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# lec14 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Topic #14 16.31 Feedback Control Systems MIMO Systems Singular Value Decomposition
Spr 2008 16.323 14–1 Multivariable Frequency Response In the MIMO case, the system G ( s ) is described by a p × m transfer function matrix (TFM) Still have that G ( s ) = C ( sI A ) 1 B + D But G ( s ) A, B, C, D MUCH less obvious than in SISO case. Also seen that the discussion of poles and zeros of MIMO systems is much more complicated. In SISO case we use the Bode plot to develop a measure of the system “size”. Given z = Gw , where G (j ω ) = | G (j ω ) | e ( w ) Then w = | w | e j ( ω 1 t + ψ ) applied to | G (j ω ) | e ( w ) yields | w || G (j ω 1 ) | j ( ω 1 t + ψ + φ ( ω 1 )) | | j ( ω 1 t + ψ o ) z e = z e Amplification and phase shift of the input signal obvious in the SISO case. MIMO extension? Is the response of the system large or small? 10 3 /s 0 G ( s ) = 0 10 3 /s June 18, 2008

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Spr 2008 16.323 14–2 For MIMO systems, cannot just plot all of the g ij elements of G Ignores the coupling that might exist between them. So not enlightening. Basic MIMO frequency response: Restrict all inputs to be at the same frequency Determine how the system responds at that
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lec14 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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