Question

Please show all steps.

1. Begin with zero gains and then increase Kp until you observe sustained oscillations, mark this value Km. 2. Mark the approximate frequency of oscillation wm 3. Then set the gains as follows: KP 0.6Km Note that if you are able to use Root Locus methods, you can formulate these values exactly 1. Use the Ziegler-Nichols method to get an initial tuning of a PID controller that improves the response of a system represented by the following openloop dynamics 0.75 s 2. What is the cross over frequency, gain margin, and phase mar- gin of the uncompensated system? 3. Compare the cross over frequency, gain margin, and phase margin with the PID compensated system. 4. Compare the step responses of both. Indicate how you gener- ated these responses

Answer 1

PID controller design:

increase the gain for the step response of the system such that the system oscillates . that gain is Km and the frequency of oscillations is Wm. this can be obtained from the MATLAB sisotool:

s=tf('s');
g=0.75*s/(s^2*(s+1)*(s+10));
sisotool(g)

in control and estimation tool box manager go to automated tuning and fill the block paramaters as below and click update compensator.

next go to analysis plot and fill the parameters as below and click show analysis plot

PID controller design: increase the gain for the step response of the system such that the system oscillates . that gain is Km and the frequency of oscillations is Wm. this can be obtained from the MATLAB sisotool: s = tf('s'): g = 0.75*s/(s^2*(s + 1)*(s + 10)): sisotool(g) in control and estimation tool box manager go to automated tuning and fill the block parameters as below and click update compensator. next go to analysis plot and fill the parameters as below and click show analysis plot now increase the gain of the compensator in the compensator editor window till oscillations are observed in the response. that gain was found to be 60 and period is 0.3278 The PID controller is K_p = 0.6* 60 = 36 K_d = 86.25 K_i = 3.75 the controller is Gc = K_p + K_i/s + K_d s 2 . margins and cross over frequencies of uncompensated system s = tf('s'): g = 0.75*s/(s^2*(s + 1)*(s + 10)): margin(g) 3 . margins and cross over frequencies of compensated system s = tf('s'): g = 0.75*s/(s^2*(s + 1)*(s + 10)): gc = 36 + 86.25*s+(3.75/s): margin(g) figure margin(g*gc) step response of the sysetms:

now increase the gain of the compensator in the compensator editor window till oscillations are observed in the response. that gain was found to be 60 and period is 0.3278

The PID controller is

Kp= 0.6* 60 = 36

Kd= 86.25

Ki=3.75

the controller is Gc = Kp + Ki / s + Kd s

2 . margins and cross over frequencies of uncompensated system

s=tf('s');
g=0.75*s/(s^2*(s+1)*(s+10));
margin(g)

3 . margins and cross over frequencies of compensated system

s=tf('s');
g=0.75*s/(s^2*(s+1)*(s+10));
gc=36+86.25*s+(3.75/s);

margin(g)

figure
margin(g*gc)

step response of the sysetms:

s=tf('s');
g=0.75*s/(s^2*(s+1)*(s+10));
gc=36+86.25*s+(3.75/s);
step(feedback(g,1),feedback(g*gc,1))
legend('uncompensated system ' , 'compensated system')