Discrete Time Systems Part 3 ppt

On Optimal track-to-track fusion, IEEE Transactions on Aerospace and Electronic Systems, Vol... Performance evaluation of track fusion with information matrix filter, IEEE Transactions

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 Δ > in contrast to the FLP’s weights 0 (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

x =x -x takes the form

x =x -x =F x +G v (A.1) Then the prediction cross-covariance (ij) ( (i) (j) T)

w are mutually uncorrelated at i j≠ , we obtain observation update equation (13) for ( T)

(ij) (i) (j)

P =E x x This completes the proof of Theorem 1

Differential equation (A.4) is homogeneous with zero initial condition therefore it has zero

solution ( )(i) ( )(i) ( )

E x ≡0 or E x =E x , t ≤ ≤τ t+Δ

Since the local predictors (i)

t+Δ

ˆx , i 1, ,N= are unbiased, then we have

( )FLP N (i) ( )(i) N (i) ( ) ( )

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 ˆxPFFt+Δ is proved by using the same method as in

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

(i) t,t ,Δ

B satisfy the identical equations To show that let consider differential equation for the difference (ijN) (ij) (iN)

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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 mathematicalmodels derived by engineers are very accurate representations of real systems, we do nothave to consider systems with unknown inputs However, in practice, the models derived byengineers often contain modelling errors which greatly increase state estimation errors as ifthe models have unknown disturbances

The most frequently discussed problem on state estimation is the optimal filtering problem

variance based on the observation Yt of the outputs {y0, y1,· · ·, y t}, i.e., Yt = σ{y s , s =

0, 1,· · ·, t} ( the smallest σ-field generated by {y0, y1,· · ·, y t} (see e.g., Katayama (2000),Chapter 4)) It is well known that the standard Kalman filter is the optimal linear filter inthe sense that it minimizes the mean-square error in an appropriate class of linear filters (seee.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 unknowndisturbances and modelling errors, many research papers have been published based on thedisturbance 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)) Theyutilized some transformations to make the original systems with unknown inputs into somesingular systems without unknown inputs The most important preceding study related tothis paper was done by Chen and Patton (Chen & Patton (1996)) They proposed the simpleand 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 theauthor in Tanikawa (2006) (see Tanikawa & Sawada (2003) also)

We here consider smoothing problems which allow us time-lags for computing estimates of

types For the first problem, the fixed-point smoothing, we investigate the optimal estimate

ˆx k/t of the state x k for a fixed k based on the observations{Yt , t=k+1, k+2,· · · } Algorithms

{y0, y1,· · ·, y N} Fixed-interval smoothers are algorithms for computing ˆx t/N , t=0, 1,· · ·, N

recursively The third problem, the fixed-lag smoothing, is to investigate the optimal estimate

are algorithms for computing ˆx 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 earlyresearch works on smoothers More recent papers have been published based on differentapproaches such as stochastic realization theory (e.g., Badawi; Lindquist & Pavon (1979) andFaurre; 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

in a comrehensive and self-contained manner as much as possible Namely, after somepreliminary results in Section 2, we derive the fixed-point smoothing algorithm given inTanikawa (2006) in Section 3 for the system with unknown inputs explicitly by applying theoptimal filter with disturbance decoupling property obtained in Tanikawa & Sawada (2003)

In Section 4, we construct the fixed-interval smoother given in Tanikawa (2008) from thefixed-point smoother obtained in Section 3 In Section 5, we construct the fixed-lag smoothergiven 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 thebest of our knowledge, no attempt has been made to investigate optimal fixed-interval andfixed-lag smoothers for systems with unknown inputs explicitly (see the stochastic systemgiven by (1)-(2)) before Tanikawa (2006) and Tanikawa (2008) Our smoothing algorithms havesimilar 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 areconsistent with the known smoothing algorithms for systems without unknown inputs

u t ∈Rr the known input vector,

matrices Q t and R t Let A t , B t , C t and E tbe known matrices with appropriate dimensions

which was proposed by Chen and Patton (Chen & Patton (1996) and Chen & Patton (1999))with the following structure:

z t+1=F t+1z t+T t+1B t u t+K t+1y t, (3)

ˆx t +1/t+1=z t+1+H t+1y t+1, (4)

for t=0, 1, 2,· · · Here, ˆx0/0is chosen to be z0for a fixed z0 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−ˆx 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 e0,{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+1and K t+1of the forms:

The next lemma on equality (6) was obtained and used by Chen and Patton (Chen & Patton

matrix Notice that this assumption is not an essential restriction

Lemma 2.1. Equality (6) holds if and only if

When the matrix K1t+1has the form

then the optimal gain matrix K1

t+1which makes the variance of the state estimation error e t+1minimum

is determined by (13) Hence, we obtain the optimal filtering algorithm:

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) inKatayama (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

the state x kbased on the observation Yt , t=k+1, k+2,· · ·, with Yt=σ{y s , s=0, 1,· · ·, t}

We define state vectorsθ t , t=k, k+1,· · ·, by

θ t+1=θ t , t=k, k+1,· · ·; θ k=x k (20)

It is easy to observe that the optimal estimate ˆθ t/tof the stateθ tbased on the observation Yt

is identical to the optimal smoother ˆx k/tin view of the equalitiesθ t=x k , t=k, k+1,· · ·

In order to derive the optimal fixed-point smoother, we consider the following augmented



u t+



E t O



d t+



I 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+1O

, T t+1=

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:

(i) the fixed-point smoother



+



T t+1B t u t O

.Thus, we have



+R t

−



H t O

Thus, equalities (27)-(28) can be obtained from (33) due to ˆθ t/t= ˆx k/t

Equalities (29)-(30) follow from (38)-(40) Finally, we have equalities P k(2,1)=P k(2,2)=P k(1,1)=

We thus have derived the fixed-point smoothing algorithm for the state-space model whichexplicitly contains the unknown inputs We can indicate that the algorithm has a rather simpleform and also has consistency with both the Kalman filter and the standard optimal smootherderived 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

Here, we note that the state estimate ˆx t +1/t+1 reduces to the state estimate ˆx t +1/tin Katayama

and ˆx t/t replaced respectively by ˆx t +1/t and ˆx t/t−1are identical to those for the pair of thestandard Kalman filter and the optimal fixed-point smoother in Katayama (2000) Thus, it hasbeen shown that this algorithm reduces to the well known optimal smoother derived fromthe Kalman filter when the unknown inputs disappear This indicates that our smoothingalgorithm is a natural extension of the standard optimal smoother to linear systems possiblywith unknown inputs

Let us introduce some notations:

We then have the following results due to (27)

Corollary 3.3. We have the equalities:

Proof It is straightforward to show the first equality from (27) For the second equality, it is

sufficient to prove the equality

4 The fixed-interval smoothing

We consider the fixed-interval smoothing problem in this section Namely, we investigate the

optimal estimate ˆx t/N of the state x t at all times t=0, 1,· · ·, N based on the observation Y Nofall the states{y0, y1,· · ·, y N} Applying equality (49), we easily obtain the following equality

ˆx t/N= ˆx t/t+1+P t L t T P t+1−1(ˆx t +1/N−ˆx t +1/t+1) (57)

ˆx t/N= ˆx t/t+1+P t L t T P t+1−1(ˆx t +1/N−ˆx t +1/t+1).

It is a simple task to obtain the following Fraser-type algorithm from (57)

Theorem 4.2. We obtain the fixed-interval smoother

for t=N−1, N−2,· · ·, 1, 0 Here, we have λ N=0.

Proof For t=0, 1,· · ·, N, we put

which follows from (27) in Tanikawa & Sawada (2003), we obtain

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 fromthe standard Kalman filter (see e.g., Katayama (2000)) Thus, our algorithm is consistent withthe known fixed-interval smoothing algorithm for systems without unknown inputs This

Propositin 2.4) Note that in (59), i.e.,

smoother and the optimal filter obtained from the standard Kalman filter Then, the aboveequality is identical to (7.18) in Katayama (2000) Since the rest of the proof can be done in thesame way as in Katayama (2000), we obtain the same smoother

5 The fixed-lag smoothing