ABSTRACT
What can be done with respect to controlling system states?
System model: ( ) ( ) ( )X Xk k u k+ = +1 Φ Γ (6.3a)
Measurement model known initial condition
: ( ) ( ) ( )
y k C k= =
X
X 0
(6.3b)
The equivalent discrete system matrices can be shown as
Φ Γ= = ∫e e d BAh h A; 0
σ σ
With a corresponding state response given by
X X( ) ( ) ( )k u ik
= + =
− −∑Φ Φ Γ0 0
(6.4)
Now consider k = n: Can we find u(0), u(1), . . ., u(n − 1), so that x(n) = ξ = arbitrary vector, starting at an initial
ξ – ( ) ( )
( ) ( ) ( )
Φ Φ Γ
Γ ΦΓ Φ Γ
u i
u n u n u
X 0
1 2 0 0
=
= − + − + +
(6.5)
ξ – ( ) |
|
|
|
|
|
( ) ( )
( )
Φ Γ ΦΓ Φ Γk n
u n
u n
u
X 0
1 2
1 = …
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
(6.6)
where
Hc n= … ⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
|
|
|
|
|
| Γ ΦΓ Φ Γ1
If Hc is invertible, it is possible to find the requisite {u(i)}. Note: state may not necessarily stay at ξ for k > n.
