Solve the second equation for z plug the result into

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online