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MEM255Su06-07.9.Character.studs

# MEM255Su06-07.9.Character.studs - MEM 255 Introduction to...

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MEM 255: Introduction to Controls (Response characterization) Dr. Ajmal Yousuff Dept. MEM Drexel University

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Yousuff MEM 255 Controls 2 Reduction of complexity Suppose G(s) has n distinct real poles {-p 1 ,…-p 2 } The, the output y(t) for an impulse input u(t) is 1 2 2 1 1 2 1 1 1 ( ) . ( ) ( sin co . ... ) . s n d n n p t p t d n t d As y t s p s p e B s e c t t e c σ σ ϖ α α ϖ α ϖ α - - - - = + + + + + + + + = + + + + L (zeros determine the constants α i … )
Yousuff MEM 255 Controls 3 dominant mode Dominant poles 1 ( ) ( ) ( ) ( ) { ( ) ( )} Y s G s U s c t G s U s - = = L 2( 2) Let ( ) and ( ) 1. ( 1)( 3) s G s U s s s + = = + + 1 3 1 1 ( ) 1 3 t t c t s s e e - - - = + + + = + L e -3t

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Yousuff MEM 255 Controls 4 Characterization of System response Based on 1. 1 st order systems 2. 2 nd order systems 1. pole/eigenvalue : real, stable; {-a} 2. pole/eigenvalue : complex, stable; 2 { 1 } n n i ζϖ ϖ ζ - - { } d i σ ϖ -
0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 t y(t) a=2 t Yousuff MEM 255 Controls 5 1 st order systems ( ) 1 ; 0 ( ) Y s a U s s a = +

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Yousuff MEM 255 Controls 6 Settling-time Settling-time (T s ) : time to reach, and stay within, 2% of y ss . 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 a=2 t w/in 2% of y ss y ss (2%) T s
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 a=2 t Yousuff MEM 255 Controls 7 Time-constant, rise-time y ss Time constant ( τ ) : time to reach 63% of y ss . 0.63

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