You will choose three values of gain for this controller to demonstrate different response types, overdamped, critically damped, and underdamped. You will choose the gains to meet settling time and/or percent overshoot overshoot requirements. You will use MATLAB (in the room next-door) to make the calculations. You will then calculate the appropriate resistor values to give you the appropriate gains. For this section you may find the MATLAB tutorials to be helpful. Refer to www.me.cmu.edu/matlab/html and go to the Root Locus tutorial. The Motor Speed Example is useful too.
|Define the variables num and den to be the numerator and denominator polynomials of the open-loop transfer function, including the 1/s in the controller (but not Ki).|
|Type rlocus(num,den). The root locus, similar to that shown below will be
drawn. Print it out. |
When the system is overdamped, even though the system is second order, one pole is much ``faster'' (i.e. further to the left) than the other. We can approximate the system as a first order system with a single pole (the slower pole). Choose this pole to give the appropriate 98% settling time.
The critically damped system will have as fast a Ts as possible while maintaining no overshoot. The damping ratio of the second order system will be 1, and there will be two equal real roots.
In the underdamped system, you will have two complex-conjugate poles and a damping ratio of less than 1. It turns out that in this system, Ts does not change from the critically damped Ts. Determine the damping ratio to a achieve 25% overshoot and the corresponding poles
|With the root locus plot on the screen, type [K,poles]=rlocfind(num,den) at the command prompt.|
|Click with the mouse on the root locus plot at the desired pole location.|
|Matlab will return the pole location you selected and the corresponding gain, Ki. If you did not get quite the poles you were looking for, try again.|
Tue Apr 20 13:30:59 EST 1999
Next-Closed Loop Response