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Unformatted text preview: 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]. Fall 2007 16.31 231 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 ) =  G ( j )  e j ( w ) Then w =  w  e j ( 1 t + ) applied to  G ( j )  e j ( w ) yields  w  G (j 1 )  e j ( 1 t + + ( 1 )) =  z  e j ( 1 t + 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 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 16.31 232 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 jt , where w c C m Then we get z = G ( s )  s = j w , z = z c e jt and z c C p We need only analyze z c = G ( j ) 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 ) 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 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]. Fall 2007 16.31 233 Singular Value Decomposition Must perform SVD of the matrix G ( s ) at each frequency s = j G ( j ) C p m U C p p R p m V C m m G = U V H U H U = I , UU H = I , V H V = I , V V H = I , and is diagonal....
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This note was uploaded on 11/07/2011 for the course AERO 16.31 taught by Professor Jonathanhow during the Fall '07 term at MIT.
 Fall '07
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