Page 450 - Mechanical Engineers' Handbook (Volume 2)

14 Future Trends in Control Systems 441

14.1 Fuzzy Logic Control
In classical set theory, an object’s membership in a set is clearly defined and unambiguous.
Fuzzy logic control is based on a generalization of classical set theory to allow objects to
belong to several sets with various degrees of membership. Fuzzy logic can be used to
describe processes that defy precise definition or precise measurement, and thus it can be
used to model the inexact and subjective aspects of human reasoning. For example, room
temperature can be described as cold, cool, just right, warm, or hot. Development of a fuzzy
logic temperature controller would require the designer to specify the membership functions
that describe ‘‘warm’’ as a function of temperature, and so on. The control logic would then
be developed as a linguistic algorithm that models a human operator’s decision process (for
example, if the room temperature is ‘‘cold,’’ then ‘‘greatly’’ increase the heater output; if the
temperature is ‘‘cool,’’ then increase the heater output ‘‘slightly’’).
Fuzzy logic controllers have been implemented in a number of applications. Proponents
of fuzzy logic control point to its ability to convert a human operator’s reasoning process
into computer code. Its critics argue that because all the controller’s fuzzy calculations must
eventually reduce to a specific output that must be given to the actuator (e.g., a specific
voltage value or a specific valve position), why not be unambiguous from the start, and
define a ‘‘cool’’ temperature to be the range between 65 and 68 , for example? Perhaps the
proper role of fuzzy logic is at the human operator interface. Research is active in this area,
and the issue is not yet settled. 11,12

14.2 Nonlinear Control
Most real systems are nonlinear, which means that they must be described by nonlinear
differential equations. Control systems designed with the linear control theory described in
this chapter depend on a linearized approximation to the original nonlinear model. This
linearization can be explicitly performed, or implicitly made, as when we use the small-
angle approximation: sin . This approach has been enormously successful because a
well-designed controller will keep the system in the operating range where the linearization
was done, thus preserving the accuracy of the linear model. However, it is difficult to control
some systems accurately in this way because their operating range is too large. Robot arms
are a good example. 13,14 Their equations of motion are very nonlinear, due primarily to the
fact that their inertia varies greatly as their configuration changes.
Nonlinear systems encompass everything that is ‘‘not linear,’’ and thus there is no general
theory for nonlinear systems. There have been many nonlinear control methods proposed—
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too many to summarize here. Lyapunov’s stability theory and Popov’s method play a central
role in many such schemes. Adaptive control is a subcase of nonlinear control (see below).
The high speeds of modern digital computers now allow us to implement nonlinear
control algorithms not possible with earlier hardware. An example is the computed-torque
method for controlling robot arms, which was discussed in Section 11 (see Fig. 47).

14.3 Adaptive Control
The term adaptive control, which unfortunately has been loosely used, describes control
systems that can change the form of the control algorithm or the values of the control gains
in real time, as the controller improves its internal model of the process dynamics or in
response to unmodeled disturbances. 16 Constant control gains do not provide adequate re-
sponse for some systems that exhibit large changes in their dynamics over their entire op-