A First Course in FUZZY and NEURAL CONTROL

2.7. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL 75

Gc(s)=KP+

KI

s

+KDs=

KDs^2 +KPs+KI

s

The objective in the controller design is to select the appropriate parameters
forKP,KI,andKDto satisfy all design criteria. Once again, we follow a
systematic approach to selecting the parameters. It is important to note that in
most cases a PI or PD controller may suffice, and there is really no need for the
development of a PID controller. As we will see in this example, a PD controller
could adequately bring the system response to within tolerable limits. We do,
however, show how a full PID controller may be implemented.

Figure 2.43. Response with proportional gain

To decrease the settling time, we need to add a proportional gain. The
proportional gain increases the rate of rise in the system response. If we add
only a proportional gainKP> 0 ,withKI=KD=0,thenGc(s)=KP.We
cannowsetupthem-fileand proceed withMatlabsimulations as follows:

%Add proportional control
Kp=10; %This is an arbitraryfirst choice
numcf=[Kp]; %Numeratorofcontrollertransferfunction
dencf=[1]; %Denominatorofcontrollertransferfunction
numf=conv(numcf,num);
%Convolve the numerators of the controller and plant
denf=conv(dencf,den);
%Convolve the denominators of the controller and plant
sysf=tf(numf,denf);
%Form the forward transfer function for controller and plant
sys=feedback(sysf,sys2); %Obtain the overall closed-loop transfer function
step(sys,0:0.001:0.2); %Obtain the step response
Notice in Figure 2.43 that the addition of a proportional gain, while sharply
increasing the rise time, causes significant overshoot in the response and still
does not meet the settling time criteria. Note that the choice of proportional
gain is arbitrary. Large values cause excessive oscillation, and small values
do not allow the settling time to be met. Some trial and error solutions are
needed to obtain what may be considered satisfactory. This is illustrated in