can,at times, have a significant impact on the quality of the solutio Radial basis Function Neural Networks Some radial basis function neural networks are equivalent to some standard fuzzy systems in the sense that they are functionally equivalent (ie, given the same inputs, they will produce the same outputs). To see this, suppose that in Equation(5.4)we let M=R(i.e, the number of receptive field units equal to the number of rules ), y,=b, (i.., the receptive field unit strengths equal to the output membership function centers ), and choose the receptive field units R(x)=(x) (.e, choose the receptive field units to be the same as the premise membership functions). In this case we see that the radial basis function neural network is identical to a certain fuzzy system that uses center-average defuzzification. This fuzzy system is then given by y=f(x) bu (x) (x) It is also interesting to note that the functional fuzzy system(the more general version of the Takagi-Sugeno fuzzy stem)is equivalent to a class of two-layer neural networks (2001 The equivalence between this type of fuzzy system and a radial basis function neural network shows that all the echniques in this book for the above type of fuzzy system work in the same way for the above type of radial basis function neural network (or, using [200, the techniques for the Takagi fuzzy system can be used for a type of multilayer radial basis function neural network) Due to the above relationships between fuzzy systems and neural networks, some would like to view fuzzy sys d neural networks as identical areas This is however not the case for the following reasons There are classes of neural networks(e.g, dynamic neural networks) that may have a fuzzy system analog, but if so it would have to include not only standard fuzzy components but some form of a differential equation component There are certain fuzzy systems that have no clear neural analog. Consider, for example, certain"fuzzy dynamic systems".We can, however, envision how you could go about "designing a neural analog to such fuzzy systems using gradient methods like back-propagation)using data, often without using extra heuristic knowledge w? ip The neural network has traditionally been a"black box"approach where the weights and biases are trained(e often have. In fuzzy systems you can incorporate heuristic information and use data to train them. This last difference is often quoted as being one of the advantages of fuzzy systems over neural networks, at least for some applications Regardless of the differences, it is important to note that many methods in neural control (i.e, when we use a neural network for the control of a system) are quite similar to those in adaptive fuzzy control. For instance, since the fuzzy system and radial basis function neural network can be linearly parameterized, we can use them as the identifier structures in direct or indirect adaptive control schemes and use gradient or least squares methods to update the parameters. Indeed, we could have used neural networks as the structure that we trained for all of the identification methods. In this sense we can use neural networks in system identification, estimation, and prediction, and as a direct (fixed) controller that is trained with input-output data. Basically, to be fluent with the methods of adaptive fuzzy systems and control, you must know the methods of neural control-and vice versa.can, at times, have a significant impact on the quality of the solution. Radial Basis Function Neural Networks Some radial basis function neural networks are equivalent to some standard fuzzy systems in the sense that they are functionally equivalent (i.e., given the same inputs, they will produce the same outputs). To see this, suppose that in Equation (5.4) we let M =R (i.e., the number of receptive field units equal to the number of rules), i y b = i (i.e., the receptive field unit strengths equal to the output membership function centers), and choose the receptive field units as () () Ri i x x = μ (i.e., choose the receptive field units to be the same as the premise membership functions). In this case we see that the radial basis function neural network is identical to a certain fuzzy system that uses center-average defuzzification. This fuzzy system is then given by 1 1 ( ) ( ) ( ) R i i i R i i b x y fx x μ μ = = = = ∑ ∑ It is also interesting to note that the functional fuzzy system (the more general version of the Takagi-Sugeno fuzzy system) is equivalent to a class of two-layer neural networks [200]. The equivalence between this type of fuzzy system and a radial basis function neural network shows that all the techniques in this book for the above type of fuzzy system work in the same way for the above type of radial basis function neural network (or, using [200], the techniques for the Takagi-Sugeno fuzzy system can be used for a type of multilayer radial basis function neural network). Due to the above relationships between fuzzy systems and neural networks, some would like to view fuzzy systems and neural networks as identical areas. This is, however, not the case for the following reasons: ƒ There are classes of neural networks (e.g., dynamic neural networks) that may have a fuzzy system analog, but if so it would have to include not only standard fuzzy components but some form of a differential equation component. ƒ There are certain fuzzy systems that have no clear neural analog. Consider, for example, certain "fuzzy dynamic systems". We can, however, envision how you could go about "designing a neural analog to such fuzzy systems. ƒ The neural network has traditionally been a "black box" approach where the weights and biases are trained (e.g., using gradient methods like back-propagation) using data, often without using extra heuristic knowledge we often have. In fuzzy systems you can incorporate heuristic information and use data to train them. This last difference is often quoted as being one of the advantages of fuzzy systems over neural networks, at least for some applications. Regardless of the differences, it is important to note that many methods in neural control (i.e., when we use a neural network for the control of a system) are quite similar to those in adaptive fuzzy control. For instance, since the fuzzy system and radial basis function neural network can be linearly parameterized, we can use them as the identifier structures in direct or indirect adaptive control schemes and use gradient or least squares methods to update the parameters. Indeed, we could have used neural networks as the structure that we trained for all of the identification methods. In this sense we can use neural networks in system identification, estimation, and prediction, and as a direct (fixed) controller that is trained with input-output data. Basically, to be fluent with the methods of adaptive fuzzy systems and control, you must know the methods of neural control—and vice versa