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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 – Can’t 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 λ λ λ … – 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: we’ve shown that controllability is necessary (otherwise, there is a pole that can’t 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 α α α α = +  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 α α α α =  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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 Linear Systems, Douglas Looze

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