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
- 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
- Chapter 3 SYSTEM DYNAMICS 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 2
- 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
- The concept of system dynamics 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 4
- 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
- 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 1Y ( s ) L 1G ( 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
- 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 1Y ( 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Dynamics of basic factors 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 16
- 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
- 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
- The proportional gain – Frequency response Bode diagram Nyquist plot 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 19
- 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
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