Topic # 2 First-order Systems Reference textbook : Control Systems, - PowerPoint PPT Presentation

ME 779 Control Systems Topic # 2 First-order Systems Reference textbook : Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012 1

Control Systems: First-order Systems Learning Objectives • Differential equation • System transfer function • Pole-zero map • Normalized reponse • Impulse response • Step response • Ramp response • Sinusoidal response • Frequency response 2

Control Systems: First-order Systems DIFFERENTIAL EQUATIONS     y y K x ( t ) Governing differential equation  Time constant, sec Static sensitivity (units depend on the input K and output variables) y(t) Response of the system x(t) Input excitation 3

Control Systems: First-order Systems System transfer function     y y K x ( t ) K   Y s ( ) K (1 s )   G s ( )   X s ( ) (1 s ) Block diagram representation of transfer function 4

Control Systems: First-order Systems Pole-zero map K  G s ( )   (1 s ) 5

Control Systems: First-order Systems Normalized response • Static components are taken out leaving only the dynamic component • The dynamic components converge to the same value for different physical systems of the same type or order • Helps in recognizing typical factors of a system 6

Control Systems: First-order Systems Impulse response    i   y y K x ( t ) Governing differential equation     Kx Kx 1   Laplacian of the response   i i Y s ( )    1 (1 s )    s    t  Kx i e  Time-domain response  y ( t )  7

Control Systems: First-order Systems Impulse response function t  K By putting x i =1 in the impulse response  e  h t ( )  Response to any force excitation t t  K       y t ( ) e F t ( ) d  0 8

Control Systems: First-order Systems Normalized impulse response  y t ( ) Kx i t  / 9

Control Systems: First-order Systems Step response    Step input of y y K x u t ( ) level x i i Kx Kx Kx Output in the Laplace    i i i Y s ( ) domain   1 s (1 s ) s  s    t       Output in the y ( t ) Kx 1 e   i time-domain   10

Control Systems: First-order Systems Normalized step response y t ( ) Kx i t  / 11

Control Systems: First-order Systems Ramp response x(t)=t (input)     y y K t   K 1     Laplace of Y s ( )   the output 2 2 1 s (1 s ) s s  s  t  y ( t )       Time-domain response t e K y    ss Steady-state response t K 12

Control Systems: First-order Systems Normalized ramp response t  y ( t )       x t ( ) t e y t ( ) K K y t ( ) K t  / 13

Control Systems: First-order Systems Sinusoidal response      y y KA sin t Sinusoidal excitation of amplitude A and frequency ω         2 K A s 1         Y s ( )               2 2   2 2 2 2 2   (1 s ) s s 1/ s s 1 Output in the Laplace domain    Time-domain y t ( ) 1          2 t /   e cos t sin t output        2 KA 1 14

Control Systems: First-order Systems    y t ( ) 1          2 t /   e cos t sin t        2 KA 1 Sinusoidal response 15

Control Systems: First-order Systems Sinusoidal response    y ( ) t 1         ss cos t sin t      2   KA 1 y ( ) t 1 Normalized steady-state     ss sin( t ) response   KA 2 1 ( )     1 tan Phase angle between input and output 16

Control Systems: First-order Systems Sinusoidal response Steady-state response Output frequency same as Input frequency 17

Control Systems: First-order Systems Steady-state sinusoidal response from system transfer function Y s ( ) 1  Normalized system transfer function   of a first-order system X s KA ( ) 1 s By replacing s=j ω  Y j ( ) 1 Magnitude     X j ( ) KA 2 1 ( )  Y j ( ) Angle      1 tan ( )  X ( j ) KA 18

Control Systems: First-order Systems Frequency response (magnitude) in decibels 1 20log dB 10   2 1 ( ) 19

Control Systems: First-order Systems Frequency response: phase     1 tan ( ) 20