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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

A new filter named the maximum likelihood-based iterated divided difference filter (MLIDDF) is developed to improve the low state estimation accuracy of nonlinear state estimation due to large initial estimation errors and nonlinearity of measurement equations. The MLIDDF algorithm is derivative-free and implemented only by calculating the functional evaluations. The MLIDDF algorithm involves the use of the iteration measurement update and the current measurement, and the iteration termination criterion based on maximum likelihood is introduced in the measurement update step, so the MLIDDF is guaranteed to produce a sequence estimate that moves up the maximum likelihood surface. In a simulation, its performance is compared against that of the unscented Kalman filter (UKF), divided difference filter (DDF), iterated unscented Kalman filter (IUKF) and iterated divided difference filter (IDDF) both using a traditional iteration strategy. Simulation results demonstrate that the accumulated mean-square root error for the MLIDDF algorithm in position is reduced by 63% compared to that of UKF and DDF algorithms, and by 7% compared to that of IUKF and IDDF algorithms. The new algorithm thus has better state estimation accuracy and a fast convergence rate.

The problem of estimating the state of a nonlinear stochastic system from noisy measurement data has been the subject of considerable research interest during the past few years. Up to now the extended Kalman filter (EKF) has unquestionably been the dominating state estimation technique [

Recently, there has been development in derivative-free state estimators. The finite difference has been used in the Kalman filter framework and the resulting filter is referred to as the finite difference filter (FDF) [

The DDF also shows its weakness in the state estimation due to the large initial error and high nonlinearity in the application for state estimation of maneuvering target in the air-traffic control and ballistic re-entry target. Emboldened by the superiority of DDF, the basic idea of the IEKF and the iteration termination condition based on maximum likelihood, we propose a new filter named the maximum likelihood based iterated divided difference Kalman filter (MLIDDF). The performance of the state estimation for MLIDDF is greatly improved when involving the use of the iteration measurement update in the MLIDDF and the use of the current measurement. The remainder of this paper is organized as follows: in Section 2, we develop the maximum likelihood surface based iterated divided difference Kalman filter (MLIDDF). Section 3 presents the applications of the MLIDDF to state estimation for maneuvering targets in air-traffic control and ballistic target re-entry applications and discuss the simulation results. Finally, Section 4 concludes the paper and presents our outlook on future work.

Consider the nonlinear function:

Assuming that the random variable ^{nx}_{x}

The transformation matrix _{x}_{x}

The multidimensional Stirling interpolation formula of

The divided difference operators _{Δ}_{z}_{j}

The partial operators _{j}

We can obtain the approximate mean, covariance and cross-covariance of _{x,j}_{x}

Consider the state estimation problem of a nonlinear dynamics system with additive noise, the _{x}_{k}_{k}_{−1} and _{k}_{k}_{−1} ∼
_{k−1)}, _{k}_{k}

Suppose the state distribution at _{k}_{−1}∼
_{k−1}, _{k}_{−1)}, and a square Cholesky factor of _{k}_{−1} is _{x,k}_{−1}. The divided difference filter (DDF) obtained with

Step 1. Time update

Calculate matrices containing the first- and second- divided difference on the estimated state _{k}_{−1} at

Evaluate the predicted state and square root of corresponding covariance:
_{x,j} is _{x,k}_{−1}. _{w,k}_{−1} denotes a square-root factor of _{k}_{−1} such that

Step 2. Measurement update

Calculate matrices containing the first- and second-divided difference on the predicted state _{k}_{x,j}_{x,k}

Evaluate the predicted measurement, square root of innovation covariance and cross-covariance:
_{v,k}_{k}

Evaluate the gain, state estimation and square root of corresponding covariance at

Consider _{k}_{k}_{k}_{k}, _{k}_{k}_{k}_{k}_{k}_{k}^{T}

Here:

The update measurement problem becomes the one that computing the optimal state estimation and corresponding covariance given

Defining the objective function:

Assuming the

We know the sequence of iterates generated by the IEKF and that generated by the Gauss-Newton method were identical, thus globally convergent was guaranteed. The initial state _{k}_{k}_{k}_{k}

Compared to

Now we consider the gain:

The terms of

Hence, we can obtain the following iterative formula:

In the measurement update step of the IEKF algorithm, the inequality

Consider
_{k}_{k}_{k}_{k}_{k}_{k}

Meanwhile, the likelihood surface is defined as follows:

We know that the solution that maximizes the likelihood function is equivalent to minimizing the cost function _{k}_{k}

We say that
_{k}

The sequence generated is guaranteed to go up the likelihood surface using

We have now arrived at the central issue of this paper, namely, the maximum likelihood based iterated divided difference filter. Enlightened by the development of IEKF and the superiority of DDF, we can derive the maximum likelihood based iterated divided difference filter (MLIDDF) which involves the use of the iteration measurement update and the current measurement. But in view of the potential problems exhibited by the IEKF, we shall refine the covariance and cross-covariance based on divided difference and use the termination criterion which guarantees the sequence obtained moves up the maximum likelihood surface. The MLIDDF is described as follows:

Step 1. Time update

Evaluate the predicted state _{k}_{k}_{k}

Step 2. Measurement update

Let

Evaluate the first- and second-order difference matrices using

Evaluate the square root of innovation covariance and cross-covariance:

Evaluate the gain:

Evaluate the state and the square root of corresponding covariance:

Step 3. If the following inequality holds:

The iteration returns to Step 2; otherwise continue to Step 4.

Step 4. If the inequality is not satisfied or if _{max}), and the ultimate state estimation and square root of corresponding covariance at

The MLIDDF algorithm has the virtues of free-derivative and better numerical stability. The measurement update of MLIDDF algorithm is transformed to a nonlinear least-square problem; the optimum state estimation and covariance are solved using Gauss-Newton method, so the MLIDDF algorithm has the same global convergence as the Gauss-Newton method. Moreover, the iteration termination condition that makes the sequence move up the maximum likelihood surface is used in the measurement update process.

In this section, we reported the experimental results obtained by applying the MLIDDF to the nonlinear state estimation of a maneuvering target in an air-traffic control scenario and a ballistic target re-entry scenario. To demonstrate the performance of the MLIDDF algorithm we compared its performance against the UKF, DDF, and the iterated UKF (IUKF) and iterated DDF (IDDF), both using a traditional iteration strategy.

We consider a typical air-traffic control scenario, where an aircraft executes a maneuvering turn in a horizontal plane at a constant, but unknown turn rate Ω. The kinematics of the turning motion can be modeled by the following nonlinear process equation [

The process nose _{k}_{−1}∼

The parameters _{1} and _{2} are related to process noise intensities. A radar is fixed at the origin of the place and equipped to measure the range, _{k}

The parameters used in this simulation were the same as those in [

Owing to the fact that the filters is sensitive to initial state estimation,

In order to analyze the impact of iteration numbers on performance of the MLIDDF algorithm, the MLIDDFs with various iteration numbers are applied to position estimation of maneuvering target.

The relative location of the ballistic target re-entry (BTR) and the radar are shown in _{I}y_{I}z_{I}_{I}_{I}_{I}y_{I}

Assume that the radar is situated at the surface of the Earth and considering the orthogonal coordinate reference system named East-North-Up coordinates system (ENU-CS) _{s}xyz_{k}_{k} ẋ_{k} y_{k} ẏ_{k} z_{k} ż_{k} β_{k}^{2}), _{e} are the Earth's gravitational constant and average Earth radius, respectively. Below 90 km at height, the air density _{1}^{−c2h} (_{1}, _{2} are constant, specifically, _{1} = 1.227, _{2} = 1.093 × 10^{−4} for h < 9,144 m, and _{1} = 1.754, _{2} = 1.49 × 10^{−4} for

Process noise _{k}_{1} (in m^{2}/s^{3}) and _{2} (in kg^{2}/m^{4}s) are the intensity of noise.

According to relative geometry, the measurement equation in the ENU-CS is described as:
_{k}_{k} E_{k} A_{k}^{T}

The measurement noise _{k}_{R}, σ_{E}, σ_{A}_{k}_{0}.

The parameters used in simulation were: _{1} = 5 m^{2}/s^{3}, _{2} = 5 kg^{2}/m^{4}s. The initial position and magnitude of the velocity: _{0} = 232 km, _{0} = 232 km, _{0} = 90 km and _{0} = 3,000 m/s, and the initial elevation and azimuth angle: _{0} = 7π/6 and _{0} = π/4. The ballistic coefficient was selected as ^{2}. The error standard deviations of the measurements were selected as _{R}_{E}_{A}_{max} = 8 was predetermined.

From the above parameters given, we can obtain the initial true state:
_{0} = ^{2} 50^{2} 100^{2} 50^{2} 100^{2} 50^{2} 200^{2}].

To compare the performance of the various filter algorithms, we also use

_{0} is chosen randomly from _{0} ∼
_{0}, _{0}) in each run. All the filters are initialized with the same condition in each run. We make 100 independent Monte Carlo runs.

From

As to the estimation of the ballistic coefficient, in the

Meanwhile, we observe from

For further comparison of performance for the various filters, the average accumulated position mean-square root error (_{p}

From

From

In this study, we provide the maximum likelihood based iterated divided difference filter which inherits the virtues of the divided difference filter and contains the iteration process in the measurement update step. The sequence obtained is guaranteed to move up the likelihood surface using the iteration termination condition based on the maximum likelihood surface. The maximum likelihood based iterated divided difference is implemented easily and is derivative-free. We apply the new filter to state estimation for a ballistic target re-entry scenario and compare its performance against the unscented Kalman filter, divided difference filter, iterated unscented Kalman filter and iterated divided difference filter with the traditional termination criteria. Simulation results demonstrate that the maximum likelihood-based iterated divided difference is much more effective than the other filters. The maximum likelihood-based iterated divided difference greatly improves the performance of state estimation and has a shorter convergence time.

Future work may focus on the applications of the maximum likelihood iteration divided difference filter to remove the outliers which is a serious deviation from the sample and caused by blink and subjective eye movement in video nystagmus signal samples of pilot candidates.

This work was partially supported by Grants No. 30871220 from National Natural Science Foundation of China and No. 2007AA010305 from Subprojects of National 863 Key Projects.

RMSE in position for DDF and MLIDDF.

RMSE in velocity for DDF and MLIDDF.

RMSE in turn rate for DDF and MLIDDF.

RMSE in position for MLIDDFs with various iterate numbers (MLIDDF_{1} with iterate numbers 2, MLIDDF_{2} with 5, MLIDDF_{3} with 8, MLIDDF_{4} with 10).

Geometry of radar and BTR.

Position

Velocity

Ballistic coefficient

Pro and cons for various filters.

EKF | Less runtime | Calculation of Jacobian |

Second-order EKF | high accuracy | Calculation of Jacobian and Hessian |

IEKF | high accuracy | Calculation of Jacobian |

UKF | Derivative-free | more runtime |

IUKF-VS | Derivative-free, high accuracy | more runtime |

FDF | Derivative-free, Less runtime | Lower accuracy |

DDF | Derivative-free | Low accuracy |

PF | Derivative-free | Heavy computational burden |

_{p}(m) |
_{v}(m/s) |
_{β}(kg/m^{2}) | |
---|---|---|---|

UKF | 2521.684 | 329.911 | 155.735 |

DDF | 2521.573 | 329.903 | 155.276 |

IUKF | 1035.340 | 259.173 | 149.603 |

IDDF | 1035.273 | 260.771 | 149.756 |

MLIDDF | 968.746 | 255.916 | 144.953 |

Runtimes of various filters.

UKF | 1.0840 |

DDF | 0.2888 |

IUKF | 1.9918 |

IDDF | 0.5133 |

MLIDDF | 1.3074 |