1. a) The block diagram below is for an angular positioning system. The setpoint is a desired angle, which is converted to a desired voltage. This is compared to a feedback voltage from a potentiometer. A PID controller is used to generate an output voltage to drive a DC motor. Simplify the block diagram.
b) Given the transfer function below, select values for Kp, Ki and Kd that will result in a second order response that has a damping factor of 0.125 and a natural frequency of 10rad/s. (Hint: eliminate Ki).
c) The function below has a step input of magnitude 1.0. Find the output as a function of time using numerical methods. Give the results in a table OR graph.
d) The function below has a step input of magnitude 1. Find the output as a function of time by integrating the differential equation (i.e., using the homogeneous and particular solutions).
e) The function below has a step input of magnitude 1. Find the output as a function of time using Laplace transforms.
f) Given the transfer function below; a) apply a phasor/Fourier transform and express the gain and phase angle as a function of frequency, b) calculate a set of values and present them in a table, c) use the values calculated in step b) to develop a frequency response plot on semi-log paper, d) draw a straight line approximation of the Bode plot on semi-log paper.
2. Select a controller transfer function, Gc, that will reduce the system to a first order system with a time constant of 0.5s, as shown below.
