ps 6 604 - Let G s = n G s d G s = s 2 s 1 s 5 s 4 s 3 s 2...

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ECE 604 LINEAR SYSTEMS Problem Set #6 Issued: Thursday, November 12, 2009 Due: Thursday, November 19, 2009 Problem 1: Consider the p -output homogeneous system ! x t ( ) = Ax t ( ) x 0 ( ) = x 0 y t ( ) = Cx t ( ) We would like to determine the initial state from samples of the output y k ! ( ) for k ! 0 . a) Define ! = e A " . Show that we determine the initial state uniquely from y k ( ) if and only if: rank C C ! C ! 2 ! C ! n " 1 # $ % % % % % % ( ( ( ( ( ( = n i.e., the sampled-data system is observable ( e A , C ( ) is observable). b) Show that e A , C ( ) (O) implies A , C ( ) is (O). c) Show (by a counterexample) that e A , C ( ) may not be (O) if A , C ( ) is (O). Problem 2:
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Unformatted text preview: Let G s ( ) = n G s ( ) d G s ( ) = s 2 ! s + 1 s 5 + s 4 + s 3 + s 2 + s + 1 Give the controllable canonical realization of G s ( ) that has dimension 5. What is the dimension of the minimal realization? Problem 3: Rugh, Problem 10.3, p. 178. Problem 4: Rugh, Problem 10.8, p. 179. Problem 5: Consider the transfer function G s ( ) = s + 1 ( ) 2 s ! 1 ( ) s 2 ! 4 ( ) s s ! 1 " # $ $ $ $ $ $ % & ’ ’ ’ ’ ’ ’ Determine a minimal realization for G ( s ) in controllable canonical form....
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ps 6 604 - Let G s = n G s d G s = s 2 s 1 s 5 s 4 s 3 s 2...

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