Which Algorithm "pidtune()" function used for finding PID gains?

I am working on tunning the PID gains for a flyback converter. I am curious about what methods out of the popular lmethods ( Ziegler–Nichols, Aström's AMIGO, Skogestad's Internal Model Control, and Chien-Hrones-Reswick) the "pidtune()" function used to give the PID gains.

I also go throw the pidtune.m file using the "edit pidtune" command in the MATLAB script to have a look what is going on inside the pidtune function. After going through several iteration line by line I am still unable to figure out the answer. To me its look like a kind of heuristic algorithm.

I am seeking assistance from all the community.

Regards

Answer 1

[Sam Chak]

Hi @Hafiz Hamza

The pidtune() command utilizes a proprietary algorithm for tuning the PID gains to achieve a balance between performance and robustness. If you do not specify the desired properties, namely the target phase margin and the closed-loop performance objective, for tuning a PID controller using the pidtuneOptions() command, then the algorithm will determine a crossover frequency based on the given plant dynamics, and then design for a target phase margin of 60 degrees.

https://www.mathworks.com/help/control/getstart/pid-tuning-algorithm.html

[Sam Chak]

If the nominal model of the plant is available or identified, I do not use any of the Ziegler–Nichols, Aström's AMIGO, Skogestad's Internal Model Control, or Chien-Hrones-Reswick methods.

For stable deterministic 2nd-order single-input, single-output (SISO) systems, I use my own intellectual formulas to directly determine the PID gains that fully meet the control performance objectives.

For high-order systems or those that cannot be reduced to a 2nd-order form, I employ a Genetic Algorithm with a custom objective function to search for the "best" PID gains.

[Hafiz Hamza]

@Sam Chak Thank you for your response. But I am confused here "proprietary algorithm".

What are these algorithms?. Yes I have my system model derived. Could you please elaborate why one should not use Ziegler–Nichols, Aström's AMIGO, Skogestad's Internal Model Control, or Chien-Hrones-Reswick methods when he/she has derived the system matehmatical model.

I shall be gratefull if you share your approach ( intellectual formulas) to determine the PID gains.

Regarding genetic algorithims how would you come up with a objective function if let suppose my system is continuously subjected to varying loads?

I am looking forwrad to your response.

Regards

[Sam Chak]

Hi @Hafiz Hamza

The "proprietary algorithm" could be utilizing the in-house 'fmincon()' algorithm or a similar optimizer to tune the PID gains until the optimization objectives are met (e.g., the target phase margin and the closed-loop performance).

You may have misinterpreted my previous comment. However, you can definitely use any of the Ziegler–Nichols, Åström's AMIGO, Skogestad's Internal Model Control, or Chien-Hrones-Reswick methods to tune the PID gains, even when the Plant's transfer function is known. These autotuning methods are model-free, reactive approaches. In other words, you inject an input signal and measure the output response. From the data analysis, you can obtain the response characteristics such as time constant, dead-time, etc.

For example, if you select the Chien-Hrones-Reswick method, these characteristics serve as the parameters to estimate the PID gains using the specified formulas that guarantee 0% (conservative) and 20% (aggressive) overshoot. No complex control design and stability proof is required!!!

Unfortunately, I am unfamiliar with the flyback converter, so I cannot comment on how to construct the objective function. However, I can share my unpublished PID formulas for 0% overshoot, if that would be helpful.

Gp  = tf(20, [1 4.5 64])        % Plant's transfer function (stable)

Gp =
 
         20
  ----------------
  s^2 + 4.5 s + 64
 
Continuous-time transfer function.

Ts  = 0.561;                    % Desired Settling Time of the Closed-loop System
[Gc, Gh] = zeropid(Gp, Ts)      % Scroll down to the end of the script

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

  with Kp = 4.86, Ki = 17, Kd = 0.349
 
Continuous-time PID controller in parallel form.

Gh =
 
  0.821 s^2 + 4.094 s + 16.95
  ----------------------------
  0.3487 s^2 + 4.863 s + 16.95
 
Continuous-time transfer function.

Gcl = feedback(Gc*Gp, Gh)       % Closed-loop System

Gcl =
 
        2.431 s^4 + 67.82 s^3 + 709.4 s^2 + 3298 s + 5749
  -------------------------------------------------------------
  0.3487 s^5 + 12.16 s^4 + 169.5 s^3 + 1182 s^2 + 4122 s + 5749
 
Continuous-time transfer function.

S   = stepinfo(Gcl)             % Performances

S = struct with fields:
         RiseTime: 0.3150
    TransientTime: 0.5610
     SettlingTime: 0.5610
      SettlingMin: 0.9066
      SettlingMax: 0.9999
        Overshoot: 0
       Undershoot: 0
             Peak: 0.9999
         PeakTime: 1.2900

step(Gcl, round(4*Ts, 0)), hold on
step(Gp), grid on, ylim([-0.2, 1.2])
xline(S.SettlingTime, '--', sprintf('Settling Time: %.3f sec', S.SettlingTime), 'color', '#7F7F7F', 'LabelVerticalAlignment', 'bottom')
legend('Closed-loop', 'Open-loop Plant', 'location', 'east')

%% The 0% overshoot method
function [C, H] = zeropid(P, Ts)
    %  Gp is a stable 2nd-order Plant
    %  Ts is the desired settling time

    [numP, denP] = tfdata(minreal(P), 'v');
    a1  = denP(2);
    a2  = denP(3);
    b   = numP(3);
    Tc  = -log(0.02)/Ts;                % Desired time constant

    % Formulas
    k1  = (3*(b^2)*((Tc/b)^2) - a2)/b;      
    k2  = (b^2)*((Tc/b)^3);
    k3  = (3*b*(Tc/b) - a1)/b;       
    k4  = 2*b*((Tc/b)^2);               % P-gain of PID controller
    k5  = k2;                           % I-gain of PID controller
    k6  = Tc/b;                         % D-gain of PID controller
    
    % PID controller and the Compensator
    C   = pid(k4, k5, k6);              % PID control in the forward path
    H   = tf([k3 k1 k2], [k6 k4 k5]);   % Compensator in the feedback path
end