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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 and quality of industrial processes. However, they exhibit considerable complexities related with their behavior as large uncertainties at a structure and parameter levels, multidimensionality, and strong mutual interactions. With these multiplying complexities, the usual models are becoming not accurate enough. It is necessary to complement them with other information-processing techniques that allow a better recognition of their behavior. The aim of this paper is to analyze common features, and the potential, but also the drawbacks that fuzzy logic and formal language theories show when used for recognition of patterns in experimental drives. Two prototype systems are used: an electrohydraulic drive and an induction motor drive. We underline the similarities and various aspects of both recognition methodologies, despite their use on different systems. A set of experimental learning situations with critical effects on their performance are presented and discussed. Index Terms—Fuzzy logic, fuzzy systems, identification, learning systems, modeling, pattern recognition.

I. INTRODUCTION

D

YNAMIC performance of drive systems has benefited from the recent and constant progresses in new materials, power electronics, microelectronics, and informatics. These progresses have allowed the increase of their performance demands. Models resulting from the mathematical formalization of the different physical phenomena that take place in drives usually support description of these drives dynamic. However, the increased analytical complexity of these models makes them very difficult 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 secondary at a command and control levels of drive systems, are today very important since the quality demands are increasing. Equally significant are the strong nonlinear relations between the drive’s state variables and the exact knowledge of some parameters, which makes even more difficult to establish functional relations representative of their dynamic behavior. The inherent difficulties within a complete drive system’s modeling, 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 and with the Laboratório de Mecatrónica, Instituto Superior Técnico (I.S.T.), 1096 Lisboa Codex, Portugal (e-mail: [email protected]). 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: [email protected]). Publisher Item Identifier S 1063-6706(01)01365-0.

such as fuzzy logic and formal language methodologies in automatic recognition of drives (dynamic) behavior. These recognition procedures have been extensively applied in image and speech processing areas. Their principles, however, can be extended to drives pattern recognition in order to complete or substitute their usual mathematical models, and therefore increase their prediction performance. Before introduce this approach, it is important to underline the difference between system identification and pattern recognition. System identification, as stated by Sage and Melsa [24], means “the process of determining a difference or differential equation (or the coefficient parameters of such an equation) such that it describes a physical process in accordance with some predetermined criterion.” Therefore, in a system identification problem, the structure and order of the model are established in advance. Identification is related to the system’s parameters that are unknown. In drive systems, for instance, techniques as Kalman filtering [25] or observer-based algorithms [26] have been applied with relative and limited success when incorporated in a control system. Pattern recognition, on the other hand, is a classifier system, grouping patterns into categories [27]. The pattern recognition problem, in the case of recognizing the behavior of drive systems, consists of identifying the relationships between the system state conditions and its observed output variables. Both in fuzzy logic and in the formal language approach, patterns are viewed as sentences in a language: IF-THEN rules in fuzzy logic and a grammar in the formal language. The main drawback regarding the use of these methodologies is the loss of some physical meaning in the obtained drive’s model. The main benefit, however, is the use of these models and associated learning algorithms in the design of new drive control schemes with learning properties. The existent mathematical models and designed controllers can also be expanded with the incorporation of these new models in the control system, thus completing its structure. In the operation of modern industrial plants, drive systems play an essential role in increasing the productivity and quality demands, and mainly in reducing energy and equipment maintenance costs at all stages of the process. The configuration of a drive, which is by far more complex, contains several motors, power converters, hydraulic and/or pneumatic elements, sensors, and digital control systems. Typical features of drive systems involve considerable complexities related with their behavior. They have a highly nonlinear coupling, presenting large uncertainties at a structural and parameter level, they are 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

drive processes originates mutual interactions between them, presenting, in some cases, internal feedback mechanisms. With those multiplying complexities, there are problems in applying their usual mathematical models since they are becoming either too many complexes to work with reasonable computational times, or their present design is not sufficient to handle the actual drive system uncertainties. The use of the same mathematical model for all system’s operating regions does not allow modifying the functional relation between the system variables in such a way that it cancels new operating modes. Consequently, the existing models are usually not accurate enough to fulfill the description of specific situations. Considering the previous difficulties within the actual drive systems, it becomes necessary to complement, and even correct, their classical modeling with other information processing techniques that allow better recognition of their behavior [3], [4]. In a previous work [7], [9], we began studying the application of fuzzy logic and neural networks to automatically recognize the (drive) systems operation. Those were different approaches, but it was verified that they share common problems when applied to a practical modeling process. Following that research in this paper, we investigate only linguistic approaches in experimental drive recognition. For this, fuzzy logic and formal language theories are used. Both approaches, in spite of representing different concepts concerning linguistic approximations, present memory, learning, and generalization skills that have to be necessarily used in a recognition system [5], [6]. The fundamental contribution in the formal language area was made by Chomsky [13], whose theory of formal grammars has had a major influence in the development of the subject. Grammatical inference is a concept that goes back to Gold’s work [14], and is defined as a way in which a system tries “to guess” general rules from examples. Since then much work as been done, which can be found in several excellent surveys [15]–[17]. The aim of this paper is to describe the application of fuzzy logic and grammatical inference techniques, based on formal language theory, in modeling two experimental drive systems. Both approaches manipulate drive information, but their objects of reasoning are fuzzy sets in the case of fuzzy logic, and an alphabet in the case of grammatical inference. In this perspective, both construct a model of the considered dynamical system. For comparison purposes between fuzzy logic and formal language, each method is applied to a drive system. Fuzzy logic in an electrohydraulic system and formal language in an induction motor drive. It is our objective, with this decision, to make more relevant their mutual modeling aspects, although using both methodologies in systems with very different dynamic characteristics. The following modeling aspects will be discussed: • How can we acquire a representative set of patterns from drive systems? • What does a representative set of patterns and their influence in the learning mechanisms mean? • How can we overcome possible lack of information in our data? • How can both approaches recognize the behavior of the drive system?

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The paper is organized as follows. Section II outlines the basic concepts of both linguistic techniques and points out their similarities. Section III describes the learning algorithms. In Section IV, we describe the drives used as experimental systems to our study. Section V analyzes the acquisition of the training data set issue. Section VI presents some recognition results. Section VII discusses the problem of lack of information in the available set of patterns. The interpolation techniques used to minimize the previous problem are described and discussed in Section VIII. Section IX discusses results concerning the generalization aspect. The conclusions and future work are explained in Section X. II. BASIC CONCEPTS Fuzzy logic and formal language methodologies are natural language approaches able to represent a system’s dynamics. Both can describe complex relations between the system variables through linguistic relations [18]. In this section, the concepts that characterize each approach and form the basis of both linguistic algorithms are summarized. A. Fuzzy Logic Fuzzy logic in pattern recognition [1] constitutes, in its final approach, a classifying system that condenses a large amount of patterns, represented by numerical data, into a rule-base structure [5]. Classification is obtained by using fuzzy IF-THEN rules that are formed by three main structures: linguistic variables, fuzzy propositions, and truth-values. 1) Linguistic Variable: In the classification rules, linguistic variables represent the feature space. For each feature, a number of words are used. These are called fuzzy variables, or fuzzy numbers, 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 of the subject, and designates the word that characterizes a certain property of the object (such as BIG, HIGH, etc.) fuzzifying each feature. In fuzzy classifier systems, which in our case are modeling systems, an th proposition has a general form given by feature feature

is

and feature is

is (1)

In a drive system, the features are signals from the system that, together, can characterize its dynamic condition. For example, pressures, electrical currents, voltages and temperature are simple examples of the features that we are interested in classifying to predict the drive’s behavior. 3) Truth-Values: Features are mathematically characterized by fuzzy sets defined in [0, 1]. The truth-value of a proposition that is defined to be a like is is denoted by a value value in [0, 1]. B. Formal Language In order to apply grammatical inference procedures, a dynamical system must be considered as an entity (linguistic source) able to generate a certain language. To characterize this entity,

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Fig. 1.

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

Fuzzy logic quantification.

which generates all the words in the language, a grammar can be used [15]. This grammar defines the structural features of the words produced by the linguistic source and, in this way, models the source itself. To specify this grammar denoted as in (2), , a nonone must specify four structures: a terminal alphabet , a start symbol , and a set of productions terminal alphabet [19] (2) 1) Terminal Alphabet: It is constituted by symbols that . make up the resulting words, denoted by 2) Nonterminal Alphabet: It is constituted by symbols that . are used to generate the patterns, denoted by 3) Start Symbol: It is a special nonterminal symbol that is used 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 of words, denoted by . For example, consider a simple grammar with a two-symbol , a one-symbol nonterminal alterminal alphabet , a start symbol , and a set of production phabet . Berules defined by ginning with the start symbol and applying the first production of , the word “ ” is obtained. When applying, for in) in the prestance, three times the second production ( ,” “ ,” and vious word (“ ”), the following strings “ ” are produced. At last, the final terminal word “ ” “ ) in the last is reached by applying the third production ( ”). One can easily verify that this grammar prostring (“ duces all words that consist of a terminal symbol “ ” followed by any number of symbols “ .” This language, denoted as , denotes the -concatenation can be represented as (3) where of symbol “ ” (3) Since the set of productions commands the generation of terminal words, they will be used to encode the dynamics of the system that generates the language. Any word regarded as a sequence of terminal symbols, derived from the start symbol by a sequence of productions of the grammar, is said to be in the language generated by the dynamical system.

The grammatical inference procedure represents a way in which a grammar is directly inferred from a set of sample words (experimental patterns) produced by the dynamical system considered as the linguistic source [17]. A basic idea in any grammatical inference process is that there is not a unique relationship 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 contained in that finite data sample, and the others that are not within that sample but are of the same nature. III. LINGUISTIC ALGORTIHMS We now present the steps in both linguistic approaches that should be observed when a linguistic algorithm is developed. These steps are: quantification of the involved variables, the rules and productions that establish the relations between the objects of reasoning, and the learning algorithm itself. Each step is formalized for fuzzy logic and formal language theories. When possible, we point out their common concepts but that use different formalisms. A. Quantification 1) Fuzzy Logic: Each feature from the system is quantified by fuzzy sets. This quantification includes the number of fuzzy sets, their distribution through each universe of discourse , the type of function used to represent each fuzzy set, and the width of each function that defines its fuzziness. Therefore, numerical data is projected onto each fuzzy feature by the membership , as shown in Fig. 1. functions, i.e., 2) Formal Language: Quantification in formal language is related with the creation of the alphabets by establishing a relation between them and the variables of the dynamical system (4). In fuzzy logic theory, the alphabet is composed by fuzzy sets. Within the formal languages formalism, the terminal alis associated with the output variable, denoted by , phabet with the input variable of the and the nonterminal alphabet dynamical system denoted by

(4)

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES

Fig. 2.

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Formal language quantification.

In order to specify both alphabets, terminal and nonterminal, a number of symbols must be assumed for each one. We will as the consider as the number of terminal symbols and number of nonterminal symbols. Then, a discrete quantification process is applied to the variables where the relation (5) is defined between the dynamical system variables and the alphabet symbols

(5)

Unlike the fuzzy approach shown in Fig. 1, the information codification in a formal language is crisp, as shown in Fig. 2. The alphabets are created dividing the dynamical systems variable range in equal intervals, and associating each interval to a symbol in the alphabet.

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

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

and

is

is

and

is

Each rule is composed by a fuzzy implication degree calculated by (8). In this expression, the operator “ ” is the inference operator, which connects the antecedent to the consequent part of the rule

(8) (6)

The fuzzy rules divide the feature space of the system in subspaces, each one constituting a rule, as illustrated in Fig. 3. The firing degree of each rule is calculated by (7). The membership values of each variable defined for every linguistic term of , are combined by a fuzzy logic AND operator rule , . Variables denoted by and are the numerand . The forical values (patterns) of features denote the membership functions mulas attributed to each fuzzy set (7)

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

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C. Learning Procedures

TABLE I TYPE PRODUCTIONS

TABLE II CORESPONDENCE BETWEEN FUNCTIONAL RELATIONSHIPS PRODUCTIONS

AND

TYPE

symbol, and is a special nonterminal symbol that can be replaced by any other nonterminal symbol (9) Table I illustrates some typical productions. A 0-type production does not have any terminal symbol in the left part of the production. A 1-type production has one terminal symbol in the left part of the production. A 2-type production has two terminal symbols in the left part of the production. A p-type production has terminal symbols in the left part. Given this general form production and since the left part of any production contains at least one nonterminal symbol, this grammar can be classified as context 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 of productions (10). This set can be concatenated as (11) where the symbol “ ” denotes the concatenation of similar productions to denote all the nonterconsidered in (10). The terms minal symbols and denotes the empty symbol

.. .

(10)

1) Fuzzy Logic: Quantification was applied to establish the classification rules that are not known in advance. Therefore, a learning procedure by which the fuzzy rules could be extracted from a data set is needed. In this study, a simple learning algorithm introduced by Wang [2] was chosen as the “basic” fuzzy learning mechanism. The learning algorithm considers, for simplicity, each variable equally partitioned by symmetric membership functions of triangular type. The use of same number of fuzzy sets in the input variables is only a simplifying option in this algorithm. This choice makes faster the trial-and-error process to attribute the best number of fuzzy sets to the fuzzy model and minimizing the quadratic error mean. The learning algorithm uses the t-conorm max to select the degree to which two fuzzy sets match, and extracts the conclusion part of each rule as a real number (fuzzy singleton). The antecedent fuzzy sets are combined by the algebraic product operator and the inference operator product is used. To extract the linguistic relations modeling the dynamic system, we proceed as follows. For each pattern value and coming from the system sensors, we calculate their membership grades in each attributed fuzzy set. Therefore, a vector with grades corresponding to the number of fuzzy sets that divide the universes is attributed to each pattern. Follow, we look for the highest degree of each vector. The corresponding is selected as the linguistic description of the fuzzy set or . This process groups the corresponding pattern value pattern values with the same antecedent fuzzy sets, extracting each rule that composes the fuzzy model. The fuzzy implication degree (8) is now calculated using the ) of the product operation rule for each pattern pair ( extracted rule. That pair with the highest implication degree is of chosen to define the pre-defuzzified output value for the rule being extracted, attributing its numerical value to . 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 the data patterns. The inference method applied to the fuzzy rule base uses the centroid-defuzzification formula and combines all rule contributions in a weighted form given by

(12)

(11) Any p-type production codifies the evolution of the output ) and on variable depending on its previous values ( the value of the input variable ( ). As presented in Table II, a correspondence is assumed between the p-type productions and representing the system a set of functional relationships dynamics. These functional relationships become then representative of the dynamical system behavior.

In (12), is the inferred output value from the fuzzy model, is the extracted response of the respective rule ( ), is the rule firing degree, and is the total number of rules composing the model. 2) Formal Language: To get a data sample of the language generated by a dynamical system, an input signal is imposed so that the output variable should assume values in the operating region of the system. This input/output signal evolution is

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This production, however, is in conflict with the first production obtained . In this case, a 1-type production is established. Both 0-type productions and are then eliminated. Similarly, the fourth symbol yields a 1-type . The productions obtained from the analysis production , and of the hypothetical data sample will be: . Obviously, in order to infer a suitable grammar, a larger data sample must be used. Note that, unlike the IF-THEN rules in fuzzy logic, the structure of the formal language productions is not established in advance. According to the samples involved, one can get different types of productions (0-type, 1-type, ) in the resulting grammar. This feature will permit modeling the different behaviors detected in a dynamical system, constituting the pattern recognition process of the formal language methodology. IV. THE EXPERIMENTAL DRIVES The majority of drive systems in the industry are today either electrical and electrohydraulic drive systems. Therefore, instead use only one type of drive, two experimental prototypes in our laboratory were used to verify the application of both linguistic methods to recognize drives behavior. The first prototype was an electrohydraulic actuator and it was used with fuzzy logic. The second one was an electronic-fed squirrel-cage induction machine system and it is used regarding the formal language theory. A. Electrohydraulic Drive System (Fuzzy Logic)

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

then quantified as described in Section III. After analyzing these sample words, the learning algorithm establishes the grammar productions. The basic learning algorithm, represented by a flowchart in Fig. 4, has a simple paradigm with two principles in order to establish the set of productions. They are: • A 0-type production is taken into consideration for every symbol of the alphabet that occurs in the sample. -type production is taken into consideration if • A the established n-type production already exists. As an example, consider the following words obtained from a hypothetical data sample, whose letters reflect the quantification of the input/output variables: Input variable: Output variable: First, the learning algorithm analyzes the leading symbol of ,” both words. The symbol “ ,” within the terminal word “ denotes the initial value of the output variable. Since no other is first information is available, a 0-type production assumed. After analyzing the second input symbol (“ ”), the . The algorithm establishes another 0-type production . third symbol (“ ”) yields a third 0-type production

The electrohydraulic system can be divided in three subsystems, 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 a permanent-magnet motor (P.M. Motor) coupled with a hydraulic pump; 2) the second subsystem in Fig. 5 is the hydraulic actuator shown in Fig. 6(b). It is connected with the first subsystem by the link between the motor and the hydraulic pump; 3) the last subsystem is a coarse position control implemented by an analogic proportional controller. A detailed description of the electrohydraulic system components is given as follows: 1) a power inverter with IGBTs and current control constituting the electrical drive system; 2) the synchronous P.M. Motor is commanded by the electrical drive system and has the following nominal parameters: 220 V, 1.2 Nm, and 3000 rpm; 3) the speed control system of the motor is composed by a proportional-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 signals: piston position ( ), piston speed ( ), pressure difference in the piston by two pressure sensors named and , and the measure of the P.M. machine speed .

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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 power inverter. These systems are modeled using elements of the electromechanical power conversion theory [10]. With some simplifying assumptions, a 12-equation state model usually represents the electrical drive system. This large number of variables, however, obstructs any attempt to perform an on-line learning process with associated huge computational costs. Therefore, there is a need to further simplify the electrical drive model. Considering a previous current control-loop [22] for command purposes, it is possible to assume that the electrical drive stator currents are controlled. The drive mathematical model can

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

(13)

(14)

MARTINS et al.: NEW TRENDS IN RECOGNIZING EXPERIMENTAL DRIVES

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Fig. 7. Electrical drive system.

In this model, we assume that components of the rotor fluxes ) and the rotor speed ( ) are state variables, and ( ) are input variables. the stator current components ( denotes the rotor time constant, is the mutual The term (stator–rotor) induction coefficient, is the inertia coefficient, is the rotor self-inductance coefficient, is the friction is the load torque. coefficient, and current is imIn our experimental results, a constant input posed in order to obtain a suitable magnetization level within the induction motor. However, fundamental problems such as accurate measurement of some system parameters, the self-inductances and the internal fluxes, for instance, are present. A detailed description of the electrical drive system components is given as follows: 1) a current controlled three-phase PWM inverter IGBTs is used as power inverter; 2) the power inverter dc-voltage can assume up to 400 V; 3) the squirrel-cage induction machine presents the folW (electrical lowing nominal characteristics: V (input voltage), A (input power), (power factor), and current), rpm/min (mechanical speed). V. ACQUISITION OF THE TRAINING DATA To obtain relevant training data, two types of information, qualitative and quantitative, can be acquired regarding the systems behavior. In drive systems, qualitative information is obtained from their mathematical models. This information is present in the mathematical relations between system variables, which permits choosing first the most relevant variables to the description of system’s behavior. In order to obtain quantitative information, one can use a previous model-based simulation study or, if possible, experimental data can be acquired from the system. To obtain this data, an excitation signal must be chosen. A typical approach is to use a pseudorandom binary signal (PRBS) that is injected in the dynamical system. However, this signal is not the best choice to

excite drive systems since it will be filtered by their mechanical time-constant [11]. A better excitation signal is thus the use of a sinusoidal signal composed by different magnitudes and frequencies. In this way, the magnitude and frequency values can be set within the limits of drive’s response, thus avoiding the filtering problem, and allowing the collected data to be better distributed in the normal system’s operating domain. Next, the problem of how to obtain a relevant training set is presented to the two drive systems. A. Electrohydraulic Drive System If the actuator is interpreted as a black-box and some hypotheses 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 ( at the piston can be neglected be( ). The pressure signal cause the load is an inertial one and so the pressure information becomes not important, as demonstrated in [12]. The final functional relation of the electrohydraulic drive system is presented in (15) with the system variables that better characterize its behavior (15) To obtain a relevant training set, an experimental essay was designed. We first considered obtaining the training set with the 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. 8 by its position signal. The piston moves more in one direction than in the other. Therefore, after some sinusoidal periods, the piston at the end of its course halts in 0.20 m. To surpass this, a closed-loop composed by a proportional controller was used to have a coarse control of the piston position and thus eliminate its asymmetric behavior. In this way, we intend the acquired data could cover a representative part of the typical functioning domain of the electrohydraulic system. An essay was

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

made consisting in the application of a sinusoidal reference position signal formed by different combinations of amplitude and frequency values. Since the piston position has a maximum displacement of 0.20 m, four equally spaced amplitude values were chosen. They were 0.05, 0.10, 0.15, and 0.20 m. Another experimental test determined the cutoff frequency of the electrohydraulic system in about 1 Hz. For higher frequencies, the evolution of the piston position starts to present significant attenuation. Therefore, a set of six frequency values was set for our essay in 0.2, 0.4, 0.6, 0.8, and 1.0 Hz. From (15), the training set of the electrohydraulic system , , , and . The four signals obmust include the signals tained from the experimental system by the previous essay are shown in Fig. 9. B. Electrical Drive System As aforementioned, to model the electrical drive system one must consider the measurement of the motor internal magnetic fluxes, which is a difficult task. To overcome this drawback, a dynamic input–output modeling approach is considered instead of a static model approach. This prevents the measurements of the internal fluxes and still achieves an accurate representation of the electrical drive’s behavior. In this way, the considered electrical drive system can be described as a functional relationship between the input and output variables defined by

A grammar of the drive language can be inferred using only available input–output system’s experimental information. This grammar will consider a nonterminal alphabet established from the quantification of the input current ( ) and a terminal alphabet established from the output speed signal ( ). The incodifies ferred productions are of the general form (9), where . In the present paper, the input variable or its evolution we simplify this codification by considering only the input variable, thus simplifying the nonterminal alphabet. This alphabet will be established only from the codification of the input variable rather from the codification of its evolution. To obtain a representative training set from this system, a sinusoidal reference signal with a combination of amplitudes and frequencies was also imposed on the input system variable ), as was made for the electrohydraulic system. Fig. 10 ( shows the evolution of the training and test sets, with the test data being displayed in bold. C. Discussion The acquisition of a training data set is an essential issue in order to obtain a good knowledge of the system to be recognized. Since the PRBS technique is not appropriate in drive systems, the training set must cover a representative part of the system’s working domain. The basic idea is not to learn the extended behavior of the drives but to learn its behavior in the areas where it usually works. If the drive changes its course of action, the learning algorithms should adapt their training data sets in order to acquire this new information. The choice of variables is also important. This choice is usually supported by the knowledge of the theoretical model of the drives. A wrong choice will compromise the learning and recognition processes, since the existence of a functional relationship representative of the drive behavior could not be assured [12], [20]. In the electrohydraulic actuator, all the fundamental state variables are accessible. Therefore, a static function relationship is assumed and learned by the fuzzy-logic algorithm. However, in the electrical drive, not all of the major state variables are accessible. In this case, the extracted functional relationship considers the evolution through time of only the input–output variables. This situation is, however, suitable for the formal language algorithm since the general form of the productions (which enables the establishment of various types of productions) takes past information into account.

(16) VI. RECOGNITION RESULTS where output variable; initial value; previous evolution; evolution depth; evolution of the input variable; functional relationship considering as the initial condition of the system. Formal language techniques can be established to analyze the input/output information that contains valuable information about the system behavior, and also to identify the referred . functional relationship

A. Electrohydraulic Drive System The results presented in this section show the recognition ability of the extracted fuzzy model. Using the training set that was shown in Fig. 9, the fuzzy-learning algorithm is applied to these data. A number of 11 fuzzy sets were attributed to each antecedent variable, and 13 fuzzy sets were attributed to the consequent variable. In this paper, the fuzzy sets number was attributed on a trial-and-error basis, minimizing the quadratic error mean. Fig. 11 shows the error signal between the fuzzy model prediction and the measured piston position. In spite of the good approximation, with errors between 5%, Fig. 11

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

B. Electrical Drive System

Fig. 10. sets.

Electrical drive system training (solid Line) and test data (bold line)

Fig. 11.

Fuzzy logic recognition results.

shows oscillations in the error signal. These may have originated from the information supplied in the learning process to be incomplete, from noise interference, or from the fact that the learning data do not cover a significant part of the system’s working domain.

In order to verify the validity of the recognition process using the formal language in the electrical drive system, two alphabets were considered. In the first one, a quantification interval of 0.01 pu units is assumed for both input and output variables, which yields a 60-symbol alphabet. The grammar inferred from the training data contains 183 productions distributed in the following way: five 0-type productions, 156 1-type productions, and 22 2-type productions. Applying this grammar with 60 symbols to the test set, we get the recognition results shown in Fig. 12. These can be considered satisfactory. However, we must point out that there are some situations where the grammar does not provide any answer, as indicated by the white arrows in Fig. 12. This happens when any production representing that particular dynamical input/output symbol relationship has not been inferred. In Section VIII, a method for establishing the inexistent productions is proposed. To avoid the influence of noise over the inferred productions, the following procedure was observed. A production is not considered as a valid one if it is inferred only once. Only when is repeatedly inferred from the data sample it is considered. This procedure works as a filter over possible noise perturbations. In the second alphabet considered, a quantification interval ten times higher (0.1 pu units) is assumed for both input and output variables, yielding now a 6-symbol alphabet, instead of the previous 60 symbols. The grammar inferred from data contains 36 productions distributed in the following way: zero 0-type productions, four 1-type productions, and 32 2-type productions. Applying this grammar to the test data, we get the recognition results presented in Fig. 13. The white arrows denote, as before, the absence of productions that can be applied. Clearly, the quantitative modeling performance deteriorates since a much smaller alphabet was considered. However, the increased error does not imply any fail in a qualitative recognition process. This increased error it is only due to the larger quantification interval imposed by the limited alphabet considered.

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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 drive evolution, besides the use of a much smaller number of symbols and productions. The limited alphabet considered is not a drawback in the qualitative recognition process. As in the previous grammar with 60 symbols, the inexistent productions problem is also present. Both grammars recognize the drive-speed evolution. However, the second one is much simpler since it presents smaller alphabets and a smaller set of productions. If the main feature is a quantitative recognition, only the first grammar presents an acceptable performance since its quantification interval is of the same order as the sensors numerical precision. C. Discussion Both methodologies present good qualitative recognition abilities. 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 recognition abilities. The reason for this goes back to the quantification of the variables. The discrete quantification performed within the formal language algorithm implies a discrete response when recognition of the drive is performed. In order to improve the quantitative recognition abilities of the formal language algorithm, two solutions can be adopted: increase the number of symbols in the alphabet (the quantification becomes more accurate), or to perform a numerical interpolation of the productions. The first solution increases the number of productions, slowing the recognition process. The second solution,

however, does not increase the number of productions and could be applied when a qualitative recognition was required, as will be shown in Section VIII. Comparing fuzzy logic and formal language structures, quantification in the first linguistic approach is made by the fuzzy sets, whereas in the formal language, quantification is made by the terminal and nonterminal alphabets. These alphabets are attributed by specifying a certain number of symbols, with the same occurring in fuzzy logic when a certain number of fuzzy sets is attributed. Clustering techniques such as Fuzzy -means could be used to determine the fuzzy sets in the antecedents. In formal language, however, no any data preprocessing is used to determine its alphabets. Therefore, to obtain a better-quality comparison between both methods, no preprocessing technique was used with the fuzzy logic method either. VII. LACK OF INFORMATION Lack of information is a problem closely related with the “quality” of the training set. To cover a significant part of the system’s working domain, a large data set is needed. To overcome this problem, the domain can be previously filled using theoretical values from possible system simulations. On the other hand, if an on-line functioning of the system is considered, one can complete the empty domain regions as soon as the system operates in these regions. Since the regions that did not receive experimental data are always considered during the learning phase, this fact provokes large errors in the learning

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Fig. 14.

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Domain filling by the training set of the electrohydraulic system.

algorithms. A good solution for this problem is to implement an interpolation algorithm. Next, the two drive systems with the corresponding learning algorithms are investigated concerning the 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 the electrohydraulic system, the training data only covers part of the system’s domain, as shown in Fig. 14. To better visualize the consequences of the empty domain areas in the learning process, the relation representing the electrohydraulic system is simplified to (17)

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

signals. Fig. 15(a) shows the domain established by and The fuzzy learning algorithm is applied to this data using a par, seven to , and 13 fuzzy sets tition set of nine fuzzy sets to to . The quality of the extracted rules, which are represented in Fig. 15(b), is first verified in reproducing the training data. Fig. 16 shows the error signal between the piston position and the value predicted by the fuzzy model. Note in Fig. 15(a) that there are more data in the inner zone of the domain, while the data are sparser in the borders. This distribution increases the possibility of acquiring empty rules in the model (rules without a conclusion), and also increases the possibility of extracting some rules which conclusion was computed using only a small number of examples. The results in Fig. 16 show a periodic behavior of the large errors. They appear in two situations when the electrohydraulic system operates in the domain borders. They are: when the piston is close to the limits of its displacement zone, and when the piston speed reaches its maximum or minimum values. Since a sinusoidal signal was used, these two situations are periodic and so justify the periodic large errors shown in Fig. 16.

To make the training set sparser and induce the appearing of large gaps in the domain, a second data set was built, which contains 10% examples of the original data. Fig. 17(a) shows the 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. The rules extracted are shown in Fig. 17(c), which for the used partition set they still cover all data, thus avoiding the appearing of empty rules in the fuzzy model. However, as there were no collected data placed in the domain borders, the learning algorithm could not infer any conclusion value from these regions. When inferring in these regions, the model will present large prediction errors. This situation, which is similar to that shown in Fig. 16, is described in Fig. 18. The error oscillations in the figure indicate that the fuzzy model presents large errors always when an inference occurs near the limits of the domain. This situation becomes even more critic because, despite the presence of empty rules, there are few collected examples in the limit regions. Therefore, apart from the low quality of the rules extracted in these regions, the inference mechanism also uses a

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Fig. 16.

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Position error signal from the fuzzy model.

Usually, increasing the number of fuzzy sets attributed to each variable can lead to a better prediction performance of the fuzzy model. However, increasing the partitions can induce more empty rules in the fuzzy model. This, containing more empty rules, will cause the inference mechanism to give position values that show a large deviation from the correct ones. To investigate these effects, the fuzzy partitions of the model were 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 of the training set, this leads to the results shown in Fig. 20. When the inference process fails completely and a null value is inferred from the fuzzy model, we decided to consider the last nonzero inferred value as the valid model response. To analyze the large modeling errors, the time interval indicated in Fig. 20 is expanded. The interval is shown in Fig. 21(a) indicating two critical instances. The first instance occurs when the inference mechanism reaches the empty regions of the rule-base (domain borders), corresponding in Fig. 21(b) to the largest errors with values over 40%, and characterizing a significant deviation of the inferred position from the measured one. Note that, in this critical instance, the domain borders that were reached were those of the piston speed limit values. Therefore, large errors occur in the high frequency values of the reference signal. The second critical instance occurs when the reference signal gets values near the piston limits. In this case, as can be verified in Fig. 19, the inference process uses only a small number of rules and the inferred position values begin to diverge from the measured ones. The inference, for instance, using just one rule, makes constant the fuzzy model output, as shown in the second critical situation in Fig. 21(a) by the “flat” predictions. B. Electrical Drive System

Fig. 17.

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

small number of valid rules (nonempty) to infer the piston position.

The problem of lack of information is similar in both methodologies. Therefore, this problem in the formal language will be presented without a wide range of situations. Technically, the recognition procedure in the formal language fails when it is not possible to find a production that is suitable for a symbol in a test word. In this case, the word is considered not belonging to the language described by the inferred grammar. Figs. 12 and 13 show when the recognition process fails. This happens when the number of available patterns is too small for a suitable grammatical inference.

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

ously filled with other data, if possible. For instance, data from simulation, linguistic rules in the case of fuzzy logic, or productions in the formal language. VIII. INTERPOLATION

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

Let us consider the case of an alphabet with 60 symbols and a reduced training set that is half of the one shown in Fig. 10. After inferring the correspondent grammar, the recognition results obtained are presented in Fig. 22. The measured drive speed is presented as a dotted line and the grammar response as a continuous one. It is easily observed that the recognition process presents considerable faults. Since a smaller training set was considered, the number of valid productions in the inferred grammar was considerably smaller, similar to the empty rules observed in the fuzzy model. C. Discussion The lack of information is a very important issue when algorithms that learn from examples are applied, since “one cannot learn what one cannot see.” In order to get a good performance in the recognition process of the drive behavior, a sufficiently and representative training set must be used. When the data does not cover important areas of the system’s working domain, the recognition results deteriorate. In the fuzzy logic algorithm, this deterioration of the performance is due to the use of a smaller number of rules, when no other information is available. In this case, the response of the algorithm diverges from the experimental values and the recognition process fails. The formal language algorithm does not give any response when no other production was established by the learning process, causing the recognition process also to fail. From the previous statements, one concludes that to attenuate the problem of lack of information, the domain could be previ-

Both results showed that the training data must cover in the best possible way the functioning domain to avoid large gaps in it. These gaps can be filled through some interpolation mechanism, can be filled using some theoretical values obtained from simulation tests or, when operating in an on-line mode, the gaps can be completed using new data acquired during the system’s operation. This section describes how a simple interpolation algorithm can be used to fill the possible gaps that can appear in the fuzzy model (empty rules) and in the grammar (null productions). A. Electrohydraulic Drive System One simple solution to complete the empty fuzzy rules is the application of some interpolation mechanism in the rule base. The interpolation mechanism used was introduced in [7]. It replaces the null values established as conclusion of the empty rules by the conclusion inferred from the rules initially extracted and located around the gaps. After filling the empty rule, the inference process is repeated to generate the final inferred value. To exemplify the interpolation process, consider the simple fuzzy model represented in Fig. 23(a). This is characterized by and that are divided by the four two antecedent variables and , membership functions respectively. It was assumed in the example that six valid rules were obtained after applying the learning algorithm to the training set. These rules are denoted in the Fig. 23(a) by , and , with the white square regions representing the empty rules as before. Suppose that during a certain inference step, the fuzzy sets marked with a circle in the figure were activated. From the four activated rules, only those , and are valid for the inference with conclusions has a null value since process. The rule with conclusion it could not be extracted from the training data. Necessarily, the inferred model response will become incorrect because the inference used only three valid rules instead of the four , the ones. To estimate a valid value for the rule conclusion interpolation algorithm will use the value inferred from the

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Fig. 20.

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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 replacing , the inference process the null value with the inferred one is repeated again, but now using the previous extracted three rules plus the rule which conclusion was estimated. In this example, the interpolation mechanism applied to compresented three already extracted neighplete the empty rule borhood rules. However, if these rules were in a smaller number, the interpolation would have little information to estimate a more correct conclusion. In that case, it becomes necessary to acquire more data placed in these domain regions to extract a higher number of neighborhood rules; even to better define the rules already extracted. In this way, the inferred conclusion will be approximated further to its “correct” value.

Fig. 23(b) shows the results when the interpolation algorithm is applied to complete the rules in Fig. 19. Testing again the completed fuzzy model in the training set, Fig. 24 shows the new error signal after the interpolation. Note that, when comparing with the anterior results in Fig. 21, the interpolation removed the large error values in some regions. However, some errors remained high despite the interpolation. These regions correspond to regions in the domain zones with few and very sparse examples. Therefore, since only a small quantity of examples was applied to compute the rule conclusions located in those regions, the interpolation results based on these rules will continue to remain far from the correct values, maintaining a large prediction error signal, as shown in Fig. 24.

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

B. Electrical Drive System The previous section showed that a small training set could lead to a faulty grammar, which fails to recognize some words. However, those words may still be in the language, within a certain distance from the words belonging to the data sample initially used to infer the grammar. In this case, it may be useful to establish a new production out from the ones inferred from the smaller sample. This feature is called grammatical interpolation. The grammatical interpolation is applied to establish a new production considering a structural matching procedure. The main idea of a structural matching is based on a formal measure of the similarity between the unknown input pattern and the available data structures. This measure of similarity consists in the distance between the inexistent production and the nearest similar productions. Several methods for structural matching have been reported in [21]. The basic algorithm states that the distance between two words is related with the sequence of edit operations (substitution, insertion, and deletion) required to transform one word into another. For any sequences of edit operations, a cost function defined by

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

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

quantification

(20)

(18) can be considered. It denotes the cost of a particular word made denotes the cost of a particular edit by sequence , and operation. The distance between two words is defined as the minimum cost necessary to transform a word into another and given by is a sequence of edit operations which transforms

into

(19)

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

and is based on the average distance between similar productions. Assuming the general form of a production introduced in (9), the coefficients denote the distance between produc,” and tions and , considering their respective words “ 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. The erroneous influence of the inexistent productions is reduced, and the number of symbols that were not recognized in the test words is substantially smaller. In Fig. 25, the grey arrows denote the fulfillment of inexistent productions, while the white arrows denote the absence of applicable productions that could not fulfilled by the grammatical interpolation procedure.

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Fig. 24.

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Error position signal after applying the interpolation mechanism.

Fig. 25. Formal language recognition results after the grammatical interpolation procedure.

Fig. 26. system.

C. Discussion

distinct from those observed during the training phase. In order to verify the generalization ability of the algorithms, a contracted training set was considered. This new set does not cover the entire working domain but only part of it. The further away from areas with data the algorithms are able of recognizing the systems, the better their generalization abilities are conceived.

From the previous analysis, it is clear that the use of an interpolation mechanism improves the performance of the algorithms. Both base their interpolation mechanism in the knowledge of the information gathered around the empty space. The fuzzy logic approach infers a value for the empty rules using the values extracted for the nearby rules. Similarly, the proposed grammatical interpolation algorithm, in the presence of an inex, as defined istent production, considers the sequence in (9) as a word, and completes the production according to the linguistic distance from the nearest productions. Similarly to the fuzzy logic procedure, the establishment of this missing production is weighted according to the distance from those already inferred productions located around. The use of the previous interpolation mechanisms only presents good results when filling small areas of unextracted information. Because they base their procedure on the information around those empty areas, if a large empty area is to be fulfilled, the errors will be huge. IX. GENERALIZATION By generalization, we mean the ability of the algorithms in well recognizing the systems even under working conditions

Representation of the restricted training set from the electrohydraulic

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 restricted region of the functioning domain, as shown in Fig. 26. To verify the generalization ability of the fuzzy model, a test set was used with 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 position error signal. The results show that, when the electrohydraulic system operates inside the domain regions covered by the training set, the model has low error values, as seen in Fig. 27. On the other hand, when the system operates out of the data area, the fuzzy model has to extrapolate as far as the rules extracted in the borders of the training region allow it to. Fig. 27 shows that the error values increase as the system operates out of the data set area, resulting only in a limited generalization.

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Fig. 27.

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Generalization results using the fuzzy logic methodology.

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

B. Electrical Drive System The generalization ability of the grammatical model for the electrical drive also decreases if the training data only includes some part of the working domain, as shown in Fig. 28. In this case, the training set only considers positive speed values. Otherwise, the considered test data set, displayed in bold, assumes positive and negative drive speeds. Fig. 29 shows the modeling results using the restricted data, showing that the recognition fails for the domain areas not covered by the training set. Namely, the symbols that codify negative drive speeds are not recognized. As for the fuzzy modeling of the electrohydraulic system, the results show the local characteristic of the generalization ability for this modeling situation. C. Discussion The results presented in this section show that both methodologies give a good approximation of respective drive system, even at points not contained in the training set. Therefore, the two learning techniques have generalization ability. However, it must be noted that generalization is a complex phenomenon and that there is no global requirement for a successful generalization. In the two algorithms, generalization has a local effect. Therefore, this demands that the experimental training data must cover 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 domain is filled by the training data, the generalization process fails. X. CONCLUSION This paper has investigated the recognition of drive system’s behavior using fuzzy logic and formal language theories. This work has been devoted to an electrohydraulic drive and an induction motor drive modeling, but can be applied to other drive systems presenting similar features. Since both methods are linguistic based, several similarities were pointed out. We emphasized the quantification by dividing the feature space into clusters, the learning procedures, and emphasized the acquisition of the training patterns. On the other hand, some aspects enhanced the particularities between both approaches: 1) Unlike the fuzzy approach, the creation of a formal language alphabet is made in a crispy way without the use of fuzzy sets. The linguistic relations within the fuzzy logic approach are established regarding the fuzzy grade of each considered variable. In the formal language approach, the structure of the productions is not set up in advance. Instead, different types of productions are established according to the incoming words from the linguist source (dynamic system).

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2) Both methodologies presented good qualitative recognition results. Due to the characteristics of the considered formal language’s alphabets, the fuzzy approach is naturally more suitable for a quantitative recognition procedure. Solutions for this formal language drawback were presented and discussed. 3) We also showed that the lack of information could deteriorate the recognition performance of both algorithms. In order to minimize this deterioration, interpolation mechanisms were proposed. The fuzzy logic approach fills the empty rules weighting the values of the nearby already extracted rules. Considering a slightly different approach within the formal language, a linguistic interpolation algorithm based in the concept of distance between words, was proposed to establish inexistent productions. 4) The use of the previous interpolation strategies improved the generalization abilities of both procedures. However, for areas far away from the training set coverage of the working domain, results showed that the generalization deteriorates. Generalization has a local characteristic. This paper represents the basis for our present research. New applications regarding fault diagnosis of drive systems are being developed that apply these techniques. For this purpose, optimization techniques, such as immunity-based learning mechanisms, are also being considered.

[15] K. S. Fu and T. L. Booth, “Grammatical inference: Introduction and survey—Part I,” IEEE Trans. Syst., Man, Cybern., vol. 5, Jan. 1975. , “Grammatical inference: Introduction and survey—Part II,” IEEE [16] Trans. Syst., Man, Cybern., vol. 5, July 1975. [17] Y. Sakakibara, “Recent advances in grammatical inference,” Theoretical Comput. Sci., no. 185, pp. 15–45, 1997. [18] R. C. Gonzalez and M. G. Thomason, Syntactic Pattern Recognition—An Introduction. Reading, MA: Addison-Wesley, 1978. [19] A. Saloma, Computation and Automata. Cambridge, U.K.: Cambridge Univ. Press, 1985. [20] J. F. Martins, A. J. Pires, and J. A. Dente, “A choice of variables in automatic modeling of AC-drive systems,” in Proc. PEMC’98, Prague, Czech Republic, Sept. 1998, pp. 5.7–5.12. [21] H. Bunke, “String matching for structural pattern recognition,” in Syntactic and Structural Pattern Recognition—Theory and Applications, H. Bunke and A. Sanfelui, Eds. New York: World Scientific, 1990, (Ser. Comput. Sci.). [22] J. F. Martins, A. J. Pires, and J. F. Silva, “A novel and simple current controller for three-phase PWM power inverters,” IEEE Trans. Indust. Electron., vol. 45, pp. 802–805, Oct. 1998. [23] J. F. Martins, A. J. Pires, and J. A. Dente, “Automatic input/output modeling of a squirrel-cage induction motor drive system using neural network,” in Proc. EPE’97, Trondheim, Norway, Sept. 1997, pp. 4.632–4.637. [24] A. P. Sage and J. L. Melsa, System Identification. New York: Academic, 1971. [25] R. Dhaouadi, S. Bolognani, and M. Zigliotto, “Effective estimation of speed and rotor position of a PM synchronous motor drive by a Kalman filtering technique,” in Proc. IEEE Power Electronic Specialists Conf. (PESC’92), Toledo, Spain, 1992, pp. 951–957. [26] M. Bodson, J. Chiasson, and R. T. Novotnak, “Nonlinear speed observer for high-performance induction motor control,” IEEE Trans. Indust. Electron., vol. 42, Aug. 1995. [27] L. Kanal, “Patterns in pattern recognition: 1968–1974,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 697–722, Nov. 1974.

REFERENCES [1] J. C. Bezdek and S. K. Pal, Fuzzy Models for Pattern Recognition. Piscataway, NJ: IEEE Press, 1992. [2] L. X. Wang and J. Mendel, “Generating fuzzy rules by learning from examples,” IEEE Trans. Syst., Man, Cybern., vol. 22, pp. 1414–1427, July 1992. [3] S. G. Romaniuk and L. O. Hall, “Learning fuzzy control rules from examples,” in Fuzzy Control Systems, A. Kandel and G. Langholz, Eds. Boca Raton, FL: CRC, 1994. [4] P. J. Costa Branco and J. A. Dente, “The application of fuzzy logic in automatic modeling electromechanical systems,” Fuzzy Sets Syst., vol. 95, no. 3, pp. 273–293, 1998. [5] W. Pedrycz, “Fuzzy sets in pattern recognition: Accomplishments and challenges,” Fuzzy Sets Syst., vol. 90, pp. 171–176, 1997. [6] N. R. Pal, “Soft computing for feature analysis,” Fuzzy Sets Syst., vol. 103, pp. 201–222, 1999. [7] P. J. Costa Branco and J. A. Dente, “An experiment in automatic modeling an electrical drive system using fuzzy logic,” IEEE Trans. Syst., Man, Cybern. C, vol. 28, pp. 254–262, May 1998. [8] L. X. Wang and J. Mendel, “Generating fuzzy rules by learning from examples,” IEEE Trans. Syst., Man, Cybern., vol. 22, pp. 1414–1427, July 1992. [9] P. J. Costa Branco, J. Martins, A. Pires, and J. A. Dente, “Recognizing patterns in electromechanical systems,” Pattern Recogn. Lett. (PRL)—Special Issue “Pattern Recognition in Practice”, vol. 18, pp. 1335–1346, 1998. [10] A. E. Fitzgerald, C. Kingsley, and S. D. Umans, Electric Machinery, 4th ed. New York: McGraw-Hill, 1985. [11] T-S. Low, T. H. Lee, and H. K. Lim, “A methodology for neural network training from control of drives with nonlinearities,” IEEE Trans. Indust. Electron., vol. 40, pp. 243–249, Apr. 1993. [12] P. J. Costa Branco, N. Lori, and J. A. Dente, “New approaches on structure identification of fuzzy models: Case study in an electromechanical system,” in Fuzzy Logic, Neural Networks, and Evolutionary Computation, T. Furuhashi and Y. Uchikawa, Eds. Berlin, Germany: SpringerVerlag, 1996, LNCS/Lecture Notes Artificial Intell., pp. 104–143. [13] N. Chomsky, Aspects of the Theory of Syntax. Cambridge, MA: MIT Press, 1965. [14] E. Mark Gold, “Language identification in the limit,” Inform. Contr., vol. 10, no. 5, pp. 447–474, May 1967.

J. F. Martins (M’96) graduated in electrical engineering at Instituto Superior Técnico (IST), Technical University of Lisbon, in 1990. He obtained the M.Sc. degree in electrical engineering at the same institute in 1996. He is finishing his Ph.D. dissertation in electrical engineering, in the application of grammatical inference learning algorithms and cellular automata within drive systems, at the IST. He is currently Adjoint Professor in the Department of Electrical Engineering at Escola Superior de Tecnologia/Instituto Politécnico de Setúbal. He is also with the Mechatronics Laboratory. His research areas are in control of electrical drives, advanced learning control techniques for electromechanical systems and nonlinear systems. He has published articles in international scientific journals such as the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS and PATTERN RECOGNITION LETTERS.

P. J. Costa Branco (M’92) is currently an Assistant Professor in the Department of Electrical and Computing Engineering, Section of Electrical Machines and Power Electronics, in the Instituto Superior Técnico (IST), Lisbon, Portugal, and has been with the Mechatronics Laboratory/IST since 1992. His research areas are in control of electrical drives, systems modeling and control using soft computing techniques, and he is presently engaged in research on advanced learning control techniques for electromechanical systems. Dr. Costa Branco is the author of published articles in international scientific journals such as the IEEE TRANSACTIONS ON MAGNETICS, IEEE TRANSACTIONS ON SYSTEM, MAN, AND CYBERNETICS, PATTERN RECOGNITION LETTERS, FUZZY SETS AND SYSTEMS, and EUROPEAN TRANSACTIONS ON ELECTRICAL POWER ENGINEERING. He has been a Referee of international scientific journals, participated in boards of international meetings, and cited in the Who’s Who in the World.

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A. J. Pires graduated in electrical engineering at Instituto Superior Técnico, Technical University of Lisbon (TUL), in 1985. He obtained the M.Sc. degree in 1988 and the Ph.D. degree in 1994 in electrical engineering at the same Institute. He is currently Coordinator Professor in the area of Electrical Engineering at Escola Superior de Tecnologia, Polytechnic Institute of Setúbal and Invited Associated Professor at the Physics Department, University of Evora. He is also with the Mechatronics Laboratory at the TUL. His research areas are in electrical machines, power electronics, and intelligent control systems for electrical drives. Dr. Pires has authored papers in international journals such as the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS and PATTERN RECOGNITION LETTERS.

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J. A. Dente graduated in electrical engineering at Instituto Superior Técnico (I.S.T.), Technical University of Lisbon, in 1974/1975. He received the Ph.D. degree in electrical engineering at the same institute in 1986, and was Associated Professor among 1989 and 1993. He is currently Full Professor in the area of Electrical Machines at Department of Electrical and Computing Engineering, Section of Electrical Machines and Power Electronics at I.S.T. He has been with the Mechatronics Laboratory as the Scientific Coordinator since 1993. He has published more than 30 scientific articles in refereed journals and books, and more than 40 articles in refereed conference proceedings. His primary areas of interest are in electrical machines, motion control, and presently is engaged in research on advanced learning control techniques for electromechanical systems.