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Unformatted text preview: 1 ME ME 5659 5659 : : Control and Control and Mechatronics Mechatronics (Dr. N. Jalili) (Dr. N. Jalili) End of Semester End of Semester Course Review Course Review 2 Important Properties of Laplace Transforms Important Properties of Laplace Transforms 3 ) ( ) ( ) ( ) ( y y t y t y ) ( ) ( . ) ( 1 1 ) ( 1 1 ) ( 1 1 ) ( 1 1 ) ( 1 ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 1 2 2 1 1 2 2 1 2 2 2 2 2 t Sin t Cos t y s L s s L s Y L s s s L s Y L s s s s Y L s s s s Y s s s Y s s s Y s Y s s Y s s Y t y L t y L t y L L t y L t y L t y t y L ) ( 1 . ) ( ) ( ) ( t t Sin t Cos t y Homogeneous ODE 4 Steady Steady State Error State Error • A steadystate error is the difference between the input and the output for a prescribed test input as time approaching to infinite. • In the “Closedloop” system: Error = Input – Output Output = (Closedloop Transfer Function) x Input Error = (1– Closedloop Transfer Function) x Input. Error/Input = (1– Closedloop Transfer Function). 5 Steady Steady State Error for Unity Feedback State Error for Unity Feedback System System • Using the Final Value Theorem: ] ) ( 1 ) ( [ lim )] ( [ lim ) ( s T s sR s sE E s s ] ) ( 1 ) ( [ lim ) ( s G s sR E s 6 Example 1 Example 1 Find the transfer function, Y ( s ) / R ( s ) 7 Example 2 Example 2 C ( s ) _ + + + R ( s ) G 4 G 1 G 2 H G 3 G 5 Find the transfer function, C ( s ) / R ( s ) 8 Example 3 • Determine the transfer function Y ( s ) / U ( s ) 9 Transient Response and SteadyState Response Transient response + SteadyState response = System response c ( t ) = c tr ( t ) + c ss ( t ) Transient response, c tr ( t ) = the response goes from the initial state to the final state. SteadyState response, c ss ( t ) = the system output behaves as time approaches infinity. 10 at at at e a a t t h s s U e t h s s U e a t h s U a τ a s s H 1 1 ) ( : 1 ) ( Response Ramp 1 ) ( : 1 ) ( Response Step ) ( : 1 ) ( Response Impulse 1 : Constant Time 1 ) ( 2 First–Order System 11 Exponential Response for a Unit–Step function input 12 Unit–Impulse Response for a First – Order System 13 Unit–Ramp Response of a First–Order System 14 Design Example 1 Determine the steady state error for a unit step input in terms of K and K 1 ....
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This note was uploaded on 05/26/2010 for the course ME 5659 taught by Professor Jalili during the Spring '10 term at Northeastern.
 Spring '10
 Jalili
 Mechatronics, Laplace

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