Discrete Time Systems Part 3 ppt

Distributed Fusion Prediction for Mixed Continuous-Discrete Linear Systems

49
5. Conclusions
In this chapter, two fusion predictors (FLP and PFF) for mixed continuous-discrete linear
systems in a multisensor environment are proposed. Both of these predictors are derived by
using the optimal local Kalman estimators (filters and predictors) and fusion formula. The
fusion predictors represent the optimal linear combination of an arbitrary number of local
Kalman estimators and each is fused by the MSE criterion. Equivalence between the two
fusion predictors is established. However, the PFF algorithm is found to more significantly
reduce the computational complexity, due to the fact that the PFF’s weights
k
(i)
t
b do not
depend on the leads 0
Δ
> in contrast to the FLP’s weights
(i)
t+Δ
a .
Appendix
Proof of Theorem 1
(a), (c) Equation (12) and formula (14) immediately follow as a result of application of the
general fusion formula [20] to the optimization problem (10), (11).
(b) In the absence of observations differential equation for the local prediction error
(i) (i)
τττ
ˆ
xx-x
=


takes the form

(i) (i) (i)
τττττ ττ
ˆ
x=x-x=Fx+Gv.


 
(A.1)
Then the prediction cross-covariance
(
)
T
(i
j
)(
j
)
(i)
τττ
P=Exx

associated with the
(i)
τ
x

and

(
j
)
τ
x


satisfies the time update Lyapunov equation (see the first and third equations in (13)). At
k
t=t the local error
k
(i)
t
x

can be written as
(
)
- -
kk k
k k kkkkk kk k kkk kkkkk
(i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i)
tt t n
t t t t t tt t t t t tt t t t t t
ˆˆ ˆ ˆ
x=x-x=x-x -L y-Hx =x -L Hx+w-Hx I-LH x -Lw.
⎡⎤⎡ ⎤
=
⎢⎥⎢ ⎥
⎣⎦⎣ ⎦

 
(A.2)
Given that random vectors
kk
(i) (i)
tt
x,w

and
k
(
j
)
t
w
are mutually uncorrelated at
i
j
≠
, we obtain
observation update equation (13) for
(
)
T
kkk
(i
j
)(
j
)

(i)
ttt
P=Exx

.
This completes the proof of Theorem 1.
Proof of Theorem 2
It is well known that the local Kalman filtering estimates
(i)
τ
ˆ
x
are unbiased, i.e.,
(i)
ττ
ˆ
E(x )=E(x )

or
(
)
(
)
(i) (i)
τττ
ˆ
Ex =Ex-x =0

at
k

0 τ t
≤
≤ . With this result we can prove unbiased property at
k
t τ t+Δ<≤ . Using (8) we obtain

kk
(i) (i)
(i) (i) (i)
τττττ ττ k
τ=t t
ˆ
x=x-x=Fx+Gv,x =x,t τ t+Δ ,
≤≤


   
(A.3)
or

(
)
(
)
(
)
(
)
kk
(i) (i)

(i) (i)
d
τττ k
τ=t t
dτ
Ex =FEx , Ex =Ex =0, t τ t+Δ.≤≤

(A.4)
Differential equation (A.4) is homogeneous with zero initial condition therefore it has zero
solution
()
(
)
(
)
(i) (i)
τττk
ˆ
Ex 0or Ex =Ex , t τ t+Δ.≡≤≤


Since the local predictors
(i)
t+Δ
ˆ
x,i1, ,N=
are unbiased, then we have

()
()

()()
NN
(i) (i) (i)
FLP
t+Δ t+Δ t+Δ
t+Δ t+Δ t+Δ
i=1 i=1
ˆˆ
Ex = a Ex = a Ex =Ex .
⎡
⎤
⎢⎥
⎣
⎦
∑∑
(A.5)
Discrete Time Systems

50
This completes the proof of Theorem 2.
Proof of Theorem 3
a., c. Equations (18) and (19) immediately follow from the general fusion formula for the
filtering problem (Shin et al., 2006)
b. Derivation of observation update equation (13) is given in Theorem 1.
d. Unbiased property of the fusion estimate
PFF
t+Δ
ˆ
x
is proved by using the same method as in

Theorem 2.
This completes the proof of Theorem 3.
Proof of Theorem 4
By integrating (8) and (17), we get

(
)
k
k
(i) (i)
PFF FF
kt+Δ kt
t+Δ
t
ˆˆ ˆˆ
x=Φ t+Δ,t x , i 1, ,N, x =Φ(t+Δ,t )x ,
=
(A.6)
where
Φ(t,s) is the transition matrix of (8) or (17). From (10) and (16), we obtain

kkk
k
kk k k
NN N
(i) (i) (i) (i) (i) (i)
FLP
t+Δ k
t+Δ t+Δ t+Δ
t t,t ,Δ t

i=1 i=1 i=1
NN
(i) (i) (i) (i)
PFF FF
t+Δ kt k
t t t,t ,Δ t
i=1 i=1
ˆˆ ˆ ˆ
x=ax=aΦ(t+Δ,t )x = A x ,
ˆˆ ˆˆ
x=Φ(t+Δ,t )x = Φ(t+Δ,t )b x = B x ,
∑∑ ∑
∑∑
(A.7)
where the new weights take the form:

(
)
(
)
kkk
(i) (i) (i) (i)
kk
t+Δ
t,t ,Δ t,t ,Δ t
A=aΦ t+Δ,t , B =Φ t+Δ,t b .
(A.8)
Next using (12) and (18) we will derive equations for the new weights (A.8). Multiplying the
first (N-1) homogeneous equations (18) on the left hand side and right hand side by the
nonsingular matrices

Φ(t+Δ,t
k
) and Φ(t+Δ,t
k
)
T
, respectively, and multiplying the last non-
homogeneous equation (18) by
Φ(t+Δ,t
k
) we obtain

() ()
()
kk k
k
N
T
(ij)
(i) (iN)
kk
ttt
i=1
N
(i)
kk
t
i=1
Φ t+Δ,t b P -P t+Δ,t =0,
j

=1, ,N-1;
Φ t+Δ,t b =Φ(t+Δ,t ).
⎡⎤
Φ
⎣⎦
∑
∑
(A.9)
Using notation for the difference
(ijN) (ij)
(iN)
sss
δP =P -P we obtain equations for
k
(i)
t,t ,Δ
B,i1, ,N= such that

()
kk k
NN
T
(ijN)
(i) (i)
kk
t,t ,Δ tt,t,Δ
i=1 i=1
B δPt+Δ,t =0,
j
=1, ,N-1; B =Φ(t+Δ,t ).Φ

∑∑
(A.10)
Analogously after simple manipulations equation (12) takes the form

()() ()
k
k
NN
11
(ij) (ijN)
(i) (iN) (i)
kk k
t+Δ t+Δ t+Δ t+Δ
t,t ,Δ
i=1 i=1
NN
(i) (i)
kk
t+Δ
t,t ,Δ
i=1 i=1
a Φ t+Δ,t Φ t+Δ,t P -P = A Φ t+Δ,t δP=0,
a Φ(t+Δ,t )= A =Φ(t+Δ,t ).
−−
⎡⎤
⎣⎦
∑∑
∑∑
(A.11)
Distributed Fusion Prediction for Mixed Continuous-Discrete Linear Systems

51
or

()
k k
NN
1
(ijN)
(i) (i)
kk
t+Δ
t,t ,Δ t,t ,Δ
i=1 i=1
A Φ t+Δ,t δP=0,
j
1, ,N-1; A =Φ(t+Δ,t ).
−
=
∑∑
(A.12)
As we can see from (A.10) and (A.12) if the equality

()()
k
T-1
(i
j
N) (i
j

N)
kk
t+Δ
t
δP Φ t+Δ,t =Φ t+Δ,t δP
(A.13)
will be hold then the new weights
k
(i)
t,t ,Δ
A and
k
(i)
t,t ,Δ
B satisfy the identical equations. To
show that let consider differential equation for the difference
(ijN) (ij)
(iN)
sss
δP=P-P. Using (13)
we obtain the Lyapunov homogeneous matrix differential equation

(
)
(
)
(ijN) (ij) (ij) (ij) (ijN) (ijN)
(iN) (iN) (iN)
TT
s ss sss ss sss ssk

δP =P -P =F P -P + P -P F =F δP+δPF,tst+Δ,≤≤

(A.14)
which has the solution

() ()
k
T
(ijN) (ijN)
kk
t+Δ
t
δP=Φ t+Δ,t δP Φ t+Δ,t .
(A.15)
By the nonsingular property of the transition matrix
k
(t+Δ,t )
Φ
the equality (A.13) holds,
then
kk
(i) (i)
t,t ,Δ t,t ,Δ
AB= , and finally using (A.7) we get

kk kk
NN
(i) (i) (i) (i)
FLP PFF
t+Δ t+Δ

t,t ,Δ tt,t,Δ t
i=1 i=1
ˆˆˆˆ
x=A x B x x.
==
∑∑
(A.16)
This completes the proof of Theorem 4.
6. References
Alouani, A. T. & Gray, J. E. (2005). Theory of distributed estimation using multiple
asynchronous sensors,
IEEE Transations on Aerospace and Electronic Systems, Vol. 41,
No. 2, pp. 717-722.
Bar-Shalom, Y. & Campo, L. (1986). The effect of the common process noise on the two-
sensor fused track covariance,
IEEE Transactions on Aerospace and Electronic Systems,
Vol. 22, No. 6, pp. 803–805.
Bar-Shalom, Y. (1990).
Multitarget-multisensor tracking: advanced applications, Artech House,
Norwood, MA.
Bar-Shalom, Y. & Li, X. R. (1995).
Multitarget-multisensor tracking: principles and techniques,
YBS Publishing.
Bar-Shalom, Y. (2006). On hierarchical tracking for the real world,
IEEE Transactions on
Aerospace and Electronic Systems
, Vol. 42, No, 3, pp. 846–850.
Berg, T. M. & Durrant-Whyte, H. F. (1994). General decentralized Kalman filter,
Proceedings
of American Control Conference

, pp. 2273-2274, Maryland.
Chang, K. C.; Saha, R. K. & Bar-Shalom, Y. (1997). On Optimal track-to-track fusion,
IEEE
Transactions on Aerospace and Electronic Systems
, Vol. 33, No. 4, pp. 1271–1275.
Discrete Time Systems

52
Chang, K. C.; Tian, Z. & Saha, R. K. (2002). Performance evaluation of track fusion with
information matrix filter,
IEEE Transactions on Aerospace and Electronic Systems, Vol.
38, No. 2, pp. 455–466.
Deng, Z. L.; Gao, Y.; Mao, L. & Hao, G. (2005). New approach to information fusion steady-
state Kalman filtering,
Automatica, Vol. 41, No, 10, pp. 1695-1707.
Gelb, A. (1974).
Applied Optimal Estimation, MIT Press, Cambridge, MA.
Hall, D. L. (1992).
Mathematical techniques in multisensor data Fusion, Artech House, London.
Hashemipour, H. R.; Roy, S. & Laub, A. J. (1998). Decentralized structures for parallel
Kalman filtering,
IEEE Transactions on Automatic Control, Vol. 33, No. 1, pp. 88-94.
Jannerup, O. E. & Hendricks, E. (2006).
Linear Control System Design, Technical University of
Denmark.
Lee, S. H. & Shin, V. (2007). Fusion Filters Weighted by Scalars and Matrices for Linear
Systems,
World Academy of Science, Engineering and Technology, Vol. 34, pp. 88-93.
Lewis, F. L. (1986).
Optimal Estimation with an Introduction to Stochastic Control Theory, John

Wiley & Sons, New York.
Li, X. R.; Zhu, Y. M.; Wang, J. & Han, C. (2003). Optimal Linear Estimation Fusion - Part I:
Unified Fusion Rules,
IEEE Transations on Information Theory, Vol. 49, No. 9, pp.
2192-2208.
Ren, C. L. & Kay, M. G. (1989). Multisensor integration and fusion in intelligent systems,
IEEE Transactions on Systems, Man, and Cybernetics, Vol. 19, No. 5, pp. 901-931.
Roecker, J. A. & McGillem, C. D. (1998). Comparison of two-sensor tracking methods based
on state vector fusion and measurement fusion,
IEEE Transactions on Aerospace and
Electronic Systems
, Vol. 24, No. 4, pp. 447–449.
Shin, V.; Lee, Y. & Choi, T. (2006). Generalized Millman’s formula and its applications for
estimation problems,
Signal Processing, Vol. 86, No. 2, pp. 257–266.
Shin, V.; Shevlyakov, G. & Kim, K. S. (2007). A new fusion formula and its application to
continuous-time linear systems with multisensor environment,
Computational
Statistics & Data Analysis
, Vol. 52, No. 2, pp. 840-854.
Song, H. R.; Joen, M. G.; Choi, T. S. & Shin, V. (2009). Two Fusion Predictors for Discrete-
Time Linear Systems with Different Types of Observations,
International Journal of
Control, Automation, and Systems
, Vol. 7, No. 4, pp. 651-658.
Sun, S. L. (2004). Multi-sensor optimal information fusion Kalman filters with applications,
Aerospace Science and Technology, Vol. 8, No. 1, pp. 57–62.
Sun, S. L. & Deng, Z. L. (2005). Multi-sensor information fusion Kalman filter weighted by
scalars for systems with colored measurement noises,
Journal of Dynamic Systems,

Measurement and Control, Vol. 127, No. 4, pp. 663–667.
Zhou, J.; Zhu, Y.; You, Z. & Song, E. (2006). An efficient algorithm for optimal linear
estimation fusion in distributed multisensor systems, IEEE Transactions on System,
Man, Cybernetics, Vol. 36, No. 5, pp.1000–1009.
Zhu,Y. M. & Li, X. R. (1999). Best linear unbiased estimation fusion,
Proceeding of
International Conference on Multisource-Multisensor Information Fusion
, Sunnyvale,
CA, pp. 1054-1061.
Zhu,Y. M.; You, Z.; Zhao, J.; Zhang, K. & Li, X. R. (2001) The optimality for the distributed
Kalman filtering fusion with feedback,
Automaica, Vol. 37, No. 9, pp.1489–1493.
Zhu, Y. M. (2002)
Multisensor decision and estimation fusion, Kluwer Academic, Boston.
1. Introduction
We consider discrete-time linear stochastic systems with unknown inputs (or disturbances)
and propose recursive algorithms for estimating states of these systems. If mathematical
models derived by engineers are very accurate representations of real systems, we do not
have to consider systems with unknown inputs. However, in practice, the models derived by
engineers often contain modelling errors which greatly increase state estimation errors as if
the models have unknown disturbances.
The most frequently discussed problem on state estimation is the optimal filtering problem
which investigates the optimal estimate of state x
t
at time t or x
t+1
at time t + 1 with minimum
variance based on the observation Y
t
of the outputs {y

0
, y
1
, ···,y
t
}, i.e., Y
t
= σ{y
s
, s =
0, 1, ···,t} ( the smallest σ-field generated by {y
0
, y
1
, ···,y
t
} (see e.g., Katayama (2000),
Chapter 4)). It is well known that the standard Kalman filter is the optimal linear filter in
the sense that it minimizes the mean-square error in an appropriate class of linear filters (see
e.g., Kailath (1974), Kailath (1976), Kalman (1960), Kalman (1963) and Katayama (2000)). But
we note that the Kalman filter can work well only if we have accurate mathematical modelling
of the monitored systems.
In order to develop reliable filtering algorithms which are robust with respect to unknown
disturbances and modelling errors, many research papers have been published based on the
disturbance decoupling principle. Pioneering works were done by Darouach et al. (Darouach;
Zasadzinski; Bassang & Nowakowski (1995) and Darouach; Zasadzinski & Keller (1992)),
Chang and Hsu (Chang & Hsu (1993)) and Hou and Müller (Hou & Müller (1993)). They
utilized some transformations to make the original systems with unknown inputs into some
singular systems without unknown inputs. The most important preceding study related to
this paper was done by Chen and Patton (Chen & Patton (1996)). They proposed the simple

and useful optimal filtering algorithm, ODDO (Optimal Disturbance Decoupling Observer),
and showed its excellent simulation results. See also the papers such as Caliskan; Mukai; Katz
& Tanikawa (2003), Hou & Müller (1994), Hou & R. J. Patton (1998) and Sawada & Tanikawa
(2002) and the book Chen & Patton (1999). Their algorithm recently has been modified by the
author in Tanikawa (2006) (see Tanikawa & Sawada (2003) also).
We here consider smoothing problems which allow us time-lags for computing estimates of
the states. Namely, we try to find the optimal estimate
ˆ
x
t−L/t
of the state x
t−L
based on the
observation Y
t
with L > 0. We often classify smoothing problems into the following three
types. For the first problem, the fixed-point smoothing, we investigate the optimal estimate
Akio Tanikawa
Osaka Institute of Technology
Japan

New Smoothers for Discrete-time
Linear Stochastic Systems with
Unknown Disturbances
4
ˆ
x
k/t
of the state x
k

for a fixed k based on the observations {Y
t
, t = k + 1, k + 2,···}. Algorithms
for computing
ˆ
x
k/t
, t = k + 1, k + 2, ···, recursively are called fixed-point smoothers. For
the second problem, the fixed-interval smoothing, we investigate the optimal estimate
ˆ
x
t/N
of the state x
t
at all times t = 0, 1, ···, N based on the observation Y
N
of all the outputs
{y
0
, y
1
, ···,y
N
}. Fixed-interval smoothers are algorithms for computing
ˆ
x
t/N
, t = 0, 1, ···, N
recursively. The third problem, the fixed-lag smoothing, is to investigate the optimal estimate
ˆ

x
t−L/t
of the state x
t−L
based on the observation Y
t
for a given L ≥ 1. Fixed-lag smoothers
are algorithms for computing
ˆ
x
t−L/t
, t = L + 1, L + 2, ···, recursively. See the references such
as Anderson & Moore (1979), Bryson & Ho (1969), Kailath (1975) and Meditch (1973) for early
research works on smoothers. More recent papers have been published based on different
approaches such as stochastic realization theory (e.g., Badawi; Lindquist & Pavon (1979) and
Faurre; Clerget & Germain (1979)), the complementary models (e.g., Ackner & Kailath (1989a),
Ackner & Kailath (1989b), Bello; Willsky& Levy (1989), Bello; Willsky; Levy & Castanon (1986)
Desai; Weinert & Yasypchuk (1983) and Weinert & Desai (1981)) and others. Nice surveys can
be found in Kailath; Sayed & Hassibi (2000) and Katayama (2000).
When stochastic systems contain unknown inputs explicitly, Tanikawa (Tanikawa (2006))
obtained a fixed-point smoother for the first problem. The second and the third problems
were discussed in Tanikawa (2008). In this chapter, all three problems are discussed
in a comrehensive and self-contained manner as much as possible. Namely, after some
preliminary results in Section 2, we derive the fixed-point smoothing algorithm given in
Tanikawa (2006) in Section 3 for the system with unknown inputs explicitly by applying the
optimal filter with disturbance decoupling property obtained in Tanikawa & Sawada (2003).
In Section 4, we construct the fixed-interval smoother given in Tanikawa (2008) from the
fixed-point smoother obtained in Section 3. In Section 5, we construct the fixed-lag smoother
given in Tanikawa (2008) from the optimal filter in Tanikawa & Sawada (2003).
Finally, the new feature and advantages of the obtained results are summarized here. To the

best of our knowledge, no attempt has been made to investigate optimal fixed-interval and
fixed-lag smoothers for systems with unknown inputs explicitly (see the stochastic system
given by (1)-(2)) before Tanikawa (2006) and Tanikawa (2008). Our smoothing algorithms have
similar recursive forms to the standard optimal filter (i.e., the Kalman filter) and smoothers.
Moreover, our algorithms reduce to those known smoothers derived from the Kalman filter
(see e.g., Katayama (2000)) when the unknown inputs disappear. Thus, our algorithms are
consistent with the known smoothing algorithms for systems without unknown inputs.
2. Preliminaries
Consider the following discrete-time linear stochastic system for t = 0,1,2, ···:
x
t+1
= A
t
x
t
+ B
t
u
t
+ E
t
d
t
+ ζ
t
,(1)
y
t
= C
t

x
t
+ η
t
,(2)
where
x
t
∈ R
n
the state vector,
y
t
∈ R
m
the output vector,
54
Discrete Time Systems
u
t
∈ R
r
the known input vector,
d
t
∈ R
q
the unknown input vector.
Suppose that ζ
t

and η
t
are independent zero mean white noise sequences with covariance
matrices Q
t
and R
t
.LetA
t
, B
t
, C
t
and E
t
be known matrices with appropriate dimensions.
In Tanikawa & Sawada (2003), we considered the optimal estimate
ˆ
x
t+1/t+1
of the state x
t+1
which was proposed by Chen and Patton (Chen & Patton (1996) and Chen & Patton (1999))
with the following structure:
z
t+1
= F
t+1
z
t

+ T
t+1
B
t
u
t
+ K
t+1
y
t
,(3)
ˆ
x
t+1/t+1
= z
t+1
+ H
t+1
y
t+1
,(4)
for t
= 0,1,2, ···. Here,
ˆ
x
0/0
is chosen to be z
0
for a fixed z
0

. Denote the state estimation error
and its covariance matrix respectively by e
t
and P
t
. Namely, we use the notations e
t
= x
t
−
ˆ
x
t/t
and P
t
= E{e
t
e
t
T
}for t = 0, 1, 2, ···. Here, E denotes expectation and T denotes transposition
of a matrix. We assume in this paper that random variables e
0
, {η
t
}, {ζ
t
} are independent. As
in Chen & Patton (1996), Chen & Patton (1999) and Tanikawa & Sawada (2003), we consider
state estimate (3)-(4) with the matrices F

t+1
, T
t+1
, H
t+1
and K
t+1
of the forms:
K
t+1
= K
1
t
+1
+ K
2
t
+1
,(5)
E
t
= H
t+1
C
t+1
E
t
,(6)
T
t+1

= I −H
t+1
C
t+1
,(7)
F
t+1
= A
t
− H
t+1
C
t+1
A
t
−K
1
t
+1
C
t
,(8)
K
2
t
+1
= F
t+1
H
t

.(9)
The next lemma on equality (6) was obtained and used by Chen and Patton (Chen & Patton
(1996) and Chen & Patton (1999)). Before stating it, we assume that E
k
is a full column rank
matrix. Notice that this assumption is not an essential restriction.
Lemma 2.1. Equality (6) holds if and only if
rank
(
C
t+1
E
t
)
=
rank
(
E
t
)
. (10)
When this condition holds true, matrix H
t+1
which satisfies (6) must have the form
H
t+1
= E
t

(

C
t+1
E
t
)
T
(
C
t+1
E
t
)

−1
(
C
t+1
E
t
)
T
. (11)
Hence, we have
C
t+1
H
t+1
= C
t+1
E

t

(
C
t+1
E
t
)
T
(
C
t+1
E
t
)

−1
(
C
t+1
E
t
)
T
(12)
which is a non-negative definite symmetric matrix.
55
New Smoothers for Discrete-time Linear Stochastic Systems with Unknown Disturbances
When the matrix K
1

t+1
has the form
K
1
t
+1
= A
1
t
+1

P
t
C
t
T
− H
t
R
t

C
t
P
t
C
t
T
+ R
t


−1
, (13)
A
1
t
+1
= A
t
− H
t+1
C
t+1
A
t
, (14)
we obtained the following result (Theorem 2.7 in Tanikawa & Sawada (2003)) on the optimal
filtering algorithm.
Proposition 2.2. If C
t
H
t
and R
t
are commutative, i.e.,
C
t
H
t
R

t
= R
t
C
t
H
t
, (15)
then the optimal gain matrix K
1
t
+1
which makes the variance of the state estimation error e
t+1
minimum
is determined by (13). Hence, we obtain the optimal filtering algorithm:
ˆ
x
t+1/t+1
= A
1
t
+1
{
ˆ
x
t/t
+ G
t
(

y
t
−C
t
ˆ
x
t/t
)}
+
H
t+1
y
t+1
+ T
t+1
B
t
u
t
, (16)
P
t+1
= A
1
t
+1
M
t
A
1

t
+1
T
+ T
t+1
Q
t
T
t+1
T
+ H
t+1
R
t+1
H
t+1
T
, (17)
where
G
t
=

P
t
C
t
T
− H
t

R
t

C
t
P
t
C
t
T
+ R
t

−1
, (18)
and
M
t
= P
t
− G
t

C
t
P
t
− R
t
H

t
T

. (19)
Remark 2.3. If the matrix R
t
has the form
R
t
= r
t
I
with some positive number r
t
for each t = 1, 2, ···, then it is obvious to see that condition (15)
holds.
Finally, we have the following proposition which indicates that the standard Kalman filter is
a special case of the optimal filter proposed in this section (see e.g., Theorem 5.2 (page 90) in
Katayama (2000)).
Proposition 2.4. Suppose that E
t
≡ O holds for all t (i.e., the unknown input term is zero). Then,
Lemma 2.1 cannot be applied directly. But, we can choose H
t
≡ O for all t in this case, and the optimal
filter given in Proposition 2.2 reduces to the standard Kalman filter.
3. The fixed-point smoothing
Let k be a fixed time. We study an iterative algorithm to compute the optimal estimate
ˆ
x

k/t
of
the state x
k
based on the observation Y
t
, t = k + 1, k + 2, ···,withY
t
= σ{y
s
, s = 0, 1, ···, t}.
We define state vectors θ
t
, t = k, k + 1, ···,by
θ
t+1
= θ
t
, t = k, k + 1, ···; θ
k
= x
k
. (20)
56
Discrete Time Systems
It is easy to observe that the optimal estimate
ˆ
θ
t/t
of the state θ

t
based on the observation Y
t
is identical to the optimal smoother
ˆ
x
k/t
in view of the equalities θ
t
= x
k
, t = k, k + 1, ···.
In order to derive the optimal fixed-point smoother, we consider the following augmented
system for t
= k, k + 1, ···:

x
t+1
θ
t+1

=

A
t
O
OI

x
t

θ
t

+

B
t
O

u
t
+

E
t
O

d
t
+

I
O

ζ
t
, (21)
y
t+1
=

[
C
t+1
O
]

x
t+1
θ
t+1

+ η
t+1
. (22)
Denote these equations respectively by

x
t+1
=

A
t

x
t
+

B
t
u

t
+

E
t
d
t
+

J
t
ζ
t
, (23)
y
t+1
=

C
t+1

x
t+1
+ η
t+1
, (24)
where

x
t

=

x
t
θ
t

,

A
t
=

A
t
O
OI

,

B
t
=

B
t
O

,


E
t
=

E
t
O

,

J
t
=

I
O

and

C
t+1
=
[
C
t+1
O
]
.
Here, I and O are the identity matrix and the zero matrix respectively with appropriate
dimensions. By making use of the notations


H
t+1
=

H
t+1
O

,

T
t+1
=

IO
OI

−

H
t+1

C
t+1
,
we have the equalities:

C
t+1


E
t
= C
t+1
E
t
,

T
t+1
=

T
t+1
O
OI

,

A
1
t
+1
=

T
t+1

A

t
=

A
1
t
+1
O
OI

.
We introduce the covariance matrix

P
t
of the state estimation error of the augmented system
(23)-(24):

P
t
=

P
(1,1)
t
P
(1,2)
t
P
(2,1)

t
P
(2,2)
t

= E


x
t
−
ˆ
x
t/t
θ
t
−
ˆ
θ
t/t

x
t
−
ˆ
x
t/t
θ
t
−

ˆ
θ
t/t

T

. (25)
Notice that P
(1,1)
t
is equal to P
t
. Applying the optimal filter given in Proposition 2.2 to the
augmented system (21)-(22), we obtain the following optimal fixed-point smoother.
Theorem 3.1. If C
t
H
t
and R
t
are commutative, i.e.,
C
t
H
t
R
t
= R
t
C

t
H
t
, (26)
then we have the optimal fixed-point smoother for (21)-(22) as follows:
57
New Smoothers for Discrete-time Linear Stochastic Systems with Unknown Disturbances
(i) the fixed-point smoother
ˆ
x
k/t+1
=
ˆ
x
k/t
+ D
t
(k)
[
y
t
−C
t
ˆ
x
t/t
]
, (27)
(ii) the gain matrix
D

t
(k)=P
(2,1)
t
C
t
T

C
t
P
t
C
t
T
+ R
t

−1
, (28)
(iii) the covariance matrix of the mean-square error
P
(2,1)
t+1
=

P
(2,1)
t
− P

(2,1)
t
C
t
T

C
t
P
t
C
t
T
+ R
t

−1

C
t
P
t
− R
t
H
t
T


A

1
t
+1
T
, (29)
P
(2,2)
t+1
= P
(2,2)
t
− P
(2,1)
t
C
t
T

C
t
P
t
C
t
T
+ R
t

−1
C

t
P
(2,1)
t
T
. (30)
Here, we note that P
(2,1)
k
= P
(2,2)
k
= P
k
.Wenoticethat
ˆ
x
t/t
is the optimal filter of the original system
(1)-(2) given in Tanikawa & Sawada (2003).
Proof Applying the optimal filter given by (16)-(17) in Proposition (2.2) to the augmented
system (23)-(24), we have


x
t+1/t+1
=

A
t+1

1



x
t/t
+

G
t

y
t
−C
t


x
t/t

+

H
t+1
y
t+1
+

T
t+1


B
t
u
t
. (31)
This can be rewritten as

ˆ
x
t+1/t+1
ˆ
θ
t+1/t+1

=

A
1
t
+1
O
OI

⎧
⎨
⎩

ˆ
x

t/t
ˆ
θ
t/t

+

P
(1,1)
t
C
t
T
− H
t
R
t
P
(2,1)
t
C
t
T

×

C
t
P
t

C
t
T
+ R
t

−1
(
y
t
−C
t
ˆ
x
t/t
)
⎫
⎬
⎭
+

H
t+1
y
t+1
O

+

T

t+1
B
t
u
t
O

.
Thus, we have
ˆ
x
t+1/t+1
= A
1
t
+1

ˆ
x
t/t
+

P
(1,1)
t
C
t
T
− H
t

R
t

C
t
P
t
C
t
T
+ R
t

−1
(
y
t
−C
t
ˆ
x
t/t
)

+H
t+1
y
t+1
+ T
t+1

B
t
u
t
(32)
and
ˆ
θ
t+1/t+1
=
ˆ
θ
t/t
+ P
(2,1)
t
C
t
T

C
t
P
t
C
t
T
+ R
t


−1
(
y
t
−C
t
ˆ
x
t/t
)
. (33)
Here, we used the equalities

C
t

P
t

C
t
T
+ R
t
=
[
C
t
O
]


P
(1,1)
t
P
(1,2)
t
P
(2,1)
t
P
(2,2)
t


C
t
T
O

+ R
t
= C
t
P
t
C
t
T
+ R

t
(34)
58
Discrete Time Systems
and

G
t
=


P
t

C
t
T
O

−

H
t
R
t



C
t


P
t

C
t
T
+ R
t

−1
=

P
(1,1)
t
P
(1,2)
t
P
(2,1)
t
P
(2,2)
t


C
t
T

O

−

H
t
O

R
t



C
t

P
t

C
t
T
+ R
t

−1
=

P
(1,1)

t
C
t
T
− H
t
R
t
P
(2,1)
t
C
t
T


C
t
P
t
C
t
T
+ R
t

−1
. (35)
Thus, equalities (27)-(28) can be obtained from (33) due to
ˆ

θ
t/t
=
ˆ
x
k/t
.
By using the notation

M
t
for the augmented system (23)-(24)which corresponds to the matrix
M
t
in Proposition (2.2), we have

M
t
=

M
(1,1)
t
M
(1,2)
t
M
(2,1)
t
M

(2,2)
t

=

P
t
−

G
t


C
t

P
t
−R
t

H
t
T
O

=

P
(1,1)

t
P
(1,2)
t
P
(2,1)
t
P
(2,2)
t

−

P
(1,1)
t
C
t
T
− H
t
R
t
P
(2,1)
t
C
t
T



C
t
P
t
C
t
T
+ R
t

−1
×

[
C
t
O
]

P
(1,1)
t
P
(1,2)
t
P
(2,1)
t
P

(2,2)
t

−

R
t
H
t
T
O


.
Thus, we have
M
(1,1)
t
= P
(1,1)
t
−

P
(1,1)
t
C
t
T
− H

t
R
t

C
t
P
t
C
t
T
+ R
t

−1

C
t
P
(1,1)
t
− R
t
H
t
T

, (36)
M
(1,2)

t
= P
(1,2)
t
−

P
(1,1)
t
C
t
T
− H
t
R
t

C
t
P
t
C
t
T
+ R
t

−1
C
t

P
(1,2)
t
, (37)
M
(2,1)
t
= P
(2,1)
t
− P
(2,1)
t
C
t
T

C
t
P
t
C
t
T
+ R
t

−1

C

t
P
(1,1)
t
− R
t
H
t
T

, (38)
and
M
(2,2)
t
= P
(2,2)
t
− P
(2,1)
t
C
t
T

C
t
P
t
C

t
T
+ R
t

−1
C
t
P
(1,2)
t
. (39)
It follows from (17) in Proposition 2.2 that

P
t+1
=

A
1
t
+1

M
t

A
1
t
+1

T
+

T
t+1

J
t+1
Q
t+1

J
t+1
T

T
t+1
+

H
t+1
R
t+1

H
t+1
T
=

A

1
t
+1
O
OI


M
(1,1)
t
M
(1,2)
t
M
(2,1)
t
M
(2,2)
t

A
1
t
+1
T
O
OI

+


T
t+1
O
OI

I
O

Q
t+1
[
IO
]

T
t+1
T
O
OI

+

H
t+1
O

R
t+1

H

t+1
T
O

. (40)
59
New Smoothers for Discrete-time Linear Stochastic Systems with Unknown Disturbances
Equalities (29)-(30) follow from (38)-(40). Finally, we have equalities P
(2,1)
k
= P
(2,2)
k
= P
(1,1)
k
=
P
k
by the definition of

P
k
.
We thus have derived the fixed-point smoothing algorithm for the state-space model which
explicitly contains the unknown inputs. We can indicate that the algorithm has a rather simple
form and also has consistency with both the Kalman filter and the standard optimal smoother
derived from the Kalman filter as shown in the following remark.
Remark 3.2. Suppose that E
t

≡ O holds for all t (i.e., the unknown input term is zero) and
that H
t
≡ O for all t(as in Proposition 2.4). In this case, it follows from Theorem 3.1 that
ˆ
x
t+1/t+1
= A
t

ˆ
x
t/t
+ P
t
C
t
T

C
t
P
t
C
t
T
+ R
t

−1

(
y
t
−C
t
ˆ
x
t/t
)

+ B
t
u
t
, (41)
ˆ
θ
t+1/t+1
=
ˆ
θ
t/t
+ P
(2,1)
t
C
t
T

C

t
P
t
C
t
T
+ R
t

−1
(
y
t
−C
t
ˆ
x
t/t
)
, (42)
P
(2,1)
t+1
=

P
(2,1)
t
− P
(2,1)

t
C
t
T

C
t
P
t
C
t
T
+ R
t

−1
C
t
P
t

A
t
T
, (43)
and
P
(2,2)
t+1
= P

(2,2)
t
− P
(2,1)
t
C
t
T

C
t
P
t
C
t
T
+ R
t

−1
C
t
P
(2,1)
t
T
. (44)
Here, we note that the state estimate
ˆ
x

t+1/t+1
reduces to the state estimate
ˆ
x
t+1/t
in Katayama
(2000) when H
t
≡ O holds. Moreover, Equalities (37)-(40) with the state estimates
ˆ
x
t+1/t+1
and
ˆ
x
t/t
replaced respectively by
ˆ
x
t+1/t
and
ˆ
x
t/t−1
are identical to those for the pair of the
standard Kalman filter and the optimal fixed-point smoother in Katayama (2000). Thus, it has
been shown that this algorithm reduces to the well known optimal smoother derived from
the Kalman filter when the unknown inputs disappear. This indicates that our smoothing
algorithm is a natural extension of the standard optimal smoother to linear systems possibly
with unknown inputs.

Let us introduce some notations:
ν
t
= y
t
−C
t
ˆ
x
t/t
, (45)
L
t
= A
1
t
+1
(
I − G
t
C
t
)
, (46)
Ψ
(t, τ)=

L
t−1
L

t−2
···L
τ
, t > τ
I , t
= τ ,
(47)
where the matrix G
t
was defined by (18), i.e.,
G
t
=

P
t
C
t
T
− H
t
R
t

C
t
P
t
C
t

T
+ R
t

−1
. (48)
We then have the following results due to (27).
Corollary 3.3. We have the equalities:
ˆ
x
k/t+1
=
ˆ
x
k/k
+
t
∑
i=k
D
i
(k)ν
i
=
ˆ
x
k/k
+ P
k
t

∑
i=k
Ψ(i, k)
T
C
i
T

C
i
P
i
C
i
T
+ R
i

−1
ν
i
. (49)
60
Discrete Time Systems
Proof It is straightforward to show the first equality from (27). For the second equality, it is
sufficient to prove the equality
D
t
(k)=P
k

Ψ(t, k)
T
C
t
T

C
t
P
t
C
t
T
+ R
t

−1
(50)
for t
≥ k. By virtue of (46), equality (29) can be rewritten as
P
(2,1)
t
= P
(2,1)
t−1

I
−C
t−1

T
G
t−1
T

A
1
t
T
= P
(2,1)
t−1
L
t−1
T
. (51)
By using this equality recursively, we have
P
(2,1)
t
= P
(2,1)
t−2
L
t−2
T
L
t−1
T
= ······= P

(2,1)
k
L
k
T
L
k+1
T
···L
t−1
T
= P
k
Ψ(t, k)
T
. (52)
Substituting this equality into (28), we obtain
D
t
(k)=P
k
Ψ(t, k)
T
C
t
T

C
t
P

t
C
t
T
+ R
t

−1
, (53)
i.e., (50).
Finally, we study the reduction of the estimation error by the fixed-point smoothing over the
optimal filtering. Due to (27), we have
P
(2,2)
t
= E

(
x
k
−
ˆ
x
k/t
)(
x
k
−
ˆ
x

k/t
)
T

. (54)
Denote this matrix simply by P
k/t
. It then follows from (30) that
P
k/t+1
= P
k/t
− P
(2,1)
t
C
t
T

C
t
P
t
C
t
T
+ R
t

−1

C
t
P
(2,1)
t
T
. (55)
Summing up these equalities for t
= k, k + 1, ···, s,wehave
P
k/k
− P
k/s+1
=
s
∑
i=k
P
(2,1)
i
C
i
T

C
i
P
i
C
i

T
+ R
i

−1
C
i
P
(2,1)
i
T
. (56)
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
x
t/N
of the state x
t
at all times t = 0, 1, ···, N based on the observation Y
N
of
all the states
{y
0
, y
1

, ···,y
N
}. Applying equality (49), we easily obtain the following equality.
Lemma 4.1. The equality
ˆ
x
t/N
=
ˆ
x
t/t+1
+ P
t
L
t
T
P
t+1
−1
(
ˆ
x
t+1/N
−
ˆ
x
t+1/t+1
)
(57)
61

New Smoothers for Discrete-time Linear Stochastic Systems with Unknown Disturbances
holds for t = 0, 1, ···, N −1.
Proof Using the notation
˜
ν
i
= C
i
T

C
i
P
i
C
i
T
+R
i

−1
ν
i
, (58)
we have
ˆ
x
k/t+1
=
ˆ

x
k/k
+ P
k
t
∑
i=k
Ψ(i, k)
T
˜
ν
i
(59)
for k
≤ t due to (49). In view of (59) , we also have
ˆ
x
k/t+1
=
ˆ
x
k/k
+ P
k
˜
ν
k
+ P
k
t

∑
i=k+1
Ψ(i, k)
T
˜
ν
i
=
ˆ
x
k/k+1
+ P
k
t
∑
i=k+1
Ψ(i, k)
T
˜
ν
i
(60)
for k
+ 1 ≤ t. Putting t + 1 = N and k = t + 1 in equality (59), we have
ˆ
x
t+1/N
=
ˆ
x

t+1/t+1
+ P
t+1
N
−1
∑
i=t+1
Ψ(i, t + 1)
T
˜
ν
i
. (61)
Putting t
+ 1 = N and k = t in equality (60), we have
ˆ
x
t/N
=
ˆ
x
t/t+1
+ P
t
N
−1
∑
i=t+1
Ψ(i, t)
T

˜
ν
i
=
ˆ
x
t/t+1
+ P
t
L
t
T
N
−1
∑
i=t+1
Ψ(i, t + 1)
T
˜
ν
i
. (62)
Substituting (61) into (62), we have
ˆ
x
t/N
=
ˆ
x
t/t+1

+ P
t
L
t
T
P
t+1
−1
(
ˆ
x
t+1/N
−
ˆ
x
t+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/N
=
ˆ
x

t/t+1
+ P
t
L
t
T
λ
t+1
, (63)
λ
t
= L
t
T
λ
t+1
+ C
t
T

C
t
P
t
C
t
T
+R
t


−1
ν
t
. (64)
for t
= N −1, N −2, ···,1,0. Here, we have λ
N
= 0.
Proof For t = 0, 1, ···, N,weput
λ
t
= P
t
−1
(
ˆ
x
t/N
−
ˆ
x
t/t
)
. (65)
We then have λ
N
= 0. Substituting (65) into (57), we obtain equality (63). Then, by utilizing
(63) and (65), we have
λ
t

= P
t
−1

ˆ
x
t/t+1
+ P
t
L
t
T
λ
t+1
−
ˆ
x
t/t

. (66)
In view of the equality
ˆ
x
t/t+1
−
ˆ
x
t/t
= P
t

˜
ν
t
(67)
62
Discrete Time Systems
which follows from (27) in Tanikawa & Sawada (2003), we obtain
λ
t
= L
t
T
λ
t+1
+
˜
ν
t
= L
t
T
λ
t+1
+ C
t
T

C
t
P

t
C
t
T
+R
t

−1
ν
t
. (68)
Thus, we proved (64).
Remark 4.3. When E
t
≡ 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 E
t
= O,wehaveH
t
= O for t = 0, 1, ···, N (see
Propositin 2.4). Note that in (59), i.e.,
ˆ
x
k/t+1
=
ˆ
x

k/k
+ P
k
t
∑
i=k
Ψ(i, k)
T
˜
ν
i
ˆ
x
k/t+1
and
ˆ
x
k/k
respectively reduce to
ˆ
x
k/t
and
ˆ
x
k/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
x
t−L/t
of the state x
t−L
based on
the observation Y
t
.
We consider the following augmented system:
⎡
⎢
⎢
⎢
⎣
x
t+1
x
t
.
.
.
x
t−L+1
⎤
⎥
⎥

⎥
⎦
=
⎡
⎢
⎢
⎢
⎣
A
t
O O
IO O
.
.
.
OIO
⎤
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎣
x
t
x
t−1

.
.
.
x
t−L
⎤
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎣
B
t
O
.
.
.
O
⎤
⎥
⎥
⎥
⎦
u
t

+
⎡
⎢
⎢
⎢
⎣
E
t
O
.
.
.
O
⎤
⎥
⎥
⎥
⎦
d
t
+
⎡
⎢
⎢
⎢
⎣
I
O
.
.

.
O
⎤
⎥
⎥
⎥
⎦
ζ
t
, (69)
y
t+1
=
[
C
t+1
O O
]
⎡
⎢
⎢
⎢
⎣
x
t+1
x
t
.
.
.

x
t−L+1
⎤
⎥
⎥
⎥
⎦
+ η
t+1
. (70)
Denote these equations respectively by

x
t+1
=

A
t

x
t
+

B
t
u
t
+

E

t
d
t
+

J
t
ζ
t
, (71)
y
t+1
=

C
t+1

x
t+1
+ η
t+1
, (72)
63
New Smoothers for Discrete-time Linear Stochastic Systems with Unknown Disturbances
where

x
t
=
⎡

⎢
⎢
⎢
⎣
x
t
x
t−1
.
.
.
x
t−L
⎤
⎥
⎥
⎥
⎦
,

A
t
=
⎡
⎢
⎢
⎢
⎣
A
t

O O
IO O
.
.
.
OIO
⎤
⎥
⎥
⎥
⎦
,

B
t
=
⎡
⎢
⎢
⎢
⎣
B
t
O
.
.
.
O
⎤
⎥

⎥
⎥
⎦
,

E
t
=
⎡
⎢
⎢
⎢
⎣
E
t
O
.
.
.
O
⎤
⎥
⎥
⎥
⎦
,

J
t
=

⎡
⎢
⎢
⎢
⎣
I
O
.
.
.
O
⎤
⎥
⎥
⎥
⎦
and

C
t+1
=
[
C
t+1
O O
]
.
Here, I and O are the identity matrix and the zero matrix respectively with appropriate
dimensions. By making use of the notations


H
t+1
=
⎡
⎢
⎢
⎢
⎣
H
t+1
O
.
.
.
O
⎤
⎥
⎥
⎥
⎦
and

T
t+1
= I −

H
t+1

C

t+1
,
we have the equalities:

C
t+1

E
t
=
[
C
t+1
O O
]
⎡
⎢
⎢
⎢
⎣
E
t
O
.
.
.
O
⎤
⎥
⎥

⎥
⎦
= C
t+1
E
t
,

T
t+1
= I −
⎡
⎢
⎢
⎢
⎣
H
t+1
O
.
.
.
O
⎤
⎥
⎥
⎥
⎦
[
C

t+1
O O
]
=
⎡
⎢
⎢
⎢
⎣
T
t+1
O O
OI O
.
.
.
OO I
⎤
⎥
⎥
⎥
⎦
,

A
1
t
+1
=


T
t+1

A
t
=
⎡
⎢
⎢
⎢
⎣
T
t+1
O O
OI O
.
.
.
OO I
⎤
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎣
A

t
O O
IO O
.
.
.
OIO
⎤
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎣
A
1
t
+1
O O
IO O
.
.
.
OIO
⎤
⎥

⎥
⎥
⎦
.
We introduce the covariance matrix

P
t
of the state estimation error of augmented system
(71)-(72):

P
t
= E
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
⎡
⎢
⎢
⎢
⎣

x
t
−
ˆ
x
t/t
x
t−1
−
ˆ
x
t−1/t
.
.
.
x
t−L
−
ˆ
x
t−L/t
⎤
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢

⎣
x
t
−
ˆ
x
t/t
x
t−1
−
ˆ
x
t−1/t
.
.
.
x
t−L
−
ˆ
x
t−L/t
⎤
⎥
⎥
⎥
⎦
T
⎫
⎪

⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
. (73)
64
Discrete Time Systems
By using the notations
P
t−i,t−j/t
= E

(
x
t−i
−
ˆ
x
t−i/t
)

x
t−j
−
ˆ

x
t−j/t

T

,
P
t−i/t
= P
t−i,t−i/t
,
we can write

P
t
=
⎡
⎢
⎢
⎢
⎣
P
t/t
P
t,t−1/t
P
t,t−L/t
P
t−1,t/t
P

t−1/t
P
t−1,t−L/t
.
.
.
.
.
.
.
.
.
P
t−L,t/t
P
t−L,t−1/t
P
t−L/t
⎤
⎥
⎥
⎥
⎦
. (74)
Here, it is easy to observe that P
t/t
= P
t
holds. We also note that


C
t

P
t

C
t
T
+ R
t
= C
t
P
t/t
C
t
T
+ R
t
. (75)
From now on, we use the following notation for brevity:
C
t
:= C
t
P
t
C
t

T
+ R
t
. (76)
Applying the optimal filter given in Proposition 2.2 to augmented system (71)-(72), we have


x
t+1/t+1
=

A
1
t
+1



x
t/t
+

G
t

y
t
−

C

t


x
t/t

+

H
t+1
y
t+1
+

T
t+1

B
t
u
t
, (77)
where

G
t
=


P

t

C
t
T
−

H
t
R
t


C
t

P
t

C
t
T
+ R
t

−1
=
⎡
⎢
⎢

⎢
⎢
⎣
P
t/t
C
t
T
− H
t
R
t
P
t−1,t/t
C
t
T
.
.
.
P
t−L,t/t
C
t
T
⎤
⎥
⎥
⎥
⎥

⎦
C
t
−1
. (78)
Identifying the component matrices of (77)-(78), we have the following optimal fixed-lag
smoother.
Theorem 5.1. If C
t
H
t
and R
t
are commutative, i.e.,
C
t
H
t
R
t
= R
t
C
t
H
t
, (79)
then we have the optimal fixed-lag smoother for (1)-(2) as follows:
(i) the fixed-lag smoother
ˆ

x
t−j/t+1
=
ˆ
x
t−j/t
+ S
t
(j)
(
y
t
−C
t
ˆ
x
t/t
)(
j = 0, 1, ···, L − 1
)
, (80)
(ii) the optimal filter
ˆ
x
t+1/t+1
= A
1
t
+1
{

ˆ
x
t/t
+ G
t
(
y
t
−C
t
ˆ
x
t/t
)}
+
H
t+1
y
t+1
+ T
t+1
B
t
u
t
, (81)
with G
t
defined by (18) in Proposition 2.2,
(iii) the gain matrices

S
t
(j)=

P
t−j,t/t
C
t
T
−δ
0,j
H
t
R
t

C
t
−1
(
j = 0, 1, ···, L − 1
)
, (82)
65
New Smoothers for Discrete-time Linear Stochastic Systems with Unknown Disturbances
where δ
i,j
stands for the Kronecker’s delta, i.e.,
δ
i,j

=

1 for i
= j
0 for i
= j
, (83)
(iv) the covariance matrix of the mean-square error
P
t+1/t+1
= A
1
t
+1
M
(0,0)
t
A
1
t
+1
T
+ T
t+1
Q
t
T
t+1
T
+ H

t+1
R
t+1
H
t+1
T
, (84)
P
t+1,t−j/t+1
= A
1
t
+1
M
(0,j)
t
(
j = 0, 1, ···, L − 1
)
, (85)
P
t−j,t+1/t+1
=

P
t+1,t−j/t+1

T
(
j = 0, 1, ···, L − 1

)
, (86)
P
t−i,t−j/t+1
= M
(i,j)
t
(
i, j = 0,1,···, L −1
)
, (87)
and
M
(i,j)
t
= P
t−i,t−j/t
−

P
t−i,t/t
C
t
T
−δ
0,i
H
t
R
t


C
t
−1

C
t
P
t,t−j/t
−δ
0,j
R
t
H
t
T

(
i, j = 0, 1,···, L
)
. (88)
Remark 5.2. Since the equalities
P
t/t
= P
t
( in Proposition 2.2 )
and
M
(0,0)

t
= M
t
( in Proposition 2.2 )
hold, the part of the optimal filter in Theorem 5.1 is identical to that in Proposition 2.2. When
E
t
≡ O holds for all t (i.e., the unknown input term is zero), we shall see that fixed-lag
smoother (80)-(88) is identical to the well known fixed-lag smoother (see e.g. Katayama (2000))
obtained from the standard Kalman filter. Thus, our algorithm is consistent with the known
fixed-lag smoothing algorithm for systems without unknown inputs. This can be readily
shown as in Remark 4.3.
Proof of Theorem 5.1 Rewriting (77)-(78) with the component matrices explicitly, we have
⎡
⎢
⎢
⎢
⎢
⎢
⎣
ˆ
x
t+1/t+1
ˆ
x
t/t+1
ˆ
x
t−1/t+1
.

.
.
ˆ
x
t−L+1/t+1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
A
1
t
+1

ˆ
x

t/t
+

P
t/t
C
t
T
− H
t
R
t

C
t
−1
(
y
t
−C
t
ˆ
x
t/t
)

ˆ
x
t/t
+


P
t/t
C
t
T
− H
t
R
t

C
t
−1
(
y
t
−C
t
ˆ
x
t/t
)
ˆ
x
t−1/t
+ P
t−1,t/t
C
t

T
C
t
−1
(
y
t
−C
t
ˆ
x
t/t
)
.
.
.
ˆ
x
t−L+1/t
+ P
t−L+1,t/t
C
t
T
C
t
−1
(
y
t

−C
t
ˆ
x
t/t
)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎢
⎣
H
t+1
y
t+1
+ T
t+1

B
t
u
t
O
O
.
.
.
O
⎤
⎥
⎥
⎥
⎥
⎥
⎦
. (89)
66
Discrete Time Systems
The statements in (i)-(iii) easily follow from (89).
Let

M
t
be defined by

M
t
=


P
t
−

G
t


C
t

P
t
− R
t

H
t
T

=

P
t
−
⎡
⎢
⎢
⎢

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P
t/t
C
t
T
−H
t
R
t
P
t−1,t/t
C
t
T
P
t−2,t/t
C
t
T
.
.
.

P
t−L,t/t
C
t
T
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
C
t
−1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎢
⎣
P
t/t
C
t
T
−H
t
R
t
P
t−1,t/t
C
t
T
P
t−2,t/t
C
t
T
.
.
.
P
t−L,t/t
C
t
T
⎤

⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
T
.
We also introduce component matrices of

M
t
as follows:

M
t
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎢
⎢
⎢
⎢
⎣
M
(0,0)
t
M
(0,1)
t
M
(0,2)
t
M
(0,L)
t
M
(1,0)
t
M
(1,1)
t
M
(1,2)
t
M
(1,L)
t
M

(2,0)
t
M
(2,1)
t
M
(2,2)
t
M
(2,L)
t
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
M
(L,0)
t
M

(L,1)
t
M
(L,2)
t
M
(L,L)
t
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
Concerning

P
t+1
,wehave

P
t+1

=

A
1
t
+1

M
t

A
1
t
+1
T
+

T
t+1

J
t
Q
t

J
t
T

T

t+1
T
+

H
t+1
R
t+1

H
t+1
T
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
A
1
t
+1
M

(0,0)
t
A
1
t
+1
T
A
1
t
+1
M
(0,0)
t
A
1
t
+1
M
(0,1)
t
A
1
t
+1
M
(0,L−1)
t
M
(0,0)

t
A
1
t
+1
T
M
(0,0)
t
M
(0,1)
t
M
(0,L−1)
t
M
(1,0)
t
A
1
t
+1
T
M
(1,0)
t
M
(1,1)
t
M

(1,L−1)
t
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
M
(L−1,0)
t
A
1
t
+1
T
M
(L−1,0)
t
M
(L−1,1)

t
M
(L−1,L−1)
t
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
T
t+1
Q

t
T
t+1
T
+H
t+1
R
t+1
H
t+1
T
OO O
OOO O
OOO O
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
OOO O

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
The final part (iv) can be obtained from the last three equalities.
67
New Smoothers for Discrete-time Linear Stochastic Systems with Unknown Disturbances
6. Conclusion
In this chapter, we considered discrete-time linear stochastic systems with unknown inputs
(or disturbances) and studied three types of smoothing problems for these systems. We
derived smoothing algorithms which are robust to unknown disturbances from the optimal
filter for stochastic systems with unknown inputs obtained in our previous papers. These
smoothing algorithms have similar recursive forms to the standard optimal filters and
smoothers. Moreover, since our algorithms reduce to those known smoothers derived from
the Kalman filter when unknown inputs disappear, these algorithms are consistent with the
known smoothing algorithms for systems without unknown inputs.
This work was partially supported by the Japan Society for Promotion of Science (JSPS) under
Grant-in-Aid for Scientific Research (C)-22540158.
7. References
Ackner, R. & Kailath, T. (1989a). Complementary models and smoothing, IEEE Trans.
Automatic Control, Vol. 34, pp. 963–969
Ackner, R. & Kailath, T. (1989b). Discrete-time complementary models and smoothing, Int. J.
Control, Vol. 49, pp. 1665–1682

Anderson, B. D. O. & Moore, J. B. (1979). Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ
Badawi, F. A.; Lindquist, A. & Pavon, M. (1979). A stochastic realization approach to the
smoothing problem, IEEE Trans. Automatic Control, Vol. 24, pp. 878–888
Bello, M. G.; Willsky, A. S. & Levy, B. C. (1989). Construction and applications of discrete-time
smoothing error models, Int. J. Control, Vol. 50, pp. 203–223
Bello, M. G.; Willsky, A. S.; Levy, B. C. & Castanon, D. A. (1986). Smoothing error dynamics
and their use in the solution of smoothing and mapping problems, IEEE Trans. Inform.
Theory, Vol. 32, pp. 483–495
Bryson, Jr., A. E. & Ho, Y. C. (1969). Applied Optimal Control, Blaisdell Publishing Company,
Waltham, Massachusetts
Caliskan, F.; Mukai, H.; Katz, N. & Tanikawa, A. (2003). Game estimators for air combat
games with unknown enemy inputs, Proc. American Control Conference, pp. 5381–5387,
Denver, Colorado
Chang, S. & Hsu, P. (1993). State estimation using general structured observers for linear
systems with unknown input, Proc. 2nd European Control Conference: ECC’93,pp.
1794–1799, Groningen, Holland
Chen, J. & Patton, R. J. (1996). Optimal filtering and robust fault diagnosis of stochastic systems
with unknown disturbances, IEE Proc. of Control Theory Applications, Vol. 143, No. 1,
pp. 31–36
Chen, J. & Patton, R. J. (1999). Robust Model-based Fault Diagnosis for Dynamic Systems,Kluwer
Academic Publishers, Norwell, Massachusetts
Chen, J.; Patton, R. J. & Zhang, H. -Y. (1996). Design of unknown input observers and robust
fault detection filters, Int. J. Control, Vol. 63, No. 1, pp. 85–105
Darouach, M.; Zasadzinski, M.; Bassang, O. A. & Nowakowski, S. (1995). Kalman filtering
with unknown inputs via optimal state estimation of singular systems, Int. J. Systems
Science, Vol. 26, pp. 2015–2028
68
Discrete Time Systems
Darouach, M.; Zasadzinski, M. & Keller, J. Y. (1992). State estimation for discrete systems with
unknown inputs using state estimation of singular systems, Proc. American Control

Conference, pp. 3014–3015
Desai, U. B.; Weinert, H. L. & Yasypchuk, G. (1983). Discrete-time complementary models and
smoothing algorithms: The correlated case, IEEE Trans. Automatic Control,Vol.28,pp.
536–539
Faurre, P.; Clerget, M. & Germain, F. (1979). Operateurs Rationnels Positifs,Dunod,Paris,France
Frank, P. M. (1990). Fault diagnosis in dynamic system using analytical and knowledge based
redundancy: a survey and some new results, Automatica, Vol. 26, No. 3, pp. 459–474
Hou, M. & Müller, P. C. (1993). Unknown input decoupled Kalman filter for time-varying
systems, Proc. 2nd European Control Conference: ECC’93, Groningen, Holland, pp.
2266–2270
Hou, M. & Müller, P. C. (1994). Disturbance decoupled observer design: a unified viewpoint,
IEEE Trans. Automatic Control, Vol. 39, No. 6, pp. 1338–1341
Hou, M. & R. J. Patton, R. J. (1998). Optimal filtering for systems with unknown inputs, IEEE
Trans. Automatic Control, Vol. 43, No. 3, pp. 445–449
Kailath, T. (1974). A view of three decades of linear filtering theory, IEEE Trans. Inform. Theory,
Vol. 20, No. 2, pp. 146–181
Kailath, T. (1975). Supplement to a survey to data smoothing, Automatica, Vol. 11, No. 11, pp.
109–111
Kailath, T. (1976). Lectures on Linear Least-Squares Estimation,Springer
Kailath, T.; Sayed, A. H. & Hassibi, B. (2000). Linear Estimation,PrenticeHall
Kalman, R. E. (1960). A new approach to linear filtering and prediction problems, in Trans.
ASME, J. Basic Eng., Vol. 82D, No. 1, pp. 34–45
Kalman, R. E. (1963). New methods in Wiener filtering theory, Proc.ofFirstSymp.Eng.Appl.of
Random Function Theory and Probability (J. L. Bogdanoff and F. Kozin, eds.), pp. 270-388,
Wiley
Katayama, T. (2000). Applied Kalman Filtering, New Edition, in Japanese, Asakura-Shoten,
Tokyo, Japan
Meditch, J. S. (1973). A survey of data smoothing for linear and nonlinear dynamic systems,
Automatica, Vol. 9, No. 2, pp. 151–162
Patton, R. J.; Frank, P. M. & Clark, R. N. (1996). Fault Diagnosis in Dynamic Systems: Theory and

Application,PrenticeHall
Sawada, Y. & Tanikawa, A. (2002). Optimal filtering and robust fault diagnosis of stochastic
systems with unknown inputs and coloredobservation noises, Proc. 5th IASTED Conf.
Decision and Control, pp. 149-154, Tsukuba, Japan
Tanikawa, A. (2006). On a smoother for discrete-time linear stochastic systems with unknown
disturbances, Int. J. Innovative Computing, Information and Control, Vol. 2, No. 5, pp.
907–916
Tanikawa, A. (2008). On new smoothing algorithms for discrete-time linear stochastic systems
with unknown disturbances, Int. J. Innovative Computing, Information and Control,Vol.
4, No. 1, pp. 15–24
Tanikawa, A. & Mukai, H. (2010). Minimum variance state estimators with disturbance
decoupling property for optimal filtering problems with unknown inputs and fault
detection (in preparation)
69
New Smoothers for Discrete-time Linear Stochastic Systems with Unknown Disturbances
Tanikawa, A. & Sawada, Y. (2003). Minimum variance state estimators with disturbance
decoupling property for optimal filtering problems with unknown inputs, Proc. of
the 35th ISCIE Int. Symp. on Stochastic Systems Theory and Its Appl., pp. 96-99, Ube,
Japan
Weinert, H. L. & Desai, U. B. (1981). On complementary models and fixed-interval smoothing,
IEEE Trans. Automatic Control, Vol. 26, pp. 863–867
70
Discrete Time Systems
Eduardo Rohr, Damián Marelli, and Minyue Fu
University of Newcastle
Australia
1. Introduction
The fast development of network (particularly wireless) technology has encouraged its use
in control and signal processing applications. Under the control system’s perspective, this
new technology has imposed new challenges concerning how to deal with the effects of

quantisation, delays and loss of packets, leading to the development of a new networked
control theory Schenato et al. (2007). The study of state estimators, when measurements are
subject to random delays and losses, finds applications in both control and signal processing.
Most estimators are based on the well-known Kalman filter Anderson & Moore (1979). In
order to cope with network induced effects, the standard Kalman filter paradigm needs to
undergo certain modifications.
In the case of missing measurements, the update equation of the Kalman filter depends on
whether a measurement arrives or not. When a measurement is available, the filter performs
the standard update equation. On the other hand, if the measurement is missing, it must
produce open loop estimation, which as pointed out in Sinopoli et al. (2004), can be interpreted
as the standard update equation when the measurement noise is infinite. If the measurement
arrival event is modeled as a binary random variable, the estimator’s error covariance (EC)
becomes a random matrix. Studying the statistical properties of the EC is important to
assess the estimator’s performance. Additionally, a clear understanding of how the system’s
parameters and network delivery rates affect the EC, permits a better system design, where
the trade-off between conflicting interests must be evaluated.
Studies on how to compute the expected error covariance (EEC) can be dated back at least
to Faridani (1986), where upper and lower bounds for the EEC were obtained using a constant
gain on the estimator. In Sinopoli et al. (2004), the same upper bound was derived as the
limiting value of a recursive equation that computes a weighted average of the next possible
error covariances. A similar result which allows partial observation losses was presented
in Liu & Goldsmith (2004). In Dana et al. (2007); Schenato (2008), it is shown that a system in
which the sensor transmits state estimates instead of raw measurements will provide a better
error covariance. However, this scheme requires the use of more complex sensors. Most of
the available research work is concerned with the expected value of the EC, neglecting higher
order statistics. The problem of finding the complete distribution function of the EC has been
recently addressed in Shi et al. (2010).

On the Error Covariance Distribution for
Kalman Filters with Packet Dropouts

5
This chapter investigates the behavior of the Kalman filter for discrete-time linear systems
whose output is intermittently sampled. To this end we model the measurement arrival event
as an independent identically distributed (i.i.d.) binary random variable. We introduce a
method to obtain lower and upper bounds for the cumulative distribution function (CDF) of
the EC. These bounds can be made arbitrarily tight, at the expense of increased computational
complexity. We then use these bounds to derive upper and lower bounds for the EEC.
2. Problem description
In this section we give an overview of the Kalman filtering problem in the presence of
randomly missing measurements. Consider the discrete-time linear system:

x
t+1
= Ax
t
+ w
t
y
t
= Cx
t
+ v
t
(1)
where the state vector x
t
∈ R
n
has initial condition x
0

∼ N(0, P
0
), y ∈ R
p
is the measurement,
w
∼ N(0, Q) is the process noise and v ∼ N(0, R) is the measurement noise. The goal of the
Kalman filter is to obtain an estimate
ˆ
x
t
of the state x
t
, as well as providing an expression for
the covariance matrix P
t
of the error
˜
x
t
= x
t
−
ˆ
x
t
.
We assume that the measurements y
t
are sent to the Kalman estimator through a network

subject to random packet losses. The scheme proposed in Schenato (2008) can be used to
deal with delayed measurements. Hence, without loss of generality, we assume that there is
no delay in the transmission. Let γ
t
be a binary random variable describing the arrival of a
measurement at time t. We define that γ
t
= 1wheny
t
was received at the estimator and γ
t
= 0
otherwise. We also assume that γ
t
is independent of γ
s
whenever t = s. The probability to
receive a measurement is given by
λ
= P(γ
t
= 1).(2)
Let
ˆ
x
t|s
denote the estimate of x
t
considering the available measurements up to time s.Let
˜

x
t|s
= x
t
−
ˆ
x
t|s
denote the estimation error and Σ
t|s
= E{(
˜
x
t|s
− E{
˜
x
t|s
})(
˜
x
t|s
− E{
˜
x
t|s
})

}
denote its covariance matrix. If a measurement is received at time t (i.e., if γ

t
= 1), the estimate
and its EC are recursively computed as follows:
ˆ
x
t|t
=
ˆ
x
t|t−1
+ K
t
(y
t
−Cx
t
) (3)
Σ
t|t
=(I −K
t
C)Σ
t|t−1
(4)
ˆ
x
t+1|t
= A
ˆ
x

t|t
(5)
Σ
t+1|t
= AΣ
t|t
A

+ Q,(6)
with the Kalman gain K
t
given by
K
t
= Σ
t|t−1
C

(CΣ
t|t−1
C

+ Q)
−1
.(7)
On the other hand, if a measurement is not received at time t (i.e., if γ
t
= 0), then (3) and (4)
are replaced by
ˆ

x
t|t
=
ˆ
x
t|t−1
(8)
72
Discrete Time Systems
Σ
t|t
= Σ
t|t−1
.(9)
We will study the statistical properties of the EC Σ
t|t−1
. To simplify the notation, we define
P
t
= Σ
t|t−1
. Then, the update equation of P
t
can be written as follows:
P
t+1
=

Φ
1

(P
t
), γ
t
= 1
Φ
0
(P
t
), γ
t
= 0
(10)
with
Φ
1
(P
t
)=AP
t
A

+ Q − AP
t
C

(CP
t
C


+ R)
−1
CP
t
A

(11)
Φ
0
(P
t
)=AP
t
A

+ Q. (12)
We point out that when all the measurements are available, and the Kalman filter reaches its
steady state, the EC is given by the solution of the following algebraic Riccati equation
P
= AP A

+ Q − APC

(CPC

+ R)
−1
CPA

. (13)

Throughout this chapter we use the following notation. For given T
∈ N and 0 ≤ m ≤ 2
T
−1,
the symbol S
T
m
denotes the binary sequence of length T formed by the binary representation
of m.WealsouseS
T
m
(i), i = 1,···, T to denote the i-th entry of the sequence, i.e.,
S
T
m
= {S
T
m
(1), S
T
m
(2), , S
T
m
(T)} (14)
and
m
=
T
∑

k=1
2
k−1
S
T
m
(k). (15)
(Notice that S
T
0
denotes a sequence of length T formed exclusively by zeroes.) We use |S
T
m
| to
denote the number of ones in the sequence S
T
m
, i.e.,
|S
T
m
| =
T
∑
k=1
S
T
m
(k). (16)
For a given sequence S

T
m
, and a matrix P ∈ R
n×n
,wedefinethemap
φ
(P, S
T
m
)=Φ
S
T
m
(T)
◦ Φ
S
T
m
(T−1)
◦ Φ
S
T
m
(1)
(P) (17)
where
◦ denotes the composition of functions (i.e. f ◦ g(x)= f (g(x))). Notice that if m is
chosen so that
S
T

m
= {γ
t−1
, γ
t−2
, ,γ
t−T
}, (18)
then the map φ
(·, S
T
m
) updates P
t−T
according to the measurement arrivals in the last T
sampling times, i.e.,
P
t
= φ(P
t−T
, S
T
m
)=Φ
γ
t−1
◦Φ
γ
t−1
◦ Φ

γ
t−T
(P
t−T
). (19)
73
On the Error Covariance Distribution for Kalman Filters with Packet Dropouts