218 CHAPTER 6. NEURAL CONTROL
in practice, system identification by neural networks works remarkably well for
reasonably well-behaved dynamical systems.
A neural network that is trained to classify plant behavior by observing the
output of the plant must be trained based on the errors that might occur in the
plant response compared to a desired response. The error function that results
from training the neural network must very closely approximate the inverse
behavior of the plant. A neural network can therefore be used to approximate
control laws that govern the controlling inputs to the plant. Neural networks
are particularly appropriate for system identification when the mathematical
model is unknown and the behavior of the system is known only in the form of
sample data. Neural networks can be used for nonlinear systems, and they can
also be used for optimal control.
Thefirststepinsystemidentification is experimentation with the plant to
develop physical insight. A good experiment will result in a set of data that
includes responses to input over the entire range of operation of the plant. This
data set will be used to train the model.
The next step is to select a neural network model. This choice includes both
selecting the inputs to the network and selecting an internal network topology.
The two most widely used families of models for this purpose are multi-layer
perceptrons and the so-called radial basis function neural networks, a family of
neural network models that we do not treat in this book.
The model can be represented asy(t)=g(x(t,θ),θ)+e(t)wherex(t,θ)is a regression vector andθis a vector containing the weights.
One form the regression vector can take is
x(t,θ)=[y(t),...,y(t−n),u(t−d),...,u(t−d−m,),e(t,θ),...,e(t−k,θ)]T
The choice of regressor will depend in part on knowledge of the plant behavior.
The model is then trained with the data that was obtained from the exper-
iment. After the training, the model should be validated to check whether it
meets the necessary criteria in terms of the intended use of the model. If the
results are not completely satisfactory, different neural network models can be
tried. If the data collected is not sufficient, it may be impossible to develop a
satisfactory model and one must start over, carrying out a new experiment to
obtain additional training data. It could be, for example, that not all actual
inputs that affect behavior of the system were recognized, and these must be
added to the experiment and to the model before achieving the desired result.
When system identification is implemented as part of the controller, the
controller is known as anadaptive controller. Such controllers are designed
to control systems whose dynamical characteristics vary with time. A more
detailed exposition of system identification can be found in advanced texts on
neural network control, such as [30] and [54].