Assignment Background

This assignment is a continuation of Assignment 1 and 2, where you have modelled the DC-motor driven positioning system. No material from Assignment 1 or 2 is required to complete this assignment.

Recall that V (s) = L {v(t)} is the voltage supplied to the DC motor and Y (s) = L {y(t)} is the dartboard’s vertical position. The transfer function GOL(s) = V Y ( ( s s ) ) is exactly

GOL(s) = s 6 + 3167s 5 + 646300s 4 + 135539583s 3 + 407750000s 2 + 13530000000s ? 0.00003/1250s 2 + 250000s + 75000000

Please run the code in MATLAB:

num = [1250 250000 75000000];

den = [1 3167 646300 135539583 407750000 13530000000 -0.00003];

G_OL = tf(num,den);

Assume zero initial conditions when required.

Part 1: Dartboard Positioning System, Sinusoidal Inputs

Q1-2. Using MATLAB, plot the Bode diagram of GOL(s) (with a grid). Observe the behaviour at low frequencies and at high frequencies (for both magnitude and phase).

(Q1) Is the gain larger at low or high frequencies? [Canvas Input: Select Low or High]

(Q2) Is the phase lag smaller at low or high frequencies? [Canvas Input: Select Low or High]

Q3-4. Apply sinusoids of varying amplitude (v = Asin(t)) and observe the steady-state behaviour of y(t) (i.e. yss(t)). For each question below, you are given a specific value A. Use this value of A to answer the question.

(Q3) What is the phase shift between yss(t) and v(t) (The answer should be in degrees and in the range [180o , 180o ))?

(Q4) What is the average value of yss(t)?

[Canvas Input: Two signed numbers - 5% tolerance allowed]

Q5-6. Now apply sinusoids of varying frequency (v = 1000 sin(ωt)) and observe the steady state behaviour of y(t) (i.e. yss(t)). For each question below, you are given a specific value ω, use this value of ω to answer the question.