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ece604 lecture 09

# ece604 lecture 09 - Lecture 9 ECE 604 State Variable...

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Oct. 6, 2009 Linear Systems © Douglas Looze 1 Lecture 9 ECE 604 State Variable Analysis Doug Looze

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Oct. 6, 2009 Linear Systems © Douglas Looze 2 Announcements PS3 available Due Thursday
Oct. 6, 2009 Linear Systems © Douglas Looze 3 Last Time Characterizations of Controllability Observability In terms of Grammians Linear independence of vector time functions

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Oct. 6, 2009 Linear Systems © Douglas Looze 4 Conditions Equivalent ( A ( t ), B ( t )) controllable Independent rows of ( 29 ( 29 , t σ B Φ Grammian invertible n × n matrix Full rank ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 , , , f t T T f t t t t t t t t t dt = W B B Φ Φ
Oct. 6, 2009 Linear Systems © Douglas Looze 5 Equivalent ( A ( t ), C ( t )) observable Independent columns of ( 29 ( 29 0 , t t t C Φ Grammian invertible n × n matrix Full rank ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 , , , f t T T f t t t t t t t t t dt = M C C Φ Φ

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Oct. 6, 2009 Linear Systems © Douglas Looze 6 Exercise Consider the linear, time-varying system ( 29 ( 29 ( 29 ( 29 0 0 0 1 d t t t u t dt = + x x b Specify whether the system is controllable on any interval for the following input matrices. If it is controllable on some interval, determine such an interval ( 29 ( 29 ( 29 1 1 1 a) b) c) 1 t t t t t e e -   = = =     b b b
Oct. 6, 2009 Linear Systems © Douglas Looze 7 The transition matrix for this system is: 0 0 0 1 1 0 0 t t t e e e = = A

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ece604 lecture 09 - Lecture 9 ECE 604 State Variable...

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