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# topic16 - Topic #16 16.31 Feedback Control Systems...

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Unformatted text preview: Topic #16 16.31 Feedback Control Systems State-Space Systems • Closed-loop control using estimators and regulators. • Dynamics output feedback • “Back to reality” 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 16–1 Combined Estimators and Regulators • Now evaluate stability and/or performance of a controller when K designed assuming u ( t ) = − K x ( t ) but implemented as u ( t ) = − K ˆ x ( t ) • Assume we have designed a closed-loop estimator with gain L ˙ ˆ x ( t ) = ( A − LC )ˆ x ( t ) + B u ( t ) + L y ( t ) ˆ y ( t ) = C ˆ x ( t ) • The closed-loop system dynamics are given by: ˙ x ( t ) = A x ( t ) + B u ( t ) ˙ ˆ x ( t ) = ( A − LC )ˆ x ( t ) + B u ( t ) + L y ( t ) y ( t ) = C x ( t ) ˆ y ( t ) = C ˆ x ( t ) u ( t ) = − K ˆ x ( t ) • Which can be compactly written as: ˙ x ( t ) ˙ ˆ x ( t ) = A − BK LC A − BK − LC x ( t ) ˆ x ( t ) ⇒ ˙ x cl ( t ) = A cl x cl ( t ) • This does not look too good at this point – not even obvious that the closed-system is stable. λ i ( A cl ) =?? November 5, 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 16–2 • Can fix this problem by introducing a new variable ˜ x = x − ˆ x and then converting the closed-loop system dynamics using the similarity transformation T ˜ x cl ( t ) x ( t ) ˜ x ( t ) = I I − I x ( t ) ˆ x ( t ) = T x cl ( t ) – Note that T = T − 1 • Now rewrite the system dynamics in terms of the state ˜ x cl ( t ) A cl ⇒ TA cl T − 1 ¯ A cl – Since similarity transformations preserve the eigenvalues we are guaranteed that λ i ( A cl ) ≡ λ i ( ¯ A cl ) • Work through the math: ¯ A cl = I I − I A − BK LC A − BK − LC I I − I = A − BK A − LC − A + LC I I − I = A − BK BK A − LC • Because ¯ A cl is block upper triangular, we know that the closed- loop poles of the system are given by det( sI − ¯ A cl ) det( sI − ( A − BK )) · det( sI − ( A − LC )) = November 5, 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 16–3 • Observation: Closed-loop poles for this system consist of the union of the regulator poles and estimator poles . • So we can just design the estimator/regulator separately and combine them at the end....
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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.

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topic16 - Topic #16 16.31 Feedback Control Systems...

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