Nonlinear Stochastic Dynamical Systems Research Articles

Identification of partially known non‐linear stochastic spatio‐temporal dynamical systems by using a novel partially linear Kernel method

The identification of non-linear stochastic spatio-temporal dynamical systems given by stochastic partial differential equations is of great significance to engineering practice, since it can always provide useful insight into the mechanism and physical characteristics of the underlying dynamics. In this study, based on the difference method for stochastic partial differential equations, a novel state-space model named multi-input–multi-output extended partially linear model for stochastic spatio-temporal dynamical system is proposed. A new Reproducing Kernel Hilbert Space-based algorithm named extended partially linear least square ridge regression is thus particularly developed for the identification of the extended partially linear model. Compared with existing identification methods available for spatio-temporal dynamics, the advantages of the proposed identification method include that (i) it can make full use of the partially linear structural information of physical models, (ii) it can achieve more accurate estimation results for system non-linear dynamics and (iii) the resulting estimated model parameters have clear physical meaning or properties closely related to the underlying dynamical system. Moreover, the proposed extended partially linear model also provide a convenient state-space model for system analysis and design (e.g. controller or filter design) of the class of non-linear stochastic partial differential dynamical systems.

Strong Tracking Filtering Algorithm of Randomly Delayed Measurements for Nonlinear Systems

This paper focuses on the filtering problems of nonlinear discrete-time stochastic dynamic systems, such as the model simplification, noise characteristics uncertainty, initial conditions uncertainty, or system parametric variation. Under these circumstances, the measurements of system have one sampling time random delay. A new method, that is, strong tracking filtering algorithm of randomly delayed measurements (STF/RDM) for nonlinear systems based on recursive operating by analytical computation and first-order linear approximations, is proposed; a principle of extended orthogonality is presented as a criterion of designing the STF/RDM, and through the residuals between available and predicted measurements, the formula of fading factor is obtained. Under the premise of using the extended orthogonality principle, STF/RDM proposed in this paper can adjust the fading factor online via calculating the covariance of residuals, and then the gain matrices of the STF/RDM adjust in real time to enhance the performance of the proposed method. Lastly, in order to prove that the performance of STF/RDM precedes existing EKF method, the experiment of tracking maneuvering aircraft is carried out.

Stochastic Averaging of Dynamical Systems with Multiple Time Scales Forced with $\alpha$-Stable Noise

Stochastic averaging allows for the reduction of the dimension and complexity of stochastic dynamical systems with multiple time scales, replacing fast variables with statistically equivalent stochastic processes in order to analyze variables evolving on the slow time scale. These procedures have been studied extensively for systems driven by Gaussian noise, but very little attention has been given to the case of $\alpha$-stable noise forcing which arises naturally from heavy-tailed stochastic perturbations. In this paper, we study nonlinear fast-slow stochastic dynamical systems in which the fast variables are driven by additive $\alpha$-stable noise perturbations and the slow variables depend linearly on the fast variables. Using a combination of perturbation methods and Fourier analysis, we derive stochastic averaging approximations for the statistical contributions of the fast variables to the slow variables. In the case that the diffusion term of the reduced model depends on the state of the slow variable, we show that this term is interpreted in terms of the Marcus calculus. For the case $\alpha = 2$, which corresponds to Gaussian noise, the results are consistent with previous results for stochastic averaging in the Gaussian case. Although the main results are derived analytically for $1 < \alpha < 2$, we provide evidence of their validity for $\alpha < 1$ with numerical examples. We numerically simulate both linear and nonlinear systems and the corresponding reduced models, demonstrating good agreement for their stationary distributions and temporal dependence properties.

A new practically oriented generation of nonlinear filtering toolbox

The paper deals with state estimation problem of nonlinear stochastic dynamic systems with a special focus on a software package supporting problem specification, system simulation, and state estimation itself. The developed package called Gaussian Filtering Toolbox is described, which is a practically oriented successor to the toolbox called Nonlinear Estimation Framework that was oriented rather towards academia with its generality and richness. The developed toolbox is described in terms of its usage and particularly the aspects leading to a better practical applicability. The key features of the toolbox, which are computational complexity reduction and numerical stability enhancement, are illustrated using two numerical examples.

Global relative controllability of fractional stochastic dynamical systems with distributed delays in control

This paper is concerned with the global relative controllability of linear and nonlinear fractional stochastic dynamical systems with distributed delays in control for finite dimensional spaces. Sufficient conditions for controllability results are obtained using the Banach fixed point theorem and the controllability Grammian matrix which is dened by the Mittag-Leffler matrix function. An example is provided to illustrate the theory.

Fitting EXPAR Models Through the Extended Kalman Filter

Exponential autoregressive (EXPAR) family of parametric nonlinear time-series models, which is a discrete-time approximation of continuous-time nonlinear stochastic dynamical system, is considered. A heartening feature of this model is that it is capable of describing those data sets that depict cyclical variations. The estimation procedure for EXPAR models is developed using extended Kalman filter (EKF). Through simulation studies, it is shown that EKF is very efficient for fitting EXPAR models. Formulae for optimal one-step and two-step ahead out-of-sample forecasts are derived analytically by recursive use of conditional expectation. Conditions for the existence of limit cycle behaviour for EXPAR models are also established. Superiority of EKF method vis-a-vis Genetic algorithms (GA) method for fitting EXPAR models is shown through simulation studies. As an illustration, EXPAR models are employed for modelling and forecasting Oil sardine, Mackerel and Bombay duck time-series landings data in India. It is shown that all the three fitted models exhibit the desirable feature of existence of limit cycle behaviour. It is concluded that the EXPAR model performs better than ARIMA methodology for both modelling and forecasting purposes for the data sets under consideration.

Information Anatomy of Stochastic Equilibria

A stochastic nonlinear dynamical system generates information, as measured by its entropy rate. Some—the ephemeral information—is dissipated and some—the bound information—is actively stored and so affects future behavior. We derive analytic expressions for the ephemeral and bound information in the limit of infinitesimal time discretization for two classical systems that exhibit dynamical equilibria: first-order Langevin equations (i) where the drift is the gradient of an analytic potential function and the diffusion matrix is invertible and (ii) with a linear drift term (Ornstein–Uhlenbeck), but a noninvertible diffusion matrix. In both cases, the bound information is sensitive to the drift and diffusion, while the ephemeral information is sensitive only to the diffusion matrix and not to the drift. Notably, this information anatomy changes discontinuously as any of the diffusion coefficients vanishes, indicating that it is very sensitive to the noise structure. We then calculate the information anatomy of the stochastic cusp catastrophe and of particles diffusing in a heat bath in the overdamped limit, both examples of stochastic gradient descent on a potential landscape. Finally, we use our methods to calculate and compare approximations for the time-local predictive information for adaptive agents.

Human Brain Networks: Spiking Neuron Models, Multistability, Synchronization, Thermodynamics, Maximum Entropy Production, and Anesthetic Cascade Mechanisms

Advances in neuroscience have been closely linked to mathematical modeling beginning with the integrate-and-fire model of Lapicque and proceeding through the modeling of the action potential by Hodgkin and Huxley to the current era. The fundamental building block of the central nervous system, the neuron, may be thought of as a dynamic element that is “excitable”, and can generate a pulse or spike whenever the electrochemical potential across the cell membrane of the neuron exceeds a threshold. A key application of nonlinear dynamical systems theory to the neurosciences is to study phenomena of the central nervous system that exhibit nearly discontinuous transitions between macroscopic states. A very challenging and clinically important problem exhibiting this phenomenon is the induction of general anesthesia. In any specific patient, the transition from consciousness to unconsciousness as the concentration of anesthetic drugs increases is very sharp, resembling a thermodynamic phase transition. This paper focuses on multistability theory for continuous and discontinuous dynamical systems having a set of multiple isolated equilibria and/or a continuum of equilibria. Multistability is the property whereby the solutions of a dynamical system can alternate between two or more mutually exclusive Lyapunov stable and convergent equilibrium states under asymptotically slowly changing inputs or system parameters. In this paper, we extend the theory of multistability to continuous, discontinuous, and stochastic nonlinear dynamical systems. In particular, Lyapunov-based tests for multistability and synchronization of dynamical systems with continuously differentiable and absolutely continuous flows are established. The results are then applied to excitatory and inhibitory biological neuronal networks to explain the underlying mechanism of action for anesthesia and consciousness from a multistable dynamical system perspective, thereby providing a theoretical foundation for general anesthesia using the network properties of the brain. Finally, we present some key emergent properties from the fields of thermodynamics and electromagnetic field theory to qualitatively explain the underlying neuronal mechanisms of action for anesthesia and consciousness.

Recursive constrained state estimation using modified extended Kalman filter

The extended Kalman filter (EKF) remains the most preferred state estimator for solving both unconstrained and constrained state estimation problems in the field of Chemical Engineering. Given, the wide spread use of EKF, we have proposed a novel optimization free recursive formulation of the EKF, to handle elegantly bounds on the estimated state variables of a stochastic non-linear dynamic system. It is well known that in the EKF, the prior and posterior distributions are approximated to be a multivariate normal distribution. In the presence of bounds imposed on the state variables, the accuracy of the first two moments of the initial state distribution and prior distribution namely the means and covariance matrices, plays a significant role in the extended Kalman filter performance. Hence, in this paper, we propose two novel schemes to modify the prior and posterior distributions of the EKF in order to satisfy the bound constraints. In addition, the initial state distribution is also suitably modified in order to satisfy the bound constraints. The efficacy of the proposed state estimation schemes using the EKF is validated on two benchmark problems reported in the literature namely a simulated gas-phase reactor and an isothermal batch reactor involving constraints on estimated state variables. Extensive simulation studies show the effectiveness of the proposed optimization free recursive constrained state estimation schemes using extended Kalman filter.

Design of Pure Propagation Unscented Kalman Filter

The paper deals with state estimation of nonlinear stochastic dynamic systems. Various unscented Kalman filter (UKF) algorithms are analyzed with the focus on computation and transformation of the σ-points for the purpose of update of state estimate moments. An algorithm of the pure propagation UKF is developed transforming the initial σ-point set forward in time without necessity of its recomputation as is usual in classical UKF algorithms. Such direct transformation of the σ-points keeps higher order moments of the σ-point set and leads consequently to an increased accuracy of the state estimate. The proposed pure propagation unscented Kalman filter is illustrated in a numerical example.

On Identifying Envelop Type Nonlinear Output Error Takagi-Sugeno Fuzzy Models for Dynamic Systems with Uncertainties

In modeling of a stochastic nonlinear dynamic system from input-output data, it may be of interest to model uncertainty in the underlying system besides predicting a most likely or average response of the system. Due to stochasticity in the system behavior, the data obtained for identification can be considered as one realization of the underlying stochastic phenomenon. In order to effectively deal with the identification of such systems, it may be advantageous to repeat the identification experiment multiple times under similar conditions. The multiple input-output time series generated in this way thus contain information about stochastic variations within the system. This paper presents one of the possible approaches to effectively deal with identification in such scenario in the framework of Nonlinear Output Error (NOE) Takagi-Sugeno (TS) fuzzy models. Based on extended Chebyshev's inequality for finite samples, the lower and upper boundaries of the output time-series are obtained using (1-α) confidence interval (envelops of the response). The proposed identification algorithm provides a model for predicting the most likely value as well as the boundary models for predicting the envelops of the output signal. The experimental results for an electro-mechanical throttle shows the applicability and validity of the proposed approach.

Probabilistic solutions of some multi-degree-of-freedom nonlinear stochastic dynamical systems excited by filtered Gaussian white noise

In this paper, the state-space-split method is extended for the dimension reduction of some high-dimensional Fokker–Planck–Kolmogorov equations or the nonlinear stochastic dynamical systems in high dimensions subject to external excitation which is the filtered Gaussian white noise governed by the second order stochastic differential equation. The selection of sub state variables and then the dimension-reduction procedure for a class of nonlinear stochastic dynamical systems is given when the external excitation is the filtered Gaussian white noise. The stretched Euler–Bernoulli beam with hinge support at two ends, point-spring supports, and excited by uniformly distributed load being filtered Gaussian white noise governed by the second-order stochastic differential equation is analyzed and numerical results are presented. The results obtained with the presented procedure are compared with those obtained with the Monte Carlo simulation and equivalent linearization method to show the effectiveness and advantage of the state-space-split method and exponential polynomial closure method in analyzing the stationary probabilistic solutions of the multi-degree-of-freedom nonlinear stochastic dynamical systems excited by filtered Gaussian white noise.

Parametric Bayesian Filters for Nonlinear Stochastic Dynamical Systems: A Survey

Nonlinear stochastic dynamical systems are commonly used to model physical processes. For linear and Gaussian systems, the Kalman filter is optimal in minimum mean squared error sense. However, for nonlinear or non-Gaussian systems, the estimation of states or parameters is a challenging problem. Furthermore, it is often required to process data online. Therefore, apart from being accurate, the feasible estimation algorithm also needs to be fast. In this paper, we review Bayesian filters that possess the aforementioned properties. Each filter is presented in an easy way to implement algorithmic form. We focus on parametric methods, among which we distinguish three types of filters: filters based on analytical approximations (extended Kalman filter, iterated extended Kalman filter), filters based on statistical approximations (unscented Kalman filter, central difference filter, Gauss-Hermite filter), and filters based on the Gaussian sum approximation (Gaussian sum filter). We discuss each of these filters, and compare them with illustrative examples.

Two-stage optimal stochastic linearization in analyzing of non-linear stochastic dynamic systems

A novel stochastic linearization approach is developed to predict the second-moment response of non-linear systems under stochastic parametric and external excitations. The present approach is realized by a two-stage optimization: the first stage of optimal linearization modeling and the second stage of parameters optimization. Five examples, including two polynomial oscillators, one hysteretic Bouc-Wen oscillator under stochastic external excitation, and two polynomial oscillators under stochastic parametric and external excitations are selected to illustrate the present approach. The validity of the present approach is validated by some approximate solutions, exact solutions, and Monte Carlo simulations. For system non-linearity, which can be approximated as a full-states linear combination in the Gaussian linearization model, the present approach offers a more accurate prediction of the second moment than that by the Gaussian linearization method. The two-stage optimal Gaussian linearization method incorporates the merits of Gaussian linearization method in the first stage and the SPEC-alternative in the second stage.

Stochastic Integration Filter

The technical note deals with state estimation of nonlinear stochastic dynamic systems. Traditional filters providing local estimates of the state, such as the extended Kalman filter, unscented Kalman filter, or the cubature Kalman filter, are based on computationally efficient but approximate integral evaluations. On the other hand, the Monte Carlo based Kalman filter takes an advantage of asymptotically exact integral evaluations but at the expense of substantial computational demands. The aim of the technical note is to propose a new local filter that utilises stochastic integration methods providing the asymptotically exact integral evaluation with computational complexity similar to the traditional filters. The technical note will demonstrate that the unscented and cubature Kalman filters are special cases of the proposed stochastic integration filter. The proposed filter is illustrated by a numerical example.

Analysis and modelling of glacial climate transitions using simple dynamical systems

Glacial climate variability is studied integrating simple nonlinear stochastic dynamical systems with palaeoclimatic records. Different models representing different dynamical mechanisms and modelling approaches are contrasted; model comparison and selection is based on a likelihood function, an information criterion as well as various long-term summary statistics. A two-dimensional stochastic relaxation oscillator model with proxy temperature as the fast variable is formulated and the system parameters and noise levels estimated from Greenland ice-core data. The deterministic part of the model is found to be close to the Hopf bifurcation, where the fixed point becomes unstable and a limit cycle appears. The system is excitable; under stochastic forcing, it exhibits noisy large-amplitude oscillations capturing the basic statistical characteristics of the transitions between the cold and the warm state. No external forcing is needed in the model. The relaxation oscillator is much better supported by the data than noise-driven motion in a one-dimensional bistable potential. Two variants of a mixture of local linear stochastic models, each associated with an unobservable dynamical regime or cluster in state space, are also considered. Three regimes are identified, corresponding to the different phases of the relaxation oscillator: (i) lingering around the cold state, (ii) rapid shift towards the warm state, (iii) slow relaxation out of the warm state back to the cold state. The mixture models have a high likelihood and are able to capture the pronounced time-reversal asymmetry in the ice-core data as well as the distribution of waiting times between onsets of Dansgaard-Oeschger events.

Unscented type Kalman filter: limitation and combination

This study investigates the shortcomings of existing unscented type Kalman filters (UTKFs) used for state estimation problem of non-linear stochastic dynamic systems. There are three kinds of UTKFs - the traditional unscented Kalman filter, the cubature Kalman filter and the transformed unscented Kalman filter. It was demonstrated in the past that these algorithms could capture the posterior mean and covariance accurately to the second order for any non-linearity when propagated through the true non-linear system. However, they yield different information on the higher order terms. Owing to the dualistic effect of higher order terms on the performance of these algorithms, it is desirable to come up with some solution to preserve the positive effect of this information in a single algorithm. Based on the assumption that the state measurements are Guassian, two methods are proposed in this work as the suitable candidates. The first one is an adaptive method that chooses the UTKF achieving the highest value of the likelihood function. The second method treats these algorithms as sub-filters and uses the partitioning approach to obtain an overall estimate. The numerical simulations show that the proposed methods can have comparable performances as one of the best UTKFs for the problems under consideration.

From Phage lambda to human cancer: endogenous molecular‐cellular network hypothesis

Experimental evidences and theoretical analyses have amply suggested that in cancer genesis and progression genetic information is very important but not the whole. Nevertheless, “cancer as a disease of the genome” is still currently the dominant doctrine. With such a background and based on the fundamental properties of biological systems, a new endogenous molecular‐cellular network theory for cancer was recently proposed by us. Similar proposals were also made by others. The new theory attempts to incorporate both genetic and environmental effects into one single framework, with the possibility to give a quantitative and dynamical description. It is asserted that the complex regulatory machinery behind biological processes may be modeled by a nonlinear stochastic dynamical system similar to a noise perturbed Morse‐Smale system. Both qualitative and quantitative descriptions may be obtained. The dynamical variables are specified by a set of endogenous molecular‐cellular agents and the structure of the dynamical system by the interactions among those biological agents. Here we review this theory from a pedagogical angle which emphasizes the role of modularization, hierarchy and autonomous regulation. We discuss how the core set of assumptions is exemplified in detail in one of the simple, important and well studied model organisms, Phage lambda. With this concrete and quantitative example in hand, we show that the application of the hypothesized theory in human cancer, such as hepatocellular carcinoma (HCC), is plausible, and that it may provide a set of new insights on understanding cancer genesis and progression, and on strategies for cancer prevention, cure, and care.

Probabilistic DHP adaptive critic for nonlinear stochastic control systems

Following the recently developed algorithms for fully probabilistic control design for general dynamic stochastic systems (Herzallah & Káarnáy, 2011; Kárný, 1996), this paper presents the solution to the probabilistic dual heuristic programming (DHP) adaptive critic method (Herzallah & Káarnáy, 2011) and randomized control algorithm for stochastic nonlinear dynamical systems. The purpose of the randomized control input design is to make the joint probability density function of the closed loop system as close as possible to a predetermined ideal joint probability density function. This paper completes the previous work (Herzallah & Káarnáy, 2011; Kárný, 1996) by formulating and solving the fully probabilistic control design problem on the more general case of nonlinear stochastic discrete time systems. A simulated example is used to demonstrate the use of the algorithm and encouraging results have been obtained.

Lyapunov function construction for nonlinear stochastic dynamical systems

Though the Lyapunov function method is more efficient than the largest Lyapunov exponent method in evaluating the stochastic stability of multi-degree-of-freedom (MDOF) systems, the construction of Lyapunov function is a challenging task. In this paper, a specific linear combination of subsystems’ energies is proposed as Lyapunov function for MDOF nonlinear stochastic dynamical systems, and the corresponding sufficient condition for the asymptotic Lyapunov stability with probability one is then determined. The proposed procedure to construct Lyapunov function is illustrated and validated with several representative examples, where the influence of coupled/uncoupled dampings and excitation intensities on stochastic stability is also investigated.