1.4. A LOOK AT CONTROLLER DESIGN 11
specific plant outputyby manipulating a plant inputuin such a way to achieve
somecontrol objective. That is to say, build a deviceCcalled acontrollerthat
will send control signalsuto the plant (uis the input to the plant) in such a
way as to achieve the given control objective (yistheoutputfromtheplant).
The functionuis referred to as acontrol law,the specification of the control
Figure 1.14. Control lawsignal. Figure 1.14 illustrates the problem. Asuccessful control lawis one that
does the job. Depending upon whether feedback information is used or not, we
have feedback or nonfeedback control laws. Theengineering problemis this.
How do youfind the functionuandhowdoyouimplementit?
Example 1.11 (Cruise control)Suppose we want to keep the speed of a car
aty 0 =65mph for allt>t 0. Thisisanexampleofaset-pointcontrol
problem. We have at our disposal a forceu(t), and we can observe the speed
y(t). We consider the open-loop case. By the nature of the control problem,
there is a relationship between the inputuand the outputy,thatis,thereisa
functionfsatisfying
y(t)=f(u(t))
Giveny 0 , the problem is tofind the control functionu 0 (t)such thatf(u 0 (t)) =
y 0 fort>t 0. It seems obvious that, without knowingf, there is no hope of
findingu 0. The functionfis referred to as a mathematical model for the plant.
From this viewpoint, standard control theory immediately focuses onfinding
suitable mathematical models for a given plant as a veryfirst task in theanalysis
andsynthesis of any control problem. Note thatanalysismeans collecting
information pertinent to the control problem at hand; whereassynthesismeans
actually constructing asuccessful control law. In most cases, a major part of
the effort is devoted to the task of developing a mathematical model for a plant.
In general, this is extremely difficult. The task requires detailed knowledge of
the plant and knowledge of physical laws that govern the interaction of all the
variables within the plant. The model is, at best, an approximate representation
of the actual physical system. So, the natural question that arises is whether
you can control the plant without knowing the relationshipfbetweenuandy
ñ that is, by using a model-free approach.
For our car example, it is straightforward to obtain a mathematical model.
From physical laws, the equation of motion (the plant dynamics) is of the form
d^2 x(t)
dt^2+
adx(t)
dt=bu(t) (1.1)