The mathematical function for the reference signal (red curve) is necessary so that we can instruct the PID controller to produce the actuated signal for the plant output, enabling it to track the reference signal.
Here is a simple demonstration:
First, the reference signal is designed. Next, the step response of the uncontrolled plant is analyzed. With a settling time of 6 seconds, the plant cannot track the reference signal effectively.
Consequently, a PID controller is tuned to achieve a settling time of approximately 1 second. Now, the closed-loop system can track the reference signal accurately.
t = linspace(0, 40, 4001);
L1 = (20 - 0)/(15 - 8)*t + ( 0 - (20 - 0)/(15 - 8)* 8);
y1 = min(max(L1, 0), 20).*(heaviside(t - 0) - heaviside(t - 17));
L2 = (10 - 20)/(22 - 20)*t + (20 - (10 - 20)/(22 - 20)*20);
y2 = min(max(L2, 10), 20).*(heaviside(t - 17) - heaviside(t - 25));
L3 = (-10 -10)/(31 - 28)*t + (10 - (-10 -10)/(31 - 28)*28);
y3 = min(max(L3,-10), 10).*(heaviside(t - 25));
plot(t, ref, 'color', '#D74967', 'linewidth', 3)
grid on, grid minor, ylim([-20, 20])
title('Reference signal')
Gp = tf(1, [1 2 1])
Gp =
1
-------------
s^2 + 2 s + 1
Continuous-time transfer function.
step(Gp, 12), grid on, grid minor
title('Step response of the Plant')
grid on, grid minor, ylim([-20, 20])
Gc = pid(Kp, Ki, Kd, Tf)
Gc =
1 s
Kp + Ki * --- + Kd * --------
s Tf*s+1
with Kp = 4.75, Ki = 2.5, Kd = 2.02, Tf = 0.1
Continuous-time PIDF controller in parallel form.
Gcl = feedback(Gc*Gp, 1)
Gcl =
25 s^2 + 50 s + 25
---------------------------------
s^4 + 12 s^3 + 46 s^2 + 60 s + 25
Continuous-time transfer function.
step(Gcl, 3), grid on, grid minor
title('Step response of the Closed-loop System')
grid on, grid minor, ylim([-20, 20])
