Robust Recursive Least-Squares Finite Impulse Response Filter in
Linear Continuous-Time Stochastic Systems with Uncertainties
SEIICHI NAKAMORI
Professor Emeritus, Faculty of Education,
Kagoshima University,
1-20-6, Korimoto, Kagoshima 891-1305,
JAPAN
Abstract: - The current research designs an original robust recursive least-squares (RLS) finite impulse
response (FIR) filter for linear continuous-time systems with uncertainties in both the system and observation
matrices. These uncertainties in the state-space model generate the degraded signal and observed value. The
robust RLS FIR filter does not account for the norm-bounded uncertainties in the system and observation
matrices. This study uses an observable companion form to represent the degraded signal state-space model.
The system and observation matrices are estimated based on the author's previous computational methods. The
robust RLS FIR filtering problem aims to minimize the mean-square errors in FIR filtering for the system state.
The robust FIR filtering estimate is formulated as an integral transformation of the degraded observations using
an impulse response function. Section 3 obtains the integral equation satisfied by the optimal impulse response
function. Theorem 1 presents the robust RLS FIR filtering algorithms for the signal and the system state. This
integral equation derives the robust RLS-FIR filtering algorithms. Numerical simulation examples show the
validity of the proposed robust RLS FIR filter.
Key-Words: - Robust RLS FIR filter, degraded signal, stochastic systems with uncertainties, observable
companion form, continuous-time stochastic systems, covariance information.
Received: March 27, 2024. Revised: October 11, 2024. Accepted: November 15, 2024. Published: December 27, 2024.
1 Introduction
Kalman filter and the finite impulse response (FIR)
filter for signal and state estimation are widely used
in the application area of navigation,
communication systems, and signal processing in
stochastic systems [1], [2]. In [3], a one-step
optimal FIR predictor is designed and applied to the
robot predictive tracking problem. In [4], an
unbiased FIR filter estimates the clock state by
measuring the time interval error based on an
interval of finite most recent past points.
Researchers have studied finite impulse response
(FIR) estimation techniques in discrete-time and
continuous-time stochastic systems, [1], [2], [3], [4],
[5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15],
[16], [17], [18], [19], [20], [21], [22], [23], [24],
[25], [26], [27], [28], [29], [30], [31], [32], [33],
[34]. In the book [35], there is a thorough discussion
of FIR estimation techniques. For state-space
models with uncertainties, FIR estimators perform
better than conventional recursive estimators in
linear discrete-time stochastic systems, [5]. Below is
a classification of FIR estimation techniques.
(1) Some references to continuous-time FIR
estimators are as follows. FIR filter and FIR
smoother in linear stochastic systems, [6].
Robust FIR filter in linear stochastic systems
with bounded uncertainties, [7]. FIR filter for
input-delayed stochastic systems, [8].
Recursive least-squares (RLS) FIR filter using
covariance information in linear continuous-
time stochastic systems, [9].
(2) Receding horizon FIR filter in linear discrete-
time stochastic systems, [10].
(3) Iterative FIR filter in linear discrete-time
stochastic systems, e.g. [11], [12], [13], [14],
[15], [16], [17], [18], [19], [20], [21].
(4) Strictly passive FIR filter in linear discrete-
time stochastic systems, [22].
(5) Fixed-lag FIR smoother in linear discrete-time
stochastic systems, [23].
(6) Robust RLS Wiener FIR filter in linear
discrete-time stochastic systems, [24].
(7) FIR filter in nonlinear discrete-time stochastic
systems, [25].
(8) Confidence set-membership FIR filter in
linear discrete time-variant stochastic systems,
[26].
(9) Unified FIR filter and smoother in linear
discrete-time stochastic systems, [27].
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(10) FIR filter for systems with delays and missing
observations in linear discrete-time stochastic
systems, [28], [29], [30], [31].
(11) Systematical analysis of batch FIR filtering
algorithms, [32].
(12) Backward FIR filter in linear discrete-time
variant stochastic systems, [33].
(13) FIR smoother estimating signal at the starting
time of fixed-interval based on algebraic
calculations by the Levinson-Durbin
algorithm, [34].
The RLS Wiener filter is designed for linear
continuous-time stochastic systems with
uncertainties, as described in [36]. This paper
extends the robust RLS Wiener filter to the robust
RLS FIR filter by utilizing covariance information
in linear continuous-time stochastic systems with
uncertainties. The robust RLS FIR filter uses the
cross-covariance function of the system state with
the degraded observed value and the auto-
covariance function of the degraded state.
The organization of this paper is as follows:
Section 2 presents he estimation method for the
system and observation matrices, [36]. As explained
in [36], the observable companion form expresses
the differential equations for uncertain states. For
robust filtering problems in linear continuous-time
stochastic systems with uncertainties, this paper
uses the state-space model of the observable
companion form for the degraded signal. Section 3
introduces the least-squares FIR filtering problem.
In Section 4, Theorem 1 presents robust RLS FIR
filtering algorithms for both the signal and the
system state. Section 5 demonstrates two numerical
simulation examples for the robust RLS FIR filter.
For finite observation intervals, we compare the
estimation accuracy of the robust RLS FIR filter.
We also compare the estimation accuracy of the
robust RLS FIR filter in Theorem 1 with that of the
robust RLS filter in Theorem 1.
2 Nominal and Degraded State-Space
Models and Degraded System
Realization
Let (1) be a nominal state-space model in linear
continuous-time stochastic systems.
(1)
Here, is the state vector, while
is the signal vector. Input noise and
observation noise are independent, zero mean,
white Gaussian noises. is the input matrix,
and is the observation matrix. The auto-
covariance functions for the input noise and
the observation noise are given by (1),
respectively. Let the state and observation equations
with uncertain parameters be given by (2).
(2)
In (2),
and
denote the degraded
system matrix and the degraded observation matrix,
respectively. In (2), and are uncertain
matrices. The initial state of the system, , is a
random vector uncorrelated with both the system
input noise and the measurement noise .
Suppose that the degraded signal is represented
as
using the degraded state vector
, where assumes components.
(3)
Let satisfies a differential equation
(4)
(4) is transformed into the observable companion
form of the state differential equations:
(5)
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In (4), denotes the residual in
approximating the degraded signal . The
degraded system matrix
is estimated by (6), [36].
,
,
=
,
(6)
Also,
is estimated by
(7)
[36].
3 Robust Finite Impulse Response
Filtering Problem
Let the FIR filtering estimate of
be given by
(8)
as a linear transformation of the degraded observed
value . Here,
represents an impulse response function. Let us
consider minimizing the mean-square value:
(9)
of the FIR filtering error .
The filtering estimate to minimize the
cost function satisfies the relationship:
,
,
(10)
from the orthogonal projection lemma [37], [38],
[39], [40]. Hence, the optimal impulse response
function satisfies the Wiener-Hopf integral
equation:
(11)
Substituting the degraded observation equation in
(2) into (11), (11) is transformed into:
.
(12)
Starting from (12), the robust RLS FIR filtering
algorithms for the signal and the system state are
derived. Consider the cross-covariance function
of with , expressed as:
.
(13)
Let
be the covariance function of ,
expressed as:
,
.
(14)
The use of covariance information , ,
, and
characterizes the current robust RLS
FIR filter in Theorem 1.
4 Robust RLS Filtering Algorithms
Theorem 1 presents the robust RLS FIR filtering
algorithms for of the signal and
of the system state .
Theorem 1 Let the state-space model for the signal
be given by (1). Let the state-space model for
the degraded signal be given by (2). Let the
cross-covariance function
of the state
with the degraded observed value be given by
(13). Let the autocovariance function
of the
degraded state be given by (14). Then robust
RLS FIR filtering algorithms for the signal
and the state using the information on the
degraded observations and the
covariances consist of the following equations (15)-
(43).
FIR filtering estimate of the signal
(15)
FIR filtering estimate of the state
(16)
(17)
(18)
(19)
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(20)
(21)
(22)
(23)
T,t)(
(24)
T,t)(
(25)
T,t)(
(26)
T,t)(
(27)
T,t)(
(28)
T,t)(
(29)
T,t)(
(30)
(31)
T,t)(
(32)
(33)
(34)
Initial condition of at :
,
(35)
Initial condition of at :
,
(36)
Initial condition of at :
,
(37)
Initial condition of at :
(38)
Initial condition of at :
(39)
Initial condition of at :
(40)
Initial condition of at :
(41)
Initial condition of at :
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(42)
Initial condition of
at :
(43)
Initial condition of the differential
equation (23) for at is calculated
by (35), starting with . Initial condition
of the differential equation (24) for
at is calculated by (36), starting with
Initial condition of the
differential equation (25) for at is
calculated by (37), starting with . Initial
condition of the differential equation (26)
for at is calculated by (38),
starting with Initial condition of
the differential equation (27) for at
is calculated by (39), starting with .
Initial condition of the differential equation
(28) for at is calculated by (40),
starting with Initial condition of
the differential equation (29) for at
is calculated by (41), starting with .
Initial condition of the differential equation
(30) for at is calculated by (42),
starting with . Initial condition
of the differential equation (31) for
at
is calculated by (43), starting with
.
From (15), the robust RLS filtering estimate
of the signal is calculated as
. From (16), the robust RLS filtering
estimate of the system state is
calculated as . (32), (34), (35),
(37), (38) and (39) compute recursively.
Section 5 provides a numerical comparison of the
estimation accuracy between the robust FIR and
robust RLS filters.
See the Appendix for proving Theorem 1.
5 A Numerical Simulation Example
Example 1
Let (44) give the state-space model for the observed
value and the nominal system state.
(44)
Let the state-space model for the degraded
observed value and the degraded state be
given by:
,
.
(45)
Here, the degraded signal is observed with
additive white Gaussian noise . denotes
an uncertain matrix additional to the system matrix
. “” denotes a scalar random number that
follows a uniform distribution in the interval .
Along with the state-space model in the observable
companion form of (3) and (5), the observation
equation for the degraded signal and the state
differential equations for the degraded state is
represented by:
.
(46)
The system matrix
for the degraded state-space
model (46) is calculated by:
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,
(47)
The estimate of
by (47) for is based on
the relationship
,
[36].
To approximate the derivatives in (47) for the
data sampling interval of , a four-point
forward difference formula with a truncation error
of is employed in the numerical
differentiation. In the calculation of the expectation
in (47), for example,
is evaluated by,
Simpson's
rule
computes the numerical integration with an
integration step size of . For the
degraded state in (46), the estimate of the
system matrix
results in:
.
(48)
Table 1. Estimates of
for the white Gaussian
observation noises , , and
White Gaussian
observation noise
Estimates of
Table 1 shows the estimates of
by (7) for the
white Gaussian observation noises
, and . , and
. The estimate of
is precisely close to
, , and .
Substituting the covariance information ,
,
, and
into the robust RLS FIR
filtering algorithm of Theorem 1, the FIR filtering
estimate of the system state is
recursively computed.
is
evaluated by
Table 2
shows the estimates of
for the white
Gaussian observation noises , ,
and .
Table 2. Estimates of
for the white
Gaussian observation noises , ,
and
White
Gaussian
obser-vation
noise
Estimates of
Figure 1 illustrates the FIR filtering estimate
, , of the state variable
vs. , , for the white Gaussian observation
noise . Figure 1 shows that
converges to as increases. Figure 2
illustrates the FIR filtering estimate ,
, of the state variable vs. ,
, for the white Gaussian observation noise
. Figure 2 shows that
gradually converges to as increases.
Fig. 1: FIR filtering estimate , , of
the state variable vs. , , for the
white Gaussian observation noise
Fig. 2: FIR filtering estimate , , of
the state variable vs. , , for the
white Gaussian observation noise
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Table 3 shows the mean-square values (MSVs)
of the filtering errors and
for
the white Gaussian observation noises ,
, and . Here, the MSVs are
calculated by
and
.
Table 3. Mean-square values of the FIR filtering
errors and ,
, for the white Gaussian observation noises
, , and
Table 4. Mean-square values of the FIR filtering
errors and ,
, for the white Gaussian observation noises
, , and
Table 4 shows the MSVs of the FIR filtering
errors and
in the case of for the white
Gaussian observation noises , ,
and . Here, the MSVs are calculated
by
and
.
Table 5 shows the MSVs of the filtering errors
and
in the case of for the white
Gaussian observation noises , ,
and . Here, the MSVs are calculated
by
and
.
Table 5. Mean-square values of the filtering errors
and
, , for the white Gaussian
observation noises , , and
From Table 3 for and Table 4 for ,
the MSVs of the robust RLS FIR filtering errors
and
for are smaller than those for
in each observation noise. This result
indicates that the estimation accuracy of the robust
RLS FIR filter improves as the fixed interval
increases. The MSVs of the robust RLS FIR filter
for in Table 4 are almost identical to those of
the robust RLS filter in Table 5 for each observation
noise. The results show that as the fixed interval
increases, the robust RLS FIR filter achieves
accuracy similar to that of the robust RLS filter.
Example 2
Consider the second-order mass-spring system
driven by zero-mean white Gaussian noise
[41].
.
(49)
Let the state-space model for the degraded
observed value and the degraded state be
given by (45). Figure 3 illustrates the FIR filtering
estimate , , of the state variable
vs. , , for the white Gaussian
observation noise . Figure 1 shows that
converges to as increases.
Figure 2 illustrates the FIR filtering estimate
, , of the state variable
vs. , , for the white Gaussian observation
noise . Figure 2 shows that
gradually converges to as increases.
White Gaussian
observation noise
MSV of
MSV of
1.17634
6.33826
1.74127
7.58909
2.39987
1.05730
White Gaussian
observation noise
MSV of
MSV of
1.13525
6.19606
1.66809
7.37079
2.14188
8.91796
White Gaussian
observation noise
MSV of
MSV of
1.11533
5.79816
1.60262
6.46205
2.00979
7.37283
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Fig. 3: FIR filtering estimate , , of
the state variable vs. , , for the
white Gaussian observation noise
Fig. 4: FIR filtering estimate , , of
the state variable vs. , , for the
white Gaussian observation noise
Figure 4 illustrates the FIR filtering estimate
, , of the state variable
vs. , , for the white Gaussian observation
noise . Figure 4 shows that
gradually converges to as
increases.Table 6 shows the MSVs of the filtering
errors and
for the white
Gaussian observation noises , ,
and . Here, the MSVs are calculated
by
and
.
Table 6. Mean-square values of the FIR filtering
errors and ,
, for the white Gaussian observation noises
, , and
Table 7. Mean-square values of the FIR filtering
errors and ,
, for the white Gaussian observation noises
, , and
Table 7 shows the MSVs of the FIR filtering
errors and
in the case of for the white
Gaussian observation noises , ,
and . Here, the MSVs are calculated
by
and
. From Table 6 for and
Table 7 for , the MSVs of the robust RLS FIR
filtering errors and
for are smaller than
those for in each observation noise. This
result indicates that the estimation accuracy of the
robust RLS FIR filter improves as the fixed interval
increases.
Table 8 shows the MSVs of the filtering errors
and
in the case of for the white
Gaussian observation noises , ,
and . Here, the MSVs are calculated
by
and
.
Table 8. Mean-square values of the filtering errors
and
, , for the white Gaussian
observation noises , , and
White Gaussian
observation noise
MSV of
MSV of
1.42786
1.00599
2.29930
9.83384
3.20115
1.10423
White Gaussian
observation noise
MSV of
MSV of
1.40065
9.80377
2.23520
9.51019
3.10346
1.07269
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The MSVs of the robust RLS FIR filter for
in Table 7 are almost identical to those of the
robust RLS filter in Table 8 for each observation
noise. The results show that as the fixed interval
increases, the robust RLS FIR filter achieves
accuracy similar to that of the robust RLS filter.
6 Conclusion
In my previous research, I designed the robust RLS
Wiener filter for linear continuous-time stochastic
systems with uncertainties. This paper presents the
robust RLS FIR filter for linear continuous-time
stochastic systems with uncertain parameters in both
the system and observation matrices. One of the
main features of the robust RLS FIR filter is the use
of cross-covariance information between the system
state and the degraded observed value, as well as the
covariance function of the degraded state. For the
robust filtering problems in linear continuous-time
stochastic systems with uncertainties, this paper
uses the state-space model of the observable
companion form for the degraded signal. (6) and (7)
give the estimates of the system and observation
matrices for the uncertain state, respectively. Usages
of the covariance information , ,
, and
characterize the current robust RLS FIR filter
as described in Theorem 1.
The numerical simulation examples have
demonstrated the effectiveness of the proposed
robust RLS FIR filtering algorithms. As the variance
of the white Gaussian observation noise increases,
the estimation accuracy of the robust RLS FIR filter
decreases. The robust RLS FIR filter estimates the
signal and the system state recursively based on
observations over the finite time interval as time
passes. As the length of the finite observation
interval increases, the estimation accuracy of the
robust RLS FIR filter improves. The MSVs of the
robust RLS FIR filter for in Table 4 and
Table 7 are almost identical to those of the robust
RLS filter in Table 5 and Table 8, respectively, for
each observation noise. The results show that as the
observation interval increases, the robust RLS FIR
filter achieves accuracy similar to that of the robust
RLS filter.
The second example is simulated on the second-
order mass-spring system driven by zero-mean
white Gaussian noise. The proposed robust FIR
filter might be applicable to control problems with
control input from the viewpoint of the separation
principle between control and estimation.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed to the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
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APPENDIX
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Proof of Theorem 1
Introducing an integral equation
(A-1)
from (2), we obtain (A-2) for the optimal impulse
response function.
(A-2)
Differentiating (A-1) with respect to , we have
(A-3)
Here, satisfies
(A-4)
satisfies
(A-5)
From (A-1), satisfies
.
(A-6)
From (14), (A-6) becomes
.
(A-7)
Introducing
(A-8)
(A-7) becomes
(A-9)
From (A-1), satisfies
(A-10)
From (14), (A-10) becomes
(A-11)
Introducing
(A-12)
(A-11) becomes
(A-13)
Differentiating (A-8) with respect to , we have
(A-14)
Substituting (A-3) into (A-14) and introducing
functions
(A-15)
and
(A-16)
we obtain
(A-17)
Differentiating (A-12) with respect to , we have
(A-18)
Substituting (A-3) into (A-18) and introducing
(A-19)
and
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(A-20)
we obtain
(A-21)
Differentiating (A-4) with respect to , we have
(A-22)
From (14) and (A-5), we obtain
(A-23)
Differentiating (A-5) with respect to , we have
(A-24)
From (14), (A-4) and (A-5), we obtain
(A-25)
From (A-5), satisfies
(A-26)
Substituting (14) into (A-26) and using (A-16), (A-
26) becomes
(A-27)
From (A-5), satisfies
(A-28)
Substituting (14) into (A-28) and introducing
(A-29)
(A-28) becomes
(A-30)
Differentiating (A-16) with respect to , we have
(A-31)
Substituting (A-25) into (A-31) and introducing
(A-32)
we obtain
(A-33)
Differentiating (A-19) with respect to , we have
(A-34)
Substituting (A-22) into (A-34) and using (A-20),
we obtain
(A-35)
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Differentiating (A-20) with respect to , we have
(A-36)
From (A-25), (A-36) becomes
(A-37)
Differentiating (A-32) with respect to , we have
(A-38)
Substituting (A-23) into (A-38) and using (A-16),
we obtain
(A-39)
From (A-4), satisfies
(A-40)
From (14) and (A-32), (A-40) becomes
(A-41)
From (A-4), satisfies
(A-42)
From (14) and (A-19), (A-42) becomes
(A-43)
The FIR filtering estimate of the state
is given by (8). Introducing a function
,
(A-44)
from (A-2), (8) becomes
(A-45)
Differentiating (A-44) with respect to , we have
(A-46)
Substituting (A-3) into (A-46) and introducing
functions
(A-47)
and
(A-48)
we obtain
(A-49)
Differentiating (A-47) with respect to , we have
(A-50)
Substituting (A-23) into (A-50) and using (A-48),
we obtain
(A-51)
Differentiating (A-48) with respect to , we have
(A-52)
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Substituting (A-25) into (A-52) and using (A-47),
we obtain
(A-53)
From (A-1), satisfies
(A-54)
Differentiating (A-54) with respect to , we have
(A-55)
By putting in (A-4), we have
(A-56)
From (14), (A-55) and (A-56), we obtain
.
(A-57)
From (A-8), is given by
(A-58)
Differentiating (A-58) with respect to we have
(A-59)
Substituting (A-57) into (A-59), we obtain
(A-60)
Here, from (A-32), is given by
(A-61)
Differentiating (A-56) with respect to , we have
(A-62)
From (A-56) and (A-62), we obtain
(A-63)
Differentiating (A-61) with respect to , we have
(A-64)
Substituting (A-63) into (A-64) and using (A-61),
we obtain
,
(A-65)
By putting in (A-54), we have
(A-66)
From (14) and (A-58), we obtain
(A-67)
By putting in (A-56), we have
(A-68)
From (14) and (A-61), we obtain
(A-69)
By putting in (A-5), we have
(A-70)
Differentiating (A-70) with respect to , we have
(A-71)
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From (A-56) and (A-71), we obtain
.
(A-72)
By putting =0 in (A-16), is given by
(A-73)
Differentiating (A-73) with respect to , we have
(A-74)
Substituting (A-72) into (A-74) and using (A-61),
we obtain
, .
(A-75)
By putting =0 in (A-19), is given by
(A-76)
Differentiating (A-76) with respect to , we have
(A-77)
Substituting (A-63) into (A-77) and using (A-76),
we obtain
, .
(A-78)
By putting =0 in (A-12), is given by
(A-79)
Differentiating (A-79) with respect to , we have
(A-80)
Substituting (A-57) into (A-80) and using (A-76),
we obtain
, .
(A-81)
By putting =0 in (A-20),
is given by
(A-82)
Differentiating (A-82) with respect to , we have
(A-83)
Substituting (A-72) into (A-83) and using (A-76),
we obtain
,
.
(A-84)
By putting in (A-26), we have
(A-85)
From (14) and (A-73), we obtain
(A-86)
From (A-45), is given by
(A-87)
From (A-44), is given by
(A-88)
Differentiating (A-88) with respect to , we have
(A-89)
Substituting (A-57) into (A-89) and introducing
(A-90)
we obtain
(A-91)
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, .
Differentiating (A-90) with respect to , we have
(A-92)
Substituting (A-63) into (A-92) and using (A-90),
we obtain
.
(A-93)
By putting in (A-48), we have
(A-94)
Differentiating (A-94) with respect to , we have
(A-95)
Substituting (A-72) into (A-95) and using (A-90),
we obtain
(A-96)
(Q.E.D.)
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