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Lec_09

# Lec_09 - Lecture 9 Time-Response Characteristics(II Time...

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1 Lecture 9: Time-Response Characteristics (II) Time Specifications of Continuous-Time Systems Mapping the s -Plane to the z -Plane Time Specifications of Sampled-Data System Reading: Chapter 6.4 Sampled-Data Systems C ( s ) H ( s ) G ( s ) R ( s ) + ¡ T Transfer function from R ( z ) to C ( z ): C ( z ) R ( z ) = G ( z ) 1+ GH ( z ) Characteristic equation: 1 + GH ( z ) = 0 whose roots are the closed-loop poles: p 1 , p 2 ,..., p n Stable systems: all closed-loop poles within the unit circle Dominant closed-loop poles: the closed-loop pole(s) furthest away from 0 Transient response largely determined by the dominant closed-loop pole(s) Problem : quantify the response characteristics based on dominant pole locations

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2 Continuous-Time Feedback Systems 1 + G ( s ) H ( s ) = 0 C ( s ) H ( s ) G ( s ) R ( s ) + ¡ Transfer function from R ( s ) to C ( s ): C ( s ) R ( s ) = G ( s ) 1+ G ( s ) H ( s ) Characteristic equation: q 1 , q 2 ,..., q n Stable systems: all closed-loop poles strictly on the left half of the complex plane Dominant closed-loop poles: the closed-loop pole(s) furthest to the right Transient response largely determined by the dominant closed-loop pole(s) whose roots are the closed-loop poles: Step Response Time (sec) Amplitude 0 5 10 15 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time Specifications of Continuous-Time Systems t d : delay time , time for s ( t ) to reach half of s ( 1 ) t r : rise time , time for s ( t ) to reach s ( 1 ) t p : peak time , time for s ( t ) to reach first peak M p : maximum overshoot t s : settling time , time for s ( t ) to settle within a range (2% or 5%) of s ( 1 ) Step response s ( t ) t d t r t p M p = s ( t p ) ¡ s ( 1 ) s ( 1 ) s ( 1 ) 0 : 5 s ( 1 ) t s