Topic # 3 Second-order Systems Reference textbook : Control - PowerPoint PPT Presentation

ME 779 Control Systems Topic # 3 Second-order Systems Reference textbook : Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012 1

Control Systems: Second-order Systems Learning Objectives • Differential equations • Normalized form of differential equations • System transfer function • Pole-zero map • Overdamping • Critical damping • Underdamping • Undamped • Impulse, step, sinusoidal repsonse • Frequency response: magnitude and phase 2

Control Systems: Second-order Systems Differential equation       m y c y k y K x ( t ) y(t) response Mass, m (kg) Damping coefficient, c (N-s/m) Stiffness, k (N/m) K static sensitivity x(t) input 3

Control Systems: Second-order Systems Differential equation: Normalized form c k Kx ( t )    By dividing throughout by m    y y y m m m Undamped natural frequency, rad/s k   n m c Damping factor   2 km Kx t ( )      2 y 2 y y n n m 4

Control Systems: Second-order Systems System transfer function Kx t ( )      2 y 2 y y n n m Y s ( ) K        2 2 X s ( ) m s 2 s n n 5

Control Systems: Second-order Systems Classification of damping factors Damping Type Property factor   1 Overdamped Exponential decay   1 Critically damped Exponential decay   1 Underdamped Oscillatory decay   Undamped Oscillatory 0 6

Control Systems: Second-order Systems Pole-zero map Y s ( ) K    ζ >1 overdamped     2 2 X s ( ) m s 2 s n n Poles       2   2 2 2 4  n n n s 1,2 2          2 s 1 1,2 n 7

Control Systems: Second-order Systems Y s ( ) K Pole-zero map    ζ =1 critically damped     2 2 X s ( ) m s 2 s n n Poles   2       2 2 2 4  n n n s 1,2 2    s 1,2 n 8

Control Systems: Second-order Systems Y s ( ) K Pole-zero map    ζ <1 underdamped     2 2 X s ( ) m s 2 s n n   2       2 2 j 4 2  n n n s 1,2 2 Damped natural      2 1 frequency d n       s j   tan 1,2 n d   2 1 Poles 9

Control Systems: Second-order Systems Pole-zero map Y s ( ) K    ζ =0 undamped   2 2 X s ( ) m s j ω j  n x n s-plane   j  s 1,2 n Poles σ  j  x n 10

Control Systems: Second-order Systems Overdamped case (ζ>1) Impulse response K       2 Differential equation y 2 y y x ( ) t n n i m   Kx 1    i Y s ( ) Laplacian of the output     2 2   m s 2 s n n     Kx 1 1     i                  2 2 2   2 m 1 ( s 1) ( s 1 n n n n n 11

Control Systems: Second-order Systems Impulse response Overdamped case (ζ>1)     Kx 1 1     i Y s ( )                2  2 2    2 m 1 ( s 1) ( s 1 n n n n n     Kx         n t 2 i y t ( ) e sinh 1 t n      2  m 1  n Time-domain response 12

Control Systems: Second-order Systems Impulse response Critically damped case ( ζ=1 )     Kx 1 Kx 1     i   i Y s ( )       2 2   2     m s 2 s m s   n n n   Kx       n t i y t ( ) te  n   m n 13

Control Systems: Second-order Systems Impulse response Undamped case ( ζ<1 )      Poles s j 12 n d Laplace output   Kx 1    i Y s ( )           m ( s j )( s j ) n d n d   Kx Time-domain      n t  i y t ( ) e sin t output  d   m d 14

Control Systems: Second-order Systems Impulse response Undamped case ( ζ<1 )   Kx      n t  i y t ( ) e sin t  d   m d K     n t Impulse response h t ( ) e sin t function  d m d t K             y t ( ) e sin F t ( ) d n  d m d 0 Duhamel’s integral 15

Control Systems: Second-order Systems Impulse response 16

Control Systems: Second-order Systems Step response Overdamped case (ζ>1)   Laplace of the    Kx 1 1  i      output Y s ( )                m s 2 2 s 1) s 1     n n n n            Kx               n t 2 2  i y t ( ) 1 e cosh 1 t sinh 1 t  n n 2     m  2    1   n Time-domain output 17

Control Systems: Second-order Systems Step response Underdamped case ( ζ<1 )     Kx 1 1  i Laplace of the     Y s ( )           output  m s ( s j )( s j ) n d n d        Kx          n t   i y t ( ) 1 e cos t sin t  d d 2     m   2   1   n Time-domain of the output 18

Control Systems: Second-order Systems Step response Underdamped case ( ζ<1 ) 19

Control Systems: Second-order Systems Step response Underdamped case ( ζ<1 )       n t j ω Kx e        i y t ( ) 1 sin( t )  x  d 2   m  2    1 n s-plane   2 1     1 tan  σ    tan   2 1 x 20

Control Systems: Second-order Systems Control Systems: Second-order Systems Step response Underdamped case ( ζ<1 )    2 t y t m ( ) e n r       r n 1 1 sin( t ) d r   Kx 2 1 i     t Rise time  r d 21

Control Systems: Second-order Systems Control Systems: Second-order Systems Step response Underdamped case ( ζ<1 )   t Peak time  p d        Kx   2   1   Peak response i y t ( ) 1 e  p 2 m     n 22

Control Systems: Second-order Systems Control Systems: Second-order Systems Step response Underdamped case ( ζ<1 ) Percentage overshoot M p     y t ( ) y      2 p 1 M e % p y  23

Control Systems: Second-order Systems Control Systems: Second-order Systems Step response Underdamped case ( ζ<1 ) Settling time 4  t 2% criterion  s n 3  5% criterion t  s n 24

Control Systems: Second-order Systems Step response Comparison of damping factors Second-order systems 25

Control Systems: Second-order Systems Sinusoidal response K Differential equation       2 y 2 y y A sin t n n m      KA 1  Laplacian of the    Y s ( )         2 2 2 2 output  m s s 2 s n n 26

Control Systems: Second-order Systems Sinusoidal response   Y s ( ) KA 1        2 2   X s ( ) m s 2 s n n  Y (j ) KA    Putting s=j ω       2 2 X (j ) m 2 j n n   Frequency ratio   2 r Y j ( ) m 1   n     2 X j ( ) KA (1 r 2 jr ) n ω forcing frequency Frequency response function 27

Control Systems: Second-order Systems Sinusoidal response   2 Y j ( ) m 1 Magnitude   n M 20log dB      X j ( ) K 2    2 2 1 r 2 r     2 r    1   tan Phase    2 1 r 28

Control Systems: Second-order Systems Magnitude Sinusoidal response 29

Control Systems: Second-order Systems Phase Sinusoidal response 30