# Lecture Fundamentals of control systems: Chapter 3 - TS. Huỳnh Thái Hoàng

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Lecture "Fundamentals of control systems - Chapter 3: System dynamics" presentation of content: The concept of system dynamics, dynamics of typical components, dynamics of control systems. Invite you to reference.

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## Nội dung Text: Lecture Fundamentals of control systems: Chapter 3 - TS. Huỳnh Thái Hoàng

1. Lecture Notes Fundamentals of Control Systems Instructor: Assoc. Prof. Dr. Huynh Thai Hoang Department of Automatic Control Faculty of Electrical & Electronics Engineering Ho Chi Minh City University of Technology Email: hthoang@hcmut.edu.vn huynhthaihoang@yahoo.com Homepage: www4.hcmut.edu.vn/~hthoang/ 6 December 2013 © H. T. Hoang - www4.hcmut.edu.vn/~hthoang/ 1
2. Chapter 3 SYSTEM DYNAMICS 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 2
3. Content  The concept of ssystem stem ddynamics namics  Time response  Frequency response  Dynamics of typical components  Dynamics y of control systems y 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 3
4. The concept of system dynamics 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 4
5. The concept of system dynamics  System dynamics is the study to understanding the behaviour of complex systems over time.  Systems described by similar mathematical model will expose similar dynamic responses.  To study st d the dynamic d namic responses responses, input inp t signals are usually s all chosen to be basic signals such as Dirac impulse signal, step signal, or sinusoidal signal.  Time response  Impulse response  Step response  Frequency response 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 5
6. Impulse response U (s) Y (s) G(s)  Impulse response: behavior of a system to a Dirac impulse Y ( s )  U ( s ).G ( s )  G ( s ) (due to U(s) = 1) y (t )  L 1Y ( s )  L 1G ( s )  g (t )  Impulse response is the inverse Laplace transform of the TF.  Impulse response is also referred as weighting function.  It is possible to calculate the response of a system to a arbitrarily input by taking convolution of the weighting function & the input. t y (t )  g (t ) * u (t )   g ( )u (t   )d 0 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 6
7. Step response U (s) Y (s) G(s)  Step response: behavior of a system to a step input G (s) Y ( s )  U ( s ).G ( s )  (because U(s) = 1/s) s   t y (t )  L 1Y ( s )  L 1  G ( s )    g ( )d  s  0  The step response is the integral of the impulse response  The step response is also referred as the transient function of the system. 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 7
8. Impulse and step response example  Calculate the impulse response and step response of the system described by the transfer function: s 1 U (s) Y (s) G(s)  G(s) ( ) s ( s  5)  Impulse response: 1  s  1  1  1 4  g (t )  L G ( s )  L  1 L     s ( s  5)   5s 5( s  5)  1 4 5t  g t   e ( ) 5 5  Step response: 1  G ( s )  1  s 1  4 1 4 h(t )  L  L  2   2  s   s ( s  5)  25s 5s 25( s  5) 1 4 5t 4  h(t )  t  e  5 25 25 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 8
9. Frequency response  Observe the response of a linear system at steady state when the input is a sinusoidal signal. 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 9
10. Frequency response definition  It can be observed that, that for linear system, system if the input is a sinusoidal signal then the output signal at steady-state is also a sinusoidal signal with the same frequency as the input, but diff different amplitude li d andd phase. h u (t)=Umsin (j) y (t)=Ymsin (j+) G(s) U (j) Y (j)  Definition: Frequency q y response p of a system y is the ratio between the steady-state output and the sinusoidal input. Y ( j ) Frequency response  U ( j ) It is proven that Frequency response  G ( s ) s  j  G ( j ) pro en that: 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 10
11. Magnitude response and phase response  general, G(j) is a complex function and it can be In general represented in algebraic form or polar form. G ( j )  P( )  jQ( )  M ( ).e j ( ) where: M ( )  G ( j )  P 2 ( )  Q 2 ( ) Magnitude response )  1  Q (  ( )  G ( j )  tg   Phase response  P ( )   Physical meaning of frequency response:  The magnitude response provides information about the gain of the system with respect to frequency .  The p phase response p p provides information about the p phase shift between the output & the input with respect to frequency 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 11
12. Graphical representation of frequency response  Bode diagram: is a graph of the frequency response of a linear system versus frequency plotted with a log-frequency axis. Bode diagram consists of two plots:  Bode magnitude plot expresses the magnitude response gain L() versus frequency  . L( )  20 lg M ( ) [dB]  Bode p phase plot expresses p p the p phase response p () versus frequency .  Nyquist N i t plot: l t is i a graph h in i polar l coordinates di t i which in hi h the th gain and phase of a frequency response G(j) are plotted when  changing g g from 0+. 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 12
13. Graphical representation of frequency response (cont’) Bode diagram Nyquist plot Gain margin Gain margin Phase margin Phase margin 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 13
14. Crossover frequency  Gain crossover frequency(c): is the frequency where the amplitude of the frequency response is 1 (or 0 dB). M (c )  1  L (c )  0  Phase crossover frequency (): is the frequency where phase shift of the frequency response is equal to 1800 (or equal to  radian).  ( )  1800   ( )   rad 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 14
15. Stability margin  Gain margin (GM): 1 GM   GM   L( ) [dB] M ( ) Physical meaning: The gain margin is the amount of additional ddi i l gain i at the h phase h crossover ffrequency required i d to bring the system to the stability boundary.  Phase margin (M) M  1800   (c ) Physical meaning: The phase margin is the amount of additional p phase lag g at the g gain crossover frequency q y required q to bring the system to the stability boundary. 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 15
16. Dynamics of basic factors 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 16
17. The proportional gain  T Transfer f function: f ti G( s)  K  Time response:  Impulse response: g (t )  K (t )  Step response: h(t )  K 1(t )  Frequency response: G ( j )  K  Magnitude response: M ( )  K  L( )  20 lg K  Phase response:  ( )  0 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 17
18. The proportional gain – Time response ((a)) Weighting g g function ((b)) Transient function 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 18
19. The proportional gain – Frequency response Bode diagram Nyquist plot 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 19
20. Integral factor 1  Transfer function: G(s)  s  Time response:  Impulse response: g (t )  1(t )  Step response: h(t )  t 1(t ) 1 1  Frequency q y response: p G ( j )  j j  1  Magnitude response: M ( )   L( )  20 lg g   Phase response:  ( )  900 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 20