Question

The parameters are as follows

k=10 a=0.50 b=0.3 c=0.6 d=9 w_1=12 w_2=15 Kv=30

θ(r) r(t) ut) elo) Figure 1: Feedback control system A pulley and belt transmission has a linearized relationship between the driven pulley angle e() in degrees and the input torque u(t) in Newton meters given by the following differential equation du(t) A feedback control system (illustrated in Figure 1) needs to be designed such that the closed-loop system is asymptotically stable and such that the following design criteria are met:

A feedback control system (illustrated in Figure 1) needs to be designed such that the closed-loop system is asymptotically stable and such that the following design criteria are met:

1. the gain crossover frequency w_c should be between w₁ and w₂.

2. the steady-state error should be zero in response to a unit step reference.

3. the velocity constant should be greater than K_v (in other words, the steady-state unit ramp error should be less than 1=K_v).

4. the phase margin should be at least 55^o.

If the four performance criteria are met, further iteration of the controller may be undertaken (if you wish) to minimise the settling time of the step response from r(t) to y(t). If you cannot meet any of the design criteria, get as close as you can while ensuring closed-loop stability, and explain where and why compromises were needed.

This task will be approached incrementally, beginning with a proportional controller and finishing with a lead-lag controller.

All graphs should be clearly labelled and legible, and all design steps
should include some justification. MATLAB or a similar computational package may be used for any of the
calculations or graphs required.

2

1. Proportional controller

Derive the transfer function G(s) from the torque to the pulley angle. Then consider first a proportional controller C(s) = K_p. Find a proportional gain K_p1 such that the constraint on the gain crossover frequency (criterion 1) is satisfied. Check, using a Bode plot, that such proportional gain does not satisfy the phase margin requirement and give the corresponding phase margin. Find a different proportional gain K_p2 that satisfies the constraints on the velocity error (criterion 3). Check, using the Bode diagram, that such propor-tional gain does not satisfy the remaining design requirements, and graph the tracking error in response to a unit ramp input.

Answer 1

2- θ(s) Jd.wuỹ.tce. 6 wỳ し12. 2 2ole di 301 B

atteaionhed 3). 9890. 13 0-148is phae mein

2 21 9-214 The phane mn met
matlab:

clc;

clear all;

close all;

s=tf('s');

g=(5*s+3)/(s*(s^2+0.6*s+9));

kp=31.989;

kv=90;

margin(kp*g);grid

figure

margin(kv*g);grid

File Edit View Insert Tools Desktop Window Help Gm Inf dB (at Inf rad/s) . Pm-0.148 deg (at 13 rad/s) 60 40 10 2 102 10 10 Frequency (rad's)

File Edit View Insert Tools Desktop Window Help Bode Diagram Gm-Inf dB (at Inf rad s), Pm-0.0321 deg (at 21.4 radis) 60 40 10 2 102 10 10 Frequency (rad's)