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1

Fractional order tube model reference adaptive control for a class of fractional order linear systems

Balaska Hanane, Ladaci Samir, Djouambi Abdelbaki, Schulte Horst, Bourouba Bachir

International Journal of Applied Mathematics and Computer Science

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2020

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Vol. 30, no. 3

501--515

EN

We introduce a novel fractional order adaptive control design based on the tube model reference adaptive control (TMRAC) scheme for a class of fractional order linear systems. By considering an adaptive state feedback control configuration, the main idea is to replace the classical reference model with a single predetermined trajectory by a fractional order performance tube guidance model allowing a set of admissible trajectories. Besides, an optimization problem is formulated to compute an on-line correction control signal within specified bounds in order to update the system performance while minimizing a control cost criterion. The asymptotic stability of the closed loop fractional order control system is demonstrated using an extension of the Lyapunov direct method. The dynamical performance of the fractional order tube model reference adaptive control (FOTMRAC) is compared with the standard fractional order model reference adaptive control (FOMRAC) strategy, and the simulation results show the effectiveness of the proposed control method.

2

Nowa wkładka igus zgodna z normami FDA i UE dla systemów liniowych

Świat Obrabiarek i Narzędzi

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2018

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R. 13, nr 9-10

22

PL

Zintegrowane, bezobsługowe i odporne na zużycie wkładki stosowane w łożyskach liniowych umożliwiają przesuwanie prowadnic liniowych bez smarowania. Firma igus aktualnie opracowała wkładkę zgodną z wymogami UE i FDA, wykonaną z wysokowydajnego tworzywa sztucznego iglidur A160, które jest idealne do specjalnych wymagań higienicznych przemysłu spożywczego. Charakteryzuje się on przede wszystkim długą żywotnością w przypadku wałków ze stali nierdzewnej zgodnej z normami FDA i EU oraz w obszarach mokrych.

3

State estimators and observers for continuous and discrete linear systems. Part 1. Differential asymptotic state estimators

Byrski J., Byrski W.

Science, Technology and Innovation

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2018

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Vol. 3, no. 2

62--68

EN

In the paper an overview of state estimators and state observers used in linear systems, will be presented. The state estimators and observers can be used in many applications like the state reconstruction for the control purposes or for the diagnosis and fault detection in technical processes or for the virtual measurements of inaccessible variables of the system as well as for the best filtration of the differential equation solution. As the standard most commonly the Kalman filter and Luenberger type observers are used. Although the Kalman filter guarantees optimal filtering quality of the state, reconstructed from the noisy measurements, both Kalman filter and the Luenberger observer guarantee only asymptotic quality of the real state changes and tracking, basing on the current measurements of the system output and input signals. Unfortunately, the value of the estimation error at any moment of time cannot be calculated. The discussion on differences between continuous and two types of discrete Kalman Filter will be presented. This paper is planned to be the introduction to presentation of another type of the state observers which have the structure given by the integral operators. Based on measurements of the system output and input signals on some predefined finite time interval, they can reconstruct, after this interval, the observed state exactly.

4

Drazin inverse matrix method for fractional descriptor discrete-time linear systems

Kaczorek T.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2016

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Vol. 64, nr 2

395--399

EN

The Drazin inverse of matrices is applied in order to find the solutions of the state equations of fractional descriptor discrete-time linear systems. The solution of the state equation is derived and the set of consistent initial conditions for a given set of admissible inputs is established. The proposed method is illustrated by a numerical example.

5

A new approach to the realization problem for fractional discrete-time linear systems

Kaczorek T.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2016

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Vol. 64, nr 1

9--14

EN

A new approach to the realization problem for fractional discrete-time linear systems is proposed. A procedure for computation of fractional realizations of given transfer matrices is presented and illustrated by numerical examples.

6

Positive and asymptotically stable time-varying continuous-time linear systems and electrical circuits

Kaczorek T.

Autobusy : technika, eksploatacja, systemy transportowe

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2016

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R. 17, nr 6

1362-1367

EN

The positivity and asymptotic stability of time-varying continuous-time linear systems and electrical circuits are addressed. Necessary and sufficient conditions for the positivity and asymptotic stability of the systems and electrical circuits are established. It is shown that there exists a large class of positive and asymptotically stable electrical circuits with time-varying parameters. Examples of positive and asymptotically stable electrical circuits are presented.

PL

W pracy są analizowane zagadnienia dodatniości i stabilności asymptotycznej ciągłych układów i obwodów elektrycznych o parametrach zmiennych w czasie. Podano warunki konieczne i wystarczające dodatniości i stabilności asymptotycznej tej klasy układów i obwodów elektrycznych. Zostanie wykazane, że istnieje obszerna klasa dodatnich i stabilnych asymptotycznie obwodów elektrycznych o zmiennych parametrach. Rozważania ogólne zostaną zilustrowane przykładami obwodów elektrycznych o zmiennych w czasie parametrach.

7

Sterowalność względem wyjścia dla systemu liniowego stacjonarnego o niepewnych parametrach

Oprzędkiewicz K.

Przegląd Elektrotechniczny

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2015

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R. 91, nr 7

133-137

PL

W pracy omówiono problem sterowalności względem wyjścia dla systemu liniowego, stacjonarnego, opisanego transmitancją operatorową z niepewnością parametryczną współczynników licznika i mianownika tej transmitancji. Sformułowano proste warunki sterowalności względem wyjścia dla rozważanej klasy systemów dynamicznych, bazujące na geometrycznej interpretacji zer i biegunów transmitancji. Wyniki zilustrowano przykładami obliczeniowymi.

EN

In the paper a controllability problem for a linear, time-invariant dynamic system with parametric uncertainty of transfer function elements is considered. Simple controllability conditions, based onto geometrical interpretation of roots and zeros of transfer function were formulated. Results were by a numerical examples depicted.

8

Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems with regular pencils

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 1

29--33

EN

The Drazin inverse of matrices is applied to find the solutions of the state equations of descriptor fractional discrete-time systems with regular pencils. An equality defining the set of admissible initial conditions for given inputs is derived. The proposed method is illustrated by a numerical example.

9

Metoda poprawy precyzji algorytmów wyznaczających gramiany sterowalności i obserwowalności

Raczyński D.

Zeszyty Naukowe. Elektryka / Politechnika Opolska

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2012

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z. 67

105-109

PL

Celem artykułu jest przedstawienie opracowanej metody poprawy precyzji algorytmów wyznaczających gramiany sterowalności i obserwowalności ciągłych stacjonarnych układów liniowych. W artykule porównano jakość opracowanych algorytmów dla metod: Bartels-Stewarta, Smitha, ADI oraz funkcji znaku.

EN

The article presents the developed methods to improve the precision of algorithms for designating the controllability and observability gramians of LTI systems. The article compares the quality of the developed algorithms for the methods: Bartels-Stewart, Smith, ADI and Sign Function.

10

Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains

Ostalczyk P.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 3

533-538

EN

Two description forms of a linear fractional-order discrete system are considered. The first one is by a fractional-order difference equation, whereas the second by a fractional-order state-space equation. In relation to the two above-mentioned description forms, stability domains are evaluated. Several simulations of stable, marginally stable and unstable unit step responses of fractional-order systems due to different values of system parameters are presented.

11

Enclosures for the solution set of parametric interval linear systems

Hladík M.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 3

561-574

EN

We investigate parametric interval linear systems of equations. The main result is a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for this case, comparing and refinement of both. We show that the latter bounds are not provable better, and that they are also sometimes too pessimistic. The presented form of both methods is suitable for combining them into one to get a more efficient algorithm. Some numerical experiments are carried out to illustrate performances of the methods.

12

An approach to the analysis of observability and controllability in nonlinear systems via linear methods

Zhirabok A., Shumsky A.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 3

507-522

EN

The paper is devoted to the problem of observability and controllability analysis in nonlinear dynamic systems. Both continuous- and discrete-time systems described by nonlinear differential or difference equations, respectively, are considered. A new approach is developed to solve this problem whose features include (i) consideration of systems with non-differentiable nonlinearities and (ii) the use of relatively simple linear methods which may be supported by existing programming systems, e.g.,Matlab. Sufficient conditions are given for nonlinear unobservability/uncontrollability analysis. To apply these conditions, one isolates the linear part of the system which is checked to be unobservable/uncontrollable and, if the answer is positive, it is examined whether or not existing nonlinear terms violate the unobservability/uncontrollability property.

13

Null-Controllability of Linear Systems on Time Scales

Pawłuszewicz E.

Acta Mechanica et Automatica

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2012

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Vol. 6, no. 4

50-55

EN

The purpose of the paper is to study the problem of controllability of linear control systems with control constrains, defined on a time scale. The obtained results extend the existing ones on any time domain. The set of values of admissible controls is a given closed and convex cone with nonempty interior and vertex at zero or is a subset of containing zero.

14

Project management in building process with uncertain tasks times

Bożejko W., Hejducki Z., Rajba P., Wodecki M.

Management and Production Engineering Review

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2011

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Vol. 2, No. 1

3-9

EN

In this paper there is presented a problem of scheduling of construction work in which certain projects must be executed. Every work consists of projects executed by separate teams. In a linear system the sequence of works is the same for every project. Uncertain tasks times are represented by fuzzy numbers or distribution of random variables. We present a tabu search algorithm and computational experiments which are aimed at checking the sustainability of set solutions.

15

Novel optimal recursive filter for state and fault estimation of linear stochastic systems with unknown disturbances

Khémiri K., Ben Hmida F., Ragot J., Gossa M.

International Journal of Applied Mathematics and Computer Science

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2011

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Vol. 21, no 4

629-637

EN

This paper studies recursive optimal filtering as well as robust fault and state estimation for linear stochastic systems with unknown disturbances. It proposes a new recursive optimal filter structure with transformation of the original system. This transformation is based on the singular value decomposition of the direct feedthrough matrix distribution of the fault which is assumed to be of arbitrary rank. The resulting filter is optimal in the sense of the unbiased minimum-variance criteria. Two numerical examples are given in order to illustrate the proposed method, in particular to solve the estimation of the simultaneous actuator and sensor fault problem and to make a comparison with the existing literature results.

16

New stability conditions for positive continuous-discrete 2D linear systems

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

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2011

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Vol. 21, no 3

521-524

EN

New necessary and sufficient conditions for asymptotic stability of positive continuous-discrete 2D linear systems are established. Necessary conditions for the stability are also given. The stability tests are demonstrated on numerical examples.

17

Stability of The Second Fornasini-Marchesini Type Model of Continuous-Discrete Linear Systems

Busłowicz M.

Acta Mechanica et Automatica

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2011

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Vol. 5, no. 4

17-21

EN

The problem of asymptotic stability of continuous-discrete linear systems is considered. Simple necessary conditions and two computer methods for investigation of asymptotic stability of the second Fornasini-Marchesini type model are given. The first method requires computation of the eigenvalue-loci of complex matrices, the second method requires computation of determinants of some matrices. Effectiveness of the methods is demonstrated on numerical example.

18

Reduction and Decomposition of Singular Fractional Discrete-Time Linear Systems

Kaczorek T.

Acta Mechanica et Automatica

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2011

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Vol. 5, no. 4

62-66

EN

Reduction of singular fractional systems to standard fractional systems and decomposition of singular fractional discrete-time linear systems into dynamic and static parts are addressed. It is shown that if the pencil of singular fractional linear discrete-time system is regular then the singular system can be reduced to standard one and it can be decomposed into dynamic and static parts The proposed procedures are based on modified version of the shuffle algorithm and illustrated by numerical examples.

19

Zagadnienia optymalnego sterowania w pracach Stanisława Trybuły

Grzybowski A., Porosiński Z., Szajowski K.J.

Matematyka Stosowana : matematyka dla społeczeństwa

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2010

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T. 11/52

97-116

PL

W pewnym okresie Trybuła zwrócił uwagę na zagadnienia adaptacyjnego sterowania (patrz [20, 21]). Wydaje się, że zainspirowała Go monografia Aoki [1]. Do tematu wrócił po dość długim czasie. Zauważył, że w literaturze zakłada się, iż zakłócenia w systemach stochastycznych mają charakter gaussowski, podczas gdy w praktyce sygnały, a więc i zakłócenia, są dyskretne. Przypominamy tutaj typowy model analizowany w tej serii prac jako, że współczesne zastosowania modeli liniowych w ekonomii i technice wymuszają sygnały zarówno typu ciągłego, jak i dyskretne. Ograniczymy się do szczegółowego przedstawienia konstrukcji sterowań bayesowskich przy kwadratowej funkcji kosztu i zakłóceniach z wykładniczej klasy, spełniających dodatkowe warunki nałożone na momenty. W konkluzji podajemy odsyłacze do prac, w których wyznaczono sterowania minimaksowe.

EN

The research of Trybuła on the Bayes and minimax control of the stochastic systems (see [20, 21]) has been inspired probably by the Aoki’s book [1]. He returned to the topic after relatively long period. He observed that the model of disturbances in the considered models by other authors was restricted to the gaussian variables. In the industrial practice however, there are linear stochastic systems with the additive noise which is discrete, well modeled by binomial and Poisson random variables. In this paper the unified approach to such systems with the disturbances belonging to the class of exponentially distributed random variables is presented. We believe that the modern economic and industrial application of control systems admit various form of noise, both continuous and discrete. The presentation will be limited to Bayes and minimax optimal controls for quadratic cost functions when the class of disturbances has some additional restriction. More advanced consideration and models closely related to presented here can be found in papers by Trybuła listed in [20] and the authors of this note. Some authors use various estimators of unknown parameters based on the observation of the process calling these approach an adapted control. The importance of the assumptions adopted by Trybuła and the coauthors have been recently recognized and further investigation is expected for the systems (see related papers [2], [3]).

20

Two implementations of the preconditioned conjugate gradient method on heterogeneous computing grids

Collignon T. P., Van Gijzen M. B.

International Journal of Applied Mathematics and Computer Science

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2010

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Vol. 20, no 1

109-121

EN

Efficient iterative solution of large linear systems on grid computers is a complex problem. The induced heterogeneity and volatile nature of the aggregated computational resources present numerous algorithmic challenges. This paper describes a case study regarding iterative solution of large sparse linear systems on grid computers within the software constraints of the grid middleware GridSolve and within the algorithmic constraints of preconditioned Conjugate Gradient (CG) type methods. We identify the various bottlenecks induced by the middleware and the iterative algorithm. We consider the standard CG algorithm of Hestenes and Stiefel, and as an alternative the Chronopoulos/Gear variant, a formulation that is potentially better suited for grid computing since it requires only one synchronisation point per iteration, instead of two for standard CG. In addition, we improve the computation-to-communication ratio by maximising the work in the preconditioner. In addition to these algorithmic improvements, we also try to minimise the communication overhead within the communication model currently used by the GridSolve middleware. We present numerical experiments on 3D bubbly flow problems using heterogeneous computing hardware that show lower computing times and better speed-up for the Chronopoulos/Gear variant of conjugate gradients. Finally, we suggest extensions to both the iterative algorithm and the middleware for improving granularity.