Control systems questions

1. (20 points)
(a) Apply the inverse Laplace transform to the following to find x(t):
X(s) =

s+5
(s + 1)(s + 3)

(b) Apply the inverse Laplace transform to the following to find y(t):
Y (s) =

2s + 1
+ 2s + 3)

s(s2

2. (25 points) Consider the simplified quarter-car model of an automobile suspension given
below where m is one quarter of the total mass of the vehicle, b is the damping in the
suspension, and k is the stiffness of the suspension.

The differential equation describing this system is
m¨
x + bx
˙ + kx = u
where x(t) is the displacement of the car body and u(t) is the displacement of the tire,

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