PID controller behaves strangely for larger Ts values

I am designing a PID controller for a rudimentary elevator system

The elevator has to take more than 5s to arrive at the next floor but no more than 6s

The transfer function is as follows:

syms Ki Kd Kp s 

M = 1019;
B = 100;
K = 19875;

num=[1];
den=[M B K];
G = tf(num, den)

G =
 
             1
  ------------------------
  1019 s^2 + 100 s + 19875
 
Continuous-time transfer function.

Zeta = 1;
Ts = 5;
Wn = 0.8;

Using the PID function

GC = (Kp*s + Ki + Kd*s^2)/s

The close loop function is found to be:

T = (Kp*s + Ki + Kd*s^2)/(M*s^3 + B*s^2 + K*s + Kp*s + Ki + Kd*s^2)

Comparing this with the ITAE ideal tranfer function for a third order system yields the following equations

ITAE_tf3 = vpa(Wn^3/(s^3 + (Wn)*s^2 + (Wn^2)*s + Wn^3))
Kp = (2.15)*(M)*(Wn^2)-(K)

Kp = -1.8473e+04

Ki = M*(Wn^3)

Ki = 521.7280

Kd = (1.75)*(M)*(Wn)-B

Kd = 1.3266e+03

However the large Kp value drives the system downwards when it should be going up and the large integrator value also causes some issues

Changing zeta and ts gives me a results that is almost what I need

overshoot = 10/100;
Zeta = (-log(overshoot))/sqrt(pi^2+(log(overshoot)^2));
% Zeta = 1
Ts = 2.4;;
Wn = 4/(Zeta*Ts);

But of course the Ts value is too high.

Any help would be greatly appreciated

Answer 1

[Sam Chak]

Hi @Wynand

There are a few things for which you can find detailed explanations in control theory books. In short, the ideal ITAE third-order transfer function is different from your actual PID-controlled third-order transfer function.

Blunder: Two coefficients are missing in your version of the ITAE third-order transfer function.

While your derived PID-controlled third-order transfer function is correct, to make an apples-to-apples comparison, we must divide both the numerator and the denominator by M.

%% Plant TF
M   = 1019;
B   = 100;
K   = 19875;
num = [1];
den = [M B K];
Gp  = tf(num, den)

Gp =
 
             1
  ------------------------
  1019 s^2 + 100 s + 19875
 
Continuous-time transfer function.

%% Ideal TF
wn  = 3;    % Tune this parameter only, DO NOT individually tune P-I-D gains (Always use single parameter tuning!)
Gi  = tf(wn^3, [1, 1.781*wn, 2.171*wn^2, wn^3])

Gi =
 
                27
  ------------------------------
  s^3 + 5.343 s^2 + 19.54 s + 27
 
Continuous-time transfer function.

figure(1)
step(Gi, 5), grid on
title('Step response of Ideal TF')

%% Pole placement
ki  =       M*wn^3;
kp  = 2.171*M*wn^2 - K;
kd  = 1.781*M*wn   - B;
Gc  = pid(kp, ki, kd)

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

  with Kp = 35.2, Ki = 2.75e+04, Kd = 5.34e+03
 
Continuous-time PID controller in parallel form.

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

Gcl =
 
    5.245 s^2 + 0.03458 s + 27
  ------------------------------
  s^3 + 5.343 s^2 + 19.54 s + 27
 
Continuous-time transfer function.

zero(Gcl)   % to be cancelled out later

ans = 
  -0.0033 + 2.2689i
  -0.0033 - 2.2689i

figure(2)
step(Gcl, 5), grid on
title('Step response of Closed-loop TF')

%% Design of Prefilter, Gf
[num, den] = tfdata(Gcl, 'v');
Gf  = tf(den(end), num)

Gf =
 
              27
  --------------------------
  5.245 s^2 + 0.03458 s + 27
 
Continuous-time transfer function.

%% Filtered Closed-loop TF
Fcl = minreal(series(Gf, Gcl))

Fcl =
 
                27
  ------------------------------
  s^3 + 5.343 s^2 + 19.54 s + 27
 
Continuous-time transfer function.

figure(3)
step(Fcl, 5), grid on
title('Step response of Filtered Closed-loop TF')

My preferred control architecture: Simple yet accurate!

%% My PID controller
Ts       = 5.5;                 % Desired Settling Time (single tuning parameter)
[Gc, Gh] = chakpid(Gp, Ts)      % Gc in forward path, Gh in feedback path

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

  with Kp = 1.03e+03, Ki = 367, Kd = 725
 
Continuous-time PID controller in parallel form.

Gh =
 
  2.862 s^2 - 25.29 s + 0.5059
  ----------------------------
     s^2 + 1.423 s + 0.5059
 
Continuous-time transfer function.

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

Gcl =
 
    0.7113 s^4 + 2.024 s^3 + 2.159 s^2 + 1.024 s + 0.1821
  ---------------------------------------------------------
  s^5 + 3.556 s^4 + 5.059 s^3 + 3.598 s^2 + 1.28 s + 0.1821
 
Continuous-time transfer function.

S        = stepinfo(Gcl)        % Performances

S = struct with fields:
         RiseTime: 3.0891
    TransientTime: 5.5000
     SettlingTime: 5.5000
      SettlingMin: 0.9002
      SettlingMax: 0.9995
        Overshoot: 0
       Undershoot: 0
             Peak: 0.9995
         PeakTime: 10.7900

figure(4)
step(Gcl, round(3*Ts, 0)), grid on, grid minor

%% Double compensation control scheme
function [C, H] = chakpid(P, Ts)
    %  Gp is a 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

    %% The 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

    C   = pid(k4, k5, k6);                      % Classical PID controller
    H   = minreal(tf([k3 k1 k2], [k6 k4 k5]));  % Compensator at the Sensor path
end