39 CHAPTER 3 FUZZY RULE BASED MODEL FOR FAULT DIAGNOSIS 3.1 INTRODUCTION Development of mathematical models is essential for many disciplines of engineering and science. Mathematical models are used for different purposes such as simulations, design and analysis of systems, and also for process monitoring and supervision. The traditional mechanistic approach to modeling is based on a thorough understanding of the nature and behavior of the actual system, and on a suitable mathematical treatment that leads to the development of a model. For incompletely understood processes, however, this approach may become laborious and inefficient. Large amount of process knowledge is qualitative and imprecise and as such cannot be readily transformed into traditional mathematical models based on differential and algebraic equations. Many physical systems are not amenable to conventional modeling approaches due to lack of precise, formal knowledge about the system, due to strong nonlinear behavior, high degree of uncertainty and due to the time varying characteristics of the system. Fuzzy systems along with other related techniques such as neural networks have been recognized as powerful tools which can facilitate the effective development of models in such cases. Modeling and control of dynamics systems belong to the field in which fuzzy
40 logic techniques have received considerable attention, not only from the scientific community but also from industry. This chapter explains the details of the FL based diagnostic system developed for pneumatic valve. It contains the procedural steps to be followed to design a FL based model for fault diagnosis. In addition to development of fuzzy logic based model the tuning of parameters of the fuzzy system are also presented here. Detailed discussions on simulation results are also presented here. 3.2 SYSTEM MODELING THROUGH FUZZY LOGIC Fuzzy logic is an extension of Boolean logic where members of the set can have varying degrees of membership. It is an approach to handle vagueness or uncertainty and, in particular, linguistic variables. It is a multivalued type of logic that allows intermediate values to be defined between conventional threshold values. Fuzzy logic was first developed by Zadeh in the mid 1960 s to provide a mathematical basis for human reasoning. Fuzzy logic uses fuzzy set theory, in which a variable is a member of one or more sets, with a specified degree of membership. Fuzzy logic when applied with the aid of computers, allows them to emulate the human reasoning process, quantify imprecise information, make decisions based on vague and incomplete data, yet by applying a defuzzification process, arrive at definite conclusions. A fuzzy logic system is unique in that it is able to simultaneously handle numerical and linguistic knowledge. A large number of applications of fuzzy logic have been performed during the 20 th century. They include control applications, medicine, manufacturing, metrology, scheduling and optimization and signal analysis for training and interpretation. The concept of fuzzy-set theory and fuzzy logic can be employed in the modeling of systems in a number of ways. Examples of fuzzy systems are
41 rule-based fuzzy systems, fuzzy linear regression model, and fuzzy models using cell structures. This report focuses on rule based fuzzy systems. Fuzzy systems are being applied successfully in an increasing number of application areas; they use linguistic rules to describe systems. These rule based systems are more suitable for complex system problems where it is very difficult, if not impossible, to describe the system mathematically. A static or dynamic system which makes use of fuzzy sets or fuzzy logic and of the corresponding mathematical framework is called as fuzzy system (Babuska and Verbruggen 1996). There are a number of ways fuzzy sets can be involved in a system, such as: In the description of the system: A system can be defined, for instance, as a collection of if-then rules with fuzzy predicates, or as a fuzzy relation. In the specification of the system s parameters: The system can be defined by algebraic or differential equations, in which the parameters are fuzzy numbers instead of real numbers. The input, output and state variables of a system may be fuzzy sets: Fuzzy inputs can be readings from unreliable sensors (noisy data), or quantities related to human perception, such as comfort, beauty, etc. Fuzzy systems can process such information, which is not the case with conventional (crisp) systems. The practical relevance of fuzzy modeling are given below: Incomplete or vague knowledge about systems: Conventional system theory relies on crisp mathematical models of systems, such as algebraic and differential or difference equations. For some systems, such as electro-mechanical systems, mathematical models can be obtained. This is because the
42 physical laws governing the systems are well understood. For a large number of practical problems, however, the gathering of an acceptable degree of knowledge needed for physical modeling is a difficult, time-consuming and expensive or even impossible task. In the majority of systems, the underlying phenomena are understood only partially and crisp mathematical models cannot be derived or are too complex to be useful. Examples of such systems can be found in the chemical or food industries, biotechnology, ecology, finance, sociology, etc. A significant portion of information about these systems is available as the knowledge of human experts, process operators and designers. This knowledge may be too vague and uncertain to be expressed by mathematical functions. It is, however, often possible to describe the functioning of systems by means of natural language, in the form of if-then rules. Fuzzy rule-based systems can be used as knowledge-based models constructed by using knowledge of experts in the given field of interest. Adequate processing of imprecise information: Precise numerical computation with conventional mathematical models only makes sense when the parameters and input data are accurately known. As this is often not the case, a modeling framework is needed which can adequately process not only the given data, but also the associated uncertainty. The stochastic approach is a traditional way of dealing with uncertainty. However, it has been recognized that not all types of uncertainty can be dealt with within the stochastic framework. Various alternative approaches have been proposed, fuzzy logic and fuzzy set theory being one of them.
43 Transparent (gray-box) modeling and identification: Identification of dynamic systems from input-output measurements is an important topic of scientific research with a wide range of practical applications. Many real-world systems are inherently nonlinear and cannot be represented by linear models used in conventional system identification. Recently, there is a strong focus on the development of methods for the identification of nonlinear systems from measured data. Artificial neural networks and fuzzy models belong to the most popular model structures used. From the input-output view, fuzzy systems are flexible mathematical functions which can approximate other functions or just data (measurements) with a desired accuracy. This property is general function approximation. The most commonly used fuzzy logic systems are: Mamdani fuzzy model: The crisp inputs are fuzzified according to a set of membership functions. The fuzzy AND and OR operators in the if-then rules are inferred using minmax or product max compositions, or other variations. The fuzzy set obtained is defuzzified to a crisp value using a strategy like the centroid or the mean of maximum methods. Sugeno Fuzzy model: It was proposed to develop a systematic approach to generating fuzzy rules from a given input-output data set. A typical fuzzy rule in a sugeno fuzzy model has the form: if x is A and y is B then z=f(x,y), where A and B are fuzzy sets in the antecedent, while z=f(x,y) is a crisp function in the consequent. Usually f(x,y) is polynomial in the input variables x and y, but it can be any function as
44 long as it can appropriately describe the output of the system within the fuzzy region specified by the antecedent of the rule. When f(x, y) is first-order polynomial, the resulting system is called a first-order sugeno fuzzy model. In this case, each rule has a crisp output, the overall output is obtained via weighted average and thus the time consuming procedure of defuzzification is avoided. Tsukamoto Fuzzy model: The consequent of each fuzzy ifthen rule is represented by a fuzzy set with monotonical membership functions. As a result, the inferred output of each rule is defined as a crisp value induced by the rule s firing strength. The overall output is taken as the weighted average at each rule s output. It avoids the time consuming process of defuzzification. 3.3 FUZZY LOGIC SYSTEM Fuzzy Logic System (FLS) is a computing framework based on the concepts of fuzzy set theory, fuzzy if-then rules, and fuzzy reasoning. The basic structure of a fuzzy inference system consists of three conceptual components: a rule base, which contains a selection of fuzzy rules, a database, which defines the membership functions used in the fuzzy rules, and a reasoning mechanism, which performs the inference procedure upon the rules and a given condition to derive a reasonable output or conclusion. Figure 3.1 shows the generic logic fuzzy system. The following subsections present the various components of a fuzzy logic system.
45 Crisp Input (X) Knowledge Base Rule Base Data Base Crisp output(y) Fuzzifier (X) Fuzzy Interface Engine (Y) Defuzzifier Figure 3.1 Generic fuzzy logic system The fuzzy system model consists of four main components namely Fuzzification interface: It can be regarded as the input interface. It has the function of modifying/scaling the inputs so that they may be compared to the rules in the rule-base. The rule-base: Here, the knowledge for the control of the system is held as a set of if-then statements. The inference mechanism: It evaluates which rules are relevant at the current time and then decides what the input to the plant should be. Defuzzification interface: It acts as an output interface. The output from the inference mechanism is converted so that it may be fed into the input for the process. When modeling a process, there will always be some uncertainty with regard to the accuracy of the process data and operating conditions. The fuzzy model provides a means of encompassing the nonlinearities and uncertainties of the process without definitive mathematical statements.
46 3.3.1 Building of Fuzzy Models Fuzzy Logic (Abe and Lan 1995) has been successfully applied in solving classification problems where boundaries between classes are not well defined. Typical fuzzy classifiers Ishibuchi et al (1995), Rajakarunakaran et al (2006) consist of interpretable if-then rules with fuzzy antecedents and class labels in the consequent part. The antecedents (if-parts) of the rules partition the input space into a number of fuzzy regions by fuzzy sets, while the consequents (then-parts) described the output of the classifier in these regions. Formation of fuzzy if-then rules and the membership functions are the important issues in the design of fuzzy classifier system. In general, the rules and membership function are formed from the experience of the human experts. Two common sources of information for building fuzzy models are the priori knowledge and data. The priori knowledge can be of a rather approximate nature (qualitative knowledge, heuristics), which usually originates from experts (Zimmermann 1987). Data are available as records of the process operation or special identification experiments can be designed to obtain the relevant data. With regard to the design of fuzzy models, two basic items are distinguished: the structure and the parameters of the model. The structure determines the flexibility of the model in the approximation of (unknown) mappings. The parameters are then tuned (estimated) to fit the data at hand. To develop a fuzzy model, the following main decisions must be made: Choose the type of fuzzy model (Takagi-Sugeno, linguistic, etc.,) Choose the inference and defuzzification methods and the particular fuzzy logic or set-theoretic operators Develop the knowledge base, i.e., the rules and the membership functions.
47 One of the most important considerations in designing any fuzzy systems is the generation of the fuzzy rules as well as the membership functions for each fuzzy set. In most of the existing applications, the fuzzy rules are generated by experts in the area, especially for control problems with only a few inputs. With an increasing number of variables, the possible number of rules for the system increases exponentionally; this makes it difficult for experts to define a complete set of rules for good system performance. To design a (linguistic) fuzzy model based on available expert knowledge, the following steps can be followed: Select the input and output variables, the structure of the rules, and the inference and defuzzification methods. Decide on the number of linguistic terms for each variable and define the corresponding membership functions. Formulate the available knowledge in terms of fuzzy if-then rules. Validate the model (typically by using data). If the model does not meet the expected performance, iterate on the above design steps. 3.4 ADAPTIVE FUZZY MODEL In the previous section, the details related to fuzzy model development using the expert knowledge is presented. The fuzzy model developed using the expert knowledge alone may not work satisfactorily under all conditions. Tuning of rule base and membership functions is very much essential to make the developed model to be suitable for all conditions. Numerical data collected from the process can be used to tune the parameters of the fuzzy system. In this section, Adaptive Neuro-Fuzzy Inference system (ANFIS) is used to tune the parameters of the fuzzy system. In general, neurofuzzy modelling (Jang 1993) is a branch of system identification and it
48 involves two phases: structure identification and parameter identification. The former is related to finding a suitable number of rules and a proper partition of feature space. The latter is concerned with the adjustment of system parameters, such as the membership functions, linear coefficients and so on. If we do not use any structure identification techniques in fuzzy modeling, we have to accept the simple grid partitioning of the input space. The architecture of the ANFIS, which is used in this work, is shown in Figure 3.2 (a). Layer 1 Layer 4 Layer 2 Layer 3 A 1 x y Layer 5 x A 2 M w 1 N w 1 w 1 f 1 f y B 1 B 2 M w 2 N w 2 x y w 2 f 2 Figure 3.2 (a) ANFIS architecture The following rules were considered for design of ANFIS model for fault diagnosis of pneumatic valve in cooler water spray system in cement industry. Rule 1 : If (x is A 1 ) and (y is B 1 ) then (f 1 = p 1 x + q 1 y + r 1 ) Rule 2 : If (x is A 2 ) and (y is B 2 ) then (f 2 = p 2 x + q 2 y + r 2 ) where x and y are the inputs, A i and B i are the fuzzy sets, f i are the outputs within the fuzzy region specified by the fuzzy rule, p i, q i and r i are the design parameters that are determined during the training process. A two input first order sugeno fuzzy model with two rules is shown in Figure 3.2 (b) and the ANFIS architecture to implement the above two rules is shown in Figure 3.2
49 (a), in which a circle indicates a fixed node, whereas a square indicates an adaptive node. A 1 B 1 w 1 f 1 =p 1 x+q 1 y+r 1 A 2 X B 2 Y f w 1 w1 f1 w f 1 1 w w2 f w 2 f 2 2 2 w 2 f 2 =p 2 x+q 2 y+r 2 x X y Y Figure 3.2 (b) A two input first order sugeno fuzzy model with two rules ANFIS model is developed using Fuzzy C-Means (FCM) clustering. Given separate sets of input and output data ANFIS generates an FIS using FCM by extracting a set of rules that models the data behavior. The rule extraction method first uses the FCM function to determine the number of rules and antecedent membership functions and then uses linear least squares estimation to determine each rule's consequent equations. The details of FCM algorithm are given in the Appendix 8. 3.4.1 Model Validation Model validation is the process by which the input vectors from input/output data sets on which the ANFIS was not trained, are presented to the trained model, to predict the corresponding output values. The statistical method can be used to calculate the Root-Mean Squared error (RMS). RMS error can be calculated as: RMS 1 ( y t ) n 2 m pre, m mea, m n (3.1)
50 where n is the number of data patterns in the independent data set, y pre,m indicates the predicted value, t mea,m is the measured value of one data point m. When the RMS value is small, the performance of ANFIS model is good. 3.5 RESULTS AND DISCUSSION This section presents the details of the simulation carried out on the developed fuzzy rule based model for fault diagnosis of pneumatic valve. Fuzzy Systems (Rajakarunakaran et al 2006) are developed using MATLAB 7.4 fuzzy logic toolbox in Pentium 4 with 2.40GHZ processor with 512 MB of RAM. While developing the fuzzy model minimum value is determined using T-norm, maximum value is determined using T- conorm and Mamdani inference was used. The developed model was tested with the data collected from the pneumatic valve. The data required are collected under normal and faulty conditions of the pneumatic valve used in cement industry. Three frequently occurring faults in the pneumatic valve used in the cement industry are considered here. The data required for the development fuzzy rule based model for fault detection were collected under normal and abnormal operating condition of pneumatic valve from the field experts, the operational log book, operating manual and maintenance record which are maintained by the operators in cement industry. The collected information includes the faultsymptom relationship for pneumatic valve and the ranges of the variables. The objective here is to capture the implicit knowledge behind the diagnosis process, which is embedded in the information collected from the technical experts, through the developed model so that it can be applied for the diagnostic process when the system is in operation. The system contains 2 input variables, which are given in Table 3.1. Flow rate and rod displacement are given as the input variables to develop the fuzzy rule based model and the output is labelled as either normal or as a fault condition.
51 Table 3.1 Input variables with operating range Name of the variable Minimum Value Maximum Value Flow (F) 2mm 3 / sec 15mm 3 / sec Rod Displacement (X) 8mm 80mm The required data were collected from the cement industry (certificate attached at Appendix-1 and data sheet in Appendix 3). From this data, the minimum value of flow rate is 2 mm 3 / sec and the maximum value of Flow rate is 15 mm 3 / sec. Similarly the minimum value of rod displacement is 8 mm and the maximum value of rod displacement is 80 mm respectively. Based on these minimum and maximum values of flow rate and the rod displacement, the fuzzy membership functions were formed. The membership functions were formed based on the guidelines given by the industrial experts. Table 3.2 shows the minimum and maximum value of flow and the rod displacement under faulty condition. Table 3.2 Range of flow and rod displacement for under faulty condition Type of fault Flow (F) Rod displacement (X) Minimum Maximum Minimum Maximum Value Value Value Value F1 2mm 3 / sec 9mm 3 / sec 8mm 40mm F3 2mm 3 / sec 4mm 3 / sec 29mm 33mm F6 7.15mm 3 / sec 15mm 3 / sec 30mm 75mm in Figure 3.3. The membership functions formed for the input variables are shown
52 1.0 Low Medium High (Flow) 0.5 0.0 2 7 15 Flow (mm3/sec) 1.0 Low Medium High (disp) 0.5 0.0 8 10 85 Rod Displacement (mm) Figure 3.3 Membership Functions for input variables in pneumatic valve Membership functions were formed for all the input variables (flow rate, rod displacement) based on their minimum and the maximum values during the normal and abnormal conditions. The combinations of triangular and trapezoidal membership functions were assigned for both the input variables (flow rate, rod displacement) and each variable was categorized into three fuzzy subsets. The expert knowledge relating to the symptoms and the various faults are formulated in the form of fuzzy if-then rules. A set of such rules constitutes the rule base of the FIS. This form of knowledge representation is appropriate because it is very close to the way the experts themselves think about the diagnosis and decision process. The if-then rules formulated for pneumatic valve in cooler water spray system are given below:
53 IF flow is medium and rod displacement is medium THEN F0 (No fault) IF flow is medium and rod displacement is low THEN the fault is F1. IF flow is low and rod displacement is medium THEN the fault is F3. IF flow is medium and rod displacement is high THEN the fault is F6. IF flow is low and rod displacement is low THEN the fault is F3. IF flow is high and rod displacement is low THEN F0 (No fault) IF flow is high and rod displacement is medium THEN the fault is F6. IF flow is low and rod displacement is high THEN F0 (No fault). IF flow is high and rod displacement is high THEN F0 (No fault). Table 3.3 refers to outputs produced by FIS for the given input values. Table 3.3 Output produced by the FIS for the given input values Input Output FLOW DISP F0 F1 F3 F6 Actual Fault (mm 3 / sec) (mm) 7.9 14.5 0 1 0 0 Valve Clogging (F1) 2.2 31.5 0 0 1 0 Valve seat sedimentation(f3) 7.8 33.5 1 0 0 0 No Fault 14.7 32.3 0 0 0 1 Internal leakage (F6)
54 The response of the FIS for four sample data corresponding to the normal condition and the three faults are given in Table 3.3. From this table it is observed that the developed FIS is able to correctly identify the fault occurring in the pneumatic valve. Next, ANFIS approach was applied to develop the fuzzy system. Availability of training data is an important requirement in the development of ANFIS model. ANFIS model was developed using the data collected from cement industry (certificate attached at Appendix 1). The binary value of normal and abnormal is taken as the output. The data set was divided into two separate data sets randomly, the training data set and the testing data set. The training data set was used to develop the ANFIS model; whereas the testing data set was used to verify the accuracy and the effectiveness of the trained ANFIS model. Three separate ANFIS models were developed for valve clogging fault, valve seat sedimentation, and internal leakage faults. Hybrid learning rule was used to train the model using the input/ output data pairs. The data contains 2 input features, which is given in Table 3.1 (data sheet attached in Appendix 3) and one output that is labeled as either normal or as a fault, with exactly one specific fault. All the input features are continuous variables while the output is represented as [0] for normal, [1] for fault. The total number of data considered is 1000, which contain 25% normal patterns and 75% of patterns with faults. Among them, 750 patterns are used for training and 250 patterns are used for testing. The testing data comprises of both normal and abnormal (faulty) data, which are totally different from the training data. In order to obtain the optimal model parameters, the fuzzy rule architecture of the ANFIS was designed by using different membership
55 functions and various number of membership functions. Hybrid learning rule was used to train the model accordingly to input/ output data pairs, and the number of iterations was 1000 although it was observed that the most of the learning was completed in the first 200 epochs. Different types of membership functions and various number of membership functions which is ranging 2-5 was used in the ANFIS model. The membership function types which were used in the ANFIS model are Gaussian and the output membership function type is linear. ANFIS was implemented using MATLAB 7.10 software package. The sugeno type fuzzy model is used and the number of fuzzy rules is generated by giving the number of clusters and the number of cluster is 3, then 3 rules are generated. The membership functions generated for 2 input variables for the valve clogging fault are shown in Figure 3.4 and Figure 3.5. 1 Membership function for Flow input in1cluster2 in1cluster3 in1cluster1 input1,cluster1 input1, cluster2 input1, cluster3 0.8 0.6 0.4 0.2 0 4 6 8 10 12 14 Flow (mm3/sec) Figure 3.4 Membership function for flow input of Valve clogging fault (F1)
56 The mean square error achieved during training is 0.02821. After training the performance of the ANFIS model is evaluated with the test data and the mean square error achieved for testing is 0.04601.The network took 11.2903 seconds to reach the error goal. The trained network classified 250 data correctly, which shows an overall detection rate 100%. Membership function for Rod displacement input 1 in2cluster2 in2cluster3 input 2, cluster1 input 2, cluster 2 input2, cluster 3 in2cluster1 0.8 0.6 0.4 0.2 0 10 20 30 40 50 60 70 Rod displacement (mm) Figure 3.5 Membership function for Rod displacement input of Valve clogging fault (F1) Table 3.4 shows the performance comparison between the various parameters of Fuzzy and ANFIS models. From the simulation results, it is found that the accuracy of fault classification is high in ANFIS models when compared with Fuzzy. The obtained results proved that ANFIS performed better than Fuzzy.
57 Table 3.4 Performance Comparison between Fuzzy and ANFIS Models Description of parameters Fuzzy ANFIS model 1 ANFIS model 2 ANFIS model 3 Number of testing patterns 250 250 250 250 Percentage of fault classified 96 % 100 100 100 3.6 SUMMARY This chapter has presented FL based approach for fault diagnosis in pneumatic valve used in cooler water spray system. The fault-symptom relationships were expressed in the form of fuzzy if-then rules. The number of input features used for the development of fuzzy model is two. Membership functions and the fuzzy rule base were formed based on the expert knowledge. Further, the numerical data collected from the system are used to fine-tune the membership functions and the fuzzy rule base. ANFIS is used to tune the parameters of the fuzzy system. The obtained result shows that developed fuzzy model produced accurate results for the given input.