A linear ODE model of the DC motor with negligible armature inductance (La = 0) and with

Chapter 4, Problem 4.22

(choose chapter or problem)

A linear ODE model of the DC motor with negligible armature inductance (La = 0) and with a disturbance torque w was given earlier in the chapter; it is restated here, in slightly different form, as where m is measured in radians. Dividing through by the coefficient of m, we obtain where With rotating potentiometers, it is possible to measure the positioning error between and the reference angle r or e = ref m. With a tachometer we can measure the motor speed m. Consider using feedback of the error e and the motor speed m. in the form where K and TD are controller gains to be determined. (a) Draw a block diagram of the resulting feedback system showing both m and m as variables in the diagram representing the motor. (b) Suppose the numbers work out so that a1 = 65, b0 = 200, and c0 = 10. If there is no load torque (w = 0), what speed (in rpm) results from va = 100 V? (c) Using the parameter values given in part (b), let the control be D = kp + kDs and find kp and kD so that, using the results of Chapter 3, a step change in ref with zero load torque results in a transient that has an approximately 17% overshoot and that settles to within 5% of steady-state in less than 0.05 sec. (d) Derive an expression for the steady-state error to a reference angle input, and compute its value for your design in part (c) assuming ref = 1 rad. (e) Derive an expression for the steady-state error to a constant disturbance torque when ref = 0 and compute its value for your design in part (c) assuming w = 1.0.