Question

Using MATLAB. We want to control output(y) using PID control in Kds? +Kps+Ki C(s) S Input(r) is a magnitude1 step. Plant is given by 1 (s+1)(s+2)(s+59) controller plant 14 y C(s) P(S) a) Calculate Closed Loop characteristics and steady-state error(unity feedback and Kp=1, Kd=1, Ki=0)) 2.Using automatic PID tuning function, reduce steady-state error=0 and report Kp=?, Kd=? And Ki=?

Answer 1

Question (a)

clc;
clear;
close all;

Kp = 1;
Kd = 1;
Ki = 0;

s = tf('s');
P = 1/((s + 1)*(s + 2)*(s + 59));

C = pid(Kp, Ki, Kd);

sys = feedback(C*P, 1);

disp('The closed loop transfer function is :');
sys

t = 0:0.0001:5;
u = ones(size(t));

y = step(sys, t);

e = u - y';

plot(t, y, 'k', 'linewidth', 2);
grid
xlabel('Time, t (sec)');
ylabel('Amplitude');
title('Step Response');

figure
plot(t, e, 'k', 'linewidth', 2);
grid
xlabel('Time, t (sec)');
ylabel('Amplitude');
title('Error');

ess = 1 - y(end);

fprintf('The steady state error will be %f \n', ess);

After executing we get

In the command window we get

The closed loop transfer function is :

sys =

s + 1
--------------------------
s^3 + 62 s^2 + 180 s + 119

Continuous-time transfer function.

The steady state error will be 0.991597
>>

10-3 Step Response 9 8 7 6 Amplitude 3 2 1 0 0 0.5 1.5 2 نيا 3.5 4.5 5 2.5 Time, t (sec)

Error 0.999 0.998 0.997 0.996 Amplitude 0.995 0.994 0.993 0.992 0.991 0 0.5 1.5 2 w 3.5 4.5 5 2.5 Time,t (sec)

Question (b)

clc;
clear;
close all;

s = tf('s');
P = 1/((s + 1)*(s + 2)*(s + 59));

pidTuner(P)

After executing we get

APID Tuner - Step Plot: Reference tracking PID TUNER VIEW 5 ? < >> Slower Response Time (seconds) Faster 0.2195 Show Reset 0.432 Plant: Type: PID Domain: P Form: Parallel Time Inspect Options A Add Plot PLANT CONTROLLER DESIGN Step Plot: Reference tracking X Export Aggressive Transient Behavior Robust Design Parameters TUNING TOOLS RESULTS Data Browser Step Plot: Reference tracking QE 1.2 Tuned response 1 0.8 Amplitude 0.6 0.4 0.2 0 0 0.2 0.4 0.6 1 1.2 1.4 1.6 0.8 Time (seconds) Controller Parameters: Kp = 2701, Ki = 3161, Kd = 515.2 21:31 D (1) ENG 31-07-2020 1 Type here to search w

The parameters

APID Tuner - Step Plot: Reference tracking PID TUNER VIEW ? < >> Slower Response Time (seconds) Faster 0.2195 Show Reset 0.432 Plant: Type: PID Domain: P Form: Parallel Time Inspect Options A Add Plot PLANT CONTROLLER DESIGN Step Plot: Reference tracking X Export Aggressive Transient Behavior Robust Design Parameters TUNING TOOLS RESULTS Data Browser Step Plot: Reference tracking QE 1.2 Tuned response 1 х Controller Parameters 0.8 Кр Ki Kd Tuned 2701.062 3161.1468 515.185 n/a Tf Amplitude 0.6 0.4 Performance and Robustness 0.2 Rise time Settling time Overshoot Peak Tuned 0.147 seconds 0.898 seconds 11.8% 1.12 Inf dB @ Inf rad/s 68 deg @ 9.13 rad/s Stable Gain margin 0 0 0.2 Phase margin Closed-loop stability 0.4 0.6 1 0.8 Time (seconds) Controller Parameters: Kp = 2701, Ki = 3161, Kd = 515.2 21:31 Dla ) ENG 31-07-2020 1 Type here to search w

We can see that the steady state error is 0

Kp = 2701, Ki = 3161, Kd = 515.2