2. We could choose R(x)= where c. and o. are defined in choice 1 There are also alternatives to how to compute the output of the radial basis function neural network. For instance rather than computing the simple sum as in Equation(5.3), you could compute a weighted average . R( y=f(x)=iel (54) ∑R(x) It is also possible to define multilayer radial basis function neural networks This completes the definition of the radial basis function neural network. Next, we explain the relationshil between multilayer perceptions and radial basis function neural networks and fuzzy systems 5.3.3 Relationships Between Fuzzy Systems and Neural Networks There are two ways in which there are relationships between fuzzy systems and neural networks. First, techniques from one area can be used in the other. Second, in some cases the functionality (i.e, the nonlinear function that they implement) is identical. Some label the intersection between fuzzy systems and neural networks with the term fuzzy-neural"or"neuro-fuzzy"to highlight that techniques from both fields are being used. Here, we avoid this terminology and simply highlight the basic relationships between the two field Multilayer Perceptrons The multilayer perceptron should be viewed as a nonlinear network whose nonlinearity can be tuned by changing the weights, biases, and parameters of the activation functions. The fuzzy system is also a tunable nonlinearity whose shape can be changed by tuning, for example, the membership functions. Since both are tunable nonlinearities, the following approaches are possible a Gradient methods can be used for training neural networks to perform system identification or to act as estimators or predictors in the same way as fuzzy systems were trained Indeed, the gradient training of neural networks, called"back-propagation training, "was introduced well before the gradient training of fuzzy systems, and the idea for training fuzzy systems this way came from the field of neural networks Hybrid methods for training can also be used for neural networks. For instance, gradient methods may be used in conjunction with clustering methods applied to neural networks a Indirect adaptive control can also be achieved with a multilayer perceptron. To do this we use two multilayer perceptrons as the tunable nonlinearities in the certainty equivalence control law and the gradient method for tuning a Gain scheduled control may be achieved by training a multilayer perceptron to map the associations between operating conditions and controller parameters This list is by no means exhaustive. It simply shows that multilayer pere networks can take on a similar role to that of a fuzzy system in performing the function of being a tunable nonlinearity. An advantage that the fuzzy system may have, however, is that it often facilitates the incorporation of heuristic knowledge into the solution to the problem, which2. We could choose ( ) 2 2 1 1 exp i i i R x x c σ = ⎛ ⎞ − + −⎜ ⎟ ⎜ ⎟ ⎝ ⎠ where i c and σ i are defined in choice 1. There are also alternatives to how to compute the output of the radial basis function neural network. For instance, rather than computing the simple sum as in Equation (5.3), you could compute a weighted average ( ) ( ) ( ) 1 1 M i i i M i i y R x y fx R x = = = = ∑ ∑ （5.4） It is also possible to define multilayer radial basis function neural networks. This completes the definition of the radial basis function neural network. Next, we explain the relationships between multilayer perceptions and radial basis function neural networks and fuzzy systems. 5.3.3 Relationships Between Fuzzy Systems and Neural Networks There are two ways in which there are relationships between fuzzy systems and neural networks. First, techniques from one area can be used in the other. Second, in some cases the functionality (i.e., the nonlinear function that they implement) is identical. Some label the intersection between fuzzy systems and neural networks with the term "fuzzy-neural" or "neuro-fuzzy" to highlight that techniques from both fields are being used. Here, we avoid this terminology and simply highlight the basic relationships between the two fields. Multilayer Perceptrons The multilayer perceptron should be viewed as a nonlinear network whose nonlinearity can be tuned by changing the weights, biases, and parameters of the activation functions. The fuzzy system is also a tunable nonlinearity whose shape can be changed by tuning, for example, the membership functions. Since both are tunable nonlinearities, the following approaches are possible: ƒ Gradient methods can be used for training neural networks to perform system identification or to act as estimators or predictors in the same way as fuzzy systems were trained. Indeed, the gradient training of neural networks, called "back-propagation training," was introduced well before the gradient training of fuzzy systems, and the idea for training fuzzy systems this way came from the field of neural networks. ƒ Hybrid methods for training can also be used for neural networks. For instance, gradient methods may be used in conjunction with clustering methods applied to neural networks. ƒ Indirect adaptive control can also be achieved with a multilayer perceptron. To do this we use two multilayer perceptrons as the tunable nonlinearities in the certainty equivalence control law and the gradient method for tuning. ƒ Gain scheduled control may be achieved by training a multilayer perceptron to map the associations between operating conditions and controller parameters. This list is by no means exhaustive. It simply shows that multilayer perceptron networks can take on a similar role to that of a fuzzy system in performing the function of being a tunable nonlinearity. An advantage that the fuzzy system may have, however, is that it often facilitates the incorporation of heuristic knowledge into the solution to the problem, which