Discrete-time Difference Equations Research Articles

A Novel Mathematical Characterization for Switched Linear Systems Based on Automata and Its Stabilizability Analysis

Mathematical characterization of switched systems based on automata is one of the most challenging problems in the context of hybrid dynamical systems. This paper investigates the mathematical representation and stabilization of a class of slowly and fastly switched linear systems with automata in the hybrid coupling framework. To address such an issue, by using the semi-tensor product of matrices, a so-called “automaton-dependent switched linear system” is established to generalize the traditional switched system, which is characterized mathematically in the new form of a hybrid dynamic system consisting of a continuous-time differential equation and a discrete-time difference equation. Moreover, a novel automaton-dependent constrained switching scheme is developed for the stabilization of the proposed model with unstable subsystems. Next, by designing slow mode-dependent average dwell time (S-MDADT), fast mode-dependent average dwell time (F-MDADT), slow cycle-dependent cycle dwell time (S-CDCDT) and fast cycle-dependent cycle dwell time (F-CDCDT) switching laws, a generalized stabilization lemma for automaton-dependent switched nonlinear systems with both stable and unstable modes inside and outside the cycle is derived in the term of digital transformations and multiple Lyapunov functions. Furthermore, based on the stabilizability criteria of nonlinear case, by setting digital transformations and linear matrix inequalities, a new stabilization theorem for automaton-dependent switched linear systems with unstable subsystems is obtained. It should be noted that the proposed scheme provides a more precise estimation of the upper and lower bounds for the dwell time of the system with unstable modes. Finally, a practical converter circuit example and a numerical example are given to show the validity of the designed techniques.

An adaptive product changeover policy for a capacitated two-product supply chain in a non-stationary demand environment

ABSTRACT The present paper investigates the effectiveness, in terms of fill rate, of a new adaptive production control policy for a two-product, two-echelon (i.e. factory and retailer) supply chain subject to a non-stationary customer demand. To model the capacity constraints more realistically, the factory node consists of an unbuffered production line subject to failures, and non-negligible changeover times. To cope with the disruptive events, we propose a new production control strategy named Adaptive Hedging Corridor Policy. A simulation model based on discrete-time difference equations is developed, and a two-step experimental campaign is conducted to compare the proposed control policy with two well-known alternatives proposed in the literature. The analysis of the results demonstrates the effectiveness of the new adaptive strategy in maximising the fill rate of supply chains subject to a non-stationary demand.

Analytical Solution for Transient Reactive Elements for DC-DC Converter Circuits

This paper develops an analytical method for modeling the inductor currents and capacitor voltages (ICCV) of a generic DC-DC converter system. The purpose of the designed methodology is to propose a new generalized modeling technique for DC-DC converter systems that accurately models the transient behavior of those systems. The modeled converter is assumed to operate over some number of circuit stages. Each circuit stage can be separately modeled as a linear time-invariant (LTI) system that is solved through the uni-lateral Laplace transform. Furthermore, the initial conditions (ICs) of these LTI systems are related through different algebraic expressions and discrete-time difference equations that originate from the continuity of the ICCV with respect to time. These discrete-time difference equations are then solved with the uni-lateral Z-transform to determine the ICs of the ICCV at each switching period. The generalized theoretical analysis is applied to the study of the transient behavior of the buck-boost converter across various different circuit parameters. This analysis justified with laboratory experimentation of the buck-boost converter, and the transient behavior of the buck-boost converter is compared for each experimental parameter set. The experimental results and the theoretical analysis provide very similar results across the different converter parameters.

Continuous-Time Difference Equations and Distributed Chaos Modelling

The article aims to attract the attention of researchers, experts and those interested in nonlinear dynamics and chaos theory to the not well known field of continuous-time difference equations, in the hopes of opening new doors into the study of chaotic system. Deterministic chaos and related notions are used in an increasing number of scientific works. There are a lot of problems associated with the mathematical aspects of the fine structure of chaos. Just as discrete-time difference equations have proven to be excellent models of temporal (discrete) chaos, so continuous-time difference equations provide new elegant mechanisms for onset and inside reconstructions of spatio-temporal (distributed) chaos. Distributed chaos is usually described by boundary value problems for partial differential equations. A number of these boundary value problems can be reduced to continuous-time difference equations, which enable one to build new chaos scenarios arising from the properties of the equations. Whereas the emergence of deterministic chaos is usually attributed to the complex structure of attractors, these new scenarios are based on a highly complex structure of spatially extended “points” of the attractor. Examples of reducible boundary value problems are set forth in the article, but the main focus is on a very elementary overview of the principal features of solutions of the simplest nonlinear continuous-time difference equations: loss of continuity, asymptotic periodicity, gradient catastrophe, fractal geometry, space-filling property, going beyond the horizon of predictability, self-stochasticity (deterministic solutions are asymptotically described by random processes), formation of hierarchical structures (down to arbitrarily small scales). Here we have a wonderful example of how very complex phenomena can be described with very simple equations. The use of continuous-time difference equations in the study of reducible and close-to-reducible boundary value problems migh help to advance in understanding possible mathematical mechanisms for distributed chaos.

STELAR: Spatio-temporal Tensor Factorization with Latent Epidemiological Regularization

Accurate prediction of the transmission of epidemic diseases such as COVID-19 is crucial for implementing effective mitigation measures. In this work, we develop a tensor method to predict the evolution of epidemic trends for many regions simultaneously. We construct a 3-way spatio-temporal tensor (location, attribute, time) of case counts and propose a nonnegative tensor factorization with latent epidemiological model regularization named STELAR. Unlike standard tensor factorization methods which cannot predict slabs ahead, STELAR enables long-term prediction by incorporating latent temporal regularization through a system of discrete-time difference equations of a widely adopted epidemiological model. We use latent instead of location/attribute-level epidemiological dynamics to capture common epidemic profile sub-types and improve collaborative learning and prediction. We conduct experiments using both county- and state-level COVID-19 data and show that our model can identify interesting latent patterns of the epidemic. Finally, we evaluate the predictive ability of our method and show superior performance compared to the baselines, achieving up to 21% lower root mean square error and 25% lower mean absolute error for county-level prediction.

Robust Interval Observer Design for Fractional-Order Models with Applications to State Estimation of Batteries

Abstract Interval observers have been investigated by many researchers during the last decade, especially for those classes of systems that can be described by finite-dimensional continuous-time ordinary differential equations, discrete-time difference equations, and sets of partial differential equations in which both, system parameters and external disturbances, may be subject to bounded uncertainty. In contrast to this, only preliminary investigations were performed for fractional-order models. Due to the fact that many electro-chemical processes such as the charging and discharging dynamics of batteries can be described in good accuracy by using fractional-order models, this paper focuses on the design and numerical validation of interval observers for such systems. Here, we present a cooperativity-enforcing observer structure leading directly to decoupled lower and upper bounding systems for the sets of reachable states. This is visualized by a battery model with interval uncertainty in the output equation.

Interval methods and contractor-based branch-and-bound procedures for verified parameter identification of quasi-linear cooperative system models

A large number of dynamic system models in engineering as well as computational physics, chemistry or biology are described after first-principle modeling approaches by sets of either discrete-time difference equations or by sets of ordinary differential equations. In both cases, the structure of those system models results from fundamental physical system properties such as conservation laws of mass, momentum and energy, Newton’s or Kirchhoff’s laws as well as a structural subdivision into subsystems or individual compartments. Regardless of the area of application, all of these system models involve parameters which are not perfectly known due to uncertainty resulting from tolerances in the construction of the respective real-life systems or which are described by an inherent uncertainty coming from model simplifications or even from a lack of knowledge about detailed microscopic dynamic effects. This is especially true for applications in engineering, where a trade-off between the model accuracy and the resulting computational complexity has to be made in order to obtain mathematical representations which can be evaluated numerically in real time. This real-time applicability is inevitable if tasks such as feedback control or model-based state estimation on the basis of incomplete state measurements are of interest. Hence, all aforementioned tasks depend on the reliable estimation of those parameters of the structurally predefined sets of state equations which are feasible with respect to the knowledge about selected measurable state variables under consideration of the associated measurement tolerances. This paper presents computational techniques for a model-based identification of system parameters under the a-priori knowledge of stability of the underlying open-loop dynamics and the structural constraint of cooperativity.

Quantitative Phenomics of Growth and Size in Animals

This study introduces a model of the growth phenotype dynamic in pigs by applying analytical methods. The model describes a transformation of the growth phenotype from 30kg live weight up to the species maximum weight.
 
 This paper focuses on a description of the determinants, which channel growth dynamic through animals' ontogeny. Theoretical notions about the growth are explicitly included in the concept. In the model, functional relations between relevant traits obtained in the experiments and field observations are analysed. Two focal variables, a feed conversion coefficient, and an invariant of growth are explicitly integrated in the model.
 
 The novelty of the research is a proposition that deterministic, analytical model is a conceivable approach to studying a phenotype of a quantitative trait. According to this proposition, processes that contribute to the growth and development continue in later life till a species maximum weight is attained. The concept is formulated as a deterministic, hybrid model. In the model, both standard continuum methods and discrete-time difference equations have been used.
 
 Applied to experimental data, the model has produced a new insight into the problem. In domestic pigs, the following three sets of ontogenetic growth phenotypes have been identified: one set with three species maximum weight phenotypes, one set with three rapid growth phenotypes, and one set with individual maximum weight phenotypes. In the study, not only the sets of growth phenotype are identified but also a possible interpretation of a conversion mode of the sets in ontogeny as well as a reading of growth trajectory dynamic are discussed.

A discrete-time dynamical system with four types of codimension-one bifurcations

Usually, several discrete-time difference equations are shown in introductory courses on dynamical systems theory, in order to illustrate the occurrence of the most common bifurcations, which are saddle-node, transcritical, pitchfork, and flip. For instance, transcritical and flip bifurcations are found in the well-known logistic map. Here, a first-order difference equation undergoing these four types of bifurcations is presented. The bifurcation diagram is analytically derived and the rationale behind the construction of this equation is explained. The main goal of this didactic work is to give tips on how to write difference equations exhibiting various types of bifurcations, which can be associated with real-world scenarios.

An Analytical Model of Animal Growth

The purpose of this study was to explain some aspects of ontogenetic growth in pigs by analysing the relationships between variables that are significant to the development of animals. The novelty of the study is a new modelling approach to the growth problem, with the attention that has been paid to both a new set of variables, and an analytical discrete-continuous hybrid model, innovative for the field. This is a first species-specific hybrid model for animal growth formulated in discrete-time difference equation technique. The efficiency of the model is not only due to the modelling technique but also due to a set of relevant variables, especially a feed conversion coefficient, which provides a link between macro and micro physiological scales. The model is based on functional relationships between relevant variables acquired from experimental data analyses, and field observations. The concept explains some aspects of growth in pigs from 30 kg to 600 kg, which is considered the maximum individual weight for a boar, and further growth up to a species maximum weight. The model predicts that boar can reach their maximum individual weight of 600 kg when 6,40 years old and are required to consume 62,51 kg of feed to put on the last kilogram. The phenotypes that can attain their maximum individual weight go through bifurcation of the growth trajectory, a transform in the growth mode. After bifurcation, the smallest number of the phenotypes go on the growth trajectory that leads to a set of species maximum weights of over 1205 kg, and the greatest number of phenotypes continue to live until aged 24,90 years, provided their maximum weights do not change. The study includes growth rate equations, identifies species maximum weight phenotypes, and produces insight into pig longevity. The results suggest that species maximum weight growth trajectories are phenotype-dependant. A modified discrete-time difference equation technique combined with standard continuum methods is an appropriate formalism to model ontogenetic growth in animals.

Semi-global persistence and stability for a class of forced discrete-time population models

We consider persistence and stability properties for a class of forced discrete-time difference equations with three defining properties: the solution is constrained to evolve in the non-negative orthant, the forcing acts multiplicatively, and the dynamics are described by so-called Lur’e systems, containing both linear and non-linear terms. Many discrete-time biological models encountered in the literature may be expressed in the form of a Lur’e system and, in this context, the multiplicative forcing may correspond to harvesting, culling or time-varying (such as seasonal) vital rates or environmental conditions. Drawing upon techniques from systems and control theory, and assuming that the forcing is bounded, we provide conditions under which persistence occurs and, further, that a unique non-zero equilibrium is stable with respect to the forcing in a sense which is reminiscent of input-to-state stability, a concept well-known in nonlinear control theory. The theoretical results are illustrated with several examples. In particular, we discuss how our results relate to previous literature on stabilization of chaotic systems by so-called proportional feedback control.

Implementing CRM models for improved oil recovery in large oil fields

This paper describes a model-based decision support system (DSS) for reservoir engineers who may wish to use simpler analytical methods to optimize waterfloods in large waterflood operations, as is common in many oil and gas companies. The engineers select a set of existing injector and producer wells, and a set of past time periods over which the model will be fitted. The model provides outputs which support waterflooding decisions, using only oil, water and gas production and bottom hole pressure at producing wells, and water injection at each injection well in each time period. The model relating injection and production is a set of discrete-time difference equations, from the family of capacitance-resistance models. We focus on the model fitting process, which uses the least squares approach, and leads to a linearly constrained nonlinear optimization problem with non-quadratic objective. The DSS is coded in the GAMS modeling language. A numerical example is provided, and some experience with the system is described. An Excel file with the data used is available as Supplementary Online material accompanying the article on the Springer website.

Designing Aviation Security Passenger Screening Systems Using Nonlinear Control

The design of aviation security systems has become increasingly important over the past decade. Passenger screening is a critical aspect of aviation security systems that employs trained personnel to operate screening devices that are designed to detect various threat objects. Several security classes can be defined with each class containing a subset of these screening devices, and in which each passenger is assigned to exactly one security class based on their perceived risk level. This paper introduces a real-time sequential binary passenger assignment model as a discrete-time difference equation. The stochastic value of the binary passenger assignment is replaced by a deterministic value capturing a fractional class assignment of each passenger. Through a probabilistic analysis, a closed-loop policy is presented to achieve desired security class occupancies for a finite set of passengers anticipated to undergo screening during a given time period, while maximizing the overall system security. The same closed-loop policy is also shown by applying feedback linearization to the fractional passenger assignment model. Closed-loop stability and convergence properties are discussed for a time-variant state feedback design. Simulation results are reported to illustrate sensitivity to parameter variation, and to demonstrate the effectiveness of control theory as an approximation to an otherwise computationally intractable dynamic programming problem. The key contribution is that passenger assignments to security classes can be performed in real time using the methodology introduced.

INVENTORY CONTROL BY MODEL PREDICTIVE CONTROL METHODS

This paper describes a model predictive control strategy to tackle the inventory control problem in Supply Chains. The problem is formulated as a receding-horizon optimization problem and the market demand is considered an external unknown disturbance. The behaviour of the Supply Chain is modelled by discrete time difference equations, both the control variables and the output variables are constrained to satisfy suitable bounds. Quantitative performance indices are introduced and the effects of the choice of the control parameters are discussed.

A First Generation Stochastic Bioeconomic Analysis of Algal Bloom Control in a Coastal Lagoon (Sacca di Goro, Po River Delta)

Abstract. In this work we have developed a very simple stochastic mathematical model of Ulva spp. growth to quantitatively evaluate the economic costs of algal harvesting and the related benefits in terms of avoided‘economic loss’of clam production due to an effective prevention of algal blooms and the consequent anoxic crises. Algal growth was simulated by means of a discrete time difference equation of Ulva biomass where the finite growth rate depends only upon water temperature. In order to explicitly include environmental variability, a seasonal autoregressive model calibrated on available data was used to simulate water temperature. Different harvesting scenarios were analysed in terms of the number of harvesting vessels employed and the threshold biomass of Ulva at which vessels start to operate. Costs of algal harvesting and disposal, as well as monetary damages resulting from the collapse of clam production as a consequence of algal blooms, were assessed by interviewing the managers of the Clam Fishermen's Union of Goro. A Monte Carlo approach was used to estimate the mean and total statistical distribution of costs and benefits of different harvesting strategies. Our analysis shows that the most cost‐effective management policy is attained with 4–6 vessels operating at low algal density able to harvest as much Ulva as possible with intensive and short interventions at the beginning of the seasonal growth.

Changes in populations of the eyespot fungiTapesia yallundaeandT. acuformisunder different fungicide regimes in successive crops of winter wheat, 1984–2000

Effects of regular treatments with the fungicides carbendazim and prochloraz applied to whole plots divided into subplots with different initial population mixtures of carbendazim‐sensitive or carbendazim‐resistantTapesia yallundaeorT. acuformiswere studied in successive crops of winter wheat from 1984/85 to 1999/2000. In unsprayed and carbendazim‐sprayed whole plots, a stable coexistence of about 50% each ofT. yallundaeandT. acuformisdeveloped within five seasons, but in whole plots sprayed with prochloraz or prochloraz plus carbendazim, the proportion ofT. acuformisincreased to > 80%. A discrete time difference equation model was derived from knowledge of the biology of eyespot and competition theory to describe the population changes. The model was fitted to the data from treatments where coexistence occurred [subplots in unsprayed (1985–92) and carbendazim‐sprayed (1985–89) whole plots], using nonlinear least squares regression. The optimized value of the resource overlap coefficient was small, suggesting niche differences between the two species. Populations were nearly 100% carbendazim‐resistant in carbendazim‐sprayed whole plots by July 1985 (one season) and in whole plots sprayed with prochloraz plus carbendazim by July 1986 (two seasons). In prochloraz‐sprayed whole plots, the proportion of carbendazim‐resistant isolates decreased more rapidly than in unsprayed whole plots in the 1980s, but by July 1992 a shift in populations in unsprayed and prochloraz‐sprayed whole plots towards predominantly carbendazim‐resistant strains had occurred.

Modelling the dynamics of supply chains

A summary of the important issues arising in supply chain research is presented. A review of the various mathematical methods used to model and analyse supply chains is then undertaken. They are categorized as: continuous time differential equation models, discrete time difference equation models, discrete event simulation models and operational research techniques. Each method is then appraised from a system dynamics perspective. To conclude, a judgement is made on whether the merits of operational research techniques justify their predominance in industry over the other methods presented here.

Comparative study of two algorithms for the calculation of the extreme eigenvalues of large matrices

Two algorithms for the calculation of extreme eigenvalues of large matrices recently presented are compared. The first one is a modification of the well-known power method with Chebyshev iterations to accelerate convergence and an auxiliary procedure capable of automatically setting all the external parameters, which was developed by us during the year 1991. The second algorithm is an iterative procedure obtained from the discrete-time difference equations for a system of coupled harmonic oscillators. The analysis presented here allows to demonstrate that this second algorithm is essentially identical to ours.

A general framework for concurrent simulation on neural network models

The analysis of complex neural network models via analytical techniques is often quite difficult due to the large numbers of components involved and the nonlinearities associated with these components. The authors present a framework for simulating neural networks as discrete event nonlinear dynamical systems. This includes neural network models whose components are described by continuous-time differential equations or by discrete-time difference equations. Specifically, the authors consider the design and construction of a concurrent object-oriented discrete event simulation environment for neural networks. The use of an object-oriented language provides the data abstraction facilities necessary to support modification and extension of the simulation system at a high level of abstraction. Furthermore, the ability to specify concurrent processing supports execution on parallel architectures. The use of this system is demonstrated by simulating a specific neural network model on a general-purpose parallel computer.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Qualitative analysis and control of a DC-to-DC boost converter operating in discontinuous mode

The boost cell operating in discontinuous conduction mode based on an approximate discrete-time difference equation is investigated. A qualitative discussion of the steady-state and open-loop dynamical behavior is presented. A linearized small-signal equation that leads to a linear feedforward control law for regulating this type of converter is derived. The conventional linear scheme provides satisfactory control in the neighborhood of the operating point, but ceases to meet the requirement as soon as the small-signal assumption is violated. A nonlinear feedforward control law whose validity extends over a wider range of fluctuation of the variables about the operating point is proposed, along with some simulation results that confirm the superiority of the proposed nonlinear control over its linear counterpart.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>