12 CHAPTER 1. A PRELUDE TO CONTROL THEORY
wherex(t)denotes the carís position.
The velocity is described by the equationy(t)=dx(t)/dt, so Equation 1.1,
written in terms ofy,is
dy(t)
dt
+ay(t)=bu(t) (1.2)This equation gives rise to the needed relation between the inputu(t)and output
y(t), namelyy(t)=f(u(t)). Thisisdonebysolvingforu(t)for a giveny(t).
This equation itself provides the control law immediately. Indeed, from it you
see that, in order fory(t)=y 0 ,fort> 0 , the accelerationdy(t)/dtshould be
equal to zero, so it is sufficient to takeu(t)=(a/b)y 0 for allt> 0.
To solve the second-order linear differential equation in Equation 1.2, you
can use Laplace transforms. This yields thetransfer functionF(s)of the
plant and puts you in thefrequency domainñthatis,youareworking
with functions of the complex frequencys. Taking inverse Laplace transforms
returnsu(t), putting you back in thetime domain. These transformations
often simplify the mathematics involved and also expose significant components
of the equations. You will see some examples of this in Chapter 2. Note that this
example is not realistic for implementation, but it does illustrate the standard
control approach.
The point is that to obtain a control law analytically, you need a mathemati-
cal model for the plant. This might implythat if you donít have a mathematical
model for your plant, you cannotfind a control law analytically. So, how can
you control complicated systems whose plant dynamics are difficult to know?
A mathematical model may not be a necessary prerequisite for obtaining a suc-
cessful control law. This is precisely the philosophy of the fuzzy and neural
approaches to control.
To be precise, typically, as in several of the preceding examples, feedback
control is needed for a successful system. These closed-loop controls are closely
related to the heuristics of ìIf...then...îrules. Indeed, if you feed back the plant
outputy(t)to the controller, then the controlu(t)should be such that the error
y(t)−y 0 =e(t)goes to zero. So, apparently, the design of the control lawu(t)
is reduced to another box with inpute(t)and outputu(t).Thus,
u(t)=g(e(t)) =h(y(t),y 0 )The problem is tofind the functiongor to approximate it from observable
values ofu(t)andy(t). Even thoughy(t)comes out from the plant, you donít
need the plantís mathematical model to be able to observey(t).Thus,where
does the mathematical model of the plant come to play in standard control
theory, in the context of feedback control? From a common-sense viewpoint,
we can often suggest various obvious functionsg. This is done for the so-called
proportional integral derivative(PID) types of controllers discussed in the next
chapter. However, these controllers are not automatically successful controllers.
Just knowing the forms of these controllers is not sufficient information to make
them successful. Choosing good parameters in these controllers is a difficult
design problem, and it is precisely here that the mathematical model is needed.