EEL5173 Fall 2009 Lecture _14

EEL5173 Fall 2009 Lecture _14 - IV. Controllability and...

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IV. Controllability and Observability f x r 1. Controllability and Reachability (Zak) i. DT System (1) Reachability x 1 x 2 ) ( ) ( ) 1 ( k Bu k x A k x + = + r r 0 r Theorem. The system is reachable if and only if {} [ ] B A AB B B A S n def 1 , , , , = L is of rank n .
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Ex. 0.5 2 3 0.7 D D u ( k ) y ( k ) x 1 ( k ) x 1 ( k+ 1) x 2 ( k ) x 2 ( k+ 1) {} [] = = = = = 0 0 5 . 0 1 , , 2 0 1 , 7 . 0 0 0 5 . 0 AB B B A S n B A S { A , B } is singular. Hence, the system is not reachable.
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Ex. A system is governed by the following state equations. Determine whether the system is reachable or not. ) ( 0 0 1 ) ( 0 1 0 0 0 1 4 6 9 ) 1 ( k u k x k x + = + r r S { A , B } is non-singular. Hence, the system is reachable. {} [] {} 1 , 1 0 0 9 1 0 75 9 1 , , , 3 2 = = = = B A S B A AB B B A S n
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i x r (2) Controllability x 1 x 2 0 r Given there exists an integer q > 0 and a control sequence , i x r { u ( k ) u ( k ) = 0 for k > q } that transfers . 0 ) ( ) 0 ( r r r r = = q x to x x i Definition (Controllability)
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Theorem. The system is controllable if and only if {} [] { } [ ] n A B A S rank B A S rank , , , = Theorem. Reachability Controllability. Theorem. If the matrix A is non-singular, then Reachability Controllability. Ex. (continue) ; 0 1 , 7 . 0 0 0 5 . 0 = = B A - not reachable. Since the matrix A is invertible and the system is proven to be not reachable, it is not controllable either.
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Ex. A system is governed by the following state equations. Determine the reachability and the controllability of the system. ) ( 0 1
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This note was uploaded on 07/15/2011 for the course EEL 5173 taught by Professor Tung during the Fall '09 term at FSU.

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EEL5173 Fall 2009 Lecture _14 - IV. Controllability and...

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