
8.8 Show that the asymptotic expression for the fractional error in the discrete-time transfer
function H(e
jvT
) resulting from explicit Euler integration of the first-order system
_
x ¼ lx þ u
is given by
e
H
¼
H(e
jvT
)
H( jv)
1
vl
2(v
2
þ l
2
)
vT j
v
2
2(v
2
þ l
2
)
vT, vT 1
What does the system reduce to when l ¼0? Comment on what happens to the real and
imaginary components.
8.9 Verify the curves plotted in Figures 8.14 and 8.15 for the fractional gain and phase errors based
on explicit Euler integration by using the MATLAB funct ions ‘‘real,’’ ‘‘imag,’’ ‘‘abs,’’ and
‘‘angle,’’ that is,
Fractional gain error e
M
¼ Re(e
H
) ¼ Re
H(e
jvT
)
H( jv)
1
¼ Re
jvT
e
jvT
1
1