A quasi-Gaussian Kalman filter
A Quasi-Gaussian Kalman Filter
Suman Chakravorty, Mrinal Kumar and Puneet Singla
Abstract— In this paper, we present a Gaussian approximation to the nonlinear filtering problem, namely the quasiGaussian Kalman filter. Starting with the recursive Bayes filter,
we invoke the Gaussian approximation to reduce the filtering
problem into an optimal Kalman recursion. We use the moment
evolution equations for stochastic dynamic equations to evaluate
the prediction terms in the Kalman recursions. We propose
two methods, one based on stochastic linearization and the
other based on a direct evaluation of the innovations terms,
to perform the measurement update in the Kalman recursion.
We test our filter on a simple two dimensional example, where
the nonlinearity of the system dynamics and the measurement
equations can be varied, and compare its performance to that
of an extended Kalman filter.
I. I NTRODUCTION
In this paper, we present a Gaussian approximation to the
exact nonlinear filtering problem, namely the quasi-Gaussian
Kalman filter. We consider a problem with continuous time
state dynamics and discrete measurement updates. We use
a Gaussian approximation to the nonlinear filtering problem
in order to reduce the problem of nonlinear filtering to one
of maintaining the estimates of the first two moments of
the state variable based on the observations, i.e., an optimal
Kalman recursion. We use the moment evolution equations
for stochastic dynamical systems in order to evaluate the
prediction terms in the nonlinear filtering recursions. Then,
we propose two methods, one based on stochastic linearization and another based on direct evaluation, to evaluate the
innovations terms. We test the proposed filter on a simple two
dimensional example, where the nonlinearity of the system
dynamics and the measurement equations can both be varied,
and compare its performance to that of an extended Kalman
filter (EKF).
Nonlinear filtering has been the subject of intensive research ever since the publication of Kalman’s celebrated
work in 1960 [1]. It is known that the solution to the general
nonlinear filtering problem results in an infinite dimensional
filter, in the sense that an infinite system of differential
equations need to be solved in real time in order to solve
the filtering problem exactly [2]–[5]. Hence, the nonlinear
filtering problem is computationally tractable only if we consider finite dimensional approximations of the true infinite
Suman Chakravorty
neering, Texas A&M
is with Faculty of Aerospace
University, College Station, TX,
EngiUSA.
schakrav@aero.tamu.edu
Mrinal Kumar is a Graduate Assistant Research in Dept. of
Aerospace Engineering at Texas A&M University, College Station, USA.
mrinal@neo.tamu.edu
Puneet Singla is a Graduate Assistant Research in Dept. of
Aerospace Engineering at Texas A&M University, College Station, USA.
puneet@neo.tamu.edu
dimensional filter [6]–[10]. However, even these methods are
difficult to implement because of the complicated nature of
the involved computations.
The Bayes filter [4] offers the optimal recursive solution to
the nonlinear filtering problem. However, the implementation
of the Bayes filter in real time is computationally intractable
because of the multi-dimensional integrals involved in the recursive equations. We note however that researchers have reported attempts at tackling this problem for low-dimensional
systems [11], [12]. It is known that the implementation of
the Bayes filter is rendered tractable in the case when all the
involved random variables are assumed to remain Gaussian
[13], [14], the obtained filter reducing to the Kalman filter in
the case of linear dynamics and measurements. The primary
concern of this class of methods is to predict the mean
and covariance of the state variable between measurement
updates, while evaluating the covariance of the measurement
and the cross-covariance between the state and the measurement after an update, in order to evaluate the prediction
and measurement update terms in a Kalman-type recursion.
The extended Kalman Filter (EKF) [13], [14] linearizes the
system dynamics about the current estimate of the state and
the measurement equations about the current predicted estimate of the state variable, in order to evaluate the prediction
and measurement update terms. The EKF by far has been
the most popular nonlinear filter in practical applications in
past several decades. In recent years, there has been renewed
interest in the topic of nonlinear filtering with the discovery
of the “sigma-point” Kalman filters [15]–[18], [20]. These
nonlinear filters consider the nonlinear filtering problem in
a discrete time setting. The different terms of interest in the
optimal Kalman recursions are obtained using a semi-global
technique known as stochastic or statistical linearization [19].
The method essentially consists of discretizing the domain
of the random variable into a set of weighted sigma points
and transforming these through the nonlinear map in order to
obtain the distribution characteristics of the transformed random variable. The various sigma point algorithms differ from
each other in their choice of the sigma points, for instance the
unscented Kalman filter (UKF) [15], the central difference
Kalman filter (CDKF) [21], the square root form UKF and
CDKF [20] and so on. In our approach to the nonlinear
filtering problem, we use a Gaussian approximation on the
random variables of interest. However, we use a continuous
time model for the system dynamics while assuming that the
measurements are made in discrete time. This allows us to
use the elegant formalism of the Fokker-Planck-Kolmogorov
equations [?], [22] to evaluate the prediction terms in the
nonlinear filtering recursions. We use two methods: one
based on stochastic linearization and the other based on
direct evaluation of the measurement update terms, in order
to evaluate the innovations terms in the optimal recursions,
i.e., to linearize the measurement equations. Our method
does not require the choice of a set of sigma points and
thus, is free from this design aspect. However, central to the
implementation of our filter is the evaluation of integrals of
the form of a product of a function and a Gaussian function.
In the case of polynomial nonlinearities, these are the higher
order moments of a Gaussian random variable. The rest of
the paper is organized as follows.
In section 2, we introduce the Bayes filter. In section 3,
we discuss the topic of Gaussian approximations in nonlinear
filtering problems and present the optimal Kalman recursions. In section 4, we present our quasi-Gaussian Kalman
filter and the methods we use to evaluate the optimal terms
in the Kalman recursions for the approximate nonlinear
filtering problem. In section 5, we present a two dimensional
example, with variable nonlinearity in both system dynamics
and measurement, and compare the performance of our filter
with that of an EKF. In section 6, we present our conclusions
and directions for further research.
II. BAYES F ILTER
In this section, we introduce the recursive Bayes filter.
Consider a system whose dynamics are governed by the
Stochastic Ito differential equation,
dx = f (x)dt + g(x)dW,
(1)
where W represents a standard Wiener process. The system
state is measured in a discrete fashion according to the
equation,
yk = h(xk ) + vk ,
(2)
where vk is the measurement noise at instant k. Let F k =
{y0 , · · · , yk } denote the history of the observations till the
time instant k. Then, the Bayes filter recursively updates
the probability density of the state variable according to the
following equations:
Z
k−1
p(xk /F
) = P (xk /xk−1 )p(xk−1 /F k−1 )dxk−1 , (3)
p(xk /F k ) = ηp(yk /xk )p(xk /F k−1 )
(4)
where η is a normalization factor, p(xk /F k ) and
p(xk /F k−1 ) represent the probability density of the system
state at the k th time instant, based on observations till
the time instants k and (k − 1) respectively, P (xk /xk−1 )
represents the transition probability from state xk−1 to state
xk corresponding to equation (1), and p(yk /xk ) represents
the probabilistic observation model corresponding to the
observation equation (2). The prediction step in the Bayes
filter, i.e. Eq.(3) makes use of the Chapman-Kolmogorov
equations [22] for propagation in between measurements,
while the update step (Eq. (4)) utilizes the Bayes rule. Note
that in general, evaluating the multi-dimensional integral in
the above equations is a formidable challenge. The predicted
probability density function, p(xk /F k−1 ), can also be ob-
tained by solving the Fokker-Planck-Kolmogorov equation
[22] corresponding to Eq.1, given by:
X ∂p
1 X X ∂ 2 (gQg t )ij p
∂p
fi
+
,
(5)
=−
∂t
∂xi
2 i j
∂xi ∂xj
i
where Q represents the intensity of the Wiener process W .
Now let
X
1 X X ∂ 2 (gQg t )ij
∂
+
,
(6)
LF P ≡ −
fi
∂xi
2 i j
∂xi ∂xj
i
where LF P denotes the differential operator corresponding to
the Fokker-Planck-Kolmogorov equation. Thus, the integral
in the Bayes filter corresponding to the prediction equation
(Eq.3) can be evaluated by solving the above partial differential equation. The prediction step in the Bayes filter can
be solved by either evaluating the integral in the prediction
equation through a Monte-Carlo type method (the subject
matter of particle filters [23]) or by solving the above partial
differential equation. However, solving the Fokker Planck
equation is a tough proposition.
III. G AUSSIAN A PPROXIMATIONS IN N ONLINEAR
F ILTERING
One method for obtaining a finite dimensional approximation to the infinite dimensional filter corresponding to
a general nonlinear filtering problem is to approximate the
state, the noise and the observation terms as Gaussian random
variables. The Bayesian recursion shown above can then
be greatly simplified, since the posterior distribution of the
state always remains Gaussian under these approximations.
Thus, estimates of only the conditional mean, E(xt /F t ),
and covariance, Px need to be maintained. It can be shown
that under the Gaussian approximation, we obtain following
recursive filter [4], [13]:
x
¯−
k
−
x
¯k = x
¯−
k + Lk (yk − yk ),
Pxk = Px−k − Lk Pyk Ltk ,
(7)
(8)
Lk = Pxk yk Py−1
,
k
(9)
where
and x
¯k represent the predicted and updated values
of the mean of the random vector xk , Px−k , and Pxk represent
the predicted and the updated values of the covariance matrix
of xk , Pyk is the covariance matrix of the observation yk and
Pxk yk is the cross-covariance matrix between the state and
observation. The key to implementing the above recursion is
−
to evaluate the quantities x
¯−
k , Pxk , Pyk and Pxk yk . Different
filters use different ways to evaluate these quantities.
In the EKF, the predicted mean and covariance of the
variable xk are evaluated by linearizing the system dynamics
about the previous estimate xk−1 and using linear Gaussian
analysis to propagate the mean and the covariance, while
the quantities Pyk and Pxk yk are evaluated by linearizing
the measurement equation about the current predicted value
of xk and using linear Gaussian analysis.
In recent years there has been a lot of interest in sigma
point Kalman filters which use statistical linearization methods to find the quantities of interest in the recursive formulas
[17], [20]. The sigma point filters perform linearization of
the system dynamics and the measurement equations in a
statistical sense as opposed to the Jacobean linearization
which is performed in the case of the EKF.
In our approach, we adopt a different approach to evaluate the prediction and innovation terms in the filtering
recursions. We first use the moment evolution equations for
the stochastic dynamical system along with the Gaussian
approximation, and then perform a stochastic linearization of
the measurement equation in order to evaluate the innovation
terms. The details of the method are presented in the next
section.
IV. A Q UASI -G AUSSIAN K ALMAN F ILTER
In this section, we present our approach to nonlinear
filtering based on Gaussian approximations of the prediction
and innovation terms. As mentioned previously, we use the
moment evolution equations for the prediction step. Then
we propose two methods to evaluate the innovation terms:
(a) stochastic linearization to linearize the measurement
equations and (b) a direct method to evaluate the innovations
terms.
A. Prediction
Recall the nonlinear filtering equations (7)-(9), and note
that:
∂p
= LF P (p),
(10)
∂t
where LF P is the spatial operator of the Fokker Planck
equation. We shall now find the time evolution equations
for the first and second moments of the state variable, given
the prior probability distribution. Let the prior probability
distribution at the k th instant be given by N (¯
xk , Pxk ) and
the mean and the correlation of the state between the time
instants be x
¯ and Rx respectively. By definition, we have:
Z
Z
x
¯ = xp(x)dx, Rx = xxt p(x)dx,
(11)
Then, it follows that the first and second moments of x evolve
according to the following differential equations:
Z
Z
Z
d
∂p
˙x
¯=
xp(x)dx = x dx = xLF P (p(x))dx, (12)
dt
∂t
Z
Z
d
R˙x =
xxt p(x)dx = xxt LF P (p(x))dx, (13)
dt
with the initial conditions x
¯k and Rxk respectively. The
above equations are integrated between the k th and (k +1)th
time instants. In absence of the Gaussian approximation,
the evaluation of the right hand side of the above equation
involves higher moments of the random variable x because of
the Fokker Planck operator. Due to the complicated nature of
this operator, the above equations are difficult to handle. We
present below an alternate method for deriving the moment
evolution equations, and also the evolution equations of the
correlation of the state x and its mean. Consider again
the system equation (1) and an infinitesimal change in the
correlation of the state x, dE(xxt ), given by:
dE[xxt ] = E[(x + dx)(x + dx)t ] − E[xxt ].
(14)
Let dx = y, and recall the following definitions:
Z
E[(x + y)(x + y)t ] = (x + y)(x + y)t p(x, y)dxdy, (15)
t
E[xx ]
Z
=
Z
=
t
xx p(x)dx =
Z
Z
xx ( p(y/x)dy)p(x)dx
t
xxt p(x, y)dxdy,
(16)
where p(x, y) represents the joint distribution function of
the random variables x and y, and Eq.16 follows from the
definition of conditional probability. Thus we have:
Z
dE[xxt ] = (xy t + yxt + yy t )p(x, y)dxdy.
(17)
It can be shown after some work that the following equations
hold:
Z
Z
xy t p(x, y)dxdy = xf t (x)p(x)dx = E[xf t (x)]dt,
Z
and
Z
yxt p(x, y)dxdy = E[f (x)xt ]dt.
(18)
yy t p(x, y)dxdy = E[f (x)f t (x)]dt2
+E[g(x)Qg t (x)]dt.
(19)
Using Eqs.18-19 in Eq.17, we obtain the following relation:
dE[xxt ]
= (E[xf t (x)] + E[f (x)xt ] +
E[g(x)Qg t (x)])dt + (E[f (x)f t (x)])dt2(20)
Dividing by dt and neglecting the O(dt) term leads us to the
following evolution equations:
t
˙
E(xx
) = R˙x = E[xf t (x)] + E[f (x)xt ] + E[g(x)Qg t (x)],
˙
E(x)
=x
¯˙ = E[f (x)].
(21)
The initial conditions for the above equations are (¯
xk , Rxk ).
Following the Gaussian approximation, we substitute p(x) =
N (¯
x, Rx ). Further, if the functions f and g are polynomials
or can be well approximated by them, the right hand sides
involve evaluation of the higher order moments of the
(Gaussian) state vector x.
B. Measurement Update
In this section, we present two methods for evaluating the
innovations terms in the nonlinear filtering equations - statistical (or stochastic) linearization to linearize the measurement
equation, and a direct method (without linearization).
1) Stochastic Linearization: Consider a general nonlinear
vector function, h(x) ∈ ℜm , of a Gaussian random vector
x ∈ ℜn . The stochastically linearized gain, Keq for the
function h, is given by:
Keq = arg min E||h(x) − Kx||2 .
K
(22)
Details on how the above expression is obtained and further
literature on statistical linearization can be found in [24].
Now, in order to evaluate the innovation terms in the
nonlinear filtering equations, we need to evaluate the measurement covariance matrix, Pyk , and the state-measurement
cross-covariance matrix, Pxk yk . Using the equivalent gain for
the nonlinear measurement function h(x) at time instant k
according to the above expression, we obtain:
x˙ 1
x˙ 2
= x2
= −αx1 − βx31 − γx2 + w
(36)
where w represents the process noise vector with covariance
matrix Q. The truth measurements are assumed to be the
total energy of the system given by following equation:
t
Pyk = Keq
Px−k Keq + Rv ,
(23)
y˜ = αx21 + βx41 + γx22 + ν
t
Px−k Keq
,
(24)
where, ν represents the measurement noise vector with
covariance matrix R. Note that the factor β can be changed
to vary nonlinearity of the dynamics and the measurement
equations. The factor γ controls how fast the system achieves
steady state. Both these quantities, (β,γ) play an important
role in the estimation of state vector x = {x1 x2 }T . For the
comparison of performance of QGKF with EKF, we consider
two test cases:
1) Low nonlinearity case with β = 0.25
2) High nonlinearity case with β = 5.
Other simulation parameters (common for both the test cases)
are listed in Table I. All the simulations are performed for
200 seconds with a sampling rate of 1 second.
Pxk yk =
Note that the gain Keq in general is time varying.
2) Direct Method: We now present a direct method for
the evaluation of the Pyk and Pxk yk matrices. Recall the
measurement model (equation 2). Note that since vk is zeromean and independent of the state xk , we have:
y¯k = E[yk ] = E[h(xk ) + vk ] = E[h(xk )],
(25)
Then, using the predicted distribution of the state x at the
k th instant (x−
k ), we can ‘directly’ obtain:
Pyk = E[h(xk )ht (xk )] − {E[h(xk )]}2 + Rvk ,
Pxk yk = E[xk ht (xk )] − x
¯k E t [h(xk )].
(26)
(27)
To summarize, the nonlinear quasi-Gaussian filtering recursions can be written as follows:
• Stochastic linearization based:
x
¯˙ = E[f (x)], (28)
R˙x = E[f (x)x ] + E[xf (x)] + E[g(x)Qg t (x)], (29)
t
Px−k Keq + Rv , (30)
Pyk = Keq
t
t
t
Pxk yk = Px−k Keq
, (31)
•
on a two dimensional problem. The results presented demonstrate how the nonlinear propagation of mean and covariance
of the assumed Gaussian pdf significantly enhances the
performance of the EKF algorithm. For testing purposes, we
consider the following nonlinear duffing oscillator:
Direct evaluation based:
x
¯˙ = E[f (x)],
t
t
˙
Rx = E[f (x)x ] + E[xf (x)] + E[g(x)Qg t (x)],
−
t −
2
Pyk = E[h(x−
k )h (xk )] − {E[h(xk )]} + Rvk ,
− t −
t
¯k E [h(x−
Pxk yk = E[xk h (xk )] − x
k )].
(32)
(33)
(34)
(35)
The initial conditions for the above equations are given by
the mean and correlation of the state x at the (k − 1)th time
instant and the integration is carried out between the (k−1)th
and the k th instants.
Note that the evaluation of both the prediction and measurement update terms involves the computation of higher
orderer moments of a Gaussian random vector, in the case
of polynomial nonlinearities. Thus, the real-time implementation of these filters depends critically on the efficient
computation of these moments.
V. I LLUSTRATIVE E XAMPLE
In this section, we compare the quasi-Gaussian Kalman
Filter (QGKF) against the Extended Kalman Filter (EKF)
α
γ
1
0.05
Q
10
−4
R
P
25
10−3
(37)
TABLE I
VARIOUS S IMULATION PARAMETERS FOR C ASE 1 AND 2
In Fig.1, we show the performance of the different filters
for case 1, i.e. with β = 0.25. Fig.1(a) shows the true system
trajectory and various estimated trajectories, while figs.1(b),
1(c) and 1(d) show the plots of state estimation error for EKF,
QGKF with stochastic linearization and QGKF with direct
evaluation, respectively. From these figures, it is clear that all
3 filters converge equally well and state estimation errors lie
with in corresponding 3-σ outliers. Figs.1(e) and 1(f) show
the various components of state estimation covariance matrix
for all 3 filters. From this figure, it is clear that although all
3 filters converge well, the state estimates confidence bound
is higher for the QGKF filters than the EKF filter.
Fig.2 illustrates the filters’ performance for case 2, with
β = 5. Notice that the estimation error for EKF is a order
of magnitude higher than both the QGKF filters. Also, from
the covariance plots, it is clear that state estimation error
covariance for EKF starts diverging after 120 seconds. These
observations can be attributed to the high nonlinearity and
low damping. Also, from Figs.2(c) and 2(d), it is clear
that the transient response of the QGKF filter based on
direct evaluation is marginally better than the one based on
stochastic linearization.
(a) True and Estimated States
(d) Estimated QGKF (DE) States
(b) Estimated EKF States
(e) Diagonal Components of State Estima- (f) Off-Diagonal Components of State Estition Error Covariance Matrix
mation Error Covariance Matrix
Fig. 1.
(a) True and Estimated States
(d) Estimated QGKF (DE) States
(c) Estimated QGKF (SL) States
Simulation Results for Case 1
(b) Estimated EKF States
(c) Estimated QGKF (SL) States
(e) Diagonal Components of State Estima- (f) Off-Diagonal Components of State Estition Error Covariance Matrix
mation Error Covariance Matrix
Fig. 2.
Simulation Results for Case 2
The above discussion brings out the fact that for the
case of high and low nonlinearity in system dynamics and
measurement equations, the QGKF always outperforms the
EKF. Finally, we mention that although we have not proved
the positive definiteness of updated state covariance matrix
in the QGKF, we did not face difficulties associated with
such issues.
VI. C ONCLUSIONS
In this paper, we have presented a quasi-Gaussian Kalman
filter for continuous dynamical systems. We have used the
moment evolution equations for stochastic continuous dynamical systems in order to evaluate the prediction terms
involved in the nonlinear filtering recursions. For the innovations terms, we have proposed two methods, one based
on stochastic linearization and another based on direct evaluation of the measurement covariance and the state measurement cross covariance matrices. We have implemented
the filters on a duffing oscillator example and compared the
performance to that of an EKF.
It was noted that the complexity of the calculations involved in evaluating the prediction and innovations terms
are of the same order as that of finding any general higher
order moment of a Gaussian random vector, when the
nonlinearities are of polynomial type. As such these can be
efficiently calculated using Gauss-Hermite quadratures or the
Wick formulas. Further, unlike in sigma point filters there is
no design criterion for forming the set of sigma points to
evaluate the prediction or the innovations terms. In general,
higher order moments of the state variable are involved in
evaluating the prediction and innovations terms which seems
to lead to better performance of the filter. Also, given the
dynamics and the measurement equations, the evaluation of
these higher order moments can be “hardwired” in order to
implement the filter in real time. However, any real time
application of this filter would be dictated by the efficiency
with which the moment calculations can be accomplished.
Hence, there is need for further research into this aspect of
the problem.
We also mention that proving the positive definiteness
of the updated covariance matrix for a nonlinear system
(using any filtering algorithm for that matter) is an open
problem in the field of filtering theory. This is also a
significant drawback with the proposed filter as it can lead
to oscillatory/unstable behavior. However, we did not face
such behavior in the example treated in this paper. It is seen
from the example that the QGKF performs better than the
EKF even for a highly nonlinear system. Thus, the stability
properties of the proposed filter could be another direction of
further research. Finally, we appreciate the truth that results
from any test are difficult to extrapolate, however, testing
the new filtering algorithm on a reasonable nonlinear system
and providing comparisons to the most obvious competing
algorithms does provide compelling evidence and a basis for
optimism.
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5, pp. 910-927, May 2000.
[22] H. Risken, The Fokker Planck Equation: Methods of Solution and
Applications, vol. 18, Springer Series in Synergetics, SpringerVerlag, Berlin, 1989.
[23] A. Doucet, N. de Freitas and N. Gordon, Sequential Monte-Carlo
Methods in Practice, Springer-Verlag, April 2001.
[24] R. C. Nigam, Introduction to Random Vibrations, MIT Press,
Cambridge, MA, 1983.
[25] J.C. Naylor and A. F. M. Smith, “ Application of a Method
for the Efficient Computation of Posterior Distributions”’, Applied
Statistics, vol. 31, pp. 214-225, 1982.
Suman Chakravorty, Mrinal Kumar and Puneet Singla
Abstract— In this paper, we present a Gaussian approximation to the nonlinear filtering problem, namely the quasiGaussian Kalman filter. Starting with the recursive Bayes filter,
we invoke the Gaussian approximation to reduce the filtering
problem into an optimal Kalman recursion. We use the moment
evolution equations for stochastic dynamic equations to evaluate
the prediction terms in the Kalman recursions. We propose
two methods, one based on stochastic linearization and the
other based on a direct evaluation of the innovations terms,
to perform the measurement update in the Kalman recursion.
We test our filter on a simple two dimensional example, where
the nonlinearity of the system dynamics and the measurement
equations can be varied, and compare its performance to that
of an extended Kalman filter.
I. I NTRODUCTION
In this paper, we present a Gaussian approximation to the
exact nonlinear filtering problem, namely the quasi-Gaussian
Kalman filter. We consider a problem with continuous time
state dynamics and discrete measurement updates. We use
a Gaussian approximation to the nonlinear filtering problem
in order to reduce the problem of nonlinear filtering to one
of maintaining the estimates of the first two moments of
the state variable based on the observations, i.e., an optimal
Kalman recursion. We use the moment evolution equations
for stochastic dynamical systems in order to evaluate the
prediction terms in the nonlinear filtering recursions. Then,
we propose two methods, one based on stochastic linearization and another based on direct evaluation, to evaluate the
innovations terms. We test the proposed filter on a simple two
dimensional example, where the nonlinearity of the system
dynamics and the measurement equations can both be varied,
and compare its performance to that of an extended Kalman
filter (EKF).
Nonlinear filtering has been the subject of intensive research ever since the publication of Kalman’s celebrated
work in 1960 [1]. It is known that the solution to the general
nonlinear filtering problem results in an infinite dimensional
filter, in the sense that an infinite system of differential
equations need to be solved in real time in order to solve
the filtering problem exactly [2]–[5]. Hence, the nonlinear
filtering problem is computationally tractable only if we consider finite dimensional approximations of the true infinite
Suman Chakravorty
neering, Texas A&M
is with Faculty of Aerospace
University, College Station, TX,
EngiUSA.
schakrav@aero.tamu.edu
Mrinal Kumar is a Graduate Assistant Research in Dept. of
Aerospace Engineering at Texas A&M University, College Station, USA.
mrinal@neo.tamu.edu
Puneet Singla is a Graduate Assistant Research in Dept. of
Aerospace Engineering at Texas A&M University, College Station, USA.
puneet@neo.tamu.edu
dimensional filter [6]–[10]. However, even these methods are
difficult to implement because of the complicated nature of
the involved computations.
The Bayes filter [4] offers the optimal recursive solution to
the nonlinear filtering problem. However, the implementation
of the Bayes filter in real time is computationally intractable
because of the multi-dimensional integrals involved in the recursive equations. We note however that researchers have reported attempts at tackling this problem for low-dimensional
systems [11], [12]. It is known that the implementation of
the Bayes filter is rendered tractable in the case when all the
involved random variables are assumed to remain Gaussian
[13], [14], the obtained filter reducing to the Kalman filter in
the case of linear dynamics and measurements. The primary
concern of this class of methods is to predict the mean
and covariance of the state variable between measurement
updates, while evaluating the covariance of the measurement
and the cross-covariance between the state and the measurement after an update, in order to evaluate the prediction
and measurement update terms in a Kalman-type recursion.
The extended Kalman Filter (EKF) [13], [14] linearizes the
system dynamics about the current estimate of the state and
the measurement equations about the current predicted estimate of the state variable, in order to evaluate the prediction
and measurement update terms. The EKF by far has been
the most popular nonlinear filter in practical applications in
past several decades. In recent years, there has been renewed
interest in the topic of nonlinear filtering with the discovery
of the “sigma-point” Kalman filters [15]–[18], [20]. These
nonlinear filters consider the nonlinear filtering problem in
a discrete time setting. The different terms of interest in the
optimal Kalman recursions are obtained using a semi-global
technique known as stochastic or statistical linearization [19].
The method essentially consists of discretizing the domain
of the random variable into a set of weighted sigma points
and transforming these through the nonlinear map in order to
obtain the distribution characteristics of the transformed random variable. The various sigma point algorithms differ from
each other in their choice of the sigma points, for instance the
unscented Kalman filter (UKF) [15], the central difference
Kalman filter (CDKF) [21], the square root form UKF and
CDKF [20] and so on. In our approach to the nonlinear
filtering problem, we use a Gaussian approximation on the
random variables of interest. However, we use a continuous
time model for the system dynamics while assuming that the
measurements are made in discrete time. This allows us to
use the elegant formalism of the Fokker-Planck-Kolmogorov
equations [?], [22] to evaluate the prediction terms in the
nonlinear filtering recursions. We use two methods: one
based on stochastic linearization and the other based on
direct evaluation of the measurement update terms, in order
to evaluate the innovations terms in the optimal recursions,
i.e., to linearize the measurement equations. Our method
does not require the choice of a set of sigma points and
thus, is free from this design aspect. However, central to the
implementation of our filter is the evaluation of integrals of
the form of a product of a function and a Gaussian function.
In the case of polynomial nonlinearities, these are the higher
order moments of a Gaussian random variable. The rest of
the paper is organized as follows.
In section 2, we introduce the Bayes filter. In section 3,
we discuss the topic of Gaussian approximations in nonlinear
filtering problems and present the optimal Kalman recursions. In section 4, we present our quasi-Gaussian Kalman
filter and the methods we use to evaluate the optimal terms
in the Kalman recursions for the approximate nonlinear
filtering problem. In section 5, we present a two dimensional
example, with variable nonlinearity in both system dynamics
and measurement, and compare the performance of our filter
with that of an EKF. In section 6, we present our conclusions
and directions for further research.
II. BAYES F ILTER
In this section, we introduce the recursive Bayes filter.
Consider a system whose dynamics are governed by the
Stochastic Ito differential equation,
dx = f (x)dt + g(x)dW,
(1)
where W represents a standard Wiener process. The system
state is measured in a discrete fashion according to the
equation,
yk = h(xk ) + vk ,
(2)
where vk is the measurement noise at instant k. Let F k =
{y0 , · · · , yk } denote the history of the observations till the
time instant k. Then, the Bayes filter recursively updates
the probability density of the state variable according to the
following equations:
Z
k−1
p(xk /F
) = P (xk /xk−1 )p(xk−1 /F k−1 )dxk−1 , (3)
p(xk /F k ) = ηp(yk /xk )p(xk /F k−1 )
(4)
where η is a normalization factor, p(xk /F k ) and
p(xk /F k−1 ) represent the probability density of the system
state at the k th time instant, based on observations till
the time instants k and (k − 1) respectively, P (xk /xk−1 )
represents the transition probability from state xk−1 to state
xk corresponding to equation (1), and p(yk /xk ) represents
the probabilistic observation model corresponding to the
observation equation (2). The prediction step in the Bayes
filter, i.e. Eq.(3) makes use of the Chapman-Kolmogorov
equations [22] for propagation in between measurements,
while the update step (Eq. (4)) utilizes the Bayes rule. Note
that in general, evaluating the multi-dimensional integral in
the above equations is a formidable challenge. The predicted
probability density function, p(xk /F k−1 ), can also be ob-
tained by solving the Fokker-Planck-Kolmogorov equation
[22] corresponding to Eq.1, given by:
X ∂p
1 X X ∂ 2 (gQg t )ij p
∂p
fi
+
,
(5)
=−
∂t
∂xi
2 i j
∂xi ∂xj
i
where Q represents the intensity of the Wiener process W .
Now let
X
1 X X ∂ 2 (gQg t )ij
∂
+
,
(6)
LF P ≡ −
fi
∂xi
2 i j
∂xi ∂xj
i
where LF P denotes the differential operator corresponding to
the Fokker-Planck-Kolmogorov equation. Thus, the integral
in the Bayes filter corresponding to the prediction equation
(Eq.3) can be evaluated by solving the above partial differential equation. The prediction step in the Bayes filter can
be solved by either evaluating the integral in the prediction
equation through a Monte-Carlo type method (the subject
matter of particle filters [23]) or by solving the above partial
differential equation. However, solving the Fokker Planck
equation is a tough proposition.
III. G AUSSIAN A PPROXIMATIONS IN N ONLINEAR
F ILTERING
One method for obtaining a finite dimensional approximation to the infinite dimensional filter corresponding to
a general nonlinear filtering problem is to approximate the
state, the noise and the observation terms as Gaussian random
variables. The Bayesian recursion shown above can then
be greatly simplified, since the posterior distribution of the
state always remains Gaussian under these approximations.
Thus, estimates of only the conditional mean, E(xt /F t ),
and covariance, Px need to be maintained. It can be shown
that under the Gaussian approximation, we obtain following
recursive filter [4], [13]:
x
¯−
k
−
x
¯k = x
¯−
k + Lk (yk − yk ),
Pxk = Px−k − Lk Pyk Ltk ,
(7)
(8)
Lk = Pxk yk Py−1
,
k
(9)
where
and x
¯k represent the predicted and updated values
of the mean of the random vector xk , Px−k , and Pxk represent
the predicted and the updated values of the covariance matrix
of xk , Pyk is the covariance matrix of the observation yk and
Pxk yk is the cross-covariance matrix between the state and
observation. The key to implementing the above recursion is
−
to evaluate the quantities x
¯−
k , Pxk , Pyk and Pxk yk . Different
filters use different ways to evaluate these quantities.
In the EKF, the predicted mean and covariance of the
variable xk are evaluated by linearizing the system dynamics
about the previous estimate xk−1 and using linear Gaussian
analysis to propagate the mean and the covariance, while
the quantities Pyk and Pxk yk are evaluated by linearizing
the measurement equation about the current predicted value
of xk and using linear Gaussian analysis.
In recent years there has been a lot of interest in sigma
point Kalman filters which use statistical linearization methods to find the quantities of interest in the recursive formulas
[17], [20]. The sigma point filters perform linearization of
the system dynamics and the measurement equations in a
statistical sense as opposed to the Jacobean linearization
which is performed in the case of the EKF.
In our approach, we adopt a different approach to evaluate the prediction and innovation terms in the filtering
recursions. We first use the moment evolution equations for
the stochastic dynamical system along with the Gaussian
approximation, and then perform a stochastic linearization of
the measurement equation in order to evaluate the innovation
terms. The details of the method are presented in the next
section.
IV. A Q UASI -G AUSSIAN K ALMAN F ILTER
In this section, we present our approach to nonlinear
filtering based on Gaussian approximations of the prediction
and innovation terms. As mentioned previously, we use the
moment evolution equations for the prediction step. Then
we propose two methods to evaluate the innovation terms:
(a) stochastic linearization to linearize the measurement
equations and (b) a direct method to evaluate the innovations
terms.
A. Prediction
Recall the nonlinear filtering equations (7)-(9), and note
that:
∂p
= LF P (p),
(10)
∂t
where LF P is the spatial operator of the Fokker Planck
equation. We shall now find the time evolution equations
for the first and second moments of the state variable, given
the prior probability distribution. Let the prior probability
distribution at the k th instant be given by N (¯
xk , Pxk ) and
the mean and the correlation of the state between the time
instants be x
¯ and Rx respectively. By definition, we have:
Z
Z
x
¯ = xp(x)dx, Rx = xxt p(x)dx,
(11)
Then, it follows that the first and second moments of x evolve
according to the following differential equations:
Z
Z
Z
d
∂p
˙x
¯=
xp(x)dx = x dx = xLF P (p(x))dx, (12)
dt
∂t
Z
Z
d
R˙x =
xxt p(x)dx = xxt LF P (p(x))dx, (13)
dt
with the initial conditions x
¯k and Rxk respectively. The
above equations are integrated between the k th and (k +1)th
time instants. In absence of the Gaussian approximation,
the evaluation of the right hand side of the above equation
involves higher moments of the random variable x because of
the Fokker Planck operator. Due to the complicated nature of
this operator, the above equations are difficult to handle. We
present below an alternate method for deriving the moment
evolution equations, and also the evolution equations of the
correlation of the state x and its mean. Consider again
the system equation (1) and an infinitesimal change in the
correlation of the state x, dE(xxt ), given by:
dE[xxt ] = E[(x + dx)(x + dx)t ] − E[xxt ].
(14)
Let dx = y, and recall the following definitions:
Z
E[(x + y)(x + y)t ] = (x + y)(x + y)t p(x, y)dxdy, (15)
t
E[xx ]
Z
=
Z
=
t
xx p(x)dx =
Z
Z
xx ( p(y/x)dy)p(x)dx
t
xxt p(x, y)dxdy,
(16)
where p(x, y) represents the joint distribution function of
the random variables x and y, and Eq.16 follows from the
definition of conditional probability. Thus we have:
Z
dE[xxt ] = (xy t + yxt + yy t )p(x, y)dxdy.
(17)
It can be shown after some work that the following equations
hold:
Z
Z
xy t p(x, y)dxdy = xf t (x)p(x)dx = E[xf t (x)]dt,
Z
and
Z
yxt p(x, y)dxdy = E[f (x)xt ]dt.
(18)
yy t p(x, y)dxdy = E[f (x)f t (x)]dt2
+E[g(x)Qg t (x)]dt.
(19)
Using Eqs.18-19 in Eq.17, we obtain the following relation:
dE[xxt ]
= (E[xf t (x)] + E[f (x)xt ] +
E[g(x)Qg t (x)])dt + (E[f (x)f t (x)])dt2(20)
Dividing by dt and neglecting the O(dt) term leads us to the
following evolution equations:
t
˙
E(xx
) = R˙x = E[xf t (x)] + E[f (x)xt ] + E[g(x)Qg t (x)],
˙
E(x)
=x
¯˙ = E[f (x)].
(21)
The initial conditions for the above equations are (¯
xk , Rxk ).
Following the Gaussian approximation, we substitute p(x) =
N (¯
x, Rx ). Further, if the functions f and g are polynomials
or can be well approximated by them, the right hand sides
involve evaluation of the higher order moments of the
(Gaussian) state vector x.
B. Measurement Update
In this section, we present two methods for evaluating the
innovations terms in the nonlinear filtering equations - statistical (or stochastic) linearization to linearize the measurement
equation, and a direct method (without linearization).
1) Stochastic Linearization: Consider a general nonlinear
vector function, h(x) ∈ ℜm , of a Gaussian random vector
x ∈ ℜn . The stochastically linearized gain, Keq for the
function h, is given by:
Keq = arg min E||h(x) − Kx||2 .
K
(22)
Details on how the above expression is obtained and further
literature on statistical linearization can be found in [24].
Now, in order to evaluate the innovation terms in the
nonlinear filtering equations, we need to evaluate the measurement covariance matrix, Pyk , and the state-measurement
cross-covariance matrix, Pxk yk . Using the equivalent gain for
the nonlinear measurement function h(x) at time instant k
according to the above expression, we obtain:
x˙ 1
x˙ 2
= x2
= −αx1 − βx31 − γx2 + w
(36)
where w represents the process noise vector with covariance
matrix Q. The truth measurements are assumed to be the
total energy of the system given by following equation:
t
Pyk = Keq
Px−k Keq + Rv ,
(23)
y˜ = αx21 + βx41 + γx22 + ν
t
Px−k Keq
,
(24)
where, ν represents the measurement noise vector with
covariance matrix R. Note that the factor β can be changed
to vary nonlinearity of the dynamics and the measurement
equations. The factor γ controls how fast the system achieves
steady state. Both these quantities, (β,γ) play an important
role in the estimation of state vector x = {x1 x2 }T . For the
comparison of performance of QGKF with EKF, we consider
two test cases:
1) Low nonlinearity case with β = 0.25
2) High nonlinearity case with β = 5.
Other simulation parameters (common for both the test cases)
are listed in Table I. All the simulations are performed for
200 seconds with a sampling rate of 1 second.
Pxk yk =
Note that the gain Keq in general is time varying.
2) Direct Method: We now present a direct method for
the evaluation of the Pyk and Pxk yk matrices. Recall the
measurement model (equation 2). Note that since vk is zeromean and independent of the state xk , we have:
y¯k = E[yk ] = E[h(xk ) + vk ] = E[h(xk )],
(25)
Then, using the predicted distribution of the state x at the
k th instant (x−
k ), we can ‘directly’ obtain:
Pyk = E[h(xk )ht (xk )] − {E[h(xk )]}2 + Rvk ,
Pxk yk = E[xk ht (xk )] − x
¯k E t [h(xk )].
(26)
(27)
To summarize, the nonlinear quasi-Gaussian filtering recursions can be written as follows:
• Stochastic linearization based:
x
¯˙ = E[f (x)], (28)
R˙x = E[f (x)x ] + E[xf (x)] + E[g(x)Qg t (x)], (29)
t
Px−k Keq + Rv , (30)
Pyk = Keq
t
t
t
Pxk yk = Px−k Keq
, (31)
•
on a two dimensional problem. The results presented demonstrate how the nonlinear propagation of mean and covariance
of the assumed Gaussian pdf significantly enhances the
performance of the EKF algorithm. For testing purposes, we
consider the following nonlinear duffing oscillator:
Direct evaluation based:
x
¯˙ = E[f (x)],
t
t
˙
Rx = E[f (x)x ] + E[xf (x)] + E[g(x)Qg t (x)],
−
t −
2
Pyk = E[h(x−
k )h (xk )] − {E[h(xk )]} + Rvk ,
− t −
t
¯k E [h(x−
Pxk yk = E[xk h (xk )] − x
k )].
(32)
(33)
(34)
(35)
The initial conditions for the above equations are given by
the mean and correlation of the state x at the (k − 1)th time
instant and the integration is carried out between the (k−1)th
and the k th instants.
Note that the evaluation of both the prediction and measurement update terms involves the computation of higher
orderer moments of a Gaussian random vector, in the case
of polynomial nonlinearities. Thus, the real-time implementation of these filters depends critically on the efficient
computation of these moments.
V. I LLUSTRATIVE E XAMPLE
In this section, we compare the quasi-Gaussian Kalman
Filter (QGKF) against the Extended Kalman Filter (EKF)
α
γ
1
0.05
Q
10
−4
R
P
25
10−3
(37)
TABLE I
VARIOUS S IMULATION PARAMETERS FOR C ASE 1 AND 2
In Fig.1, we show the performance of the different filters
for case 1, i.e. with β = 0.25. Fig.1(a) shows the true system
trajectory and various estimated trajectories, while figs.1(b),
1(c) and 1(d) show the plots of state estimation error for EKF,
QGKF with stochastic linearization and QGKF with direct
evaluation, respectively. From these figures, it is clear that all
3 filters converge equally well and state estimation errors lie
with in corresponding 3-σ outliers. Figs.1(e) and 1(f) show
the various components of state estimation covariance matrix
for all 3 filters. From this figure, it is clear that although all
3 filters converge well, the state estimates confidence bound
is higher for the QGKF filters than the EKF filter.
Fig.2 illustrates the filters’ performance for case 2, with
β = 5. Notice that the estimation error for EKF is a order
of magnitude higher than both the QGKF filters. Also, from
the covariance plots, it is clear that state estimation error
covariance for EKF starts diverging after 120 seconds. These
observations can be attributed to the high nonlinearity and
low damping. Also, from Figs.2(c) and 2(d), it is clear
that the transient response of the QGKF filter based on
direct evaluation is marginally better than the one based on
stochastic linearization.
(a) True and Estimated States
(d) Estimated QGKF (DE) States
(b) Estimated EKF States
(e) Diagonal Components of State Estima- (f) Off-Diagonal Components of State Estition Error Covariance Matrix
mation Error Covariance Matrix
Fig. 1.
(a) True and Estimated States
(d) Estimated QGKF (DE) States
(c) Estimated QGKF (SL) States
Simulation Results for Case 1
(b) Estimated EKF States
(c) Estimated QGKF (SL) States
(e) Diagonal Components of State Estima- (f) Off-Diagonal Components of State Estition Error Covariance Matrix
mation Error Covariance Matrix
Fig. 2.
Simulation Results for Case 2
The above discussion brings out the fact that for the
case of high and low nonlinearity in system dynamics and
measurement equations, the QGKF always outperforms the
EKF. Finally, we mention that although we have not proved
the positive definiteness of updated state covariance matrix
in the QGKF, we did not face difficulties associated with
such issues.
VI. C ONCLUSIONS
In this paper, we have presented a quasi-Gaussian Kalman
filter for continuous dynamical systems. We have used the
moment evolution equations for stochastic continuous dynamical systems in order to evaluate the prediction terms
involved in the nonlinear filtering recursions. For the innovations terms, we have proposed two methods, one based
on stochastic linearization and another based on direct evaluation of the measurement covariance and the state measurement cross covariance matrices. We have implemented
the filters on a duffing oscillator example and compared the
performance to that of an EKF.
It was noted that the complexity of the calculations involved in evaluating the prediction and innovations terms
are of the same order as that of finding any general higher
order moment of a Gaussian random vector, when the
nonlinearities are of polynomial type. As such these can be
efficiently calculated using Gauss-Hermite quadratures or the
Wick formulas. Further, unlike in sigma point filters there is
no design criterion for forming the set of sigma points to
evaluate the prediction or the innovations terms. In general,
higher order moments of the state variable are involved in
evaluating the prediction and innovations terms which seems
to lead to better performance of the filter. Also, given the
dynamics and the measurement equations, the evaluation of
these higher order moments can be “hardwired” in order to
implement the filter in real time. However, any real time
application of this filter would be dictated by the efficiency
with which the moment calculations can be accomplished.
Hence, there is need for further research into this aspect of
the problem.
We also mention that proving the positive definiteness
of the updated covariance matrix for a nonlinear system
(using any filtering algorithm for that matter) is an open
problem in the field of filtering theory. This is also a
significant drawback with the proposed filter as it can lead
to oscillatory/unstable behavior. However, we did not face
such behavior in the example treated in this paper. It is seen
from the example that the QGKF performs better than the
EKF even for a highly nonlinear system. Thus, the stability
properties of the proposed filter could be another direction of
further research. Finally, we appreciate the truth that results
from any test are difficult to extrapolate, however, testing
the new filtering algorithm on a reasonable nonlinear system
and providing comparisons to the most obvious competing
algorithms does provide compelling evidence and a basis for
optimism.
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