Two Stage Kalman Estimators with Probabilistically Weighted Average

With spherical c oordinate, the ad aptive e stimation using multiple model fil tering is enhanced in this pap er. The enhancement is achieved by using just two depended parallel Kalman filters, in stead of multiple models, with the p robabilistically weighted average, which p rovides the ad aptive me chanism. The f irst filter i s constant velocity filter (CVF ) which is used to estimate the position and velocity of the moving target in non maneuvering course. The second filter calculates th e acceleration and th e new adjustment for the CVF. T he second filter is referred as variable v elocity filter (VVF) . Monte Ca rlo computer simulation results are included to demonstrate the effectiveness of the proposed algorithm in enhancement the multiple model adaptive filtering


I. Introduction
Target tracking systems operating in a track-while-scan mode have great difficulty maintaining contact with targets performing unpredictable maneuvers methods for solving this problem there are variety of approaches.One of these methods utilizes a bank of N Kalman filter each designed to model different maneuver [1,2,9,11].The estimate of the state of the target is either the output of the filter having an innovation sequence closet to a white noise sequence or is a weighted sum of all N filter outputs.Computational time constraints generally limit the usefulness of this method.Maybeck and Suize [3] also addressed the problem and solved it by using multiple module filtering.The multiple are created by tuning filters for best performance.Bar-Shalom [4] present interacting multiple module algorithms.However, in these algorithms two or three Kalman filters are operating in parallel.By deriving a transition probability matrix with conditional probability, from the innovation sequence of each filter, the output will be the i probability weight sum of each filter.These algorithm gives an improvement over other algorithms such that Bogler algorithm [5].Djouadi and Berkany[10] use the IMM algorithm with the unscented Kalman filter to deal with non linear model.This paper investigates adaptation algorithm, which is enhancement of the two multiple adaptive filtering [3].It is named as enhancement multiple adaptive filtering [EMAF].In this algorithm, two parallel stage Kalman filters are used to generate state estimates from the shared sensor.Adaptive expansion is attained by generating the probabilistically weighted average of the two filter state estimates.

II. Two Stage Kalman Estimator and the Proposed Algorithm
The idea of using a two stage filter to implement an augmented state filter was introduced in[8].The idea is to decouple the central filter into two parallel filter.The first filter, the "bias-free" filter, is based on the assumption that the bias is nonexistent.The second filter, the bias filter, produces an estimate of the bias vector.The output of the first filter is then corrected with the output of the second filter [6].
in this paper, the proposed tracking algorithm (Fig. 1) is consist of two parallel filter worked together and from the property of the innovation sequence and state estimates of these filters, the adaptation detector switch can be worked depending on the probabilistically weighted average.moreover, the first filter is two state Kalman (constant velocity) filter that used to best performance for estimating the position and velocity of the target in case of non-maneuvering targets tracking, while the second filter(the acceleration filter) which depended on the residual sequence of the first filter, is single Kalman filter and it is used in parallel of the first Kalman filter to estimate the acceleration and the new estimate to the position and velocity of the maneuvered targets without modifying the operation of the first Kalman filter.The details about the operations and the simulation examples for the proposed PDF created with pdfFactory Pro trial version www.pdffactory.comtracking algorithm is given in the next sections

III. The Constant Velocity Filter
In the absence of maneuver, the target is modeled as a constant velocity object in a plant with some process noise that models slight changes in the velocity.The target process model, discretized over time Where, using spherical co ordinates and for one-dimensional range tracking situation, the state vector is given as And With Φ as the transition matrix and W(k) is white Gaussian noise sequence with, The covariance matrix where Where T is the sampling time, q is the spectral density of the continuous white noise change in acceleration process and 2 m σ is the variance of the change in acceleration noise.It is assumed that only range measurement is a variable where Filter state estimate: The two state Kalman is initialized as PDF created with pdfFactory Pro trial version www.pdffactory.comwhere y(0) and y(1) are, respectively, the first and second received sensor measurements The initial estimation error covariance for this coordinate is then The state measurement model which define in Eq.( 2) is become, Where H & V(k) is define in previous section.while C is defined by, The algorithm for compensating the output of the two state Kalman filter with the output of single Kalman filter is given by PDF created with pdfFactory Pro trial version www.pdffactory.com

IV. Weighting Coeficients Computation
As given in [1], let b denote the vector of uncertain parameters in a given model; here it is composed of the strength of the white noise driving the target acceleration model.In order to make identification of b tractable, its continues range values is discretized into L representative values.If we define the hypothesis conditional probability j P (for j=1,2,…………..,L) conditioned on the observed measurement history to time k, i.e.
and m is measurement dimension the Bayesian estimate of the state is the probabilistically weighted average: 24

) ( i j k b
will converge to unity for the coefficient corresponding to the true process and to zero for the others.

V. Simulation Results
The performance of EMAF algorithm is compared with other tracking methods.The filter tracking performance is evaluated by doing a Monte Carlo simulation of 50 runs and then tacking the average of these runs this is called ;time average root mean square(TARMS) error and given by [7] [ ] N is the number of samples for the trajectory, 2 N is the number of Monte Carlo Runs.The EMAF is compared with the single CVF and two stage Kalman filter estimator.Two maneuver scenarios considered for performance analysis.In the first maneuver scenarios, we assume that target is on a constant course and velocity until time t=120 second, when it maneuvers a slow 90° turn with acceleration input 30 m/sec 2 .It completes PDF created with pdfFactory Pro trial version www.pdffactory.coma turn at t=150 sec.Remaining course is constant velocity.The track filter parameters for the two scenarios are given by the following: The standard deviation of the observation additive white Gaussian noise σ r : [100 m] The anticipated standard deviation of the plant noise disturbance σ m : 5/sec 2 .The constant target radial velocity V: 300m/sec.A Monte Carlo simulation of 50 runs was obtained for each algorithm and the roots mean square (rms) values of the estimation errors were computed.Probability for each filter is shown in Fig. 2, while the rms range and velocity errors for two stage Kalman filter and EMAF are shown in Fig. 3 &Fig.4.It can be seen from the simulation results that EMAF not only yields improved performance during the maneuvering period, but also provides for better estimation during the non maneuvering period than the tracking filter using two stage Kalman filter.
The second maneuver scenarios is considered to test and compare performance of the EMAF with performance of Two-stage Kalman filter.In this scenarios, we assume that the target moves in a plane on constant course with constant velocity until time t=50 sec, when it maneuvers a 90° turn with acceleration 20 m/sec.It completes the turn at t=80 sec.Then non maneuvering course continuous for 40 sec, followed by the second 90° turn which stars at y = 120 with acceleration of 50 m/sec 2 and is completed at t = 150 sec.

VI. Conclusions
The enhanced multiple model adaptive filter EMMAF algorithm for tracking the non maneuvering and maneuvering targets has been presented and illustrated.The proposed algorithm is improve and enhance the MMAF because it used two depended parallel Kalman filter instead of multiple model with the probabilistically weighted average, which provide the adaptation mechanism.

And 2 rσ
is the variance of the observation channel noise (the error of the measured range).The noise process W(k) & ) (k V are uncorrected.The recursive two state Kalman filter equations are: Second(Acceleration) Kalman FilterWhen a maneuver occurs, the target process model in Eq.(1) Becomes[6]    Where B is acceleration transition vector defined by, B=[T 2 /2 T] T u(k) is the unknown acceleration term, this term can be deemed as either a random signal or a deterministic signal.Here we consider it as a random signal that influences the system dynamics.Gaussian sequence with zero mean and variance given by[6] The Equations for the acceleration filter and the correction steps are[ 6]: Where and R(k) is the residual of the first twostate filter.

Fig. 5
Fig.5 shows the probability for each filter and Fig. 6& Fig. 7, shows the rms estimation errors over 50 Monte Carlo runs.This figure show that both of the methods have similar

Fig
Fig.3c:The rms error of the proposed system of scenario1

Fig. 6c :
Fig.6c: The rms error of the proposed system of scenario2 .