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1

The mathematical modelling of transient processes in a drive system including asynchronous and synchronous drives of susceptible motion transmission

Chaban Andriy, Lis Marek, Szafraniec Andrzej, Levoniuk Vitaliy

Przegląd Elektrotechniczny

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2022

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R. 98, nr 1

66--70

EN

A mathematical model is developed of an electric drive system including a power transformer, deep bar cage induction asynchronous motors, and long shaft synchronous drives of distributed mechanical parameters. The model of a deep bar cage induction asynchronous motor addresses the skin-effects across the rotor cage bars using the theory of electromagnetic field. The motion transmission of the drives is described with a mixed problem of the third type Poincaré boundary conditions. The general system of differential equations is given in the Cauchy form, integrated by means of the latent Euler method. Results of computer simulations are shown with graphics, which are analysed.

PL

W pracy niniejszej opracowano model matematyczny zespołu elektrycznego obciążenia, który zawiera transformator mocy, głębokożobkowe napędy asynchroniczne i napędy synchroniczne z długimi wałami o mechanicznych parametrach rozłożonych. Model głębokożłobkowych silników asynchronicznych uwzględnia zjawisko naskórkowości w prętach klatki wirników za pomocą teorii pola elektromagnetycznego. Transmisję ruchu napędów opisuje się za pomocą zadania mieszanego o warunkach brzegowych trzeciego rodzaju Poincaré. Ogólny układ równań różniczkowych podany jest w postaci Cauchy, który całkuje się za pomocą ukrytej metody Eulera. Wyniki symulacji komputerowej przedstawione są w postaci rysunków, które są analizowane.

2

Approximate state-space and transfer function models for 2x2 linear hyperbolic systems with collocated boundary inputs

Bartecki Krzysztof

International Journal of Applied Mathematics and Computer Science

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2020

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Vol. 30, no. 3

475--491

EN

Two approximate representations are proposed for distributed parameter systems described by two linear hyperbolic PDEs with two time- and space-dependent state variables and two collocated boundary inputs. Using the method of lines with the backward difference scheme, the original PDEs are transformed into a set of ODEs and expressed in the form of a finite number of dynamical subsystems (sections). Each section of the approximation model is described by state-space equations with matrix-valued state, input and output operators, or, equivalently, by a rational transfer function matrix. The cascade interconnection of a number of sections results in the overall approximation model expressed in finite-dimensional state-space or rational transfer function domains, respectively. The discussion is illustrated with a practical example of a parallel-flow double-pipe heat exchanger. Its steady-state, frequency and impulse responses obtained from the original infinite-dimensional representation are compared with those resulting from its approximate models of different orders. The results show better approximation quality for the “crossover” input–output channels where the in-domain effects prevail as compared with the “straightforward” channels, where the time-delay phenomena are dominating.

3

Analysis of transient processes in a power supply system of concentrated and distributed parameters based on variational approaches

Chaban A., Lis M., Szafraniec A., Chrzan M., Levoniyk V.

Przegląd Elektrotechniczny

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2018

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R. 94, nr 12

154--157

EN

Starting with variational approaches using a modified Hamilton-Ostrogradsky principle, a mathematical model of a power system is developed and analysed as a concentrated parameters system for power autotransformer feeder reactors and capacitors and as a distributed parameters system for supply lines. The final discretised state equations of the power system are represented in Cauchy format. Results of computer simulations are presented as drawings.

PL

Korzystając z podejść wariacyjnych przy użyciu zmodyfikowanej zasady Hamiltona-Ostrogradzkiego, opracowano matematyczny model systemu elektroenergetycznego, który analizuje się jako układ o parametrach rozproszonych. Końcowe, dyskretyzowane równania stanu systemu elektroenergetycznego są przedstawione w formacie Cauchy'ego. Wyniki symulacji komputerowych przedstawiono na rysunkach.

4

Distributed scheduling of measurements in a sensor network for parameter estimation of spatio-temporal systems

Patan M., Kowalów D.

International Journal of Applied Mathematics and Computer Science

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2018

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Vol. 28, no. 1

39--54

EN

The main aim of the paper is to develop a distributed algorithm for optimal node activation in a sensor network whose measurements are used for parameter estimation of the underlying distributed parameter system. Given a fixed partition of the observation horizon into a finite number of consecutive intervals, the problem under consideration is to optimize the percentage of the total number of observations spent at given sensor nodes in such a way as to maximize the accuracy of system parameter estimates. To achieve this, the determinant of the Fisher information matrix related to the covariance matrix of the parameter estimates is used as the qualitative design criterion (the so-called D-optimality). The proposed approach converts the measurement scheduling problem to a convex optimization one, in which the sensor locations are given a priori and the aim is to determine the associated weights, which quantify the contributions of individual gaged sites to the total measurement plan. Then, adopting a pairwise communication scheme, a fully distributed procedure for calculating the percentage of observations spent at given sensor locations is developed, which is a major novelty here. Another significant contribution of this work consists in derivation of necessary and sufficient conditions for the optimality of solutions. As a result, a simple and effective computational scheme is obtained which can be implemented without resorting to sophisticated numerical software. The delineated approach is illustrated by simulation examples of a sensor network design for a two-dimensional convective diffusion process.

5

The approximation function of bridge deck vibration derived from the measured eigenmodes

Sokol M., Komorníková M., Bacigál T., Rodríguez M. X.

International Journal of Applied Mathematics and Computer Science

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2017

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Vol. 27, no. 4

799--814

EN

This article deals with a method of how to acquire approximate displacement vibration functions. Input values are discrete, experimentally obtained mode shapes. A new improved approximation method based on the modal vibrations of the deck is derived using the least-squares method. An alternative approach to be employed in this paper is to approximate the displacement vibration function by a sum of sine functions whose periodicity is determined by spectral analysis adapted for non-uniformly sampled data and where the parameters of scale and phase are estimated as usual by the least-squares method. Moreover, this periodic component is supplemented by a cubic regression spline (fitted on its residuals) that captures individual displacements between piers. The statistical evaluation of the stiffness parameter is performed using more vertical modes obtained from experimental results. The previous method (Sokol and Flesch, 2005), which was derived for near the pier areas, has been enhanced to the whole length of the bridge. The experimental data describing the mode shapes are not appropriate for direct use. Especially the higher derivatives calculated from these data are very sensitive to data precision.

6

Feedback design of differential equations of reconstruction for second-order distributed parameter systems

Maksimov V. I., Mordukhovich B. S.

International Journal of Applied Mathematics and Computer Science

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2017

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Vol. 27, no. 3

467--475

EN

The paper aims at studying a class of second-order partial differential equations subject to uncertainty involving unknown inputs for which no probabilistic information is available. Developing an approach of feedback control with a model, we derive an efficient reconstruction procedure and thereby design differential equations of reconstruction. A characteristic feature of the obtained equations is that their inputs formed by the feedback control principle constructively approximate unknown inputs of the given second-order distributed parameter system.

7

Transmitancje operatorowe pewnej klasy układów typu hiperbolicznego z wejściami brzegowymi

Bartecki K.

Pomiary Automatyka Kontrola

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2014

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R. 60, nr 1

49--52

PL

W artykule przedstawiono ogólną postać transmitancji operatorowych pewnej klasy układów o parametrach rozłożonych, opisanych dwoma równaniami różniczkowymi cząstkowymi typu hiperbolicznego. Zakładając istnienie w układzie dwóch wejść o charakterze wymuszeń brzegowych typu Dirichleta oraz dwóch wyjść rozłożonych wzdłuż osi zmiennej przestrzennej, przedstawiono wyrażenia opisujące transmitancje operatorowe układu dla dwóch różnych konfiguracji sygnałów wejściowych. Rozważania zilustrowano praktycznym przykładem wymiennika ciepła pracującego w układach: współ- oraz przeciwprądowym.

EN

Transfer function models for a class of distributed parameter systems described by the two hyperbolic partial differential equations defined on a one-dimensional finite spatial domain are considered. Assuming two boundary inputs of Dirichlet type, the closed-form expressions for the individual elements of the 22 transfer function matrix are proposed based on the decoupled canonical representation of the system. The influence of the location of the boundary inputs on the transfer function representation is demonstrated for two different input configurations. The first one is the so-called congruent arrangement, for which both inputs act on the system at the same spatial position, l=0 (Fig. 1). The second one is the incongruent arrangement, where both inputs act on the system at its opposite ends, l=0 and l=L, respectively (Fig. 2). The considerations are illustrated with a practical example of a shell and tube heat exchanger operating in parallel- and countercurrent-flow modes (Fig. 3), which correspond to the two abovementioned boundary input configurations. Based on the transfer function model, both frequency and time responses of the system can be determined, which can be useful e.g. in the case of the model-based fault detection scheme.

8

A general transfer function representation for a class of hyperbolic distributed parameter systems

Bartecki K.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 2

291--307

EN

Results of transfer function analysis for a class of distributed parameter systems described by dissipative hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of two boundary inputs, the closed-form expressions for the individual elements of the 2×2 transfer function matrix are derived both in the exponential and in the hyperbolic form, based on the decoupled canonical representation of the system. Some important properties of the transfer functions considered are pointed out based on the existing results of semigroup theory. The influence of the location of the boundary inputs on the transfer function representation is demonstrated. The pole-zero as well as frequency response analyses are also performed. The discussion is illustrated with a practical example of a shell and tube heat exchanger operating in parallel- and countercurrent-flow modes.

9

Steady-state analysis for a class of hyperbolic systems with boundary inputs

Bartecki K.

Archives of Control Sciences

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2013

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Vol. 23, no. 3

295--310

EN

Results of a steady-state analysis performed for a class of distributed parameter systems described by hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of the system with two state variables and two boundary inputs, the analytical expressions for the steady-state distribution of the state variables are derived, both in the exponential and in the hyperbolic form. The influence of the location of the boundary inputs on the steady-state response is demonstrated. The considerations are illustrated with a practical example of a shell and tube heat exchanger operating in parallel- and countercurrent-flow modes.

10

Sensor network scheduling for identification of spatially distributed processes

Uciński D.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 1

25-40

EN

The work treats the problem of fault detection for processes described by partial differential equations as that of maximizing the power of a parametric hypothesis test which checks whether or not system parameters have nominal values. A simple node activation strategy is discussed for the design of a sensor network deployed in a spatial domain that is supposed to be used while detecting changes in the underlying parameters which govern the process evolution. The setting considered relates to a situation where from among a finite set of potential sensor locations only a subset of them can be selected because of the cost constraints. As a suitable performance measure, the Ds-optimality criterion defined on the Fisher information matrix for the estimated parameters is applied. The problem is then formulated as the determination of the density of gauged sites so as to maximize the adopted design criterion, subject to inequality constraints incorporating a maximum allowable sensor density in a given spatial domain. The search for the optimal solution is performed using a simplicial decomposition algorithm. The use of the proposed approach is illustrated by a numerical example involving sensor selection for a two-dimensional diffusion process.

11

Distributed scheduling of sensor networks for identification of spatio-temporal processes

Patan M.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 2

299-311

EN

An approach to determine a scheduling policy for a sensor network monitoring some spatial domain in order to identify unknown parameters of a distributed system is discussed. Given a finite number of possible sites at which sensors are located, the activation schedule for scanning sensors is provided so as to maximize a criterion defined on the Fisher information matrix associated with the estimated parameters. The related combinatorial problem is relaxed through operating on the density of sensors in lieu of individual sensor positions. Then, based on the adaptation of pairwise communication algorithms and the idea of running consensus, a numerical scheme is developed which distributes the computational burden between the network nodes. As a result, a simple exchange algorithm is outlined to solve the design problem in a decentralized fashion.

12

PCA-based approximation of a class of distributed parameter systems: classical vs. neural network approach

Bartecki K.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2012

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Vol. 60, nr 3

651-660

EN

In this article, an approximation of the spatiotemporal response of a distributed parameter system (DPS) with the use of the principal component analysis (PCA) is considered. Based on a data obtained by the numerical solution of a set of partial differential equations, a PCA-based approximation procedure is performed. It consists in the projection of the original data into the subspace spanned by the eigenvectors of the data covariance matrix, corresponding to its highest eigenvalues. The presented approach is carried out using both the classical PCA method as well as two different neural network structures: two-layer feed-forward network with supervised learning (FF-PCA) and single-layer network with unsupervised, generalized Hebbian learning rule (GHA-PCA). In each case considered, the effect of the approximation model structure represented by the number of eigenvectors (or, in the neural case, units in the network projection layer) on the mean square approximation error of the spatiotemporal response and on the data compression ratio is analysed. As shown in the paper, the best approximation quality is obtained for the classical PCA method as well as for the FF-PCA neural approach. On the other hand, an adaptive learning method for the GHA-PCA network allows to use it in e.g. an on-line identification scheme.

13

Regional observation and sensors

El Jai A., Hamzaoui H.

International Journal of Applied Mathematics and Computer Science

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2009

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Vol. 19, no 1

5-14

EN

The purpose of this short paper is to provide original results related to the choice of the number of sensors and their supports for general distributed parameter systems. We introduce the notion of extended sensors and we show that the observation error decreases when the support of a sensor is widened. We also show that the observation error decreases when the number of sensors increases.

14

Mobile sensor routing for parameter estimation of distributed systems using the parallel tunneling method

Zięba T., Uciński D.

International Journal of Applied Mathematics and Computer Science

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2008

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Vol. 18, no 3

307-318

EN

The paper deals with the problem of optimal path planning for a sensor network with mutliple mobile nodes, whose measurements are supposed to be primarily used to estimate unknown parameters of a system modelled by a partial differential equation. The adopted framework permits to consider two- or three-dimensional spatial domains and correlated observations. Since the aim is to maximize the accuracy of the estimates, a general functional defined on the relevant Fisher information matrix is used as the design criterion. Central to the approach is the parameterization of the sensor trajectories based on cubic B-splines. The resulting finite-dimensional global optimization problem is then solved using a parallel version of the tunneling algorithm. A numerical example is included to clearly demonstrate the idea presented in the paper.

15

Configuring a sensor network for fault detection in distributed parameter systems

Patan M., Uciński D.

International Journal of Applied Mathematics and Computer Science

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2008

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Vol. 18, no 4

513-524

EN

The problem of fault detection in distributed parameter systems (DPSs) is formulated as that of maximizing the power of a parametric hypothesis test which checks whether or not system parameters have nominal values. A computational scheme is provided for the design of a network of observation locations in a spatial domain that are supposed to be used while detecting changes in the underlying parameters of a distributed parameter system. The setting considered relates to a situation where from among a finite set of potential sensor locations only a subset can be selected because of the cost constraints. As a suitable performance measure, the Ds-optimality criterion defined on the Fisher information matrix for the estimated parameters is applied. Then, the solution of a resulting combinatorial problem is determined based on the branch-and-bound method. As its essential part, a relaxed problem is discussed in which the sensor locations are given a priori and the aim is to determine the associated weights, which quantify the contributions of individual gauged sites. The concavity and differentiability properties of the criterion are established and a gradient projection algorithm is proposed to perform the search for the optimal solution. The delineated approach is illustrated by a numerical example on a sensor network design for a two-dimensional convective diffusion process.

16

Approximate controllability of infinite dimensional systems of the n-th order

Respondek J. S.

International Journal of Applied Mathematics and Computer Science

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2008

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Vol. 18, no 2

199-212

EN

The objective of the article is to obtain general conditions for several types of controllability at once for an abstract differential equation of arbitrary order, instead of conditions for a fixed order equation. This innovative approach was possible owing to analyzing the n-th order linear system in the Frobenius form which generates a Jordan transition matrix of the Vandermonde form. We extensively used the fact that the knowledge of the inverse of a Jordan transition matrix enables us to directly verify the controllability by Chen’s theorem. We used the explicit analytical form of the inverse Vandermonde matrix. This enabled us to obtain more general conditions for different types of controllability for infinite dimensional systems than the conditions existing in the literature so far. The methods introduced can be easily adapted to the analysis of other dynamic properties of the systems considered.