< FLP = PFF . The CPU times for CP, FLP and PFF are represented in Fig. 4, where it
1,
P t+Δ
1,
P t+Δ
1,
P t+Δ
is shown that FLP requires considerably more CPU time than PFF, but CPU time of PFF is
similar to CP.
Thus, from Examples 4.1 and 4.2 we can confirm that PFF is preferable to FLP in terms of
computation efficiency.
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 (i)
b do no t
tk
depend on the leads Δ > 0 in contrast to the FLP’s weights (i)
at+Δ .
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)
x = x -ˆ
τ
τ xτ takes the form
(i)
(i)
(i)
x =x -ˆ
τ
τ xτ = τ
F xτ +Gτvτ. (A.1)
T
Then the prediction cross-covariance (ij)
τ
P =E( (i) (j)
xτ xτ ) associated with the (i)
xτ and (j)
xτ
satisfies the time update Lyapunov equation (see the first and third equations in (13)). At
t=t
(i)
k the local error x
can be written as
tk
-
-
-
-
(i)
(i)
(i)
(i) ⎡ (i)
(i) (i) ⎤
(i)
(i) ⎡ (i)
(i)
(i) (i) ⎤
(i)
(i)
(i)
(i)
(i)
x =x -ˆx =x -ˆx -L
y -H ˆx
=x -L
H x +w -H ˆx
= I -L H x -L w . (A.2)
t
t
t
t
t
t ⎢ t
t
t
⎣
⎥
t
t
⎦
⎢ t t
t
t
t
n
k
k
k
k
k
k
k
k
k
k
k
k
k
⎣
⎥ (
tk
tk )
-
k
k
k
tk
tk
tk
⎦
Given that random vectors (i)
(i)
x
(j)
≠ , we obtain
t , w
and w are mutually uncorrelated at i j
k
tk
tk
observation update equation (13) for (ij)
t
P =E(
T
(i) (j)
x x
.
k
tk tk )
This completes the proof of Theorem 1.
Proof of Theorem 2
It is well known that the local Kalman filtering estimates (i)
ˆx
ˆ
τ are unbiased, i.e.,
(i)
E(xτ )=E(xτ )
or E( (i)
x )=E(
(i)
x -ˆ
≤ ≤
τ
τ xτ )=0 at 0
τ tk . With this result we can prove unbiased property at
t < ≤
k
τ t+Δ . Using (8) we obtain
(i)
(i)
(i)
(i)
(i)
x =x -ˆ
≤ ≤
(A.3)
τ
τ xτ = τ
F xτ +Gτvτ , xτ=t =xt , tk τ t+Δ ,
k
k
or
d E x =F E x , E x
=E x
=0 , t ≤ τ ≤ t+Δ. (A.4)
dτ
( (i)τ) τ ( (i)τ) ( (i)
(i)
τ=t
t
k
k )
( k )
Differential equation (A.4) is homogeneous with zero initial condition therefore it has zero
solution E( (i)
x ) ≡ 0 or E( (i)
ˆ
≤ ≤
τ
xτ )=E(xτ ), tk τ t+Δ.
Since the local predictors (i)
ˆx
=
t+Δ , i
1,...,N are unbiased, then we have
N
N
⎡
⎤
E( FLP
ˆx
=∑a E ˆ
(A.5)
t+Δ )
(i)
t+Δ ( (i)
xt+Δ )
(i)
=⎢∑at+Δ ⎥E(xt+Δ )=E(xt+Δ ).
i=1
⎣i=1
⎦
50
Discrete Time Systems
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
ˆxt+Δ 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
(i)
ˆx =Φ t+Δ,t ˆx , i = 1,...,N , ˆx
=Φ(t+Δ,t )ˆ
(A.6)
t+Δ
(
) (i)
PFF
FF
k
t
t+Δ
k xt ,
k
k
where Φ(t,s) is the transition matrix of (8) or (17). From (10) and (16), we obtain
N
N
N
FLP
(i)
(i)
(i)
(i)
(i)
(i)
ˆx
=∑a ˆx =∑a Φ(t+Δ,t )ˆx =∑A
ˆ
t+Δ
t+Δ t+Δ
t+Δ
k
x ,
tk
t,tk ,Δ tk
i=1
i=1
i=1
(A.7)
N
N
PFF
FF
(i) (i)
(i)
(i)
ˆx
=Φ(t+Δ,t ˆ
)x =∑Φ(t+Δ,t )b ˆx =∑B
ˆ
t+Δ
k
t
x ,
k
k
tk tk
t,tk ,Δ tk
i=1
i=1
where the new weights take the form:
(i)
(i)
A
=a
Φ t+Δ,t , B
=Φ t+Δ,t b . (A.8)
t,t ,Δ
t+Δ
(
) (i)t,t ,Δ (
) (i)
k
k
k
k
tk
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,tk) and Φ(t+Δ,tk)T, respectively, and multiplying the last non-
homogeneous equation (18) by Φ(t+Δ,tk) we obtain
N
∑Φ(t+Δ,t ) (i) (ij) (iN)
⎡
⎤Φ
k b
P -P
t+Δ,t
=0, j=1,...,N-1;
t ⎣ t
t
(
k )T
k
k
k
⎦
i=1
(A.9)
N
∑Φ(t+Δ,t ) (i)
k b =Φ(t+Δ,t ).
t
k
k
i=1
Using notation for the difference
(ijN)
(ij)
(iN)
δ s
P
= s
P - s
P
we obtain equations for
(i)
B
, i = 1,...,N such that
t,tk ,Δ
N
N
(i)
(ijN)
∑B
Φ
∑
(A.10)
t,t ,Δδ t
P
(t+Δ,t )T
(i)
k
=0, j=1,...,N-1;
Bt,t ,Δ=Φ(t+Δ,tk ).
k
k
k
i=1
i=1
Analogously after simple manipulations equation (12) takes the form
N
N
(i)
∑a
−
−
⎡
⎤
t+ΔΦ ( t+Δ,t ) Φ ( t+Δ,t ) 1
(ij)
(iN)
(i)
k
k
t+
P Δ - t+
P Δ =∑A
⎣
⎦
t,t ,ΔΦ ( t+Δ,tk ) 1
(ijN)
δP
=0,
k
t+Δ
i=1
i=1
(A.11)
N
N
(i)
(i)
∑at+ΔΦ(t+Δ,tk)=∑At,t ,Δ=Φ(t+Δ,tk).
k
i=1
i=1
Distributed Fusion Prediction for Mixed Continuous-Discrete Linear Systems
51
or
N
N
(i)
∑A
−
=
∑
(A.12)
t,t ,ΔΦ ( t+Δ,t ) 1
(ijN)
(i)
k
δ t+
P Δ =0, j 1,...,N-1;
At,t ,Δ=Φ(t+Δ,tk ).
k
k
i=1
i=1
As we can see from (A.10) and (A.12) if the equality
(ijN)
δ t
P
Φ(t+Δ,t )T =Φ(t+Δ,t )-1
(ijN)
k
k
δP
(A.13)
k
t+Δ
will be hold then the new weights (i)
A
a nd
(i)
B
satisfy the identical equations. To
t,tk ,Δ
t,tk ,Δ
show that let consider differential equation for the difference
(ijN)
(ij)
(iN)
δ s
P
= s
P - s
P
. Usin g (1 3)
we obtain the Lyapunov homogeneous matrix differential equation
(ijN)
(ij)
(iN)
δP
=P -P
=F ( (ij) (iN)
P -P
)+( (ij) (iN)
P -P
) T
(ijN)
(ijN) T
≤ ≤
s
s
s
s
s
s
s
s
s
F = s
F δ s
P
+δ s
P
s
F , tk s t+Δ, (A.14)
which has the solution
(ijN)
δ t+
P Δ =Φ(t+Δ,t ) (ijN)
k δ t
P
Φ(t+Δ,tk )T . (A.15)
k
By the nonsingular property of the transition matrix Φ(t+Δ,tk ) the equality (A.13) holds,
then (i)
(i)
A
= B
, and finally using (A.7) we get
t,tk ,Δ
t,tk ,Δ
N
N
FLP
(i)
(i)
(i)
(i)
PFF
ˆx
=∑A
ˆx = ∑B
ˆx = ˆ
t+Δ
(A.16)
t,t ,Δ t
t,t ,Δ t
xt+Δ .
k
k
k
k
i=1
i=1
This completes the proof of Theorem 4.
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4
New Smoothers for Discrete-time
Linear Stochastic Systems with
Unknown Disturbances
Akio Tanikawa
Osaka Institute of Technology
Japan
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 xt at time t or xt+1 at time t + 1 with minimum
variance based on the observation Y t of the outputs { y 0, y 1, · · · , yt}, i.e., Y t = σ{ ys, s =
0, 1, · · · , t} ( the smallest σ-field generated by { y 0, y 1, · · · , yt} (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 xt− L/ t of the state xt− 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
54
Discrete Time Systems
ˆ xk/ t of the state xk for a fixed k based on the observations {Y t, t = k + 1, k + 2, · · · }. Algorithms for computing ˆ xk/ t, t = k + 1, k