A First Course in FUZZY and NEURAL CONTROL

66 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

a largeKpwill have the effect of reducing the rise time and will reduce (but
never eliminate) the steady-state error. Integral controlKi, will have the effect
of eliminating the steady-state error,but it will make the transient response
worse. If integral control is to be used, a smallKishould always be triedfirst.
Derivative control will have the effect of increasing the stability of the system,
reducing the overshoot, and improving the transient response. The effects on
the closed-loop response of adding to the controller termsKp,Ki,andKdare
listed in Table 2.1.

Table 2.1. Effects on closed-loop response

Controller Steady-state

Term Rise Time Overshoot Settling Time Error

Kp Decreases Increases No change Decreases

Ki Decreases Increases Increases Eliminates

Kd No change Decreases Decreases No change

It is important to note that these correlations are to be used only as a guide
and not to imply the exact relations between the variables and their effect on
each other. Changing one of these variables can change the effect of the other
two. The table serves as a guide to allow the designer to adjust the parameters
by trial and error.

Simulating the open-loop step response Many PID controllers are de-
signed by the trial and error selection of the variablesKp,Ki,andKd.There
are some rules of thumb that you can consult to determine good values to start
from; see your controls book for some explanations of these recommendations.
Suppose we have a second-order plant transfer function

G(s)=

1

s^2 +10s+20

Let usfirst view the open-loop step response. To model this system inMatlab,
create a newm-fileand add in the following code:

numsys=1;
densys=[1 10 20];
step(numsys,densys);
A plot of the response is shown in Figure 2.32. From the given transfer
function, it is easy to see that thefinal value ofG(s), applying the Final Value
theorem, is 1 / 20 , as is illustrated in the step response. This value of 0.05, in
relation to the unit step input amounts to a steady-state error of 0.95. Further-
more, the rise time is about 1 second and the settling time is nearly 2 seconds.
While it is not clear as to what the control criteria should be, it is evident that
at least the steady-state error must be reduced to within some tolerable lim-
its. We proceed now to design a PID controller in a step-by-step manner that
illustrates the changes in response that occur as a result of various parameter
adjustments.