A First Course in FUZZY and NEURAL CONTROL

56 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

The objective in the controller design then is to select the appropriate parame-
ters forKP,KI,andKDto satisfy all design criteria.
To design a PID controller for a given system, follow the steps shown below
to obtain a desired response.

1. Obtain the open-loop response and determine what needs to be improved.


2. Add proportional control to improve the rise timeKP> 0.


3. Add derivative control to improve the overshootKD> 0.


4. Add integral control to eliminate the steady-state errorKI> 0.


5. Adjust each ofKP,KI,andKDuntil a desired overall response is ob-
   tained.

Working through each of the steps listed above, we have already seen that
the open-loop response is not satisfactory in that the rise time is inadequate for
the automobile to reach a velocity of 10meters/second in less than 5 seconds.
We must therefore provide some proportional gain,KP> 0 , such that the rise
time is smaller ñ that is, the velocity reaches 10 meters/second in less than
5 seconds. In general, it is intuitive to think that if the rise time is made too
small, the velocity might reach the desired value at the expense of creating
a large overshoot before settling down to the desired value. Naturally then,
we need to be concerned about how we can best control the overshoot as a
consequence of reducing the rise time.
To achieve only proportional control action, we selectKP> 0 and setKD=
KI=0in the controller transfer functionGc(s). Selecting values for any of the
constants is on a purely trial and error basis. However, we should keep in mind
that the proportional gainKPaffects the time constant and consequently the
rate of rise.
From the control system diagram, we see that the controller and plant can
be combined into a single transfer function, namely,

G(s)=Gc(s)Gp(s)=KP

μ

1

ms+b

∂

The closed-loop transfer function therefore is determined as

Y(s)

F(s)

=

G(s)

1+G(s)H(s)

whereH(s)is the feedback transfer function. In this case, since we have chosen
a unity feedback system, we setH(s)=1. We therefore obtain the closed-loop
transfer function of the system as

Y(s)

F(s)

=

G(s)

1+G(s)

=

KP

ms+b

1+msKP+b

=

KP

ms+b+KP