68 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL
Figure 2.34. Step response withKd= 500the uncontrolled function. We now take the liberty to increase the proportional
gain, say to 500, and try the simulation. Note that our objective is to bring the
system response as close to the step input value as possible, in this case 1.0, as
fast as we can without causing much overshoot. Figure 2.34 illustrates the effect
of increasing the proportional gain. The overshoot is definitely unsatisfactory.
It must be reduced to something less than 5%. As far as the settling time is
concerned, in this example we are really not concerned because the system we
are dealing with has no real-world connection and as such, all we can say is
that the performance with control is better than without control. We will see
later on, in examples that have a physical meaning, that settling time becomes
a very important criteria in the design of controllers. For now, it is important
to realize what the proportional, derivative, and integral parameters can do to
improve the system response.
We proceed further to examine what we need to do to bring the overshoot
down to some tolerable value, typically to less than 5% as we suggested earlier.
For this, we need to add derivative control into the system. The following
Matlabcode illustrates how this can be done:
kp=500; %Proportional gain
kd=10; %Derivative gain
numc=[kd kp] %Define the numerator polynomial (sKd+Kp)
sysopenloop=tf(numc,densys); %Open-loop TF
sysfeedback=[1]; %Unity feedback
sysclosedloop=feedback(sysopenloop,sysfeedback);%Closed-loop TF
step(sysclosedloop,0:0.001:2.0); %Step response
As shown in Figure 2.35, the derivative control parameter has most definitely
provided damping effect and shows marked improvement in the overshoot com-
pared to the previous case with no derivative control. We also see that the
system has settled down more rapidly. So our conclusion is that we can further
increase the derivative control so that the response can be brought closer to a
critically damped response. Let us try one more simulation in which we increase
the derivative gain to 100. The result of this is illustrated in Figure 2.36.