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Unformatted text preview: MEEN 651 Control System Design Homework 1: Preliminary Mathematics and Modeling of Mechanical Systems Assigned: Tuesday, 2 Sept. 2007 Due: Tuesday, 9 Sept. 2007, 5:00 pm Problem 1) Show the following: a. That the Laplace Transform, ( ) ( ) dt e t x s X st − ∞ ∫ = , is a linear operator. In other words, show that ( ) ( ) [ ] ( ) ( ) s bY s aX t by t ax + = + L . b. That for the convolution integral, ( ) ( ) ( ) ( ) ∫ ∞ ∞ − − = ∗ λ λ λ d t y x t y t x , the order of convolution does not matter. In other words, show that ( ) ( ) ( ) ( ) t x t y t y t x ∗ = ∗ . c. That ( ) ( ) ( ) t tri t rect t rect = ∗ . Problem 2) Given a dynamic system described by the ODE ( ) ( ) ( ) ( ) t x t x t y t y 3 2 + = + & & and an input signal of ( ) ( ) t s te t x t 5 3 − = , where s(t) is the unit step function, ( ) ⎩ ⎨ ⎧ > < = 1 t t t s . Assume a relaxed system (zero initial conditions), and solve for the response, y(t) , using: a. Convolution b. Laplace Transform Problem 3)...
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