The objective of this problem is to use the Ziegler and Nichol approach to develop the PID controller equations that allow an embedded system to control the DC motor presented in Exercise D13.10 (i.e., work through the steps of Exercise 13.10 for a PID system).

Exercise 13.10

The objective of this problem is to use the Ziegler and Nichol approach to develop the PI controller equations that allow an embedded system to control a DC motor. The state variable is speed, which is measured using 16-bit input capture and has a measurement resolution of 1 rpm. The input capture device driver repeatedly updates a global variable, called Speed. This 16-bit unsigned variable has units of rpm and a range of 0 to 20000. The microcontroller uses pulse-width modulation to control power to the motor. The controller software writes to a global variable, called Duty, which ranges from 0 (0%) to 10000 (100%). The following plot shows an experimental measurement obtained when Duty is changed from 2500 to 5000. The desired speed is stored in the global variable, Desired, which has the same units as Speed. Design a fixed-point PI controller that takes Speed and Desired as inputs and calculates Duty as an output. From the response graph in Figure 13.30, estimate the L and R parameters of the Ziegler and Nichol method. How often should the controller be executed? Show just the equations (no software or hardware is required), calculating Duty as a function of Speed and Desired.

Figure 13.30