1
2
0
2
2
3.32 In aircraft control systems, an ideal pitch response (
) versus a pitch command (
) is
described by the transfer function
( )
(
1/
)
( )
2
The
o
c
n
c
n
n
q
q
Q
s
s
Q
s
s
s
τω
τ
ζω
ω
+
=
+
+
actual aircraft response is more complicated than this ideal transfer function; nevertheless,
the ideal model is used as a guide for autopilot design. Assume that
is the desired rise time,
r
t
and that
1.789
1
1.6
0.89
Show that this ideal response possesses a fast settling time and minimal overshoot by plotting
n
r
r
t
t
ω
τ
ζ
=
=
=
the step response for
0.8, 1.0, 1.2, and 1.5 sec.
r
t
=
Solution: The following program statements in MATLAB produce the following plots:
% Problem 3.32
tr = [0.8 1.0 1.2 1.5];
t=[1:240]/30;
tback=fliplr(t);
clf;
for I=1:4,
wn=(1.789)/tr(I); %Rads/second
tau=tr(I)/(1.6); %tau
zeta=0.89; %
b=tau*(wn^2)*[1 1/tau];
a=[1 2*zeta*wn (wn^2)];
y=step(b,a,t);
subplot(2,2,I);
plot(t,y);
titletext=sprintf('tr=%3.1f seconds',tr(I));
title(titletext);
xlabel('t (seconds)');
ylabel('Qo/Qc');
ymax=(max(y)-1)*100;
msg=sprintf('Max overshoot=%3.1f%%',ymax);
text(.50,.30,msg);
yback=flipud(y);
yind=find(abs(yback-1)>0.01);
ts=tback(min(yind));
msg=sprintf('Settling time =%3.1f sec',ts);
text(.50,.10,msg);
grid;
end

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