Discrete-time Approach Research Articles

A novel discrete-time approach to the continuous-time SIS dynamics

A novel discrete-time approach to the continuous-time SIS dynamics

Discrete-Time Sliding Mode Control Strategies—State of the Art

Variable structure control systems are known to provide a high level of robustness to external disturbances and modeling uncertainties with comparably low computational complexity. Thanks to these features, they have found applications in various fields, such as power engineering, electronics, robotics, and aviation. In recent decades, the field of sliding mode control has developed significantly. Therefore, this study aims to discuss the basic concepts and design methodology of such strategies. Although in the 20th century, continuous-time sliding mode control has been the center of the control engineering society’s attention, it has certain major shortcomings. In particular, such control schemes result in undesirable high-frequency oscillations when applied digitally. Therefore, the more recent discrete-time approach to sliding mode control has gained recognition in the 21st century. Since the introduction of discrete-time sliding mode control strategies, the reaching law-based controller design method has been designed, within which two main paradigms may be named: the switching type and the nonswitching type quasi-sliding mode. This paper presents a broad review of the discrete-time sliding mode control strategies, starting from the definition of sliding mode through the controller design procedures and up to potential applications. The aim of this study is to provide an up-to-date state of the art and introduce readers to the newest trends and achievements in the field of sliding mode control.

Control of a Wind Turbine Working in the Intermediate Velocity Zone: A Model Free Discrete Time Approach

Control of a Wind Turbine Working in the Intermediate Velocity Zone: A Model Free Discrete Time Approach

Seismic assessment of cylindrical storage tanks to records with different frequency contents considering fluid–structure–soil/foundation interaction

This study aims to primarily assess the dynamic behavior of ground-supported cylindrical tanks to real near-fault earthquake records with different frequency contents including soil-structure interaction effects. A cylindrical tank’s fluid-tank-soil interaction is modeled using a three-degree-of-freedom lumped mass model for the fluid. A mass-spring-dashpot model with frequency independence is used to model the soil/foundation system. Broad, medium, and slender tank models are developed considering and ignoring the soil flexibility effect. An algorithm based on discrete-time state-space approaches is developed for performing numerical simulations of the modeled tanks excited by the low-, medium-, and high-frequency contents. The obtained results for the modeled tanks in terms of base shear, overturning moment, and connective and impulsive mass displacements incorporating the influence of supporting soils as well as the frequency contents of excitation records are evaluated and compared with the corresponding results for modeled tanks with a fixed base. In addition, the liquid sloshing height for all aspect ratios is also evaluated and compared. The study’s findings clearly indicate that records with low-frequency contents require significantly more seismically demanding than other records for the considered soil types. Remarkably, soil-tank interaction under earthquake motions substantially amplifies the induced response.

A Constraint Programming Approach for Discrete Time–Cost Tradeoff Problems in a Time-Constrained Activity Network

The discrete time-cost tradeoff problem (DTCTP) is a well-researched topic in the field of operations research. The majority of existing DTCTP models are based on traditional activity networks, which permit the execution of an activity as soon as all its predecessors have been completed. This assumption is reasonable, but it is important to note that there are always exceptions. The main work of this study was threefold. Firstly, we expanded the analysis of the DTCTP to encompass time-constrained activity networks (DTCTPTC), which encompassed three different types of time constraints. The first constraint was the time-window constraint, which limited the time interval during which an activity could be executed. The second constraint was the time-schedule constraint, which specified the times at which an activity could begin execution. The third constraint was the time-switch constraint, which required project activities to start at specific times and remain inactive during designated time periods. Secondly, a constraint programming (CP) model was developed for the purpose of solving the DTCTPTC. The model employed interval variables to define the activity and its potential time constraints, while CP expressions were utilized to ensure the feasibility of the solution. The objective was to identify the optimal execution mode for each activity, the optimal start times for time-scheduled activities, and the optimal work/rest patterns for time-switch activities, with the aim of minimizing the total cost of the project. Finally, the efficacy of the proposed CP model was validated through two case studies based on two illustrative projects of varying sizes. The outcomes were then compared against existing algorithms. The results demonstrated that time constraints were important factors affecting schedule optimization, and the proposed CP model had the ability to solve large-scale DTCTPTC.

Analysis of sampled-data hybrid integrator-gain systems: A discrete-time approach

Hybrid integrator-gain systems (HIGS) are hybrid control elements used to overcome fundamental performance limitations of linear time-invariant feedback control, and have enjoyed recent successes in engineering applications such as high-precision motion systems. However, despite the relevance of digital implementations, the creation of sampled-data versions of HIGS and their formal analysis have not been addressed in the literature so far, and will form the topic of the present paper. Thereto, we present discrete-time HIGS elements, which preserve the main philosophy behind the operation of HIGS in continuous time. Moreover, stability criteria are presented that can be used to certify input-to-state stability of discrete-time and sampled-data HIGS-controlled systems based on both (i) (measured) frequency response data, and (ii) linear matrix inequalities (LMIs). A comparison between these stability criteria is presented as well. A numerical case study is provided to illustrate the application of the main results.

Quick prediction of thinning in stretch forming of hat-shaped profiles during multi-stage hot sheet metal forming

A novel quick model predicting the thinning distribution in hot stretch forming of hat-shaped profiles as a building block for a closed-loop control of a multi-stage forming process operated at a stroke rate of up to 12 strokes/min (5 s per stroke) is presented. The key characteristic of this new model is that it considers both a spatially as well as time-variant temperature distribution within the blank during forming. This is achieved by using temperature sensors and applying the information as input for the model, which is based on force equilibria and the theory of plasticity. The model predicts the geometry of the formed part which can be used as crucial input information to the process control. For the quick model, an element-based time-discrete approach was chosen. By assuming a plane strain condition and a decoupling of thermal and mechanical computation as well as by adapting further assumptions, the calculation time on a current desktop PC is 4 s. The achieved calculation time is sufficiently short to realize the control at the given stroke rates. The model is validated by experiments using a hot stretch forming setup designed to simulate the multi-stage process. The model successfully replicates the influence of various stroke rates on the final thinning distribution and predicts the effects of diverse pre-cooling scenarios.

On-policy and off-policy Q-learning strategies for spacecraft systems: An approach for time-varying discrete-time without controllability assumption of augmented system

This article investigates two On-policy and Off-policy Q-learning algorithms for time-varying linear discrete-time systems (DTSs) in the presence of complete dynamic uncertainties. To handle the challenge of time-varying description, the lifting method is employed to transform the original time-varying linear DTS into time-invariant linear DTS in the absence of the conventional controllability condition, which affects to the convergence of traditional Q-learning algorithms. Based on theoretical analysis of the structure in the obtained time-invariant linear DTS, On-policy and Off-policy algorithms are proposed to guarantee the convergence of Q-learning algorithms. Both On-policy and Off-policy Q-learning algorithms guarantee model-free consideration under the data collection. Especially, the Off-policy technique is able to develop the algorithm with high data efficiency because the collected data can be utilized again after each iteration. Finally, the simulation results of two-dimensional systems and spacecraft control systems are presented to validate the effectiveness of the two proposed control schemes.

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 Adaptive Control for Uncertain Scalar Multiagent Systems with Coupled Dynamics: A Lyapunov-Based Approach

Discrete-time architectures offer a distinct advantage over their continuous counterparts, as they can be seamlessly implemented on embedded hardware without the necessity for discretization processes. Yet, because of the difficulty of ensuring Lyapunov difference expressions, their designs, which are based on quadratic Lyapunov-based frameworks, are highly complex. As a result, various existing continuous-time results using adaptive control methods to deal with system uncertainties and coupled dynamics in agents of a multiagent system cannot be directly applied to the discrete-time context. Furthermore, compared to their continuous-time equivalent, discrete-time information exchange based on periodic time intervals is more practical in the control of multiagent systems. Motivated by these standpoints, in this paper, we first introduce a discrete-time adaptive control architecture designed for uncertain scalar multiagent systems without coupled dynamics as a preliminary result. We then introduce another discrete-time adaptive control approach for uncertain multiagent systems in the presence of coupled dynamics. Our approach incorporates observer dynamics to manage unmeasurable coupled dynamics, along with a user-assigned Laplacian matrix to induce cooperative behaviors among multiple agents. Our solution includes Lyapunov analysis with logarithmic and quadratic Lyapunov functions for guaranteeing asymptotic stability with both controllers. To demonstrate the effectiveness of the proposed control architectures, we provide an illustrative example. The illustrative numerical example shows that the standard discrete-time adaptive control in the absence of observer dynamics cannot guarantee the reference state vector tracking, while the proposed discrete-time adaptive control can ensure the tracking objective.

The impact of dispersal and allee effects on tick invasion: a spatially-explicit discrete-time modelling approach

We develop a spatially-explicit discrete-time model that describes tick population dynamics and incorporates the developmental stages for an individual tick. The model allows for two types of movements across patches: (1) active tick movement and (2) movement of ticks through a host. We analyse the dynamics of the model under a nonlinear fecundity term that exhibits an Allee effect. We show that the system has bounded solutions and, under certain conditions on the parameters, is monotone. Using the next generation matrix approach, we define the net reproductive number of the multi-patch model, R ( 0 ) , and establish conditions under which R ( 0 ) determines the local and global asymptotic stability of the extinction equilibrium. We then show that the model exhibits rich dynamics including a monostable equilibrium, multi-stable equilibria, and cycles. Finally, we perform some numerical studies to further examine how dispersal and Allee effects impact tick survival and system stability.

Secure message recovery in presence of the cyber-attack using discrete-time chaotic modulation approach

The secure message recovery problem in presence of cyber-attack is considered in this paper. The message is modulated by a master chaotic system which may be vulnerable to a cyber-attack and then transmitted to a receiver. By designing a novel observer at the receiver side, the message is recovered correctly. All the design process is performed in discrete-time domain which is suitable for digital communications. Besides, there are no restrictions on the message signal and nonlinear term signature matrix which makes the proposed approach more suitable in the real applications. Numerical simulations are introduced to show the feasibility and the superior performance of the proposed scheme in comparison with the conventional methods.

Discrete Acoustics: ARMA-modeling of Time Processes, Theory

In physics, in particular, acoustics, time is traditionally considered as a continuous coordinate. Some exception is signal processing, where sampling is necessary for calculations on computers. But all acoustic problems are formulated and solved using time-continuous models described by differential equations and their solutions in the form of continuous time functions. Meanwhile, these problems can be formulated and solved in an equivalent way using discrete-time models described by finite-difference equations and their solutions in the form of time series. As the experience of some other fields of science, for example, control theory, shows, the discrete approach has a number of advantages over the continuous approach, the useof which greatly facilitates the solution of many problems. This paper aims to partially fill the gap in acoustics that exists here and is aimed at creating the theoretical foundations of a discrete-time approach to solving acoustic problems. The paper is limited to the consideration of one oscillatory system widely used in acoustics— a linear structure with N degrees of freedom consisting of lumped inertial, elastic and dissipative elements, to which, in particular, the finite element method leads. For several continuous models of this system, equivalentdiscrete-time models are constructed in the paper, finite-difference equations are derived and their solutions are obtained. The criterion of equivalence of continuous and discrete models in the paper is the mathematically exact equality of the corresponding solutions at all discrete points in time. Based on this criterion, analytical relations have been established between the parameters of continuous and discrete models and their equations, which make it possible to build its discrete-time model based on a continuous model of the systemand, conversely, to build its continuous model based on a known discrete model. Special attention is paid in the paper to the forced vibrations of the system under the action of kinematic excitation, which is important in many acoustic problems, whereas in the literature only force excitation is considered. The paper also discussesone of the most useful properties of discrete modeling—the simplicity of constructing discrete models based on experimentally measured signals. A corresponding example is given. Note that the term “ARMAmodel” is an abbreviation for “autoregressive and moving average model”, generally accepted in control theory, systems theory and other fields of science.

Discrete-time scale-shift approach for wavelet construction and analysis

We provide a sufficient condition to construct a wavelet family from a purely discrete-time signal. The construction is based on an exact mathematical definition of the scale-shift operator in discrete-time and strongly motivated by a strict equivalence between voice transform and wavelet coefficients of a discrete-time signal. Particular classes of signals which satisfy this condition are given and furthers analysis are also considered.

Strategic Optimization of the Flushing Operations in Lubricant Manufacturing and Packaging Facilities.

Commercial lubricant industries use a complex pipeline network for the sequential processing of thousands of unique products annually. Flushing is conducted between changeovers to ensure the integrity of each production batch. An upcoming product is used for cleaning the residues of the previous batch, resulting in the formation of a commingled/mixed oil that does not match the specifications of either of the two batches. The existing operations are based on the operator's experience and trial and error. After a selected flush time, the samples are tested for their viscosity to determine the success of a flush. The approach results in long downtime, the generation of large commingled oil volumes, and huge economic losses. Hence, to overcome the drawback, our work introduces a solution strategy for systematically optimizing flushing operations and making more informed decisions to improve the resource-management footprint of these industries. We use the American Petroleum Institute-Technical Data Book (API-TDB) blending correlations for calculating the mixture viscosities in real-time. The blending correlations are combined with our first-principles models and validated against well-designed experimental data from the partnered lubricant facility. Next, we formulate an optimal control problem for predicting the optimum flushing times. We solve the problem using two solution techniques viz. Pontryagin's maximum principle and discrete-time nonlinear programming. The results from both approaches are compared with well-designed experimental data, and the economic and environmental significance are discussed. The results illustrate that with the application of a discrete-time nonlinear programming solution approach, the flushing can be conducted at a customized flow rate, and the necessary flushing volume can be reduced to over 30% as compared to the trial-and-error mode of operation.

Large deviations for the two-time-scale stochastic convective Brinkman-Forchheimer equations

The convective Brinkman-Forchheimer (CBF) equations are employed to characterize the motion of incompressible fluid in a saturated porous medium. This work investigates the small noise asymptotic of two-time-scale stochastic CBF equations in two and three dimensional bounded domains. More precisely, we establish a Wentzell-Freidlin type large deviation principle for stochastic partial differential equations that have slow and fast time-scales. The slow component is the stochastic CBF equations in two or three dimensions perturbed by a small multiplicative Gaussian noise, while the fast component is a stochastic reaction-diffusion equation with damping. The results are obtained by using a variational method (based on weak convergence approach) developed by Budhiraja and Dupuis, Khasminkii's time discretization approach and stopping time arguments. In particular, the findings from this study are also applicable to two-dimensional stochastic Navier-Stokes equations.

Model-Based and Data-Driven Control of Event-and Self-Triggered Discrete-Time Linear Systems.

The present paper considers the model-based and data-driven control of unknown discrete-time linear systems under event-triggering and self-triggering transmission schemes. To this end, we begin by presenting a dynamic event-triggering scheme (ETS) based on periodic sampling, and a discrete-time looped-functional approach, through which a model-based stability condition is derived. Combining the model-based condition with a recent data-based system representation, a data-driven stability criterion in the form of linear matrix inequalities (LMIs) is established, which also offers a way of co-designing the ETS matrix and the controller. To further alleviate the sampling burden of ETS due to its continuous/periodic detection, a self-triggering scheme (STS) is developed. Leveraging precollected input-state data, an algorithm for predicting the next transmission instant is given, while achieving system stability. Finally, numerical simulations showcase the efficacy of ETS and STS in reducing data transmissions as well as practicality of the proposed co-design methods.

A continuous time model for observer-based frequency tracking

Frequency tracking has been the focus of several research works for decades. One common method is to utilize a state observer to dynamically estimate the frequency of the measured signal. Most of existing works rely on a discrete time dynamic model for frequency and make use of discrete time observers. This paper introduces a new continuous time model which facilitates the use of several continuous time state observers from the literature. Consequently, the paper compares performance of discrete time approaches to continuous time one. The simulation results show efficacy of the proposed continuous time model specially when combined with specific nonlinear continuous time observers. Some of discrete time models as well as a recent continuous time model used in combination with a finite time observer are shown to yield poor results in most of simulative scenarios.

Optimal scheduling of vessels passing a waterway bottleneck

We develop a novel schedule optimization model for vessels passing a waterway bottleneck. From the system-optimal perspective, the model aims to minimize the total vessel bunker cost and delay penalties at destinations by incorporating the nonlinear relationship between bunker consumption and sailing speed into its calculations. The nonlinear model is linearized via two commonly used approximation techniques. The first one linearizes the bunker consumption function using a piecewise linear lower bound, while the second does so by discretizing the time. Numerical case studies are conducted for a real-world waterway bottleneck, the Three Gorges Dam Lock. Results reveal how the optimal cost components, vessel schedules, and delays are affected by key operating parameters, including the fuel prices, delay penalty rates, and the tightness of sailing time windows. Comparison against two simpler benchmark scheduling strategies (one with no vessel coordination and the other adopting a naïve coordination) manifests the sizeable benefit of optimal vessel scheduling.This paper presents the first investigation into the system-optimal scheduling strategy for vessels navigating a shared bottleneck, considering bunker costs, schedule delay penalties, and varying sailing speeds. The results highlight the significant potential of system-optimal scheduling and potential coordination strategies that enable approximation of the system-optimal solution. Additionally, our numerical experiments uncover the limitations of the outer-approximation method, while demonstrating that the discrete-time approach surpasses it in terms of both solution quality and computational efficiency.

The interplay between multiple control mechanisms in a host–parasitoid system: a discrete-time stage-structured modelling approach

We propose a discrete-time host–parasitoid model with stage structure in both species. For this model, we establish conditions for the existence and global stability of the extinction and parasitoid-free equilibria as well as conditions for the existence and local stability of an interior equilibrium and system persistence. We study the model numerically to examine how pesticide spraying may interact with natural enemies (parasitoids) to control the pest (host) species. We then extend the model to an impulsive difference system that incorporates both periodic pesticide spraying and augmentation of the natural enemies to suppress the pest population. For this system, we determine when the pest-eradication periodic solution is globally attracting. We also examine how varying the control measures (pesticide concentration, natural enemy augmentation and the frequency of applications) may lead to different pest outbreak or persistence outcomes when eradication does not occur.