Question

Problem 51: (25 points) Figure 5 is an example of a feedback control system that is designed to regulate the angular position θ(t) of a motor shaft to a desired value θr(t). The signal e(t) represents the error between the measured shaft angle θ(t) and the desired shaft angle θ (t). The Laplace transforms ofa,(t), θ(t), and e(t) are denoted as ΘR(s), θ(s), and E(s), respectively. The control gains Ki and K2 are chosen by the control engineer to achieve a desired transient response and steady-state accuracy (s + 5) K2 Figure 5: Block diagram of a feedback control system. I. (6 points) Calculate the closed-loop transfer function Θ(s)/9R(s). 2. (6 points) Calculate the closed-loop transfer function E(s)/OR(S 3. (4 points) Assume that the controller gains Ki and K2 are chosen so that the closed-loop system is BIBO stable. Use the final value theorem to determine the steady-state value of the error e(t) for the following command inputs (a) The unit-step input 0r(t) - u(t) (b) The unit-ramp input 6, (t) = t u(t). 4. (11 points) Assume that the controller gains Ki and K2 are chosen so that the closed-loop system is BIBO stable. In response to the input θ(t) = πυ(t) and some initial conditions θ(0) and θ(0), it is observed that the closed-loop response has the form where Co, G, T, and wo are constants. (a) (3 points) Use your answer from part 3(a) to determine the value of Co (b) (8 points) Use your answer from part 1 and the Laplace transform pair to express the parameters τ and wo in terms of the controller gains K1 and K2·Note that by appropriately choosing the values of Ki and K2, the control engineer can set the frequency (wo) and time constant (T) of the exponentially decaying transient response Ce cos(wot)u(t).

Answer 1

多女 ers) Sts+3 EG) 3k S(3+5+ 3b) GR(S) 34 3K1 ss+5+ 3k2)

37 e(e) nal value is eCo 3%