Continuous-time Nonlinear Dynamic Systems Research Articles

A Sampling Theorem for Exact Identification of Continuous-Time Nonlinear Dynamical Systems

A Sampling Theorem for Exact Identification of Continuous-Time Nonlinear Dynamical Systems

Novel command-filtered Nussbaum design for continuous-time nonlinear dynamical systems with multiple unknown high-frequency gains

Novel command-filtered Nussbaum design for continuous-time nonlinear dynamical systems with multiple unknown high-frequency gains

Effective computational discretization scheme for nonlinear dynamical systems

Numerical methods are essential to investigate and apply nonlinear continuous-time dynamical systems in many fields of science and engineering and discretization schemes are required to obtain the solutions of such dynamical systems. Although computing power has been speedily growing in recent decades, embedded and large-scale problems have motivated significant research to improve the computational efficiency. Nevertheless, few studies have focused on finite precision limitation on discretization schemes due to round-off effects in floating-point number representation. In this paper, a computational effective discretization scheme for nonlinear dynamical systems is introduced. By means of a theorem, it is shown that high-order terms in the Runge-Kutta method can be neglected with no accuracy loss. The proposed approach is illustrated using three well-known systems, namely the Rössler systems, the Lorenz equations and the Sprott B system. The number of mathematical operations and simulation time have reduced up to 81.1% and 90.7%, respectively. Furthermore, as the step-size decreases, the number of neglected terms increases due to the precision of the computer. Yet, accuracy, observability of dynamical systems and the largest Lyapunov are preserved. The adapted scheme is effective, reliable and suitable for embedded and large-scale applications.

Exponential stabilization of fractional-order continuous-time dynamic systems via event-triggered impulsive control

Exponential stabilization of fractional-order continuous-time dynamic systems via eventtriggered impulsive control (EIC) approach is investigated in this paper. Nonlinear and linear fractional-order continuous-time dynamic systems are studied, respectively. The impulsive instants are determined by some given event-triggering function and event-triggering condition, which are dependent on the state of the systems. Sufficient conditions on exponential stabilization for nonlinear and linear cases are presented, respectively. Moreover, the Zeno-behavior of impulsive instants is excluded. Finally, the validity of theoretical results are also illustrated by some numerical simulation examples including the synchronization control of fractional-order jerk chaotic system.

Neural ordinary differential equation control of dynamics on graphs

We study the ability of neural networks to calculate feedback control signals that steer trajectories of continuous-time nonlinear dynamical systems on graphs, which we represent with neural ordinary differential equations (neural ODEs). To do so, we present a neural ODE control (NODEC) framework and find that it can learn feedback control signals that drive graph dynamical systems toward desired target states. While we use loss functions that do not constrain the control energy, our results show, in accordance with related work that NODEC produces low energy control signals. Finally, we evaluate the performance and versatility of NODEC against well-known feedback controllers and deep reinforcement learning. We use NODEC to generate feedback controls for more than one thousand coupled, nonlinear ODEs that represent epidemic processes and coupled oscillators.

Kleene Algebra to Compute Invariant Sets of Dynamical Systems

In this paper, we show that a basic fixed point method used to enclose the greatest fixed point in a Kleene algebra will allow us to compute inner and outer approximations of invariant-based sets for continuous-time nonlinear dynamical systems. Our contribution is to provide the definitions and theorems that will allow us to make the link between the theory of invariant sets and the Kleene algebra. This link has never be done before and will allow us to compute rigorously sets that can be defined as a combination of positive invariant sets. Some illustrating examples show the nice properties of the approach.

Short-term dependency of a class of nonlinear continuous time dynamic systems

Dynamic systems usually have long-term dependency. Recent work, however, shows that the long-term memory models like recurrent neural networks and short-term memory models like feed-forward neural networks have comparable performance in approximating the behavior of dynamic systems. While the pioneering researchers try to understand the short-term dependency in discrete time dynamic systems, this paper focuses on a class of continuous time dynamic systems characterized by ordinary differential equations. In this class of continuous time dynamic systems, only the variable can be observed, and its derivatives with respect to time cannot be measured directly. By analyzing the observability of the continuous time dynamic systems and the sampled systems, we show that the current output only depends on finite steps of history information when sampling fast. If the sampling is fast enough such that the Euler approximation can approximate the sampled system well, the current output only relies on the most recent n steps of history information. Here, n is the order of the ordinary differential equation. Later, we verified our results with the NARMAX method.

Applications of the Poincaré–Hopf Theorem: Epidemic Models and Lotka–Volterra Systems

This article focuses on properties of equilibria and their associated regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincaré–Hopf theorem is used to derive a general result providing a sufficient condition for the system to have a unique equilibrium. The condition involves the Jacobian of the system at possible equilibria and ensures that the system is in fact locally exponentially stable. We apply this result to the susceptible–infected–susceptible (SIS) networked epidemic model, and a generalized Lotka–Volterra system. We use the result further to extend the SIS model via the introduction of decentralized feedback controllers, which significantly change the system dynamics, rendering existing Lyapunov-based approaches invalid. Using the Poincaré–Hopf approach, we identify a necessary and sufficient condition, under which the controlled SIS system has a unique nonzero equilibrium (a diseased steady state), and monotone systems theory is used to show that this nonzero equilibrium is attractive for all nonzero initial conditions. A counterpart condition for the existence of a unique equilibrium for a nonlinear discrete-time dynamical system is also presented.

Data-driven modeling of the chaotic thermal convection in an annular thermosyphon

dentifying accurate and yet interpretable low-order models from data has gained a renewed interest over the past decade. In the present work, we illustrate how the combined use of dimensionality reduction and sparse system identification techniques allows us to obtain an accurate model of the chaotic thermal convection in a two-dimensional annular thermosyphon. Taking as guidelines the derivation of the Lorenz system, the chaotic thermal convection dynamics simulated using a high-fidelity computational fluid dynamics solver are first embedded into a low-dimensional space using dynamic mode decomposition. After having reviewed the physical properties the reduced-order model should exhibit, the latter is identified using SINDy, an increasingly popular and flexible framework for the identification of nonlinear continuous-time dynamical systems from data. The identified model closely resembles the canonical Lorenz system, having a similar structure and exhibiting the same physical properties. Finally, extensions to other flow configurations with or without control are discussed.

Модель нейронної схеми слідкуючого керування нелінійними динамічними системами неперервного часу

Модель нейронної схеми слідкуючого керування нелінійними динамічними системами неперервного часу

Iterative statistical linear regression for Gaussian smoothing in continuous-time non-linear stochastic dynamic systems

Abstract This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary Gaussian process. Two methods are developed based on (1) taking the limit of statistical linear regression of the discretised process and (2) minimising an upper bound to a cost functional. Their difference is manifested in the diffusion of the approximate processes. This in turn gives novel derivations of pre-existing Gaussian smoothers when Method 1 is used and a new class of Gaussian smoothers when Method 2 is used. Furthermore, based on the aforementioned development the iterative Gaussian smoothers in discrete-time are generalised to the continuous-time setting by iteratively re-linearising the stochastic differential equation with respect to the current Gaussian process approximation to the smoothed process. The method is verified in two challenging tracking problems, a reentry problem and a radar tracked coordinated turn model with state dependent diffusion. The results show that the method has competitive estimation accuracy with state-of-the-art smoothers.

On the effectiveness of Monte Carlo for initial uncertainty forecasting in nonlinear dynamical systems

This paper considers the problem of initial uncertainty forecasting in deterministic nonlinear continuous-time dynamical systems via particle ensembles. The popular Monte Carlo method, which while simple to implement, faces fundamental issues. In particular, it is not clear how well the propagated particles continue to represent the true state-uncertainty at future times. This paper evaluates the performance of Monte Carlo forecasting by analyzing it in the context of Markov chain Monte Carlo (MCMC) theory. The propagated ensemble is viewed as the realization of a Markov chain at each time instant, generated by an associated instantaneous transition kernel. It is shown that for a special class of nonlinear systems that have zero divergence, the propagated kernel is in detailed balance with the true state probability density function. This guarantees statistical consistency of the Monte Carlo ensemble with the truth at all times for such systems. On the other hand, no such guarantee is possible for systems with non-zero divergence.

An Interval Approach to Compute Invariant Sets

This paper proposes an original interval-based method to compute an outer approximation of all invariant sets (such as limit cycles) of a continuous-time nonlinear dynamic system, which are included inside a prior set of the state space. Contrary to all other existing approaches, our method has the following properties: first, it is guaranteed (a solution cannot be lost); second, it is applicable to a large class of systems without any specific assumption such as the knowledge of a Lyapunov function or any partial linearity; and third, there is no need to integrate the system.

A novel neural optimal control framework with nonlinear dynamics: Closed-loop stability and simulation verification

In this paper, we focus on developing adaptive optimal regulators for a class of continuous-time nonlinear dynamical systems through an improved neural learning mechanism. The main objective lies in that establishing an additional stabilizing term to reinforce the traditional training process of the critic neural network, so that to reduce the requirement with respect to the initial stabilizing control, and therefore, bring in an obvious convenience to the adaptive-critic-based learning control implementation. It is exhibited that by employing the novel updating rule, the adaptive optimal control law can be obtained with an excellent approximation property. The closed-loop system is constructed and its stability issue is handled by considering the improved learning criterion. Experimental simulations are also conducted to verify the efficient performance of the present design method, especially the major role that the stabilizing term performed.

Robust Control for Nonlinear Systems with Nonlinear Characteristics

본 연에서는 미지의 데드존을 갖는 시 연속 비선형 동적 시스템에 대해서 견실한 적응제어 법칙을 제안하였다. 액츄에터의 비선형특성은 제어 시스템에서 흔하게 존재하게 된다. 이러한 여러 비선형 특성은 데드존, 백래쉬, 히스테리시스, 또는 피이스와이스 선형에 대해서 부드럽지 않은(넌 스무스)한 특성을 가지게 된다. 이러한 비선형특성을 갖는 제어 시스템에 대해서 많은 연구가들이 제안한 제어법칙들은 대부분 역 데드존을 활용해서 제어법칙들을 제안하였다. 본 연구에서 제안하는 견실한 적응제어 법칙은 역 데드존을 사용하지 않고 제안하였으며, 데드존에 대해서 직관과 수학적인 관점에서의 모델을 활용하였다. 제안된 법칙에 대해서 리아프노프 정리를 활용하여 대역적 안정도에 대해서 증명을 통해서 제시 하였다. 제안된 제어기에 대해서 수치적 예를 통한 결과로부터 제안된 알고리즘의 타당성을 검증 하였다.

CollocInfer: Collocation Inference in Differential Equation Models

This monograph details the implementation and use of the CollocInfer package in R for smoothing-based estimation of continuous-time nonlinear dynamic systems. These routines represent an extension of the generalized profiling methods in Ramsay, Hooker, Campbell, and Cao (2007) for estimating parameters in nonlinear ordinary differential equations. An interface to the fda package is included. The package also supports discretetime systems. We describe the methodological and computational framework and the necessary steps to use the software. Equivalent functionality is available in MATLAB.

Lebesgue approximation model of continuous-time nonlinear dynamic systems

Traditional model-based approaches are based on periodic iterations, where the continuous-time model is discretized with a fixed period. Despite the easiness in analysis and design, such periodic approximation model may be undesirable from the computation-efficiency point of view. This paper presents the Lebesgue-approximation model (LAM) of continuous-time nonlinear systems, where the iteration is activated on an “as-needed” basis, but not periodically. We show that the proposed LAM behaves exactly the same as a specific event-triggered feedback system, through which the properties of the LAM can be studied. We provide a sufficient condition to ensure asymptotic stability of the LAM and derive theoretical bounds on the difference between the states of the LAM and the original continuous-time system. The LAM is then integrated in the particle-filtering approach for fault prognosis. Simulation results show that the LAM can dramatically reduce the number of iterations in prognosis without sacrificing accuracy and precision.

Bal: A library for the brute-force analysis of dynamical systems

This paper describes the functionality and usage of bal, a C/C++ library with a Python front-end for the brute-force analysis of continuous-time dynamical systems described by ordinary differential equations (ODEs). bal provides an easy-to-use wrapper for the efficient numerical integration of ODEs and, by detecting intersections of the trajectory with appropriate Poincaré sections, allows to classify the asymptotic trajectory of a dynamical system for bifurcation analysis. Some examples of application are discussed, concerning two-dimensional bifurcation diagrams, Lyapunov exponents and finite-time Lyapunov exponents, basins of attraction, simulation of switching ODE systems, and integration with AUTO, a software package for continuation analysis. Program summaryProgram title: BAL (Library)Catalogue identifier: AEYY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEYY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 65256No. of bytes in distributed program, including test data, etc.: 438133Distribution format: tar.gzProgramming language: C/C++, Python.Computer: PCs and Workstations.Operating system: Any Unix based system such as Linux and Mac OSX.RAM: Variable, depending on the number of dimensions of the dynamical system and duration of the simulation.Classification: 4.3, 4.12.External routines: CVODE, Boost, HDF5Nature of problem: The numerical analysis of continuous-time nonlinear dynamical systems often requires the computation of a large number of solutions of the system, for varying parameter sets, and the subsequent classification of the steady state solution.Solution method: This library provides a convenient interface to the library CVODE for the integration of sets of differential equations and at the same time implements all the machinery required for integrating and classifying the behaviour of the system for varying parameter sets and initial conditions. A high-level Python interface makes the usage of the library amenable also to users who are not familiar with the C/C++ programming languages.

Delimiting hyperchaotic regions in parameter planes of a 5D continuous-time dynamical system

Different ways to numerically characterize hyperchaotic regions in parameter planes of a 5D continuous-time nonlinear dynamical system are utilized in this work. The method considers the three largest Lyapunov exponents, to construct two-dimensional parameter planes colorful plots. In some these plots different levels of hyperchaos are represented by a continuously changing yellow to red scale, while in other different colors mean different number of positive Lyapunov exponents.

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.