T
2,1)
k/ k − Pk/ s+1 = ∑ P
C
C
+ R
C
.
(56)
i
i
iPiCi
i
i Pi
i= k
Thus, the right hand side indicates the amount of the reduction of the estimation error by the
fixed-point smoothing over the optimal filtering.
4. The fixed-interval smoothing
We consider the fixed-interval smoothing problem in this section. Namely, we investigate the
optimal estimate ˆ xt/ N of the state xt at all times t = 0, 1, · · · , N based on the observation Y N of all the states { y 0, y 1, · · · , yN}. Applying equality (49), we easily obtain the following equality.
Lemma 4.1.
The equality
ˆ x
T
−1
t/ N = ˆ
xt/ t+1 + Pt Lt Pt+1
( ˆ xt+1/ N − ˆ xt+1/ t+1)
(57)
62
Discrete Time Systems
holds for t = 0, 1, · · · , N − 1 .
Proof Using the notation
−1
˜
ν
T
T
i = Ci
CiPiCi + Ri
νi,
(58)
we have
t
ˆ xk/ t+1 = ˆ xk/ k + Pk ∑ Ψ( i, k) T ˜ νi
(59)
i= k
for k ≤ t due to (49). In view of (59) , we also have
t
t
ˆ xk/ t+1 = ˆ xk/ k + Pk ˜ νk + Pk ∑ Ψ( i, k) T ˜ νi = ˆ xk/ k+1 + Pk ∑ Ψ( i, k) T ˜ νi (60)
i= k+1
i= k+1
for k + 1 ≤ t. Putting t + 1 = N and k = t + 1 in equality (59), we have
N−1
ˆ xt+1/ N = ˆ xt+1/ t+1 + Pt+1 ∑ Ψ( i, t + 1) T ˜ νi.
(61)
i= t+1
Putting t + 1 = N and k = t in equality (60), we have
N−1
N−1
ˆ x
T
t/ N = ˆ
xt/ t+1 + Pt ∑ Ψ( i, t) T ˜ νi = ˆ xt/ t+1 + PtLt ∑ Ψ( i, t + 1) T ˜ νi.
(62)
i= t+1
i= t+1
Substituting (61) into (62), we have
ˆ x
T
−1
t/ N = ˆ
xt/ t+1 + Pt Lt Pt+1
( ˆ xt+1/ N − ˆ xt+1/ t+1) .
The above derivation is valid for t = 0, 1, · · · , N − 2. It is easy to observe that equality (57)
also holds for t = N − 1.
It is a simple task to obtain the following Fraser-type algorithm from (57).
Theorem 4.2.
We obtain the fixed-interval smoother
ˆ x
T
t/ N = ˆ
xt/ t+1 + Pt Lt λt+1 ,
(63)
−1
λ
T
T
T
t = Lt λt+1 + Ct
CtPtCt + Rt
νt .
(64)
for t = N − 1, N − 2, · · · , 1, 0 . Here, we have λN = 0 .
Proof For t = 0, 1, · · · , N, we put
λ
−1
t = Pt
( ˆ xt/ N − ˆ xt/ t) .
(65)
We then have λN = 0. Substituting (65) into (57), we obtain equality (63). Then, by utilizing
(63) and (65), we have
λ
−1
T
t = Pt
ˆ xt/ t+1 + Pt Lt λt+1 − ˆ xt/ t .
(66)
In view of the equality
ˆ xt/ t+1 − ˆ xt/ t = Pt ˜ νt
(67)
New Smoothers for Discrete-time Linear Stochastic Systems with Unknown Disturbances
63
which follows from (27) in Tanikawa & Sawada (2003), we obtain
λ
T
t = Lt
λt+1 + ˜ νt
−1
= L T
T
T
t
λt+1 + Ct
CtPtCt + Rt
νt.
(68)
Thus, we proved (64).
Remark 4.3.
When Et ≡ O holds for all t (i.e., the unknown input term is zero), we shall see
that fixed-interval smoother (63)-(64) is identical to the fixed-interval smoother obtained from
the standard Kalman filter (see e.g., Katayama (2000)). Thus, our algorithm is consistent with
the known fixed-interval smoothing algorithm for systems without unknown inputs. This
can be shown as follows. Assuming that Et = O, we have Ht = O for t = 0, 1, · · · , N (see
Propositin 2.4). Note that in (59), i.e.,
t
ˆ xk/ t+1 = ˆ xk/ k + Pk ∑ Ψ( i, k) T ˜ νi
i= k
ˆ xk/ t+1 and ˆ xk/ k respectively reduce to ˆ xk/ t and ˆ xk/ k−1 which are respectively the optimal smoother and the optimal filter obtained from the standard Kalman filter. Then, the above
equality is identical to (7.18) in Katayama (2000). Since the rest of the proof can be done in the
same way as in Katayama (2000), we obtain the same smoother.
5. The fixed-lag smoothing
We study the fixed-lag smoothing problem in this section. For a fixed L > 0, we investigate
an iterative algorithm to compute the optimal state estimate ˆ xt− L/ t of the state xt− L based on
the observation Y t.
We consider the following augmented system:
⎡
⎤
⎡
⎤ ⎡
⎤ ⎡
⎤
⎡
⎤
⎡ ⎤
xt+1
At O . . . O
xt
Bt
Et
I
⎢
⎢ x
⎥
⎢
⎥ ⎢
⎥ ⎢
⎥
⎢
⎥
⎢ ⎥
t
I O . . . O
xt−1
O
O
O
⎢
⎥ = ⎢
⎥ ⎢
⎥ + ⎢ ⎥
⎢
⎥
⎢ ⎥ ζ
⎣
.
⎥
⎢
⎥ ⎢
⎥ ⎢
⎥
⎢
⎥
⎢ ⎥
.
.
.
.
.
.
t,
(69)
.
⎦
⎣
. .
⎦ ⎣ .. ⎦ ⎣ .. ⎦ ut + ⎣ .. ⎦ dt + ⎣ .. ⎦
xt− L+1
O
I O
xt− L
O
O
O
⎡
⎤
xt+1
⎢
⎢ x
⎥
t
⎥
yt+1 = [ Ct+1 O . . . O] ⎢
+ η
⎣
.
⎥
.
t+1.
(70)
.
⎦
xt− L+1
Denote these equations respectively by
xt+1 = At xt + Bt ut + Et dt + Jt ζt,
(71)
yt+1 = Ct+1 xt+1 + ηt+1,
(72)
64
Discrete Time Systems
where
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
xt
At O . . . O
Bt
Et
⎢
⎢ x
⎥
⎢
⎥
⎢
⎥
⎢
⎥
t−1 ⎥
⎢ I O . . . O ⎥
⎢ O ⎥
⎢ O ⎥
xt = ⎢
⎣ . ⎥
⎢
⎥
⎢
⎥
⎢
⎥
.
.
.
.
.
⎦ , At = ⎣
. .
⎦ , Bt = ⎣ .. ⎦ , Et = ⎣ .. ⎦ ,
xt− L
O
I O
O
O
⎡ ⎤
I
⎢
⎢ O ⎥
⎥
Jt = ⎢
⎣ . ⎥
.. ⎦ and Ct+1 = [ Ct+1 O . . . O] .
O
Here, I and O are the identity matrix and the zero matrix respectively with appropriate
dimensions. By making use of the notations
⎡
⎤
Ht+1
⎢
⎢ O ⎥
⎥
Ht+1 = ⎢
⎣ . ⎥
.. ⎦ and Tt+1 = I − Ht+1 Ct+1,
O
we have the equalities:
⎡
⎤
Et
⎢
⎢ O ⎥
⎥
Ct+1 Et = [ Ct+1 O . . . O] ⎢
=
⎣ . ⎥
C
.
t+1 Et,
. ⎦
O
⎡
⎤
⎡
⎤
Ht+1
Tt+1 O . . . O
⎢
⎢ O ⎥
⎥
⎢
⎢ O I . . . O ⎥
⎥
Tt+1 = I − ⎢
[
⎣ . ⎥ C
⎢
⎥
.
t+1 O . . . O] =
.
.
⎦
⎣
. .
⎦ ,
O
O O . . . I
⎡
⎤ ⎡
⎤
⎡
⎤
Tt+1 O . . . O
At O . . . O
A 1
O . . . O
⎢
t+1
⎢ O I . . . O ⎥
⎥ ⎢
⎢ I O . . . O ⎥
⎥
⎢
⎢ I
O . . . O ⎥
⎥
A 1
= T
⎢
⎥ ⎢
⎥ = ⎢
⎥
t+1
t+1 At = ⎣
. .
.
.
.
⎦ ⎣
. .
⎦
⎣
. .
⎦ .
O O . . . I
O
I O
O
I O
We introduce the covariance matrix Pt of the state estimation error of augmented system
(71)-(72):
⎧
⎫
⎪⎡
⎤ ⎡
⎤
⎪
T
⎪
x
x
⎪
⎪
t − ˆ
xt/ t
t − ˆ
xt/ t
⎪
⎨⎢
⎪
⎪
⎢ x
⎥ ⎢
⎥ ⎬
t−1 − ˆ
xt−1/ t ⎥ ⎢ xt−1 − ˆ xt−1/ t ⎥
Pt = E ⎪⎢
.
⎥ ⎢
.
⎥
.
(73)
⎪
⎪⎣
.
⎦ ⎣
.
⎦ ⎪
⎪
.
.
⎪
⎩
⎪
⎪
x
⎭
t− L − ˆ
xt− L/ t
xt− L − ˆ xt− L/ t
New Smoothers for Discrete-time Linear Stochastic Systems with Unknown Disturbances
65
By using the notations
T
Pt− i, t− j/ t = E ( xt− i − ˆ xt− i/ t) xt− j − ˆ xt− j/ t
,
Pt− i/ t = Pt− i, t− i/ t ,
we can write
⎡
⎤
Pt/ t
Pt, t−1/ t . . . Pt, t− L/ t
⎢
⎢ P
⎥
t−1, t/ t
Pt−1/ t . . . Pt−1, t− L/ t ⎥
Pt = ⎢
⎣
.
⎥
.
.
.
.
. .
..
⎦ .
(74)
Pt− L, t/ t Pt− L, t−1/ t . . . Pt− L/ t
Here, it is easy to observe that Pt/ t = Pt holds. We also note that
T
C
T
t Pt Ct
+ Rt = Ct Pt/ t Ct + Rt.
(75)
From now on, we use the following notation for brevity:
C
T
t := Ct Pt Ct
+ Rt.
(76)
Applying the optimal filter given in Proposition 2.2 to augmented system (71)-(72), we have
xt+1/ t+1 = A 1
x
t+1
t/ t + Gt
yt − Ct xt/ t
+ Ht+1 yt+1 + Tt+1 Bt ut,
(77)
where
⎡
⎤
P
T − H
⎢ t/ t Ct
t Rt
T
⎥