202 CHAPTER 6. NEURAL CONTROL
for adaptive control where controllers need to adapt to changing environment,
such as for time-variant systems. In practice, neural network controllers have
proved to be most useful for time-invariant systems.
Basically, to build a neural network-based controller that can force a plant
to behave in some desirable way, we need to adjust its parameters from the ob-
served errors that are the difference between the plantís outputs and the desired
outputs. Adjustment of the controllerís parameters will be done by propagating
back these errors across the neural network structure. This is possible if the
mathematical model of the plant is known. When the mathematical model of
the plant is not known, we need to know at least an approximate model of the
plant in order to do the above. An approximate (known) model of the plant is
called anidentified model. When we use input-output data from the plant to
train a neural network to provide an approximate model to the plant, we obtain
aneural network identified modelof the plant. Neural network identified
models are used in indirect neural control designs. After a general discussion
of inverse dynamics, we willfirst discuss direct neural control designs and then
indirect control.
6.2 Inverse dynamics
An ideal control law describes theinverse dynamicsof a plant. For a simple
example, consider plant dynamics of the form
x ̇(t)=f(x(t),u(t))A control law for this plant has the form
u(t)=g(x(t))Even when a plant possesses inverse dynamics, the solution might not have a
closed form, in which case approximations are needed.
In general, it is difficult to check the existence of the inverse dynamics of
a plant. However, in the case of linear systems, this existence can be checked
easily. In this case, in fact, the existence of inverse dynamics is equivalent to
the controllability of the plant, which was discussed in Section 2.3.
Consider, for example, the time-invariant linear system described in discrete
time by the equations
x(k+1)=Ax(k)+Bu(k),k=0, 1 , 2 ,...whereAis ann◊nmatrix andBisn◊ 1 .Thenwehave
x(k+2) = Ax(k+1)+Bu(k+1)
= A(Ax(k)+Bu(k)) +Bu(k+1)
= A^2 x(k)+ABu(k)+Bu(k+1)