Discrete-time Equation Research Articles

Robust adaptive predictive digital controller with integral action for set-point regulation of saturated uncertain systems

A robust adaptive digital controller is systematically investigated on set-point regulation in uncertain systems under input saturation. So, an optimization strategy will be derived to solve the reference regulation robustly in the discrete-time systems. To this purpose, a discrete-time integral controller is considered through a fixed state-feedback structure. Then, its coefficients would be numerically quantified at each sample time by means of a minimization. The formulation and merit of the addressed predictive control method are theoretically proved in the paper. Then the control technique is simulated in a discrete-time equation. Additionally, some numerical results are executed to declare the remarkability and vigor of the recommended algorithm against uncertainties, actuator saturation, and disturbance in contrast to the existing control ways.

Modeling the population dynamics and movement of Zebra mussels

Zebra mussels have caused significant damage in many lakes and rivers. By using a hybrid population model with discrete-time equations and ordinary differential equations, we represent the zebra mussel's life cycle, growth, and population movement. The dynamics of the larvae (unsettled and settled larvae) are represented during the summer months in a system of two ordinary differential equations, while the juvenile, small adult, and large adult stages are represented by a discrete model with yearly time steps. The goal is to investigate the effects of zebra mussel movement between three different spatial locations and possible control measures. Zebra mussel data and temperature values from three locations in the Hudson River over several years were used to estimate model parameters. We investigate the stability and net reproductive number for our model. We illustrate numerically the hybrid population model for several scenarios of movement and cleaning boats to remove mussels.

Data-driven modelling of brain activity using neural networks, diffusion maps, and the Koopman operator.

We propose a machine-learning approach to construct reduced-order models (ROMs) to predict the long-term out-of-sample dynamics of brain activity (and in general, high-dimensional time series), focusing mainly on task-dependent high-dimensional fMRI time series. Our approach is a three stage one. First, we exploit manifold learning and, in particular, diffusion maps (DMs) to discover a set of variables that parametrize the latent space on which the emergent high-dimensional fMRI time series evolve. Then, we construct ROMs on the embedded manifold via two techniques: Feedforward Neural Networks (FNNs) and the Koopman operator. Finally, for predicting the out-of-sample long-term dynamics of brain activity in the ambient fMRI space, we solve the pre-image problem, i.e., the construction of a map from the low-dimensional manifold to the original high-dimensional (ambient) space by coupling DMs with Geometric Harmonics (GH) when using FNNs and the Koopman modes per se. For our illustrations, we have assessed the performance of the two proposed schemes using two benchmark fMRI time series: (i) a simplistic five-dimensional model of stochastic discrete-time equations used just for a "transparent" illustration of the approach, thus knowing a priori what one expects to get, and (ii) a real fMRI dataset with recordings during a visuomotor task. We show that the proposed Koopman operator approach provides, for any practical purposes, equivalent results to the FNN-GH approach, thus bypassing the need to train a non-linear map and to use GH to extrapolate predictions in the ambient space; one can use instead the low-frequency truncation of the DMs function space of L2-integrable functions to predict the entire list of coordinate functions in the ambient space and to solve the pre-image problem.

Parameter Estimation and Predictive Speed Control of Chopper-Fed Brushed DC Motors

This paper presents an effective speed control method for brushed DC motors fed by a DC chopper using the concept of Finite Control Set-Model Predictive Control (FCS-MPC). As this control algorithm requires the parameters of the controlled object, the estimation of motor parameters is first performed by using two types of data. The first data includes the output speed response corresponding to the step input voltage to obtain the transfer function in the no-load regime. The second data consists of the motor speed and armature current when a load torque is applied to the motor shaft. The discrete-time equation of the motor armature circuit is used to obtain the future values of the armature circuit current and the motor speed. A cost function is defined based on the difference between the reference and predicted motor speed. The optimal switching states of the DC chopper are selected corresponding to the maximum value of the cost function. The performance of the proposed speed control algorithm is validated on an experimental system. The simulation and experimental results obtained show that the MPC controller can outperform the conventional proportional-integral (PI) controller.

Numerically efficient and robust interior-point algorithm for finite strain rate-independent crystal plasticity

The challenges of classic rate-independent crystal plasticity models regarding the determination of active slip systems and computation of solutions due to linear dependent slip systems are still a field of active research. In that regard, interior-point methods are promising approaches because a logarithmic barrier term naturally avoids the combinatorial active set search, and these methods are applicable even when the gradient of inequality constraints is linearly dependent. In this paper, a new interior-point algorithm suitable for rate-independent crystal plasticity at finite strains is developed. For that purpose, the postulate of maximum dissipation is recast as a constrained optimization problem whose optimality conditions are derived by utilizing interior-point approaches. This optimization problem is discretized in time by means of an exponential time integration of the flow rule, and the time-discrete equations are solved by a Newton scheme. The focus of this paper is the efficiency and robustness of the implementation of this approach. In particular, the effect of different line search procedures, favorable residual forms, preconditioning methods, barrier parameter updates and the consistent algorithmic tangent modulus are discussed and further elaborated. In addition, aspects regarding uniqueness and well-posedness of the algorithm are analyzed in a geometrically linearized setting. Efficiency and robustness of the developed algorithm are shown by numerical examples of an FCC crystal including finite element simulations without stabilizing hardening or rate-dependent effects. Furthermore, the results are compared to the well-established augmented Lagrangian formulation.

Development of an explicit time integration algorithm based on optimal integration parameter updating

AbstractIntegration algorithm is an important mean to solve the discrete time equations of structural dynamics in earthquake engineering. In this paper, a new model‐based explicit integration algorithm, featured by optimization‐based integration parameter updating, is developed for a more accurate and efficient solution of structural dynamic problems. The new integration algorithm is designed to yield minimum error and controllable numerical dispersion under the premise of stability by defining objective functions and constraints. Numerical properties of the new integration algorithm are investigated in details using the amplification matrix method for a linear elastic single degree‐of‐freedom system. Meanwhile, representative numerical examples and complex structure examples are analyzed and compared with other representative algorithms. As a result, the proposed algorithm yields comparable numerical characteristics with other methods, but exhibits pronounced superiority in controllable algorithmic dissipation, higher precision, and better efficiency. Therefore, the new integration algorithm possesses extensive application prospect in solving complicated linear and nonlinear structural dynamic problems.

Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy

We explore the simultaneous exact controllability of mean and variance of an insurance policy by utilizing the benefit St and premium Pt as control inputs to manage the policy value tV and the variance 2σt of future losses. The goal is to determine whether there exist control inputs that can steer the mean and variance from a prescribed initial state at t=0 to a prescribed final state at t=T, where the initial–terminal pair of states (0V,TV) and (2σ0,2σT) represent the mean and variance of future losses at times t=0 and t=T, respectively. The mean tV and variance 2σt are governed by Thiele’s and Hattendorff’s differential equations in continuous time and recursive equations in discrete time. Our study focuses on solving the problem of exact controllability in both continuous and discrete time. We show that our result can be used to devise control inputs St,Pt in the interval [0,T] so that the mean and variance partially track a specified curve tV=a(t) and 2σt=b(t), respectively, i.e., at a fine sampling of points in the time interval [0,T].

Efficient numerical strategies for an implicit volume fraction transfer scheme for single crystal plasticity including twinning and secondary plasticity on the example of magnesium

AbstractIn this work an efficient algorithm for a fully implicit single crystal plasticity routine including twinning and secondary plasticity is presented and implemented for the example of magnesium. The material model uses the volume fraction transfer scheme for the evolution of twinning, that is, plastic slip in newly formed twins (“secondary plasticity”) is resolved. This is considered particularly important, if the volume fractions of some twins reach the order of magnitude of the parent phase. However the resolution of secondary plasticity also implies a large number of unknowns, which is significantly reduced by a newly proposed algorithm. For magnesium a hardening model based on basal, prismatic, pyramidal a and pyramidal c+a slip modes as well as a tension and a compression twinning mode and yields a total of 18 slip systems and 12 twinning systems. As a special feature of the proposed algorithm, the total amount of 246 unknowns, due to simultaneous slip in the parent phase and the twinned phases, is reduced to only 31 unknowns. Additionally, thermodynamic consistency is ensured by including the second law with the Clausius–Duhem equation, which considers the change of free energy upon twinning as an additional driving force for twinning. Further, the setup of the time discrete nonlinear equation system using midpoint rule, as well as the analytical solution of the algorithmic tangent are given in detail. Finally, the implemented model is tested in finite element simulations and compared to single‐ and polycrystal compression and tension experiments.

Controllability of a simplified time-discrete stabilized Kuramoto-Sivashinsky system

<p style='text-indent:20px;'>In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- and second-order parabolic equations. This system can be regarded as a simplification of the well-known stabilized Kumamoto-Sivashinsky equation. Unlike the continuous case, we can prove only a relaxed observability inequality which yields a <inline-formula><tex-math id="M1">\begin{document}$ \phi(\triangle t) $\end{document}</tex-math></inline-formula>-controllability result. This result tells that we cannot reach exactly zero but rather a small target whose size goes to 0 as the discretization parameter <inline-formula><tex-math id="M2">\begin{document}$ \triangle t $\end{document}</tex-math></inline-formula> goes to 0. The proof relies on a known Carleman estimate for second-order time-discrete parabolic operators and a new Carleman estimate for the time-discrete fourth-order equation.</p>

A Mathematical Model of Susceptible Diabetes Complication (SDC) Model in Discrete Time Fuzzy and Crisp Environment

In this study, we examined the mathematical model of the discrete-time equation system with susceptible diabetes complication (SDC), which is known to be caused by environmental and genetic factors in a fuzzy environment. From the diabetes complication (DC) model, the susceptible diabetes complication (SDC) model is being developed. It was obtained using definitions of how the behavior of this model changes in a fuzzy environment. A nonlinear differential equation system transforms the sensitive diabetes complication (SDC) model into a discrete time equation system. Stability analysis of the model with jury criterion was examined. In addition, numerical solutions and graphics of the analysis of the discrete model in fuzzy environment are obtained by using the MATLAB package program.

Dynamical analysis of discrete time equations with a generalized order

Some natural phenomena that arise in nonlinear sciences can be sometimes discussed using discrete-time systems. In this work, we investigate the periodicity, boundedness, oscillation, stability, and some exact solutions of nonlinear difference equations with generalized order. Exact solutions are extracted using the traditional iteration and the modulus operator. The stability of the equilibrium points is examined using some well-known theorems. Some numerical examples are also introduced to confirm the validity of the theoretical work. We use MATLAB to implement the numerical part. The used technique can be easily utilized to deal with other rational recursive problems.

A Class of Boundary Conditions for Time-Discrete Green--Naghdi Equations with Bathymetry

This work is devoted to the structure of the time-discrete Green-Naghdi equations including bathymetry. We use the projection structure of the equations to characterize homogeneous and inhomogeneous boundary conditions for which the semi-discrete equations are well-posed. This structure allows us to propose efficient and robust numerical treatment of the boundary conditions that ensures entropy stability of the scheme by construction. Numerical evidence is provided to illustrate that our approach is suitable for situations of practical interest that are not covered by existing theory.

A novel time-frequency model, analysis and parameter estimation approach: Towards multiple close and crossed chirp modes

In this paper, a novel model for online time-frequency representation and analysis with multiple close and crossed chirp modes is proposed. It is shown that, when a signal is described by a discrete time equation, its highest time domain resolution is given by its sample interval. According to the uncertainty principle, the frequency resolution of the signal on the interval would meanwhile be nothing. However, when the amplitude and the phase of each chirp mode are taken as the state variables whose evolutions are described by a polynomial prediction model, a difference is made here. Namely, the frequency resolution can be governed by the model states and their evolutions. Based on the proposed model, it is then exemplified that an unscented Kalman filter can be used to obtain the time-frequency representation of the signal. For situations where the variance of Gaussian measurement noise is unknown or variable, a new alternative nonlinear Bayesian filtering with a simple closed form and low computational cost for doing the time-frequency analysis of the signal is presented. Simulation results verify the effectiveness of our model and analysis approach, especially in analyzing multiple close and crossed chirp modes.

Nonuniform input-output criteria for exponential expansiveness of discrete dynamical systems and applications

The aim of this paper is to present a new and complete study based on input-output methods for exponential expansiveness of discrete variational systems, treating both the nonuniform behavior and the uniform one. We continue the line of studies initiated in Dragičević et al. (2020) [19] and Dragičević et al. (2022) [20], providing a new approach in exploring the expansiveness properties. We associate to a discrete variational system a control type system that consists of a family of input-output discrete-time equations defined between two sequence spaces whose norms are translations invariant and we introduce new concepts of exact admissibility that rely on nonuniform local conditions. First, by imposing some minimal hypotheses on the input space or respectively on the output space, we establish connections between the exact admissibilities and the nonuniform exponential expansiveness. Moreover, we give for the first time a complete description of the structure of exactly admissible pairs that can be considered to explore nonuniform expansive behaviors in the discrete variational case. In addition, all the hypotheses on the classes from which we choose the input or the output spaces are motivated by illustrative examples. Next, we present characterizations for uniform exponential expansiveness via certain nonuniform exact admissibilities and we generalize the recent criteria for expansiveness of dynamical systems from Dragičević et al. (2020) [19]. Furthermore, we show via relevant examples that the results are the most general and the working hypotheses cannot be weakened or removed. As application, we give general criteria for uniform exponential expansiveness of skew-product semiflows in terms of nonuniform admissibility properties with sequence spaces invariant under translations.

Suboptimal finite-time control of singular large scale discrete-time equations with delayed interconnections

The problem of suboptimal finite-time control for a class of large scale singular discrete-time equations with delayed interconnections is studied in this paper. Making use of singular value theory and Lyapunov function method, new delay-dependent conditions are established for solving the finite-time H∞ control and guaranteed cost control problems of such equations. The conditions are presented in the form of tractable linear matrix inequalities (LMIs). Illustrated examples with simulations show the feasibility and effectiveness of the proposed results.

Distributed consensus problem with caching on federated learning framework

Federated learning framework facilitates more applications of deep learning algorithms on the existing network architectures, where the model parameters are aggregated in a centralized manner. However, some of federated learning participants are often inaccessible, such as in a power shortage or dormant state. That will force us to explore the possibility that the parameter aggregation is operated in an ad hoc manner, which is based on consensus computing. On the contrary, since caching mechanism is indispensable to any federated learning mobile node, it is necessary to investigate the connection between it and consensus computing. In this article, we first propose a novel federated learning paradigm, which supports an ad hoc operation mode for federated learning participants. Second, a discrete-time dynamic equation and its control law are formulated to satisfy the demands from federated learning framework, with a quantized caching scheme designed to mask the uncertainties from both asynchronous updates and measurement noises. Then, the consensus conditions and the convergence of the consensus protocol are deduced analytically, and a quantized caching strategy to optimize the convergence speed is provided. Our major contribution is to give the basic theories of distributed consensus problem for federated learning framework, and the theoretical results are validated by numerical simulations.

A Convex Dynamic Approach for Globally Optimal Profit in Supply Chains

Supply chain finance aims to coordinate multiple stakeholders to maximize the flow of cash and internal and external funding along the supply chain, as shown in prior research. From a regulatory standpoint, the goal of this paper is to maximize the profitability of an entire supply chain. As a result, a constrained finite time Linear Quadratic Regulation (LQR) approach is provided for determining an entity’s optimal profit state in a supply chain. The framework is represented by discrete-time linear dynamical equations for each entity in the supply chain network, taking state and input variables into account. The problem is formulated in terms of a convex quadratic programming optimization for which several numerically efficient algorithms are readily available. In order to validate the approach, it was tested on two topologies. The first topology is a fully connected supply chain with six nodes; the second is a simple topology based on the Iranian pharmaceutical supply chain. The results indicate that the proposed approach successfully planned production and financing decisions within the simulated supply chain and obtained globally optimal profit for all supply chain stakeholders.

Quadrotor Attitude and Altitude Tracking Control Using Finite Discrete-Time Linear Quadratic Tracking Controller

In this paper, an optimal finite discrete-time linear quadratic tracking (LQT) control method is proposed to control the altitude and attitude of a quadrotor. First, the dynamic model of the quadrotor is derived using Newton-Euler equations. Next, non-linear equations of the quadrotor are written in the state space form and linearized around an equilibrium point. Then, continuous-time linear state-space equations are converted into discrete-time equations considering a specific sampling time. Moreover, the controller design process is completed by determining the performance index and the weighting matrices, and the optimal control input is acquired for the closed-loop system. In the end, the simulation results are shown to demonstrate the robustness of the controller against parameter uncertainties and show its performance in attenuating the external disturbance effect.

Conservative finite volume method on deforming geometries: The case of protein aggregation in dividing yeast cells

Motivated by experimental observations of asymmetric protein aggregates distributions in dividing yeast cells, we present a conservative finite volume approach for reaction-diffusion systems defined over deforming geometries. The key idea of our approach is to use spatio-temporal control volumes instead of integrating the time-discretized equations in space, as it is common practice. Both our theoretical and computational results demonstrate the convergence of our method and highlight how traditional approaches can lead to inaccurate solutions. We employ this novel approach to investigate the partitioning of protein aggregates in dividing yeast cells, leveraging the flexibility of the level set method to construct realistic biological geometries. Using a simple reaction-diffusion model, we find that spatial heterogeneity in yeast cells during division can alone create asymmetries in the concentration of protein aggregates. Moreover, we find that obstructing intracellular entities, such as nuclei or insoluble protein compartments, amplify these asymmetries, suggesting that they may play an essential role in regulating molecular partitioning. Beyond these findings, our results illustrate the flexibility of our approach and its potential to design realistic predictive tools to explore intracellular bio-mechanisms.

A population model for Triatoma infestans in chicken coops

Despite the strenuous efforts devoted to its elimination, Chagas disease is still prevalent in large regions of Latin America. Its main vector, Triatoma infestans (kissing bug), feeds on people and domestic animals and finds refuge in the peripheral structures of rural and semi-rural houses. Here we propose a mathematical model based on discrete-time equations that describe the evolution of the populations of all the stages of T. infestans residing in a chicken coop, a typical bug reservoir in non-urban habitats. The chicken coop has also been used as a suitable location for observations of T. infestans populations under realistic, controlled conditions. The variables that govern the model are the time of the year, the temperature, and the blood intake by the triatomines. Using the available data, we design fitting functions, such as maximum blood intake, development time, mortality rate, egg fertility, etc., for each population group. Then, we validate our model using D. Gorla's experimental work on T. infestans populations. Finally, we analyze the minimum conditions of temperature and food availability necessary for the survival of T. infestans, showing that average daily temperatures below 12 °C or average yearly temperatures below 16 °C or above 32 °C lead to colony extinction. Feeding also regulates the colony size, and we find that if a few triatomines (5% of the colony) are well fed, and the rest occasionally get a meager meal, the colony size can be, on average, constant over the years. By quantifying what happens in structures crucial to the species survival, our model is an important step forward towards the formulation of a general population model for T. infestans in the Grand Chaco under current conditions.