ece604 lecture 21

ece604 lecture 21 - Lecture 21 ECE 604 Linear Systems Doug...

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Dec. 3, 2009 Linear Systems © Douglas Looze 1 Lecture 21 ECE 604 Linear Systems Doug Looze
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Dec. 3, 2009 Linear Systems © Douglas Looze 2 Announcements Problem set 8 available Due Thursday, Dec. 10 Final exam Take home Due Wednesday, Dec. 16
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Dec. 3, 2009 Linear Systems © Douglas Looze 3 Last Time Sufficient condition for (ES) using Lyapunov functions
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Dec. 3, 2009 Linear Systems © Douglas Looze 4 Today LTI systems Necessary and sufficient conditions for linear systems BIBO stability Definition Fundamental relationship to impulse response Relationship of stability concepts Reading Rugh 123–125, 203–214
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Dec. 3, 2009 Linear Systems © Douglas Looze 5 Linear, Time-Invariant Systems System ( 29 ( 29 ( 29 0 0 d t t dt = = x Ax x x Note constant system matrix: ( 29 n n t × = A A R
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Dec. 3, 2009 Linear Systems © Douglas Looze 6 “Constant” quadratic Lyapunov function ( 29 ( 29 , T V t V = x x x Qx Note: ( 29 (iii) 0 0 d V t dt = = x Q Conditions
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Dec. 3, 2009 Linear Systems © Douglas Looze 7 Stability of LTI Systems Results Existence of Lyapunov function implies (S) Existence of Lyapunov function and ( 29 ( 29 min min 0 T λ = - - W A Q QA implies (ES) Procedure
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Dec. 3, 2009 Linear Systems © Douglas Looze 8 We can do more Requires positive definite derivative for (ES) Trial and error Theorem: (Invariance) Let V ( x ) = x T Qx be a Lyapunov function for ( 29 ( 29 ( 29 0 0 d t t dt = = x Ax x x Assume that = T T = - - W C C A Q QA and that ( A , C ) is (O). Then the system is (ES)
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Dec. 3, 2009 Linear Systems © Douglas Looze 9 Proof: Lyapunov function implies (S) ( 29 ( 29 ( 29 2 0 0 0 t V t V e d σ - = - A x x C x Integrating ( 29 0 t x t e x = A
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Dec. 3, 2009 Linear Systems © Douglas Looze 10 Assume ( 29 ( 29 Stable Re 0 i λ A Let
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Dec. 3, 2009 Linear Systems © Douglas Looze 11 1
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Dec. 3, 2009 Linear Systems © Douglas Looze 12 All eigenvalues of A have negative real parts ( 29 ( 29 Re 0 i i λ < 2200 A System is (ES) Note: This result ( Invariance ) implies we can substitute negative semi-definite and observable for negative definite
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Dec. 3, 2009 Linear Systems © Douglas Looze 13 Theorem: Given
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ece604 lecture 21 - Lecture 21 ECE 604 Linear Systems Doug...

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