HW1s - MEEN 651 Control System Design Homework 1:...

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MEEN 651 Control System Design Homework 1: Preliminary Mathematics and Modeling of Mechanical Systems Solution Assigned: Tuesday, 4 Sept. 2007 Due: Tuesday, 11 Sept. 2007, 5:00 pm Problem 1) Show the following: a. That the Laplace Transform, () () dt e t x s X st = 0 , is a linear operator. In other words, show that ( ) ( ) [ ] ( ) ( ) s bY s aX t by t ax + = + L . Solution: [] + = + 0 )) ( ) ( ( dt e t by t ax t by t ax st L ∫∫ ∞∞ + = 00 )) ( ( )) ( ( dt e t by dt e t ax st st + = ) ( ) ( dt e t y b dt e t x a st st ) ( ) ( s bY s aX + = b. That for the convolution integral, () () ( ) ( ) = 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 = . Solution: = d t y x t y t x Define ξ = t-x λ = t- ξ d ξ = -d λ as λ goes from - to , ξ goes from to -
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() ( ) −∞ + = ) ( ξ d y t x ()( ) ) ( * ) ( t x t y d t x y = = c. That () () ( ) t tri t rect t rect = . Solution: Rewriting ) 5 . 0 ( ) 5 . 0 ( + = t s t s t rect And since ( ) ( ) α = t y t h t x (time shift property) And () () () t r t s t s = (convolution of step with a step is a ramp) () () [] [ ] [ ] [ ] ) ( ) 1 ( ) ( ) ( ) 1 ( ) 5 . 0 ( * ) 5 . 0 ( ) 5 . 0 ( * ) 5 . 0 ( ) 5 . 0 ( * ) 5 . 0 ( ) 5 . 0 ( * ) 5 . 0 ( ) 5 . 0 ( ) 5 . 0 ( * ) 5 . 0 ( ) 5 . 0 ( t tri t r t r t r t r t s t s t s t s t s t s t s t s t s t s t s t s 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, > < = 0 1 0 0 t t t s .
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HW1s - MEEN 651 Control System Design Homework 1:...

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