Problem Assignment

Re-consider the DC motor drive system ( K_m =147, τ_m =0.1) that you examined in problem 1.

The goal is to design a feedback controller G_c(s) to meet the following performance requirements:

P1. The steady-state error caused by a unit constant disturbance torque T_L (s) should be zero (Steady-state performance specifications).

P2. The unit step response of the closed-loop system to the reference input should have an overshoot of 15% and a 2% setting time of 0.1 sec (Transient performance specifications).

a. Using the closed-loop transfer function G_YT(s) from the disturbance torque TL (s)to the output Y(s), obtain the steady-state error caused by a unit constant disturbance torque T_L(s) when a PD feedback controller G_c(s)= K_P + K_Ds is used. If you are doing correctly, you will find that the steady-state error obtained is not zero and thus a PD controller cannot satisfy the steady-state performance requirement P1.

b. To satisfy the steady-state performance requirement Pl, we can use the following proportional plus integral and derivative (PID) feedback controller:

G_c(s)= K_P + K_Ds + K₁(1/S)

which can implemented by applying the following physical control input to the plant:

u(t) = K_Pe(t) + K_De^·(t)+ K_{1 0}∫¹(t)dt or U(s) = [K_P + K_Ds + K₁(1/S)] E(s)

Obtain the closed-loop transfer functions G_YR(s) and G_YT(s) when the above PID feedback controller is used.

c. Show that with the above PID controller, as long as the closed-loop system is stable, the steady-state performance requirement P1 will be satisfied.

d. To determine the PID gains K_P, K_D and K₁ to satisfy the transient performance requirement P2, we will use the pole placement technique. Your denominator of the closed loop transfer function should be third order so you will need to place three poles. Determine a desired form for the denominator of the closed loop transfer function such that the dominant poles will be second order with an overshoot of 15% and a 2% setting time of 0.1 sec (as prescribed by requirement P2). Place a third pole at -100.

e. Determine the PID gains K_P, K_D and K₁ which will produce the desired pole locations from part d.