HA2.1 A unity feedback control system has an open loop transfer function G(S) = 9/S(S+3). Find the rise time, percentage over shoot, peak time and settling time. HA2.2 Given the following differential equation: y'(t) = -5y(t) + u(t) with initial value y(0) = 4. Assume that the input variable u(t) is a step of amplitude 1 at time t=0. a. Calculate the response in the output variable, y(t), using the Laplace transform. b. Calculate the steady-state value of y(t) using the Final Value Theorem. c. Also calculate the steady-state value, ys, from y(t) directly. Are all these values of ys the same?
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1. Given the system shown in Figure 1, determine: a. The overall transfer function. b. The damping ratio when the percentage of overshoot in the unit-step response is 10%. c. The values of K and Kf when the peak time is 1.5 sec. d. The rise time. e. The settling time for 2% and 5%. f. Compare the peak time and rise time by using VisSim simulation. 2. By using the Routh-Hurwitz tabulation method, determine whether the unity feedback system of Figure 2 is stable if: a. How many poles are in the right half-plane and left half-plane in the system. b. Verify the system stability by using VisSim simulation.
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Q2. (a) For a first-order process plant with a time constant of 5 seconds and a gain of 2, determine a controller gain K(s) = K in a unity feedback system such that the closed-loop time constant is placed at TCL = 0.5. (b) Sketch the open-loop and closed-loop time responses of the system in (a) to a unit step input, noting carefully the steady state and time constant of both systems. (c) Given K(s) = K/s, find the closed-loop transfer function and calculate the damping factor and natural frequency for K = 1. Sketch the closed loop response to a unit step for this gain. (d) Find the range of gain K for which the closed-loop system in (c) is underdamped, critically damped, or overdamped.
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