Multi-Pass Optimal FIR Filtering for Processes with Unknown Initial States and Temporary Mismatches

Abstract

The multi-pass optimal finite impulse response (OFIR) filtering approach is developed for industrial processes with unknown initial conditions under temporary model mismatches. The forward and backward OFIR filters are derived in the batch and fast iterative forms using recursions. The doublepass OFIR (DOFIR) filter supported by the unbiased FIR (UFIR) filter and triple-pass OFIR (TOFIR) filter starting with some initial values are designed and extensively investigated using simulations and experimental data. It is shown that the DOFIR and TOFIR filters are able to essentially improve the performance close to the initial values and are more robust against temporary model mismatches than the Kalman, OFIR, and UFIR filters.

Full text

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XX XXXX 1
Multi-Pass Optimal FIR Filtering for Processes with
Unknown Initial States and Temporary Mismatches
Shunyi Zhao, Senior Member, IEEE, Yuriy S. Shmaliy, Fellow, IEEE, Jose A. Andrade-Lucio, Member, IEEE, and
Fei Liu
Abstract—The multi-pass optimal finite impulse response
(OFIR) filtering approach is developed for industrial processes
with unknown initial conditions under temporary model mis-
matches. The forward and backward OFIR filters are derived in
batch and fast iterative forms using recursions. The double-pass
OFIR (DOFIR) filter supported by the unbiased FIR (UFIR) filter
and triple-pass OFIR (TOFIR) filter starting with some initial
values are designed and extensively investigated using simulations
and experimental data. It is shown that the DOFIR and TOFIR
filters are able to essentially improve the performance close to
the initial values and are more robust against temporary model
mismatches than the Kalman, OFIR, and UFIR filters.
Index Terms—Industrial environments, optimal FIR filter,
Kalman filter, temporary uncertainty, state estimation.
I. INTRODUCTION
THE well-known and yet annoying specific of many indus-
trial processes is the practical inability to keep operation
conditions constant and avoid systems faults [1], [2] caused
by power surges, mechanical shock, jumps in velocity, and
vibrations, just a few to mention [3], [4]. Given that a control
system using a Kalman filter (KF) often does not demonstrate
an optimal performance under the short-time uncertain impacts
[5]–[7], more robust solutions are required [8], [9].
A better protection against short-time mismodeling caused
by the above effects can be found in finite horizon (FH)
optimal control [10], [11]. An idea was originally expressed
by Jazwinski [12] and later reformulated by Schweppe [13]
as an old estimate updated in discrete time not over all data,
but over most resent observations. Maybeck then stated in [14]
that it is preferable to rerun the growing memory filter over the
FH data in what was called the limited memory filter (LMF).
The problem one meets here is in the initial values, which
are required to be more accurate than produced by the KF.
Since for Gaussian processes there is no estimator better than
the KF, the idea of LMF has not been implemented. A short
historical review is given in [15].
Another FH approach was developed in optimal finite
impulse response (OFIR) filtering [16]. The FIR estimator
operates in discrete time index non a FH [m, n]of Nmost
resent data points, from m=n−N+ 1 to n, and, unlike the
This work was supported in part by the National Natural Science Foundation
of China (61973136, 61991402, and 61833007), 111 Project (B12018), and
Mexican CONACyT-SEP Project A1-S-10287, Funding CB2017-2018.
S. Zhao. and Fei Liu are with the Key Laboratory of Advanced Process
Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi
214122, China (e-mail: shunyi.s.y@gmail.com, fliu@jiangnan.edu.cn).
Y. S. Shmaliy and J. A. Andrade-Lucio are with the Department of
Electronic Engineering, Universidad de Guanajuato, Salamanca, Gto, 36855,
Mexico (e-mail: ahmaliy@ugto.mx, andrade@ugto.mx).
Fig. 1. Strategy of TOFIR filtering: (a) triple-pass chain, (b) improving the
initial values, and (c) improving the robustness against temporary mismatch.
KF and LMF, requires all data on [m, n]at once. Note that
the earlier derived receding horizon (RH) FIR filters [18]–[20]
operating similarly on [n−N , n−1] in model predictive control
[21], [22] are not optimal. A distinctive difference between the
OFIR filter and LMF is that the former has the convolution-
based batch form, while the latter utilizes Kalman recursions
and has thus the infinite impulse response (IIR).
Given FH [m, n]data, the OFIR filter can be applied forward
as F-OFIR to produce a state estimate ˆxn,ˆxn|nand error
covariance Pn. It can also be applied backward as B-OFIR to
estimate the initial state as ˆxm,ˆxm|nand error covariance
as Pm. A seemingly obvious solution suggests that, provided
ˆxmand Pmby the B-OFIR filter, ˆxnand Pncan be updated
by rerunning the F-OFIR filter again and such a procedure can
be multiply repeated in what can be said to be a multi-pass
OFIR filter. As an example, the triple-pass OFIR (TOFIR)
filter structure is sketched in Fig. 1a to solve two keyproblems:
1) improving the initial values in the second pass (Fig. 1b)
and 2) improving the robustness against the temporary model
mismatch in the third pass (Fig. 1c) [15]. Note that the
concepts employed in Fig. 1 differ from smoothing, since
the forward and backward computations relate to the horizon
[m, k]endpoints, mand k.
Overall, the multi-pass filter is able to make the transients
shorter, thereby reducing dynamic errors [23], [25]. Even so,
the discrete convolution-based OFIR algorithm suffers from a
crucial drawback: it is computationally inefficient (the time-
varying extended matrices have large dimensions) and has thus
limited applications, especially when N≫1.

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XX XXXX 2
The issue can be circumvented if we compute the OFIR
batch using recursions, similarly to the KF. However, no
recursive forms were developed so far for OFIR structures,
especially in a backward computation manner. To show right
away a necessity of designing fast OFIR algorithms, we give
an example. If the KF is implemented to consume 1 ms, then
the recursive OFIR will require about Nms, and the TOFIR
filter 3Nms. That is quite acceptable for tracking with a
sampling time of 1 s with a relatively large N= 100. Note
that the batch OFIR filter was shown in [26] to consume about
4 s against the KF requiring 5 ms.
In this paper, we design the F-OFIR and B-OFIR filters in
the batch and fast recursive forms. We also design the double-
pass OFIR (DOFIR) and TOFIR filters and show their better
performance against the KF, OFIR filter, and unbiased FIR
(UFIR) filter [15]. The main contributions of this paper are
1) the derivation of the forward and backward OFIR filters in
the batch and recursive forms and 2) formulation and design
of different multiple-pass OFIR filters. The trade-off between
the estimators designed is investigated by simulations and
using experimental data. The rest of this paper is organized as
follows. In section II, we consider the discrete-time state-space
model of an industrial process and formulate the problem.
The F-OFIR and B-OFIR filtering algorithms are derived in
the batch and recursive forms in section III. The DOFIR
and TOFIR iterative filtering algorithms using recursions are
designed in section IV. In section V, we test the proposed
algorithm by several experimental and simulated examples.
Finally, conclusions can be found in section VI.
Notations: The following notations will be used: RKis the
K-dimensional Euclidean space, E{x}denotes the expectation
of x,N(¯x0, P0)is a Gaussian distribution with dummy mean
¯xand covariance P0,diag(a1··· an)is a diagonal matrix
with elements a1,··· , an,Iand Orespectively denote the
identity matrix and zero matrix with appropriate dimensions.
II. MODEL AND PROBLEM FORMULATIONS
We consider an industrial process represented in discrete-
time state-space with the linear state and observation equa-
tions, respectively,
xn=Fnxn−1+Enun+Bnwn,(1)
yn=Hnxn+vn,(2)
where xn∈ RKis the state vector, yn∈ RPis the measure-
ment vector, un∈ RLis the input vector, wn∼ N (0, Qn)and
vn∼ N (0, Rn)are zero mean white and uncorrelated noise
sources with covariances Qnand Rn, and Fn,En,Bn, and
Hnare given matrices. To design the F-OFIR and B-OFIR
state estimators, the following extended models are required.
1) Forward-in-Time Extended Model: On a horizon [m, n],
where m=n−N+ 1 and Ndenotes the horizon length, the
extended state-space model can be written as [15]
Xm,n =Fm,nxm+Sm,n Um,n +Dm,n Wm,n ,(3)
Ym,n =Hm,nxm+Lm,n Um,n +Gm,nWm,n +Vm,n ,(4)
where the augmented vectors are defined as Xm,n =
[xT
mxT
m+1 . . . xT
n]T,Um,n = [ uT
muT
m+1 . . . uT
n]T,Wm,n =
[wT
mwT
m+1 . . . wT
n]T,Ym,n = [ yT
myT
m+1 . . . yT
n]T, and
Vm,n = [ vT
mvT
m+1 . . . vT
n]Tand the extended matrices are
Fm,n =I F T
m+1 ... (Fm+1
n−1)T(Fm+1
n)TT,(5)
¯
S(N−q)
m,n =Fm+1
n−qEmFm+2
n−qEm+1 ... Fn+1
n−qEn,(6)
Fg
r=




FrFr−1...Fg, g < r + 1 ,
I , g =r+ 1
0, g > r + 1
,(7)
Hm,n =¯
Hm,nFm,n ,Lm,n =¯
Hm,nSm,n ,Gm,n =
¯
Hm,nDm,n . Matrix ¯
S(N−q)
m,n ,q∈[0, N −1], is the (N−q)th
block raw vector in matrix Sm,n and so is ¯
D(N−q)
m,n in
Dm,n if we substitute Enwith Bn, and matrix ¯
Hm,n =
diag (HmHm+1 . . . Hn)is diagonal.
2) Back-in-Time Extended Model: On [m, n], the extended
state-space model can also be written from nto mas
Xn,m =Fb
n,mxn−Sb
n,mUn,m −Db
n,mWn,m ,(8)
Yn,m =Hb
n,mxn−Lb
n,mUn,m −Gb
n,mWn,m +Vn,m ,(9)
where the components in vectors Xn,m,Un,m,Wn,m ,Yn,m,
and Vn,m are rearranged in opposite directions, from nto m,
and the extended matrices are given by
Fb
n,m = [IXnT
n· · · Xm+2T
nXm+1T
n]T,(10)
¯
Sb(N−q)
m,n =Xm+1
n−qEmXm+2
n−qEm+1 ... Xn+1
n−qEn,(11)
Xg
r=((FrFr−1...Fg)−1, g 6r+ 1
0, g > r + 1 ,(12)
Hb
n,m =¯
Hn,mFb
n,m,Lb
n,m =¯
Hn,mSb
n,m,Gb
n,m =
¯
Hn,mDb
n,m. Matrix ¯
Sb(N−q)
n,m ,q∈[0, N −1], is the (N−
q)th block raw vector in Sb
n,m and so is ¯
Db(N−q)
n,m in
Db
n,m if we substitute Enwith Bn, and matrix ¯
Hn,m =
diag (HnHn−1. . . Hm)is a diagonal matrix.
III. F-OFIR AND B-OFIR FILTERS
In [16] [17], the OFIR filter was originally designed for
systems without input. In this section, we derive more general
F-OFIR and B-OFIR filters for control systems and find
recursive forms.
A. F-OFIR Filter
Given a data vector (9), the F-OFIR filtering estimate ˆxn,
ˆxn|ncan be defined as [21]
ˆxn=Hh
m,nYm,n +Hf
m,nUm,n ,(13)
where gains Hh
m,n and Hf
m,n are required to be optimal. By
representing state xnwith the last raw vector in Xm,n as
xn=Fm+1
nxm+¯
Sm,nUm,n +¯
Dm,nWm,n ,(14)
where ¯
Sm,n is the last raw vectors in Sm,n and so is ¯
Dm,n
in Dm,n, and defining the estimation error as εn=xn−ˆxn,
the optimal Hh
m,n and Hf
m,n can be found by satisfying the
orthogonality condition [16]
E{(xn− Hh
m,nYm,n − Hf
m,nUm,n )YT
m,n}= 0 ,(15)

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XX XXXX 3
which, by providing the averaging, can be transformed to
Bm,nχmHT
m,n +Wm,nQm,n GT
m,n − Vm,nRm,n
= (Hh
m,nLm,n −¯
Sm,n +Hf
m,n)Ψm,n LT
m,n ,(16)
where χm=E{xmxT
m},Ψm,n =E{Um,nUT
m,n},Bm,n =
Fm+1
n− Hh
m,nHm,n ,Wm,n =¯
Dm,n − Hh
m,nGm,n ,Vm,n =
Hh
m,n,Qm,n =E {Wm,n WT
m,n}, and Rm,n =E {Vm,nVT
m,n}.
For zero input, Ψm,n = 0, (16) returns Hh
m,n. Then, for
zero initial conditions, it gives Hf
m,n and we finally have
Hh
m,n = (Fm+1
nχmHT
m,n +Z1)
×(Zχ+Z2+Rm,n)−1,(17)
Hf
m,n =¯
Sm,n − Hh
m,nLm,n ,(18)
where Zχ=Hm,nχmHT
m,n,Z1=¯
Dm,nQm,n GT
m,n, and
Z2=Gm,nQm,n GT
m,n. The batch forward OFIR filter (13)
thus becomes
ˆxn=Hh
m,nYm,n + ( ¯
Sm,n − Hh
m,nLm,n )Um,n ,(19)
where Ym,n and Um,n contain real data. The error covariance
Pn=E{εnεT
n}can be found for (19) as
Pn=Bm,nχmBT
m,n +Wm,nQm,n WT
m,n
+Vm,nRm,n VT
m,n ,(20)
where the bias and random errors are balanced optimally.
1) Recursive Forms and Algorithm: Given the batch F-
OFIR filter (19) with initial ˆxmand Pm, the iterative compu-
tation on [m, n]can be provided using a pseudo code listed as
Algorithm 1, which derivation is postponed to Appendix A.
The iterative F-OFIR filter operates as follows: an auxiliary
Algorithm 1: Iterative F-OFIR Filtering Algorithm
Data:yn,un,ˆxm,Pm,Qn,Rn,N
1begin
2for n= 1,2,··· do
3m=n−N+ 1 if n > N −1and m= 0
otherwise;
4for i=m+ 1, m + 2,··· , n do
5ˆx−
i=Fiˆxi−1+Eiui;
6P−
i=FiPi−1FT
i+BiQiBT
i;
7zi=yi−Hiˆx−
i;
8Si=HiP−
iHT
i+Ri;
9Ki=P−
iHT
iS−1
i;
10 ˆxi= ˆx−
i+Kizi;
11 Pi= (I−KiHi)P−
i;
12 end for
13 end for
Result:ˆxn,Pn
14 end
time-index igrows from m+ 1 to nand the outputs ˆxnand
Pnare taken when i=n.
B. B-OFIR Filter and Recursive Forms
Reasoning along similar lines as for the F-OFIR filter with
no essential specifics, the B-OFIR applied to model (8) and
(9) can be shown to be
˜xm=Hbh
n,mYn,m + (Hbh
n,mLn,m −¯
Sb
n,m)Un,m ,(21)
where the backward homogeneous gain Hbh
n,m is given by
Hbh
n,m = (Xm+1
nχnHbT
n,m +¯
Db
n,mQn,m GbT
n,m)
×(Hb
n,mχnHbT
n,m +Gb
n,mQn,m GbT
n,m
+Rn,m)−1.(22)
Here, Qn,m =E{Wn,mWT
n,m}and Rn,m =E {Vn,m VT
n,m}
are the augmented noise covariances corresponding to the
back-in-time extended state-space models (8) and (9).
1) Recursive Forms and Algorithm: The recursive forms for
the B-OFIR filter can be found similarly to the F-OFIR filter
and we postpone it to Appendix B. A pseudo code of the
iterative B-OFIR filtering algorithm is listed as Algorithm 2.
Given the initial values at n, this algorithm updates iteratively
Algorithm 2: Iterative B-OFIR Filtering Algorithm
Data:yn,un,˜xn,Pn,Qn,Rn,N
1begin
2for n= 1,2,··· do
3m=n−N+ 1 if n > N −1and m= 0
otherwise;
4for i=n−1, n ··· , m do
5P−
i−1=F−1
i(Pi+BiQiBT
i)F−T
i;
6˜x−
i−1=F−1
i(˜xi−Eiui);
7Si−1=Hi−1P−
i−1HT
i−1+Ri−1;
8Ki−1=P−
i−1HT
i−1S−1
i−1;
9˜zi−1=yi−1−Hi−1˜x−
i−1;
10 ˜xi−1= ˜x−
i−1+Ki−1˜zi−1;
11 Pi−1= (I−Ki−1Hi−1)P−
i−1;
12 end for
13 end for
Result:ˆxm= ˜xm,Pm
14 end
all vectors and matrices and produces the final estimates ˆxm
and Pmat the start horizon point m. Comparing Algorithm 2
and Algorithm 1, it can be noticed that they are optimal on
[m, k]but act in different directions. Specifically, Algorithm
1 produces ˆxi,i∈[m, k], forward, while Algorithm 2 does it
backward. Herewith only the final estimates on [m, k ]go to
the outputs of these filters.
IV. MULTI-PASS OFIR FILTERING ALGORITHMS
It follows that a multi-pass OFIR filter can be designed
and implemented in different forms aimed at optimizing the
structure and reducing the computation complexity, compared
with the conventional batch computation forms. In this section,
we consider two possible solutions: the TOFIR filter (Fig. 1a)
and the double-pass OFIR (DOFIR) filter employing the UFIR
filter.

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XX XXXX 4
A. TOFIR Filtering Algorithms
The TOFIR filtering algorithm is a straightforward imple-
mentation of the triple-pass chain (Fig. 1a), which pseudo code
is listed as Algorithm 3. Given the initial ˆxmand Pm, a sub-
Algorithm 3: TOFIR Filtering Algorithm
Data:ˆxm,Pm,yn,un,Qn,Rn,N
1begin
2Run Algorithm 1 to get ˆxnand Pn;
3Run Algorithm 2 to get ˆxmand Pm;
4Run Algorithm 1 to get ˆxnand Pn;
Result:ˆxn,Pn
5end
Algorithm 1 computes ˆxnand Pn, which are next used to
initiate the sub-Algorithm 2 and improve ˆxmand Pm. Finally,
a sub-Algorithm 1 is rerun again to update ˆxnand Pn. To
run Algorithm 3, the initial ˆxmand Pmmust be available as
a necessary condition. Although there were proposed several
approaches of how to estimate the initial conditions and reduce
their effect on the output [16], [22], [24], the effectiveness and
accuracy of these methods remain insufficient. Below we will
show how to remove this requirement employing the batch
FIR form.
B. UFIR-supported DOFIR Filtering Algorithms
To avoid the first pass in Fig. 1a, the initial ˆxmand Pmcan
be provided using a backward UFIR (B-UFIR) filter, which
does not require ˆxnand Pnfor the initialization. Referring to
the backward model (8) and (9) and following the derivation
of the forward UFIR filter [15], the B-UFIR estimate ˜xmcan
be defined as
˜xm=ˆ
Hbh
n,mYn,m +ˆ
Hbf
n,mUn,m .(23)
By satisfying the unbiasedness condition E{ˆxm}=E{xm}
for the state model
xm=Xm+1
nxn−¯
Sn,mUn,m −¯
Dn,mWn,m ,(24)
which is the last Nth row vector in (8), two unbiasedness
constraints can be derived,
Xm+1
n=ˆ
Hbh
n,mHb
n,m ,(25)
ˆ
Hbf
n,m =ˆ
Hbh
n,mLn,m −¯
Sn,m .(26)
The first constraint (25) returns the homogeneous gain
ˆ
Hbh
n,m =Xm+1
n(HbT
n,mHb
n,m)−1HbT
n,m (27)
and, referring to (26), the B-UFIR estimate (23) becomes
˜xm=ˆ
Hbh
n,mYn,m + ( ˆ
Hbh
n,mLn,m −¯
Sn,m)Un,m (28)
with the error covariance
Pm=Wb
n,mQn,m WbT
n,m +Vb
n,mRn,m VbT
n,m ,(29)
where the error residual matrices are given by Vb
n,m =ˆ
Hbh
n,m
and Wb
n,m =¯
Db
n,m −ˆ
Hbh
n,mGb
n,m.
A pseudo code of the DOFIR filtering algorithm is listed
as Algorithm 4. The initial ˆxmand Pmare computed here
Algorithm 4: DOFIR Filtering Algorithm
Data:yn,un,ˆx0,P0,Qn,Rn,N
1begin
2for n= 1,2,··· do
3m=n−N+ 1 if n > N −1and m= 0
otherwise;
4if m= 0 then
5Run Kalman recursions;
6end if
7else
8Compute ˆxmby (28) and Pmby (29) ;
9Run Algorithm 1 ;
10 end if
11 end for
Result:ˆxn,Pn
12 end
in batch forms (28) and (29). However, if the computational
complexity in some applications is still an issue, the B-UFIR
estimate can also be represented with a fast iterative algorithms
as shown in [15]. To reduce the computation time, the B-UFIR
filter is run in a small batch form [α, n]to produce ˆxαand
Pαwhen m < α < n. Then the backward recursions are run
from ˆxαand Pαto produce ˆxmand Pm.
Note that although recursions are used within the horizon,
both TOFIR and DOFIR filters are inherently bounded input-
bounded output stable [15].
V. APPLICATIONS
To show the efficiency of multi-pass OFIR filtering, in this
section we test the TOFIR filter (Algorithm 3) and DOFIR
filter (Algorithm 4) by several simulated and experimental
examples. Since the initial condition is required by the TOFIR
filter and its performance is similar to the DOFIR filter, we
treat the TOFIR filter as an illustrative method of the multiple-
pass FIR filters and omit it in some examples for clarity.
All the way, we will compare the algorithms to the KF,
OFIR filter, and UFIR filter. The estimation errors and average
computation time will be used as performance criteria.
A. Moving Vehicle Tracking
We first consider a practical example of the moving vehicle
tracking in a suborn industrial area, which is a growing
market [27]. The vehicle moves in a two-dimensional region,
but we are only interested in the north direction, whose
coordinates are measured using a Global Positioning System
(GPS) navigation unit. The two-state tracking model (1) and
(2) is used with
F="1τ
0 1#, B ="τ /2
1#, H =h1 0i, E = 0 .
By analyzing the vehicle trajectory and referring to GPS
service errors, the tuning factors were set as: τ= 1 s,N= 5,
σ2w= 2 m/sin the vehicle velocity, and σv= 3.75 m. The
initial vehicle location was voluntary set as y0= 300 m and

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XX XXXX 5
0 10 20 30
50
-
0
50
100
n
28 30 32 34
75
80
85
90
KF
UFIR
TOFIR
Ground truth
KF
TOFIR
(a)
(b)
UFIR
m,10´
n
y
DOFI R
DOFI R
Fig. 2. Vehicle tracking in the north direction with TOFIR filter, DOFIR filter,
UFIR filter, and KF: (a) improved initial state estimates and (b) improved
robustness against temporary model mismatch.
then all filters run to produce estimates shown in Fig. 2. Two
typical effects can be observed here with a reference to Fig. 1:
1) a significant improvement of estimates close to the initial
state (Fig. 2a) and 2) a higher robustness against temporary
model mismatch (Fig. 2b). The root MSEs (RMSEs) produced
by the filters over a thousand of discrete-time points excluding
the transients compute 2.24 m for the UFIR filter, 1.83 m
for KF, and 1.69 m for TOFIR that confirms advantages of
the TOFIR approach. To produce a single estimate, the KF
consumed about 5 ms, OFIR 25 ms, TOFIR 80 ms, and DOFIR
78 ms that is quite acceptable for tracking with τ= 1 s.
B. Localization of a Flying Ball
In the second example, we simulate a ball flying in the 3-
D surveillance region, which is commonly used as the core
part of a ball-catching robot [28], [29]. Accurate ball position
estimation at each time instant is crucial for the robot to fulfil
the task. The state and observation equations (1) and (2) are
specified with B=Iand
F=











1 0 0 τ0 0
0 1 0 0 τ0
0 0 1 0 0 τ
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1











, E =











0
0
τ2
2
0
0
τ











,
H=


1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0 

, un=−g ,
where g= 10 m/s2and τ= 1 s. The first three components in
the state vector xn= [ x1nx2nx3nx4nx5nx6n]Trepresent
the ball positions along axes x,y, and zand the remaining ones
the corresponding velocities. We assume that the process starts
with x10 = 1 m,x20 = 2 m,x30 = 3 m,x40 = 2 m/s,x50 =
1 m/s, and x60 = 1 m/shaving the initial error covariance
(I) (I)
T
T
T
T
Fig. 3. Localization errors produced by the TOFIR, DOFIR, UFIR, OFIR,
and KF along the x-axis in the presence of wrong initial conditions and a
temporary model mismatch: (a) position and (b) velocity.
100 102 104 106 108
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
100 102 104 106 108
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
100 102 104 106 108
-10
-8
-6
-4
-2
0
2
410-3
100 102 104 106 108
-8
-6
-4
-2
0
2
410-3
UFIR
Time steps Time steps
Time steps
Time steps
Estimation errors
OFIR
KF
DOFIR
KF
OFIR
DOFIR
UFIR
(a) (b)
( )c (d)
KF
OFIR
UFIR
DOFIR
KF
UFIR
OFIR
DOFIR
Estimation errors
Estimation errors
Estimation errors
N=8 N=20
N=45 N=85
Fig. 4. Effect of the horizon length Non KF, UFIR filter, OFIR filter, and
TOFIR filter performances: (a) N= 8, (b) N= 20, (c) N= 45, and (d)
N= 85.
P0=I. The system and measurement noise variances are
chosen as Q= 10−3Iand R= 0.1I, respectively. For all FIR-
type filters, the horizon is set as N= 9. To investigate the filter
response to a temporary uncertainty, an unknown disturbance
dn= [0,0,0,3,0,0]Tis injected into xnas xn=F xn−1+
Eun+dn+wnwhen 20 6n622.
The estimation errors are sketched in Fig. 3. Ideally, all
filters must produce zero errors in the position and velocity.
What can be seen is that the DOFIR filter and TOFIR filter do
it better than the other ones. Indeed, these filters significantly
improve estimates close to the start points and respond with a
shorter excursion to the temporary uncertainty. The effect in
the velocity turned out to be less appreciable (Fig. 3b). Even
so, the DOFIR filter and the TOFIR filter also demonstrate
here shorter excursions and return faster to normal operation.
Note that the difference between the TOFIR filter and DOFIR
filter is indistinguishable, although only the former requires
the initial values. Due to this observation, we further omit the
TOFIR estimates.
In addition, we investigate the effect of Non the filter
performance as shown in Fig. 4. Since the KF is N-invariant,
its accuracy remains almost constant with an increase in N.
On the contrary, all FIR-type filters are N-sensitive and their

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XX XXXX 6
&RXQWHUZHLJKW
%DF NPR WRU
)UR QWP RWR U
$UP
+HOLFRSWHUERG\
%DV H
7UD YH O$[ LV
(OH YD WL RQ$[ LV
3LW FK$ [L V
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Fig. 5. Setup of a 3-DOF helicopter system produced by Quanser.
performances improve with an increase in N, except for the
UFIR filter, which requires an optimal N[15]. It follows from
Fig. 4 that no one FIR structure is able to produce acceptable
estimates on short horizons. However, the difference between
the DOFIR and OFIR outputs remains insignificant under the
normal operation conditions when the initial values are set
correctly and the model matches the process.
C. 3-DOF Helicopter Tracking
To verify the previously made conclusions, we further test
the DOFIR and other filters by the three degree-of-freedom
(3-DOF) helicopter system produced by Quanser [6], [30] as
pictured in Fig. 5. In this system, two motors are mounted at
one end of the helicopter frame to drive propellers and at the
other end there is placed a counterweight to be carried. The
whole frame is suspended from an instrumented joint mounted
at the end of a long arm and is free to pitch about its centre.
The arm is installed on a 2-DOF joint and can move in the
elevation and travel directions.
By selecting the voltages V1nand V2napplied to the front
and back motors as inputs un= [V1nV2n]T, and defining
the state vector as xn= [ x1nx2nx3nx4nx5nx6n]T, where
xin,i∈[1,3], denotes the elevation, pitch, and travel angles,
respectively, and xin,i∈[4,6], the corresponding angle
velocities, the discrete-time state-space model (1) and (2) is
specified for τ= 0.01 s with B=Iand
F=











1 0 0 0.1 0 0
0 1 0 0 0.1 0
0−6.2α1 0 −0.2α0.1
0 0 0 1 0 0
0 0 0 0 1 0
0−123α0 0 −6.2α1











,
where α= 10−3,H= [I3×303×3], and E= [E1E2]with
E1= [ 0.4 2.9 0 −8.6 58.1−0.1 ]Tαand E2= [ 0.4−
2.9 0 −8.6−58.1 0.1 ]Tα.
To investigate filtering errors, we run the whole process
from x0= 0 for 30 s. Three encoders of high accuracy are
mounted to measure the elevation angular, pitch angular, and
Fig. 6. Noisy measurements and ground truth values: (a) elevation angular, (b)
pitch angular, (c) travel angular, (d) angular velocity of elevation, (e) angular
velocity of pitch, and (f) angular velocity of travel.
0 10 20 30
-6
-4
-2
0
2
4
6
0 10 20 30
-4
-2
0
2
4
6
8
0 10 20 30
-2
0
2
4
6
8
10
0 10 20 30
-10
-5
0
5
10
15
20
0 10 20 30
-6
-4
-2
0
2
4
6
8
10
0 10 20 30
-5
-4
-3
-2
-1
0
1
2
3
4
Time steps
(f)
Time steps
Velocity of travel angular, in deg/s
(a)
DOFIR
UFIR
Elevation angular, in deg
Pitch angular, in deg
Time steps Time steps
Time steps Time steps
Travel angular, in deg
Velocity of pitch angular, in deg/s
Velocity of elevation angular, in deg/s
(b) ( )c
(d) (e)
OFIR
KF
Ground truth
Ground truth
Ground truth
DOFIR
Ground truth
KF
UFIR
DOFIR
OFIR
UFIR
DOFIR
KF
OFIR
KF
UFIR
OFIR
Ground truth
OFIR
UFIR
KF
DOFIR
Ground truth
OFIR
KF
UFIR
DOFIR
Fig. 7. Estimation errors produced by the filtering algorithms for the 3-DOF
helicopter systems as caused by wrongly set initial values: (a) elevation, (b)
pitch, (c) travel, (e) elevation velocity, (f) pitch velocity, and (d) travel velocity.
travel angular. We thus accept their outputs as the ground
truth for the first three components in xn. For our purposes,
we also draw noise sampled vnfrom a zero-mean Gaussian
distribution with R= 10−4and add to the measurement as
y′
n=yn+vnto mimic noisy measurements. In Fig. 6 we
sketch noisy measurements y′
nalong with the ground truth
yn. The reference velocity trajectories are computed using the
KF and ynfor Qn= 10−4I.
The estimation errors produced for the ground truth by the
filtering algorithms applied to y′
nare shown in Fig. 7 for a
wrongly set initial state 5x0and N= 25. No new features
can be revealed here and we arrive at similar conclusions as
for Fig. 3. Namely, the KF produces largest errors and the
DOFIR filter improves the performance significantly over the
KF and OFIR filter. The UFIR filter, which does not require
the initial values, has a similar performance as the DOFIR
filter, but with an inherent dead zone limited with Npoints.

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XX XXXX 7
100 110 120
-3
-2
-1
0
1
100 110 120
-10
-8
-6
-4
-2
0
2
4
100 110 120
-4
-3
-2
-1
0
1
100 110 120
-4
-3
-2
-1
0
1
100 110 120
-10
-8
-6
-4
-2
0
2
4
100 110 120
-4
-2
0
2
4
6
8
Time steps Time steps Time steps
Estimation errors
Estimation errors
Estimation errors
Estimation errors
Estimation errors
Estimation errors
Time steps Time steps Time steps
(a) (b) ( )c
(d) (e) (f)
DOFIR
OFIR
KF
UFIR
KF
UFIR
DOFIR
OFIR
KF
UFIR
DOFIR
OFIR
DOFIR
KF
UFIR
OFIR
UFIR
KF
DOFIR
OFIR
OFIR
KF
DOFIR
UFIR
Fig. 8. Estimation performances of filtering algorithms for a 3-DoF helicopter
system in the presence of temporary model mismatching: (a) elevation, (b)
pitch, (c) travel, (e) elevation velocity, (f) pitch velocity, and (d) travel velocity.
Robustness of the filtering algorithms against temporary
model mismatching is investigated as shown in Fig. 8 by
injecting an unspecified disturbance dn= [1,1,2,2,5,3]T,
100 6n6101, into the process and ignoring dnin the algo-
rithms. The results confirm the previously made observations
and we can conclude that the proposed DOFIR filter has a
better performance than the KF, UFIR filter, and OFIR filter.
What left behind is to measure the computational time. We
do it using Matlab R2019a software operating in a computer
with Intel Core i9 CPU (2.40 GHz) and 32.00 GB RAM.
It shows that the TOFIR filter 0.644 s and the DOFIR filter
0.484 s consume the largest computation time, compared with
the OFIR filter 0.057 s, the UFIR filter 0.041 s, and the KF
0.004 s. The consumed-time difference between the TOFIR
and DOFIR filters is caused by three passes in the TOFIR
filter and supporting batch UFIR operation in the DOFIR filter.
Note that the above-measured values are typically acceptable
for object tracking under the industrial conditions and can be
further reduced in practical implementations.
VI. CONCLUSIONS
The multi-pass OFIR filtering approach developed in this
paper in the TOFIR and DOFIR structures has demonstrated
a better performance than the KF, OFIR filter, and UFIR
filter under unknown initial values and when the process
temporary undergoes unspecified changes that is typical for
many industrial applications. The effect was achieved by
deriving the batch F-OFIR and B-OFIR filters and finding
their recursive forms. Several examples employing simulated
and experimental data have confirmed the higher robustness of
the TOFIR and DOFIR algorithms, which have also demon-
strated acceptable computational complexities. Because the
improved accuracy is obtained by processing a larger number
of measurements, we now consider optimizing the multi-pass
OFIR filtering structure to reduce the computation time, while
retaining the achieved robust performance.
APPENDIX A
RECURSIVE FORMS FOR F-OFIR FILTER
Consider the first term H1h
m,n =Fm+1
nχmHT
m,n +Z1in
(17) and decompose matrices as Fm+1
n=FnFm+1
n−1,
¯
Dm,n = [Fn¯
Dm,n−1Bn],
Gm,n ="Gm,n−10
HnFn¯
Dm,n−1HnBn#,
take into account that Qm,n = diag(Qm,n−1Qk)and
Fn¯
Dm,n−1Qm,n−1¯
DT
m,n−1FT
n+BnQnBT
n=¯
Dm,nQm,n ¯
DT
m,n ,
and represent H1h
m,n recursively for n>m+ 1 as
H1h
m,n = [FnH1h
m,n−1MnHT
n],(A.1)
where Mn=Fm+1
nχmFm+1T
n+¯
Dm,nQm,n ¯
DT
m,n.
To derive a recursive form for the second term H2h
m,n =
(Zχ+Z2+Rm,n)−1in (17), transform Zχ+Z2as
Zχ+Z2="H2h−1
m,n−1− Rm,n−1˜
H2hT
m,n−1
˜
H2h
m,n−1HnMnHT
n#,
where ˜
H2h
m,n−1=HnFnH1h
m,n−1, refer to Rm,n =
diag (Rm,n−1Rn), and represent Zχ+Z2+Rm,n as
Zχ+Z2+Rm,n ="H2h−1
m,n−1˜
H2hT
m,n−1
˜
H2h
m,n−1HnMnHT
n+Rn#,
separate it into ¯
Zm,n = diag (H2h−1
m,n−1Rn)and
˜
Zm,n ="0˜
H2hT
m,n−1
˜
H2h
m,n−1HnMnHT
n#,
and decompose H2h
m,n as
H2h
m,n =¯
Z−1
m,n ˆ
Z−1
m,n ,(A.2)
where ˆ
Zm,n =I+˜
Zm,n ¯
Z−1
m,n. By the Schur complement [31],
[32], represent ˆ
Z−1
m,n as
ˆ
Z−1
m,n ="Z11 Z12
Z21 Z22 #,(A.3)
where Z11 =I+¯
Ωn,Z12 =−˜
H2hT
m,n−1R−1
nΩ−1
n,
Z21 =−Ω−1
n˜
H2h
m,n−1H2h
m,n−1,Z22 = Ω−1
n,¯
Ωn=
˜
H2hT
m,n−1R−1
nΩ−1
n˜
H2h
m,n−1H2h
m,n−1, and
Ωn=I+HnΛ−
nHT
nR−1
n,(A.4)
Λ−
n=Mn−FnHh
m,n−1H1hT
m,n−1FT
n.(A.5)
Transform the product of (A.1) and (A.2) and provide a
recursion for Hh
m,n as
Hh
m,n = [(Fn−KnHnFn)Hh
m,n−1Kn],(A.6)
where
Kn= Λ−
nHT
n(HnΛ−
nHT
n+Rn)−1.(A.7)
Substitute (A.6) into (25) and arrive at the recursive estimate
ˆxh
n=Fnˆxh
n−1+Kn(yn−HnFnˆxh
n−1).(A.8)

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XX XXXX 8
Next, consider ˆxf
ndefined by (25), refer to ¯
Sm,n =
[Fn¯
Sm,n−1En]and
Lm,n ="Lm,n−10
HnFn¯
Sm,n−1HnEn#,
and come up with another recursion
ˆxf
n= (I−KnHn)Fnˆxf
n−1+ (I−KnHn)Enun.(A.9)
Combining (A.8) and (A.9) gives the F-OFIR estimate
ˆxn= ˆx−
n+Kn(yn−Hnˆx−
n),(A.10)
where ˆx−
n=Fnˆxn−1+Enun, and similar manipulations with
Λ−
ngiven by (A.5) allows writing it recursively with
Λ−
n=FnΛn−1FT
n+BnQnBT
n,(A.11)
Λn= (I−KnHn)Λ−
n.(A.12)
Finally, rename Λ−
n=P−
nand Λn=Pn, substitute nwith
i, and arrive at the recursive forms used in Algorithm 1.
APPENDIX B
RECURSIVE FORMS FOR B-OFIR FILTER
Consider (22), introduce Hb1h
n,m =Xm+1
nχnHbT
n,m +
¯
Db
n,mQn,m GbT
n,m and Hb2h
n,m = (Hb
n,mχnHbT
n,m +
Gb
n,mQn,m GbT
n,m +Rn,m)−1, and rewrite the B-OFIR
filter gain Hbh
n,m as
Hbh
n,m =Hb1h
n,mHb2h
n,m .(B.1)
Following the derivations of (A.1) and using Xm+1
n=
F−1
m+1Xm+2
n, represent Hb1h
n,m recursively as
Hb1h
n,m = [ F−1
m+1Hb1h
n,m+1 Mb
mHT
m],(B.2)
where Mb
m=Xm+1
nχnXm+1T
n+¯
Db
n,mQn,m ¯
DbT
n,m.
Observe that Hb2h
n,m with H2h
m,n, have similar structures
and replace the forward-in-time matrices with the backward-
in-time matrices. Next, refer to (A.6) and represent Hbh
n,m
recursively as
Hbh
n,m =(I−Kb
mHm)F−1
m+1Hbh
n,m+1 Kb
m,(B.3)
where Kb
m= Λb−
mHT
m(HmΛb−
mHT
m+Rm)−1and Λb−
m=
Mb
m−F−1
m+1Hbh
n,m+1HbhT
n,m+1F−T
m+1.
Note that (B.3) and (A.6) have similar structures and thus
the remaining batch forms can be represented recursively
by substituting all forward-in-time matrices in (A.8)–(A.12)
with the backward-in-time matrices. Finally, substitute Λb−
m
with P−
mand Kb
mwith Km, introduce an iterative variable i
descending from n−1to m, and end up with the iterative
B-OFIR filtering Algorithm 2.
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Shunyi Zhao (M’13, SM’20) received the Ph.D.
degree in control theory and application from the
Key Laboratory of Advanced Process Control for
Light Industry (Ministry of Education), Institute of
Automation, Jiangnan University, Wuxi, China, in
2015. From 2013 to 2014, he has been a Visiting
Student and, from 2015 to 2018, a Postdoctoral
Fellow in the Department of Chemical and Materials
Engineering, University of Alberta, Edmonton, AB,
Canada. In 2015, he joined the Jiangnan University
as an Associate Professor, where he is currently a
Professor. Dr. Zhao is the recipient of Alexander von Humboldt Research
Fellowship in Germany, the excellent Ph.D. thesis award (2016) in Jiangsu
Province, China, and a nomination of excellent doctoral thesis from Chinese
Association of Automation (CAA) in 2016. His research interests include
statistical signal processing, Bayesian estimation theory, and fault detection
and diagnosis.
Yuriy S. Shmaliy (M’96, SM’00, F’11) received
the B.S., M.S., and Ph.D. degrees in 1974, 1976
and 1982, respectively, from the Kharkiv Aviation
Institute, Ukraine, all in Electrical Engineering. He
serves as Full Professor beginning in 1986. In 1992
he received the Dr.Sc. degree in Electrical Engi-
neering from the Kharkiv Railroad Institute. Since
1985 to 1999, he had been with the Kharkiv Mili-
tary University. In 1992, he founded the Scientific
Center “Sichron” and served as a director by 2002.
Since 1999, he has been with the Universidad de
Guanajuato of Mexico, where, from 2012 to 2015, headed the Department of
Electronics Engineering.
Dr. Shmaliy has 469 Journal and Conference papers and 81 patents. He has
authored the books Continuous-Time Signals (Springer, 2006), Continuous-
Time Systems (Springer, 2007), and GPS-Based Optimal FIR Filtering of
Clock Models (New York: Nova Science Publ., 2009). He also edited the book
Probability: Interpretation, Theory and Applications (New York: Nova Science
Publ., 2012). He was rewarded a title, Honorary Radio Engineer of the USSR,
in 1991. He was listed in Outstanding People of the 20th Century, Cambridge,
U.K., in 1999. He has received the Royal Academy of Engineering Newton
Research Collaboration Program Award in 2015. He was invited many times
to give tutorial, seminar, and plenary lectures. His current interests include
optimal and robust state estimation.
Jose A. Andrade-Lucio (S’93, M’95) was born
on December 15th, 1969 in Chiapas, Mexico. He
received the B.S. degree in 1993 in Communica-
tions and Electrical Engineering and M.E. degree
in Electrical Engineering in 1995, both from the
Universidad de Guanajuato, Mexico. He received
the Ph.D. degree in optoelectronics sciences in 1999
from the National Institute for Astrophysics Optics
and Electronics, Puebla, Mexico. Since 1999 he
has been an associate professor and since 2001
a titular professor in the Department of Electrical
Engineering of Universidad de Guanajuato. His research interests are focused
on optical signal processing, non linear optics, and photonics. He is a Member
of the Mexican Academy of Sciences.
Fei Liu received the B.S. degree in Electrical Tech-
nology from Wuxi Institute of Light Industry, China,
in 1987; the M.S. degree in Industrial Automation
from Wuxi Institute of Light Industry, China, in
1990; and the Ph.D. degree in Control Science
and Control Engineering from Zhejiang University,
China, in 2002. From 1990 to 1999, he has been
an Assistant, Lecturer, and Associate Professor in
Wuxi Institute of Light Industry. Since 2003, he
has been a Professor of the Institute of Automation,
Jiangnan University. From 2005 to 2006, he was
a Visiting Professor with the University of Manchester, UK. His research
interests include advanced control theory and applications, batch process
control engineering, statistical monitoring and diagnosis in industrial process,
and intelligent technique with emphasis on fuzzy and neural systems.