Control Systems Assignment

Question 1. The block diagram of a feedback control system is shown in Figure 1.

1) Find the closed-loop transfer function Y(s)/R(s).

2) Determine the poles and zeros of the closed-loop system.

3) Calculate the output response y(t) to a unit step input R(s) = 1/s.

Question 2. Consider the signal-flow graph in Figure 2. Applying Mason's gain formula, obtain the transfer function Z(s)/X(s).

(Note: you need to show the steps of your work)

Question 3.

A control system has the transfer function

Y(s)/R(s) = (3(s+3)(s+4))/((s+1)(s+2)(s+5)) = (3s² + 21s + 36)/(s³ + 8s²+ 17s +10)

(1) Draw the signal-flow graph state model in

(a) phase variable canonical form
(b) input feedforward canonical form
(c) diagonal canonical form

(2) Find the state space representation x.(t) = Ax(t) + Br(t), y(t) = Cx(t) + Dr(t) for each canonical form in (1).

Question 4. Consider the feedback control system in Figure 3, where R(s) is the reference input and D(s) represents the disturbance.

(a) Derive the transfer function Y(s)/D(s).

(b) Assume the reference input r(t) = 0 and that the parameter K operates within the range that ensures closed-loop stability. Use the final value theorem to find the steady-state response of y(t) to a unit step disturbance.

(c) Let the disturbance d (t) = 0. Find the steady-state error e_ss to a unit ramp input r(t) = t.

(d) Let the disturbance d (t) = 0. Determine the sensitivity of the closed-loop system to the parameter K.

(e) Let the disturbance d (t) = 0. Using the Routh-Hurwitz criterion, determine the range of K such that the closed-loop system is stable.