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Script d2017_10_25_gyak6_PID

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File: d2017_10_25_gyak6_PID.m

Author: Peter Polcz ppolcz@gmail.com

Created on 2017. October 25.

Reference tracking using PID controller. Signal attributes

$$H(s) = \frac{s^2 + 3s + 2}{s^3 + 2s^2 - 6s + 8}$$

H = tf([1 3 2],[1 2 -6 8]);

We design a PID controller $C(s)$.

C = pidtune(H,'pid');

The resulting system: $H_e(s) = \frac{C(s) H(s)}{1 + C(s) H(s)}$.

He = C*H / (1 + C*H);

Generate the step response function.

T = 10;
t = linspace(0,T,1000);

u = t*0;
u(t > T/20) = 1;

Simulate

y = lsim(He,u,t);

Rise time plot

figure, risetime(y,t), xlim([0 T/5]), title 'Rise time plot'

Output:

ans =
    0.0810

Settling time plot

figure, settlingtime(y,t,2.68), title 'Settling time plot'

Output:

ans =
    2.4259

Overshoot plot

figure, overshoot(y,t), title 'Overshoot plot'

Output:

ans =
   15.6977

Undershoot plot

figure, undershoot(y,t), title 'Undershoot plot'

Output:

ans =
   -1.0401

Step info

stepinfo(y,t)

Output:

ans = 
  struct with fields:

        RiseTime: 0.0816
    SettlingTime: 3.0669
     SettlingMin: 0.9335
     SettlingMax: 1.1536
       Overshoot: 15.3693
      Undershoot: 0
            Peak: 1.1536
        PeakTime: 0.7407

Oscillating system. Undershoot

$$H(s) = \frac{1}{(s+1)(s^2 + 0.6s + 1.09)}$$

s = tf('s');
H = 1 / ( (s+0.3-1i)*(s+0.3+1i)*(s+1) );
[y,t] = step(H);

undershoot(y,t)
title 'Undershoot plot'

Output:

ans =
    9.4668

Settling time of a noisy signal

load('transitionex.mat', 'x')
[s,slev,sinst] = settlingtime(x,10)
settlingtime(x,10)

Output:

s =
    1.8901
slev =
    2.2400
sinst =
   23.3901
ans =
    1.8901

PID Tuner

$$H(s) = \frac{s^2 + 3s + 2}{s^3 + 2s^2 - 6s + 8}$$

The resulting system: $H_e(s) = \frac{C(s) H(s)}{1 + C(s) H(s)}$.

H = tf([1 3 2],[1 2 -6 8]);
figure(1), pzmap(H)
C = pidtune(H,'pid');
pidTuner(H,C)