2) Find the necessary and sufficient condition with respect to b1,b2,L,bn so that Sdiag can be controllable

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システム制御工学およびダイナミックシステム制御論

(阿部健一 教授・吉澤 誠 教授)

レポート課題 (2014 年 5 月 1 日出題)

提出期限:2014 年 5 月 8 日(木)9:00 提出先:通研 408 講義室

To print out this document, refer to http://www.abe.ecei.tohoku.ac.jp/?Lecture Problem

Consider a system Sdiag represented by the diagonal canonical form as follows:

=

+

= ) ( )

(

) ( ) ( ) (

Tx t c t y

t bu t Ax t Sdiag x&

where

T 2

1

T 2

1

2 1

] , , , [

] , , , [

] , , , [ diag

n n

n

c c c c

b b b b

A

L L

L

=

=

= λ λ λ

and λ1, λ2, L ,λn are eigenvalues of A that are distinct from one another. Let V

denote Vandermonde matrix defined by

=

1 2

1 2 2

2 2

1 1 2

1 1

1 1 1

n n n

n

n n

V

λ λ

λ

λ λ

λ

λ λ

λ

L M M

M M

L L

1) Find the determinant of V .

2) Find the necessary and sufficient condition with respect to b1,b2,L,bn so that Sdiag can be controllable. In the same way, find the necessary and sufficient condition with respect to

cn

c

c1, 2,L, so that Sdiag can be observable.

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