Question

Please solve as a MATLAB code.

A unity feedback closed loop control system is displayed in Figure 4. (a) Assume that the controller is given by G (s) 2. Based on the lsim function of MATLAB, calculate and obtain the graph of the response for (t) at. Here a 0.5°/s. Find the height error after 10 seconds, (b) In order to reduce the steady-state error, substitute G (s) with the following controller This is a Proportional-Integral (PI) controller. Repeat part (a) in the presence of Pl controller, and juxtapose the steady state error of the previous controller with the PI controller. An+ Desired attitude 酬r) -10 (s2+3.5s+6)ctual s+10 attitude Figure 4: A unity feedback closed loop control system

Answer 1

with proportional controller:

k=2;

matlab code:

clc;clear all;close all;
t=0:0.001:20;
num=[10 50];
d1=[1 10];
d2=[1 3.5 6 0];
denum=conv(d1,d2)
sys1=tf(num,denum)
sys=feedback(sys1,1)
ip=0.5*t;
lsim(sys,ip,t),grid

output:

denum =

1.0000 13.5000 41.0000 60.0000 0

sys1 =

10 s + 50
------------------------------
s^4 + 13.5 s^3 + 41 s^2 + 60 s

Continuous-time transfer function.

sys =

10 s + 50
-----------------------------------
s^4 + 13.5 s^3 + 41 s^2 + 70 s + 50

Continuous-time transfer function.

output response:

with proportional control:

Linear Simulation Results Input: In(1) Time (seconds): 11 Amplitude: 5.48 System: sys Time (seconds): 11 Amplitude: 4.92 20 10 12 14 16 18 Time (seconds)

g(s)=2+(1/s);

matlab code:

clc;clear all;close all;
t=0:0.001:20;
n1=[2 1];
n2=[10 50];
num=conv(n1,n2);
d1=[1 10];
d2=[1 3.5 6 0 0];
denum=conv(d1,d2)
sys1=tf(num,denum)
sys=feedback(sys1,1)
ip=0.5*t;
lsim(sys,ip,t),grid

output:

denum =

1.0000 13.5000 41.0000 60.0000 0 0

sys1 =

20 s^2 + 110 s + 50
--------------------------------
s^5 + 13.5 s^4 + 41 s^3 + 60 s^2

Continuous-time transfer function.

sys =

20 s^2 + 110 s + 50
---------------------------------------------
s^5 + 13.5 s^4 + 41 s^3 + 80 s^2 + 110 s + 50

Continuous-time transfer function..

response:

Linear Simulation Results System: sys Time (seconds): 11 Amplitude: 5.51 20 10 12 14 16 18 Time (seconds)