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Fig. 2. A Takagi-Sugeno fuzzy model as a piece-wise linear approximation of a nonlinear system. Example 2. Consider a static characteristic of an actuator with a dead-zone and a non-symmetrical response for posi- tive and negative inputs. Such a system can conveniently be represented by a TS model with three rules each covering a subset of the operating domain that can be approximated by a local linear model, see Fig. 2. The corresponding rules

Figure 2 A Takagi-Sugeno fuzzy model as a piece-wise linear approximation of a nonlinear system. Example 2. Consider a static characteristic of an actuator with a dead-zone and a non-symmetrical response for posi- tive and negative inputs. Such a system can conveniently be represented by a TS model with three rules each covering a subset of the operating domain that can be approximated by a local linear model, see Fig. 2. The corresponding rules

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Common fuzzy logic operators In fuzzy systems with multiple inputs, the antecedent oroposition is usually represented as a combination of terms with univariate membership functions, by using logic oper- ators ‘and’ (conjunction), ‘or’ (disjunction) and ‘not’ (com- olement). In fuzzy set theory, several families of operators rave been introduced for these logical connectives. Table 1 shows the two most common ones.
Common fuzzy logic operators In fuzzy systems with multiple inputs, the antecedent oroposition is usually represented as a combination of terms with univariate membership functions, by using logic oper- ators ‘and’ (conjunction), ‘or’ (disjunction) and ‘not’ (com- olement). In fuzzy set theory, several families of operators rave been introduced for these logical connectives. Table 1 shows the two most common ones.
where a; is the consequent parameter vector, b; is a scalar offset and i = 1,..., K. This model combines a linguis- tic description with standard functional regression: the an- tecedents describe fuzzy regions in the input space in which the consequent functions are valid. The output y is computed by taking the weighted average of the individual rules’ con- tributions:
where a; is the consequent parameter vector, b; is a scalar offset and i = 1,..., K. This model combines a linguis- tic description with standard functional regression: the an- tecedents describe fuzzy regions in the input space in which the consequent functions are valid. The output y is computed by taking the weighted average of the individual rules’ con- tributions:
A feedforward multi-layer neural network (MNN) has one input layer, one output layer and an number of hidden layers between them. For illustration purposes, consider a MNN with one hidden layer (Fig. 3). ~ In the modeling of dynamic systems, fuzzy models are used to parameterize of the nonlinear functions f in (1) org and h in (3). Consider, for instance, the TS NARX model:
A feedforward multi-layer neural network (MNN) has one input layer, one output layer and an number of hidden layers between them. For illustration purposes, consider a MNN with one hidden layer (Fig. 3). ~ In the modeling of dynamic systems, fuzzy models are used to parameterize of the nonlinear functions f in (1) org and h in (3). Consider, for instance, the TS NARX model:
Fig. 5. Radial basis function network. Fig. 4. Neural network models of dynamic systems. where w(n) is the weight vector at iteration n, a(n) is a (vari- able) learning rate (a parameter) and VJ (w) is the Jacobian of the network: Another possibility is to use recurrent networks in which neurons are arranged in one or more layers and feedback is introduced either internally in the neurons, to other neurons in the same layer, or to neurons in preceding layers. Exam- ples of these networks are the Elman network (Fig. 4b) or the Hopfield network.
Fig. 5. Radial basis function network. Fig. 4. Neural network models of dynamic systems. where w(n) is the weight vector at iteration n, a(n) is a (vari- able) learning rate (a parameter) and VJ (w) is the Jacobian of the network: Another possibility is to use recurrent networks in which neurons are arranged in one or more layers and feedback is introduced either internally in the neurons, to other neurons in the same layer, or to neurons in preceding layers. Exam- ples of these networks are the Elman network (Fig. 4b) or the Hopfield network.
Fig. 7. An example of a first-order TS fuzzy model with two mules represented as a neuro-fuzzy network called ANFIS.
Fig. 7. An example of a first-order TS fuzzy model with two mules represented as a neuro-fuzzy network called ANFIS.
Fig. 6. An example of a zero-order TS fuzzy model with two rules represented as a neuro-fuzzy network.
Fig. 6. An example of a zero-order TS fuzzy model with two rules represented as a neuro-fuzzy network.
Fig. 8. The result of local (top) and global (bottom) estimation of the consequent parameters. The dashed line is the output of the model.
Fig. 8. The result of local (top) and global (bottom) estimation of the consequent parameters. The dashed line is the output of the model.
with input-dependent ‘parameters’ a(x), b(x) which are con- vex linear combinations of the individual consequent param- eters a; and bj, i.e.:
with input-dependent ‘parameters’ a(x), b(x) which are con- vex linear combinations of the individual consequent param- eters a; and bj, i.e.:
Fig. 10. Antecedent space decomposition by a heuristic search algorithm. The dark areas represent rules with the worst local fit in the given step. Construction methods based on fuzzy clustering originate from data analysis and pattern recognition, where the con- cept of fuzzy membership is used to represent the degree to which a given data object is similar to some prototypical object. The degree of similarity can be calculated by using a suitable distance measure. Based on the similarity, data
Fig. 10. Antecedent space decomposition by a heuristic search algorithm. The dark areas represent rules with the worst local fit in the given step. Construction methods based on fuzzy clustering originate from data analysis and pattern recognition, where the con- cept of fuzzy membership is used to represent the degree to which a given data object is similar to some prototypical object. The degree of similarity can be calculated by using a suitable distance measure. Based on the similarity, data
Fig. 11. Identification of membership functions through fuzzy clustering.
Fig. 11. Identification of membership functions through fuzzy clustering.
Fig. 12. Approximation by the initial TS fuzzy model. The root-mean- squared error is RMS = 0.0258. vectors are clustered such that the data within a cluster are as similar as possible, and data from different clusters are as dissimilar as possible. WUTLS GLY USLIVULOU LLUilh LULL WW LIYtib VY fA UbUuUYyll fly. Note that this initial model can easily be interpreted in terms of the local behavior (the rule consequents) and it is already reasonably accurate (the root-mean-squared error is RMS = 0.0258). However, by using the ANFIS method, the model parameters can be fine-tuned and the approxi- mation accuracy can be significantly improved. A model
Fig. 12. Approximation by the initial TS fuzzy model. The root-mean- squared error is RMS = 0.0258. vectors are clustered such that the data within a cluster are as similar as possible, and data from different clusters are as dissimilar as possible. WUTLS GLY USLIVULOU LLUilh LULL WW LIYtib VY fA UbUuUYyll fly. Note that this initial model can easily be interpreted in terms of the local behavior (the rule consequents) and it is already reasonably accurate (the root-mean-squared error is RMS = 0.0258). However, by using the ANFIS method, the model parameters can be fine-tuned and the approxi- mation accuracy can be significantly improved. A model
Fig. 14. Membership functions before (left) and after training (right). The membership functions are denoted from left to right by ‘Low,’ ‘Medium’ and ‘High.’ Note that the consequent model in the first rule has a negative coefficient for Q(k). As this is a physically impossible value, not interpretation can be given to these parameters and this trained model has become a complete black-box. Also notice in Fig. 14 that the membership functions were adjusted in The process is approximated as a first-order discrete time NARX model: A neutralization tank with three influent streams (acid, buffer and base) and one effluent stream is considered. The identification and validation data sets are obtained by sim- ulating the model by Hall and Seborg (1989) for random changes of the influent base stream flow rate Q. The influent buffer stream and the influent acid stream are kept constant. The output is the pH in the tank. The identification data set, containing N = 499 samples with the sampling time of 15s, is shown in Fig. 13. This data set was obtained from Johansen (1994).
Fig. 14. Membership functions before (left) and after training (right). The membership functions are denoted from left to right by ‘Low,’ ‘Medium’ and ‘High.’ Note that the consequent model in the first rule has a negative coefficient for Q(k). As this is a physically impossible value, not interpretation can be given to these parameters and this trained model has become a complete black-box. Also notice in Fig. 14 that the membership functions were adjusted in The process is approximated as a first-order discrete time NARX model: A neutralization tank with three influent streams (acid, buffer and base) and one effluent stream is considered. The identification and validation data sets are obtained by sim- ulating the model by Hall and Seborg (1989) for random changes of the influent base stream flow rate Q. The influent buffer stream and the influent acid stream are kept constant. The output is the pH in the tank. The identification data set, containing N = 499 samples with the sampling time of 15s, is shown in Fig. 13. This data set was obtained from Johansen (1994).
The initial rule base, with the consequent estimated by weighted local least squares (31), is given by:
The initial rule base, with the consequent estimated by weighted local least squares (31), is given by:
Fig. 15. Performance of the initial and trained model on the validation data set (solid line: data; dashed line: model).
Fig. 15. Performance of the initial and trained model on the validation data set (solid line: data; dashed line: model).
Related topics:
Fuzzy SystemNonlinear system identificationNonlinear identificationElectrical and Electronic Engineering
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