Discrete Integration Research Articles

Discrete Time Approximation of Continuous Time Nonlinear State Space Models

In this paper we discuss the problem of approximating a continuous time nonlinear state space model with an explicit discrete time state space model. Two steps are made to do so. The first one is to replace the classical zero-order-hold (ZOH) excitation signal assumption by a more general concept of low pass (LP) signals. It will be shown that opposed to the ZOH-property, the LP-property is maintained for a wide class of (non)linear systems, including cascaded and closed loop systems. The second step is based on a result from the signal processing theory that states that a BL-signal can be exactly predicted from its past samples by a fixed causal linear filter. This result is generalized to the discrete time integration of LP-signals by introducing an error bound that can be made arbitrarily small by increasing the sample frequency. Both ideas are then combined to bound the approximation errors of an explicit discrete time nonlinear state space approximation for a continuous time nonlinear state space model. The order of the decay of the approximation error as a function of the sample frequency is given. It is shown that eventually the aliasing error is the dominating error, at a cost of a (slightly) increased model complexity. These results are directly applicable to nonlinear system identification, and will be experimentally verified on the identification of a closed loop nonlinear system (the silverbox) using a discrete time nonlinear state space model.

Numerical analysis and investigation of short- and long-term behavior of unidirectional composites

This study gives a detailed analysis on estimating the ultimate tensile strength of unidirectional fiber reinforced composites and its creep behavior under sustained tension load. We develop two different micromechanical models that allow us to estimate the longitudinal tensile strength and the evolution with time of fiber and matrix stresses around arbitrary array of fiber breaks. The first model is based on the shear-lag theory while the second one is developed using the software Abaqus. The comparison of the above models allowed to validate the fundamental assumptions of the shear-lag theory (first model) as well as several numerical issues related to time integration and spatial discretization. The Monte–Carlo method was used in order to account for the stochastic fiber strength and its impact on the ultimate tensile strength (short-term) and creep (long-term behavior) of unidirectional composites. Finally, a parametric investigation on the fiber type and the load level on the long-term behavior of unidirectional composites was performed showing an accelerating creep effect for fibers of inferior quality such as glass fibers compared to carbon fibers.

Evaluating Wave Random Path Using Multilevel Monte Carlo

Wind waves are important due to their high energy and impact on marine activities. This phenomenon is affects directly or indirectly the construction of coastal infrastructure, shipping and recreational activities. Due to the issues presented, marine parameters are very important. In this study, we try to pay attention to wave as one of the most important marine parameters. As the movements of waves have high uncertainty, financial models can be used to simulate the wave's paths. We use the Monte Carlo method for this purpose. The Monte Carlo simulation is a flexible and simple tool that is widely used in the evaluation of random paths. To compute a random path, we require an integral discretization. In this paper, we study the valuation of European options using Monte Carlo simulation and then compare this result with multi-level Monte Carlo approach and other antithetic variables. Then, we use the multi-level Monte Carlo approach proposed by (M. B. Giles, 2008) for pricing under the two-factor stochastic volatility model. We show that the multi-level Monte Carlo method reduces the computational complexity and also cost of the two-factor stochastic volatility model when compared with the standard Monte Carlo method. Also, we compare the multi-level Monte Carlo method and standard Monte Carlo method using an Euler discretization scheme and then, analyze the numerical results.

From quantitative to qualitative orness for lattice OWA operators

This paper deals with OWA (ordered weighted average) operators defined on an arbitrary finite lattice endowed with a t-norm and a t-conorm. A qualitative orness measure for any OWA operator is suggested, based on its proximity to the OR operator that yields the maximum of the given data. In the particular case of a finite distributive lattice, considering the t-norm given by the meet and the t-conorm given by the join, this qualitative measure agrees with the value that some discrete Sugeno integral takes on the vector consisting of all the members of the lattice. Some applications of the qualitative orness of OWA operators to decision-making problems are shown. In addition, OWA operators defined on a finite product lattice are also applied in image processing. We analyze the effect of several OWA operators with respect to their orness.

Accelerating equilibrium isotope effect calculations. I. Stochastic thermodynamic integration with respect to mass

Accurate path integral Monte Carlo or molecular dynamics calculations of isotope effects have until recently been expensive because of the necessity to reduce three types of errors present in such calculations: statistical errors due to sampling, path integral discretization errors, and thermodynamic integration errors. While the statistical errors can be reduced with virial estimators and path integral discretization errors with high-order factorization of the Boltzmann operator, here we propose a method for accelerating isotope effect calculations by eliminating the integration error. We show that the integration error can be removed entirely by changing particle masses stochastically during the calculation and by using a piecewise linear umbrella biasing potential. Moreover, we demonstrate numerically that this approach does not increase the statistical error. The resulting acceleration of isotope effect calculations is demonstrated on a model harmonic system and on deuterated species of methane.

Climate classification through recursive multivariate statistical inferences: a case study of the Athabasca River Basin, Canada

ABSTRACTIn this study, a recursive dissimilarity and similarity inferential climate classification (ReDSICC) approach is developed to provide an alternative tool for climate classification. Based on incorporation of a discrete distribution transformation (DDT) method and integration of advanced statistical inferential methods, a recursive framework of dissimilarity and similarity inferences is proposed for stepwise grouping multi‐dimensional climate‐variable observations. ReDSICC is capable of eliminating the restriction of samples being normally distributed, enabling classification of regional climates under data uncertainties and multivariate dependencies, identifying the most desired climate classification result, and avoiding subjective judgments in the classification process. To verify methodological effectiveness and facilitate related studies, ReDSICC is applied to climate classification in the Athabasca River Basin (ARB), Canada. It is revealed that the complicated dissimilarities and similarities of climatic conditions among all grids over the ARB are effectively reflected in the results of ReDSICC. A reversible transformation between an abnormal distribution and a normal distribution is achieved by DDT. The effectiveness of climate classification which is represented as the Nash coefficient for climatic features over any grid and the corresponding climate class is decreased if DDT is not employed. In comparison with daily minimum temperature, the spatial heterogeneity of daily maximum temperature is higher while that of daily cumulative precipitation is lower over the ARB. The classification result of ReDSICC varies with changes of representative climate variables and parameter values. These advantages and revelations are helpful for enhancing the reliability of climate classification results, improving the effectiveness of existing climate classification methods, and providing scientific support for the related studies in the ARB or neighbouring regions.

Downward continuation of airborne gravimetry data based on Poisson integral iteration method

The research and application of airborne gravimetry technology has become one of the hottest topics in gravity field in recent years. Downward continuation is one of the key steps in airborne gravimetry data processing, and the quality of continuation results directly influence the further application of surveying data. The Poisson integral iteration method is proposed in this paper, and the modified Poisson integral discretization formulae are also introduced in the downward continuation of airborne gravimerty data. For the test area in this paper, compared with traditional Poisson integral discretization formula, the continuation result of modified formulae is improved by 10.8 mGal, and the precision of Poisson integral iteration method is in the same amplitude as modified formulae. So the Poisson integral iteration method can reduce the discretization error of Poisson integral formula effectively. Therefore, the research achievements in this paper can be applied directly in the data processing of our country's airborne scalar and vector gravimetry.

Conservative discretization of the Landau collision integral

We describe a density-, momentum-, and energy-conserving discretization of the nonlinear Landau collision integral. The method is suitable for both the finite-element and discontinuous Galerkin methods and does not require structured meshes. The conservation laws for the discretization are proven algebraically and demonstrated numerically for an axially symmetric nonlinear relaxation problem using a finite-element implementation.

Peripheral arterial disease screening for hemodialysis patients using a fractional-order integrator and transition probability decision-making model.

Atherosclerosis and resultant peripheral arterial disease (PAD) are common complications in patients with type 2 diabetes mellitus or end-stage renal disease and in elderly patients. The prevalence of PAD is higher in patients receiving haemodialysis therapy. For early assessment of arterial occlusion using bilateral photoplethysmography (PPG), such as changes in pulse transit time and pulse shape, bilateral timing differences could be used to identify the risk level of PAD. Hence, the authors propose a discrete fractional-order integrator to calculate the bilateral area under the systolic peak (AUSP). These indices indicated the differences in both rise-timing and amplitudes of PPG signals. The dexter and sinister AUSP ratios were preliminarily used to separate the normal condition from low/high risk of PAD. Then, transition probability-based decision-making model was employed to evaluate the risk levels. The joint probability could be specified as a critical threshold, < 0.81, to identify the true positive for screening low or high risk level of PAD, referring to the patients' health records. In contrast to the bilateral timing differences and traditional methods, the proposed model showed better efficiency in PAD assessments and provided a promising strategy to be implemented in an embedded system.

Variational Integrators for Interconnected Lagrange–Dirac Systems

Interconnected systems are an important class of mathematical models, as they allow for the construction of complex, hierarchical, multiphysics, and multiscale models by the interconnection of simpler subsystems. Lagrange--Dirac mechanical systems provide a broad category of mathematical models that are closed under interconnection, and in this paper, we develop a framework for the interconnection of discrete Lagrange--Dirac mechanical systems, with a view towards constructing geometric structure-preserving discretizations of interconnected systems. This work builds on previous work on the interconnection of continuous Lagrange--Dirac systems (Jacobs and Yoshimura 2014) and discrete Dirac variational integrators (Leok and Ohsawa 2011). We test our results by simulating some of the continuous examples given in Jacobs and Yoshimura 2014.

Real time simulation of inextensible surgical thread using a Kirchhoff rod model with force output for haptic feedback applications

We discuss the dynamics of an inextensible thin Kirchhoff rod used in the modeling of surgical threads, and demonstrate a very efficient scheme to not only simulate the motion of the thread in real-time (up to 1 ms per frame) but also obtain the constraint axial forces which can be fed back to a haptic system. The numerical scheme is based on a family of schemes called geometric or discrete variational integrators guaranteeing that the momentum and energy are exactly conserved over long periods of time for conservative systems. Besides, we report on an efficient numerical procedure to handle the inextensibility of the thread through physically based Lagrange multipliers, as well as the internal dissipation of the thread. We have performed simulations to verify the capabilities of our model to conserve momentum and energy, accurately calculate the axial constraint forces along the thread for haptic feedback, and capture bending-torsion coupling leading to the formation of plectonemes. While many of the ideas are well known in the computer graphics community (especially in hair modeling), we have implemented several improvements for the specific purpose of speeding up the computations for developing physically based haptic interfaces for knot tying and suturing.

Stability Analysis of Discrete Integral Sliding Mode Control for Input–Output Model

The sliding mode control (SMC) has several advantages in terms of good transient performance and system robustness. However, the sensitivity of this control technique to disturbance before reaching the sliding surface and the existence of the chattering phenomenon can be considered as the major problems in its implementation. To overcome these problems, we propose a new discrete integral sliding mode control (DISMC) for input–output model and compare it with the classical SMC and a recent version of DISMC. Then, a stability analysis is presented. Simulation results are given to illustrate the effectiveness of the proposed method.

Research on Displacement Calculation of Dynamometer Card Based on Kaiman Filter and Discrete Numerical Integration

In order to calculate the dynamometer card of oil well using acceleration sensor, the algorithm which combined by Kalman filter and discrete numerical integration is proposed. It can be applied to calculate the displacement and precipitation displacement period of oil well dynamometer card. The Kalman filter not only filters out the noise of the acceleration signal, but also maintains the original shape feature. The accurate precipitation of the displacement period ensures the correctness of displacement. The discrete numerical integration algorithm can make the relative error of displacement measurement less than 1%, which meets the requirement for dynamometer card accuracy. It is suitable for different types of oil wells.

Algebro-Geometric Solutions for a Discrete Integrable Equation

With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.

Моделирование самообучающейся системы управления инвертором преобразователя частоты для подавления высших гармоник

Simulation of a three-phase frequency converter based on three single-phase full-bridge voltage inverters with a sine-wave output filter is considered. A new self-learning control system for the converter’s inverters is proposed. The previously developed inverter control system uses the discrete Fourier transform (DFT) to suppress the output voltage higher harmonic components. Individual regulators are used to control each harmonic component (including the fundamental and higher-order ones). The DFT algorithm produces signals proportional to the amplitudes of the fundamental and a number of higher-order harmonic components, which are applied to the regulators as feedbacks. The main drawback of the existing system is that it poses rather high demand to the microcontroller computation capacities. The newly proposed system is based on the self-learning principle and uses a periodic integrator (a P-integrator) as the output voltage regulator. The P-integrator is represented in discrete form by an array of discrete integrators, each of which operates only in the 1/64 part of the fundamental harmonic component cycle. The new control system's software code is much smaller in size and contains a fewer number of “slow” arithmetic operations, due to which it is easier to implement and less demanding with respect to computation resources. The control system analysis results demonstrate its good performance in terms of suppressing higher harmonic components: the output voltage total harmonic distortion factor does not exceed 5% for the cases of operation on linear and nonlinear loads. The article describes the modernized control system structure and its implementation in MATLAB Simulink software package. Advantages of the self-learning algorithm as compared with the algorithm based on the discrete Fourier transform are demonstrated. The simulation results confirm the applicability of the new control method. The algorithm is implemented in the C language in the MATLAB S-function block and is suitable for being implemented in real-time control systems based on dedicated microcontrollers without the need of significantly modifying the software code.

Design of Computer Composition Principle Experiment Board Based on CPLD

In accordance with the problem that the traditional experiment board of computer composition principle can only be used for functional realization and verification, which cannot satisfy the students’ design and innovation, this paper proposes an improved experiment board method of computer composition principle. This method has used the complex programmable logic device (CPLD) chip to replace the original discrete small-scale integration chip; the Quartus II is used to complete the circuit description by drawing schematic diagram; in the meantime, the control signal board and the data path board are integrated, which form a complete computer system with the microprogrammed board, which can be used in the experiment teaching. The experimental test results show that the new experiment board based on CPLD has stable and reliable operation, which can realize functions consistent with the original experiment instrument, and it can be used as the EDA experiment board.

Long and Short Memory in Economics: Fractional-Order Difference and Differentiation

&lt;div&gt;&lt;p&gt;&lt;em&gt;Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the &lt;/em&gt;&lt;em&gt;Grunwald-Letnikov fractional differences of non-integer order d. Equations of ARIMA(p,d,q) and ARFIMA(p,d,q) models are the fractional-order difference equations with the Grunwald-Letnikov differences of order d. We prove that the long and short memory with power law should be described by the exact fractional-order differences, for which &lt;/em&gt;&lt;em&gt;the Fourier transform &lt;/em&gt;&lt;em&gt;demonstrates the power law exactly. The fractional differencing and the Grunwald-Letnikov fractional differences cannot give exact results for the long and short memory with power law, since &lt;/em&gt;&lt;em&gt;the Fourier transform&lt;/em&gt;&lt;em&gt; of these discrete operators satisfy the &lt;/em&gt;&lt;em&gt;power law in the neighborhood of zero only. We prove that the economic processes with t&lt;/em&gt;&lt;em&gt;he continuous time long and short memory, which is characterized by the power law, should be described by the fractional differential equations.&lt;/em&gt;&lt;/p&gt;&lt;/div&gt;

Variational Integrators for Structure-Preserving Filtering

Estimation and filtering are important tasks in most modern control systems. These methods rely on accurate discrete-time approximations of the system dynamics. We present filtering algorithms that are based on discrete mechanics techniques (variational integrators), which are known to preserve system structures (momentum, symplecticity, and constraints, for instance) and have stable long-term energy behavior. These filtering methods show increased performance in simulations and experiments on a real digital control system. The particle filter as well as the extended Kalman filter benefits from the statistics-preserving properties of a variational integrator discretization, especially in low bandwidth applications. Moreover, it is shown how the optimality of the Kalman filter can be preserved through discretization by means of modified discrete-time Riccati equations for the covariance updates. This leads to further improvement in filter accuracy, even in a simple test example.

Mass, momentum, and energy flux conservation for nonlinear wave-wave interaction

A fully nonlinear solution for bi-chromatic progressive waves in water of finite depth in the framework of the homotopy analysis method (HAM) is derived. The bi-chromatic wave field is assumed to be obtained by the nonlinear interaction of two monochromatic wave trains that propagate independently in the same direction before encountering. The equations for the mass, momentum, and energy fluxes based on the accurate high-order homotopy series solutions are obtained using a discrete integration and a Fourier series-based fitting. The conservation equations for the mean rates of the mass, momentum, and energy fluxes before and after the interaction of the two nonlinear monochromatic wave trains are proposed to establish the relationship between the steady-state bi-chromatic wave field and the two nonlinear monochromatic wave trains. The parametric analysis on ε1 and ε2, representing the nonlinearity of the bi-chromatic wave field, is performed to obtain a sufficiently small standard deviation Sd, which is applied to describe the deviation from the conservation state (Sd = 0) in terms of the mean rates of the mass, momentum, and energy fluxes before and after the interaction. It is demonstrated that very small standard deviation from the conservation state can be achieved. After the interaction, the amplitude of the primary wave with a lower circular frequency is found to decrease; while the one with a higher circular frequency is found to increase. Moreover, the highest horizontal velocity of the water particles underneath the largest wave crest, which is obtained by the nonlinear interaction between the two monochromatic waves, is found to be significantly higher than the linear superposition value of the corresponding velocity of the two monochromatic waves. The present study is helpful to enrich and deepen the understanding with insight to steady-state wave-wave interactions.

H-Index and Other Sugeno Integrals: Some Defects and Their Compensation

The famous Hirsch index has been introduced just ca. ten years ago. Despite that, it is already widely used in many decision-making tasks, like in evaluation of individual scientists, research grant allocation, or even production planning. It is known that the $h$ -index is related to the discrete Sugeno integral and the Ky Fan metric introduced in the 1940s. The aim of this paper is to propose a few modifications of this index as well as other fuzzy integrals—also on bounded chains—that lead to better discrimination of some types of data that are to be aggregated. All of the suggested compensation methods try to retain the simplicity of the original measure.