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# topic23 - Topic#23 16.31 Feedback Control Systems MIMO...

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Topic #23 16.31 Feedback Control Systems MIMO Systems Singular Value Decomposition Multivariable Frequency Response Plots Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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Fall 2007 16.31 23–1 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 sys- tems 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 ) | e j ( ω 1 t + ψ + φ ( ω 1 )) = | z | e j ( ω 1 t + ψ o ) z Amplification and phase shift of the input signal obvious in the SISO case. MIMO extension? Is the response of the system large or small? G ( s ) = 10 3 /s 0 0 10 3 /s November 18, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 2007 16.31 23–2 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 frequency See how this response changes with frequency So inputs are w = w c e jωt , where w c C m Then we get z = G ( s ) | s = j ω w , z = z c e jωt and z c C p We need only analyze z c = G ( j ω ) w c As in the SISO case, we need a way to establish if the system response is large or small . How much amplification we can get with a bounded input. Consider z c = G ( j ω ) w c and set w c 2 = w H c w c 1 . What can we say about the z c 2 ? Answer depends on ω and on the direction of the input w c Best found using singular values. November 18, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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Fall 2007 16.31 23–3 Singular Value Decomposition Must perform SVD of the matrix G ( s ) at each frequency s = j ω G ( j ω ) C p × m U C p × p Σ R p × m V C m × m G = U Σ V H U H U = I , UU H = I , V H V = I , V V H = I , and Σ is diagonal. Diagonal elements σ k 0 of Σ are the singular values of G . σ i = λ i ( G H G ) or σ i = λ i ( GG H ) the positive ones are the same from both formulas.
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• Fall '07
• jonathanhow
• Feedback Control Systems, Massachusetts Institute of Technology, Singular value decomposition, DD Month YYYY

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