Question

Design a digital PID controller based on the Ziegler_Nichols method for for the following system:

G(s)=1/((s+1)^4 )

use the sampling time of T=0.1 seconds and simulate the resulting controlled system by applying a unit

step and then changing the value of input to another value after some appropriate time

Discus the effect of set point change.

Answer 1

Hello,
          Please find the answer attached as under. Please give a thumbs up rating if you find the answer useful! Have a rocking day ahead!

********** Matlab Code *********

s = tf('s');
h = 1/(s+1)^4;                                  % given transfer function
rlocus(h);                                      % root locus to apply to zeigler nichols tuning method
v=[-4 4 -4 4]; axis(v);
Kp = 2.28;Ti = pi;Td = pi/4;
gc = (Kp/Ti)*(Td*Ti*s*s + Ti*s + 1)/(s);        % PID controller
Ts = 0.1;
Gd = c2d(gc,Ts,'tustin');                       % discrete versions
G = c2d(h,Ts);

sys = feedback(G*Gd,1);                         % closed loop system
t = 0:Ts:40;                                    % time vector
x1 = ones(1,length(t));                         % unit step
x1(1)=0;
x2 = double(t>20);                              % delayed step
x = x1+x2;

figure;
lsim(sys,x,t)                                   % time response
grid;

**** Output ****

?The set point change simply changes the input to the system and acts as if a new step input has been applied to the system. The system will behave with the same dynamics as with a normal step input.