Question

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 Isim function of MATLAB, calculate and obtain the graph of the response for 6, (t)-at. Here a : 0.5%, Find the height error after 10 seconds, G) -2 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 Pl controller. (s +5)Actual 10 s+10 Desired-+ attitude 3+3.5s+6) attitude

PLEASE solve it with MATLAB code

Answer 1

with proportional controller:

t=0:0.001:25;
k=2*(-10)*(-1);
z=[ -5];
p=[ 0 -10 -1.7500+1.7139*i -1.7500-1.7139*i ];
s1=zpk(z,p,k) %open loop transfer function
sys=feedback(s1,1) % close loop transfer function
ip=0.5*t; % ramp input with 0.5 as amplitude.
lsim(sys,ip,t)

output:

s1 =

20 (s+5)
-------------------------
s (s+10) (s^2 + 3.5s + 6)

Continuous-time zero/pole/gain model.

sys =

20 (s+5)
------------------------------------------
(s+10.14) (s+2.239) (s^2 + 1.122s + 4.405)

Continuous-time zero/pole/gain model.

with proportional + integral control:

Linear Simulation Results 14 12 10 Input: In(1) Time (seconds): 10.8 Amplitude: 5.38 System: sys Time (seconds): 10.8 Amplitude: 5.11 10 15 20 25 Time (seconds)

t=0:0.001:25;
k=2*(-10)*(-1);
z=[-0.5 -5];
p=[0 0 -10 -1.7500+1.7139*i -1.7500-1.7139*i ];
s1=zpk(z,p,k) %open loop transfer function
sys=feedback(s1,1) % close loop transfer function
ip=0.5*t; % ramp input with 0.5 as amplitude.
lsim(sys,ip,t)

output:

s1 =

20 (s+0.5) (s+5)
---------------------------
s^2 (s+10) (s^2 + 3.5s + 6)

Continuous-time zero/pole/gain model.

sys =


20 (s+5) (s+0.5)

------------------------------------------------------
  
(s+10.13) (s+1.904) (s+0.7302) (s^2 + 0.7337s + 3.549)
  
  

Continuous-time zero/pole/gain model.

response:

Linear Simulation Results 14 12 10 10 15 20 25 Time (seconds)