How can I adjust a controller?

I am trying to adjust a controller according to the method above.

Wikipedia

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.

Answer 1

[Ced]

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

[Ced]

I just had a look at my code from last night, there was a typo in the gain calculations, please update them in your code!

The procedure itself should work, however the two zeros at s = 0 are bound to give you problems. Just give it a try! If it doesn't work, ZN also have rules for e.g. PI controllers, you could try this instead.

[Bob]

I have tried to apply, according to your example, to system of 2 DOF but I get errors.

I can attach the code if you want to.

Results

Displacement of Mass 1, X1:

Ku = 14.2481 
WKu = 23.3366 
fKu = 3.71413 
Tu = 0.269242 
Kp = 8.54883 
Ki = 63.5029 
Kd = 0.287713

Dispalcement of Mass2 2, X2:

I have got error

Ku = Inf 
WKu = NaN 
fKu = NaN 
Tu = NaN 
Kp = Inf 
Ki = NaN 
Kd = NaN

Deflection, X1-X2:

Warning: The closed-loop system is unstable. 
> In ctrlMsgUtils.warning (line 25)
  In DynamicSystem/margin (line 65)
  In Untitled (line 35) 
Ku = 0.892531 
WKu = 16.5515 
fKu = 2.63426 
Tu = 0.379613 
Kp = 0.535519 
Ki = 2.82139 
Kd = 0.0254113

[Ced]

Hi

Sorry, totally swamped at the moment. As a general rule, ZN should be applicable to 4th order systems, but it depends a bit on the shape of your dynamics. From what you show me, you have an infinite gain margin in X2, meaning that you can increase Kp as much as you want, it will (in theory) never destabilize the system, and the basic idea of ZN does not work anymore.

There is quite a lot of information out there though if you google a bit:

Ziegler Nichols on Active Suspension System

Details about ZN and it's applicability

Why is ZN not applicable to 2nd order systems

Have you tried using X1 and the deflection as states?

For the record, suspensions are quite a tricky business, that's why there are thousands of publications on it. There are also a lot of control suggestions. Unless you are forced (assignment or so) to use ZN, I would suggest picking one of them for starters. The first link above could be interesting for you.

[Bob]

Thank you for your answer.

It's not compulsory to use ZN method, I have read some papers about suspensions system and they used this method and I thought it is the most appropriate. I just need to use a controller to improve the suspension through a method and not by myself (manually). I will read the links you gave me. The first one seems very good.

I don't understand what you mean here "Have you tried using X1 and the deflection as states?" You refer to state space? If you mean state space, no I haven't tried that. My knowledge is limited. I know a few things about transfer function and simulink.

Thanks again for your help. Can I contact to you via email if I have any question?

[Ced]

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.