I/O SNR and Noise Covariances Norm Ratio Relation in Kalman Filter
NICHOLAS ASSIMAKIS1, MARIA ADAM2,
1Department of Digital Industry Technologies,
National and Kapodistrian University of Athens,
34400 Psachna Evias,
GREECE
2Department of Computer Science and Biomedical Informatics,
University of Thessaly,
2-4 Papasiopoulou Str., 35131, Lamia,
GREECE
Abstract: - Kalman filters are used with great success to solve filtering problems in many fields of science and
engineering. The ignorance of state noise covariance or the measurement noise covariance often creates
difficulties in the practical application of Kalman filters. In this paper, the relation between the Input/Output
signal-to-noise ratio (I/O SNR) and the noise covariance norm ratio for the discrete-time steady-state Kalman
filter is established. The state or measurement noise covariance can be tuned via the I/O SNR. This result can
be applied in time-varying systems and in steady-state systems, without the a priori knowledge of the state or
measurement noise covariance.
Key-Words: - Kalman filter, Discrete-time, Steady-state, Riccati equation, Lyapunov equation, State Noise
Covariance, Measurement Noise Covariance, Signal-to-Noise Ratio.
1 Introduction
State estimation uses measurements to
estimate/predict the system states. A popular
algorithm for this purpose is the Kalman filter [1],
which has been successfully used in various fields:
object detection and tracking [2], robotic
applications [3], electric load estimation [4], stock
price prediction [5], weather forecasts [6], satellite
orbit determination [7], power generation prediction
[8], cases prediction of Covid-19 [9], multi-
observation fusion applications related to timescale
[10], DC-Drives and sensors applications [11],
estimation with unlimited sensing measurements
[12], applications where the measurement noise is
correlated with the state noise [13], multi-target
localization [14].
The discrete-time Kalman filter is associated
with discrete-time state space systems, which
describe the relation between the state vector
and the measurement vector , at
time . In the time-invariant case, all the Kalman
filter parameters are constant real matrices: is the
transition matrix, is the output matrix, is the
state noise covariance matrix and is the
measurement noise covariance matrix.
Kalman filter computes the mean and
covariance of estimation, as well as the
mean and covariance of
prediction and the Kalman filter gain .
Time-invariant Kalman filter takes the form of
steady-state Kalman filter, when well-defined
conditions [15] are satisfied. In the steady-state
case, the estimation error covariance matrix, the
prediction error covariance matrix, and the Kalman
filter gain remain constant. The steady-state
prediction error is the unique solution of the
associated Riccati equation:
(1)
Note that the existence of the inverse in the
Riccati equation is guaranteed when the
measurement noise covariance matrix is positive
definite (which means that no measurement is
exact).
Because of the importance of the Riccati
equation, there is considerable literature on its
solution, [15], [16], [17], [18], [19], [20].
5HFHLYHG0D\5HYLVHG2FWREHU$FFHSWHG1RYHPEHU3XEOLVKHG'HFHPEHU
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Nicholas Assimakis, Maria Adam
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In the infinite measurement noise case, the
Riccati equation takes the form of the Lyapunov
equation:
(2)
Due to the importance of the Lyapunov
equation, there exists considerable literature on its
solution, [16], [20].
Using the matrix inversion lemma, the Riccati
equation is written as:
(3)
The nonsingularity of and (which then are
positive definite matrices) ensures the
nonsingularity of .
The steady-state estimation error covariance
matrix is:
(4)
Note that the steady-state prediction error
covariance matrix and the steady-state estimation
error covariance matrix are real square symmetric
positive definite matrices.
The steady-state Kalman filter gain is:
(5)
The steady-state Kalman filter produces the
state estimation using the previous state estimation
and the actual measurement:
(6)
2 Noise Covariance Matrices
Kalman filter assumes the knowledge of all Kalman
filter parameters, i.e. the matrices are
known. In the case where the noise covariances are
unknown, the identification of the noise covariances
of the Kalman filter is discussed in [21]. Kalman
filter statistics ( ) tuning is discussed in [22]. In
fact, can be estimated by computing the
covariance of measurements, but cannot be easily
estimated, due to the fact that a) the state is not
measured directly and b) the state noise covariance
functions as a “waste basket” for unknown
modeling errors. As explained in [23], if we choose
a too small , then the Kalman filter will converge
too slowly, while if we choose a too large , then
will become large, and the filter becomes over-
sensitive, [23]. The idea proposed in [23] is to make
so large that it just about matches the effects of
the measurement noise covariance
(7)
where is a scalar positive tuning factor.
Then, an acceptable choice for is:
(8)
where denotes the Moore-Penrose
pseudoinverse of .
Note that it is required that the output matrix
is full rank [23]; if not, then it the can be replaced
by a proper output matrix by using the observability
matrix [23].
Thus, the desired is derived:
(9)
The noise covariances norm ratio is the ratio
defined by the state noise covariance norm divided
by the measurement noise covariance norm:
(10)
where the subscript indicates the Frobenius norm.
In [23] it is depicted that if the ratio is known,
then the tuning factor is derived:
(11)
(11)
where is the largest singular value of [23].
Then, obviously, the desired can be derived
via the ratio :
(12)
where
(13)
3 SNR Definitions
The input signal-to-noise ratio (input SNR) and
output signal-to-noise ratio (output SNR) are
defined in [16]:
The input signal-to-noise ratio (input SNR) is:
(14)
The output signal-to-noise ratio (output SNR) is:
(15)
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where the following signal-to-noise ratio
improvement property holds:
(16)
The I/O SNR is the ratio defined by the output
SNR divided by the input SNR:
(17)
Obviously,
(18)
4 Relation between SNR and Noise
Covariances
In the following, the relation between the I/O SNR
and the noise covariances norm ratio is
established.
From (14) and (15) we get:
From (4) and (7) we get:
Then, we have:
and using (17) we get:
(19)
or
(20)
Finally, using (11) we get the relation between and
:
(21)
or
(22)
5 Noise Covariances Estimation
5.1 State Noise Covariance Estimation
When the state noise covariance is unknown, we
are able to estimate it via the I/O SNR. In fact, we
rewrite (12) and (13) using (22). Then, the desired
can be derived via the ratio :
(23)
where
(24)
5.2 Measurement Noise Covariance
Estimation
When the state noise covariance is unknown, we
are able to estimate it via the I/O SNR. In fact, we
rewrite (23) as:
(25)
The solution of this Lyapunov equation depends
on the known parameters . It is worth to note
that a proper selection of is prerequisite for the
existence of the unique solution of this Lyapunov
equation.
Then, the desired can be derived via the ratio by
rewriting (7) as:
(26)
6 Application in Steady-State
Kalman Filter
In both the above cases where the state or the
measurement noise covariance is unknown, the
steady-state Kalman filter gain (5) becomes [23]:
Then, using (26) we get:
(27)
Hence, the steady-state Kalman filter becomes:
Steady-State Kalman Filter
It is obvious that the steady-state Kalman filter
parameters depend on .
The resulting steady-state Kalman filter is
suboptimal [23], but it can be implemented without
the a priori knowledge of the state or measurement
noise covariance.
Note that in the time-invariant case, all the
Kalman filter parameters constant. In the steady-
state case, the Kalman filter gain in (27) is constant
as well. The estimation error covariance and the
prediction error covariance are also constant. The
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DOI: 10.37394/23203.2024.19.40
Nicholas Assimakis, Maria Adam
E-ISSN: 2224-2856
375
Volume 19, 2024
state estimation is derived by the steady-state
Kalman filter equation. The state prediction is:
(28)
7 Application in Time-Varying
Kalman Filter
Consider the time-varying case, where all the
Kalman filter parameters are time-varying, i.e.
.
Then we use the time-varying noise covariances
norm ratio :
(29)
Following the ideas in [23], we define the time-
varying scalar positive tuning factor :
(30)
with
(31)
Then, the Kalman filter gain becomes:
(32)
since
The estimation and the estimation error
covariance matrix are:
(33)
(34)
The prediction and the prediction error
covariance matrix are:
(35)
(36)
or
(37)
Finally, we define the time-varying factor :
(38)
Of course
(39)
Then, from (32) and (39) we get:
(40)
Thus, we are able to use the time-varying noise
covariances norm ratio in order to derive the
time-varying Kalman filter:
Time-Varying Kalman Filter
8 Conclusions
Kalman filters are successfully used for solving
filtering problems in many different areas of science
and engineering. Especially in the field of electrical
engineering and electric controls, Kalman filters are
an integral part of many states of the art of electric
controls.
However, the practical implementation of
Kalman filters often presents difficulties due to the
ignorance of noise covariances. In this paper, the
relation between the I/O SNR and the noise
covariances norm ratio for the discrete-time steady-
state Kalman filter has been determined and it is
shown that when the state or measurement noise
covariance is unknown, it can be tuned via the I/O
SNR.
This result can be applied in time-varying
systems and in steady-state systems, without the a
priori knowledge of the state or measurement noise
covariance. The impact of this result on Kalman
filtering, combined with AI techniques to estimate
the noise covariances, can be to derive reliable
estimates, in the absence of noise covariances.
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Future work includes solving real-world
electrical engineering and electronic problems using
the proposed approach and investigating the
extension of the proposed method in nonlinear
prediction and estimation applications.
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