PracticeProblemsMidterm_solu

Solve the second equation for z plug the result into

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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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