L7-1_xferfcns - CDS 101: Lecture 6.1 Transfer Functions...

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CDS 101: Lecture 6.1 Transfer Functions Richard M. Murray and Steven Low 3 November 2003 Goals: y Motivate and define the input/output transfer function of a linear system y Understand the relationships among frequency response (Bode plot), transfer function, and state-space model y Introduce block diagram algebra for transfer functions of interconnected systems Reading: y Packard, Poola, Horowitz, Chapters 5-6 y Optional : Astrom, Section 5.1-5.3 y Advanced : Lewis, Chapters 3-4
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3 Nov 03 RMM and HM, Caltech CDS 2 Review: Frequency Response and Bode Plots Defn. The frequency response of a linear system is the relationship between the gain and phase of a sinusoidal input and the corresponding steady state (sinusoidal) output. Bode plot (1940; Henrik Bode) y Plot gain and phase vs input frequency y Gain is plotting using log-log plot y Phase is plotting with log-linear plot y Can read off the system response to a sinusoid – in the lab or in simulations y Linearity can construct response to any input (via Fourier decomposition) sin( ) uA t ω = sin( ) yB t φ = + (rad/sec) Frequency Response Frequency (rad/sec) Phase (deg) Magnitude (dB) -60 -50 -40 -30 -20 -10 0 10 20 0.1 1 10 -360 -270 -180 -90 0 () A B φω (deg) (rad/sec) 0 5 10 -1 0 1 0 5 10 -1 0 1
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3 Nov 03 RMM and HM, Caltech CDS 3 Transfer Functions Example: single “integrator” () H s sin( ) uA t ω = 0 5 10 -1 0 1 0 5 10 -1 0 1 22 | | Hj j H j αβ =+ = + x u yx = = & 2 sin( ) (/) s i n ( ) t yA t π ωω = =− ¯ | | 1/ = 1 s u y '' ( ) '' yH s u = y = |H(j ω )|A sin( ω t+ ] H(j ω )) ] H(j ω ) = - π /2 ] H(j ω ) = tan -1 ( β / α )
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3 Nov 03 RMM and HM, Caltech CDS 4 Transfer functions and frequency response H(j ω ) is like a complex function representation of the Bode plot… Frequency (rad/sec) Phase (deg) Magnitude (dB) -60 -50 -40 -30 -20 -10 0 10 20 0.1 1 10 -360 -270 -180 -90 0 | H(j ) | ] H(j ) One way to determine the transfer function of a given system is to fit the frequency response by a (rational) complex function. This works well in practice for so-called “minimum phase” systems, but otherwise can be tricky…
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3 Nov 03 RMM and HM, Caltech CDS 5 Transfer functions from state-space models Thm. The transfer function for a linear system Σ =( A , B , C , D ) is given by Thm.
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L7-1_xferfcns - CDS 101: Lecture 6.1 Transfer Functions...

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