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MIT16_30F10_lab02_app

# MIT16_30F10_lab02_app - By then repeating this process with...

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16.30/31, Fall 2010 Lab #2 Appendix Consider a system driven by multiple controllers in parallel; a block diagram representing this scenario for two controllers is provided below. The plant and all controllers are spec- ified as state-space models; our objective is to identify the state-space model for the loop dynamics L ( s ) = G ( s ) G c ( s ), with inputs ( y 1 c , y 2 c ) and outputs ( y 1 , y 2 ). The state-space models for the plant and controllers are as follows: G ( s ) : x ˙ ( t ) = A x ( t ) + [ B 1 B 2 ] u u 1 2 ( ( t t ) ) , y 1 ( t ) C 1 y 2 ( t ) = C 2 x ( t ) , G 1 c ( s ) : x ˙ c 1 ( t ) = A 1 c x c 1 ( t ) + B c 1 e 1 ( t ) , u 1 ( t ) = C c 1 x 1 c ( t ) + D c 1 e 1 ( t ) , G 2 c ( s ) : x ˙ c 2 ( t ) = A 2 c x c 2 ( t ) + B c 2 e 2 ( t ) , u 2 ( t ) = C c 2 x c 2 ( t ) + D c 2 e 2 ( t ) , where e i ( t ) = y c ( t ) y i ( t ). First, form the composite dynamics for the plant G ( s ) and i controller G 1 c ( s ): � � x ˙ A B 1 C 1 x B 1 D 1 B 2 x ˙ 1 = 0 A 1 c x 1 + B 1 c e 1 + 0 u 2 , c c c c � � y 1 C 1 0 x y 2 = C 2 0 x c 1 . We can “close the loop” by simply applying the fact that e 1 ( t ) = y c ( t ) y 1 ( t ): 1 � � � � �� x ˙ A B 1 C 1 x B 1 D 1 c x B 2 1 = A 1 c 1 + B 1 c y 1 [ C 1 0 ] 1 + u 2 x ˙ 0 x x 0 c c c c c � � = A B 1 D c 1 C 1 B 1 C c 1 x + B 1 D c 1 y c B 2 B c 1 C 1 A c 1 x c

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Unformatted text preview: By then repeating this process with the second input u 2 , only the references y 1 c and y 2 c will remain as inputs, yielding the desired state-space model. G c ( s ) 1 G c ( s ) 2 y 2 c y 1 c G( s ) y 1 y 2 u 1 e 1 u 2 e 2 _ _ Image by MIT OpenCourseWare. MIT OpenCourseWare http://ocw.mit.edu 16.30 / 16.31 Feedback Control Systems Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT16_30F10_lab02_app - By then repeating this process with...

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