68 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 1, FEBRUARY 2001

New Trends in Recognizing Experimental Drives:Fuzzy Logic and Formal Language Theories

J. F. Martins, Member, IEEE, P. J. Costa Branco, Member, IEEE, A. J. Pires, and J. A. Dente

Abstract—Drive systems today determine the productivity andquality of industrial processes. However, they exhibit considerablecomplexities related with their behavior as large uncertainties ata structure and parameter levels, multidimensionality, and strongmutual interactions. With these multiplying complexities, the usualmodels are becoming not accurate enough. It is necessary to com-plement them with other information-processing techniques thatallow a better recognition of their behavior. The aim of this paperis to analyze common features, and the potential, but also the draw-backs that fuzzy logic and formal language theories show whenused for recognition of patterns in experimental drives. Two proto-type systems are used: an electrohydraulic drive and an inductionmotor drive. We underline the similarities and various aspects ofboth recognition methodologies, despite their use on different sys-tems. A set of experimental learning situations with critical effectson their performance are presented and discussed.

Index Terms—Fuzzy logic, fuzzy systems, identification, learningsystems, modeling, pattern recognition.

I. INTRODUCTION

DYNAMIC performance of drive systems has benefitedfrom the recent and constant progresses in new materials,

power electronics, microelectronics, and informatics. Theseprogresses have allowed the increase of their performancedemands.

Models resulting from the mathematical formalization of thedifferent physical phenomena that take place in drives usuallysupport description of these drives dynamic. However, the in-creased analytical complexity of these models makes them verydifficult to quantify and restricts their use in many applications.Several aspects, such as thermal behavior, magnetic saturation,elasticity, viscosity, and dead-times, usually considered as sec-ondary at a command and control levels of drive systems, aretoday very important since the quality demands are increasing.Equally significant are the strong nonlinear relations betweenthe drive’s state variables and the exact knowledge of some pa-rameters, which makes even more difficult to establish func-tional relations representative of their dynamic behavior.

The inherent difficulties within a complete drive system’smodeling, along with an increasing performance demand,led us to the application of pattern recognition techniques,

Manuscript received April 6, 2000; revised February 29, 2000.J. F. Martins and A. J. Pires are with the Escola Superior de Tecnologia de

Setúbal, Instituto Politécnico de Setúbal, Estefanilha 2910 Setúbal, Portugal andwith the Laboratório de Mecatrónica, Instituto Superior Técnico (I.S.T.), 1096Lisboa Codex, Portugal (e-mail: jmartins@est.ips.pt).

P. J. Costa Branco and J. A. Dente are with the Laboratório de Mecatrónica,Instituto Superior Técnico (I.S.T.), 1096 Lisboa Codex, Portugal (e-mail:pbranco@alfa.ist.utl.pt).

Publisher Item Identifier S 1063-6706(01)01365-0.

such as fuzzy logic and formal language methodologies inautomatic recognition of drives (dynamic) behavior. Theserecognition procedures have been extensively applied in imageand speech processing areas. Their principles, however, canbe extended to drives pattern recognition in order to completeor substitute their usual mathematical models, and thereforeincrease their prediction performance. Before introduce thisapproach, it is important to underline the difference betweensystem identification and pattern recognition.

System identification, as stated by Sage and Melsa [24],means “the process of determining a difference or differentialequation (or the coefficient parameters of such an equation)such that it describes a physical process in accordance withsome predetermined criterion.” Therefore, in a system iden-tification problem, the structure and order of the model areestablished in advance. Identification is related to the system’sparameters that are unknown. In drive systems, for instance,techniques as Kalman filtering [25] or observer-based algo-rithms [26] have been applied with relative and limited successwhen incorporated in a control system. Pattern recognition, onthe other hand, is a classifier system, grouping patterns intocategories [27]. The pattern recognition problem, in the case ofrecognizing the behavior of drive systems, consists of identi-fying the relationships between the system state conditions andits observed output variables. Both in fuzzy logic and in theformal language approach, patterns are viewed as sentences ina language:IF-THEN rules in fuzzy logic and a grammar in theformal language.

The main drawback regarding the use of these methodolo-gies is the loss of some physical meaning in the obtained drive’smodel. The main benefit, however, is the use of these models andassociated learning algorithms in the design of new drive con-trol schemes with learning properties. The existent mathemat-ical models and designed controllers can also be expanded withthe incorporation of these new models in the control system,thus completing its structure.

In the operation of modern industrial plants, drive systemsplay an essential role in increasing the productivity and qualitydemands, and mainly in reducing energy and equipment main-tenance costs at all stages of the process. The configuration ofa drive, which is by far more complex, contains several mo-tors, power converters, hydraulic and/or pneumatic elements,sensors, and digital control systems. Typical features of drivesystems involve considerable complexities related with theirbehavior. They have a highly nonlinear coupling, presentinglarge uncertainties at a structural and parameter level, theyare multidimensional, and contain unknown nonlinearities.Furthermore, the existing great interconnection between all

1063–6706/01$10.00 © 2001 IEEE

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES 69

drive processes originates mutual interactions between them,presenting, in some cases, internal feedback mechanisms.

With those multiplying complexities, there are problems inapplying their usual mathematical models since they are be-coming either too many complexes to work with reasonablecomputational times, or their present design is not sufficientto handle the actual drive system uncertainties. The use of thesame mathematical model for all system’s operating regionsdoes not allow modifying the functional relation between thesystem variables in such a way that it cancels new operatingmodes. Consequently, the existing models are usually not accu-rate enough to fulfill the description of specific situations.

Considering the previous difficulties within the actual drivesystems, it becomes necessary to complement, and even correct,their classical modeling with other information processing tech-niques that allow better recognition of their behavior [3], [4]. Ina previous work [7], [9], we began studying the application offuzzy logic and neural networks to automatically recognize the(drive) systems operation. Those were different approaches, butit was verified that they share common problems when appliedto a practical modeling process. Following that research in thispaper, we investigate only linguistic approaches in experimentaldrive recognition. For this, fuzzy logic and formal language the-ories are used. Both approaches, in spite of representing dif-ferent concepts concerning linguistic approximations, presentmemory, learning, and generalization skills that have to be nec-essarily used in a recognition system [5], [6].

The fundamental contribution in the formal language areawas made by Chomsky [13], whose theory of formal grammarshas had a major influence in the development of the subject.Grammatical inference is a concept that goes back to Gold’swork [14], and is defined as a way in which a system tries“to guess” general rules from examples. Since then much workas been done, which can be found in several excellent surveys[15]–[17].

The aim of this paper is to describe the application of fuzzylogic and grammatical inference techniques, based on formallanguage theory, in modeling two experimental drive systems.Both approaches manipulate drive information, but their objectsof reasoning are fuzzy sets in the case of fuzzy logic, and analphabet in the case of grammatical inference. In this perspec-tive, both construct a model of the considered dynamical system.For comparison purposes between fuzzy logic and formal lan-guage, each method is applied to a drive system. Fuzzy logicin an electrohydraulic system and formal language in an in-duction motor drive. It is our objective, with this decision, tomake more relevant their mutual modeling aspects, althoughusing both methodologies in systems with very different dy-namic characteristics. The following modeling aspects will bediscussed:

• How can we acquire a representative set of patterns fromdrive systems?

• What does a representative set of patterns and their influ-ence in the learning mechanisms mean?

• How can we overcome possible lack of information in ourdata?

• How can both approaches recognize the behavior of thedrive system?

The paper is organized as follows. Section II outlines the basicconcepts of both linguistic techniques and points out their sim-ilarities. Section III describes the learning algorithms. In Sec-tion IV, we describe the drives used as experimental systemsto our study. Section V analyzes the acquisition of the trainingdata set issue. Section VI presents some recognition results. Sec-tion VII discusses the problem of lack of information in theavailable set of patterns. The interpolation techniques used tominimize the previous problem are described and discussed inSection VIII. Section IX discusses results concerning the gener-alization aspect. The conclusions and future work are explainedin Section X.

II. BASIC CONCEPTS

Fuzzy logic and formal language methodologies are naturallanguage approaches able to represent a system’s dynamics.Both can describe complex relations between the system vari-ables through linguistic relations [18]. In this section, the con-cepts that characterize each approach and form the basis of bothlinguistic algorithms are summarized.

A. Fuzzy Logic

Fuzzy logic in pattern recognition [1] constitutes, in its finalapproach, a classifying system that condenses a large amount ofpatterns, represented by numerical data, into a rule-base struc-ture [5]. Classification is obtained by using fuzzyIF-THEN rulesthat are formed by three main structures: linguistic variables,fuzzy propositions, and truth-values.

1) Linguistic Variable: In the classification rules, linguisticvariables represent the feature space. For each feature, a numberof words are used. These are called fuzzy variables, or fuzzynumbers, that are represented by fuzzy sets.

2) Fuzzy Proposition:Propositions are sentences that have,in general, a canonical form like is where is a feature ofthe subject, and designates the word that characterizes a cer-tain property of the object (such asBIG, HIGH, etc.) fuzzifyingeach feature. In fuzzy classifier systems, which in our case aremodeling systems, anth proposition has a general form givenby

feature is and feature is

feature is (1)

In a drive system, the features are signals from the systemthat, together, can characterize its dynamic condition. For ex-ample, pressures, electrical currents, voltages and temperatureare simple examples of the features that we are interested in clas-sifying to predict the drive’s behavior.

3) Truth-Values: Features are mathematically characterizedby fuzzy sets defined in [0, 1]. The truth-value of a propositionlike is is denoted by a value that is defined to be avalue in [0, 1].

B. Formal Language

In order to apply grammatical inference procedures, a dynam-ical system must be considered as an entity (linguistic source)able to generate a certain language. To characterize this entity,

70 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 1, FEBRUARY 2001

Fig. 1. Fuzzy logic quantification.

which generates all the words in the language, a grammar canbe used [15]. This grammar defines the structural features of thewords produced by the linguistic source and, in this way, modelsthe source itself. To specify this grammar denoted asin (2),one must specify four structures: a terminal alphabet, a non-terminal alphabet , a start symbol , and a set of productions

[19]

(2)

1) Terminal Alphabet:It is constituted by symbols thatmake up the resulting words, denoted by.

2) Nonterminal Alphabet:It is constituted by symbols thatare used to generate the patterns, denoted by.

3) Start Symbol:It is a special nonterminal symbol that isused to begin the generation of words, denoted by.

4) Set of Productions:It is a set of rules (in the form ,where and are strings) that determines the generation ofwords, denoted by .

For example, consider a simple grammar with a two-symbolterminal alphabet , a one-symbol nonterminal al-phabet , a start symbol , and a set of productionrules defined by . Be-ginning with the start symbol and applying the first produc-tion of , the word “ ” is obtained. When applying, for in-stance, three times the second production ( ) in the pre-vious word (“ ”), the following strings “ ,” “ ,” and“ ” are produced. At last, the final terminal word “ ”is reached by applying the third production ( ) in the laststring (“ ”). One can easily verify that this grammar pro-duces all words that consist of a terminal symbol “” followedby any number of symbols “.” This language, denoted as,can be represented as (3) wheredenotes the -concatenationof symbol “ ”

(3)

Since the set of productions commands the generation ofterminal words, they will be used to encode the dynamics ofthe system that generates the language. Any word regarded as asequence of terminal symbols, derived from the start symbol bya sequence of productions of the grammar, is said to be in thelanguage generated by the dynamical system.

The grammatical inference procedure represents a way inwhich a grammar is directly inferred from a set of sample words(experimental patterns) produced by the dynamical system con-sidered as the linguistic source [17]. A basic idea in any gram-matical inference process is that there is not a unique relation-ship between a language and a grammar used to generate it [15].A finite sample does not serve to uniquely define a language.The inferred grammar can only recognize the words containedin that finite data sample, and the others that are not within thatsample but are of the same nature.

III. L INGUISTIC ALGORTIHMS

We now present the steps in both linguistic approaches thatshould be observed when a linguistic algorithm is developed.These steps are: quantification of the involved variables, therules and productions that establish the relations between the ob-jects of reasoning, and the learning algorithm itself. Each step isformalized for fuzzy logic and formal language theories. Whenpossible, we point out their common concepts but that use dif-ferent formalisms.

A. Quantification

1) Fuzzy Logic: Each feature from the system is quantifiedby fuzzy sets. This quantification includes the number of fuzzysets, their distribution through each universe of discourse, thetype of function used to represent each fuzzy set, and the widthof each function that defines its fuzziness. Therefore, numericaldata is projected onto each fuzzy feature by the membershipfunctions, i.e., , as shown in Fig. 1.

2) Formal Language:Quantification in formal language isrelated with the creation of the alphabets by establishing a re-lation between them and the variables of the dynamical system(4). In fuzzy logic theory, the alphabet is composed by fuzzysets. Within the formal languages formalism, the terminal al-phabet is associated with the output variable, denoted by,and the nonterminal alphabet with the input variable of thedynamical system denoted by

(4)

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES 71

Fig. 2. Formal language quantification.

In order to specify both alphabets, terminal and nonterminal,a number of symbols must be assumed for each one. We willconsider as the number of terminal symbols and as thenumber of nonterminal symbols. Then, a discrete quantificationprocess is applied to the variables where the relation (5) is de-fined between the dynamical system variables and the alphabetsymbols

(5)

Unlike the fuzzy approach shown in Fig. 1, the informationcodification in a formal language is crisp, as shown in Fig. 2.The alphabets are created dividing the dynamical systems vari-able range in equal intervals, and associating each interval to asymbol in the alphabet.

B. Rules and Productions

1) Fuzzy Logic: Fuzzy rules are expressed as (6) whereis the th rule, are the features expressing system’s condition,and is the output variable of the system. Symbols are thefuzzy sets previously attributed to each feature, and is therule consequent

is and is and is

is (6)

The fuzzy rules divide the feature space of the system in sub-spaces, each one constituting a rule, as illustrated in Fig. 3. Thefiring degree of each rule is calculated by (7). The member-ship values of each variable defined for every linguistic term ofrule , , are combined by a fuzzy logicAND operator

. Variables denoted by and are the numer-ical values (patterns) of features and . The for-mulas denote the membership functions

attributed to each fuzzy set

(7)

Fig. 3. Partition of the feature space by the fuzzy sets.

Each rule is composed by a fuzzy implication degree calcu-lated by (8). In this expression, the operator “” is the inferenceoperator, which connects the antecedent to the consequent partof the rule

(8)

2) Formal Language:As aforementioned, the generation oflanguage words is determined by the application of the produc-tions contained in set . After establishing the alphabets, thelearning algorithm must infer this set of productions from a setof sample words obtained from the source. Considering the pre-vious discrete quantification, a specific set of productions mustbe established in order to settle the relations between both alpha-bets (input and output information), and thus produce terminalwords that denote the output variable evolution. In this way,p-type productions are introduced assuming the general form(9). The sequence is constituted by terminal symbolsof length is any nonterminal symbol, is a terminal

72 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 1, FEBRUARY 2001

TABLE ITYPE PRODUCTIONS

TABLE IICORESPONDENCEBETWEEN FUNCTIONAL RELATIONSHIPS AND TYPE

PRODUCTIONS

symbol, and is a special nonterminal symbol that can be re-placed by any other nonterminal symbol

(9)

Table I illustrates some typical productions. A0-type produc-tion does not have any terminal symbol in the left part of theproduction. A1-type production has one terminal symbol in theleft part of the production. A2-type production has two terminalsymbols in the left part of the production. Ap-type productionhas terminal symbols in the left part. Given this general formproduction and since the left part of any production contains atleast one nonterminal symbol, this grammar can be classified ascontext sensitive in the Chomsky hierarchy [13].

The nonterminal symbol is used to allow the conclusion,or not, of a generated word by the use of the special set ofproductions (10). This set can be concatenated as (11) wherethe symbol “” denotes the concatenation of similar productionsconsidered in (10). The terms to denote all the nonter-minal symbols and denotes the empty symbol

... (10)

(11)

Any p-type production codifies the evolution of the outputvariable depending on its previous values ( ) and onthe value of the input variable ( ). As presented in Table II, acorrespondence is assumed between thep-type productions anda set of functional relationships representing the systemdynamics. These functional relationships become then represen-tative of the dynamical system behavior.

C. Learning Procedures

1) Fuzzy Logic: Quantification was applied to establish theclassification rules that are not known in advance. Therefore, alearning procedure by which the fuzzy rules could be extractedfrom a data set is needed. In this study, a simple learning algo-rithm introduced by Wang [2] was chosen as the “basic” fuzzylearning mechanism. The learning algorithm considers, for sim-plicity, each variable equally partitioned by symmetric mem-bership functions of triangular type. The use of same numberof fuzzy sets in the input variables is only a simplifying optionin this algorithm. This choice makes faster the trial-and-errorprocess to attribute the best number of fuzzy sets to the fuzzymodel and minimizing the quadratic error mean. The learningalgorithm uses the t-conormmaxto select the degree to whichtwo fuzzy sets match, and extracts the conclusion part of eachrule as a real number (fuzzy singleton). The antecedent fuzzysets are combined by the algebraic product operator and the in-ference operator product is used.

To extract the linguistic relations modeling the dynamicsystem, we proceed as follows. For each pattern valueand

coming from the system sensors, we calculate their mem-bership grades in each attributed fuzzy set. Therefore, a vectorwith grades corresponding to the number of fuzzy sets thatdivide the universes is attributed to each pattern. Follow, welook for the highest degree of each vector. The correspondingfuzzy set is selected as the linguistic description of thecorresponding pattern value or . This process groups thepattern values with the same antecedent fuzzy sets, extractingeach rule that composes the fuzzy model.

The fuzzy implication degree (8) is now calculated using theproduct operation rule for each pattern pair ( ) of theextracted rule. That pair with the highest implication degree ischosen to define the pre-defuzzified output value offor the rule being extracted, attributing its numerical valueto

.Previous procedures are repeated for each subspace that was

pre-determined by the fuzzy set, as illustrated in Fig. 3. After,a group of rules forming the fuzzy model is obtained from thedata patterns.

The inference method applied to the fuzzy rule base uses thecentroid-defuzzification formula and combines all rule contri-butions in a weighted form given by

(12)

In (12), is the inferred output value from the fuzzy model,is the extracted response of the respective rule (),

is the rule firing degree, andis the total number of rules com-posing the model.

2) Formal Language:To get a data sample of the languagegenerated by a dynamical system, an input signal is imposedso that the output variable should assume values in the oper-ating region of the system. This input/output signal evolution is

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES 73

Fig. 4. Flowchart of the grammatical inference learning algorithm.

then quantified as described in Section III. After analyzing thesesample words, the learning algorithm establishes the grammarproductions.

The basic learning algorithm, represented by a flowchart inFig. 4, has a simple paradigm with two principles in order toestablish the set of productions. They are:

• A 0-type production is taken into consideration for everysymbol of the alphabet that occurs in the sample.

• A -type production is taken into consideration ifthe establishedn-type production already exists.

As an example, consider the following words obtained from ahypothetical data sample, whose letters reflect the quantificationof the input/output variables:

Input variable:Output variable:

First, the learning algorithm analyzes the leading symbol ofboth words. The symbol “,” within the terminal word “ ,”denotes the initial value of the output variable. Since no otherinformation is available, a0-type production is firstassumed. After analyzing the second input symbol (“”), thealgorithm establishes another0-type production . Thethird symbol (“ ”) yields a third 0-type production .

This production, however, is in conflict with the first productionobtained . In this case, a1-type productionis established. Both0-type productions andare then eliminated. Similarly, the fourth symbol yields a1-typeproduction . The productions obtained from the analysisof the hypothetical data sample will be: , and

. Obviously, in order to infer a suitable grammar, alarger data sample must be used.

Note that, unlike theIF-THEN rules in fuzzy logic, the struc-ture of the formal language productions is not established inadvance. According to the samples involved, one can get dif-ferent types of productions (0-type,1-type, ) in the resultinggrammar. This feature will permit modeling the different be-haviors detected in a dynamical system, constituting the patternrecognition process of the formal language methodology.

IV. THE EXPERIMENTAL DRIVES

The majority of drive systems in the industry are today eitherelectrical and electrohydraulic drive systems. Therefore, insteaduse only one type of drive, two experimental prototypes in ourlaboratory were used to verify the application of both linguisticmethods to recognize drives behavior. The first prototype wasan electrohydraulic actuator and it was used with fuzzy logic.The second one was an electronic-fed squirrel-cage inductionmachine system and it is used regarding the formal languagetheory.

A. Electrohydraulic Drive System (Fuzzy Logic)

The electrohydraulic system can be divided in three subsys-tems, as shown in Fig. 5, as follows:

1) the first subsystem is made of an electrical drive,shown in Fig. 6(a), that commands the speed of apermanent-magnet motor (P.M. Motor) coupled with ahydraulic pump;

2) the second subsystem in Fig. 5 is the hydraulic actuatorshown in Fig. 6(b). It is connected with the first subsystemby the link between the motor and the hydraulic pump;

3) the last subsystem is a coarse position control imple-mented by an analogic proportional controller.

A detailed description of the electrohydraulic system compo-nents is given as follows:

1) a power inverter with IGBTs and current control consti-tuting the electrical drive system;

2) the synchronous P.M. Motor is commanded by the elec-trical drive system and has the following nominal param-eters: 220 V, 1.2 Nm, and 3000 rpm;

3) the speed control system of the motor is composed by aproportional-integral (PI) controller;

4) the hydraulic pump has a fixed displacement;5) the hydraulic actuator controls a linear piston with 0.2 m

as maximal displacement;6) a mechanic inertial load can be imposed to the piston.7) a sensor set allows the acquisition of the following sig-

nals: piston position (), piston speed (), pressure dif-ference in the piston by two pressure sensors namedand , and the measure of the P.M. machine speed.

74 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 1, FEBRUARY 2001

Fig. 5. Schematic diagram of the electrohydraulic system.

Fig. 6. Electrohydraulic drive system. (a) Electrical drive. (b) Hydraulic actuator.

B. Electrical Drive System (Formal Language)

The second experimental system is illustrated in Fig. 7.It is composed of an induction motor driven by a powerinverter. These systems are modeled using elements of theelectromechanical power conversion theory [10]. With somesimplifying assumptions, a 12-equation state model usuallyrepresents the electrical drive system. This large number ofvariables, however, obstructs any attempt to perform an on-linelearning process with associated huge computational costs.Therefore, there is a need to further simplify the electrical drivemodel.

Considering a previous current control-loop [22] for com-mand purposes, it is possible to assume that the electrical drivestator currents are controlled. The drive mathematical model can

then be reduced from a 12th- to a third-order system (13) asshown in [23], and thus described in a rotor flux reference frame(14)

(13)

(14)

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES 75

Fig. 7. Electrical drive system.

In this model, we assume that components of the rotor fluxes( ) and the rotor speed () are state variables, andthe stator current components ( ) are input variables.The term denotes the rotor time constant, is the mutual(stator–rotor) induction coefficient, is the inertia coefficient,

is the rotor self-inductance coefficient, is the frictioncoefficient, and is the load torque.

In our experimental results, a constant inputcurrent is im-posed in order to obtain a suitable magnetization level withinthe induction motor. However, fundamental problems such asaccurate measurement of some system parameters, the self-in-ductances and the internal fluxes, for instance, are present.

A detailed description of the electrical drive system compo-nents is given as follows:

1) a current controlled three-phase PWM inverter IGBTs isused as power inverter;

2) the power inverter dc-voltage can assume up to 400 V;3) the squirrel-cage induction machine presents the fol-

lowing nominal characteristics: W (electricalpower), V (input voltage), A (inputcurrent), (power factor), andrpm/min (mechanical speed).

V. ACQUISITION OF THETRAINING DATA

To obtain relevant training data, two types of information,qualitative and quantitative, can be acquired regarding thesystems behavior. In drive systems, qualitative information isobtained from their mathematical models. This information ispresent in the mathematical relations between system variables,which permits choosing first the most relevant variables to thedescription of system’s behavior.

In order to obtain quantitative information, one can use aprevious model-based simulation study or, if possible, experi-mental data can be acquired from the system. To obtain this data,an excitation signal must be chosen. A typical approach is to usea pseudorandom binary signal (PRBS) that is injected in the dy-namical system. However, this signal is not the best choice to

excite drive systems since it will be filtered by their mechanicaltime-constant [11]. A better excitation signal is thus the use ofa sinusoidal signal composed by different magnitudes and fre-quencies. In this way, the magnitude and frequency values canbe set within the limits of drive’s response, thus avoiding thefiltering problem, and allowing the collected data to be betterdistributed in the normal system’s operating domain. Next, theproblem of how to obtain a relevant training set is presented tothe two drive systems.

A. Electrohydraulic Drive System

If the actuator is interpreted as a black-box and some hy-potheses are established to its operation, the piston position ()can be defined as a function of the reference position signal( ), the motor speed (), and the linear speed of the piston( ). The pressure signal at the piston can be neglected be-cause the load is an inertial one and so the pressure informationbecomes not important, as demonstrated in [12]. The final func-tional relation of the electrohydraulic drive system is presentedin (15) with the system variables that better characterize its be-havior

(15)

To obtain a relevant training set, an experimental essay wasdesigned. We first considered obtaining the training set withthe electrohydraulic system operating in an open-loop mode,and applying a sinusoidal signal to the motor speed. However,the piston revealed an asymmetric behavior, as shown in Fig. 8by its position signal. The piston moves more in one directionthan in the other. Therefore, after some sinusoidal periods, thepiston at the end of its course halts in 0.20 m. To surpass this,a closed-loop composed by a proportional controller was usedto have a coarse control of the piston position and thus elim-inate its asymmetric behavior. In this way, we intend the ac-quired data could cover a representative part of the typical func-tioning domain of the electrohydraulic system. An essay was

76 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 1, FEBRUARY 2001

Fig. 8. Electrohydraulic operating in an open-loop mode. Piston positionsignal (y).

made consisting in the application of a sinusoidal reference po-sition signal formed by different combinations of amplitude andfrequency values. Since the piston position has a maximum dis-placement of 0.20 m, four equally spaced amplitude values werechosen. They were 0.05, 0.10, 0.15, and 0.20 m. Another exper-imental test determined the cutoff frequency of the electrohy-draulic system in about 1 Hz. For higher frequencies, the evo-lution of the piston position starts to present significant attenu-ation. Therefore, a set of six frequency values was set for ouressay in 0.2, 0.4, 0.6, 0.8, and 1.0 Hz.

From (15), the training set of the electrohydraulic systemmust include the signals , , , and . The four signals ob-tained from the experimental system by the previous essay areshown in Fig. 9.

B. Electrical Drive System

As aforementioned, to model the electrical drive system onemust consider the measurement of the motor internal magneticfluxes, which is a difficult task. To overcome this drawback, adynamic input–output modeling approach is considered insteadof a static model approach. This prevents the measurements ofthe internal fluxes and still achieves an accurate representationof the electrical drive’s behavior. In this way, the consideredelectrical drive system can be described as a functional relation-ship between the input and output variables defined by

(16)

whereoutput variable;initial value;previous evolution;evolution depth;evolution of the input variable;functional relationship considering as the initialcondition of the system.

Formal language techniques can be established to analyze theinput/output information that contains valuable informationabout the system behavior, and also to identify the referredfunctional relationship .

A grammar of the drive language can be inferred using onlyavailable input–output system’s experimental information. Thisgrammar will consider a nonterminal alphabet established fromthe quantification of the input current () and a terminal al-phabet established from the output speed signal (). The in-ferred productions are of the general form (9), wherecodifiesthe input variable or its evolution . In the present paper,we simplify this codification by considering only the input vari-able, thus simplifying the nonterminal alphabet. This alphabetwill be established only from the codification of the input vari-able rather from the codification of its evolution.

To obtain a representative training set from this system, asinusoidal reference signal with a combination of amplitudesand frequencies was also imposed on the input system variable( ), as was made for the electrohydraulic system. Fig. 10shows the evolution of the training and test sets, with the testdata being displayed in bold.

C. Discussion

The acquisition of a training data set is an essential issuein order to obtain a good knowledge of the system to be rec-ognized. Since the PRBS technique is not appropriate in drivesystems, the training set must cover a representative part of thesystem’s working domain. The basic idea is not to learn the ex-tended behavior of the drives but to learn its behavior in the areaswhere it usually works. If the drive changes its course of action,the learning algorithms should adapt their training data sets inorder to acquire this new information.

The choice of variables is also important. This choice is usu-ally supported by the knowledge of the theoretical model of thedrives. A wrong choice will compromise the learning and recog-nition processes, since the existence of a functional relationshiprepresentative of the drive behavior could not be assured [12],[20].

In the electrohydraulic actuator, all the fundamental statevariables are accessible. Therefore, a static function relation-ship is assumed and learned by the fuzzy-logic algorithm.However, in the electrical drive, not all of the major statevariables are accessible. In this case, the extracted functionalrelationship considers the evolution through time of only theinput–output variables. This situation is, however, suitable forthe formal language algorithm since the general form of theproductions (which enables the establishment of various typesof productions) takes past information into account.

VI. RECOGNITION RESULTS

A. Electrohydraulic Drive System

The results presented in this section show the recognitionability of the extracted fuzzy model. Using the training set thatwas shown in Fig. 9, the fuzzy-learning algorithm is applied tothese data. A number of 11 fuzzy sets were attributed to eachantecedent variable, and 13 fuzzy sets were attributed to theconsequent variable. In this paper, the fuzzy sets number wasattributed on a trial-and-error basis, minimizing the quadraticerror mean. Fig. 11 shows the error signal between the fuzzymodel prediction and the measured piston position. In spite ofthe good approximation, with errors between5%, Fig. 11

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES 77

Fig. 9. Training set obtained from the electrohydraulic system.

Fig. 10. Electrical drive system training (solid Line) and test data (bold line)sets.

Fig. 11. Fuzzy logic recognition results.

shows oscillations in the error signal. These may have origi-nated from the information supplied in the learning process tobe incomplete, from noise interference, or from the fact thatthe learning data do not cover a significant part of the system’sworking domain.

B. Electrical Drive System

In order to verify the validity of the recognition process usingthe formal language in the electrical drive system, two alpha-bets were considered. In the first one, a quantification intervalof 0.01 pu units is assumed for both input and output vari-ables, which yields a 60-symbol alphabet. The grammar inferredfrom the training data contains 183 productions distributed inthe following way: five0-type productions, 1561-type produc-tions, and 222-type productions. Applying this grammar with60 symbols to the test set, we get the recognition results shownin Fig. 12. These can be considered satisfactory. However, wemust point out that there are some situations where the grammardoes not provide any answer, as indicated by the white arrowsin Fig. 12. This happens when any production representing thatparticular dynamical input/output symbol relationship has notbeen inferred. In Section VIII, a method for establishing the in-existent productions is proposed.

To avoid the influence of noise over the inferred productions,the following procedure was observed. A production is not con-sidered as a valid one if it is inferred only once. Only when isrepeatedly inferred from the data sample it is considered. Thisprocedure works as a filter over possible noise perturbations.

In the second alphabet considered, a quantification intervalten times higher (0.1 pu units) is assumed for both input andoutput variables, yielding now a 6-symbol alphabet, insteadof the previous 60 symbols. The grammar inferred from datacontains 36 productions distributed in the following way:zero 0-type productions, four1-type productions, and 322-type productions. Applying this grammar to the test data,we get the recognition results presented in Fig. 13. The whitearrows denote, as before, the absence of productions that canbe applied. Clearly, the quantitative modeling performancedeteriorates since a much smaller alphabet was considered.However, the increased error does not imply any fail in aqualitative recognition process. This increased error it is onlydue to the larger quantification interval imposed by the limitedalphabet considered.

78 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 1, FEBRUARY 2001

Fig. 12. Formal language recognition results using an alphabet with 60 symbols.

Fig. 13. Formal language recognition results using an alphabet with 6 symbols.

The grammar response quantitatively accompanies the driveevolution, besides the use of a much smaller number of symbolsand productions. The limited alphabet considered is not a draw-back in the qualitative recognition process. As in the previousgrammar with 60 symbols, the inexistent productions problemis also present.

Both grammars recognize the drive-speed evolution. How-ever, the second one is much simpler since it presents smalleralphabets and a smaller set of productions. If the main featureis a quantitative recognition, only the first grammar presents anacceptable performance since its quantification interval is of thesame order as the sensors numerical precision.

C. Discussion

Both methodologies present good qualitative recognitionabilities. The results with fuzzy logic were obtained in a more“continuous” way than with the formal language algorithm,making fuzzy logic more suitable for quantitative recognitionabilities. The reason for this goes back to the quantification ofthe variables. The discrete quantification performed within theformal language algorithm implies a discrete response whenrecognition of the drive is performed. In order to improvethe quantitative recognition abilities of the formal languagealgorithm, two solutions can be adopted: increase the numberof symbols in the alphabet (the quantification becomes moreaccurate), or to perform a numerical interpolation of theproductions. The first solution increases the number of produc-tions, slowing the recognition process. The second solution,

however, does not increase the number of productions andcould be applied when a qualitative recognition was required,as will be shown in Section VIII.

Comparing fuzzy logic and formal language structures, quan-tification in the first linguistic approach is made by the fuzzysets, whereas in the formal language, quantification is made bythe terminal and nonterminal alphabets. These alphabets are at-tributed by specifying a certain number of symbols, with thesame occurring in fuzzy logic when a certain number of fuzzysets is attributed. Clustering techniques such as Fuzzy-meanscould be used to determine the fuzzy sets in the antecedents.In formal language, however, no any data preprocessing is usedto determine its alphabets. Therefore, to obtain a better-qualitycomparison between both methods, no preprocessing techniquewas used with the fuzzy logic method either.

VII. L ACK OF INFORMATION

Lack of information is a problem closely related with the“quality” of the training set. To cover a significant part ofthe system’s working domain, a large data set is needed. Toovercome this problem, the domain can be previously filledusing theoretical values from possible system simulations. Onthe other hand, if an on-line functioning of the system is con-sidered, one can complete the empty domain regions as soon asthe system operates in these regions. Since the regions that didnot receive experimental data are always considered during thelearning phase, this fact provokes large errors in the learning

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES 79

Fig. 14. Domain filling by the training set of the electrohydraulic system.

algorithms. A good solution for this problem is to implementan interpolation algorithm. Next, the two drive systems with thecorresponding learning algorithms are investigated concerningthe problem of lack of information.

A. Electrohydraulic Drive System

Fig. 14 shows the three signals forming the training set,, and , and composing the domain of the relation expressed

in (15). Even using a coarse closed-loop position control in theelectrohydraulic system, the training data only covers part of thesystem’s domain, as shown in Fig. 14.

To better visualize the consequences of the empty domainareas in the learning process, the relationrepresenting the electrohydraulic system is simplified to

(17)

Fig. 15(a) shows the domain established byand signals.The fuzzy learning algorithm is applied to this data using a par-tition set of nine fuzzy sets to , seven to , and 13 fuzzy setsto . The quality of the extracted rules, which are representedin Fig. 15(b), is first verified in reproducing the training data.Fig. 16 shows the error signal between the piston position andthe value predicted by the fuzzy model. Note in Fig. 15(a) thatthere are more data in the inner zone of the domain, while thedata are sparser in the borders. This distribution increases thepossibility of acquiring empty rules in the model (rules withouta conclusion), and also increases the possibility of extractingsome rules which conclusion was computed using only a smallnumber of examples. The results in Fig. 16 show a periodic be-havior of the large errors. They appear in two situations when theelectrohydraulic system operates in the domain borders. Theyare: when the piston is close to the limits of its displacementzone, and when the piston speed reaches its maximum or min-imum values. Since a sinusoidal signal was used, these two situ-ations are periodic and so justify the periodic large errors shownin Fig. 16.

Fig. 15. (a) Domain filling by the training data fory = f(y ; v). (b)Simplified representation of the extracted fuzzy rules covering the domain.

To make the training set sparser and induce the appearing oflarge gaps in the domain, a second data set was built, whichcontains 10% examples of the original data. Fig. 17(a) showsthe original data set and Fig. 17(b) shows the reduced set.

Considering for the fuzzy model the same structure (9, 7, 13),the learning algorithm is now applied to the reduced set. Therules extracted are shown in Fig. 17(c), which for the used par-tition set they still cover all data, thus avoiding the appearingof empty rules in the fuzzy model. However, as there were nocollected data placed in the domain borders, the learning algo-rithm could not infer any conclusion value from these regions.When inferring in these regions, the model will present largeprediction errors. This situation, which is similar to that shownin Fig. 16, is described in Fig. 18. The error oscillations in thefigure indicate that the fuzzy model presents large errors alwayswhen an inference occurs near the limits of the domain. This sit-uation becomes even more critic because, despite the presenceof empty rules, there are few collected examples in the limitregions. Therefore, apart from the low quality of the rules ex-tracted in these regions, the inference mechanism also uses a

80 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 1, FEBRUARY 2001

Fig. 16. Position error signal from the fuzzy model.

Fig. 17. (a) Original training data set. (b) Reduced set. (c) Rules extracted.

small number of valid rules (nonempty) to infer the piston po-sition.

Usually, increasing the number of fuzzy sets attributed toeach variable can lead to a better prediction performance ofthe fuzzy model. However, increasing the partitions can inducemore empty rules in the fuzzy model. This, containing moreempty rules, will cause the inference mechanism to give posi-tion values that show a large deviation from the correct ones.To investigate these effects, the fuzzy partitions of the modelwere increased to 13 fuzzy sets. Using the reduced training set,the learning algorithm extracted the rule set shown in Fig. 19.Testing again the extracted fuzzy model in the reproduction ofthe training set, this leads to the results shown in Fig. 20. Whenthe inference process fails completely and a null value is inferredfrom the fuzzy model, we decided to consider the last nonzeroinferred value as the valid model response.

To analyze the large modeling errors, the time interval indi-cated in Fig. 20 is expanded. The interval is shown in Fig. 21(a)indicating two critical instances. The first instance occurswhen the inference mechanism reaches the empty regions ofthe rule-base (domain borders), corresponding in Fig. 21(b) tothe largest errors with values over 40%, and characterizing asignificant deviation of the inferred position from the measuredone. Note that, in this critical instance, the domain bordersthat were reached were those of the piston speed limit values.Therefore, large errors occur in the high frequency values ofthe reference signal.

The second critical instance occurs when the reference signalgets values near the piston limits. In this case, as can be verifiedin Fig. 19, the inference process uses only a small number ofrules and the inferred position values begin to diverge from themeasured ones. The inference, for instance, using just one rule,makes constant the fuzzy model output, as shown in the secondcritical situation in Fig. 21(a) by the “flat” predictions.

B. Electrical Drive System

The problem of lack of information is similar in both method-ologies. Therefore, this problem in the formal language will bepresented without a wide range of situations. Technically, therecognition procedure in the formal language fails when it isnot possible to find a production that is suitable for a symbol ina test word. In this case, the word is considered not belongingto the language described by the inferred grammar. Figs. 12 and13 show when the recognition process fails. This happens whenthe number of available patterns is too small for a suitable gram-matical inference.

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES 81

Fig. 18. Position error signal obtained with the reduced training set.

Fig. 19. Rules extracted from the reduced training set after increasing thenumber of fuzzy partitions to 13.

Let us consider the case of an alphabet with 60 symbols and areduced training set that is half of the one shown in Fig. 10. Afterinferring the correspondent grammar, the recognition results ob-tained are presented in Fig. 22. The measured drive speed is pre-sented as a dotted line and the grammar response as a continuousone. It is easily observed that the recognition process presentsconsiderable faults. Since a smaller training set was considered,the number of valid productions in the inferred grammar wasconsiderably smaller, similar to the empty rules observed in thefuzzy model.

C. Discussion

The lack of information is a very important issue when algo-rithms that learn from examples are applied, since “one cannotlearn what one cannot see.” In order to get a good performancein the recognition process of the drive behavior, a sufficientlyand representative training set must be used. When the data doesnot cover important areas of the system’s working domain, therecognition results deteriorate. In the fuzzy logic algorithm, thisdeterioration of the performance is due to the use of a smallernumber of rules, when no other information is available. In thiscase, the response of the algorithm diverges from the experi-mental values and the recognition process fails. The formal lan-guage algorithm does not give any response when no other pro-duction was established by the learning process, causing therecognition process also to fail.

From the previous statements, one concludes that to attenuatethe problem of lack of information, the domain could be previ-

ously filled with other data, if possible. For instance, data fromsimulation, linguistic rules in the case of fuzzy logic, or produc-tions in the formal language.

VIII. I NTERPOLATION

Both results showed that the training data must cover in thebest possible way the functioning domain to avoid large gaps init. These gaps can be filled through some interpolation mecha-nism, can be filled using some theoretical values obtained fromsimulation tests or, when operating in an on-line mode, the gapscan be completed using new data acquired during the system’soperation. This section describes how a simple interpolation al-gorithm can be used to fill the possible gaps that can appear inthe fuzzy model (empty rules) and in the grammar (null produc-tions).

A. Electrohydraulic Drive System

One simple solution to complete the empty fuzzy rules isthe application of some interpolation mechanism in the rulebase. The interpolation mechanism used was introduced in[7]. It replaces the null values established as conclusion ofthe empty rules by the conclusion inferred from the rulesinitially extracted and located around the gaps. After fillingthe empty rule, the inference process is repeated to generatethe final inferred value.

To exemplify the interpolation process, consider the simplefuzzy model represented in Fig. 23(a). This is characterized bytwo antecedent variables and that are divided by the fourmembership functions and ,respectively. It was assumed in the example that six validrules were obtained after applying the learning algorithm tothe training set. These rules are denoted in the Fig. 23(a) by

, and , with the white square regionsrepresenting the empty rules as before. Suppose that during acertain inference step, the fuzzy sets marked with a circle in thefigure were activated. From the four activated rules, only thosewith conclusions , and are valid for the inferenceprocess. The rule with conclusion has a null value sinceit could not be extracted from the training data. Necessarily,the inferred model response will become incorrect becausethe inference used only three valid rules instead of the fourones. To estimate a valid value for the rule conclusion, theinterpolation algorithm will use the value inferred from the

82 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 1, FEBRUARY 2001

Fig. 20. Position error signal using the reduced set and the finest fuzzy model (13, 13, 13).

Fig. 21. (a) Expanded interval showing the two critical fuzzy modeling instances. (b) Position error signal.

three valid responses , and . Therefore, after replacingthe null value with the inferred one , the inference processis repeated again, but now using the previous extracted threerules plus the rule which conclusion was estimated.

In this example, the interpolation mechanism applied to com-plete the empty rule presented three already extracted neigh-borhood rules. However, if these rules were in a smaller number,the interpolation would have little information to estimate amore correct conclusion. In that case, it becomes necessary toacquire more data placed in these domain regions to extract ahigher number of neighborhood rules; even to better define therules already extracted. In this way, the inferred conclusionwill be approximated further to its “correct” value.

Fig. 23(b) shows the results when the interpolation algorithmis applied to complete the rules in Fig. 19. Testing again thecompleted fuzzy model in the training set, Fig. 24 shows thenew error signal after the interpolation. Note that, when com-paring with the anterior results in Fig. 21, the interpolation re-moved the large error values in some regions. However, someerrors remained high despite the interpolation. These regionscorrespond to regions in the domain zones with few and verysparse examples. Therefore, since only a small quantity of ex-amples was applied to compute the rule conclusions located inthose regions, the interpolation results based on these rules willcontinue to remain far from the correct values, maintaining alarge prediction error signal, as shown in Fig. 24.

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES 83

Fig. 22. Formal language recognition results using an alphabet with 60symbols and a reduced training set.

B. Electrical Drive System

The previous section showed that a small training set couldlead to a faulty grammar, which fails to recognize some words.However, those words may still be in the language, within acertain distance from the words belonging to the data sampleinitially used to infer the grammar. In this case, it may be usefulto establish a new production out from the ones inferred from thesmaller sample. This feature is called grammatical interpolation.

The grammatical interpolation is applied to establish a newproduction considering a structural matching procedure. Themain idea of a structural matching is based on a formal mea-sure of the similarity between the unknown input pattern and theavailable data structures. This measure of similarity consists inthe distance between the inexistent production and the nearestsimilar productions.

Several methods for structural matching have been reportedin [21]. The basic algorithm states that the distance between twowords is related with the sequence of edit operations (substitu-tion, insertion, and deletion) required to transform one word intoanother. For any sequences of edit operations, a cost function

defined by

(18)

can be considered. It denotes the cost of a particular word madeby sequence, and denotes the cost of a particular editoperation. The distance between two words is defined as theminimum cost necessary to transform a word into another andgiven by

is a sequence of edit operations

which transforms into (19)

where and denote any two words. When a word cannot berecognized due to an inexistent production necessary to estab-lish some symbol in that word, a grammatical interpolation for-

Fig. 23. (a) Interpolation applied to a hypothetical fuzzy model. (b) Rule-baseafter applying the interpolation mechanism.

mula is proposed and applied in order to find an estimate of thatunreachable symbol. The interpolation formula is expressed by

quantification (20)

and is based on the average distance between similar produc-tions. Assuming the general form of a production introducedin (9), the coefficients denote the distance between produc-tions and , considering their respective words “ ,” and

as the last terminal symbol in production.Applying the grammatical interpolation in the anterior results

of Fig. 22, the new recognition results are shown in Fig. 25. Theerroneous influence of the inexistent productions is reduced,and the number of symbols that were not recognized in the testwords is substantially smaller. In Fig. 25, the grey arrows de-note the fulfillment of inexistent productions, while the whitearrows denote the absence of applicable productions that couldnot fulfilled by the grammatical interpolation procedure.

84 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 1, FEBRUARY 2001

Fig. 24. Error position signal after applying the interpolation mechanism.

Fig. 25. Formal language recognition results after the grammaticalinterpolation procedure.

C. Discussion

From the previous analysis, it is clear that the use of an in-terpolation mechanism improves the performance of the algo-rithms. Both base their interpolation mechanism in the knowl-edge of the information gathered around the empty space. Thefuzzy logic approach infers a value for the empty rules usingthe values extracted for the nearby rules. Similarly, the proposedgrammatical interpolation algorithm, in the presence of an inex-istent production, considers the sequence , as definedin (9) as a word, and completes the production according to thelinguistic distance from the nearest productions. Similarly to thefuzzy logic procedure, the establishment of this missing produc-tion is weighted according to the distance from those alreadyinferred productions located around.

The use of the previous interpolation mechanisms onlypresents good results when filling small areas of unextractedinformation. Because they base their procedure on the infor-mation around those empty areas, if a large empty area is to befulfilled, the errors will be huge.

IX. GENERALIZATION

By generalization, we mean the ability of the algorithms inwell recognizing the systems even under working conditions

Fig. 26. Representation of the restricted training set from the electrohydraulicsystem.

distinct from those observed during the training phase. In orderto verify the generalization ability of the algorithms, a con-tracted training set was considered. This new set does not coverthe entire working domain but only part of it. The further awayfrom areas with data the algorithms are able of recognizing thesystems, the better their generalization abilities are conceived.

A. Electrohydraulic Drive System

Using the initial structure (11, 11, 11, 13) of the fuzzy model,a second training set was built, which was located in a restrictedregion of the functioning domain, as shown in Fig. 26. To verifythe generalization ability of the fuzzy model, a test set was usedwith unlimited examples, as shown also in Fig. 26.

Using the learning algorithm with the limited training set,Fig. 27 shows its generalization ability through the positionerror signal. The results show that, when the electrohydraulicsystem operates inside the domain regions covered by thetraining set, the model has low error values, as seen in Fig. 27.On the other hand, when the system operates out of the dataarea, the fuzzy model has to extrapolate as far as the rulesextracted in the borders of the training region allow it to. Fig. 27shows that the error values increase as the system operates outof the data set area, resulting only in a limited generalization.

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES 85

Fig. 27. Generalization results using the fuzzy logic methodology.

Fig. 28. Restricted training (solid line) and test (bold line) data sets to theelectrical drive system.

B. Electrical Drive System

The generalization ability of the grammatical model for theelectrical drive also decreases if the training data only includessome part of the working domain, as shown in Fig. 28. In thiscase, the training set only considers positive speed values. Oth-erwise, the considered test data set, displayed in bold, assumespositive and negative drive speeds.

Fig. 29 shows the modeling results using the restricted data,showing that the recognition fails for the domain areas not cov-ered by the training set. Namely, the symbols that codify nega-tive drive speeds are not recognized. As for the fuzzy modelingof the electrohydraulic system, the results show the local charac-teristic of the generalization ability for this modeling situation.

C. Discussion

The results presented in this section show that both method-ologies give a good approximation of respective drive system,even at points not contained in the training set. Therefore, thetwo learning techniques have generalization ability. However,it must be noted that generalization is a complex phenomenonand that there is no global requirement for a successful general-ization. In the two algorithms, generalization has a local effect.Therefore, this demands that the experimental training data mustcover a significant part of the system’s working domain. When

Fig. 29. Generalization results using the formal language methodology.

this is not assured and only a small zone of the working domainis filled by the training data, the generalization process fails.

X. CONCLUSION

This paper has investigated the recognition of drive system’sbehavior using fuzzy logic and formal language theories. Thiswork has been devoted to an electrohydraulic drive and an in-duction motor drive modeling, but can be applied to other drivesystems presenting similar features.

Since both methods are linguistic based, several similaritieswere pointed out. We emphasized the quantification by dividingthe feature space into clusters, the learning procedures, and em-phasized the acquisition of the training patterns.

On the other hand, some aspects enhanced the particularitiesbetween both approaches:

1) Unlike the fuzzy approach, the creation of a formal lan-guage alphabet is made in a crispy way without the useof fuzzy sets. The linguistic relations within the fuzzylogic approach are established regarding the fuzzy gradeof each considered variable. In the formal language ap-proach, the structure of the productions is not set up inadvance. Instead, different types of productions are estab-lished according to the incoming words from the linguistsource (dynamic system).

86 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 1, FEBRUARY 2001

2) Both methodologies presented good qualitative recogni-tion results. Due to the characteristics of the consideredformal language’s alphabets, the fuzzy approach is natu-rally more suitable for a quantitative recognition proce-dure. Solutions for this formal language drawback werepresented and discussed.

3) We also showed that the lack of information could deteri-orate the recognition performance of both algorithms. Inorder to minimize this deterioration, interpolation mech-anisms were proposed. The fuzzy logic approach fills theempty rules weighting the values of the nearby alreadyextracted rules. Considering a slightly different approachwithin the formal language, a linguistic interpolation al-gorithm based in the concept of distance between words,was proposed to establish inexistent productions.

4) The use of the previous interpolation strategies improvedthe generalization abilities of both procedures. However,for areas far away from the training set coverage of theworking domain, results showed that the generalizationdeteriorates. Generalization has a local characteristic.

This paper represents the basis for our present research. Newapplications regarding fault diagnosis of drive systems are beingdeveloped that apply these techniques. For this purpose, opti-mization techniques, such as immunity-based learning mecha-nisms, are also being considered.

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J. F. Martins (M’96) graduated in electricalengineering at Instituto Superior Técnico (IST),Technical University of Lisbon, in 1990. He obtainedthe M.Sc. degree in electrical engineering at thesame institute in 1996. He is finishing his Ph.D. dis-sertation in electrical engineering, in the applicationof grammatical inference learning algorithms andcellular automata within drive systems, at the IST.

He is currently Adjoint Professor in the Depart-ment of Electrical Engineering at Escola Superiorde Tecnologia/Instituto Politécnico de Setúbal. He

is also with the Mechatronics Laboratory. His research areas are in control ofelectrical drives, advanced learning control techniques for electromechanicalsystems and nonlinear systems.

He has published articles in international scientific journals such as the IEEETRANSACTIONS ON INDUSTRIAL ELECTRONICS and PATTERN RECOGNITION

LETTERS.

P. J. Costa Branco(M’92) is currently an AssistantProfessor in the Department of Electrical andComputing Engineering, Section of ElectricalMachines and Power Electronics, in the InstitutoSuperior Técnico (IST), Lisbon, Portugal, and hasbeen with the Mechatronics Laboratory/IST since1992. His research areas are in control of electricaldrives, systems modeling and control using softcomputing techniques, and he is presently engagedin research on advanced learning control techniquesfor electromechanical systems.

Dr. Costa Branco is the author of published articles in internationalscientific journals such as the IEEE TRANSACTIONS ON MAGNETICS, IEEETRANSACTIONS ONSYSTEM, MAN, AND CYBERNETICS, PATTERN RECOGNITION

LETTERS, FUZZY SETS AND SYSTEMS, and EUROPEAN TRANSACTIONS ON

ELECTRICAL POWER ENGINEERING. He has been a Referee of internationalscientific journals, participated in boards of international meetings, and citedin theWho’s Who in the World.

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES 87

A. J. Pires graduated in electrical engineering atInstituto Superior Técnico, Technical Universityof Lisbon (TUL), in 1985. He obtained the M.Sc.degree in 1988 and the Ph.D. degree in 1994 inelectrical engineering at the same Institute.

He is currently Coordinator Professor in the areaof Electrical Engineering at Escola Superior de Tec-nologia, Polytechnic Institute of Setúbal and InvitedAssociated Professor at the Physics Department, Uni-versity of Evora. He is also with the MechatronicsLaboratory at the TUL. His research areas are in elec-

trical machines, power electronics, and intelligent control systems for electricaldrives.

Dr. Pires has authored papers in international journals such as the IEEETRANSACTIONS ON INDUSTRIAL ELECTRONICS and PATTERN RECOGNITION

LETTERS.

J. A. Dentegraduated in electrical engineering at In-stituto Superior Técnico (I.S.T.), Technical Univer-sity of Lisbon, in 1974/1975. He received the Ph.D.degree in electrical engineering at the same institutein 1986, and was Associated Professor among 1989and 1993.

He is currently Full Professor in the area ofElectrical Machines at Department of Electrical andComputing Engineering, Section of Electrical Ma-chines and Power Electronics at I.S.T. He has beenwith the Mechatronics Laboratory as the Scientific

Coordinator since 1993. He has published more than 30 scientific articles inrefereed journals and books, and more than 40 articles in refereed conferenceproceedings. His primary areas of interest are in electrical machines, motioncontrol, and presently is engaged in research on advanced learning controltechniques for electromechanical systems.