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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 34

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online