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Unformatted text preview: Dec. 10, 2009 Linear Systems Douglas Looze 1 Lecture 23 ECE 604 Linear Systems Doug Looze Dec. 10, 2009 Linear Systems Douglas Looze 2 Announcements Problem Set 8 due today Final Exam available Take Home Due Wednesday, Dec. 16 Dec. 10, 2009 Linear Systems Douglas Looze 3 Last Time State feedback Definition and concept State feedback does not affect controllability State feedback can affect observability Dec. 10, 2009 Linear Systems Douglas Looze 4 Today Pole placement Observers State estimate Feedback with observers Dec. 10, 2009 Linear Systems Douglas Looze 5 SISO Pole Placement Concept Pole location important to dynamic response of system Cant affect zeros with feedback Can we place the system poles where we want? Same motivation for MIMO systems For flexibility of state feedback in MIMO systems, see: B.C. Moore, On the flexibility offered by state feedback in multivariable systems beyond closed loop eigenvalue assignment, IEEE TAC , Oct. 1976 Dec. 10, 2009 Linear Systems Douglas Looze 6 Problem statement Desired poles (complex conjugate pairs) 1 2 , , , n Define Dec. 10, 2009 Linear Systems Douglas Looze 7 Want ( 29 ( 29 q s q s = That is, find K such that ( 29 ( 29 det s q s + = I A BK Theorem: The eigenvalues of a controllable SISO system can be arbitrarily assigned by state feedback. Note: weve shown that controllability is necessary (otherwise, there is a pole that cant be moved) Dec. 10, 2009 Linear Systems Douglas Looze 8 Proof: We will follow the steps Show for systems that are in standard controllable form Show all controllable systems can be transformed to standard controllable form system Use these results to construct a feedback gain for an arbitrary controllable system Dec. 10, 2009 Linear Systems Douglas Looze 9 Assume the system is in standard controllable form Note that we need only consider the state equation ( 29 ( 29 { ( 29 1 2 1 1 1 1 1 n d t t u t dt  = +  B A x x L L M M M M M L L 1 4 4 4 4 442 4 4 4 4 4 43 Dec. 10, 2009 Linear Systems Douglas Looze 10 Closed loop system (without the reference) ( 29 ( 29 1 1 2 2 3 1 1 1 1 n n d t t dt k k k k  =  A BK x x L L M M M M L L 1 4 4 4 4 4 4 4 442 4 4 4 4 4 4 4 4 43 ( 29 ( 29 t + Br Dec. 10, 2009 Linear Systems Douglas Looze 11 Desired closed loop system (without the reference) ( 29 ( 29 1 2...
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 Spring '10

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