84 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL
Synthesize a set of PID parameters for different values ofαin the interval
[0.1; 20]. Discuss your results.
- For the inverted pendulum problem example in Section 2.1.2, consider the
following parameters
M =10.5kg
m =0.5kg
b =0.01 N/m/s
l =0.6m
I =0.016 kg m^2
Consider the effects of changing proportional gainKP over a range of
0 − 1000 , the derivative gainKDover a range of 0 − 500 ,andtheintegral
gainKIover the range of 0 − 10. Obtain a suitable set of PID parameters
that will meet the standard criteria of less than5%overshoot and a two
second settling time.
- For the inverted pendulum problem, suppose that it is necessary for the
cart to return to the same position or to a desired position. Include cart
control in your model and obtain simulations using Simulink.
17.Project: For the ardent student, modeling the ancient control system
developed by Hero should be a challenge. While there are no specific
guidelines that can be given, commonsense and reasonableness account for
obtaining the mathematical model of the system. Some hints in modeling,
however, can be useful. A simple lag, for example, is given by
G 1 (s)=
1
s+T 1
Note that the inverse Laplace transform yields
g 1 (t)=e−T^1 tu(t)
This should provide the insight into choosing the appropriate time con-
stants, likeT 1 , so that fast and slow responding actions can be appro-
priately modeled. The goal, of course, is to design a controller that will
perform the opening and closing of the temple doors.
