Robust Recursive Least-Squares Wiener Filter for Linear Continuous-
Time Uncertain Stochastic Systems
Abstract: - For linear continuous-time systems with uncertainties in the system and observation matrices, an
original robust RLS Wiener filter is designed in this study. The robust RLS Wiener filter does not assume
norm-bounded uncertainty for the system and observation matrices, in contrast to the robust Kalman filter. In
the design of the robust RLS Wiener filter, the degraded signal, affected by the uncertainties in the system and
observation matrices, is modeled by an autoregressive (AR) model. The system and observation matrices for
the degraded signal are formulated from the relationship between the AR model of the degraded signal and the
state-space model. Estimation formulas for the system and observation matrices are proposed in Section 2. The
robust filtering problem is introduced based on the minimization of the mean-square value of the filtering errors
for the system states. The robust filtering estimate is given as an integral transformation of the degraded
observations using the impulse response function. The integral equation that an optimal impulse response
function satisfies is given in Section 3. Theorem 1 presents a robust RLS Wiener filtering algorithm starting
from this integral equation. The proposed robust RLS Wiener filter outperforms the existing robust Kalman
filter regarding estimate accuracy, as shown by a numerical simulation example.
Key-Words: - Robust RLS Wiener filter, degraded observations, stochastic systems with uncertainties,
autoregressive model, continuous-time stochastic systems.
Received: June 14, 2022. Revised: August 17, 2023. Accepted: September 19, 2023. Published: October 4, 2023.
1 Introduction
For both continuous-time and discrete-time
uncertain stochastic systems, robust filters have
been studied during the past few decades, e.g., [1]-
[8]. For linear continuous-time stochastic systems
with norm-bounded uncertainties in the system and
observation matrices, robust Kalman filters [1], [2]
are developed. In [3], norm-bounded uncertainties
are assumed in the four matrices, including the input
and observation noise matrices in the continuous-
time state-space model. In [4], the filtering,
prediction, and smoothing problems are considered
for the system with uncertain matrices and known
input. The discrete-time robust Kalman filter is
investigated in [5] and [6]. In [7], robust Kalman
filters are described for continuous and discrete-time
stochastic systems with uncertainties. For linear
discrete-time stochastic systems with uncertainties,
the robust recursive least-squares (RLS) Wiener
filter is proposed [8]. A specific characteristic is that
the degraded signal affected by uncertain parameters
is expressed in terms of an autoregressive (AR)
model of finite order. Unlike the robust Kalman
filter, the robust RLS Wiener filter does not use
knowledge of norm-bounded uncertainties.
This paper designs a novel robust RLS Wiener
filter for linear continuous-time stochastic systems
with uncertainties in the system and observation
matrices. This paper does not assume norm-bounded
uncertainties for the system and observation
matrices. The AR model [9] and the autoregressive
moving average (ARMA) model [10] have been
investigated in conjunction with modeling for
continuous-time stochastic processes. In the design
of the robust RLS Wiener filter, the degraded signal,
influenced by the uncertainties in the system and
observation matrices, is modeled by an AR model.
The system and observation matrices for the
degraded signal are formulated from the relationship
between the AR model of the degraded signal and
the state-space model, as shown in Section 2.
The vehicle tracking problem with model
uncertainty is an example where the proposed robust
RLS Wiener filter can be applied similarly to the
robust Kalman filter. Also, the H-infinity tracking
control algorithm in [11] is designed for linear,
discrete-time stochastic systems with uncertain
parameters. It includes a practical example of
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Seiichi Nakamori
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SEIICHI NAKAMORI
Professor Emeritus, Faculty of Education
Kagoshima University
1-20-6, Korimoto, Kagoshima 890-0065
JAPAN
tracking control for an F-16 aircraft. In [11], the
robust RLS Wiener filter of [12] and [13] estimates
signal and state based on the separation principle
between control and estimation. The H-infinity
tracking control algorithm in [14] is designed for
linear, continuous-time deterministic systems.
The estimates of the system and observation
matrices are formulated in Section 2. Section 3
introduces a robust filtering problem. In Section 4,
Theorem 1 presents the robust RLS Wiener filtering
algorithm. Section 5 demonstrates a numerical
simulation example of the robust RLS Wiener filter
in comparison with the robust Kalman filter [1].
2 Nominal and Degraded State-Space
Models and Degraded System
Realization
Let (1) be a discrete-time state-space model of the
linear stochastic system.
(1)
Here, is the state vector, and
is the signal vector. The input noise and
the observation noise are mutually uncorrelated
white Gaussian noise of mean zero. is the
input matrix, and is the observation
matrix. The auto-covariance functions for the input
noise and the observation noise are given
in (1). This paper considers the state and
observation equations with uncertain parameters in
(2).
(2)
In (2), the degraded system matrix and the
degraded observation matrix are introduced
instead of the system matrix and the observation
matrix in (1), respectively. Here, the matrix
elements of and contain uncertain
variables. The initial system state is a random
vector that is uncorrelated with both system and
measurement noise processes.
Assume that the degraded signal is expressed by
with a state vector having
components.
.
(3)
Let satisfy a differential equation
(4)
(4) is transformed into the state differential
equations.
(5)
in (4) is the residual in approximating the
degraded signal . It is recommended that the
order of the differential equation of (4) is and the
variance of the random residual is set to zero
from the viewpoint of least mean squares estimation
for the system matrix . A numerical simulation
example will verify these two suggestions. For
,
is valid. Hence, satisfies
and is estimated
by the relationship of (6).
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,
,
, ,
=
,
(6)
Also,
is estimated by
(7)
3 Robust Filtering Problem
Let the 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 filtering error . The filtering
estimate to minimize the cost function
satisfies the relationship
, ,
(10)
from the orthogonal projection lemma [16].
Therefore, we get an integral equation
(11)
Substituting the degraded observation equation in
(2) into (11), (11) is transformed into
.
(12)
Starting with (12), the robust RLS Wiener filtering
algorithm is derived. Assume that the cross-
covariance function of with is
expressed as
(13)
Let the covariance function of be
expressed by
(14)
Apart from the current approach, the Kalman
filter is utilized for state estimation with accurate
information on the state-space model. When dealing
with systems that have uncertain parameters, the
Kalman filter with artificial intelligence (AI) based
on neural networks (NN) can be classified into four
groups [17].
4 Robust RLS Wiener Filtering
Algorithm
Theorem 1 presents the robust RLS Wiener filtering
algorithm for .
Theorem 1 For the nominal system (1), the robust
RLS Wiener filtering algorithm with the degraded
observed value in (2) consists of (15)-(20).
(15)
: Filter gain for .
,
: Cross-variance function of
with .
(16)
: Cross-variance function of with t),
(17)
: Filtering estimate of .
(18)
: Filter gain for .
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(19)
: Variance function of t),
(20)
For the robust RLS Wiener filter of Theorem 1 to
be stable, must be a positive definite matrix:
. In addition, the asymptotic stability condition for
the robust RLS Wiener filter is that all eigenvalues
of the matrices and
have negative
real parts.
Proof of Theorem 1 is deferred to the Appendix.
4 A Numerical Simulation Example
Let the observation equation for and the state
differential equations for be given by
(21)
Let the observation equation for the degraded signal
, and the state differential equations for the
degraded state be given by
,
.
(22)
Here, is the additional uncertain matrix to the
system matrix . “” represents a scalar random
number chosen from a uniform distribution in the
interval . From (3) and (5), let the observation
equation for the degraded signal and the state
differential equations for the degraded state be
given by
.
(23)
It should be noted that the robust RLS Wiener
filtering algorithm of Theorem 1 does not require
information on the input noise variance . Provided
that from (6), is calculated by
.
(24)
The other expression for is given by
.
(25)
In (25), , , is one candidate. The
estimates of by (24) and (25) for are an
exact coincidence. (25) is based on the relationship
, [11].
For the data sampling interval , a four-
point forward-difference formula with a truncation
error of is utilized in numerical
differentiation to approximate the derivatives in
(24). For computing the expectation
as
an example,
is
approximately computed using Simpson's
rule's
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ds
ds
ds
ds
numerical integration method. The integral step size
in this case is . For the state vector with
two components, the estimate of the system matrix
is calculated by (24) as
.
(26)
For the state vector with three and four
components, respectively, the estimates of the
system matrices are calculated as (27) and (28).
(27)
(28)
The estimates of the system matrices in (27) and
(28) are unavailable, since the third row in (27) and
the fourth row in (28) display large values of orders
and , respectively. Therefore, is the
appropriate order for the estimate of . Table 1
shows the estimates of
from (7) for the white
Gaussian observation noises , ,
and . For the degraded signal
in (23), the estimate of
is very close to
Substituting , ,
, ,
and the observed value into the robust
RLS Wiener filtering algorithm of Theorem 1, the
Table 1. Estimates of
for the white Gaussian
observation noises , , and
.
White
Gaussian
observation
noise
Estimates of
filtering estimate of the state is recursively
computed. Fig. 1 illustrates the filtering estimate
of the state variable vs. for the white
Gaussian observation noise .
Fig. 1: Filtering estimate of the state variable vs. for the white Gaussian observation noise
.
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Fig. 2: Filtering estimate of the state variable vs. for the white Gaussian observation noise
.
As time advances, the filtering estimate
approaches the state gradually. Fig. 2
illustrates the filtering estimate of the state
variable vs. for the white Gaussian
observation noise . As time advances,
the filtering estimate approaches the state
gradually. Table 1 shows mean-square values
(MSVs) of the filtering errors and
for the white Gaussian observation
noises , , and . Here,
the MSVs are calculated by
and
respectively.
Table 2. Mean-square values of the filtering errors
and for the white
Gaussian observation noises , ,
and .
Table 3 shows the MSVs of the filtering errors
and for the white
Gaussian observation noises , ,
and by the robust Kalman filter [1].
Based on the robust Kalman filter [1], we employ
the following parameters:
, a scaling parameter
Table 2 and Table 3 show that the proposed
robust RLS Wiener filter is superior in estimation
accuracy to the robust Kalman filter [1].
In the simulation example, the numerical
integration computations were performed using a
fourth-order Runge-Kutta-Gill method with a
sampling interval of h=0.001.
Table 3. Mean-square values of the filtering errors
and for the white
Gaussian observation noises , ,
and [1].
4 Conclusion
This paper has proposed a novel robust RLS Wiener
filter for linear continuous-time systems with
uncertainties in the system and observation matrices.
Under the uncertainties in the system and
observation matrices, the degraded signal is fitted to
the AR model of finite order. The AR model of the
degraded signal is related to the state-space model,
White Gaussian
observation
noise
MSV of
MSV of
6.707662
3.209815
4.526449
2.479447
8.558077
3.611747
White Gaussian
observation
noise
MSV of
MSV of
0.125156
7.160740
0.128165
7.178225
0.133048
7.222379
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and the system and observation matrices for the
degraded signal are formulated. Estimation formulas
for the system and observation matrices were
suggested in Section 2. Therefore, unlike the robust
Kalman filter, the robust RLS Wiener filter of
Theorem 1 does not need to assume norm-bounded
uncertainties for the uncertain system and
observation matrices. The robust RLS Wiener filter
does not use the information of input matrix and
the input noise variance in (1). The robust
filtering problem is introduced based on the
minimization of the mean-square value of the
filtering errors for the nominal system states. The
robust filtering estimate is given as the integral
transformation of the degraded observations using
the impulse response function. The integral equation
that the optimal impulse response function satisfies
is given in Section 3. Theorem 1 presented the
robust RLS Wiener filtering algorithm starting from
this integral equation.
The numerical simulation example has shown
that the robust RLS Wiener filter has better
estimation accuracy than the robust Kalman filter.
The new design of an H-infinity tracking
controller is desirable for linear continuous
stochastic systems with uncertain parameters as a
future challenge. By combining the robust RLS
Wiener filter proposed in this paper with the new H-
infinity tracking control algorithm, tracking control
is implementable in linear continuous-time
stochastic systems with uncertain parameters.
Appendix: Proof of Theorem 1
Substituting in
(13) into (12), (12) is rewritten as
(A-1)
Let us introduce an auxiliary function , which
satisfies
(A-2)
From (A-1) and (A-2), is given by
(A-3)
Differentiating (A-2) with respect to , we have
(A-4)
Introducing
(A-5)
and using (14),
satisfies
(A-6)
Putting in (A-2), we have
(A-7)
Introducing
(A-8)
and using (14), we have
.
(A-9)
Differentiating (A-8) with respect to , we have
(A-10)
Substituting (A-6) into (A-10), we have
(A-11)
Here, we introduced the function given by
.
(A-12)
Differentiating (A-5) with respect to , we have
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(A-13)
From (A-5) and (A-13), we obtain
(A-14)
Differentiating (A-12) with respect to , we have
.
(A-15)
Substituting (A-14) into (A-15), we obtain
(A-16)
From (8) and (A-3), the filtering estimate is
written as
(A-17)
Differentiating (A-17) with respect to , using (A-3)
and (A-6) with
from (13), and
introducing
(A-18)
we have
(A-19)
From (A-3) and (A-9), is given by
(A-20)
Using (13) and introducing a function
, is written as
(A-21)
Let be given by
(A-22)
From (5) and (14), it is clear that
(A-23)
Differentiating (A-18) with respect to and using
(A-14), we obtain
(A-24)
Here, is given by
.
(A-25)
Differentiating with respect to , we have
.
(A-26)
From, (13), (14), (A-3), and (A-11), (A-26) is
rewritten as
.
(A-27)
Here,
.
(A-28)
Differentiating (A-28) with respect to , using (A-
16), and introducing
(A-29)
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we obtain
(A-30)
From (A-5), satisfies
(A-31)
From (14) and (A-12), (A-31) is rewritten as
.
(A-32)
Hence, we obtain an expression for as
(A-33)
(Q.E.D.)
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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
that are relevant to the content of this article.
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