Discrete Time Scale Research Articles

Time‐Coordinated Motion Planning for Multiple Unmanned Vehicles Using Distributed Model Predictive Control and Sequential Convex Programming

ABSTRACTThis paper introduces an integrated approach for time‐coordinated motion planning of multiple unmanned vehicles using distributed model predictive control (DMPC) and sequential convex programming (SCP). This approach employs a unified framework that integrates trajectory planning and tracking into a single optimization problem, effectively expanding the domain of attraction for the MPC controller and addressing the challenge of time‐coordination among multiple vehicles. Non‐uniform discrete time scales are introduced to mitigate the dimensionality of the optimization problem, thereby enhancing computational efficiency. By combining the ability of DMPC to distribute computational efforts across multiple vehicles with the iterative convexification method of SCP, our approach efficiently handles the complexities of non‐linear optimization. Theoretical analysis has confirmed the feasibility and stability of the proposed method. Based on this approach, the time‐coordinated sequential convex programming‐based distributed model predictive control (TC‐SCP‐DMPC) algorithm is proposed. Numerical simulations are conducted to validate the effectiveness and efficiency of the proposed algorithm in achieving time‐coordinated control of multiple unmanned vehicles.

Fixation-related potentials during a virtual navigation task: The influence of image statistics on early cortical processing.

Historically, electrophysiological correlates of scene processing have been studied with experiments using static stimuli presented for discrete timescales where participants maintain a fixed eye position. Gaps remain in generalizing these findings to real-world conditions where eye movements are made to select new visual information and where the environment remains stable but changes with our position and orientation in space, driving dynamic visual stimulation. Co-recording of eye movements and electroencephalography (EEG) is an approach to leverage fixations as time-locking events in the EEG recording under free-viewing conditions to create fixation-related potentials (FRPs), providing a neural snapshot in which to study visual processing under naturalistic conditions. The current experiment aimed to explore the influence of low-level image statistics-specifically, luminance and a metric of spatial frequency (slope of theamplitude spectrum)-on the early visual components evoked from fixation onsets in a free-viewing visual search and navigation task using a virtual environment. This research combines FRPs with an optimized approach to remove ocular artifacts and deconvolution modeling to correct for overlapping neural activity inherent in any free-viewing paradigm. The results suggest that early visual components-namely, the lambda response and N1-of the FRPs are sensitive to luminance and spatial frequency around fixation, separate from modulation due to underlying differences in eye-movement characteristics. Together, our results demonstrate the utility of studying the influence of image statistics on FRPs using a deconvolution modeling approach to control for overlapping neural activity and oculomotor covariates.

A novel prognostic method for degrading devices with nonlinear degradation processes indexed by both continuous and discrete time scales

Remaining useful life (RUL) prediction techniques based on degradation modeling to ensure the safety and reliability of stochastic degraded devices are of critical value. However, although the performance degradation of various devices in engineering practice is affected by multi-timescales, there are few RUL prediction methods for stochastic degraded devices with dual time scales in the existing literature. The research of prediction methods considering continuous and discrete time scales is blank. Toward this end, this paper proposes a novel prognostic method for nonlinear degradation devices based on the continuous-discrete dual time scales. First, the performance degradation with a discrete timescale is determined with a compound Poisson process and integrated into the diffusion process model to establish a nonlinear dual-timescale degradation model. Subsequently, the joint expression of the dual-timescale RUL distribution of the device in the first hitting time sense solves with threshold transformation. Finally, the effectiveness and superiority of the proposed method are verified through a numerical simulation and a gyroscope example. The experimental results show that the proposed method can provide more comprehensive information about the life distribution and effectively improve the accuracy of the RUL prediction.

Conditional score approaches to errors-in-variables competing risks data in discrete time.

Analysis of competing risks data has been an important topic in survival analysis due to the need to account for the dependence among the competing events. Also, event times are often recorded on discrete time scales, rendering the models tailored for discrete-time nature useful in the practice of survival analysis. In this work, we focus on regression analysis with discrete-time competing risks data, and consider the errors-in-variables issue where the covariates are prone to measurement errors. Viewing the true covariate value as a parameter, we develop the conditional score methods for various discrete-time competing risks models, including the cause-specific and subdistribution hazards models that have been popular in competing risks data analysis. The proposed estimators can be implemented by efficient computation algorithms, and the associated large sample theories can be simply obtained. Simulation results show satisfactory finite sample performances, and the application with the competing risks data from the scleroderma lung study reveals the utility of the proposed methods.

On Grüss, Ostrowski and trapezoid-type inequalities via nabla integral on time scales

AbstractOstrowski inequality gives the absolute deviation of the function from its integral mean. Delta and nabla calculi are first two approaches to study time scales calculus. This article presents the Ostrowski inequality for univariate first order nabla differentiable function by using Montgomery identity established for nabla integrals. Some extensions of dynamic Ostrowski-type inequality are investigated with the help of integration by parts for nabla integrals, properties of the modulus and polynomials on time scales. Furthermore, dynamic Grüss and trapezoid-type inequalities are established in their generalized form for twice nabla differentiable functions by utilizing the Montgomery identity. In addition, the obtained inequalities are discussed for continuous and discrete time scales.

Diamond-Type Dirac Dynamic System in Mathematical Physics

In order to merge continuous and discrete analyses, a number of dynamic derivative equations have been put out in the process of developing a time-scale calculus. The investigations that incorporated combined dynamic derivatives have led to the proposal of improved approximation expressions for computational application. One such expression is the diamond alpha (⋄α) derivative, which is defined as a linear combination of delta and nabla derivatives. Several dynamic equations and inequalities, as well as hybrid dynamic behavior—which does not occur in the real line or on discrete time scales—are analyzed using this combined concept. In this study, we consider a ⋄α Dirac system under boundary conditions on a uniform time scale. We examined some basic spectral properties of the problem we are considering, such as the simplicity, the reality of eigenvalues, orthogonality of eigenfunctions, and self adjointness of the operator. Finally, we construct an expression for the eigenfunction of the ⋄α Dirac boundary value problem (BVP) on a uniform time scale.

Log-periodic oscillations as real-time signatures of hierarchical dynamics in proteins.

The time-dependent relaxation of a dynamical system may exhibit a power-law behavior that is superimposed by log-periodic oscillations. D. Sornette [Phys. Rep. 297, 239 (1998)] showed that this behavior can be explained by a discrete scale invariance of the system, which is associated with discrete and equidistant timescales on a logarithmic scale. Examples include such diverse fields as financial crashes, random diffusion, and quantum topological materials. Recent time-resolved experiments and molecular dynamics simulations suggest that discrete scale invariance may also apply to hierarchical dynamics in proteins, where several fast local conformational changes are a prerequisite for a slow global transition to occur. Employing entropy-based timescale analysis and Markov state modeling to a simple one-dimensional hierarchical model and biomolecular simulation data, it is found that hierarchical systems quite generally give rise to logarithmically spaced discrete timescales. By introducing a one-dimensional reaction coordinate that collectively accounts for the hierarchically coupled degrees of freedom, the free energy landscape exhibits a characteristic staircase shape with two metastable end states, which causes the log-periodic time evolution of the system. The period of the log-oscillations reflects the effective roughness of the energy landscape and can, in simple cases, be interpreted in terms of the barriers of the staircase landscape.

Deciding Equations in the Time Warp Algebra

Join-preserving maps on the discrete time scale $\omega^+$, referred to as time warps, have been proposed as graded modalities that can be used to quantify the growth of information in the course of program execution. The set of time warps forms a simple distributive involutive residuated lattice -- called the time warp algebra -- that is equipped with residual operations relevant to potential applications. In this paper, we show that although the time warp algebra generates a variety that lacks the finite model property, it nevertheless has a decidable equational theory. We also describe an implementation of a procedure for deciding equations in this algebra, written in the OCaml programming language, that makes use of the Z3 theorem prover.

On a Special Two-Person Dynamic Game

The asymptotical properties of a special dynamic two-person game are examined under best-response dynamics in both discrete and continuos time scales. The direction of strategy changes by the players depend on the best responses to the strategies of the competitors and on their own strategies. Conditions are given first for the local asymptotical stability of the equilibrium if instantaneous data are available to the players concerning all current strategies. Next, it is assumed that only delayed information is available about one or more strategies. In the discrete case, the presence of delays has an effect on only the order of the governing difference equations. Under continuous scales, several possibilities are considered: each player has a delay in the strategy of its competitor; player 1 has identical delays in both strategies; the players have identical delays in their own strategies; player 1 has different delays in both strategies; and the players have different delays in their own strategies. In all cases, it is assumed that the equilibrium is asymptotically stable without delays, and we examine how delays can make the equilibrium unstable. For small delays, the stability is preserved. In the cases of one-delay models, the critical value of the delay is determined when stability changes to instability. In the cases of two and three delays, the stability-switching curves are determined in the two-dimensional space of the delays, where stability becomes lost if the delay pair crosses this curve. The methodology is different for the one-, two-, and three-delay cases outlined in this paper.

Estimated energy consumption and billing modelling using power availability recorder

Accurate estimation and billing of unmetered facilities and homes by the energy providers in Nigeria has been a serious nightmare to the rural and other consumers. The unjust billing due to inaccurate and wrong estimations have resulted to rancor, facilities damage and economic hardship among the people due to poor equipment and unreliable consumption data. This research work presents accurate energy consumption and billing model using power availability recorder (PAR) which keeps track of how long electricity lasts in a building or facility at discrete time scale. Two residential buildings situated at the University of Nigeria, Nsukka, and one facility each in two chosen communities in Nsukka, were used as commercial and residential electricity consumers respectively for this study. The readings from the PAR were obtained in discrete scale (mini-seconds), and were later converted to seconds and hours. Similarly, the readings from the Energy Consumption meter were taken in kWh. These daily results taken simultaneously at 6 am and 6 pm were used to calculate the Availability Factor (AF), Load Use Factor (LUF) and Average Load (AL) in kW. With PAR as energy meter in significant number of houses within the chosen communities, AF, LUF, and AL for different facilities in the community were obtained and used for the energy billing model development. A relationship, which depends on availability and load use factor, has been suggested in this study with a proportionality constant value of 14.78 kWh/day/customer for use in obtaining better estimates of the bills chargeable to unmetered customers. This will control the black-out relay period usually encountered by consumers instead of the grossly unjust estimated bills currently being used by Nigeria's electricity distribution companies.

Finite driving rate effects in the nonequilibrium athermal random field Ising model of thin systems

Thin disordered ferromagnetic systems driven by varying magnetic field represent a challenge from theoretical, experimental and technological perspective. Understanding their temporal evolution under the mutual influence of finite driving rate with which the external magnetic field changes, geometry of the sample, and the amount of disorder is a complex task. It is studied here within the framework of the nonequilibrium athermal random field Ising model which has a dynamical behavior suitable for this kind of analysis both in statistical physics and physics of magnetism. Our results, obtained by means of numerical simulations performed at a discrete time scale of the model in a wide range of driving rates and domains of disorder, unveil the rate sensitivity exhibited by the response signal and the underlying avalanche distributions, correlation functions of spin-flip events, average avalanche shapes and the values of coercive field. Due to the rate-imposed spreading of multiple avalanches during the same/partially shared intervals of time, the pertaining effective exponents turn out to be rate-dependent as well. Our findings provide new insights on some features of field driven nonequilateral systems that could be applicable for investigations of a variety of thin systems driven at finite driving rates.

A case for chaos theory inclusion in neuropsychoanalytic modeling

ABSTRACT This paper provides a review of the current evidence of chaoticity at various scales of the brain–mind as well as the application of nonlinear tools in clinical practice. Based on these data, a hypothesis is formulated that the brain–mind at various scales can operate in linear, nonlinear, or hybrid modes, such as chaotic functioning accompanied by noise. A thesis formulated by Mark Solms that living systems must minimize Shannon’s entropy of physical states (sensory entropy) is considered. Based on the data presented in this paper, minimization of entropy in that sense appears to be describing only a part of the complex brain–mind dynamics. Studies evaluating measures of entropy specifically developed for real living systems such as discrete timescale entropy (ApEn) suggest that a decrease in EEG entropy can be observed in some neuronal processes (e.g. progression from wakefulness to deep sleep); however, EEG entropy is observed to be increasing at other times and in other modes of brain–mind functioning (e.g. progression from deep sleep to REM to wakefulness; and from vegetative state to wakefulness). The clinical implications are discussed. This paper proposes that it would be theoretically and clinically beneficial for future revisions of neuropsychoanalytic models to consider including the chaos theory framework.

Ligands selectively tune the local and global motions of neurotensin receptor 1 (NTS1)

Nuclear magnetic resonance (NMR) studies have revealed that fast methyl sidechain dynamics can report on entropically-driven allostery. Yet, NMR applications have been largely limited to the super-microsecond motional regimes of G protein-coupled receptors (GPCRs). We use13Cε-methionine chemical shift-based global order parameters to test if ligands affect the fast dynamics of a thermostabilized GPCR, neurotensin receptor 1 (NTS1). We establish that the NTS1 solution ensemble includes substates with lifetimes on several, discrete timescales. The longest-lived states reflect those captured in agonist- and inverse agonist-bound crystal structures, separated by large energy barriers. We observe that the rapid fluctuations of individual methionine residues, superimposed on these long-lived states, respond collectively with the degree of fast, global dynamics correlating with ligand pharmacology. This approach lends confidence to interpreting spectra in terms of local structure and methyl dihedral angle geometry. The results suggest a role for sub-microsecond dynamics and conformational entropy in GPCR ligand discrimination.

FLOQUET MULTIPLIERS AND THE STABILITY OF PERIODIC LINEAR DIFFERENTIAL EQUATIONS: A UNIFIED ALGORITHM AND ITS COMPUTER REALIZATION

Floquet multipliers (characteristic multipliers) play significant role in the stability of the periodic equations. Based on the iterative method, we provide a unified algorithm to compute the Floquet multipliers (characteristic multipliers) and determine the stability of the periodic linear differential equations on time scales unifying discrete, continuous, and hybrid dynamics. Our approach is based on calculating the value of $\mathcal{A}$ and $\mathcal{B}$ (see Theorem 3.1), which are the sum and product of all Floquet multipliers (characteristic multipliers) of the system, respectively. We obtain an explicit expression of $\mathcal{A}$ (see Theorem 4.1) by the method of variation and approximation theory (iterative method), and an explicit expression of $\mathcal{B}$ by Liouville's formula. Furthermore, a computer program is designed to realize our algorithm. Specifically, you can determine the stability of a second order periodic linear system, whether they are discrete, continuous or hybrid, as long as you enter the program codes associated with the parameters of the equation. In fact, few literatures have dealt with the algorithm to compute the Floquet multipliers, not mention to design the program for its computer realization. Our algorithm gives the explicit expressions of all Floquet multipliers and our computer program is based on the approximations of these explicit expressions. In particular, on an arbitrary discrete periodic time scale, we can do a finite number of calculations to get the explicit value of Floquet multipliers (see Theorem 4.2). Therefore, for any discrete periodic system, we can accurately determine the stability of the system by our algorithm even without computer! Finally, in Section 6, several examples are presented to illustrate the effectiveness of our algorithm.

Разработка имитационной модели определения координат воздушных объектов по отраженному от воздушного объекта и принятому на антенну навигационного приемника навигационному сигналу ГНСС

This article is devoted to the development of a simulation model for determining the coordinates of air objects based on the GNSS navigation signal reflected from the air object and received on the antenna of the navigation receiver. To solve the problem of radar of aerial objects using bistatic location based on the use of GNSS satellites, it was necessary to pre-evaluate the energy or energy potential of the satellite navigation signal reflected from the aerial object, which then comes to the receiving antenna of the navigation equipment of consumers. The results obtained in the form of the developed computational schemes will form the basis of the algorithmic and software of the passive radar system and will allow determining the requirements for its technical support carried out within the framework of the grant project of the Ministry of Education and Science of the Republic of Kazakhstan No. AP09260581. In this paper, simulation modeling is understood as such an approach to modeling complex systems, which allows us to develop a single meta-model that combines into an integral set of models of objects and processes of their interaction based on a single information and event space in a single discrete time scale, and implement the resulting meta-model in an electronically equivalent form on an electronic computer. One of the main tasks of the study of complex radio engineering systems by modeling is the choice of modeling principles that should allow the construction of the simulated system in accordance with the goals and objectives of the study in such a way that the model adequately reflects the processes and patterns of behavior of the system under study with the necessary degree of reliability and detail. One of the most productive methods of developing a software and hardware complex of simulation modeling is object-oriented modeling.

Forecasting of SPI and Meteorological Drought Based on the Artificial Neural Network and M5P Model Tree

Climate change has caused droughts to increase in frequency and severity worldwide, which has attracted scientists to create drought prediction models to mitigate the impacts of droughts. One of the most important challenges in addressing droughts is developing accurate models to predict their discrete characteristics, i.e., occurrence, duration, and severity. The current research examined the performance of several different machine learning models, including Artificial Neural Network (ANN) and M5P Tree in forecasting the most widely used drought measure, the Standardized Precipitation Index (SPI), at both discrete time scales (SPI 3, SPI 6). The drought model was developed utilizing rainfall data from two stations in India (i.e., Angangaon and Dahalewadi) for 2000–2019, wherein the first 14 years are employed for model training, while the remaining six years are employed for model validation. The subset regression analysis was performed on 12 different input combinations to choose the best input combination for SPI 3 and SPI 6. The sensitivity analysis was carried out on the given best input combination to find the most effective parameter for forecasting. The performance of all the developed models for ANN (4, 5), ANN (5, 6), ANN (6, 7), and M5P models was assessed through the different statistical indicators, namely, MAE, RMSE, RAE, RRSE, and r. The results revealed that SPI (t-1) is the most sensitive parameters with highest values of β = 0.916, 1.017, respectively, for SPI-3 and SPI-6 prediction at both stations on the best input combinations i.e., combination 7 (SPI-1/SPI-3/SPI-4/SPI-5/SPI-8/SPI-9/SPI-11) and combination 4 (SPI-1/SPI-2/SPI-6/SPI-7) based on the higher values of R2 and Adjusted R2 while the lowest values of MSE values. It is clear from the performance of models that the M5P model has higher r values and lesser RMSE values as compared to ANN (4, 5), ANN (5, 6), and ANN (6, 7) models. Therefore, the M5P model was superior to other developed models at both stations.

Conditional Degradation of UNC-31/CAPS Enables Spatiotemporal Analysis of Neuropeptide Function.

Neuropeptide release from dense-core vesicles in Caenorhabditis elegans is promoted by UNC-31, ortholog of the calcium-dependent activator protein for secretion (CAPS). Loss of UNC-31 causes multiple phenotypes in C. elegans including reduced motility, retention of late-stage eggs, and reduction in evoked synaptic release. However, the ability to analyze UNC-31 function over discrete timescales and in specific neurons is lacking. Here, we generated and validated a tool to enable UNC-31 expression and spatiotemporal functional analysis. We show that endogenously tagged UNC-31 is expressed in major ganglia and nerve cords from late embryonic stages through to adult. Using the auxin-inducible degradation system, we depleted UNC-31 postembryonically from the hermaphrodite nervous system and revealed defects in egg laying, locomotion, and vesicle release that were comparable to those in unc-31 null mutant animals. In addition, we found that depleting UNC-31 specifically from the BAG sensory neurons causes increased intestinal fat storage, highlighting the spatial sensitivity of this system. Together, this protein degradation tool may be used to facilitate studies of neuropeptide function at precise cellular and temporal scales.SIGNIFICANCE STATEMENT Animal behavior and physiology is controlled by neuropeptides that are released from specific neuronal sources. The ability to dissect discrete neuropeptide functions requires precise manipulation of neuropeptide release. We have developed and validated a tool that enables precise spatiotemporal regulation of neuropeptide release that will enable researchers to examine neuropeptide function at exceptional resolution.

A structure-preserving algorithm for time-scale non-shifted Hamiltonian systems

The variational calculus of time-scale non-shifted systems includes both the traditional continuous and traditional significant discrete variational calculus. Not only can the combination of Δ and ∇ derivatives be beneficial to obtaining higher convergence order in numerical analysis, but also it prompts the time-scale numerical computational scheme to have good properties, for instance, structure-preserving. In this letter, a structure-preserving algorithm for time-scale non-shifted Hamiltonian systems is proposed. By using the time-scale discrete variational method and calculus theory, and taking a discrete time scale in the variational principle of non-shifted Hamiltonian systems, the corresponding discrete Hamiltonian principle can be obtained. Furthermore, the time-scale discrete Hamilton difference equations, Noether theorem, and the symplectic scheme of discrete Hamiltonian systems are obtained. Finally, taking the Kepler problem and damped oscillator for time-scale non-shifted Hamiltonian systems as examples, they show that the time-scale discrete variational method is a structure-preserving algorithm. The new algorithm not only provides a numerical method for solving time-scale non-shifted dynamic equations but can be calculated with variable step sizes to improve the computational speed.

Normal Forms, Holomorphic Linearization and Generic Bifurcations of Dynamic Equations on Discrete Time Scales

Normal Forms, Holomorphic Linearization and Generic Bifurcations of Dynamic Equations on Discrete Time Scales

The chaotic monopolist revisited with bounded rationality and delay dynamics

Two types of boundedly rational monopolists are studied when the marginal revenue is not necessarily negative sloping. A limited monopolist (ℓ-monopolist) knows only the price and output values in two previous periods. A knowledgeable monopolist (k-monopolist) knows the analytic expression of the marginal profit function and forms sales expectations based on past market information. The k-monopolist adjusts its output levels according to the usual gradient process, while the ℓ-monopolist approximates the marginal profit with a two-point finite difference formula. Stability conditions and complex dynamics are studied in discrete and continuous time scales. In the discrete model, the two monopolists exhibit different dynamics after the stability loss. The ℓ-monopolist has a cascade of a Neimark-Sacker bifurcation. In contrast, the k-monopolist has two different routes, a period-doubling and a Neimark-Sacker bifurcation, depending on how to form sales expectations. In the continuous case, the discrete models are transformed into continuous models via Euler transformation. The stability switching curves are analytically constructed and the directions of stability switching are characterized by computing the stability index for each point of the curves. It is numerically demonstrated that two monopolists exhibit similar dynamics and the k-monopolist approximates the ℓ-monopolist under a special circumstance.