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
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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.