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Editor-in-Chief Scientific Secretary

Józef KORBICZ Dariusz UCI ´NSKI

University of Zielona Góra, University of Zielona Góra,

Institute of Control and Computation Engineering, Institute of Control and Computation Engineering, Poland, e-mail: J.Korbicz@issi.uz.zgora.pl Poland, e-mail: D.Ucinski@issi.uz.zgora.pl

Associate Editors Marian ADAMSKI

University of Zielona Góra, Poland Robert BABUŠKA

Delft University of Technology, The Netherlands

Stanisław BA ´NKA

Technical University of Szczecin, Poland Mikołaj BUSŁOWICZ

Technical University of Białystok, Poland Michael A. DEMETRIOU

Worcester Polytechnic Institute, USA Andras EDELMAYER

Hungarian Academy of Sciences, Budapest, Hungary

Abdelhaq EL JAI

University of Perpignan, France Krzysztof GAŁKOWSKI University of Zielona Góra, Poland Adam GRZECH

Wrocław University of Technology, Poland George W. IRWIN

Queen’s University Belfast, UK Janusz KACPRZYK

Polish Academy of Sciences, Warsaw, Poland Jerzy KLAMKA

Silesian University of Technology, Gliwice, Poland

Witold KOSI ´NSKI

Polish-Japanese Institute of Information Technology, Warsaw, Poland

Jan M. KO ´SCIELNY

Warsaw University of Technology, Poland Adam KOWALEWSKI

AGH University of Science and Technology, Cracow, Poland

Zdzisław KOWALCZUK

Gda´nsk University of Technology, Poland

Krzysztof KOZŁOWSKI

Pozna´n University of Technology, Poland Marek KURZY ´NSKI

Wrocław University of Technology, Poland Irena LASIECKA

University of Virginia, Charlottesville, USA Vyacheslav MAKSIMOV

Russian Academy of Sciences, Ural Branch, Ekaterinburg, Russia Krzysztof MALINOWSKI

Warsaw University of Technology, Poland Wojciech MITKOWSKI

AGH University of Science and Technology, Cracow, Poland

Henk NIJMEIJER

Eindhoven University of Technology, The Netherlands

Wilém NOVÁK

University of Ostrava, Czech Republic Stanisław OSOWSKI

Warsaw University of Technology, Poland Thomas PARISINI

University of Trieste, Italy Ronald J. PATTON University of Hull, UK Witold PEDRYCZ

University of Alberta, Edmonton, Canada Andrzej PIEGAT

Technical University of Szczecin, Poland Marios POLYCARPOU

University of Cyprus, Nicosia, Cyprus Ewaryst RAFAJŁOWICZ

Wrocław University of Technology, Poland Eric ROGERS

University of Southampton, UK Leszek RUTKOWSKI

Technical University of Cz˛estochowa, Poland

Jose SÁ da COSTA

Technical University of Lisbon, Portugal Dominique SAUTER

University of Henri Poincaré, Vandoeurve, France

Alexey E. SHUMSKY Pacific State Economic University, Vladivostok, Russia

Roman SŁOWI ´NSKI

Pozna´n University of Technology, Poland Pilar SOBREVILLA

Technical University of Catalonia, Barcelona, Spain

Jan SOKOŁOWSKI

University of Nancy, Vandoeuvre, France Tomasz SZMUC

AGH University of Science and Technology, Cracow, Poland

Andrzej ´SWIERNIAK Silesian University of Technology, Gliwice, Poland

Ryszard TADEUSIEWICZ

AGH University of Science and Technology, Cracow, Poland

Yonghong TAN

Guilin Universtity of Electronic Technology, China

Piotr TATJEWSKI

Warsaw University of Technology, Poland Krzysztof TCHO ´N

Wrocław University of Technology, Poland Donghua ZHOU

Tsinghua University, Beijing, China Enrique ZUAZUA

University Autonomous of Madrid, Spain Jacek M. ZURADA

University of Louisville, KY, USA

Frequency: Published four issues per year; two regular and two special issues prepared by invited editor(s).

Subscription: Annual Institutional Subscription Rate 2006: Europee 180.00. All other countries e 200.00. Prices include postage and are subject to change without notice. Subscriptions are accepted on a prepaid basis only and are entered on a calendar year basis. Issues are sent by standard mail (surface within Europe, and air delivery outside Europe).

To subscribe, send a cheque payable to the University of Zielona Góra or make a payment by a bank transfer to Kredyt Bank S.A., Zielona Góra, Poland, account No. (IBAN): 46150018101218100386550000, SWIFT-Adr. (BIC): KRDBPLPW (Postscript:

AMCS’2006 subscription).

Internet: The International Journal of Applied Mathematics and Computer Science is currently available on-line.

Author’s Kit, Scope & Profile, Subscription Notes, Editorial Information and Summaries of recent issues can be found at:

http://www.issi.uz.zgora.pl/amcs

Publishing and Editorial Office: University of Zielona Góra, Int. J. of Applied Mathematics and Computer Science, ul. Podgórna 50, 65–246 Zielona Góra, Poland

E-mail: amcs@issi.uz.zgora.pl Fax: (+48 68) 324 72 95

Tel.: (+48 68) 328 25 06

Copyright ©University of Zielona Góra, Poland. All rights reserved.

Copyright ©Lubuskie Scientific Society in Zielona Góra. All rights reserved.

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Aims and Scope

The INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND COMPUTER SCIENCE (AMCS)strives to meet the demand for the presentation of interdisciplinary research concerned with applications of mathematical methods to computer science and engineering. AMCS publishes high quality original research results in the following areas:

• mathematical methods in computer science and engineering,

• modern control theory and applications,

• artificial intelligence methods and their applications,

• applied mathematics and mathematical optimization approaches.

The scope of the journal is extensive. Topics include:

Control systems:optimal control, distributed parameter systems, system identification, adap- tive and robust control, non-linear systems, multi-dimensional systems.

Mathematical and numerical modelling:ordinary and partial differential equations, numerical algorithms, simulation, applications.

Artificial intelligence:artificial neural networks, fuzzy and expert systems, rough sets theory, evolutionary and genetic algorithms, search methods, cellular automata, soft computing, applications.

Knowledge processing: knowledge elicitation and acquisition, knowledge representation, expert systems, data mining, data basis, applications.

Fault detection, fault analysis and diagnostics:model-based systems, soft computing tech- niques, pattern recognition, state and parameter estimation, signal processing, applications.

Optimization: mathematical optimization techniques, global optimization, evolutionary and genetic algorithms.

Robotics: modelling and simulation, optimal control, controllability, parameter and state estimation, optimal trajectory planning, singularities.

TheInternational Journal of Applied Mathematics and Computer Science is issued quarterly. It is indexed by Zentralblatt MATH, Mathematical Reviews, Current Mathematical Publications, Math- SciNet, INSPEC, VINITI, and Applied Mechanics Reviews.

Financial support: This journal is supported by State Committee for Scientific Research in Poland.

TheInternational Journal of Applied Mathematics and Computer Scienceis published jointly by the University of Zielona Góra and the Lubuskie Scientific Society in Zielona Góra, Poland.

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Information for Authors

Types of contribution. Research articles, review papers and book reviews.

Submission by the Internet. Submitting the paper using the Internet considerably speeds up the submission procedure.

OnlyPDFfiles that are ready to be printed can be accepted. Go to our website http://www.issi.uz.zgora.pl/amcs select Submit now and follow the instructions.

Review process. All manuscript are sent to at least two independent reviewers to ensure both accuracy and relevance to the journal. The final decision on acceptance will be made by the Editor-in-Chief.

Text. Typewrite with double space. Headings and subheadings for different sections of the paper should be clearly indi- cated and numbered using Arabic numerals. The title page should include the name and affiliation of each author, and a full correspondence address, including fax and e-mail, for the contact author. Provide an informative 100- to 250-words abstract at the head of the manuscript, and 5–7 keywords.

References. All publications cited in the text should be included in the list of references following the text of the manu- script. In the text please refer to the author’s name and year of publication, e.g. (Pandolfi, 1998; 1999). Up to two authors can be mentioned in text references, e.g. (Burns and King, 1995; Rosenblum and Rovnyak, 1985); more than two au- thors should be shortened to the first author’s name with et al., e.g. (Banks et al., 2000). References should be listed alphabetically at the end of the manuscript and should conform to the examples below.

Journals: Banks H.T., del Rosario R.C.H. and Smith R.C. (2000): Reduced order model feedback control de- sign: Numerical implementation in a thin shell model. — IEEE Trans. Automat. Contr., Vol. 45, No. 7, pp. 1312–1324.

Pandolfi L. (1998): Dissipativity and Lur’e problem for parabolic boundary control systems. — SIAM J.

Contr. Optim., Vol. 36, No. 6, pp. 2061–2081.

Books: Burns J.A. and King B.B. (1995): Representation of feedback operators for hyperbolic systems, In: Compu- tation and Control IV (K.L. Bowers and J. Lund, Eds.). — Boston: Birkhäuser, pp. 57–74.

Rosenblum M. and Rovnyak J. (1985): Hardy Classes and Operator Theory. — New York: Oxford U.P.

Proceedings:Pandolfi L. (1999): Recent results on the Kalman-Popov-Yakubovich problem. — Proc. Int. Conf.Mathe- matics and its Applications, Yogyakarta, Indonesia, pp. 47–60.

Electronic publishing. In order to accelerate editing and printing, the authors of the accepted manuscript will be required to provide its final version by the Internet (source file). Most word processing formats are accepted but LATEX is preferred.

Voluntary page charge. Papers selected for publication in the Int. J. AMCSare subject to a voluntary Page Charge, which will be invoiced through the author to the author’s company, institution, or agency. Publication is not dependent on the payment of the voluntary Page Charge. 15 reprints and one sample copy of the journal are provided when the paper is published. Additional reprints and journal copies can be ordered from the Editorial Office.

Editorial Office Prof. Józef Korbicz

Int. J. Applied Mathematics and Computer Science

University of Zielona Góra Phone: (+48 68) 328 24 22 or 324 72 95

Institute of Control and Computation Engineering Fax: (+48 68) 324 72 95

ul. Podgórna 50, 65–246 Zielona Góra, Poland E-mail: amcs@issi.uz.zgora.pl

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Contents

Special Section

Tatjewski P. and Ławry ´nczuk M. Soft computing in model-based predictive control . . . 7

Ko´scielny J.M. and Syfert M. Fuzzy diagnostic reasoning that takes into account the uncertainty of the relation between faults and symptoms . . . 27

Piegat A. What is not clear in fuzzy control systems? . . . 37

Kosi ´nski W. On fuzzy number calculus . . . 51

Kowalczuk Z. and Białaszewski T. Niching mechanisms in evolutionary computations . . . 59

Witczak M. Advances in model-based fault diagnosis with evolutionary algorithms and neural networks . . . 85

Kasi ´nski A. and Ponulak F. Comparison of supervised learning methods for spike time coding in spiking neural networks . . . 101

Regular Section Shumsky A. and Zhirabok A. Nonlinear diagnostic filter design: Algebraic and geometric points of view . . . 115

Sierociuk D. and Dzieli ´nski A. Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation . . . 129

Kaczorek T. An extension of the Cayley-Hamilton theorem for nonlinear time-varying systems . . . 141

Kisilewicz J.W. and Grzybowski A. Optimal erasures in decision-feedback equalization for the Gaussian noise . . . 147

Hnatkowska B., Huzar Z. and Tuzinkiewicz L. Refinement of UML collaborations . . . 155

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