1.4. A LOOK AT CONTROLLER DESIGN 13
In the case of linear and time-invariant systems, the mathematical model can
be converted to the so-called transfer functions of the plant and of the controller
to be designed. As we will see, knowledge of the poles of these transfer functions
is necessary for designing state-variable feedback controllers or PID controllers
that will perform satisfactorily.
Even for linear and time-invariant plants, the modern view of control is
feedback control. From that viewpoint, a control law is a function of the error.
Proposing a control law, or approximating it from training data (a curvefitting
problem), are obvious ways to proceed. The important point to note is that
the possible forms of a control law are not derived from a mathematical model
of the plant, but rather fromheuristics. What the mathematical model does is
help in a systematic analysis leading to the choice of good parameters in the
proposed control law, in order to achieve desirable control properties. In other
words,with a mathematical model for the plant, there exist systematic ways to
design successful controllers.
In the absence of a mathematical model for the plant, we can always approx-
imate a plausible control law, either from a collection of ìIf... then... î rules or
from training data. When we construct a control law by any approximation
procedures, however, we have to obtain a good approximation. There are no
parameters, per se, in this approximation approach to designing control laws.
There are of course ìparametersî in weights of neural networks, or in the mem-
bership functions used by fuzzy rules, but they will be adjusted by training
samples or trial and error. There is no need for analytical mathematical models
in this process. Perhaps that is the crucial point explaining the success of soft
computing approaches to control.
Let us examine a little more closely the prerequisite for mathematical models.
First, even in the search for a suitable mathematical model for the plant, we
can only obtain, in most cases, a mathematical representation that approximates
the plant dynamics. Second, from a common sense point of view, any control
strategyisreallybaseduponìIf...then...îrules. Theknowledgeofafunctional
relationshipfprovides specific ìIf... then... î rules, often more than needed.
The question is: Can wefind control laws based solely on ìIf... then... î rules?
If yes, then obviously we can avoid the tremendous task of spending the major
part of our effort infinding a mathematical model. Of course, if a suitable
mathematical model is readily available, we generally should use it.
Our point of view is that a weaker form of knowledge, namely a collection
of ìIf...then...î rules, might be sufficient for synthesizing control laws. The
rationale is simple: we are seeking an approximation to the control law ñ
that is, the relationship between input and output of the controller directly,
and not the plant model. We are truly talking aboutapproximating functions.
The many ways of approximating an unknown function include using training
samples (neural networks) and linguistic ìIf... then... î rules (fuzzy logic).^2
(^2) In both cases, the theoretical foundation is the so-calleduniversal approximation capabil-
ity, based on the Stone-Weierstrass Theorem, leading to ìgoodî models for control laws.