Discrete Time Research Articles

High order conservative Lagrangian schemes for two-dimensional radiation hydrodynamics equations in the equilibrium-diffusion limit

Radiation hydrodynamics equations (RHE) refer to the study of how interactions between radiation and matter influence thermodynamic states and dynamic flow, which has been widely applied to high temperature hydrodynamics, such as inertial confinement fusion (ICF) and astrophysical gaseous stars. Solving RHE accurately and robustly even under the equilibrium diffusion approximation is a challenging task. To address this, we develop two types of high order conservative Lagrangian schemes for RHE in the equilibrium-diffusion limit for the two dimensional case on the Lagrangian moving mesh. Based on the multi-resolution WENO reconstruction for the spatial discretization and strong stability preserving Runge-Kutta (SSP-RK) time discretization, we first develop an explicit Lagrangian scheme with the HLLC numerical flux to achieve high order accuracy in space and time. We also discuss the positivity-preserving property of the high order explicit Lagrangian scheme. To overcome the severe time step restriction arising from the nonlinear radiation diffusion term in the explicit scheme, we further present a high order explicit-implicit-null (EIN) Lagrangian scheme. By adding a sufficiently large linear diffusion term on both sides of the scheme, we treat the complicated nonlinear parts explicitly and efficiently, and treat the added linear diffusion term on the right-hand side implicitly with a relaxed time step restriction. According to our numerical experiments, these two types of Lagrangian schemes are high order accurate, conservative and can capture the interfaces automatically. Additionally, the explicit scheme is found to be non-oscillatory and can preserve positivity while maintaining the original high order accuracy.

Multi‐event‐triggered adaptive dynamic programming for non‐zero‐sum game of unknown nonlinear system

AbstractIn this article, a multi‐event‐triggered (MET) near‐optimal tracking control algorithm is developed for a discrete‐time (DT) nonlinear non‐zero‐sum game (NZSG). Firstly, the iterative dual heuristic dynamic programming (DHP) approach is utilized to obtain the optimal tracking control set. Subsequently, distinct independent triggering conditions are formulated for control inputs to improve the utilization efficiency of resources and ensure that the independence between them and the closed‐loop system has an excellent control performance. Meanwhile, input‐to‐state stability (ISS) is proven for the MET tracking plant. Additionally, neural networks (NNs) are designed in the iterative DHP algorithm to construct the model module, the critic module and the action modules, with aims to identify the unknown system, approximate costate function and estimate optimal tracking control strategy set, respectively. Finally, two numerical experimental simulations are conducted to verify the effectiveness of the proposed scheme.

Prognostic Information on Progression to Dementia: Quantification of the Impact on Quality of Life.

The increasing interest in early identification of people at risk of developing dementia, has led to the development of numerous models aimed at estimating the likelihood of progression from mild cognitive impairment (MCI) to dementia. It is important to study both the need for and possible outcomes related with such prediction models, including the impact of risk predictions on perceived quality of life (QoL). This study aimed to quantify the impact that receiving a risk prediction on progression from MCI to dementia has on QoL. A Discrete Choice Experiment (DCE) and Time Trade Off (TTO) study were performed. Participants completed choice tasks related to dementia prognosis while imagining having MCI. We collected DCE data by an online survey, and TTO data via videoconferencing interviews. DCE data were analyzed using a mixed multinomial logit model and were anchored to a health state utility scale using mean observed TTO valuations. 296 people participated in the DCE and 42 in the TTO. Moderate and high predicted dementia risks were associated with decrements in utility (-0.05 and -0.18 respectively), compared to no prognostic information. Low predicted risk was associated with an increase in utility (0.06), as well as the availability of medication or lifestyle interventions (0.05 and 0.13 respectively). This study shows a significant impact of dementia risk predictions on QoL and highlights the importance of caution when sharing information about expected MCI disease courses.

Remote tele‐operation using model reference discrete‐time sliding mode control

AbstractIn this article, a dissipativity based discrete time sliding mode controller is proposed for bilateral tele‐operation in presence of communication delays and packet losses. The controllers for both master and slave arm are designed to follow a predefined impedance dynamics with dissipative characteristics. The slave arm controller is designed as a two layered structure. The first layer uses the tracking error between master and slave to create a target velocity profile based on the user defined impedance dynamics. The second layer uses this target velocity profile, slave motion and the contact forces to design a reaching law based discrete time sliding mode controller. A model following reaching law approach is used to design the sliding mode control which alleviates the chattering in the input by removing the impact of the accumulated effects of the past disturbance from the system. Moreover, an adaptive tuning method is used for the switching gain of the sliding mode control so that the controller can be designed even when exact information on the uncertainty bounds are unknown. Numerical simulations are presented for Phantom Omni arm to show the efficacy of the proposed method.

Online state estimation of Li-ion batteries using continuous-discrete nonlinear Kalman filters based on a nonlinear simplified electrochemical model

Online monitoring of Lithium-ion batteries (LIBs) during the charge and discharge periods is imperative to characterize their performance, assess their safety, and mitigate their degradation. One way to achieve this goal is to resort to model-based state estimations. Electrochemical based modelling has demonstrated significant potential to predict both the microscopic and the macroscopic battery dynamics. Existing electrochemical models are however continuous-time systems while measurements are performed at discrete time steps. Yet, most estimators are designed based on a fully discretized model. In this study, various types of continuous-discrete nonlinear Kalman filters are developed not only to address this challenge but also to compare their performance in utilizing highly nonlinear and meaningful physical variables. The filters rest on a simplified electrochemical model that accounts for the nonlinearities that stem from the dependence of the parameters on the electrolyte concentration. The performance of the estimators in adaptive and nonadaptive modes is tested for two different driving schedule loads (FUDS and Synthetic random higher C-rate). The results reveal that the estimators can track the real states' values if the filter model matches the real battery, even in the presence of 30 % error in the states' initial conditions. Furthermore, unlike the continuous-discrete extended Kalman filter (CD-EKF), the suggested continuous-discrete unscented Kalman filter (CD-UKF) effectively predicts the actual state values, even in instances where a significant structural mismatch exists between the embedded model and the synthetically generated measurement signal.

Design of a model‐free controller based on predictive functional control

AbstractIntelligent‐PID (i‐PID) control proposed by Fliess is a simple control algorithm. The controller is designed based on ultra‐local model, and consisted of PID type controller and derivatives of reference signal and controlled variables. Authors have considered discrete i‐PID controller and its properties, and one of the result was that PD controller was sufficient in discrete time domain. In discrete i‐PID controller, PD type controller is still required to ensure convergence of controlled error to zero, and PD type parameters affects convergence property, fast‐response property and stability. In this paper, predictive functional control (PFC) based on an ultra‐local model is proposed. By introducing of PFC, controlled variable converge to reference value without PD type controller. User‐specified parameters of the proposed controller is smaller than the conventional i‐PID controller. Characteristics of the proposed controller are compared with discrete i‐PID controller by numerical examples.

Intermittent monitoring‐based adaptive fault‐tolerant control for uncertain nonlinear systems with actuator switching

AbstractIn this paper, an adaptive fault‐tolerant control (FTC) scheme is proposed for a class of uncertain nonlinear systems with actuator failure, in which a healthy actuator is available as a backup. Instead of monitoring the failure in real‐time, an intermittent monitoring mechanism (IMM) is proposed by designing a set of monitoring functions operating only at discrete time instants, which makes the proposed scheme more applicable for networked control systems for which the actuator status has to be monitored via network. An adaptive backstepping controller is designed with the errors being modified to include shifting functions. Once actuator failure is detected by the monitoring functions, the actuator is switched to its backup and the controller is reconfigured by updating the shifting functions so that the tracking error can be driven into the prescribed performance in a given time.

A posteriori error estimates of the weak Galerkin finite element methods for parabolic problems

In this paper, We propose the residual-based a posteriori error estimator of the weak Galerkin finite element method with the backward Euler time discretization for the linear parabolic partial differential equation. For the a posteriori error estimator, we introduce the Helmholtz decomposition technique to prove its reliability. We mainly study WG element (Pj(K),Pℓ(∂K),V(K,r)=RTj(K)). Numerical experiments based on the lowest order case, i.e. (P0(K),P0(∂K),RT0(K)), are provided to verify the theoretical research.

Application of the Z-transform on the deconvolution of lightning electric field waveforms obtained from single-station measurements

Fast electric field measurement systems used in the field of atmospheric electricity often have decay time constants of the order of 1 ms, which means that slower processes – such as long continuing currents – are not visible in the raw electric field records. This paper presents a practical approach to compensate for the time decay caused by the system response and regain the original lightning electric field waveforms. To accomplish this task, the transfer function of the measurement system is modeled in the Z-domain. Then, from the inverse Z-transform of the transfer function, the compensated electric field is obtained in discrete time domain. Based on data obtained at Nova Gameleira campus of CEFET-MG (Belo Horizonte, Brazil), we present and discuss results of typical electric field waveforms that include some slower lightning processes, which are not readily observable in the raw data.

Ion-acoustic solitons oblique collisions in a magnetized ultra-relativistic degenerate plasma at critical compositions and nonzero temperatures

The mechanism of ion-acoustic (IA) soliton oblique collisions is examined at nonzero temperatures and critical compositions in a magnetized ultra-relativistic degenerate plasma using the extended Poincaré–Lighthill–Kuo (EPLK) method. Two conditions, referred to as the generic and critical cases, are obtained. Korteweg-de Vries (KdV) equations that govern the excitation of IA solitons and their phase shifts have been derived for the generic case. At a critical composition, modified KdV (mKdV) equations describe the nonlinear propagation of IA solitons, and thus the corresponding phase shifts are deduced. Furthermore, for an in-depth investigation, we examined the collision of two IA solitons of the same polarity and opposite polarities arising from oblique collisions at discrete times. In these two situations, the phase shifts of IA solitons due to the oblique collisions increase with an increase in the collision angle. The phase shift curve reaches its maximum value at the perpendicular collision. This work may aid in interpreting the features of IA solitons in a degenerate plasma, such as white dwarfs, that support soliton propagation.

Quantifying spatiotemporal patterns in classical and quantum systems out of equilibrium.

A rich variety of nonequilibrium dynamical phenomena and processes unambiguously calls for the development of general numerical techniques to probe and estimate a complex interplay between spatial and temporal degrees of freedom in many-body systems of completely different nature. In this work we provide a solution to this problem by adopting a structural complexity measure to quantify spatiotemporal patterns in the time-dependent digital representation of a system. On the basis of very limited amount of data our approach allows us to distinguish different dynamical regimes and define critical parameters in both classical and quantum systems. By the example of the discrete time crystal realized in nonequilibrium quantum systems we provide a complete low-level characterization of this nontrivial dynamical phase with only processing bitstrings, which can be considered as a valuable alternative to previous studies based on the calculations of qubit correlation functions.

Existence, Blow-Up, and Blow-Up Rate of Weak Solution to Fourth-Order Non-Newtonian Polytropic Variation-Inequality Arising from Consumption-Investment Models

This article obtains the conditions for the existence and nonexistence of weak solutions for a variation-inequality problem. This variational inequality is constructed by a fourth-order non-Newtonian polytropic operator which is receiving much attention recently. Under the proper condition of the parameter, the existence of a solution is proved by constructing the initial boundary value problem of an ellipse by time discretization and some elliptic equation theory. Under the opposite parameter condition, we analyze the nonexistence of the solution. The results show that the weak solution will blow up in finite time. Finally, we give the blow-up rate and the upper bound of the blow-up time.

Expressive completeness by separation for discrete time interval temporal logic with expanding modalities

Recently we established an analog of Gabbay's separation theorem about linear temporal logic (LTL) for the extension of Moszkowski's discrete time propositional Interval Temporal Logic (ITL) by two sets of expanding modalities, namely the unary neighbourhood modalities and the binary weak inverses of ITL's chop operator. One of the many useful applications of separation in LTL is the concise proof of LTL's expressive completeness wrt the monadic first-order theory of 〈ω,<〉 it enables. In this paper we show how our separation theorem about ITL facilitates a similar proof of the expressive completeness of ITL with expanding modalities wrt the monadic first- and second-order theories of 〈Z,<〉.

On convergence of a three‐layer semi‐discrete scheme for the non‐linear dynamic string equation of Kirchhoff‐type with time‐dependent coefficients

AbstractThis paper considers the Cauchy problem for the non‐linear dynamic string equation of Kirchhoff‐type with time‐varying coefficients. The objective of this work is to develop a time‐domain discretization algorithm capable of approximating a solution to this initial‐boundary value problem. To this end, a symmetric three‐layer semi‐discrete scheme is employed with respect to the temporal variable, wherein the value of a non‐linear term is evaluated at the middle node point. This approach enables the numerical solutions per temporal step to be obtained by inverting the linear operators, yielding a system of second‐order linear ordinary differential equations. Local convergence of the proposed scheme is established, and it achieves quadratic convergence regarding the step size of the discretization of time on the local temporal interval. We have conducted several numerical experiments using the proposed algorithm for various test problems to validate its performance. It can be said that the obtained numerical results are in accordance with the theoretical findings.

Sequential activity of CA1 hippocampal cells constitutes a temporal memory map for associative learning in mice

Sequential neural dynamics encoded by time cells play a crucial role in hippocampal function. However, the role of hippocampal sequential neural dynamics in associative learning is an open question. We used two-photon Ca2+ imaging of dorsal CA1 (dCA1) neurons in the stratum pyramidale (SP) in head-fixed mice performing a go-no go associative learning task to investigate how odor valence is temporally encoded in this area of the brain. We found that SP cells responded differentially to the rewarded or unrewarded odor. The stimuli were decoded accurately from the activity of the neuronal ensemble, and accuracy increased substantially as the animal learned to differentiate the stimuli. Decoding the stimulus from individual SP cells responding differentially revealed that decision-making took place at discrete times after stimulus presentation. Lick prediction decoded from the ensemble activity of cells in dCA1 correlated linearly with lick behavior. Our findings indicate that sequential activity of SP cells in dCA1 constitutes a temporal memory map used for decision-making in associative learning. VIDEO ABSTRACT.

ACERAC: Efficient Reinforcement Learning in Fine Time Discretization.

One of the main goals of reinforcement learning (RL) is to provide a way for physical machines to learn optimal behavior instead of being programmed. However, effective control of the machines usually requires fine time discretization. The most common RL methods apply independent random elements to each action, which is not suitable in that setting. It is not feasible because it causes the controlled system to jerk and does not ensure sufficient exploration since a single action is not long enough to create a significant experience that could be translated into policy improvement. In our view, these are the main obstacles that prevent the application of RL in contemporary control systems. To address these pitfalls, in this article, we introduce an RL framework and adequate analytical tools for actions that may be stochastically dependent in subsequent time instances. We also introduce an RL algorithm that approximately optimizes a policy that produces such actions. It applies experience replay (ER) to adjust the likelihood of sequences of previous actions to optimize expected n -step returns that the policy yields. The efficiency of this algorithm is verified against four other RL methods [continuous deep advantage updating (CDAU), proximal policy optimization (PPO), soft actor-critic (SAC), and actor-critic with ER (ACER)] in four simulated learning control problems (Ant, HalfCheetah, Hopper, and Walker2D) in diverse time discretization. The algorithm introduced here outperforms the competitors in most cases considered.

Adaptive Predictive Deadbeat Current Control of Single-Phase Multi-Tuned Shunt Hybrid Active Power Filters

This paper suggests an adaptive predictive deadbeat current control method for single-phase multi-tuned shunt hybrid active power filters (HAPFs) to improve the power quality of single-phase and three-phase four-wire utility grids. The HAPF structure eliminates the resonance between the passive power filter and the grid impedance. Furthermore, it can be integrated into passive filters to enhance their filtering performance. In this paper, a digital algorithm is proposed for managing the performance of each converter leg accurately. To do so, an exact model of the high-order system is developed, and the transfer function of the plant is calculated in continuous and discrete time domains. Then, a predictive deadbeat technique for HAPF current control is presented, which benefits from high accuracy, fast dynamics, and low sensitivity to system parameter mismatches. Extensive simulation and experimental tests are conducted and the results match well to confirm the success and appropriate performance of the overall system. Also, performance comparison with conventional solutions demonstrates the superiority of the suggested filtering technique.

Averaging principle for stochastic 3D generalized Navier–Stokes equations

In this paper, the multiscale stochastic 3D generalized Navier–Stokes equations are studied. By using Khasminkii’s time discretization approach and the technique of stopping time, the strong averaging principle for stochastic 3D generalized Navier–Stokes equations is proved in the space [Formula: see text].

Discrete time crystal in an open optomechanical system

Discrete time crystal in an open optomechanical system

An asymptotic preserving and energy stable scheme for the Euler-Poisson system in the quasineutral limit

An asymptotic preserving (AP) and energy stable scheme for the Euler-Poisson (EP) system under the quasineutral scaling is designed and analysed. Appropriate stabilisation terms are introduced in the convective fluxes of mass and momenta, and the gradient of the electrostatic potential which lead to the dissipation of mechanical energy and consequently the entropy stability of solutions. The time discretisation is semi-implicit in nature, whereas the space discretisation uses a finite volume framework on a marker and cell (MAC) grid. The numerical resolution of the fully-discrete scheme involves two steps: the solution of a linear elliptic problem for the potential and an explicit evaluation for the density and velocity. The proposed scheme possesses several physically relevant attributes, such as the positivity of density, entropy stability and the consistency with the weak formulation of the continuous EP system. The AP property of the scheme, i.e. the boundedness of the mesh parameters with respect to Debye length and its consistency with the quasineutral limit system, is demonstrated. The results of numerical case studies are presented to substantiate the robustness and efficiency of the proposed method.