This problem will address the importance of FRF phase in the context of time del

Question

This problem will address the importance of FRF phase in the context of time delay. A communication satellite that links a control center to a micro-drone is shown at The right. The transmission delay time is denoted as Delta. This delay is present in sending a signal to the drone and also in receiving a signal from it. A lead compensator, designed to add PM to the system, resulted in the following forward loop transfer function: G(s) = G_c (s) G_p (s) = (0.1174s + 1/0.0155s + 1)(100 e^-s Delta/s(0.5s + 1)). The feedback is H(s) = e^-s Delta. The open loop Bode plot for Delta = 0 sec is shown at right. Determine then maximum value of Delta for which the closed loop will remain stable. Solution:

Explanation / Answer

%clearing window, variables and figures
clear all;
close all;
clf;
clc;
j=0;
s=tf('s'); dt=1.465*10^-2:.0000001:1; %reduce samplimg time starting from .01 and start dt from corresponding result next time

l=length(dt);
i=1;
while i<=l && j<=0
Gc=(0.1174*s+1)/(0.0155*s+1);
Gp=(100*exp(-s*dt(:,i)))/(s*(0.5*s+1));
sys=feedback(Gp*Gc,exp(-s*dt(:,i)));
[Pm,Gm,Wg,Wp]=margin(sys);
if Pm<=0 || Gm<=0
fprintf(' The maximum value is:%d ',dt(:,i));
j=j+1;
end
i=i+1;
end

Result;

The maximum value is:1.465080e-02

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