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Gp4_Chapter 5

# Gp4_Chapter 5 - EE2010 Systems and Control(Chapter 5...

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EE2010 Systems and Control (Chapter 5) 5-1 Consider the position control problem below Relate system parameters of underdamped 2 nd order system ( and n ) with language humans use to describe task/objective. Transient Response Specifications

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EE2010 Systems and Control (Chapter 5) 5-2 2% settling time , t s , is the time taken for the system to settle down to within 2% of the final value. The step response of an underdamped second-order system is comprises of a complex exponential signal that is bounded by the curves Hence, settling time corresponding to a ±2% tolerance band may be estimated by the time the exponential curve takes to decay to 0.02, i.e., t s t when 2 1 1  t n e K 02 . 0 1 2 s n t e
EE2010 Systems and Control (Chapter 5) 5-3 Peak time , t p , is the amount of time taken for the system output to reach its maximum value. From page 4-33, step response of an underdamped second-order system is Maximum value of y ( t ) can be found when n n s t  2 2 1 ln 4 1 ln 02 . 0 ln Time, Settling % 2     t e K t Ke K t y n t n t n n 2 2 2 1 sin 1 1 cos ) ( 0 ) ( dt t dy 6 . 0 when 4 n s t

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EE2010 Systems and Control (Chapter 5) 5-4 ,... 2 , , 0 i.e., 0 sin only when 0 ) ( ) ( , 0 Since t t t d t dy t e d d t   0 sin 0 cos sin sin cos 0 sin cos 2 t Ke t t Ke t t e K t t Ke K dt d d d d t d d d t d d d t d d d t Since t p corresponds to the first peak, d t p = 2 1 n d p t
EE2010 Systems and Control (Chapter 5) 5-5 Maximum overshoot , M p , is the amount by which the system response proceeds beyond the steady-state value, i.e., Substitute into step response expression, Note that the formulae for the maximum overshoot, M p , contains the static gain, K and the magnitude of the input step function (above derivation assumes input signal is the unit step function)   ss p y t y M ) ( max   d p t t t y when occurs ) ( max   K Ke K M Ke K t y d p d d d sin cos sin cos ) ( max 2 1  Ke M p

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EE2010 Systems and Control (Chapter 5) 5-6 The percent overshoot is Percent overshoot is independent of K and the magnitude of the input step function. The 10% to 90%
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Gp4_Chapter 5 - EE2010 Systems and Control(Chapter 5...

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