Ienkaran Arasaratnam

Cubature Kalman Filters

In this paper, we present a new nonlinear filter for high-dimensional state estimation, which we have named the cubature Kalman filter (CKF). The heart of the CKF is a spherical-radial cubature rule, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear Bayesian filter. Specifically, we derive a third-degree spherical-radial cubature rule that provides a set of cubature points scaling linearly with the state-vector dimension. The CKF may therefore provide a systematic solution for high-dimensional nonlinear filtering problems. The paper also includes the derivation of a square-root version of the CKF for improved numerical stability. The CKF is tested experimentally in two nonlinear state estimation problems. In the first problem, the proposed cubature rule is used to compute the second-order statistics of a nonlinearly transformed Gaussian random variable. The second problem addresses the use of the CKF for tracking a maneuvering aircraft. The results of both experiments demonstrate the improved performance of the CKF over conventional nonlinear filters.

Discrete-Time Nonlinear Filtering Algorithms Using Gauss–Hermite Quadrature

In this paper, a new version of the quadrature Kalman filter (QKF) is developed theoretically and tested experimentally. We first derive the new QKF for nonlinear systems with additive Gaussian noise by linearizing the process and measurement functions using statistical linear regression (SLR) through a set of Gauss-Hermite quadrature points that parameterize the Gaussian density. Moreover, we discuss how the new QKF can be extended and modified to take into account specific details of a given application. We then go on to extend the use of the new QKF to discrete-time, nonlinear systems with additive, possibly non-Gaussian noise. A bank of parallel QKFs, called the Gaussian sum-quadrature Kalman filter (GS-QKF) approximates the predicted and posterior densities as a finite number of weighted sums of Gaussian densities. The weights are obtained from the residuals of the QKFs. Three different Gaussian mixture reduction techniques are presented to alleviate the growing number of the Gaussian sum terms inherent to the GS-QKFs. Simulation results exhibit a significant improvement of the GS-QKFs over other nonlinear filtering approaches, namely, the basic bootstrap (particle) filters and Gaussian-sum extended Kalman filters, to solve nonlinear non- Gaussian filtering problems.

Cubature Kalman Filtering for Continuous-Discrete Systems: Theory and Simulations

In this paper, we extend the cubature Kalman filter (CKF) to deal with nonlinear state-space models of the continuous-discrete kind. To be consistent with the literature, the resulting nonlinear filter is referred to as the continuous-discrete cubature Kalman filter (CD-CKF). We use the Itô-Taylor expansion of order 1.5 to transform the process equation, modeled in the form of stochastic ordinary differential equations, into a set of stochastic difference equations. Building on this transformation and assuming that all conditional densities are Gaussian-distributed, the solution to the Bayesian filter reduces to the problem of how to compute Gaussian-weighted integrals. To numerically compute the integrals, we use the third-degree cubature rule. For a reliable implementation of the CD-CKF in a finite word-length machine, it is structurally modified to propagate the square-roots of the covariance matrices. The reliability and accuracy of the square-root version of the CD-CKF are tested in a case study that involves the use of a radar problem of practical significance; the problem considered herein is challenging in the context of radar in two respects- high dimensionality of the state and increasing degree of nonlinearity. The results, presented herein, indicate that the CD-CKF markedly outperforms existing continuous-discrete filters.

Square-Root Quadrature Kalman Filtering

The quadrature Kalman filter (QKF) is a recursive, nonlinear filtering algorithm developed in the Kalman filtering framework. It computes the mean and covariance of all conditional densities using the Gauss-Hermite quadrature rule. In this correspondence, we develop a square-root extension of the quadrature Kalman filter using matrix triangularizations. The square-root quadrature Kalman filter (SQKF) propagates the mean and the square root of the covariance. Although equivalent to the QKF algebraically, the SQKF exhibits excellent numerical characteristics, but at the expense of increased computational complexity. We also present possible refinements of the generic SQKF.

Cubature Kalman smoothers

The cubature Kalman filter (CKF) is a relatively new addition to derivative-free approximate Bayesian filters built under the Gaussian assumption. This paper extends the CKF theory to address nonlinear smoothing problems; the resulting state estimator is named the fixed-interval cubature Kalman smoother (FI-CKS). Moreover, the FI-CKS is reformulated to propagate the square-root error covariances. Although algebraically equivalent to the FI-CKS, the square-root variant ensures reliable implementation when committed to embedded systems with fixed precision or when the inference problem itself is ill-conditioned. Finally, to validate the formulation, the square-root FI-CKS is applied to track a ballistic target on reentry.

Cognitive tracking radar

For the first time ever, this paper presents the design and implementation of a next-generation of tracking radar systems: the cognitive tracking radar (CTR). At the heart of the CTR, we have a cognitive waveform-selection (CWS) algorithm that can optimally pick the transmit waveform from a prescribed library, in response to information fed back from the receiver to the transmitter. In accordance with dynamic programming, the waveform-selection algorithm seeks to minimize the expected tracking error over a horizon of prescribed length. Approximation of the problem is also studied to mitigate the burden of computational load. To evaluate the system, we introduce computer experiments on a classical ballistic target tracking problem, the results of which confirm the superiority of the CTR over a conventional radar with fixed waveform.

Control theoretic approach to tracking radar: First step towards cognition

In Haykin (2006) [8], the idea of Cognitive Radar was described for the first time. Four essential points were emphasized in that seminal paper: Bayesian filtering in the receiver, dynamic programming in the transmitter, memory, and global feedback to facilitate computational intelligence. This paper provides a first step towards designing a cognitive radar for tracking applications by presenting a fore-active tracking radar; a radar that utilizes its previous measurements and actions to optimize its transmitted waveform (Haykin, 2011 [11]). In our design, the emphasis is being placed on the cubature Kalman filter to approximate the Bayesian filter in the receiver, approximate dynamic programming for transmit-waveform selection in the transmitter, and global feedback embodying the transmitter, the radar environment, and the receiver all under one overall feedback loop. Simulation results, based on the tracking of an object falling in space, are presented, which substantiate practical validity of the superior performance of a fore-active tracking radar over a traditional active radar with fixed waveform.

Reduced-Order Electrochemical Model Parameters Identification and SOC Estimation for Healthy and Aged Li-Ion Batteries Part I: Parameterization Model Development for Healthy Batteries

The current phase in our transportation system represents a paradigm shift from conventional, fossil-fuel-based vehicles into the second-generation electric and hybrid vehicles. Electric vehicles (EVs) provide numerous advantages compared with conventional vehicles because they are more efficient, sustainable, greener, and cleaner. The commercial market penetration and success of EVs depend on the efficiency, safety, cost, and lifetime of the traction battery pack. One of the current key electrification challenges is to accurately estimate the battery pack state of charge (SOC) and state of health (SOH), and therefore provide an estimate of the remaining driving range at various battery states of life. To estimate the battery SOC, a high-fidelity battery model along with a robust, accurate estimation strategy is necessary. This paper provides three main contributions: 1) introducing a new SOC parameterization strategy and employing it in setting up optimizer constraints to estimate battery parameters; 2) identification of the full-set of the reduced-order electrochemical battery model parameters by using noninvasive genetic algorithm optimization on a fresh battery; and 3) model validation by using real-world driving cycles. Extensive tests have been conducted on lithium iron phosphate-based cells widely used in high-power automotive applications. Models can be effectively used onboard of battery management system.

Reduced-Order Electrochemical Model Parameters Identification and State of Charge Estimation for Healthy and Aged Li-Ion Batteries—Part II: Aged Battery Model and State of Charge Estimation

Recently, extensive research has been conducted in the field of battery management systems due to increased interest in vehicles electrification. Parameters, such as battery state of charge (SOC) and state of health, are of critical importance to ensure safety, reliability, and prolong battery life. This paper includes the following contributions: 1) tracking reduced-order electrochemical battery model parameters variations as battery ages, using noninvasive genetic algorithm optimization technique; 2) the development of a battery aging model capable of capturing battery degradation by varying the effective electrode volume; and 3) estimation of the battery critical SOC using a new estimation strategy known as the smooth variable structure filter based on reduced-order electrochemical model. The proposed filter is used for SOC estimation and demonstrates strong robustness to modeling uncertainties, which is relatively high in case of reduced-order electrochemical models. Batteries used in this research are lithium-iron phosphate cells widely used in automotive applications. Extensive testing using real-world driving cycles is used for estimation strategy application and for conducting the aging test. Limitations of the proposed strategy are also highlighted.

Nonlinear Bayesian Filters for Training Recurrent Neural Networks

In this paper, we present nonlinear Bayesian filters for training recurrent neural networks with a special emphasis on a novel, more accurate, derivative-free member of the approximate Bayesian filter family called the cubature Kalman filter. We discuss the theory of Bayesian filters, which is rooted in the state-space modeling of the dynamic system in question and the linear estimation principle. For improved numerical stability and optimal performance during training period, a number of techniques of how to tune Bayesian filters is suggested. We compare the predictability of various Bayesian filter-trained recurrent neural networks using a chaotic time-series. From the empirical results, we conclude that the performance may be greatly improved by the new square-root cubature Kalman filter.