Unformatted text preview: )s2 = U, s2 Z + (2s 10)⇥ = 0. Solve the second equation for Z , plug the result into the ﬁrst, and ﬁnd the ratio ⇥/U . Then we have
⇥
=2
U
s 1
6s + 30 1 5. Solve the following differential equation for x(t):
x + 3x + 2x = u,
¨
˙ x(0) = 1, u ( t) = e t . x(0) = 0,
˙ Solution: Taking the Laplace transform, we have
s2 X (s) sx(0) x(0) + 3(sX (s)
˙ x(0)) + 2X (s) = 1
,
s+1 from which we obtain
X (s) = s+2
1
1
=
+
(s + 1)2
s + 1 (s + 1)2 where we applied the partial fraction expansion. Taking the inverse Laplace transform,
x(t) = (1 + t)e t 6. Consider the system P (s) and input u(t) given by
P (s) = (s 1)(s + 4)
,
(s2 + 9)(s + 3)s u(t) = 3 sin(4t). Which of the following terms are expected to appear in the steady state output?
(A) 1 (B) et (C) e (D) e 3t 4t (E) cos(3t) (F) sin(4t) (G) cos(4t) (H) sin(5t) Solution: The output signal is given by the inverse Laplace transform of P (s) and U (s). Hence, each
factor of the denominator of P (s)U (s) gives a term in the output, which are A, C, E, F, G. In the steady
state, the exponential terms die out (converges to zero), and hence the a...
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 Winter '09
 IDAN
 Steady State

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