Topic # 1 Laplace transform Reference textbook : Control Systems, - PowerPoint PPT Presentation

ME 779 Control Systems Topic # 1 Laplace transform Reference textbook : Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012 1

Control Systems: Laplace transform Learning Objectives • Laplace transform of typical time-domain functions • Partial fraction expansion of Laplace transform functions • Final value theorem • Initial value theorem • System transfer function • General transfer function: poles and zeros, block diagram • Force response • Types of excitations • Impulse response function 2

Control Systems: Laplace transform System dynamics is the study of characteristic behaviour of dynamic systems First-order systems Second-order systems } Differential equations Laplace transforms convert differential equations into algebraic equations System transfer function can be defined Transient response can be obtained They can be related to frequency response 3

Control Systems: Laplace transform Basic definition    st x t e ( ) dt ℒ {x(t)}=X(s)= 0 4

Control Systems: Laplace transform No. Function Time-domain Laplace domain x(t)= ℒ -1 {X(s)} X(s)= ℒ {x(t)} e - τ s δ(t - τ) 1 Delay δ(t) 2 Unit impulse 1 3 Unit step u(t) 1 s 4 Ramp t 1 s 2 e - α t 5 Exponential decay 1   s     e   6 Exponential t 1   approach s ( s ) 5

Control Systems: Laplace transform  sin ω t 7 Sine   2 2 s cos ω t 8 Cosine s   2 2 s  sinh α t 9 Hyperbolic sine   2 2 s cosh α t 10 Hyperbolic cosine s   2 2 s  sin    11 Exponentially t e t     2 2 ( s ) decaying sine wave  cos     12 Exponentially s t e t     2 2 decaying cosine ( s ) wave 6

Control Systems: Laplace transform Partial fraction expansion of Laplace transform functions Factors of the denominator • Unrepeated factors • Repeated factors • Unrepeated complex factors 7

Control Systems: Laplace transform Unrepeated factors By equating both sides, determine A and B 8

Control Systems: Laplace transform Example Expand the following equation of Laplace transform in terms of its partial fractions and obtain its time-domain response . 2 s  Y s ( )   ( s 1)( s 2) 2 s A B       ( s 1)( s 2) ( s 1) ( s 2) 2 4      t 2 t    y t ( ) 2 e 4 e ( ) Y s   ( 1) ( 2) s s 9

Control Systems: Laplace transform Repeated factors   N s ( ) A B A B s ( a )        2 2 2 ( s a ) ( s a ) ( s a ) ( s a ) 10

Control Systems: Laplace transform EXAMPLE 2 s Expand the following Laplace transform in terms of its  Y s ( ) partial fraction and obtain its time-domain response   2 ( s 1) ( s 2) 2 s A B C         2 2 ( s 1) ( s 2) ( s 1) ( s 1) ( s 2) } 2 4 4     Y s ( )    2 ( s 1) ( s 1) ( s 2)        t t 2 t ( ) 2 4 4 y t te e e 11

Control Systems: Laplace transform Complex factors: they contain conjugate pairs in the denominator  N s ( ) As B        2 2 ( s a s )( a ) ( s ) 12

Control Systems: Laplace transform EXAMPLE Express the following Laplace transform in terms of its partial fractions and obtain its time-domain response .  2 s 1  Y s ( )     ( s 1 j )( s 1 j ) 2 s 1   Y s ( )     2 2 ( 1) 1 ( 1) 1 s s     t t y t ( ) 2 e cos t e sin t 13

Control Systems: Laplace transform Final-value theorem      lim sY s ( ) lim y t ( )  t  s 0 EXAMPLE Determine the final value of the time-domain function represented by          t t 2 t y t ( ) 2 te 4 e 4 e   2 s  Y s ( )   2 ( s 1) ( s 2) 14

Control Systems: Laplace transform Initial-value theorem      lim sY s ( ) lim y t ( ) EXAMPLE  t 0  s Determine the initial value of the time-domain response of the following equation using the initial-value theorem  2 s 1      Y s ( )   t t y t ( ) 2 e cos t e sin t       ( s 1 j )( s 1 j )  s (2 s 1)  lim 2     ( s 1 j s )( 1 j )  s 15

Control Systems: Laplace transform SYSTEM TRANSFER FUNCTION Block diagram ( ) Y s  G s ( ) X s ( ) System transfer function is the ratio of output to input in the Laplace domain 16

Control Systems: Laplace transform SYSTEM TRANSFER FUNCTION General System Transfer Function m   ( s z )     i K ( s z )( s z ) ( s z )    1 2 m i 1 ( ) G s K    n  ( s p )( s p ) ( s p )   ( s p ) 1 2 n j  j 1 z , z ... z are called zeros 1 2 m K is a constant p , p ... p are called the poles 1 2 n Number of poles n will always be greater than the number of zeros m 17 (Laplace transform is a rational polynomial )

Control Systems: Laplace transform EXAMPLE SYSTEM TRANSFER FUNCTION Obtain the pole-zero map of the following transfer function      ( 2 )( 2 4 )( 2 4 ) s s j s j  (1) G ( s )        ( s 3 )( s 4 )( s 5 )( s 1 j 5 )( s 1 j 5 ) Zeros Poles s=2 s=3 s=-2-j4 s=4 s=-2+j4 s=5 s=-1-j5 s=-1+j5 18

Control Systems: Laplace transform SYSTEM TRANSFER FUNCTION EXAMPLE Zeros Poles Zeros Poles s=2 s=3 s=-2-j4 s=4 s=-2+j4 s=5 s=-1-j5 s=-1+j5 19

Control Systems: Laplace transform Forced response     ( )( ) ( ) K s z s z s z   1 2 m C ( s ) G ( s ) R ( s ) R ( s )     ( s p )( s p ) ( s p ) 1 2 n R(s) input excitation 20

Control Systems: Laplace transform Forced response TYPES OF EXCITATIONS 1. Impulse 2. Step 3. Ramp 4. Sinusoidal 21

Control Systems: Laplace transform Forced response Impulse input    x ( t ) x ( t a ) i x i  Dirac delta function 22

Control Systems: Laplace transform Forced response Laplace transform of an impulse input  Integral property of Dirac       ( ) ( t t t ) dt ( ) t delta function o o          st sa X ( s ) e x ( t a ) dt x e i i 0 23

Control Systems: Laplace transform Forced response Step input  x     st i X s ( ) e x dt Laplace transform of step input i s 0 24

Control Systems: Laplace transform Forced response Example The following transfer function is subjected to a unit step input. Determine the response  1 ( 1) s   R s ( ) G s ( ) p 1 =-4, z 1 =-1  ( s 4) s  ( s z ) A B     1 C s ( ) R s G s ( ) ( )   s s ( p ) s s p 1 1   z z 1 3       p t 4 t 1  1  c t ( ) 1 e e 1   p p 4 4 1 1 25

Control Systems: Laplace transform Forced response Example Step response 1 3    4 t ( ) c t e 4 4 0.25 26

Control Systems: Laplace transform Forced response  Ramp input 1     st ( ) X s e tdt 2 s 0 Laplace transform of the ramp input 45 0 27

Control Systems: Laplace transform Forced response Sinusoidal input        st X s ( ) e sin t dt   2 2 s 0 28

Control Systems: Laplace transform Forced response IMPULSE RESPONSE FUNCTION Time-domain response of a system subjected to unit impulse excitation h(t)= ℒ -1 {G(s))} It is the inverse Laplace transform of the system transfer function 29

Control Systems: Laplace transform Forced response Convolution Integral Each infinitesimal strip of force defines an impulse response function ^   d  F F ( ) ^ ^ Response due to each    d  y F h ( t ) strip of the force t       y ( t ) F ( ) h ( t ) d Total response due to 0 entire force history 30