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
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
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
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)
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
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
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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