Problem 1:

Consider the system shown in the figure below, which represents control of the angle of a pendulum that has no damping.

(a) What conditions must D(s) satisfy so that the system can track a ramp reference input with constant steady-­-state errors?

(b) For a transfer function D(s) that stabilizes the system and satisfies the condition in part (a), find the class of disturbance w (t) that the system can reject with zero steady-­-state error.

Problem 2:

Consider the system shown in the figure with PI control.

(a) Determine the transfer function from R to Y.

(b) Determine the transfer function from W to Y.

(c) Use Routh's criteria to find the range of (kp, kI) for which the system is stable.

(d) What is the system type and error constant with respect to reference tracking?

(e) What is the system type and error constant with respect to disturbance rejection?

Problem 3:

Consider the automobile speed control system depicted in the figure.

(a) Find the transfer functions from W(s) and from R(s) to Y(s).

(b) Assume that the desired speed is a constant reference r, so that R(s) = ro/s. Assume that the road is level, so w(t) = 0. Compute values of the gains K, Hr, and Hy to guarantee that:

lim y(t ) = ro

t→∞

(c) Repeat part (b) assuming that a constant grade disturbance W(s) = wo/s is present in addition to the reference input. In particular, find the variation in speed owing to the grade change for both the open-­-loop ( Hy  = 0 ) and feedback ( Hy    ¹ 0 )

cases. Using your results to explain: (1) why feedback control is necessary and (2) how the kp  should be chosen to reduce steady-­-state  errors.

(d) Assume that w(t) = 0 and that the gain A undergoes the perturbation A + dA. Determine the error in speed owing to the gain change for both the open-­-loop and feedback cases. How should the gains be chosen in these cases to reduce the effects of dA?