Discrete-time Equation Research Articles

Parametric Reduced-Order Modeling of the Unsteady Vortex-Lattice Method

A method for frequency-limited balancing of the unsteady vortex-lattice equations is introduced that results in compact models suitable for computational-intensive applications in load analysis, aeroelastic optimization, and control synthesis. The balancing algorithm relies on a frequency-domain solution of the vortex-lattice equations that effectively eliminates the cost associated to the wake states. It is obtained from a transform of the underlying discrete-time equations, and it requires no additional geometrical or kinematic assumptions for the lifting surfaces. Parametric reduced-order modeling is demonstrated through interpolation over 1) projection matrices, 2) state-space realizations, and 3) transfer functions, which trade accuracy, robustness, and cost. Methods are finally exemplified in the dynamic stability of a T-tail configuration with varying incidence. Numerical studies show that a very small number of balanced realizations is sufficient to accurately capture the unconventional aeroelastic response of this system, which includes in-plane kinematics and steady loads, over a wide range of operation conditions.

Carleman estimates for time-discrete parabolic equations and applications to controllability

In this paper, we prove a Carleman estimate for a time-discrete parabolic operator under some condition relating the large Carleman parameter to the time step of the discretization scheme. This estimate is then used to obtain relaxed observability estimates that yield, by duality, some controllability results for linear and semi-linear time-discrete parabolic equations. We also discuss the application of this Carleman estimate to the controllability of time-discrete coupled parabolic systems.

Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker–Planck equations

In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker-Planck equations. In particular, for these dis-cretizations of velocity and space, we prove the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Our method uses new types of discrete Poincare inequalities for a two-direction discretization of the derivative in velocity. For the inhomogeneous problem, we adapt hypocoercive methods to the discrete cases.

A Model of Growth Trajectory Bifurcation in Animals Ontogeny

The aim of this study was to formulate and analyse a model of ontogenetic growth cessation in pigs. The cessation of growth when an animal reaches its species-specific size in ontogeny is still a problem. Systemic factors that contribute to this process are unknown. The focus of the research is an analysis of the growth dynamic that explains some aspects of the problem.
 
 The method applied to meet the purpose of the study was mathematical modelling. To enhance the understanding of the growth trajectories in ontogeny an analytical model of growth in pigs was built. The model was formulated as a hybrid dynamic system with discrete-time and continuum equations. The novelty of the study is a concept of ontogenetic growth in the pig. Both a new modelling technique, and new variables are introduced. A central theme of the study is an analysis of the growth trajectory bifurcation, and a description of the two emerged growth trajectories. A reading of a normal form of bifurcation applied to the growth trajectory bifurcation has been offered.
 
 The results suggest that ontogenetic growth in pigs is not continuous. The growth trajectory has bifurcation at the point the animals attain their individual maximum weight. At this point, two new growth trajectories emerge. On one trajectory, animals continue to grow till a species maximum weight is reached. On other trajectory, animals continue to live till obtainable life span is attained. The emerged trajectories are genetic channels that open the way to grow for the certain phenotypes. Ontogenetic growth stops when the feed conversion coefficient grows into infinity.

Research of Mapping Element Space Method for Uncertain Lurie System Stability Diagnosis

A new diagnosis method of Lurie system stability by using Chaotic time series data was proposed in order to solve stability analysis of flight control system with fault and uncertain. Firstly, the reasons for the instability of the nonlinear system of Ruri leaf was analyzed by using the small gain theory and linear matrix inequalities in the robust control theory. And the stability conditions of Rui leaf system under the condition of uncertainty and failure were proved theoretically. In order to quantify the stability of nonlinear systems in complex Ruri leaves, based on the theory of phase space reconstruction, the nonlinear Ruri system with continuous discrete characteristics was converted into an approximate time discrete equation, mapping it to a low dimensional primitive space, by introducing Q Gauss function into Kernel function, the generalization ability of neural networks are enhanced, realizing the stability analysis method based on the characteristic of primitive migration, which is suitable for various fault and uncertainty conditions, and the relative quantized stability norm can be given. The simulation shows that the present method can effectively solve the stability analysis and determination of flight control system under various factors.

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.

Method and algorithm to construct a quasi-chaotic sequence

In this work, a first order nonlinear discrete-time equation is studied in order to obtain all of its exact solutions in an analytical way, including those presenting chaotic behavior. Besides, an efficient numerical algorithm for the construction of the quasi chaotic sequences, with a large number of terms, is also proposed. It is also important to mention that the constructed sequences are distributed along the interval [0,1].

On the rate‐depentent regularization for damage modeling

AbstractA model framework for the analysis of isotropic quasi‐brittle damage, was recently presented in [1]. Within this paper, the model in [1] is significantly improved. To be more precise, and in contrast to [1], the novel model: (i) eliminates unnecessary model parameters, (ii) can be better interpreted from a physics point of view, (iii) can capture a fully softened state (zero stresses) and (iv) is characterized by a very simple evolution equation which (v) can be integrated fully implicitly and (vi) the resulting time discrete evolution equation can be solved analytically providing a numerically efficient closed form solution, cf. [2].

Existence of weighted bounded solutions for nonlinear discrete-time fractional equations

ABSTRACT Let X be a Banach space and T be a bounded linear operator defined on X. In this work we study the existence of solutions of a class of nonlinear difference equation of fractional order in the form where corresponds to the fractional difference operator of order in sense of Riemann–Liouville and is a function satisfying suitable conditions. More specifically, by using operator-theoretical methods and fixed point theory, we show the existence of solutions of such class of equations on the vector-valued weighted space of sequences

Effect of harvest timing on the dynamics of the Ricker–Seno model

The moment of intervention is a key question in harvest programmes and is currently generating an increasing interest. However, little is known about its effect on the population stability. This lack of knowledge is greater in the case of global stability, which is always desirable as it allows to predict the fate of populations regardless of the initial size. Here we use a discrete-time equation to model the dynamics of populations harvested at any time during the reproductive season. We study the effect of the time of intervention on the global stability of populations governed by the Ricker model, which is one of the most relevant models in discrete-time population dynamics. We prove that harvest timing never has a negative effect on the global stability of these populations, extending recent results in the literature. We also study the effect of delayed harvesting on the constancy stability of the controlled populations.

Moore‐Penrose‐pseudo‐inverse‐based Kalman‐like filtering methods for estimation of stiff continuous‐discrete stochastic systems with ill‐conditioned measurements

This study aims at exploring numerical stability properties of software sensors used in chemical and other engineering. These are utilised for evaluation of variables and/or parameters of plants, which are not measurable by technical devices. Software sensors are often grounded in the extended Kalman filtering (EKF) technique. A conventional continuous-discrete stochastic system consists of an It o ^ -type stochastic differential equation representing the plant's dynamics and a discrete-time equation linking the model's state to measurements. Here, the authors focus on the numerical stability of EKF-type methods, which are applicable to ill-conditioned stiff stochastic models arisen in applied science and engineering. They explore filters' accuracies when the inverse matrices are replaced with the Moore-Penrose pseudo-inverse ones in their measurement updates. This investigation is fulfilled within the authors' ill-conditioned stochastic Oregonator scenario and evidences that the pseudo-inversion indeed resolves many performance problems in some non-square-root methods when the stochastic system is sufficiently ill-conditioned. However, it fails to improve the accuracy in the mildly ill-conditioned case. Eventually, only the square-root nested implicit Runge-Kutta-based filters are found out to be accurate and robust in their examination and, hence, to be the methods of choice.

LebesguE Regularity for Nonlocal Time-Discrete Equations with Delays

Abstract In this work we provide a new and effective characterization for the existence and uniqueness of solutions for nonlocal time-discrete equations with delays, in the setting of vector-valued Lebesgue spaces of sequences. This characterization is given solely in terms of the R-boundedness of the data of the problem, and in the context of the class of UMD Banach spaces.

Fractional discrete-time diffusion equation with uncertainty: Applications of fuzzy discrete fractional calculus

This study provides some basics of fuzzy discrete fractional calculus as well as applications to fuzzy fractional discrete-time equations. With theories of r-cut set, fuzzy Caputo and Riemann–Liouville fractional differences are defined on a isolated time scale. Discrete Leibniz integral law is given by use of w-monotonicity conditions. Furthermore, equivalent fractional sum equations are established. Fuzzy discrete Mittag-Leffler functions are obtained by the Picard approximation. Finally, fractional discrete-time diffusion equations with uncertainty is investigated and exact solutions are expressed in form of two kinds of fuzzy discrete Mittag-Leffler functions. This paper suggests a discrete time tool for modeling discrete fractional systems with uncertainty.

Quasi-brittle damage modeling based on incremental energy relaxation combined with a viscous-type regularization

This paper deals with a constitutive model suitable for the analysis of quasi-brittle damage in structures. The model is based on incremental energy relaxation combined with a viscous-type regularization. A similar approach—which also represents the inspiration for the improved model presented in this paper—was recently proposed in Junker et al. (Contin Mech Thermodyn 29(1):291–310, 2017). Within this work, the model introduced in Junker et al. (2017) is critically analyzed first. This analysis leads to an improved model which shows the same features as that in Junker et al. (2017), but which (i) eliminates unnecessary model parameters, (ii) can be better interpreted from a physics point of view, (iii) can capture a fully softened state (zero stresses), and (iv) is characterized by a very simple evolution equation. In contrast to the cited work, this evolution equation is (v) integrated fully implicitly and (vi) the resulting time-discrete evolution equation can be solved analytically providing a numerically efficient closed-form solution. It is shown that the final model is indeed well-posed (i.e., its tangent is positive definite). Explicit conditions guaranteeing this well-posedness are derived. Furthermore, by additively decomposing the stress rate into deformation- and purely time-dependent terms, the functionality of the model is explained. Illustrative numerical examples confirm the theoretical findings.

Model matching of switched asynchronous sequential machines via matrix approach

ABSTRACTThis paper investigates model matching problem of switched asynchronous sequential machines (ASMs) via semi-tensor product (STP) of matrices. A switched ASM is composed of several input/state ASMs and it cut-overs between submachines in asynchronous mechanism. By using STP method, a switched ASM is depicted as a discrete-time equation form. Under this framework, stable transition matrix and admissible model-input-state transition matrix are defined. Then, two important problems, reachability and optimal control input design of switched ASMs, are discussed by these two new matrices, respectively. Finally, a necessary and sufficient condition for the existence of a controller for model matching is obtained and a practical example on asynchronous error accumulators is provided to show the feasibility of the proposed method.

GENERIC Integrators: Structure Preserving Time Integration for Thermodynamic Systems

Abstract Thermodynamically admissible evolution equations for non-equilibrium systems are known to possess a distinct mathematical structure. Within the GENERIC (general equation for the non-equilibrium reversible–irreversible coupling) framework of non-equilibrium thermodynamics, which is based on continuous time evolution, we investigate the possibility of preserving all the structural elements in time-discretized equations. Our approach, which follows Moser’s [1] construction of symplectic integrators for Hamiltonian systems, is illustrated for the damped harmonic oscillator. Alternative approaches are sketched.

Unconditionally Optimal Error Estimates of a Linearized Galerkin Method for Nonlinear Time Fractional Reaction–Subdiffusion Equations

This paper is concerned with unconditionally optimal error estimates of linearized Galerkin finite element methods to numerically solve some multi-dimensional fractional reaction–subdiffusion equations, while the classical analysis for numerical approximation of multi-dimensional nonlinear parabolic problems usually require a restriction on the time-step, which is dependent on the spatial grid size. To obtain the unconditionally optimal error estimates, the key point is to obtain the boundedness of numerical solutions in the $$L^\infty $$ -norm. For this, we introduce a time-discrete elliptic equation, construct an energy function for the nonlocal problem, and handle the error summation properly. Compared with integer-order nonlinear problems, the nonlocal convolution in the time fractional derivative causes much difficulties in developing and analyzing numerical schemes. Numerical examples are given to validate our theoretical results.

A Continuous Reconstruction Observer for Sampled-Data Linear Time Varying Systems

In this paper, a novel continuous sampled-data observer for linear time-varying systems is presented. The proposed observer is based on two different impulsive observers. The state estimates of these impulsive observers are fused in a manner such that continuous state estimation is achieved. Another significant contribution is comprehensive stability analysis of the proposed observer. The analysis establishes conditions that guarantee exponential convergence of observers. Contrary to the common understanding, it is revealed that the convergence of associated discrete-time equation for impulsive observers is not a sufficient condition for overall convergence. Contributions are illustrated by simulations.

Double-loop PI controller design of the DC-DC boost converter with a proposed approach for calculation of the controller parameters

Parameters of digital proportional–integral/proportional–integral–derivative controllers are usually calculated using commonly known conventional methods or solution of discrete-time equations. In literature, a model-based compact form formulation for calculation of discrete-time proportional–integral/proportional–integral–derivative controller parameters has not been come across yet. The proposed model-based compact form formulations are introduced to calculate the proportional–integral parameters in discrete time as a new approach. Generally, different types of control techniques are chosen in similar studies for double-loop control for direct current–direct current boost converter control except proportional–integral/proportional–integral. In this study, double-loop proportional–integral controller is used as a different control method from literature. By this way, the most important advantages of the proposed study are to reduce different design methods to a unique proportional–integral design method and shorten all calculations. The accuracy of the double-loop proportional–integral controller’s parameters calculated using the model-based compact form formulations is validated both in simulation and experimental studies under various disturbance effects. Satisfactory performance of the proposed controller under model uncertainty and other cases are comparatively shown with the predefined performance criteria.

A modal frequency-domain generalised force matrix for the unsteady Vortex Lattice method

The unsteady Vortex Lattice method is becoming an increasingly popular aerodynamic modelling method for incompressible aeroelastic problems, such as flexible low-speed aircraft, wind turbines and flapping flight. It leads to discrete time aeroelastic state space equations, which must be solved in a time-marching framework. Eigenvalue or singular value decompositions of the discrete time equations can be used in order to perform stability analysis but such procedures must be accompanied by model order reduction because the size of the equations is large. This work proposes a modal frequency domain implementation of the Vortex Lattice method, resulting in a modal generalised force matrix. Model order reduction is implicit in the modal approach and stability analysis can be carried out using industry-standard flutter analysis techniques, such as the p–k method. The approach is validated by comparison to wind tunnel flutter data obtained from rectangular cantilever flat plate wings of different aspect ratios and sweep angles. It is found that the aeroelastic model predictions follow the experimental trends for both flutter speed and frequency but tend to be moderately conservative.