In algorithms of nonlinear Kalman filter, the so-called extended Kalman filter algorithm actually uses first-order Taylor expansion approach to transform a nonlinear system into a linear system. It is obvious that this algorithm will bring some systematic deviations because of ignoring nonlinearity of the system. This paper presents two extended Kalman filter algorithms for nonlinear systems, called second-order nonlinear Kalman particle filter algorithms, by means of second-order Taylor expansion and linearization approximation, and correspondingly two recursive formulas are derived. A simulation example is given to illustrate the effectiveness of two algorithms. It is shown that the extended Kalman particle filter algorithm based on second-order Taylor expansion has a more satisfactory performance in reducing systematic deviations and running time in comparison with the extended Kalman filter algorithm and the other second-order nonlinear Kalman particle filter algorithm.
It is well known that the theory of Kalman filter has been widely used in many scientific fields. Particularly, the linear Kalman filter algorithm has received a great deal of attention due to its effectiveness in estimating the state of an underlying linear system in [
A widely used Kalman filter algorithm for linear systems is called extended Kalman filer algorithm (EKF for short), which actually uses first-order Taylor expansion approach to transform a nonlinear system into a linear system. Since this nonlinear Kalman filter algorithm is linear in nature and it does not consider the nonlinearity of the system equation, a sub-optimal state estimate for a nonlinear state-space system can be given in [
To this end, this paper will give two Kalman particle filter algorithms for nonlinear state-space models. One is based on second-order Taylor expansion of structural functions of nonlinear system (SEKPF), the other, denoted by SLEKPF, uses a linearization method to approximate the quadratic of second order Taylor expansion of struc- tural functions. Accordingly, recursive formulas of the above nonlinear Kalman filter algorithms are derived. A simulation example is given to compare EKPF with SEKPF and SLEKPF. It can be seen from running time and root mean square error of the state estimation that they have similar calculation capacity, but SLEKPF is superior to EKPF and SEKPE in accuracy of state estimation.
For a nonlinear state-space model, the extended Kalman filter is a frequently used method to estimate the system state. The key point of this algorithm is to use first-order Taylor expansion to approximate the structural functions of the model. More precisely, a general nonlinear discrete-time state-space model is given as follows:
where the system state
If
bances
continuous first-order partial derivatives and
mated by linear model can be used as follows:
Proposition 2.1 If the discrete time linear system (3) satisfies (2), and the disturbances
(1) state estimates
and
(2) mean square error matrix of system.
The proof of Proposition 2.1 can be found in [
In the extended nonlinear Kalman filter algorithm, an application of first-order Taylor expansion on
Proposition 2.2 Under the same conditions as Proposition 2.1, if
(1) state estimates are given by
and
(2) mean square error matrix of system
Proof: To guarantee normality of
Note that
An application of Taylor expansion to
Taking conditional expectations on both sides of (5), the state estimates are given by
where we have applied the following fact that, if
On the other hand, the residual is expressed as
Thus the mean square error matrix of system is
It should be pointed out that the above nonlinear Kalman filter algorithm needs to calculate Hessian matrix of
Proposition 2.3 Under the same conditions as Proposition 2.2 we have the following nonlinear Kalman filter algorithm for the nonlinear system (1):
(1) state estimates are
and
(2) mean square error matrix of system state is
Proof: Firstly, a second-order Taylor expansion of
Applying first-order Taylor expansion to
This also implies that
It is not hard to get that
Moreover, the mean square error matrix of system is given by
Note that in this nonlinear filter algorithm, we have applied first-order Taylor expansion to deal with the qua-
dratic
that this algorithm can give multi-step predictions of the system state.
The extended particle filter (EKPF) algorithm was initially given by Freitas in [
(1) Set initial value
(2) Draw N particles from prior distribution (important distribution);
(3) Apply recursive formulas of EKF, SEKF, and SLEKF to update sampling particles;
(4) Update weights;
(5) Normalize weights;
(6) Resample,
(7)
(8) If the number of particles is enough, the procedure stops.
To compare EKPF with SEKPF and SLEKPF algorithms, we use single variable non-stable growth model in [
Example 3.1 Consider the following nonlinear state-space model:
where the initial state
where
The root-mean-square error is defined as:
where T is the whole time.
the EKPF algorithm has the worst performance in estimating the system state. The reason lies in no consideration of higher-order Taylor expansion term. Means and variances of 100 root-mean-square errors and running time for the above particle filter algorithms are given in
Example 3.2 Consider the following nonlinear state-space model:
where the noises w1,t and w2,t obey Gamma distributions with mean 0.5 and variance 0.5, the noise vt is normally
distributed with mean 0 and variance 0.1. We take vector
The number of particles is 200. The whole time is 30 seconds. We have 100 independent experiments.
In many practical state-space models, the observation equations are linear, but the state equations are nonlinear; sometimes the disturbances are non-Gaussian [
Algorithms | Root-mean-square error | Running time (s) | |
---|---|---|---|
Mean | Variance | ||
EKPF | 0.295 | 0.309 | 1.87 |
SKEPF | 0.179 | 0.152 | 1.98 |
SLEKPF | 0.190 | 0.240 | 2.03 |
Algorithm | EKPF | SEKPF | SLEKPF | |
---|---|---|---|---|
Root-mean-squareerrors of X1 | Mean | 0.035 | 0.008 | 0.016 |
Variance | 0.142 | 0.005 | 0.072 | |
Root-mean-square error of X2 | Mean | 0.064 | 0.062 | 0.063 |
Variance | 0.005 | 0.001 | 0.001 | |
Running time (s) | 2.87 | 2.98 | 3.05 |
expansion of the state equation, and the other uses a linearization approach to approximate the quadratic of second-order Taylor expansion of the state equation. In order to improve the precision of state estimation, we have combined the above nonlinear Kalman filter algorithms with particle filter and given two nonlinear Kalman particle filter algorithms. Through two examples, we compare the two nonlinear Kalman particle filter algorithms with the extended Kalman particle filter algorithm. The simulation results show that our second-order Kalman particle filter algorithms are superior to the extended Kalman particle filter algorithm. On one hand, two nonlinear Kalman particle filter algorithms have stronger calculation capacity than the extended Kalman filter particle algorithm; on the other hand, they can effectively improve state estimations.