How can I adjust a controller?

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Bob
Bob on 28 Mar 2016
Commented: Ced on 6 Apr 2016
I am trying to adjust a controller according to the method above.
I can't understand the how to apply this method, this is what I've tried to do so far.
Output1 = initial signal
Output2 = Kp reaches ultimate gain Ku at which the output of the control loop has stable and consistent oscillations.
Output3 = I set the parameters.

Accepted Answer

Ced
Ced on 28 Mar 2016
Edited: Ced on 29 Mar 2016
Hi
Your basic procedure is correct, but you mixed up the parameters a bit. I admit that this is a bit confusing, but the structure that the Ziegler-Nichols table is written for is
C(s) = Kp*(1+1/(s*Ti)+s*Td)
I.e. you get Ti and Td, and not Ki and Kd.
Your measured values seem a bit off btw, did you read them from the plot?
You can check the exact values with e.g. the margin command, which gives you the gain margin and the corresponding frequency (i.e. your Ku and Tu). Note that the frequency will be given in rad/s and not Hz, so you need to convert that.
You should also be aware that ZN is a heuristic method, and will not work or return meaningful results for arbitrary plants.
Here a small script to compute the Ziegler Nichols Parameters for a system (programmatically, not with simulink):
% Ziegler nychols example
clear all
close all
clc
% spring damper system with eigenfrequency at 10 rad/s and additional
% rolloff --> Ziegler Nichols returns huge overshoot
M = 1;
D = 8;
K = 100;
% P = tf(90,[M D K])*tf(1,[0.015 1]);
% Better behaved:
s = tf('s');
P = 1/(s+1)^5;
% get gain margin
[ Ku, ~, WKu, ~ ] = margin(P);
fKu = WKu/(2*pi);
Tu = 1/fKu;
fprintf('Limit Kp: %g, Period: %g [s]\n',Ku,Tu)
%%Generate a few PID controllers for comparison
Kn = 0.01; % selecting "real" PID like in your simulation
% select some slow gains
Kp0 = 0.4;
Ki0 = 0.1;
Kd0 = 0.1;
C_PID_0 = pid(Kp0,Ki0,Kd0,Kn);
% set Kp gain limit
Kplim = 0.9*Ku;
Kilim = 0;
Kdlim = 0;
C_PID_lim = pid(Kplim,Kilim,Kdlim,Kn);
% set Ziegler Nichols gains
KpZN = 0.6*Ku;
TiZN = 0.5*Tu;
KiZN = KpZN/TiZN;
TdZN = 0.125*Tu;
KdZN = KpZN*TdZN;
C_PID_ZN = pid(KpZN,KiZN,KdZN,Kn);
fprintf('Ziegler Nichols Parameters:\n')
fprintf('Kp: %g, Ki: %g [s], Kd: %g [s]\n',KpZN,KiZN,KdZN);
%%Close the loop
G0 = feedback(P*C_PID_0,1);
G1 = feedback(P*Kplim,1);
G2 = feedback(P*C_PID_ZN,1);
%%Check Nyquist plot
figure(1)
nyquist(P*C_PID_0,P*C_PID_lim,P*C_PID_ZN)
grid on
axis([ -2.8419 3.6822 -4.7619 4.3024 ])
legend('low gains','limit','Ziegler Nichols')
%%Plot
figure(2)
step(G0,G2)
ylim([- 2 2 ])
legend('Low Gains','Ziegler Nichols','Location','SouthEast')
Cheers
EDIT: typo in ZN gain calculation, fixed
  5 Comments
Ced
Ced on 6 Apr 2016
Yes, I meant using x = [ X1 dX1 E dE ] as your state, where E is the deflection.
Sure, but feel free to continue posting here so others can tune in. I will see it.

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