2.8. NONLINEAR CONTROL SYSTEMS 77
Figure 2.46. System responsea satisfactory controller design. The basic steps in developing PID controllers
can be summarized as follows:
ïObtain the open-loop and closed-loop response of the plant and determine
what needs to be improved.ïAdd proportional controlKP> 0 to improve the rise time.ïAdd derivative controlKD> 0 to improve the overshoot.ïAdd integral control to eliminate the steady-state errorKI> 0.ïAdjust each ofKP,KI,andKDuntil you obtain a desired overall response.2.8 Nonlinearcontrolsystems
Any system for which the superposition principle does not apply is said to be
nonlinear. In this case, there is no possibility of generalizing from the response
for any class of inputs to the response for any other input. This constitutes
a fundamental and important difficulty that necessarily requires any study of
nonlinear systems to be quite specific. One can attempt to calculate the response
for a specific case of initial conditions and input, but make very little inference
based on this result regarding the response characteristics in other cases.
Despite analytical difficulties, one has no choice but to attempt to deal in
some way with nonlinear systems, because they occupy a very important place
in any practical system. Most linear systems can be thought of as piecewise
linear approximations of a nonlinear system. The inverted pendulum example
described at the beginning of this chapter is one such example. In some cases the
approximation may be very good, but most physical variables, if allowed to take
on large values, will eventually go out of their range of reasonable linearity. Most
drive systems such aselectricalandhydraulic actuatorscan only be thought of
as being linear over small ranges. Gas jets, for example, have no linear range
at all, and to achieve minimum-time control, an ON-OFF (bang-bang or relay)