate that there is a necessity of provide moreinformation to the learning algorithm and so build amore complete model.

When it is introduced the information about pumpŽ .speed, expressed in variable v, to relation 11 , we

obtain the fuzzy modelling results shown in Fig.Ž .9 a . The additional information increases the model

prediction performance, reducing the error oscilla-Ž .tions, as observed in Fig. 9 b .

4.2.2. Situation 2For the electrical drive system, it is difficult to

assure the existence of a highly non-linear relation-Ž .ship as 8 and verify the relative significance of its

arguments. The use of PCA to indicate the bestvariable set to this dynamic model is very difficultsince each acquired signal is extremely correlatedwith its previous values, and the auto-correlation ofeach signal becomes higher than the correlation be-tween different variables.

To verify the relative significance of each variableŽ .in relation 8 , a linearization of the electrical drive

Ž .model 7 is performed. The new variables in lin-Ž .earized relation 12 , v , i and i , corre-d qsref d dsref d

spond to their small variation around an operatingpoint. The relation states the linearized motor speedsignal v as a linear combination of previous timedkq 1

instants k, ky1 and ky2, from obtainable vari-ables v , i and i . The coefficients a , b , gd qsref d dsref d

Ž . Ž . Ž .Fig. 7. a Domain for relation ys f y , y . b Fuzzy partition dividing the domain in rule regions.ref


Ž . Ž . Ž ) . Ž . Ž .Fig. 8. a Piston position y and inferred position signal y by the fuzzy model to relation ys f y ,n . b Error signal evolution.ref

Ž . Ž . Ž ) . Ž . Ž .Fig. 9. a Piston position y and inferred position signal y by the fuzzy model to relation ys f y ,n ,v . b Error signal evolution.ref

Ž . Ž .Fig. 10. Evolution of a , b coefficients for different sampling intervals. a Speed coefficients – a . b Input current i coefficients –i qsref

b .i


Fig . 11 . Speed-error evolu tion w ith a sam pling in terval of 0 ,8 sec and using relation v sk q 1Ž .g v ,v ,v ,i ,i ,i ,i ,i ,i .k ky1 ky2 qsref qsref qsref dsref dsref dsrefk ky1 ky2 k ky1 ky2

Ž .in 12 are dependent on calculated linearized systemmatrices and so become dependent on each operating

Ž .point and sampling interval Takahashi et al., 1972 .

2

v s a v qb i qg i .Ž .Ýd i d i qsref d i dsref dkq 1 ky i ky i ky iis0

12Ž .

This linearized relation has a limited accuracy. How-ever, its analysis allows us to introduce further sim-plifications in the learning algorithm. Each term in

Ž .12 has a relative influence on v that dependsdkq 1

on the used sampling interval. To illustrate this, Fig.10 shows the coefficient values a and b fori i

various sampling intervals around the same operatingpoint. The evolution of the other coefficient value g ,inot displayed, is similar to the b evolution. Thei

continuous lines in Fig. 10 refer to the k instantvalues a and b , the dashed lines to ky1 instant0 0

values a and b , and the dotted-dashed lines to the1 1

ky2 instant values a and b .2 2Ž .The results for the a coefficients in Fig. 10 ai

show, for large sampling intervals, the permanent

Ž .Fig. 12. Speed-error evolution with a sampling interval of 0,8 sec and using the simplified relation v sg v ,i ,i .kq 1 k qsref dsrefk k


Fig. 13. Illustration of the limited training set and the test set forthe electro-hydraulic system.

influence of previous speed values v weighted bydk

a , and the decaying influence of v and v0 d dky 1 ky2

speed values weighted by decreasing a and a1 2Ž .coefficients. In Fig. 10 b , the evolution of the bi

coefficients shows the growing influence of iqsref d k

values weighted by b as the sampling interval0

increases. The other current values at instants ky1Ž .and ky2 i and i weighted by the bqsref qsref 1ky 1 ky2

and b coefficients show some influence only for2

sampling intervals close to the system’s flux timeŽ .constant 0.13 sec . For sufficiently large samplingŽ .intervals DT)0.6 sec , the number of arguments in

Fig. 15. Training set with a poor spanning and test data set for theelectrical drive system.

Ž .the dynamic model 8 can then be decreased obtain-Ž .ing relation 13 .

v sg v ,i ,i . 13Ž .Ž .kq1 k qsref dsrefk k

Figs. 11 and 12 show the results obtained using aŽ .sufficiently large sampling interval DTs0,8 sec .

They display the error evolution between measuredmotor speed signal and the inferred value from the

Ž .neural network model using relation 8 , Fig. 11, andŽ .using the simplified relation 13 , Fig. 12. The simi-

larity between the results allows us to confirm theŽ .simpler relation 13 for large sampling intervals,

Ž . Ž . Ž ) . Ž .Fig. 14. a Evolution of the piston position y and the inferred signal y when the training set is limited to a domain part. b Errorsignal evolution.


Fig. 16. Speed-error evolution concerning a poor spanning of theworking domain.

which represents the motor speed dynamics consider-ing less information, and consequently a simplifiedneural network structure.

4.3. Generalisation capability

In this section, we investigate the generalisationcapability for both modelling situations.

4.3.1. Situation 1For a generalisation test, the training set from the

electro-hydraulic system is restricted to a domainŽregion limiting the piston displacement to 0.1 m Fig.

.13 . The learning algorithm uses this training set anda test set that is not restricted as illustrated in Fig. 13too. The fuzzy modelling results in Fig. 14 show thatthe learning algorithm has a local generalisationcapability and not a global one, generalising only forthe situations presented in the training set.

4.3.2. Situation 2The generalisation capability of the neural net-

work model for the electrical drive decreases if thetraining data set shows a poor spanning of the work-ing domain as shown in Fig. 15. Fig. 16 shows theneural modelling results using this data where thespeed error increases for domain areas not coveredby the training data. As for the fuzzy modellingprocess of the electro-hydraulic system, the resultsshow also a local characteristic of the generalisationcapability for this modelling situation.

5. Conclusion

Pattern recognition techniques using fuzzy logicand neural networks demonstrated their potential to

automatically establish a model for electromechani-cal systems. The modelling of two experimentalsystems was presented: an electro-hydraulic systemusing fuzzy logic with a static model type; and anelectrical drive system using neural networks with adynamic model type. Although each modelling ap-proach has its particular characteristics, the papershows that they have some common problems thatappear in all practical modelizations.

Discussion

Roli: You said that a crucial point is the creation ofthe training set. In order to create a training set, youalso use traditional, numerical codes, numerical sim-ulators used by mechanical engineers even if theyare affected by uncertainties in the values of parame-ters. So in effect, you produce a noisy training set.

Costa Branco: Yes, if you have a simple model, youcan do some simulation and start with that to obtaina training set. For the real system, for example, whenit operates in an open loop, it is difficult to send thesystem to the regions where you want to acquire thetraining examples.

Roli: Just another remark: from the viewpoint ofmechanical engineers working in the automatic con-trol field, is the ‘‘opacity’’ of the neural network nota problem? The neural network is a black box, so itis very different from the traditional numerical codethat they use, which is based on explicit differentialequations. So, I could imagine that this a problem.

Costa Branco: Fuzzy logic contains explicit rules.These are useful for diagnostic purposes. In theexplicit models, you sometimes loose this informa-tion.

Roli: I can imagine that mechanical engineers arehappy with fuzzy rules, because they are ‘‘whiteboxes’’. But I would think they are less happy with‘‘black box’’ neural networks.

Costa Branco: I do not say that one is better than theother. I do only say that there are problems whichare common to both techniques and that it is time tostudy them.

Roli: I ask this question, because in other classicalfields of engineering, engineers are not satisfied with


black box tools, because they are not able to validatethe results.

Costa Branco: It depends on the application. In mycase, start by implementing the fuzzy rules. If itworks and if it is fast and simple to implement, Iprefer fuzzy logic. The neural network is sometimesa ‘‘grey box’’, rather than a ‘‘black box’’. Othersalso try to extract the information from neural net-works.

References

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ŽSets and Systems, to appear available as technical report in.http:rrrorty.ist.utl.ptrpbranco .

Costa Branco, P.J., Dente, J.A., 1997b. Control of an electro-hy-draulic system using neuro-fuzzy modelling and real-timelearning approaches. Internat. J. Knowledge-Based Intelligent

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Costa Branco, P.J., Lori, N., Dente, J.A., 1996, New approacheson structure identification of fuzzy models: Case study in anelectromechanical system, In: Furuhashi, T., Uchikawa, Y.Ž .Eds. , Fuzzy Logic, Neural Networks, and Evolutionary Com-putation, Lecture Notes in Artificial Intelligence. Springer,

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Martins, J.F., Silva, J.F., Pires, A.J., 1997b. A novel and simplecurrent controller for three-phase IGBT PWM power inverters– A comparative study. In: Proc. ISIE’97, Guimaraes, Portu-˜

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