Page 804 - Mechanical Engineers' Handbook (Volume 2)

3 Feedback Linearization Design of NN Tracking Controllers 795

which uses NNs to approximately solve the optimal control problem, and NNs in closed-
loop feedback control. Many researchers have contributed to the development of these fields.
See Section 11 and the References and Bibliography.
Several NN feedback control topologies are illustrated in Fig. 3, 10 some of which are
11
derived from standard topologies in adaptive control. Solid lines denote control signal flow
loops while dashed lines denote tuning loops. There are basically two sorts of feedback
control topologies: indirect and direct techniques. In indirect NN control there are two func-
tions; in an identifier block, the NN is tuned to learn the dynamics of the unknown plant,
and the controller block then uses this information to control the plant. Direct control is
more efficient and involves directly tuning the parameters of an adjustable NN controller.
The challenge in using NNs for feedback control purposes is to select a suitable control
system structure and then to demonstrate using mathematically acceptable techniques how
the NN weights can be tuned so that closed-loop stability and performance are guaranteed.
In this chapter, we shall show different methods of NN controller design that yield guaranteed
performance for systems of different structure and complexity. Many researchers have par-
ticipated in the development of the theoretical foundation for NNs in control applications.
See Section 11.

3 FEEDBACK LINEARIZATION DESIGN OF NN TRACKING CONTROLLERS
In this section, the objective is to design an NN feedback controller that causes a robotic
system to follow, or track, a prescribed trajectory or path. The dynamics of the robot are
unknown, and there are unknown disturbances. The dynamics of an n-link robot manipulator
may be expressed as 12

Control Output
y d ( (t) ) u(t) ) y(t)
t
y(t)
u(t
y d
NN controller
NN c o nt rol l er Plant
Desired Identification
error
output
output error
N N sy st e m ( ˆ y( ˆ y ) t) t
NN system
i identifier r Estimated
d
ti
fi
n
e
e
output
output
Control Output
(a) y d ( (t) ) NN u(t ) y(t)
y(t)
t
u(t)
y d
NN
controller 1 Plant
Desired
output
output
NN
NN
Control Output controller 2
y(t)
y d ( (t) ) u(t) ) y(t) Tracking
t
u(t
y d
o
NN c
NN controller Plant error
nt
l
er
rol
error
Desired
output
output
(c)
Tracing
error
error
(b)
Figure 3 NN control topologies: (a) indirect scheme; (b) direct scheme; (c) feedback/feedforward
scheme.