Moment Differential Equations Research Articles

NIRK-based mixed-type accurate continuous–discrete Gaussian filters with deterministically sampled expectation and covariance for state estimation in continuous-time stochastic process models with discrete measurements

This paper advances further the idea of overall continuous–discrete Gaussian filtering with deterministically sampled expectation and covariance towards efficient mixed-type accurate Kalman-like schemes based on self-regulated Ordinary Differential Equation (ODE) solvers with automatic local and global error controls intended for treating continuous-time nonlinear stochastic process models with discrete measurements. The universal state estimation algorithms of such sort designed recently by Kulikov and Kulikova (2022) possess three potential limitations, which might affect their successful applications in practice. First, these deal with complicated Moment Differential Equations (MDEs) and, hence, can be rather time-consuming in estimating stochastic process models of large size, that compromises their efficiency for online computations. Second, the aforementioned filters do not employ any kind of global error regulation and, hence, can be insufficiently accurate in solving difficult MDEs arisen. Third, these are hardly effective in treating stiff continuous–discrete stochastic systems. Below, all the listed issues are addressed and solved via utilization of the much simpler but more robust MDEs arisen in the usual continuous–discrete Extended Kalman Filter (EKF) within the mixed-type accurate filtering approach initiated by Kulikova and Kulikov (2015), which is rooted in the variable-stepsize Nested Implicit Runge–Kutta (NIRK) methods. For the sake of stability of our novel methods to the numerical integration and round-off errors, we devise square-root versions of the mixed-type Gaussian filters, which are based either on hyperbolic QR factorizations or on hyperbolic Singular Value Decompositions (SVDs). Performances of these new mixed-type square-root Gaussian filters are assessed and compared to their original (non-mixed-type) counterparts as well as to their non-square-root versions in a simulated non-stiff ill-conditioned radar tracking scenario and on a stiff stochastic Oregonator reaction.

Solutions of linear systems of moment differential equations via generalized matrix exponentials

A generalized exponential matrix based on the construction of kernel operators for generalized summability is defined and analyzing its main properties, generalizing the classical exponential matrix and fractional exponential matrix.This object serves as a practical tool to express the solutions of linear systems of moment differential equations in a compact manner, in the spirit of the classical exponential matrix.

On the summability and convergence of formal solutions of linear q-difference-differential equations with constant coefficients

AbstractWe consider the Cauchy problem for homogeneous linear q-difference-differential equations with constant coefficients. We characterise convergent, k-summable and multisummable formal power series solutions in terms of analytic continuation properties and growth estimates of the Cauchy data. We also introduce and characterise sequences preserving summability, which make a very useful tool, especially in the context of moment differential equations.

On derivative-free extended Kalman filtering and its Matlab-oriented square-root implementations for state estimation in continuous-discrete nonlinear stochastic systems

Recent research in nonlinear filtering and signal processing has suggested an efficient derivative-free Extended Kalman filter (EKF) designed for discrete-time stochastic systems. Such approach, however, has failed to address the estimation problem for continuous-discrete models. In this paper, we develop a novel continuous-discrete derivative-free EKF methodology by deriving the related moment differential equations (MDEs) and sample point differential equations (SPDEs). Additionally, we derive their Cholesky-based square-root MDEs and SPDEs and obtain several numerically stable derivative-free EKF methods. Finally, we propose the MATLAB-oriented implementations for all continuous-discrete derivative-free EKF algorithms derived. They are easy to implement because of the built-in fashion of the MATLAB numerical integrators utilized for solving either the MDEs or SPDEs in use, which are the ordinary differential equations (ODEs). More importantly, these are accurate derivative-free EKF implementations because any built-in MATLAB ODE solver includes the discretization error control that bounds the discretization error arisen and makes the implementation methods accurate. Besides, this is done in automatic way and no extra coding is required from users. The new filters are particularly effective for working with stochastic systems with highly nonlinear and/or nondifferentiable drift and observation functions, i.e. when the calculation of Jacobian matrices are either problematical or questionable. The performance of the novel filtering methods is demonstrated on several numerical tests.

Multisummability of Formal Solutions for a Family of Generalized Singularly Perturbed Moment Differential Equations

The notion of moment differentiation is extended to the set of generalized multisums of formal power series via an appropriate integral representation and accurate estimates of the moment derivatives. The main result is applied to characterize generalized multisummability of the formal solution to a family of singularly perturbed moment differential equations in the complex domain, broadening widely the range of singularly perturbed functional equations to be considered in practice, such as singularly perturbed differential equations and singularly perturbed fractional differential equations.

Universal MATLAB‐based square‐root solutions in the family of continuous‐discrete Gaussian filters for state estimation in nonlinear stochastic dynamic systems

AbstractThis article concerns with the issue of square‐rooting in continuous‐discrete Gaussian filters intended for state estimation in nonlinear stochastic dynamic systems of continuous‐discrete sort. These cover all methods devised within the quadrature, cubature, and unscented Kalman filtering approaches as well as those that can be constructed in the future. Based on the universal moment differential equations developed by Särkkä and Sarmavuori in 2013, we advance further that study and design two square‐root solutions grounded on MATLAB ODE solvers in the mentioned Gaussian filtering framework. The main problem addressed is a potential negativity of some weights utilized in calculations of the predicted and filtering means and covariances, which precludes from orthogonal square‐rooting schemes to be applied. In practice, such square‐root implementations are often requested because of their exceptional numerical robustness to round‐off and other disturbances. These also preserve the symmetry and positivity of the covariances computed, automatically. Here, we employ the recently‐devised ‐orthogonal square‐rooting technique for designing our universal MATLAB‐based square‐root solutions in the realm of quadrature, cubature, and unscented Kalman filters, which are easily adjusted to any particular method by using its weights and deterministically selected samples exploited in calculations of the sampled means and covariances. Such a ‐orthogonal square‐rooting approach is grounded on hyperbolic factorizations. It leads to two novel algorithms covering any continuous‐discrete Gaussian filter of the quadrature, cubature, or unscented Kalman‐like kind. Practical performances of our square‐root solutions are validated, assessed, and compared within two simulated ill‐conditioned scenarios in aeronautical and chemical engineering.

Entire Solutions of Linear Systems of Moment Differential Equations and Related Asymptotic Growth at Infinity

AbstractThe general entire solution to a linear system of moment differential equations is obtained in terms of a moment kernel function for generalized summability, and the Jordan decomposition of the matrix defining the problem. The growth at infinity of any solution of the system is also determined, both globally and also following rays to infinity, determining the order and type of such solutions.

Application of Complex Fractional Moment in nonlinear system with Gaussian colored noise

This paper studies the transient response of the nonlinear system with Gaussian colored noise. The one-dimensional Itô stochastic differential equation with respect to amplitude response is obtained by the stochastic averaging technique, and the Mellin transformation is applied to the related Fokker–Planck–Kolmogorov (FPK) equation to obtain a set of Complex Fractional Moment (CFM) differential equations. The semi-analytical solution of the FPK equation can be reconstructed by solving the CFM equations. Numerical method is adopted to verify the validity of the CFM method, and the dynamical temporal behaviors of the transient response Probability Density Function (PDF) are analyzed. In addition, the transient bifurcation phenomenon of the system is discussed, and the 3-D distribution diagram of peak value of transient PDF is produced.

Square-root filtering via covariance SVD factors in the accurate continuous-discrete extended-cubature Kalman filter

This paper continues our research devoted to an accurate nonlinear Bayesian filters' design. Our solution implies numerical methods for solving ordinary differential equations (ODE) when propagating the mean and error covariance of the dynamic state. The key idea is that an accurate implementation strategy implies the methods with a discretization error control involved. This means that the filters' moment differential equations are to be solved accurately, i.e. with negligible error. In this paper, we explore the continuous-discrete extended-cubature Kalman filter that is a hybrid method between Extended and Cubature Kalman filters (CKF). Motivated by recent results obtained for the continuous-discrete CKF in Bayesian filtering realm, we propose the numerically stable (to roundoff) square-root approach within a singular value decomposition (SVD) for the hybrid filter. The new method is extensively tested on a few application examples including stiff systems.

MATLAB-based general approach for square-root extended-unscented and fifth-degree cubature Kalman filtering methods

A stable square-root approach has been recently proposed for the unscented Kalman filter (UKF) and fifth-degree cubature Kalman filter (5D-CKF) as well as for the mixed-type methods consisting of the extended Kalman filter (EKF) time update and the UKF/5D-CKF measurement update steps. The mixed-type estimators provide a good balance in trading between estimation accuracy and computational demand because of the EKF moment differential equations involved. The key benefit is a consolidation of reliable state mean and error covariance propagation by using delicate discretization error control while solving the EKF moment differential equations and an accurate measurement update according to the advanced UKF and/or 5D-CKF filtering strategies. Meanwhile the drawback of the previously proposed estimators is an utilization of sophisticated numerical integration scheme with the built-in discretization error control that is, in fact, a complicated and computationally costly tool. In contrast, we design here the mixed-type methods that keep the same estimation quality but reduce a computational time significantly. The novel estimators elegantly utilize any MATLAB-based numerical integration scheme developed for solving ordinary differential equations (ODEs) with the required accuracy tolerance pre-defined by users. In summary, a simplicity of the suggested estimators, their numerical robustness with respect to roundoff due to the square-root form utilized as well as their estimation accuracy due to the MATLAB ODEs solvers with discretization error control involved are the attractive features of the novel estimators. The numerical experiments are provided for illustrating a performance of the suggested methods in comparison with the existing ones.

Stochastic analysis of asymmetric monostable harvesters driven by Gaussian white noise with moment differential equations

As an effective means to overcome the shortcomings of linear systems only performing well near the resonance frequency, monostable piezoelectric energy harvesters (MPEHs) have been investigated widely in the area of energy harvesting. However, it is difficult to achieve a perfectly symmetric potential energy function due to the asymmetries in magnets and materials. Therefore, the response characteristics of the asymmetric MPEHs with quartic potential function under Gaussian white noise excitation are explored in this paper. The method of moment differential equation is applied to approximately determine the output performance of the asymmetric potential MPEHs under the excitation of Gaussian white noise. For the symmetric MPEH, the influence of external excitation intensity and internal system parameters on the outputs is firstly analyzed theoretically and numerically. When a quadratic nonlinear coefficient is introduced and only using its variation to characterize the variation of the asymmetry, the output performance of the MPEH is enhanced and the power increases with an increase in the asymmetry. Particularly, the existence of asymmetry leads the system to have a nonzero mean value for the displacement response, which then influences the shape of the probability density function. In the condition that more coefficients are employed to represent the asymmetry, the influence of asymmetry on the output depends on the potential energy function’s shape.

Numerical robustness of extended Kalman filtering based state estimation in ill‐conditioned continuous‐discrete nonlinear stochastic chemical systems

SummaryThis paper presents a case study investigation of numerical robustness of extended Kalman filters used for estimation of stochastic chemical systems with ill‐conditioned measurements. Here, we consider both a batch reactor model and that of a continuously stirred tank reactor. Our purpose is to explore performance of extended Kalman filtering–based state estimators when the measurement model becomes increasingly ill conditioned. In this way, we determine numerically robust methods, which are suitable for accurate estimation of stochastic chemical systems in the presence of round‐off and other disturbances. We examine both conventional filters and their square‐root forms. All these algorithms are implemented by means of the conventional matrix inversion used in their measurement update steps and the Moore‐Penrose pseudoinversion as well. Furthermore, the square‐root filters under investigation are obtained in two ways, namely, by solving square‐root moment differential equations and by square rooting the filter itself. We show that only the square‐root filters grounded in the second approach (with use of stable orthogonal decompositions) are numerically robust and provide the excellent estimation accuracy within all our ill‐conditioned stochastic chemical system scenarios considered in this paper. In addition, the convectional filters and the square‐root variants based on solving moment equations are rather sensitive to round‐off and may be useful and accurate if the chemical system at hand is rather well conditioned.

Stability analysis of Extended, Cubature and Unscented Kalman Filters for estimating stiff continuous–discrete stochastic systems

This paper studies stiffness and stability properties of Extended Cubature and Unscented Kalman Filters applied to continuous–discrete stochastic systems with stiff dynamic behavior. The main part of these methods relies on numerical integration of Moment Differential Equations (MDEs). Our focus is to understand how the stiffness of MDEs influences performance of the filters. The proposed linear stability analysis shows that the MDEs that havearisen within these methods can enlarge the stiffness of continuous-time stochastic model and, hence, require solvers with advanced stability properties for their effective and accurate treatment. Besides, the proposed nonlinear stability analysis proves that such MDEs may become extremely unstable in simulation intervals. The latter raises the uncertainty of state estimation and results in two interesting implications: (i) the lower-order Extended Kalman Filter outperforms the higher-order Cubature and Unscented Kalman Filters in the accuracy of state estimation of stochastic systems with unstable MDE behavior; (ii) the methods under exploration fail when the stiffness is large enough. Our theoretical consideration is supported by numerical tests with filters based on the MATLAB code ode15s, which is a benchmark solver for stiff ODEs. These are examined on linear and nonlinear stiff continuous-time stochastic models.

Moment stability analysis of linear stochastic human controller model in visuomotor tracking task

We investigate the stability limits of linear stochastic human controller model. In our previous study, we had proposed a model by which the probability distributions of human visuomotor tracking data can be accurately reproduced. In this study, we conduct a stochastic analysis on our model by deriving a system of moment differential equations with respect to random state variables of this model. We evaluate the stability limits of our model by detecting zero-eigenvalue conditions of the Jacobian matrix of the moment differential equations. The resulting stability limits make it possible to characterize the human parameter values that were experimentally identified from the human participants in our previous study; this shows that during the visuomotor tracking experiments, the human participants generated huge fluctuations far from the stability limits while applying almost neutrally stable proportional gain.

Square‐root Kalman‐like filters for estimation of stiff continuous‐time stochastic systems with ill‐conditioned measurements

This study addresses the performance of the extended and extended-cubature Kalman filters for state estimation of ill-conditioned stiff continuous-discrete stochastic systems. Stiff models arise naturally in practise as, for instance, the Van der Pol oscillator in electrical engineering and the Oregonator reaction in chemistry. Their accurate state estimation is a challenging task, which is made even more difficult by including ill-conditioned measurements, here. The authors' goal is to study accuracy of these methods when the Van der Pol oscillator and the Oregonator reaction become increasingly ill-conditioned. Thus, the authors determine efficient tools for estimation of stiff stochastic systems in the presence of round-off and other disturbances. They examine both conventional filters and their square-root forms. The square-root methods are obtained in two ways, namely, by solving the square-root moment differential equations and by square-rooting the filter itself. They prove that only the square-root filters grounded in the second approach (with use of stable orthogonal decompositions) are numerically stable and provide the highest state estimation accuracy in the ill-conditioned stiff stochastic examples under consideration.

Accurate continuous–discrete unscented Kalman filtering for estimation of nonlinear continuous-time stochastic models in radar tracking

This paper presents a new state estimation technology grounded in the unscented Kalman filtering for nonlinear continuous-time stochastic systems. The resulting accurate continuous–discrete unscented Kalman filter is based on adaptive solvers with automatic global error control for treating numerically the moment differential equations arising in the mean and covariance calculation of propagated Gaussian density. It is intended for an accurate and robust state estimation in nonlinear continuous–discrete stochastic systems of various sorts, including in radar tracking models. This new filter is examined in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn. The latter is considered to be a challenging one for testing nonlinear filtering algorithms. For comparison, we also examine such efficient state estimators as the accurate continuous–discrete extended Kalman filter, the continuous–discrete unscented Kalman filter and the mixed-type accurate continuous–discrete extended-unscented Kalman filter designed earlier, but further modified in the present study. The comparison is fulfilled in terms of accuracy and efficiency of estimating the state in the mentioned air traffic control scenario.

The continuous–discrete extended Kalman filter revisited

Abstract This paper elaborates a new approach to nonlinear filtering grounded in an accurate implementation of the continuous–discrete extended Kalman filter for estimating stochastic dynamic systems. It implies that the moment differential equations for calculation of the predicted state mean and error covariance of propagated Gaussian density are solved accurately, i.e., with negligible errors. The latter allows the total error of the extended Kalman filter to be reduced significantly and results in a new accurate continuous–discrete extended Kalman filtering method. In addition, this filter exploits the scaled local and global error controls to avoid any comparison of different physical units. The designed state estimator is compared numerically with continuous–discrete unscented and cubature Kalman filters to expose its practical efficiency. The problem of long waiting times (i.e., infrequent measurements) arisen in chemical and other engineering is also addressed.

Accurate cubature and extended Kalman filtering methods for estimating continuous-time nonlinear stochastic systems with discrete measurements

This paper further advances the idea of accurate Gaussian filtering towards efficient cubature Kalman filters for estimating continuous-time nonlinear stochastic systems with discrete measurements. It implies that the moment differential equations describing evolution of the predicted mean and covariance of the propagated Gaussian density in time are solved accurately, i.e. with negligible error. The latter allows the total error of the cubature Kalman filtering to be reduced significantly and results in a new accurate continuous–discrete cubature Kalman filtering method. At the same time, we revise the earlier developed version of the accurate continuous–discrete extended Kalman filter by amending the involved iteration and relaxing the utilized global error control mechanism. In addition, we build a mixed-type method, which unifies the best features of the accurate continuous–discrete extended and cubature Kalman filters. More precisely, the time updates are done in this state estimator as those in the first filter whereas the measurement updates are conducted with use of the third-degree spherical-radial cubature rule applied for approximating the arisen Gaussian-weighted integrals. All these are examined in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, and compared to the state-of-the-art cubature Kalman filters.

NIRK-based accurate continuous–discrete extended Kalman filters for estimating continuous-time stochastic target tracking models

This paper presents three state estimators grounded in the variable-stepsize Gauss- and Lobatto-type Nested Implicit Runge–Kutta (NIRK) formulas of orders 4 and 6 and designed for treating continuous-time stochastic systems arisen in radar tracking. Our filters are built within the Extended Kalman Filtering (EKF) framework and based on accurate numerical integrations of the corresponding Moment Differential Equations (MDEs). Automatic local and global error regulation mechanisms implemented in these methods allow the committed discretization error to be under control and made negligible in automatic mode. The latter raises the state estimation accuracy of the constructed filters, significantly. This also leads to the advanced notion of Accurate Continuous–Discrete Extended Kalman Filtering (ACD-EKF) developed by Kulikov and Kulikova in 2013–2016. Our novel methods are constructed within the same approach, but possess the improved accuracy and efficiency in comparison to their predecessors due to both more effective error control mechanisms implemented for integrating MDEs and more accurate iterations used for treating arisen nonlinear equations in the revised filters. Numerical experiments with the updated state estimators and their comparison to the cited earlier-designed ACD-EKFs are fulfilled in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, in Matlab. This examination suggests that the novel state estimation algorithms outperform their predecessors and possess a promising potential for solving target tracking tasks in real-world applications.

Synchronous dynamics of small-world neuronal network system with spatially correlated white noise

In this paper, by using the Terman-Wang small-world neuronal network with electrical synapse coupling, we investigate the synchronous dynamics of neuronal network system subjected to spatially correlated white noise. First, the dynamical mean-field approximation theory is extended to the small-world network system under spatially correlated white noise, through which the original 2N-dimensional stochastic differential equations of the network system are transformed to 11-dimensional deterministic moment differential equations. Then, based on this set of moment differential equations, the key effects of spatially correlated noise and network structure on the synchronous firing property are discussed in the Terman-Wang neuronal network system. The results show that the synchronization ratio of this considered neuronal network system becomes higher not only as the noise correlation coefficient is increased but also as the coupling strength and the average vertex degree are added. Those results imply that the noise spatial correlation coefficient, the coupling strength, and the average vertex degree can play a positive role in inducing synchronous neuronal behaviors. Furthermore, the synchronous dynamics of the original neuronal network system, obtained by direct numerical simulations, is compared with those obtained by the dynamical mean-field approximation theory, and good consistence between them is revealed.