Smita Sadhu

Sigma point Kalman filter for bearing only tracking

Relative merits of sigma point Kalman filters (SPKF), also known as unscented Kalman filters (UKF) vis-a-vis extended Kalman filter (EKF) and iterated extended Kalman filter (IEKF) for a bearing-only target-tracking problem using rms error and robustness with respect to outlier initial conditions are explored. After establishing that the rms error performance obtainable by SPKF/UKF and IEKF for this fairly severe non-linear system is similar to those obtainable from other competing techniques, the relative robustness of IEKF, SPKF/UKF and EKF with respect to large initial condition uncertainty (a common occurrence for this class of tracking problems) is investigated. Using several versions of SPKF/UKF, it is shown that SPKF is about 20 times more robust compared to EKF. It is illustrated that the additional design freedom available with a sealed version of SPKF/UKF may be utilised for further improvement of the robustness. The main contribution of this paper is quantification of relative robustness of these non-linear filters. A simplified criterion is suggested and used for quantifying track loss and the relative occurrence of such track loss in batch Monte Carlo simulation has been used as a measure of (lack of) robustness. As the SPKF/UKF does not introduce substantial computational burden, when compared to EKF, it is argued that SPKF/UKF algorithm may become a strong candidate for on-board implementation.

Central Difference Formulation of Risk-Sensitive Filter

A numerically efficient algorithm for risk-sensitive filters (known to be robust to model uncertainties) of nonlinear plants, using central difference approximation is proposed. The proposed filter, termed central difference risk-sensitive filter (CDRSF), overcomes several disadvantages associated with the extended risk-sensitive filter (ERSF), reported earlier. The theory of formulation and the algorithm of the CDRSF are presented. With an example, it is demonstrated that the proposed new filter would give much better tracking performance compared to the ERSF for certain nonlinear systems. The CDRSF would be nearly as fast as the ERSF, thus making it more preferable for real-time applications compared to the risk-sensitive particle filter

Sight Line Rate Estimation in Missile Seeker Using Disturbance Observer-Based Technique

Filtering of base motion disturbance from the sight line rate is necessary for homing guidance of missiles. The present work proposes using a noninvasive seeker filter based on the disturbance observer concept to extract the target sight line rate signal from the raw signal corrupted with base motion disturbance. It is shown that the disturbance observer-based filter in favorable (nominal) condition can totally eliminate the platform motion from the raw sight line rate signal. The performance of the filter for off-nominal condition is also derived and exemplified. Simulation results demonstrate improved performance of the proposed filter compared to previously reported schemes.

Particle Methods for Risk Sensitive Filtering

Risk sensitive filters (RSF) are known to be robust in the presence of uncertainties in the system parameters. Unfortunately these filters only admit closed form expressions for a very limited class of models including finite state-space Markov chains and linear Gaussian models. In this paper, we present an efficient Monte Carlo particle implementation of these filters for non-linear and non-Gaussian state-space models. This non-standard particle algorithm is based on a probabilistic interpretation of the RSF recursion. This algorithm significantly extends the range of applications of risk-sensitive techniques. Simulation results demonstrate the performance of the algorithm.

Particle-method-based formulation of risk-sensitive filter

A novel particle implementation of risk-sensitive filters (RSF) for nonlinear, non-Gaussian state-space models is presented. Though the formulation of RSFs and its properties like robustness in the presence of parametric uncertainties are known for sometime, closed-form expressions for such filters are available only for a very limited class of models including finite state-space Markov chains and linear Gaussian models. The proposed particle filter-based implementations are based on a probabilistic re-interpretation of the RSF recursions. Accuracy of these filtering algorithms can be enhanced by choosing adequate number of random sample points called particles. These algorithms significantly extend the range of practical applications of risk-sensitive techniques and may also be used to benchmark other approximate filters, whose generic limitations are discussed. Appropriate choice of proposal density is suggested. Simulation results demonstrate the performance of the proposed algorithms.

Adaptive Grid Solution of Risk Sensitive Estimator Problems

An on-going work proposing a novel method for numerical computation of risk sensitive state estimates for non-linear non-Gaussian problems is reported. The algorithm is based on point mass approximation also called the grid method and utilises a modified form of information state based recursive relation, proposed and proved as a theorem. The modified form is claimed to be more efficient for numerical evaluation of risk sensitive estimate, especially for aposteriori risk sensitive state estimation. Though grid based filters are known for low numerical efficiency, heuristics for adaptive choice of grid points has been proposed to alleviate the shortcoming. The performance of this filter is demonstrated with a linear Gaussian case. Salient features of this Adaptive Grid RSF is then contrasted against the recently proposed Risk Sensitive Filters using the particle approach.

Risk Sensitive Estimators for Inaccurately Modelled Systems

Robustness of risk sensitive (RSE) estimators/filters for inaccurately modelled plant are elucidated and exemplified. A theorem which allows alternative pathway for deriving RSE filter relation and derivation of different closed form relations for RS filters in linear Gaussian cases is provided. Consequently, errors in expressions in earlier publications have been detected and rectified. Properties of RS filters are briefly reviewed and the interpretation of robustness of RS filters elaborated. Using Monte Carlo simulation, it is shown that RS filters perform significantly better compared to risk-neutral filters when (i) process noise covariance is in error (ii) the true system (truth model) contains unmodelled bias (iii) the state transition matrix is inaccurately known. Design pragmatics for the choice of the risk sensitive parameter is indicated.

Alternative Formulation of Risk-Sensitive Particle Filter (Posterior)

An algorithm for posterior risk-sensitive particle filter for nonlinear non-Gaussian system has been proposed in this paper. For Gaussian linear measurement case optimal proposal and for nonlinear Gaussian measurement case linearized version of optimal proposal for risk-sensitive particle filter is derived. The applicability of nonlinear risk-sensitive filters such as extended risk-sensitive filter (ERSF), central difference risk-sensitive filter (CDRSF) as a proposal for risk-sensitive particle filter is discussed. The proposed filter is applied to a highly nonlinear Gaussian system. Results are provided to show the comparative performance of extended risk-sensitive filter (ERSF), posterior risk-sensitive particle filter (RSPF) and adaptive grid risk-sensitive filter (AGRSF) for a representative run. Root mean square error (RMSE) of the proposed filter has also been provided and compared with ERSF and AGRSF. The computational cost of the proposed risk-sensitive estimator is studied and compared with other nonlinear risk-sensitive filters

A Risk Sensitive Estimator for Nonlinear Problems using the Adaptive Grid Method

An Adaptive Grid Method based on the well-known point-mass approximation has been developed for computation of risk-sensitive state estimates in non-linear non-Gaussian problems. Risk-sensitive estimators, believed to have increased robustness compared to their risk neutral counterparts, admit closed form expressions only for a very limited class of models including linear Gaussian models. The Extended Risk-Sensitive Filter (ERSF) which uses an EKF-like approach fails to take care of non-Gaussian problems or severe non-linearities. Recently, a particle-filter based approach has been proposed for extending the range of applications of risk-sensitive techniques. The present authors have developed the adaptive grid risk-sensitive filter (AGRSF), which was partially motivated by the need to validate the particle filter based risk-sensitive filter and uses a set of heuristics for the adaptive choice of grid points to improve the numerical efficiency. The AGRSF has been cross-validated against closed-form solutions for the linear Gaussian case and against the risk-sensitive particle filter (RSPF) for fairly severe non-linear problems which create a multi-modal posterior distribution. Root mean square error and computational cost of the AGRSF and the RSPF have been compared.

Evolving homing guidance configuration with Cramer Rao Bound

This paper advocates the use of Cramer Rao Bound (CRB) as a tool to aid decisions on guidance system configuration for homing missiles. It is argued that the CRB provides a quantitative understanding of the influence of model parameters and instrumentation/signal processing capabilities of the tracking filter performance, without going into the specifics of filter design and elaborate Monte Carlo performance analysis. The concepts have been demonstrated by a bearing only target-tracking (BOT) problem. The CRB performance for position and velocity error has been studied with respect to (i) variation of the sampling time, (ii) simplistic change in the measurement noise variance, (iii) the effect of introducing additional measurement with different noise variances, (iv) the effect of eclipsing on measurement The example demonstrates that the CRB analysis provides a good handle for tracking system design trade off. The situations where CRB results may mislead are also indicated.