Support Vector Regression-Based Adaptive Divided Difference Filter for Nonlinear State Estimation Problems

Abstract

We present a support vector regression-based adaptive divided difference filter (SVRADDF) algorithm for improving the low state estimation accuracy of nonlinear systems, which are typically affected by large initial estimation errors and imprecise prior knowledge of process and measurement noises. The derivative-free SVRADDF algorithm is significantly simpler to compute than other methods and is implemented using only functional evaluations. The SVRADDF algorithm involves the use of the theoretical and actual covariance of the innovation sequence. Support vector regression (SVR) is employed to generate the adaptive factor to tune the noise covariance at each sampling instant when the measurement update step executes, which improves the algorithm’s robustness. The performance of the proposed algorithm is evaluated by estimating states for (i) an underwater nonmaneuvering target bearing-only tracking system and (ii) maneuvering target bearing-only tracking in an air-traffic control system. The simulation results show that the proposed SVRADDF algorithm exhibits better performance when compared with a traditional DDF algorithm.

1. Introduction

The problem of state estimation for nonlinear systems has been a subject of considerable research interest in recent years, but there is still no single solution that outperforms all other approaches. Most proposed estimators are nonlinear extensions of the dominated Kalman filter (see [1]), and each approach provides a suboptimal trade-off between properties such as numerical robustness, computational burden, and estimation accuracy. The extended Kalman filter (EKF), which linearizes both nonlinear terms of a current estimated state trajectory, is based on a first-order Taylor series and displays poor performance if the system is highly nonlinear. The limitations of EKFs are enumerated in [2]. Another improved algorithm is the iterated extended Kalman filter (IEKF), which linearizes the nonlinear model around an updated state rather than the predicted state (see [3]). Although IEKFs have been proven to perform better than EKFs in addition to globally guaranteeing convergence, the algorithm still requires a Jacobian matrix just like EKFs. However, no solution exists for the Jacobian matrix in nonlinear systems for some situations, which limits the potential application of both EKFs and IEKFs.

In recent years, a new class of filter known as sigma-point Kalman filter (SPKF) has attracted a great deal of attention. In SPKFs, the algorithm propagates a cluster of points centered on the current state instead of linearizing the system dynamics to improve the approximations of the conditional mean and covariance. Unscented Kalman filters (UKF) and divided difference filters (DDF) are two kinds of SPKFs.

UKFs use a deterministic sampling technique to pick a minimal set of sample points around the mean to catch the higher order statistics of the system so as to better estimation accuracy and convergence characteristics (see [4]). In [5, 6], a UKF for a class of nonlinear discrete-time systems with correlated noises was designed to deal with the problem of nonlinear filtering failure found in conventional UKFs when system noise is correlated with measurement noise. The proposed UKF breaks the limitation of conventional UKFs that requires system noise and measurement noise to be uncorrelated Gauss white noises, thus extending the potential application of conventional UKFs. In [7], a UKF filtering algorithm with colored measurement noise was proposed. The algorithm was first derived on the basis of augmented measurement information and minimum mean square error estimation, and a filtering recursive formula of UKF with colored noise then added by applying an unscented transformation to calculate the posterior mean and covariance of the nonlinear state within the optimal framework. The proposed UKF effectively dealt with the fact that traditional UKFs fail when measurement noise is colored. In [8], a UKF was applied to multiple target tracking, with the proposed UKF shown to have improved performance versus previous EKF approaches. In [9], the paper discussed an adaptive multiuser receiver for CDMA systems in which the scaled unscented filter (SUF) and the square root unscented filter (SURF) were used for joint estimation and tracking of the code delays and multipath coefficients of the received CDMA signals. The proposed channel estimators were more near-far resistant than in conventional EKFs and presented lower complexity than conventional particle filter- (PF-) based methods. Computer simulation results demonstrated the superior performance of the proposed channel estimators, and the proposed estimators were shown to exhibit lower complexity relative to the PF-based method. Although UKFs have undergone a significant amount of meaningful theory innovation and are now used in many fields, [10–12] show that UKF accuracy is lower than that of DDF, while also having a higher computational cost.

The divided difference filter (DDF) first proposed by Nøgaard et al. (see [13]) linearizes the nonlinear terms based on Stirling’s interpolation polynomial approximations formula rather than Taylor’s approximation of nonlinear terms in an EKF. Conceptually, the implementation principle resembles that of an EKF; however, the DDF is significantly simpler as it does not need to calculate the Jacobian matrix and no derivatives are required. The DDF that Nøgaard et al. developed works on general discrete-time nonlinear models in which the noises are not assumed to be additive. In [14], the paper further formulated a DDF in terms of the innovation vector approach, the additive process, and the measurement noise sources. In [15], the paper proposed a new filter named the maximum likelihood-based iterated divided difference filter (MLIDDF), which improved the low state estimation accuracy of nonlinear state estimation that results from large initial estimation errors and the nonlinearity of the measurement equations. Simulation results showed that the MLIDDF algorithm possessed better state estimation accuracy and a faster convergence rate. In [16], the authors proposed a novel adaptive version of the DDF that was applicable to nonlinear systems with a linear output equation. In order to make the filter robust to modeling errors, upper bounds on the state covariance matrix were derived. The parameters of the upper bound were then estimated using a combination of offline tuning and online optimization with a linear matrix inequality constraint, which ensured that the predicted output error covariance was larger than the observed output error covariance. Simulation results demonstrated the superior performance of the proposed filter as compared to the standard DDF. Reference [17] presented an ensemble-based approach that handled nonlinearity based on a simplified divided difference approximation through Stirling’s interpolation formula. The algorithm used Stirling’s interpolation formula to evaluate the statistics of the background ensemble during the prediction step, employing the formula in an ensemble square root filter (EnSRF) at the filtering step to update the background for analysis. In this sense, the algorithm is a hybrid of Stirling’s interpolation formula and the EnSRF method, while the computational cost of the algorithm is less than that of EnSRF.

Different studies have focused on the application of DDFs to nonlinear state estimation problems. In [18], time delay and channel gain estimation for multipath fading code division multiple access (CDMA) signals using a DDF were investigated, and the simulation results showed that the DDF was simpler to implement and more resilient to near-far interference in CDMA networks compared with an EKF. In [19, 20], the relative kinematic states of a reentry vehicle obtained from noisy seeker measurements using a DDF were examined. The results were compared to those obtained using an EKF and a UKF and showed that the DDF was more accurate than estimators based on a Taylor approximation like the EKF. Reference [21] investigated the possibility of using a DDF for estimating the internal variables of a synchronous generator, such as the rotor angle where the data acquired is from a phasor measurement. The effectiveness of the method was tested on a single machine infinite bus system, a nine-bus system, and a 68-bus New England-New York interconnected system. In [22], a DDF using orientation estimation was considered. The fourth element of the quaternion error vector was removed from the system states to alleviate estimated error covariance matrix divergence. The measurement system was a MARG sensor, which consisted of a triaxial rate gyro, a triaxial accelerometer, and a triaxial magnetometer. The nonlinear measurement model was obtained based on the principals of operation of the magnetometer and accelerometer and the properties of the quaternion vector space. The performance of three filters, DDF, EKF, and UKF, was compared with different sampling frequencies. The work showed that the tested DDF and the UKF were more robust than the EKF under the same initial angle-error conditions. The DDF also performed better than the UKF, although the computational load for the UKF was less. In [23], a DDF-based data fusion algorithm was presented, which utilized the complementary noise profile of rate gyros and gravimetric inclinometers to extend their limits and achieve more accurate attitude estimates. In [24], a DDF-based ballistic target tracking system for the reentry phase was proposed. The paper compared DDF, EKF, and UKF algorithms using a Monte Carlo simulation approach, with the simulation results showing that the DDF outperformed both the EKF and UKF in terms of estimation accuracy and filtering credibility. In [25], a DDF with quaternion-based dynamic process modeling was applied to global positioning system (GPS) navigation to increase navigation estimation accuracy at high-dynamic regions while preserving precision at low-dynamic regions. Some properties and performance metrics were assessed and compared to those using EKF and UKF approaches.

Despite their recent popularity, DDF algorithms require that both the system model and the stochastic information must be accurate. However, this condition cannot be satisfied in many practical situations, which forces the filter to adapt itself to changing conditions. One of the problems with this requirement is that any change in the process introduces measurement noise covariance. In this work, we make use of the theoretical and actual covariance of the innovation sequence, employing SVR to generate the scale factor to tune the noise covariance at each sampling instant when the measurement update step is executed to adapt the filtering algorithm. This paper is organized as follows. Section 2 briefly introduces DDF theory and the proposed SVR-based adaptive strategy. Passive target tracking is then carried out to evaluate the performance of DDF and SVRADDF algorithms using a Monte Carlo simulation in Section 3. Finally, conclusions are provided in Section 4.

2. Development of the Support Vector Regression-Based Adaptive Divided Difference Filter

2.1. Divided Difference Filter

Consider the nonlinear function 𝐲=𝐟(𝐱), (1) where 𝐱∈𝐑^𝑛𝑥 and 𝐲∈𝐑^𝑛𝑦. If the function is analytic, then the first-order Taylor series expanded about some point 𝐱=𝐱 becomes 𝐲=𝐟(𝐱+Δ𝐱)=𝐟(𝐱)+𝐃_Δ𝐱𝐟=𝐟(𝐱)+Δ𝑥₁𝜕𝜕𝑥₁+Δ𝑥₂𝜕𝜕𝑥₂+⋯+Δ𝑥_𝑛𝜕𝜕𝑥_𝑛𝐟(𝐱)_𝐱=𝐱 (2) with (2) truncated after the first-order term. Note that (2) can achieve a better local approximation if more terms are included. However, such an expanded Taylor series requires derivatives and cannot be fulfilled in some situations. Stirling’s interpolation formula is based on a finite number of evaluations of the function and does not require derivatives, with the first-order approximation yielding 𝐲=𝐟(𝐱+Δ𝐱)=𝐟(𝐱)+𝐃_Δ𝐱𝐟(𝐱)=𝐟(𝐱)+1ℎ^𝑛_𝑖∑₌₁Δ𝑥_𝑖𝜇_𝑖𝛿_𝑖𝐟(𝐱), (3) where ℎ denotes a selected interval length and 𝛿 and 𝜇 are determined by 𝛿_𝑖𝐟(𝐱)=𝐟ℎ𝐱+2𝐞_𝑖−𝐟ℎ𝐱−2𝐞_𝑖,𝜇_𝑖𝐟(𝐱)=12𝐟ℎ𝐱+2𝐞_𝑖+𝐟ℎ𝐱−2𝐞_𝑖, (4) with 𝐞_𝑖 being the 𝑖th unit vector.

Figure 1 compares the results found by using (2) and (3). The function example is 𝑓(𝑥)=𝑒^𝑥, where ℎ=0.56. From the figure, we can see that Stirling’s interpolation provides better accuracy than the Taylor series under the same order approximations.

(a)

(b)

Figure 1 

Comparison of first-order Stirling series with first-order Taylor series results.

We now assume that the variable 𝐱 has a Gaussian density with mean 𝐱 and covariance 𝐏_𝐱. We can introduce a transformation matrix 𝐒_𝐱 which we select as a square Cholesky factor of 𝐏_𝐱, such that 𝐏_𝐱=𝐒_𝐱𝐒^𝑇_𝐱. To illustrate how others can be derived, we introduce the linear transformation of 𝐱: 𝐳=𝐒_𝐱⁻¹𝐱. (5)

This linear transformation results in a stochastic decoupling of 𝐱 as the elements of 𝐳 become mutually uncorrelated (see [13]). This changes (3) to 𝐲=𝐟𝐒_𝐱𝐳=𝐟̃(𝐳)=𝐟̃(𝐳)+𝐃_Δ𝐳𝐟,̃ (6) where 𝐃_Δ𝐳𝐟̃ is determined by 𝐃_Δ𝐳̃1𝐟=ℎ^𝑛_𝑖=1Δ𝐳_𝑖𝜇_𝑖𝛿_𝑖𝐟(𝐳). (7)

The mean 𝐲, covariance 𝐏_𝐲𝐲, and cross covariance 𝐏_𝐱𝐲 of 𝐲 are obtained from 𝐲=𝐄[𝐲]=𝐄𝐟̃(𝐳)+𝐃_Δ𝐳𝐟̃=𝐄𝐟̃(𝐳(𝐱),𝐏)=𝐟_𝐲𝐲=𝐄(𝐲−𝐲)(𝐲−𝐲)^𝑇=𝐄𝐃_Δ𝐳𝐟̃𝐃_Δ𝐳𝐟̃^𝑇=14ℎ^2𝑛_𝑖=1𝐟𝐳+ℎ𝐞_𝑖−𝐟𝐳−ℎ𝐞_𝑖×𝐟𝐳+ℎ𝐞_𝑖−𝐟𝐳−ℎ𝐞_𝑖^𝑇=14ℎ^2𝑛_𝑖=1𝐟𝐱+ℎ𝐬_𝐱,𝑖−𝐟𝐱−ℎ𝐬_𝐱,𝑖×𝐟𝐱+ℎ𝐬_𝐱,𝑖−𝐟𝐱−ℎ𝐬_𝐱,𝑖^𝑇,𝐏_𝐱𝐲=𝐄(𝐱−𝐱)(𝐲−𝐲)^𝑇=𝐄𝐒_𝐱Δ𝐳𝐃_Δ𝐳𝐟̃^𝑇=12ℎ^𝑛_𝑖=1𝐬_𝐱,𝑖𝐟𝐱+ℎ𝐬_𝐱,𝑖−𝐟𝐱−ℎ𝐬_𝐱,𝑖^𝑇, (8) where 𝐬_𝐱,𝑖 is the 𝑖th column of the matrix 𝐒_𝐱.

Consider the following nonlinear dynamic system with states to be estimated: 𝐱_𝑘+1=𝐟𝐱_𝑘+𝝎_𝑘𝐲_𝑘=𝐡𝐱_𝑘+𝝂_𝑘, (9) where 𝝎_𝑘 and 𝝂_𝑘 are assumed to be independent and identically distributed and independent of current and past states, such that 𝝎_𝑘∼𝐍(𝐨,𝐐_𝑘) and 𝝂_𝑘∼𝐍(𝟎,𝐑_𝑘).

The DDF takes the same predictor-corrector structures in the EKF and can be described as follows.

Step 1 (initialization). Suppose the state distribution at 𝑘 instant is 𝐱_𝑘∼𝐍(𝐱_𝑘,𝐏_𝑘), where 𝐱_𝑘 and 𝐏_𝑘 are obtained by 𝐱_𝑘=𝐄𝐱_𝑘,𝐏_𝑘=𝐄𝐱−𝐱_𝑘𝐱−𝐱_𝑘^𝑇. (10)

Step 2 (square Cholesky factorizations). Consider the following: 𝐏_𝑘=𝐒_𝐱𝐒^𝑇_𝐱𝐒_𝐱𝐱=1𝐟2ℎ^𝑖𝐱_𝑘+ℎ𝐬_𝐱,𝑗−𝐟_𝑖𝐱_𝑘−ℎ𝐬_𝐱,𝑗, (11) where 𝐬_𝐱,𝑖 is the 𝑗th column of 𝐒_𝐱.

Step 3 (state and covariance propagation). One has 𝐱⁻_𝑘+1=𝐟𝐱_𝑘𝐏⁻_𝑘+1=𝐒_𝐱𝐱𝐒_𝐱𝐱^𝑇+𝐐_𝑘=𝐒⁻_𝐱𝐒⁻_𝐱^𝑇𝐒_𝐲𝐱=1𝐡2ℎ^𝑖𝐱⁻_𝑘+1+𝐬⁻_𝐱,𝑗−𝐡_𝑖𝐱⁻_𝑘+1−𝐬⁻_𝐱,𝑗, (12)(13)(14) where 𝐱⁻_𝑘+1 is the predicted state and 𝐏⁻_𝑘+1 is the predicted covariance matrix.

Step 4 (observation and innovation covariance propagation). Consider 𝐲⁻_𝑘+1=𝐡𝐱⁻_𝑘+1𝐏^𝝂𝝂_𝑘+1=𝐒_𝐲𝐱𝐒_𝐲𝐱^𝑇+𝐑_𝑘𝐏^𝐱𝐲_𝑘+1=𝐒⁻_𝐱𝐒_𝐲𝐱^𝑇, (15)(16)(17) where 𝐲⁻_𝑘+1 is the predicted observation vector, 𝐏^𝝂𝝂_𝑘+1 is the innovation covariance matrix, and 𝐏^𝐱𝐲_𝑘+1 is the cross correlation matrix.

Step 5 (update). Consider the following: 𝜿_𝑘+1=𝐏^𝐱𝐲_𝑘+1𝐏^𝝂𝝂_𝑘+1⁻¹𝐏⁺_𝑘+1=𝐏⁻_𝑘+1−𝜿_𝑘+1𝐏^𝝂𝝂_𝑘+1𝜿^𝑇_𝑘+1𝝊_𝑘+1=𝐲_𝑘+1−𝐲⁻_𝑘+1𝐱⁺_𝑘+1=𝐱⁻_𝑘+1+𝜿_𝑘+1𝝊_𝑘+1, (18) where 𝜿_𝑘+1 is the gain, 𝐏⁺_𝑘+1 is the updated covariance matrix, 𝝊_𝑘+1 is the innovation vector, and 𝐱⁺_𝑘+1 is the updated estimated state.

2.2. Adaptive DDF Algorithm

As stated previously, the DDF algorithm assumes a complete prior knowledge of the process and the measurement noise statistics 𝐐_𝑘 and 𝐑_𝑘. However, 𝐐_𝑘 and 𝐑_𝑘 are unknown in most applications, and incorrect priori noise statistics can lead to performance degradation or even divergence for the solution. One of the effective ways to overcome this weakness is to use an algorithm to adapt the noise statistics. In this paper, we propose using a support vector regression adaptive scheme of the DDF to adjust 𝐐_𝑘 and 𝐑_𝑘, respectively.

2.2.1. Support Vector Regression

The principles underlying support vector regression (SVR) developed by Vapnik and are presented in several works (see [26, 27]).

Given the train set 𝑇𝑆={(𝐱₁,𝑦₁),…,(𝐱_𝑚,𝑦_𝑚)}∈(𝐑^𝑛×𝐑)^𝑚, where 𝐱_𝑖∈𝐑^𝑛, 𝑦_𝑖∈𝐑, 𝑖=1,2,…,𝑚, the SVR problem can be defined as solving for the nonlinear function 𝑔(𝐱) about 𝐱∈𝐑^𝑛 to construct a relationship between the output and an arbitrary input 𝐱: 𝑦(𝐱,𝝎)=^𝑚_𝑖=1𝜔_𝑖𝑔𝐱_𝑖+𝑏=(𝝎⋅𝐠(𝐱))+𝑏. (19)

In [26], the author shows that changing the regression estimate minimizes the risk functional by using the following form: 𝑦(𝐱,𝜶)=^𝑚_𝑖=1𝛼^∗_𝑖−𝛼_𝑖𝑔𝐱_𝑖(𝐱=⋅𝑔)+𝑏_𝑚𝑖=1𝛼^∗_𝑖−𝛼_𝑖𝐾𝐱_𝑖,𝐱+𝑏, (20) where 𝛼^∗_𝑖 and 𝛼_𝑖 are Lagrange multipliers that satisfy the condition 𝛼_𝑖,𝛼^∗_𝑖≥0, 𝛼_𝑖𝛼^∗_𝑖=0 and 𝐾(𝐱_𝑖,𝐱) is a kernel function that satisfies Mercer’s condition. In this paper, we use a translation invariant Gaussian kernel, that is, =exp(−‖𝐱_𝑖−𝐱‖²/2𝜎²).

2.2.2. Adaptive Scheme Based on SVR (𝐐_𝑘 Is Fixed)

The covariance matrix 𝐑_𝑘 represents the accuracy of the measurement instrument. If we assume that the noise covariance 𝐐_𝑘 is completely known, then we can derive the SVR algorithm to estimate the measurement noise covariance 𝐑_𝑘 by defining an adaptive factor Δ𝑟_𝑘 to get the form: 𝐑_𝑘=Δ𝑟_𝑘𝐑, (21) where 𝐑 is the constant noise covariance matrix.

This work uses such an SVR algorithm to derive the adaptive factor at time instant 𝑘, so as to estimate the value of 𝐑_𝑘 during the algorithm’s execution.

The innovation sequence 𝝊_𝑘+1 has a theoretical covariance 𝐏^𝝂𝝂_𝑘+1, as (16) shows. The actual residual covariance 𝐏^𝝂𝝂_𝑘+1 can be approximated using its sample covariance by averaging inside a moving window of size 𝑁, where 𝐂_𝑘=1𝑁^𝑘_{𝑖=𝑘−𝑁+1}𝝊_𝑘+1𝝊^𝑇_𝑘+1. (22)

If 𝐂_𝑘 differs from 𝐏^𝝂𝝂_𝑘+1, then the diagonal elements of 𝐑_𝑘 can be adjusted to minimize the difference as much as possible. The size of this difference is given by 𝐃𝐎𝐌_𝑘=diag𝐏^𝝂𝝂_𝑘+1−𝐂_𝑘∈𝐑^𝑛, (23) where the function diag denotes the diagonal elements of the matrix. If the elements of 𝐃𝐎𝐌_𝑘 are rounded with zero, the covariance matrix 𝐑_𝑘 is likely accurate; otherwise there may be a large deviation between values.

To adjust 𝐑_𝑘, we define the SVR train set 𝑇𝑆_Δ𝑟 as 𝑇𝑆_Δ𝑟={(𝐃𝐎𝐌_{𝑘−𝑁+1},Δ𝑟_{𝑘−𝑁+1}),…,(𝐃𝐎𝐌_𝑘,Δ𝑟_𝑘)}. With this form, we can then create a corresponding function about the adaptive factor Δ𝑟_𝑘 and 𝐃𝐎𝐌_𝑘 based on (19), such that when 𝐃𝐎𝐌_𝑘+1 comes, then Δ𝑟_𝑘+1 can be resolved by the function.

To avoid having any element within the train set be too large to affect the accuracy of Δ𝑟_𝑘, we use a normalized function of the train set 𝑇𝑆; that is, 𝑇𝑆^′_Δ𝑟={(𝐃𝐎𝐌^′_{𝑘−𝑁+1},Δ𝑟^′_{𝑘−𝑁+1})…,(𝐃𝐎𝐌^′_𝑘,Δ𝑟^′_𝑘)}, where 𝐃𝐎𝐌^′_𝑘 and 𝑟^′_𝑘 can be expressed as follows: 𝐃𝐎𝐌^′_𝑘=1𝑁^𝑁_𝑖=1𝐃𝐎𝐌_{𝑘−𝑖+1,𝑗}𝑗=1,2,…𝑛,Δ𝑟^′_𝑘=1𝑁^𝑁_𝑖=1Δ𝑟_{𝑘−𝑖+1}. (24)

2.2.3. Adaptive Scheme Based on SVR (𝐑_𝑘 Is Fixed)

Assuming that the noise covariance 𝐑_𝑘 is completely known, we can derive an SVR algorithm to estimate the measurement noise covariance 𝐐_𝑘. From (13), (14), and (16), we can deduce that a change in 𝐐_𝑘 will affect the covariance matrix 𝐏^𝝂𝝂_𝑘+1; if we increase 𝐐_𝑘, then 𝐏^𝝂𝝂_𝑘+1 also increases. We can adjust 𝐐_𝑘 in the SVR by deliberately mismatching 𝐏^𝝂𝝂_𝑘+1 and 𝐂_𝑘+1.

We first define an adaptive factor Δ𝑟_𝑘, where 𝐑_𝑘 has the following form: 𝐐_𝑘=Δ𝑞_𝑘𝐐, (25) where 𝐐 is the constant noise covariance matrix.

We can then define the SVR train set 𝑇𝑆_Δ𝑞 as 𝑇𝑆_Δ𝑞={(𝐃𝐎𝐌_{𝑘−𝑁+1},Δ𝑟_{𝑘−𝑁+1}),…,(𝐃𝐎𝐌_𝑘,Δ𝑟_𝑘)}. At this point, the solution process is the same as for solving Δ𝑟_𝑘.

3. Monte Carlo Simulation Results and Discussion

In this section, we report the experimental results obtained by applying SVRADDF to the nonlinear state estimation of a nonmaneuvering target in an underwater tracking control scenario and a maneuvering target in an air-traffic control scenario. To demonstrate the performance of the SVRADDF algorithm, we compare its performance against a DDF algorithm.

3.1. Underwater Nonmaneuvering Target Bearing-Only Tracking Control Scenario

We consider a bearing-only tracking control scenario, where an underwater target executes a uniform motion in a horizontal plane but unknown velocity, while a passive sonar platform performs a uniform circular motion in a horizontal plane. Figure 2 shows a representative trajectory of the target and the passive sonar platform. The kinematics of the relative motion between the target and the platform can be modeled using the following linear process equation: 𝐱_𝑘+1=𝐱10𝑇0010𝑇00100001^𝑘+𝐰_𝑘. (26)

Figure 2 

Target trajectory (I-initial position and F-final position) and sonar trajectory.

Here, the state of the equation is 𝐱=[𝑥𝑦𝑥̇𝑦]̇, where 𝑥 and 𝑦 denote position and 𝑥̇ and 𝑦̇ denote velocity in the 𝑥 and 𝑦 directions, respectively, and 𝑇 is the time interval between two consecutive measurements and the process noise 𝐰_𝑘∼𝑁(0,𝐐) with a nonsingular covariance where 𝐐=𝑞₁×𝑇²2𝑇²2𝑇𝑇. (27)

In (27), the parameter 𝑞₁ is related to process noise intensities. The measurement equation is written as follows: 𝐲_𝑘=𝜃_𝑘=tan⁻¹𝑦_𝑘𝑥_𝑘+𝜐_𝑘, (28) where the measurement noise 𝜐_𝑘∼𝑁(0,𝑅) with a nonsingular covariance.

Given the following initial conditions: 𝑞𝑇=1s₁=0.0001m²s⁻³𝑅=0.02mrad, (29)

the true initial state is 𝐱₀=0m1500m0ms⁻¹0ms^−1𝑇 (30) and the associated covariance is 𝐏₀=diag100m²2000m²1m²s⁻²1m²s⁻². (31)

The initial estimate state 𝐱₀ was chosen randomly from 𝑁(𝐱₀,𝐏₀) in each run, and the total number of scans per run was 1000.

To provide a fair comparison, we performed 50 independent Monte Carlo runs. To track the underwater target, we used both the SVRADDF and the DDF algorithms and compared their performance. The adaptive factor was set to Δ𝑞_𝑘=Δ𝑟_𝑘=0.1. Both of the filters were initialized with the same initial conditions for each run.

Performance metrics: to compare the nonlinear performance of the filters, we used the root mean square error (RMSE) of the target position and velocity. The RMSE yields a combined measure of the bias and variance of a filter estimate. The RMSE of the position at time 𝑘 was found using RSME_pos(𝑘)=1𝑁^𝑁_𝑖=1𝑥^𝑖_𝑘−𝑥^𝑖_𝑘²+𝑦^𝑖_𝑘−𝑦^𝑖_𝑘², (32) where (𝑥^𝑖_𝑘,𝑦^𝑖_𝑘) and (𝑥^𝑖_𝑘,𝑦^𝑖_𝑘) are the true and estimated positions, respectively, in the 𝑖th Monte Carlo run. The form for the RMSE of the velocity is similar.

Figures 3 and 4 show the estimated RMSE in target position and velocity. The SVRADDF uses SVR to adjust the adaptive factor during algorithm execution, which leads to a marginally better performance compared to the DDF, as seen in the figures.

Figure 3 

RMSE in position for DDF and SVRADDF.

Figure 4 

RMSE in velocity for DDF and SVRADDF.

3.2. Maneuvering Target Tracking in the Air-Traffic Control Scenario

A typical air-traffic control scenario was considered next, where an aircraft executes a maneuvering turn in a horizontal plane at a constant and known turn rate Ω. Figure 5 shows a representative trajectory of the aircraft.

Figure 5 

Aircraft trajectory and radar location.

The kinematics of the turning motion can be modeled using the following nonlinear process equation: 𝐱_𝑘=1sinΩ𝑇Ω0−1−cosΩ𝑇Ω00cosΩ𝑇0−sinΩ𝑇1−cosΩ𝑇Ω1sinΩ𝑇Ω0sinΩ𝑇0cosΩ𝑇×𝐱_𝑘−1+𝐰_𝑘−1. (33)

The state of the aircraft is given by 𝐱=[𝑥𝑥̇𝑦𝑦]̇^𝑇, where 𝑥 and 𝑦 denote position and 𝑥̇ and 𝑦̇ denote velocity in the 𝑥 and 𝑦 directions, respectively, and 𝑇 is the time interval between two consecutive measurements and the process noise 𝐰_𝑘∼𝑁(0,𝐐) with a nonsingular covariance, where 𝐐=𝑞₁×𝑇³3𝑇²2𝑇00₂2𝑇𝑇0000³3𝑇²2𝑇00²2𝑇. (34)

The parameter 𝑞₁ related to process noise intensities. A passive radar is fixed at the origin and equipped to measure the bearing 𝜃. The measurement equation is written as follows: 𝐲_𝑘=𝜃_𝑘=tan⁻¹𝑦_𝑘𝑥_𝑘+𝜐_𝑘, (35) where the measurement noise 𝜐_𝑘∼𝑁(0,𝑅).

The parameters used in this simulation were 𝑇=1s, Ω=−3^∘s⁻¹, 𝑞₁=0.1m²s⁻³, and 𝑅=√10mard. The true initial state of the aircraft was 𝐱₀=[1000m300ms⁻¹1000m0ms⁻¹]^𝑇 and the associated covariance matrix was 𝐏₀=diag[100m²10m²s⁻²100m²10m²s⁻²]. The initial state 𝐱₀ for the filters was chosen randomly from 𝑁(𝐱₀,𝐏₀) in each Monte Carlo run, and the simulation time per run was 1000.

For a fair comparison, we performed 100 independent Monte Carlo runs for each filter. To track the maneuvering aircraft, we used both the SVRADDF and the DDF algorithms and compared their performance. The adaptive factor was set to Δ𝑞_𝑘=0.3 and Δ𝑟_𝑘=0.5. Both filters were initialized with the same initial conditions for each run.

Figures 6 and 7 show the estimate RMSE in target position and velocity for the SVRADDF and DDF filters. Not surprisingly, both filters exhibit divergence due to a mismatch between the initial filter design assumption and the Gaussian noise nature of the problem. The SVRADDF filter exhibits marginally better performance compared to the DDF since it was able to adjust the statistical properties of the noise during execution.

Figure 6 

RMSE in position for DDF and SVRADDF.

Figure 7 

RMSE in velocity for DDF and SVRADDF.

4. Conclusions

This work has proposed and developed an innovation-based SVRADDF algorithm. The algorithm introduces an adaptive factor estimated using SVR, which allows for estimation of the noise statistical characteristics of nonlinear stochastic systems. The SVRADDF algorithm avoids instability and divergence in the solution which is caused by incorrect statistical characteristics of the noise. Monte Carlo simulation results of an underwater nonmaneuvering bearing-only target tracking system and a maneuvering target bearing-only tracking system in an air-traffic control setting showed that the SVRADDF algorithm provides better state estimation accuracy than a traditional DDF algorithm.

Although the SVRADDF algorithm has showed better performance under Monte Carlo simulation, there are still several challenging issues to be considered for future study. To improve its feasibility and effectiveness in the complex environment, more comprehensive and detailed studies are still needed to solve under the nonlinear and non-Gaussian noise conditions. Since the algorithm has only been tested under Monte Carlo simulation, the following work might be testing the algorithm under trial data to approve its better performance online.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research work is supported by the National Natural Science Foundation of China (Grant no. E091002/50979017), Ph.D. Programs Foundation of Ministry of Education of China and Basic Technology (Grant no. 20092304110008), and Harbin Science and Technology Innovation Talents of Special Fund Project (Outstanding Subject Leaders) (no. 2012RFXXG083).