Risk-sensitive Filtering Research Articles

Adaptive risk-sensitive filter for Markovian jump linear systems

In this paper, we present divergence-based minimax methods to derive the adaptive risk-sensitive filter for Markovian jump linear systems with parameter uncertainties. The risk-sensitive parameter is treated as a random variable rather than chosen as a prior, which reflects the hypothesis merging property. To tackle the proposed joint probability density function due to the introduction of the random variable, a method that projects orthogonally the original probability measure into a new independent conditional probability space in the Kullback–Leibler (KL) sense is then presented. The exact close-form expression can be obtained via optimization that is of a form suitable in a recursive procedure. The proposed filter can improve robustness by increasing the mixed covariance, whose risk-sensitive like parameter can be updated adaptively using the measurements to fit different situations well. Simulation examples illustrate that the proposed methods can be regarded as competitive alternatives to existing robust approaches against modeling uncertainties.

Nonlinear robust estimation for tracking of ballistic target using risk sensitive cubature quadrature Kalman filter

In this paper, a risk sensitive estimator based on cubature quadrature Kalman filter is formulated and applied for tracking a ballistic object during its re-entry phase. The formulated algorithm is named here as risk sensitive cubature quadrature Kalman filter. The process model and measurement model for two dimensional ballistic target motion is described. The expression for measurement noise covariance is derived for Cartesian coordinate from polar coordinate. Also expression for initial error covariance is derived. The performance of the risk sensitive cubature quadrature Kalman filter is compared with the cubature quadrature Kalman Filter in terms of root mean square error. The simulation results reveal that for wrongly modeled process noise parameter, the risk sensitive filter performs better than their risk neutral counterpart. Moreover, risk sensitive cubature quadrature Kalman filter performs better than risk sensitive cubature Kalman filter. Since expression for measurement noise covariance as well as initial error covariance for two dimensional ballistic target motion is derived, other Gaussian filters may also be applied for tracking of ballistic objects.Article HighlightsThe process model for two dimensional ballistic target motion is described comprehensively.The expression for measurement noise covariance matrix for linear measurement model is derived for Cartesian coordinate from polar coordinate.The expression for initial error covariance matrix for the filter initialized from the first two measurements is derived.Risk sensitive filter based on cubature quadrature Kalman filter is applied for tracking of two dimensional ballistic target motion and improvement in performance over its risk neutral counterpart is found.Improvement in performance of risk sensitive filter based on cubature quadrature Kalman filter over risk sensitive filter based on cubature Kalman filter is found.

Risk sensitive filtering with randomly delayed measurements

Conventional Bayesian estimation requires an accurate stochastic model of a system. However, this requirement is not always met in many practical cases where the system is not completely known or may differ from the assumed model. For such a system, we consider a scenario where the measurements are transmitted to a remote location using a common communication network and due to which, a delay is introduced while receiving the measurements. The delay that we consider here is random and one step maximum at a given time instant. For such a scenario, this paper develops a robust estimator for a linear Gaussian system by minimizing the risk sensitive error criterion that is defined as an expectation of the accumulated exponential quadratic error. The criteria for the stability of the risk sensitive Kalman filter (RSKF) are derived and the results are used to study the stability of the developed filter. Further, it is assumed that the latency probability related to delay is not known and it is estimated by maximizing the likelihood function. Simulation results suggest that the proposed filter shows acceptable performance under the nominal conditions, and it performs better than the Kalman filter for randomly delayed measurements and the RSKF in presence of both the model uncertainty and random delays.

Intelligent optimization in model-predictive control with risk-sensitive filtering

The uncertainty issue in real-work optimization affects the level of optimization significantly. Because most future uncertainties cannot be foreseen in advance, the optimization must take the uncertainties as a risk in an intelligent way in the process of computation algorithm. Based on our risk-sensitive filtering algorithm, this study adopts a model-predictive control to construct a risk-averse, predictable model that can be used to regulate the level of a real-world system. Our model is intelligent in that the predictive model needs not to identify the system parameters in advance, and our algorithm will learn the parameters through data. When the real-world system is under the disturbance of unexpected events, our model can still maintain suitable performance. Our results show that the intelligent model designed in this study can learn the system parameters in a real-world system and minimize unexpected real-world disturbances. Through the learning process, our model is robust, and the optimal performance can still be retained even the system parameters deviate from expected, e.g., material shortage in a supply chain due to earthquake. When parameter error risks occur, the control rules can still drive the overall system with a minimal performance drop.

The Interplay Between Missing Data and Out-of-Order Measurements using Data Fusion in Wireless Sensor Networks

Multi-data transmission is the most important processing of target detection with a reduction in delay in the transmission of the data. This may occur in certain technological circumstances, and it happens significantly often in wireless sensor networks—processing such data to keep track of and make predictions about targets of interest might result in errors due to the inherent nature of the data. The Kalman filter and other algorithms with equivalent functionality are most useful for their principal application, estimating the states of dynamic systems. This difficulty of modeling and filtering such delayed states and missing data is dealt with synergistically throughout this proposed work. This is done to ensure that the best possible results are obtained. Filtering methods similar to the optimal Kalman filter are most utilized in fusing measurement data at different levels. This relatively creative technique includes filtering delayed states while also using observations that have been randomly excluded, then putting those screened delayed states and words to use in a process that involves fusing data. One of these applications is the fusion of images. To successful the task of performance evaluation for the integrated plan, the use of numerical simulations is essential. The state delay, as well as the data that is absent at random, are both included in four distinct alternative algorithms. These algorithms are then investigated, and the results are given in this paper. Referring to the gain fusion, the H-infinity a posteriori filter, the H-infinity risk sensitive filter, and the H-infinity risk sensitive filter. To accommodate a scenario that involves MATLAB and the integration of sensor data, global filtering approaches are being updated and evaluated with the use of numerical simulations that are being carried out. In addition, we provide a nonlinear observer based on the gain of the continuous time data fusion filter. Using the Lyapunov energy function, we can conclude on asymptotic convergence in the system. These observers are presented after the previous step. Therefore, the filtering algorithms and the observers described in the current proposed work make a definite step towards improvement for controlling state delays and randomly missing data synergistically for wireless sensor networks.

Robust Kalman Filtering Under Model Perturbations

We consider a family of divergence-based minimax approaches to perform robust filtering. The mismodeling budget, or tolerance, is specified at each time increment of the model. More precisely, all possible model increments belong to a ball which is formed by placing a bound on the Tau-divergence family between the actual and the nominal model increment. Then, the robust filter is obtained by minimizing the mean square error according to the least favorable model in that ball. It turns out that the solution is a family of Kalman like filters. Their gain matrix is updated according to a risk sensitive like iteration where the risk sensitivity parameter is now time varying. As a consequence, we also extend the risk sensitive filter to a family of risk sensitive like filters according to the Tau-divergence family.

A Contraction Analysis of the Convergence of Risk-Sensitive Filters

A contraction analysis of risk-sensitive Riccati equations is proposed. When the state-space model is reachable and observable, a block-update implementation of the risk-sensitive filter is used to show that the $N$-fold composition of the Riccati map is strictly contractive with respect to the Thompson's part metric of positive definite matrices, when $N$ is larger than the number of states. The range of values of the risk-sensitivity parameter for which the map remains contractive can be estimated a priori. It is also found that a second condition must be imposed on the risk-sensitivity parameter and on the initial error variance to ensure that the solution of the risk-sensitive Riccati equation remains positive definite at all times. The two conditions obtained can be viewed as extending to the multivariable case an earlier analysis of Whittle for the scalar case.

Robust Ensemble Kalman Filter Based on Exponential Cost Function

AbstractThis paper provides an alternative point of view to the robust estimation technique for nonlinear non Gaussian systems based on exponential quadratic cost function. The proposed method, named the risk sensitive ensemble Kalman filter (RSEnKF), is based on the ensemble Kalman filter (EnKF) which may be thought of as a Monte Carlo implementation of the Kalman filter for nonlinear estimation problems. The theory and formulation of the RSEnKF are presented in this paper. The proposed method is superior to the extended risk sensitive filter (ERSF) and the quadrature based risk sensitive filters in terms of estimation accuracy, and is faster than the risk sensitive particle filter (RSPF).

Risk-sensitive filtering for discrete-time systems with time-varying delay

In this paper, the risk-sensitive filtering problem with time-varying delay is investigated. The problem is transformed into Krein space as an equivalent optimisation problem. The observations with time-varying delays are restructured as ones with multiple constant delays by defining a binary variable model with respect to the arrival process of observations, containing the same state information as the original. Finally, the reorganised innovation analysis approach in Krein space allows the solution to the proposed risk-sensitive filtering in terms of the solutions to Riccati and matrix difference equations.

Robust State Space Filtering Under Incremental Model Perturbations Subject to a Relative Entropy Tolerance

This paper considers robust filtering for a nominal Gaussian state-space model, when a relative entropy tolerance is applied to each time increment of a dynamical model. The problem is formulated as a dynamic minimax game where the maximizer adopts a myopic strategy. This game is shown to admit a saddle point whose structure is characterized by applying and extending results presented earlier in “Robust least-squares estimation with a relative entropy constraint” (B. C. Levy and R. Nikoukhah, IEEE Trans. Inf. Theory, vol. 50, no. 1, 89-104, Jan. 2004) for static least-squares estimation. The resulting minimax filter takes the form of a risk-sensitive filter with a time varying risk sensitivity parameter, which depends on the tolerance bound applied to the model dynamics and observations at the corresponding time index. The least-favorable model is constructed and used to evaluate the performance of alternative filters. Simulations comparing the proposed risk-sensitive filter to a standard Kalman filter show a significant performance advantage when applied to the least-favorable model, and only a small performance loss for the nominal model.

Quasi-equality constrained risk-sensitive filtering for nonlinear discrete-time systems

More and more data fusion models contain state constraints with valuable information in the filtering process. In this study, an optimal filter of risk-sensitive with quasi-equality constraints is formulated using the reference probability method. Through recursion processes of probability density acquired from the probability measure change, the derived algorithm is optimal in the sense of the risk-sensitive parameter. The system and constraint models are consistent in statistics. Simulation results show that it is more robust and efficient than projection filters for the worst-case of noises and model uncertainty.

Risk-sensitive filtering for nonlinear Markov jump systems on the basis of particle approximation

SUMMARY In this paper, the risk-sensitive filtering method that relaxes the dependence on model accuracy is extended to nonlinear Markov jump systems (MJSs). In the method, the so-called reference probability technique together with particle approximation is utilized to derive the risk-sensitive filter in nonlinear non-Gaussian framework. The novelty of the proposed approach is that a ‘risky’ interacting resampling step is performed to both moderate the modeling uncertainties and to solve the problem of particle explosion. A designer-chosen parameter named risk-sensitive parameter allows us to make a trade-off between the filtering accuracy for the nominal model and the robustness to uncertainties. With a meaningful example, it shows that the developed method can outperform the widely used method-particle filter and interacting multiple model-particle filter in nonlinear MJSs with uncertainties. Copyright © 2011 John Wiley & Sons, Ltd.

Bearings-only tracking with iterated extended risk sensitive filter

This paper concerns the application of an iterated extended risk sensitive filter (IERSF) to target tracking problems. The relative merits of IERSF vis-a-vis the extended risk sensitive filter (ERSF) for a bearings-only tracking problem using root mean square error (RMSE) and robustness with uncertainty in initial condition are explored. An ERSF weakness, specifically an accumulation error in the computation of innovation steps due to approximating nonlinear functions at a recently available prior estimate, is presented. By using the IERSF with proper tuning of risk factor and local iteration, the filtering divergence may be overcome, and a stable, robust, and unbiased estimation is satisfactorily obtained. With numerical simulation results, the tracking performance of IERSF is compared with the performance of ERSF and extended Kalman filter. The IERSF results in reduced estimation error without much of an increase in burden of the associated computational algorithm.

Adaptive grid risk-sensitive filter for non-linear problems

A novel adaptive grid-based method has been proposed for risk-sensitive state estimation in non-linear non-Gaussian problems. The algorithm, which is based on point-mass approximation, is called the adaptive grid risk-sensitive filter (AGRSF). Although risk-sensitive estimators have been known to be robust compared to their risk-neutral counterparts, the implementation of risk-sensitive filters (RSFs) is almost impossible except for very trivial systems like linear Gaussian systems. The existing extended risk-sensitive filter (ERSF) fails to take care of non-Gaussian problems or severe non-linearities. Recently, other variants of RSFs have been proposed for extending the range of applications of risk-sensitive techniques. The AGRSF has been formulated to act as a benchmark and aid in the validation of other RSFs. The algorithm uses a modified form of information state-based recursive relation and provides guidelines for the adaptive choice of grid points to improve the numerical efficiency. The developed filter has been applied to a single-dimensional non-linear poorly observable system and a non-linear two-dimensional bearing only tracking problem. The convergence of the algorithm has been shown by simulation. The estimation efficiency and computational load of AGRSF has been compared with other RSFs.

Optimal Risk-Sensitive Filtering for System Stochastic of Second and Third Degree

The risk-sensitive filtering design problem with respect to the exponential mean-square cost criterion is con-sidered for stochastic Gaussian systems with polynomial of second and third degree drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed-form optimal fil-tering equations are obtained using quadratic value functions as solutions to the corresponding Focker- Plank-Kolmogorov equation. The performance of the obtained risk-sensitive filtering equations for stochastic polynomial systems of second and third degree is verified in a numerical example against the optimal po-lynomial filtering equations (and extended Kalman-Bucy for system polynomial of second degree), through comparing the exponential mean-square cost criterion values. The simulation results reveal strong advan-tages in favor of the designed risk-sensitive equations for some values of the intensity parameters.

Risk-Sensitive Cubature Filtering for Jump Markov Nonlinear Systems and Its Application to Land Vehicle Positioning⋆

In this paper, a risk-sensitive filter for a class of jump Markov nonlinear systems is proposed. By using a set of weighted cubature points to approximate the intractable risk-sensitive recursions, the proposed filter could exhibit improved numerical stability with respect to the derivation of the square-root covariance, for which the unscented transform technique might halt the operations due to the presence of non-positive definite covariance. The performance of the proposed filter is illustrated through a simulation study of tracking a land vehicle with maneuvers.

Risk-sensitive filtering and smoothing for jump Markov non-linear systems based on unscented transform

This study is concerned with risk-sensitive filtering and smoothing for a class of discrete-time jump Markov non-linear systems. Using the so-called reference probability method, the authors present a general theoretical framework to yield recursions for deriving filtered and smoothed estimates through identifying the approximations made by the interacting multiple model (IMM) estimation approach. A suboptimal risk-sensitive filtering algorithm is developed by applying the unscented transform (UT) technique and the one-step fixed-lag smoothing result is also presented for such systems. The effectiveness of the proposed algorithms is demonstrated via a manoeuvering target tracking simulation study.

Risk-sensitive formulation of unscented Kalman filter

A novel method for non-linear risk-sensitive estimation based on the unscented transform has been developed. The proposed filter, referred to as risk-sensitive unscented Kalman filter (RSUKF), and would be able to overcome inherent disadvantages associated with the earlier reported extended risk-sensitive filter (ERSF). The theory and formulation of RSUKF has been presented and possible variants thereof indicated. Using two well-known non-linear examples, the superiority of RSUKF performance has been demonstrated. As the RSUKF has similar computational efficiency and better robustness as compared with the ERSF, the former may be more suitable for onboard implementation, quick exploration of risk-sensitive filtering for non-linear problems and also for generating proposal densities for more computation intensive risk-sensitive particle filters.

Risk-Sensitive Particle Filters for Mitigating Sample Impoverishment

Risk-sensitive filters (RSF) put a penalty to higher-order moments of the estimation error compared to conventional filters as the Kalman filter minimizing the mean square error (MSE). The result is a more cautious filter, which can be interpreted as an implicit and automatic way to increase the state noise covariance. On the other hand, the process of jittering, or roughening, is well known in particle filters to mitigate sample impoverishment. The purpose of this contribution is to introduce risk-sensitive particle filters (RSPF) as an alternative approach to mitigate sample impoverishment based on constructing explicit risk functions from a general class of factorizable functions. It is first shown that RSF can be done in nonlinear systems using a recursion of an infinite dimensional information state which involves general risk functions. Then, this information state calculation is carried out using particle approximations. Some alternative approaches, generalizations, specific cases, comparison to existing methods of sample impoverishment mitigation and issues related to the selection of risk functions and parameters are examined. Performance of the resulting filter using various risk functions is illustrated on a simulated scenario and compared with the roughening method.

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.