Question

System dynamics and control course.

Use only “MATLAB “to solve this.

Need a pro to help

Let a system have plant transfer function (00.2) s3 +22s 156s+232 Design a PID controller such that the closed-loop rise time is less than 0.5 seconds, the overshoot is less than 10%, and the steady-state error is zero for a step command.

Answer 1

Following is the relevant MATLAB code to fit required PID controller:

%%%%%%%%%

clear

% open loop step response of given transfer function
s = tf('s');
P = 1/(s^3 + 22*s^2 + 156*s + 232);
figure(1)
step(P)

% PID control (with tuned parameters)

Kp = 512.5;
Ki = 1145.0;
Kd = 50.86;
C = pid(Kp,Ki,Kd)
T = feedback(C*P,1);
t = 0:0.01:2;
figure(2)
step(T,t)

% Tuning GUI
pidTuner(P,'pid')

%%%%%%%%%%%

Below are parameters and metrics obtained using pid tuning above. They are inside given limits.

Controller Parameters Tuned 512.5295 1145.0258 50.8631 Кр Ki Kd Tf Performance and Robustness Tuned 0.382 seconds 1.52 seconds 5.43 % 1.05 Inf dB Inf rad/s 69.2 deq 3.67 rad/s Stable Rise time Settling time Overshoot Peak Gain margin Phase margin Closed-loop stability

Step plot of closed loop control (zero steady state error)

Step Plot: Reference tracking 1.2 Tuned response 0.8 0.6F 0.4 H 0.2 0 1.5 2.5 0.5 Time (seconds)