Real Dynamic System Research Articles

OPT-FRAC-CHN: Optimal Fractional Continuous Hopfield Network

The continuous Hopfield network (CHN) is a common recurrent neural network. The CHN tool can be used to solve a number of ranking and optimization problems, where the equilibrium states of the ordinary differential equation (ODE) related to the CHN give the solution to any given problem. Because of the non-local characteristic of the “infinite memory” effect, fractional-order (FO) systems have been proved to describe more accurately the behavior of real dynamical systems, compared to the model’s ODE. In this paper, a fractional-order variant of a Hopfield neural network is introduced to solve a Quadratic Knap Sac Problem (QKSP), namely the fractional CHN (FRAC-CHN). Firstly, the system is integrated with the quadratic method for fractional-order equations whose trajectories have shown erratic paths and jumps to other basin attractions. To avoid these drawbacks, a new algorithm for obtaining an equilibrium point for a CHN is introduced in this paper, namely the optimal fractional CHN (OPT-FRAC-CHN). This is a variable time-step method that converges to a good local minima in just a few iterations. Compared with the non-variable time-stepping CHN method, the optimal time-stepping CHN method (OPT-CHN) and the FRAC-CHN method, the OPT-FRAC-CHN method, produce the best local minima for random CHN instances and for the optimal feeding problem.

A distributionally robust optimization approach for the potassium fertilizer product transportation considering transshipment through crossdocks

Potassium fertilizer is an essential input for agricultural productivity, and plays a critical role in various plant processes, influencing water uptake, enzyme activation, and photosynthesis. The efficient delivery of potassium fertilizer products to the end-users, typically farmers and agricultural enterprises, is of utmost importance. Due to the long traveling distance between the supplier and customers, decision makers of supplier normally set up multiple crossdocks to aid in transition and storage of potassium fertilizer products. In this situation, making an optimal transportation plan as well as the inventory level of crossdocks will directly enhance the efficiency and effectiveness of long-distance transportation. In this study, we model this problem as a dynamic transportation problem with transshipment considering the uncertain time-varying customers’ demand. To characterize the demand information, we first construct an ambiguity set of customers’ demand based in limited historical data and then develop a distributionally robust optimization-based (DRO) framework to optimize the transportation plan and related inventory level of crossdocks simultaneously. We also propose a general approach to overcome the computational challenges of DRO by transforming the original DRO into a second-order cone programming based on duality theory. Additionally, we introduce linear decision rules to adjust the optimization strategy based on the new observed demand, thus lead the model to handle the dynamic information flow in real time. In case study, all involved data are collected or derived from a real potassium fertilizer company located at Western China. The results show our model reduces the cost by 18.07% compared to stochastic model (sample average approximation), indicating a significant effectiveness of our model on the improvement of delivery efficiency and cost saving in real dynamic logistic systems. Also, we conclude the managerial insight that decision-makers should develop a comprehensive strategy, including improving communication to ensure order status updates, planning rationally to evenly distribute orders, and proactively allocating resources to meet operational demands.

Self-consistent dynamical models with a finite extent – IV. Wendland models based on compactly supported radial basis functions

ABSTRACT We present a new step in our systematic effort to develop self-consistent dynamical models with a finite radial extent. The focus is on models with simple analytical density profiles allowing for analytical calculations of many dynamical properties. In this paper, we introduce a family of models, termed Wendland models, based on compactly supported radial basis functions. The family of models is characterized by a parameter k that controls the smoothness of the transition at the truncation radius. In the limit $k\rightarrow \infty$, the Wendland model reduces to a non-truncated model with a Gaussian density profile. For each Wendland model, the density, mass and gravitational potential are simple truncated polynomial functions of radius. Via the sphecow tool we demonstrate that all Wendland models can be supported by isotropic distribution functions. Surprisingly, the isotropic distribution function exhibits varied behaviour across different Wendland models. Additionally, each model can be supported by a continuum of Osipkov–Merritt orbital structures, ranging from radially anisotropic to completely tangential at the truncation radius. To the best of our knowledge, the Wendland models presented here are the first family of models accommodating both radial and tangential Osipkov–Merritt distribution functions. Using linear superposition, these models can easily be combined to generate Wendland models with even more diverse orbital structures. While the Wendland models are not fully representative of real dynamical systems due to their Gaussian-like density profile, this study lays important groundwork for constructing more realistic models with truncated density profiles that can be supported by a range of orbital structures.

Koopman neural operator as a mesh-free solver of non-linear partial differential equations

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural operators, a kind of neural-network-based PDE solver, these solvers become less accurate and explainable while learning long-term behaviors of non-linear PDE families. In this paper, we propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. With the same objective of learning an infinite-dimensional mapping between Banach spaces that serves as the solution operator of the target PDE family, our approach differs from existing models by formulating a non-linear dynamic system of equation solution. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, to act on the flow mapping of the dynamic system, we can equivalently learn the solution of a non-linear PDE family by solving simple linear prediction problems. We validate the KNO in mesh-independent, long-term, and zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes equation and the Rayleigh-Bénard convection) and three real dynamic systems (e.g., global water vapor patterns and western boundary currents). In these experiments, the KNO exhibits notable advantages compared with previous state-of-the-art models, suggesting the potential of the KNO in supporting diverse science and engineering applications (e.g., PDE solving, turbulence modeling, and precipitation forecasting).

Mathematical analysis of a real time dynamical system: Interaction between Agricultural Variables

In this study, we are interested in the investigation and analysis of a mathematical model for the growth of normal agricultural biomass. The model is divided into four compartments as normal Agricultural assets, auxiliary Agricultural assets, Industrial assets, and Ecospherical assets. We applied the Laplace Decomposition Method to obtain approximate solutions in the form of infinite series and numerical justification was performed on the model parameter values with the aid of Maple 18 software. We studied the effect of varying the growth rate of normal agriculture on the yield of agricultural assets and mathematical analyses of the model equation to its uniqueness and stability of equilibrium are carried out. From the results, it was observed that the yield of normal Agricultural assets decreases due to the decrease in the growth rate coefficient of normal agriculture from 10 % to 90 % and the yield of normal Agricultural assets increases due to the increase in the growth rate coefficient of normal agriculture from 110 % to 190 %. We observed a loss in biodiversity when the growth rate coefficients of normal agriculture decreased from 90 %, 20 %, and 10 %, and a biodiversity gain was observed due to the increase of the growth rate coefficients of normal Agriculture from 110 % to 190 %.

Secular Dynamics of Compact Three-planet Systems

The secular Laplace–Lagrange orbital solution, decomposing eccentricities into a set of uniformly precessing eigenmodes, is a classical result that is typically solved numerically. However, in the limit where orbits are closely spaced, several simplifications make it possible to make analytical progress. We derive simple expressions for the eccentricity eigenmodes in a coplanar three-planet system where the middle planet is much less massive than its neighbors, and we show that these approximate the true eigenmodes of more general systems with three massive planets in various limits. These results provide intuition for the secular dynamics of real systems, and have applications for understanding the stability boundary for compact multiplanet systems.

Analytic solutions of alpha-beta -time- derivatives complex finance chaotic dynamical system: synchronization and extended center manifold. An explicit approach

The present study focuses on a real finance nonlinear dynamic system (FNLDS), which has been shown to exhibit chaotic behavior. The solutions for such nonlinear dynamical systems (NLDSs) have typically been derived using numerical techniques. The objective of this study aims to; firstly, derive approximate analytical solutions for the complex FNLDS (CFNLDS) by constructing the Picard iterative scheme. The convergence of this scheme is proven, and the error analysis shows good tolerance, indicating the efficiency of the technique. Second, a novel criterion for synchronizing the real and imaginary parts of the system is presented, based on a necessary condition. Thirdly, a new method for constructing the extended center manifold is introduced. The 3D portrait reveals a feedback scroll pattern, while the 2D portrait, representing the mutual components, shows multiple pools. The synchronization of the real and imaginary parts of the system is demonstrated graphically. The FNLDS is tested for sensitivity dependence against tiny variations in the initial conditions, and it is found that the system components are moderately sensitive. Furthermore, the Hamiltonian and the extended center manifold establish a two-fold structure. It is observed that the effect of the α-β derivative leads to a delay in the behavior of the solutions.

A sequential excitation and simplified ant colony optimization based global extreme seeking control method for performance improvement

To date, it is lacking of an effective global extreme seeking control (GESC) approach to improve transient responses. Motivated by this engineering problem, a novel and general method of global ESC is proposed theoretically and technically in three steps: the step of a simplified ant colony optimization (SACO) algorithm for the task of global extreme seeking, the step of a sequential excitation (SE) algorithm for the improvement of both transient and steady state responses, and the step of discrete to continuous control conversion to control the real world dynamical systems. Both numerical simulations and physical experiments have been conducted to verify the effectiveness of the proposed controller with SE-SACO on the overall performance improvement in terms of transient responses, steady state responses and global extreme seeking. In comparison with the cutting-edge methods such as IABC-SHTS, SSA, BA and GA-ACO, the settling time and oscillations due to the proposed method are reduced significantly when keeping the global extreme seeking ability. It alleviates the conflict of global extreme seeking and good transient responses of the dynamical systems, an unsolved issue in the literature.

An Efficient Method for Isolating and Purifying Nuclei from Mice Brain for Single-Molecule Imaging Using High-Speed Atomic Force Microscopy.

Nuclear pore complexes (NPCs) on the nuclear membrane surface have a crucial function in controlling the movement of small molecules and macromolecules between the cell nucleus and cytoplasm through their intricate core channel resembling a spiderweb with several layers. Currently, there are few methods available to accurately measure the dynamics of nuclear pores on the nuclear membranes at the nanoscale. The limitation of traditional optical imaging is due to diffraction, which prevents achieving the required resolution for observing a diverse array of organelles and proteins within cells. Super-resolution techniques have effectively addressed this constraint by enabling the observation of subcellular components on the nanoscale. Nevertheless, it is crucial to acknowledge that these methods often need the use of fixed samples. This also raises the question of how closely a static image represents the real intracellular dynamic system. High-speed atomic force microscopy (HS-AFM) is a unique technique used in the field of dynamic structural biology, enabling the study of individual molecules in motion close to their native states. Establishing a reliable and repeatable technique for imaging mammalian tissue at the nanoscale using HS-AFM remains challenging due to inadequate sample preparation. This study presents the rapid strainer microfiltration (RSM) protocol for directly preparing high-quality nuclei from the mouse brain. Subsequently, we promptly utilize HS-AFM real-time imaging and cinematography approaches to record the spatiotemporal of nuclear pore nano-dynamics from the mouse brain.

SINDy vs Hard Nonlinearities and Hidden Dynamics: a Benchmarking Study

In this work, we analyze the effectiveness of the Sparse Identification of Nonlinear Dynamics (SINDy) technique on three benchmark datasets for nonlinear identification, to provide a better understanding of its suitability as a modeling tool for real dynamical systems. While SINDy is often portrayed as an appealing strategy for pursuing physics-based learning, our analysis highlights two weaknesses, i.e., the difficulties in applying this technique when dealing with unobserved states and non-smooth dynamics. Due to the ubiquity of these features in real systems in general, and control applications in particular, we complement our analysis with hands-on approaches to tackle these issues.

Data-Driven Predictive Control and MPC: Do we achieve optimality?

In this paper, we explore the interplay between Predictive Control and closed-loop optimality, spanning from Model Predictive Control to Data-Driven Predictive Control. Predictive Control in general relies on some form of prediction scheme on the real system trajectories. However, these predictions may not accurately capture the real system dynamics, for e.g., due to stochasticity, resulting in sub-optimal control policies. This lack of optimality is a critical issue in case of problems with economic objectives. We address this by providing sufficient conditions on the underlying prediction scheme such that a Predictive Controller can achieve closed-loop optimality. However, these conditions do not readily extend to Data-Driven Predictive Control. In this context of closed-loop optimality, we conclude that the factor distinguishing the approaches within Data-Driven Predictive Control is if they can be cast as a sequential decision-making process or not, rather than the dichotomy of model-based vs. model-free. Furthermore, we show that the conventional approach of improving the prediction accuracy from data may not guarantee optimality.

Ensuring Safety of Gas Field Infrastructure Using ALARP and a Systematic Approach

Introduction. A significant part of the global and state economy is the production and supply of hydrocarbon raw materials. The issues of safety in this area will remain important in the coming decades. The problem is actively discussed in the professional and scientific community. Theoretical and applied works are published. Local, point-based methods are calculated and implemented, which allow predicting and preventing emergencies in certain units of the considered objects. At the same time, there are no sufficiently justified and reproducible system solutions that can take into account the state of, for example, oil or gas field as a single complex and act as indicators not only of ordinary local accidents, but also of systemic accidents — catastrophes. Such a scientifically and experimentally based solution is described in this article. The approach is proposed as part of the formation of a comprehensive scientific and technical program (CSTP) to ensure the safety of natural and technogenic objects (NTO). The aim of this work was to describe the practice of its application in the conditions of specific gas fields and to justify the refusal to focus on solutions that took into account only the minimum practically acceptable risk, that is, built on the ALARP principle (as low as reasonably practicable). Materials and Methods. The article was based on the results of field tests of natural and technogenic objects (NTO) of the oil and gas complex — the Yuzhno-Russkoye field of OJSC Severneftegazprom (SNGP), LLC Gazprom Dobycha Yamburg (GDYA) and the gas pumping station (GPS) Orlovka-2 (Ukraine), conducted with the authors’ participation. Significant results were obtained and evidence-based physically interpreted during acceptance tests of an explosion-proof certified, created under the guidance of the authors of a prototype of a disaster response system (DRS) at the integrated gas treatment plant (IGTP) in LLC GDYA in 2006. At the same time, for the first time in the world practice, the fact of early detection and parrying without consequences by means of DRS on the UKPG-2 of the development of a large-scale general industrial disaster was confirmed. In the form of graphs, the revealed patterns of NTO have been visualized, which made it possible to form harbingers of the development of accidents and catastrophes at the NTO of LLC "GDYA", SNGP and GPS "Orlovka-2". Information on the high experimental reproducibility of the obtained results was presented. The technology has been developed of early detection and parrying of all types of potentially dangerous self-exciting systemic phenomena on real NTO infrastructure — self-oscillations. Three cases of their excitation in real gas fields were presented. Results. The paper shows the fragmentary nature and locality of emergency protection systems based on the ALARP principle. The consequence of this was its complete unsuitability for early detection and counteraction to the most large-scale and costly system accidents — catastrophes that were multifactorial processes, in which none of the factors was decisive. The alternative complex solution of the problem proposed by the authors and brought to working condition was based on a system approach adapted to the NTO of the oil and gas complex. The measured parameters of various NTO infrastructure — fields of LLC "GDYA" and SNGP were processed and analyzed at the moments of development of self-oscillating modes on them, due to self-sustaining nonlinear mechanisms of interaction of NTO elements with constant (non-oscillatory) sources of energy replenishment. Three such modes of self-excitation were illustrated. The most informative in this case were the transient modes of operation of the equipment. According to the technologies experimentally developed by the authors, the areas of critical operating modes of equipment with pronounced potentially dangerous bifurcation points were analyzed. The result of superimposing treatments of the measured parameters of eight full-scale tests of NTO — graphs of dimensionless amplitude-frequency characteristics of the interconnections of a real dynamic system was presented: "input — gas cooling housing" → "output — pipe casing" at the frequency of self-oscillations. Discussion and Conclusion. The capabilities of ALARP did not meet the tasks of system monitoring of the occurrence and development of dangerous incidents in gas fields. This conclusion can be attributed to all standard sizes of NTO infrastructure in Russia. Fundamentally different solutions should be used to ensure comprehensive observability, controllability, and safety of NTO. A comprehensive scientific and technical program is recommended: "Innovative hardware and software tools and technologies to ensure the observability, controllability, and safety of the NTO infrastructure of Russia".

A thrust allocation method for DP vessels equipped with rudders

The thrust allocation unit is an important part of dynamic positioning system. Many dynamic positioning vessels are equipped with propellers and rudders, but the complex hydrodynamic characteristics of the propeller-rudder combination make thrust allocation a challenging task. However, research on thrust allocation methods has primarily focused on azimuth thrusters, and only limited attention has been given to the propeller-rudder combination. In this study, we propose a model-based thrust allocation method that incorporates an interaction model between the rudder and propeller, while also considering other factors such as the energy consumption and actuator mechanics. The proposed thrust allocation method is tested by using the dynamic positioning model test in a basin and the experimental results show that our method effectively maintains vessel position even in adverse sea conditions, with the thrust allocation results following the control commands well. The experimental results validate the effectiveness of the proposed thrust allocation method, indicating its potential application in real ship dynamic positioning systems.

Dynamical analysis of nonlinear fractional order Lorenz system with a novel design of intelligent solution predictive radial base networks

In this research work, the convective flow of Lorenz attractor in the fractional domain is modeled with a nonlinear flexible structure of radial basis neural network (RBNN). The physical parameters of the complex fractional-order system are initially computed with fractional order Runge–Kutta solver for different stochastic scenarios of control constraints and designed its parametric model with RBFN under various initial conditions of the chaotic Lorenz system. Multiple chaos of the Lorenz system with the dynamical neural network structure are computed for different fractional order solutions to analyze the sensitivity of chaotic behavior by means of Lyapunov exponents. Phase diagrams of the chaotic pattern show the fractional-order Lorenz system has compact dynamical behavior and has the potential for the design application of a real dynamical system. Embedded dimension and time delay of the fractional Lorenz system are computed using the average mutual information (AMI) technique to evaluate the suitability of the time delay chaotic pattern in the fractional domain, which is affected by the numerical algorithm and fluctuated on minute time step size. The proposed design of RBFN has been confirmed as an exceptionally high-performance tool for soft computing and dynamics analysis of chaotic systems in the fractional order domain.

Using scaling-region distributions to select embedding parameters

Reconstructing state-space dynamics from scalar data using time-delay embedding requires choosing values for the delay τ and the dimension m. Both parameters are critical to the success of the procedure and neither is easy to formally validate. While embedding theorems do offer formal guidance for these choices, in practice one has to resort to heuristics, such as the average mutual information (AMI) method of Fraser & Swinney for τ or the false near neighbor (FNN) method of Kennel et al. for m. Best practice suggests an iterative approach: one of these heuristics is used to make a good first guess for the corresponding free parameter and then an “asymptotic invariant” approach is then used to firm up its value by, e.g., computing the correlation dimension or Lyapunov exponent for a range of values and looking for convergence. This process can be subjective, as these computations often involve finding, and fitting a line to, a scaling region in a plot: a process that is generally done by eye and is not immune to confirmation bias. Moreover, most of these heuristics do not provide confidence intervals, making it difficult to say what “convergence” is. Here, we propose an approach that automates the first step, removing the subjectivity, and formalizes the second, offering a statistical test for convergence. Our approach rests upon a recently developed method for automated scaling-region selection that includes confidence intervals on the results. We demonstrate this methodology by selecting values for the embedding dimension for several real and simulated dynamical systems. We compare these results to those produced by FNN and validate them against known results—e.g., of the correlation dimension—where these are available. We note that this method extends to any free parameter in the theory or practice of delay reconstruction.

Editorial: Tensor numerical methods and their application in scientific computing and data science

Editorial: Tensor numerical methods and their application in scientific computing and data science

Learning safety in model-based Reinforcement Learning using MPC and Gaussian Processes

This paper proposes a method to encourage safety in Model Predictive Control (MPC)-based Reinforcement Learning (RL) via Gaussian Process (GP) regression. The framework consists of 1) a parametric MPC scheme that is employed as model-based controller with approximate knowledge on the real system's dynamics, 2) an episodic RL algorithm tasked with adjusting the MPC parametrization in order to increase its performance, and 3) GP regressors used to estimate, directly from data, constraints on the MPC parameters capable of predicting, up to some probability, whether the parametrization is likely to yield a safe or unsafe policy. These constraints are then enforced onto the RL updates in an effort to enhance the learning method with a probabilistic safety mechanism. Compared to other recent publications combining safe RL with MPC, our method does not require further assumptions on, e.g., the prediction model in order to retain computational tractability. We illustrate the results of our method in a numerical example on the control of a quadrotor drone in a safety-critical environment.

The large key space image encryption algorithm based on modulus synchronization between real and complex fractional-order dynamical systems.

This paper constructs and analyzes the dynamical properties of a new fractional-order real hyper-chaotic system and its corresponding complex variable system. A thorough analysis was done by employing stability of equilibrium points, phase plots, Lyapunov spectrum, and bifurcation analysis for the consequences of varying fractional-order derivative and parameter values on the system. For the first time, a modulus synchronization scheme is proposed to synchronize real and complex fractional-order dynamical systems. Based on Lyapunov stability theory, non-linear controllers are designed to achieve the proposed modulus synchronization scheme. A new modulus synchronization encryption algorithm with a large key space size for digital images is introduced for the application. The experimental results and analysis validate the desired algorithm. Also, we compare our result of the new encryption algorithm with the previously published literature and verify the efficacy of the considered scheme. Numerical simulations are given to validate the theoretical analysis

Deep adaptive control with online identification for industrial robots

Derivation of control equations from data is a critical problem in numerous scientific and engineering fields. The inverse dynamic control of robot manipulators in the field of industrial robot research is a key example. Traditionally, researchers needed to obtain the robot dynamic model through physical modeling methods before developing controllers. However, the robot dynamic model and suitable control methods are often elusive and difficult to tune, particularly when dealing with real dynamical systems. In this paper, we combine an enhanced online sparse Bayesian learning (OSBL) algorithm and a model reference adaptive control method to obtain a data-driven modeling and control strategy from data containing noise; this strategy can be applied to dynamical systems. In particular, we use a sparse Bayesian approach, relying only on some prior knowledge of its physics, to extract an accurate mechanistic model from the measured data. Unmodeled parameters are further identified from the modeling error through a deep neural network (DNN). By combining the identification model with a model reference adaptive control approach, a general deep adaptive control (DAC) method is obtained, which can tolerate unmodeled dynamics. The adaptive update law is derived from Lyapunov’s stability criterion, which guarantees the asymptotic stability of the system. Finally, the Enhanced OSBL identification method and DAC scheme are applied on a six-degree-of-freedom industrial robot, and the effectiveness of the proposed method is verified.

Numerical Modeling of Biogeochemical Carbon Cycles in Swamp Ecosystems

A dynamic model of biogeochemical carbon cycles in swamp ecosystems is proposed. There are fast and slow biogeochemical cycles. Fast cycles operate in the biosphere and include photosynthesis, vegetative growth, and decomposition. Swamp ecosystems are one of the significant reservoirs of biogeochemical cycles. It is known that huge reserves of carbon in the form of slightly decomposed organic matter are preserved in the swamps. They are active sources of methane and carbon dioxide runoff from the atmosphere.
 Mathematical models of dynamic processes in ecology can be divided into two categories: quantitative and qualitative. Quantitative models, as a rule, are aimed at solving problems of predicting numerical indicators of the dynamics of real systems. They must be modified to consider specific climatic conditions, special types of swamp vegetation, and hydrological regime for their successful application.
 Qualitative models written as systems of differential equations assume the finding of singular points, their classification and study for stability, the construction of phase portraits, etc. Such models rarely lend themselves to quantitative verification, but provide important knowledge and understanding of processes in nature. A qualitative study of the system of ordinary differential equations describing carbon cycles is carried out, the types of singular points are investigated, integral curves and phase portraits are constructed.