How do I code Ziegler-Nichols Tuning Method to find PID control constants?

I need to use the Ziegler-Nichols Tuning rules to determine the PID control constants for the following system to meet a settling time ts ≤ 5 sec, an overshoot of Mp ≤ 50%, and zero steady state error to a unit step function input.

System: PID control = Kp((Ti*Td*s^2+Ti*s+1)/(Ti*s))

And plant open loop transfer function = 10/(2*s^3+12*s^2+22*s+12)

PID control and plant function are in series with negative feedback loop.

For PID control tuning rules are as follows, Kp=0.6*Kr, Ti=0.5*Pcr, Td=0.125*Pcr

I appreciate your help.

[ahmad nouri]

What command should I enter to find the PID coefficients using the Ziegler-Nichols method?

[kerem]

transfer fonksiyonu verilen bir sistem için PID katsayıları nasıl belirleyebilirim?

[Sam Chak]

Hi @kerem

I suggest posting a new question and providing the transfer function of the given Plant. However, it is important to note that not all types of transfer functions can be stabilized by a PID controller. PID controllers are typically most effective for linear systems of 2nd-order or lower.

Answer 1

[Sam Chak]

Hi @ieng100,

It is now time to provide a conclusion for this control problem. In the Ziegler-Nichols 2nd method, the critical gain can be determined from the Routh–Hurwitz criterion. Simulating the closed-loop system will cause the output to exhibit sustained oscillations. From the oscillations, the critical period can be found using the findpeaks() command.

While the Ziegler-Nichols-tuned PID gains do not strictly satisfy the settling time requirement, it is possible to employ a heuristic approach to quickly tune the PID gains. However, this heuristic approach may not always lead to the optimal solution.

Ziegler-Nichols Second method

%% Process Plant, Gp
s       = tf('s');
Gp      = 10/(2*s^3 + 12*s^2 + 22*s + 12)

Gp =
 
              10
  --------------------------
  2 s^3 + 12 s^2 + 22 s + 12
 
Continuous-time transfer function.

%% Find Critical Gain, Kcr (determined by Routh-Hurwitz criterion)
Kcr     = 12;
Gc      = pid(Kcr);
Gcl     = minreal(feedback(Gc*Gp, 1));  % closed-loop system
[y, t]  = step(Gcl, 10);
figure(1)
plot(t, y), grid on, xlabel('t'), ylabel('y(t)'), title('Step Response')

%% Find Critical Period, Pcr
[pk, lo]= findpeaks(y, t);
Pcr     = lo(2) - lo(1)

Pcr = 1.8894

%% Apply Ziegler-Nichols Tuning Rules
Kp      = 0.6*Kcr;
Ti      = 0.5*Pcr;
Td      = 0.125*Pcr;

%% PID controller in standard form
Gc      = pidstd(Kp, Ti, Td)

Gc =
 
             1      1          
  Kp * (1 + ---- * --- + Td * s)
             Ti     s          

  with Kp = 7.2, Ti = 0.945, Td = 0.236
 
Continuous-time PID controller in standard form

%% Closed-loop system
Gcl     = minreal(feedback(Gc*Gp, 1))

Gcl =
 
        8.502 s^2 + 36 s + 38.11
  -------------------------------------
  s^4 + 6 s^3 + 19.5 s^2 + 42 s + 38.11
 
Continuous-time transfer function.

figure(2)
nfo     = stepinfo(Gcl);
disp(nfo.SettlingTime)

    5.0045

disp(nfo.Overshoot)

   44.2727

step(Gp,  10), hold on
step(Gcl, 10), grid on, hold off
xline(nfo.SettlingTime, '--', sprintf('Settling Time: %.3f sec', nfo.SettlingTime), 'color', '#7F7F7F', 'LabelVerticalAlignment', 'bottom')
legend('Original System', 'Closed-loop System', '')

Rational number PID gains

%% PID Controller
kp  =  9/10;
ki  = 18/25;
kd  =  1/8;
Tf  =  5/12;
Gc  = pid(kp, ki, kd, Tf)   % PID controller

Gc =
 
             1            s    
  Kp + Ki * --- + Kd * --------
             s          Tf*s+1 

  with Kp = 0.9, Ki = 0.72, Kd = 0.125, Tf = 0.417
 
Continuous-time PIDF controller in parallel form.

Gcl  = minreal(feedback(Gc*Gp, 1))   % closed-loop system

Gcl =
 
                 6 s^2 + 14.4 s + 8.64
  ---------------------------------------------------
  s^5 + 8.4 s^4 + 25.4 s^3 + 38.4 s^2 + 28.8 s + 8.64
 
Continuous-time transfer function.

figure(3)
step(Gp),  hold on
step(Gcl), grid on
S    = stepinfo(Gcl);
xline(S.SettlingTime, '--', sprintf('Settling Time: %.3f sec', S.SettlingTime), 'color', '#7F7F7F', 'LabelVerticalAlignment', 'bottom')
legend('Original System', 'Closed-loop System', '')