System Model Bias Processing Approach for Regional Coordinated States Information Involved Filtering

In the Kalman filtering applications, the conventional dynamic model which connects the states information of two consecutive epochs by state transitionmatrix is usually predefined and assumed to be invariant. Aiming to improve the adaptability and accuracy of dynamicmodel, we proposemultiple historical states involved filtering algorithm.An autoregressivemodel is used as the dynamic model which is subsequently combined with observation model for deriving the optimal window-recursive filter formulae in the sense of minimum mean square error principle. The corresponding test statistics characteristics of system residuals are discussed in details. The test statistics of regional predicted residuals are then constructed in a time-window for model bias testing with two hypotheses, that is, the null and alternative hypotheses. Based on the innovations test statistics, we develop a model bias processing procedure including bias detection, location identification, and state correction. Finally, the minimum detectable bias and bias-tonoise ratio are both computed for evaluating the internal and external reliability of overall system, respectively.


Introduction
The Kalman filter (KF) has been widely used in various fields, such as target tracking, navigation estimation, distributed network, and multiagent consensus problem [1][2][3][4][5].Under the assumption of correct system functional and stochastic models, it generates a statistically optimal estimate of underlying system state by recursively operating on streams of noisy input data [6].However, in real applications, the mathematical system model is not perfect such that the assumption is not always satisfied.For example, false modeled noise, improper system model, or unexpected sudden changes of state existed during filtering process [7], thus degrading the solution precision and even leading to divergence of filter.
Many adaptive KF methods have been developed for dealing with abnormal system noise by compensating their influence into stochastic model where the corresponding variance of noise is inflated to reduce the contributions of dynamic model or observation model to the KF final solution [8,9].Such methods essentially reduce or even withdraw the inaccurate a priori knowledge of noise by adjusting the weights between the dynamic model and observation model.In fact, these methods are based on stochastic model, that is, variance of noises.A more straightforward strategy is to use their first moment information which can be referred to functional model modification.The unmodelled error is treated as a systematic error, namely, bias which is examined by system innovations based testing theory through generalized likelihood ratio method or the specific chi-square or consistency test [10,11], and then the bias is further identified and adapted in the ways of data snooping [12,13].Another representative method is the interactive multiple model based Kalman filtering.It is in principle a hybrid state estimation filtering where a set of models must be selected in advance to capture the complex target dynamics [14].Recently, a self-constructed trajectory based dynamic model has been constructed and successfully applied in navigation campaigns [2].By using multiple states information, it well accommodates the dynamic model bias that cannot be adequately described in the predefined invariant dynamic model of conventional KF, for instance, accelerating, turning, and maneuverings versus constant velocity dynamic model.

Mathematical Problems in Engineering
Substantially, it is an autoregressive (AR) model and this provides a new thought to establish a time-variant dynamic model.However, there is still potentially a considerable possibility of bias occurrence.Moreover, unexpected bias occurs during observation and thus also contaminates the observation model.It is therefore of great importance to develop the quality control strategy for dynamic system.
Aiming to construct a general accurate dynamic model, we conduct an optimal window-recursive filter for processing AR- dynamic model in the sense of minimum mean square error (MMSE) principle.Considering the test statistics characteristics of filtering, the regional innovation test statistics are constructed in a time-window for two hypotheses testing, that is, the null and alternative hypotheses.Based on the innovations test statistics, we further develop a model bias processing procedure including bias detection, location identification, and state correction.The minimum detectable bias and bias-to-noise ratio are both computed for the internal and external reliability analyses of overall system as well.This paper is organized as follows.Section 2 optimally derives the recursive formulae for multiple historical states involved filtering.The characteristics of local model test statistics are discussed in Section 3. The regional test statistics for system innovation are constructed in the two scenarios in Section 4. The system model bias processing procedure is developed in three stages in Section 5. Section 6 investigates the internal and external reliability of our proposed filter model.The concluding remarks are given in Section 7.

Principle of Multiple Historical States Involved Filtering
The -order autoregressive model are widely expressed by where the subscript  is the time step;  denotes the number of involved epochs; x is the actual state with dimension ; a is the autoregressive coefficient; w is the zero-mean normal distributed process noise with the variance of Q w .The intuitive representation of (1) in state space form can be given as However, as  is getting large, the computational burden of such state augmentation will be accordingly increased.For this reason, we rewrite AR- model (1) into matrix form without state augmentation: where Φ (,−:−1) = (a 1 a 2 ⋅ ⋅ ⋅ a  ) is the transition matrix transforming the states of the previous  epochs into the current one, x (−:−1) is the vector that consists of the stacked state vectors from the epoch ( − ) to the epoch ( − 1), and the corresponding covariance matrix is Q x(−:−1) .On the other hand, the linear or linearized observation model reads where l is the measurement vector with dimension ;  is the zero-mean normal distributed measurement noise with the variance of R; A is the design matrix.It is easy to compute the predicted state and its variance by where x  and Q x  denote the predicted state and its corresponding variance, respectively; Q x(−:−1) is the variance of x(−:−1) .Let us start derivations with the following defined linear expression of estimated state: where K  is the so-called gain matrix to be determined.Denoting the a priori and posterior estimate errors x,−1 = x  − x  and x = x − x  , respectively, it is rather easy to compute the estimated state variance with where (⋅) denotes the expectation operator.Alternative expression of ( 8) can be written as in which I is the identity matrix which has the same dimension with state vector.Next we will find out K  in the sense of MMSE satisfying the following condition: with the fact of where Tr(⋅) denotes the trace of matrix.Thereby the unique solution of K  is solved by Furthermore, we also obtain the state estimate covariance: Note that differing from conventional KF formulae derivation, the correlation between x and x(−:−1) exists and should be rigorously considered when the time-window moves forward.Rewrite (8) into following form: Inserting ( 5) into ( 14) yields Noting that l  is uncorrelated with x(−:−1) , the covariance matrix Q x x(−:−1) between x and x(−:−1) is then derived in terms of the error propagation law as We further symbolize the variance of x(−:) in block matrix as Thus, with the window moving forward one step, the new epoch will be introduced and the first epoch removed.It is also rather easy to derive the filtering solution x(−+1:+1) and Q x(−+1:+1) .It should be pointed out that when the timewindow length  = 1, the multiple historical states involved filtering algorithm reduces to the KF.Its implementation procedure is summarized as follows: (i) Initialize the historical state information of x(−:−1) and Q x(−:−1) of  epochs.
The above multiple states involved filtering generates optimal estimators of state only when the system noises are mutually uncorrelated and zero-mean normal distributed.However, such assumption is not always satisfied in real applications.Misspecifications in the system model and unexpected errors inevitably generate biased estimation results.It is therefore of great importance to correctly deal with these system model faults and outliers.In the next section, we will investigate the model bias processing theory to ensure the accuracy and reliability of overall system.

Test Statistics Characteristics of Local Filter Test
According to (3) and ( 4), three error sources exist in the filter solutions, that is, the system process noise w  , the observation noise   , and the errors of x(−:−1) .Based on this point of view, the system models can be reformulated as the following residual equations for pseudo-observations [7]: where z x  = Φ (,−:−1) x(−:−1) , z w  = 0, and z l  = l  with their pseudo-observation variance matrices by , Defining a new observation vector , we obtain its corresponding partitioned matrix , then the following test statistics widely used in system diagnosing of local system model can be constructed as where  is the redundancy of z  .In practice, the following alternate chi-square test is more popular: where k  is the system innovation defined as the difference between raw observation and predicted one derived from the predicted state: Evidently, it mixes up all the error sources in (18)∼(20).Further inserting ( 5) into (24), the innovation vector arrives at Accordingly, its variance matrix is derived by It should be pointed out that the two test statistics in ( 22) and ( 23) have been proved to be equivalent [15].Therefore, we use the distribution of system innovation vector to test the mathematical model of filter.

Regional Test Statistics for System Innovation
The aforementioned local test statistics can be potentially used to detect sudden failures or outliers in system model.However, there are two drawbacks: (i) the local testing is too insensitive to detection of the slowly growing sensor errors such as gyroscope drift and accelerometer zero-bias; (ii) only the current observation information is included in the test statistics, thus degrading the reliability of testing results.Therefore, in this section, we introduce a sliding window based test statistics which contain information of multiple system innovations rather than one, thereby enhancing system reliability in the aspect of model fault detection.The predicted residual vectors of  epochs can be expressed as The following null and alternative hypotheses are considered: where Δk (−+1:) is an unknown residual bias vector with dimension ∑  =−+1   .Define the following relationship between Δk (−+1:) and unknown model bias vector b: where is a projection matrix which can be interpreted as the transformation from system model bias into innovation bias.Here two typical model errors scenarios are considered.
Case A. A bias error vector b occurs in the dynamic model at epoch , and the dynamic model at arbitrary epoch is expressed as where B  is the coefficient matrix denoting how the bias vector b enters into the dynamics.For convenience, the transition matrix Φ (,−:−1) is denoted as Φ  in the latter expressions.
Then (30) can accommodate the unmodeled error in dynamic model, for example, maneuverings, accelerating for vehicles with constant velocity.The matrix C k  is conducted by where H  is the coefficient matrix of observation bias vector.Thereby the matrix C k  is computed by 4.1.Generalized Likelihood Ratio Test Statistics.Thus far, the alternative hypothesis has been specified.Next we introduce the generalized likelihood ratio test statistic for testing  0 against   as follows [13]: Reject  0 if where Q k (−+1:) is the covariance matrix of k (−+1:) .As aforementioned, the bias occurs at epoch , and then (35) can be further rewritten as where the innovation residual vectors from  to  are calculated by Evidently, the innovation residuals vectors in (38) are strongly correlated with each other.Thus, Q k (:) is not a diagonal matrix and its covariance should be rigorously considered.
Let us reformulate the above arbitrary innovation residual vector as Inserting (40) into (39), and reformulating (38) into matrix form, we obtain where Therefore, in terms of the error propagation law, the covariance matrix Q k (:) is derived by where Q U (:) is the covariance of U (:) .For the independence of subvectors, Q U (:) is written in a form of block diagonal matrix as ] . (44)

The System Model Bias Processing Procedure
In this section, based on the testing principle of previous section, we propose a system model bias processing procedure including three steps, that is, bias detection, location identification, and state correction.
Step 1 (bias detection).First we implement the regional testing with test statistic  to examine whether the bias exists in the time-window of filter.Therefore, the overall system model testing with (35) is carried out based on following  0 and   , respectively, where  is the noncentrality parameter and can be computed by 0 will be accepted if  (−+1:) <  2 (∑  =−+1   , 0); that is, no bias occurs during the filtering process.Otherwise, we will accept the alternative hypothesis   .
Step 2 (time and location identification).If the null hypothesis  0 is rejected, we need to locate the starting time and position of model bias.Since the time when the bias occurs is still unknown, the test statistics screening strategy  is proposed here to find the most likely starting time of bias from the beginning of the time-window −+1 to the current epoch.The starting time and location of bias can be identified as follows: (i) Similar to the data snooping method, we construct alternative hypotheses for potential blunders through making the elements in coefficient matrix for quantizing the influences of model bias of overall system.A scalar squared bias-to-noise ratio is defined as [13] where Δx is the influences term of model bias on state in (48) for the alternative hypothesis   and Q x is the covariance matrix of bias-free state for the null hypothesis.Further inserting ( 13) and ( 49) into (60) yields Evidently, a large value of  BNR indicates the significant influences of model bias b on state estimation, while a small value of  BNR exhibits a good external reliability of system.
It is therefore of importance to design a reasonable system model structure for minimizing the influences of model bias on solutions.

Concluding Remarks
This paper proposes a model bias processing approach for the autoregressive (AR) dynamic model involved dynamic system.The optimal time-window recursive filtering formulae are derived in the case of AR- dynamic model in the sense of minimum mean square error (MMSE) principle.
The test statistics of regional system innovation, that is, predicted residuals under the null and alternative hypotheses, are constructed for model bias testing specified in two bias occurrence scenarios.Based on the system innovation test statistics, we develop a model bias processing procedure including bias detection, location identification, and state correction.The minimum detectable bias and bias-to-noise ratio are both computed and comprehensively analyzed for evaluating the internal reliability and external reliability, respectively, of overall system.In the future, since the recursive time-window will inevitably generate the strong correlations between innovations and state estimates, the multiple hypotheses will be further constructed considering the probability of wrong decisions on the bias location when the null hypothesis is rejected.In addition, the computations for the regional testing and processing of model bias are timeconsuming; thus, an efficient computation strategy is also desired in the future work.