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 *S** _{diag}* represented by the diagonal canonical form as follows:

⎩⎨

⎧

=

+

= ) ( )

(

) ( ) ( ) (

T*x* *t*
*c*
*t*
*y*

*t*
*bu*
*t*
*Ax*
*t*
*S*_{diag}*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 λ_{1}, λ_{2}, L ,λ* _{n}* 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 *b*_{1},*b*_{2},L,*b** _{n}* so that

*S*

*can be controllable. In the same way, find the necessary and sufficient condition with respect to*

_{diag}*c**n*

*c*

*c*_{1}, _{2},L, so that *S** _{diag}* can be observable.