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Journal of the University of Zielona Góra

International Journal of Applied Mathematics and Computer Science

This paper considers a method of designing fractional-order observers for continuous-time linear fractional-order systems with unknown inputs. Conditions for the existence of these observers are given. Sufficient conditions for the asymptotical stability of fractional-order observer errors with the fractional order α satisfying 0 &lt; α &lt; 2 are derived in terms of linear matrix inequalities. Two numerical examples are given to demonstrate the applicability of the proposed approach, where the fractional order α belongs to 1≤α&lt;2 and 0&lt;α≤1, respectively. A stability analysis of the fractional-order error system is made and it is shown that the fractional-order observers are as stable as their integer order counterpart and guarantee better convergence of the estimation error.

Abstract

Faculty of Science, Technology and Communication (FSTC) University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg

Research Center for Automatic Control of Nancy (CRAN UMR, 7039, CNRS) University of Lorraine, IUT de Longwy, 186 rue de Lorraine, 54400 Cosnes et Romain, France

Keywords 

<article-title>Design of unknown input fractional-order observers for fractional-order systems</article-title>

Design of unknown input fractional-order observers for fractional-order systems

{"article-title":"Design of unknown input fractional-order observers for fractional-order systems"}

Fractional positive asymptotically stable continuous-time linear systems are approximated by fractional positive asymptotically stable discrete-time systems using a linear Padé-type approximation. It is shown that the approximation preserves the positivity and asymptotic stability of the systems. An optional system approximation is also discussed.

Faculty of Electrical Engineering Białystok Technical University, ul. Wiejska 45D, 15-351 Białystok, Poland

<article-title>Approximation of fractional positive stable continuous-time linear systems by fractional positive stable discrete-time systems</article-title>

Approximation of fractional positive stable continuous-time linear systems by fractional positive stable discrete-time systems

A simple semi-recursive routine for nonlinearity recovery in Hammerstein systems is proposed. The identification scheme is based on the Haar wavelet kernel and possesses a simple and compact form. The convergence of the algorithm is established and the asymptotic rate of convergence (independent of the input density smoothness) is shown for piecewise- Lipschitz nonlinearities. The numerical stability of the algorithm is verified. Simulation experiments for a small and moderate number of input-output data are presented and discussed to illustrate the applicability of the routine.

Institute of Computer Engineering, Control and Robotics Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, Wrocław, Poland

<article-title>A simple scheme for semi-recursive identification of Hammerstein system nonlinearity by Haar wavelets</article-title>

A simple scheme for semi-recursive identification of Hammerstein system nonlinearity by Haar wavelets

A combined, parametric-nonparametric identification algorithm for a special case of NARMAX systems is proposed. The parameters of individual blocks are aggregated in one matrix (including mixed products of parameters). The matrix is estimated by an instrumental variables technique with the instruments generated by a nonparametric kernel method. Finally, the result is decomposed to obtain parameters of the system elements. The consistency of the proposed estimate is proved and the rate of convergence is analyzed. Also, the form of optimal instrumental variables is established and the method of their approximate generation is proposed. The idea of nonparametric generation of instrumental variables guarantees that the I.V. estimate is well defined, improves the behaviour of the least-squares method and allows reducing the estimation error. The method is simple in implementation and robust to the correlated noise.

Institute of Computer Engineering, Control and Robotics Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

<article-title>Nonparametric instrumental variables for identification of block-oriented systems</article-title>

Nonparametric instrumental variables for identification of block-oriented systems

The focus of this paper is to develop reliable observer and filtering techniques for finite-dimensional battery models that adequately describe the charging and discharging behaviors. For this purpose, an experimentally validated battery model taken from the literature is extended by a mathematical description that represents parameter variations caused by aging.

Chair of Mechatronics University of Rostock, Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany

<article-title>Nonlinear state observers and extended Kalman filters for battery systems</article-title>

Nonlinear state observers and extended Kalman filters for battery systems

{"article-title":"Nonlinear state observers and extended Kalman filters for battery systems"}

This paper is concerned with actuator fault detection in nonlinear systems in the presence of disturbances. A nonlinear unknown input observer is designed and the output estimation error is used as a residual for fault detection. To deal with the problem of high Lipschitz constants, a modified mean-value theorem is used to express the nonlinear error dynamics as a convex combination of known matrices with time-varying coefficients. Moreover, the disturbance attenuation is performed using a modified H∞ criterion. A sufficient condition for the existence of an unknown input observer is obtained using a linear matrix inequality formula, and the observer gains are obtained by solving the corresponding set of inequalities. The advantages of the proposed method are that no a priori assumption on the unknown input is required and that it can be applied to a large class of nonlinear systems. Performances of the proposed approach are shown through the application to a diesel engine model.

HEI, Engineer School, 13, rue de Toul, 59046, Lille, France

Control Laboratory (LAGIS) Lille 1 University, Rue Paul Langevin, 59655, Villeneuve d’Ascq, France

Automatic Control Laboratory (LAGIS) Lille 1 University, Rue Paul Langevin, 59655, Villeneuve d’Ascq, France

<article-title>Robust nonlinear observer design for actuator fault detection in diesel engines</article-title>

Robust nonlinear observer design for actuator fault detection in diesel engines

{"article-title":"Robust nonlinear observer design for actuator fault detection in diesel engines"}

Drinking Water Distribution Systems (DWDSs) play a key role in sustainable development of modern society. They are classified as critical infrastructure systems. This imposes a large set of highly demanding requirements on the DWDS operation and requires dedicated algorithms for on-line monitoring and control to tackle related problems. Requirements on DWDS availability restrict the usability of the real plant in the design phase. Thus, a proper model is crucial. Within this paper a DWDS multi-species quality model for simulation and design is derived. The model is composed of multiple highly inter-connected modules which are introduced to represent chemical and biological species and (above all) their interactions. The chemical part includes the processes of chloramine decay with additional bromine catalysis and reaction with nitrogen compounds. The biological part consists of both heterotrophic and chemo-autotrophic bacteria species. The heterotrophic bacteria are assumed to consume assimilable organic carbon. Autotrophs are ammonia oxidizing bacteria and nitrite oxidizing bacteria species which are responsible for nitrification processes. Moreover, Disinfection By-Products (DBPs) are also considered. Two numerical examples illustrate the derived model’s behaviour in normal and disturbance operational states.

Department of Control Systems Engineering Gdańsk University of Technology, ul. G. Narutowicza 11/12, 80-233 Gdańsk, Poland

Department of Electronic, Electrical and Computer Engineering, College of Engineering and Physical Sciences University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

<article-title>A biochemical multi-species quality model of a drinking water distribution system for simulation and design</article-title>

A biochemical multi-species quality model of a drinking water distribution system for simulation and design

Considered is the control and stabilizability of a slowly rotating non-homogeneous Timoshenko beam with the aid of a torque. It turns out that the beam is (approximately) controllable with the aid of the torque if and only if it is (approximately) controllable. However, the controllability problem appears to be a side-effect while studying the stabilizability. To build a stabilizing control one needs to go through the methods of correcting the operators with functionals so that they have finally the appropriate form and the results on C0-continuous semigroups may be applied.

Institute of Mathematics Szczecin University, ul. Wielkopolska 15, 70-451 Szczecin, Poland

<article-title>Controlling a non-homogeneous Timoshenko beam with the aid of the torque</article-title>

Controlling a non-homogeneous Timoshenko beam with the aid of the torque

{"article-title":"Controlling a non-homogeneous Timoshenko beam with the aid of the torque"}

This paper deals with two important practical problems in motion control of robot manipulators: the measurement of joint velocities, which often results in noisy signals, and the uncertainty of parameters of the dynamic model. Adaptive output feedback controllers have been proposed in the literature in order to deal with these problems. In this paper, we prove for the first time that Uniform Global Asymptotic Stability (UGAS) can be obtained from an adaptive output feedback tracking controller, if the reference trajectory is selected in such a way that the regression matrix is persistently exciting. The new scheme has been experimentally implemented with the aim of confirming the theoretical results.

Division of Graduate Studies and Research Laguna Institute of Technology, Blvd Revolucion y Cuauhtemoc S/N, 27000, Torreon, Mexico

Department of Systems and Control National Polytechnic Institute (CITEDI), Avenida del Parque 1310, 22510, Tijuana, Mexico

<article-title>An adaptive output feedback motion tracking controller for robot manipulators: Uniform global asymptotic stability and experimentation</article-title>

An adaptive output feedback motion tracking controller for robot manipulators: Uniform global asymptotic stability and experimentation

In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic analysis and the theory of fuzzy sets.We assume that some parameters of the financial instrument cannot be precisely described and therefore they are introduced to the model as fuzzy numbers. Application of fuzzy arithmetic enables us to consider various sources of uncertainty, not only the stochastic one. To obtain the European call option pricing formula we use the minimal entropy martingale measure and Levy characteristics.

Systems Research Institute Polish Academy of Sciences, ul. Newelska 6, 01-447 Warsaw, Poland

The John Paul II Catholic University of Lublin, ul. Konstantynów 1H, 20-708 Lublin, Poland

<article-title>A fuzzy approach to option pricing in a Levy process setting</article-title>

A fuzzy approach to option pricing in a Levy process setting

{"article-title":"A fuzzy approach to option pricing in a Levy process setting"}

In this article, a new class of the epoch-incremental reinforcement learning algorithm is proposed. In the incremental mode, the fundamental TD(0) or TD(λ) algorithm is performed and an environment model is created. In the epoch mode, on the basis of the environment model, the distances of past-active states to the terminal state are computed. These distances and the reinforcement terminal state signal are used to improve the agent policy.

Faculty of Electrical and Computer Engineering Rzeszów University of Technology, Al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland

<article-title>Epoch-incremental reinforcement learning algorithms</article-title>

Epoch-incremental reinforcement learning algorithms

{"article-title":"Epoch-incremental reinforcement learning algorithms"}

While most of state-of-the-art image processing techniques were built under the so-called classical linear image processing, an alternative that presents superior behavior for specific applications comes in the form of Logarithmic Type Image Processing (LTIP). This refers to mathematical models constructed for the representation and processing of gray tones images. In this paper we describe a general mathematical framework that allows extensions of these models by various means while preserving their mathematical properties. We propose a parametric extension of LTIP models and discuss its similarities with the human visual system. The usability of the proposed extension model is verified for an application of contrast based auto-focus in extreme lighting conditions. The closing property of the named models facilitates superior behavior when compared with state-of-the-art methods.

Image Processing and Analysis Laboratory University Politehnica of Bucharest, Splaiul Independenţei 313, Bucharest, Romania

<article-title>Parametric logarithmic type image processing for contrast based auto-focus in extreme lighting conditions</article-title>

Parametric logarithmic type image processing for contrast based auto-focus in extreme lighting conditions

The aim of works described in this article is to elaborate and experimentally evaluate a consistent method of Language Model (LM) construction for the sake of Polish speech recognition. In the proposed method we tried to take into account the features and specific problems experienced in practical applications of speech recognition in the Polish language, reach inflection, a loose word order and the tendency for short word deletion. The LM is created in five stages. Each successive stage takes the model prepared at the previous stage and modifies or extends it so as to improve its properties. At the first stage, typical methods of LM smoothing are used to create the initial model. Four most frequently used methods of LM construction are here. At the second stage the model is extended in order to take into account words indirectly co-occurring in the corpus. At the next stage, LM modifications are aimed at reduction of short word deletion errors, which occur frequently in Polish speech recognition. The fourth stage extends the model by insertion of words that were not observed in the corpus. Finally the model is modified so as to assure highly accurate recognition of very important utterances. The performance of the methods applied is tested in four language domains.

Institute of Informatics Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Department of Systems and Computer Networks Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

<article-title>Pipelined language model construction for Polish speech recognition</article-title>

Pipelined language model construction for Polish speech recognition

{"article-title":"Pipelined language model construction for Polish speech recognition"}

The Robinson-Foulds (RF) distance is the most popular method of evaluating the dissimilarity between phylogenetic trees. In this paper, we define and explore in detail properties of the Matching Cluster (MC) distance, which can be regarded as a refinement of the RF metric for rooted trees. Similarly to RF, MC operates on clusters of compared trees, but the distance evaluation is more complex. Using the graph theoretic approach based on a minimum-weight perfect matching in bipartite graphs, the values of similarity between clusters are transformed to the final MC-score of the dissimilarity of trees. The analyzed properties give insight into the structure of the metric space generated by MC, its relations with the Matching Split (MS) distance of unrooted trees and asymptotic behavior of the expected distance between binary n-leaf trees selected uniformly in both MC and MS (Θ(n3/2)).

Department of Algorithms and System Modeling, Faculty of Electronics, Telecommunications and Informatics Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland

<article-title>On a matching distance between rooted phylogenetic trees</article-title>

On a matching distance between rooted phylogenetic trees

{"article-title":"On a matching distance between rooted phylogenetic trees"}

The Linear Canonical Transform (LCT) is a four parameter class of integral transform which plays an important role in many fields of signal processing. Well-known transforms such as the Fourier Transform (FT), the FRactional Fourier Transform (FRFT), and the FreSnel Transform (FST) can be seen as special cases of the linear canonical transform. Many properties of the LCT are currently known but the extension of FRFTs and FTs still needs more attention. This paper presents a modified convolution and product theorem in the LCT domain derived by a representation transformation in quantum mechanics, which seems a convenient and concise method. It is compared with the existing convolution theorem for the LCT and is found to be a better and befitting proposition. Further, an application of filtering is presented by using the derived results.

Electronics and Communication Engineering Section, Yadavindra College of Engineering Punjabi University Guru Kashi Campus, Talwandi Sabo-151302, Punjab, India

Department of Electronics and Communication Engineering Thapar University, Patiala-147001, Punjab, India

<article-title>A modified convolution and product theorem for the linear canonical transform derived by representation transformation in quantum mechanics</article-title>

A modified convolution and product theorem for the linear canonical transform derived by representation transformation in quantum mechanics

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Editor-in-Chief <A href="http://staff.uz.zgora.pl/jkorbicz/">Józef Korbicz</A>, University of Zielona Góra, Poland Deputy Editor <A href="http://staff.uz.zgora.pl/ducinski/viewpage.php?page_id=12">Dariusz Uciński</A>, University of Zielona Góra, Poland Associate Editors Stefan DOMEK, West Pomeranian University of Technology in Szczecin, Poland Miroslav FIKAR, Slovak University of Technology in Bratislava, Slovakia Marios M. POLYCARPOU, University of Cyprus, Nicosia, Cyprus Vicenç PUIG, Technical University of Catalonia, Barcelona, Spain Silvio SIMANI, University of Ferrara, Italy Jerzy STEFANOWSKI, Poznań University of Technology, Poland Guisheng ZHAI, Shibaura Institute of Technology, Tokyo, Japan Board Members Harald ASCHEMANN, University of Rostock, Germany Cherukuri ASWANI KUMAR, VIT University, Vellore, India Czesław BAJER, Polish Academy of Sciences, Warsaw, Poland Andrzej BARTOSZEWICZ, Technical University of Łódź, Poland Marek BODNAR, University of Warsaw, Poland Paolo CASTALDI, University of Bologna, Italy Zhaohui CEN, Qatar Enviroment and Energy Research Institute, Ar Rayyan, Qatar Jérôme CIESLAK, University of Bordeaux, France Julio CLEMPNER, National Polytechnic Institute, Mexico City, Mexico Bogusław CYGANEK, AGH University of Science and Technology, Cracow, Poland Andrzej DZIELIŃSKI, Warsaw University of Technology, Poland Anna FABIJAŃSKA, Lodz University of Technology, Poland Marcin GORAWSKI, Silesian University of Technology, Gliwice, Poland Martin GUGAT, Friedrich-Alexander University of Erlangen-Nuremberg, Germany Xiao HE, Tsinghua University, Beijing, China Bin JIANG, Nanjing University of Aeronautics and Astronautics, China Janusz KACPRZYK, Polish Academy of Sciences, Warsaw, Poland Jerzy KLAMKA, Silesian University of Technology, Gliwice, Poland Jacek KLUSKA, Rzeszów University of Technology, Poland Joanna KOŁODZIEJ, Cracow University of Technology, Poland Jan M. KOŚCIELNY, Warsaw University of Technology, Poland Zdzisław KOWALCZUK, Gdańsk University of Technology, Poland Piotr KULCZYCKI, AGH University of Science and Technology, Cracow, Poland Marek KURZYŃSKI, Wrocław University of Technology, Poland Maciej KUSY, Rzeszów University of Technology, Poland Francisco-Ronay LÓPEZ-ESTRADA, Technological Institute of Tuxtla Gutiérrez, Mexico Maciej ŁAWRYŃCZUK, Warsaw University of Technology, Poland 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 Gang NIU, Tongji University, Shanghai, China Ronald J. PATTON, University of Hull, UK Witold PEDRYCZ, University of Alberta, Edmonton, Canada Piotr PORWIK, University of Silesia in Katowice, Poland Jianbin QIU, Harbin Institute of Technology, China Ewaryst RAFAJŁOWICZ, Wrocław University of Technology, Poland Rotislav RAZUMCHIK, Russian Academy of Sciences, Moscow, Russia Leszek RUTKOWSKI, Technical University of Częstochowa, Poland Andrey V. SAVCHENKO, National Research University HSE, Nizhny Novgorod, Russia Horst SCHULTE, HTW Berlin, Germany Piotr SKRZYPCZYŃSKI, Poznań University of Technology, Poland Roman SŁOWIŃSKI, Poznań University of Technology, Poland Florin STOICAN, University POLITEHNICA of Bucharest, Romania Andrzej ŚWIERNIAK, Silesian University of Technology, Gliwice, Poland Zoltán SZABÓ, Hungarian Academy of Sciences, Budapest, Hungary Ryszard TADEUSIEWICZ, AGH University of Science and Technology, Cracow, Poland Didier THEILLIOL, University of Lorraine, Nancy, France Haoping WANG, Nanjing University of Science and Technology, China Marcin WITCZAK, University of Zielona Góra, Poland Shen YIN, Harbin Institute of Technology, China Alexey ZHIRABOK, Far Eastern Federal University, Vladivostok, Russia Teresa ZIELIŃSKA, Warsaw University of Technology, Poland Jacek M. ZURADA, University of Louisville, USA Editorial Office Agnieszka Rożewska, University of Zielona Góra, Poland – Manager Agata Wiśniewska-Kubicka, University of Zielona Góra, Poland – Technical Editor Contact AMCS Editorial Office University of Zielona Góra, Institute of Control and Computation Engineering ul. prof. Z. Szafrana 2 65-516 Zielona Góra phone: +48683282506 e-mail: <A href="mailto:amcs@uz.zgora.pl">amcs@uz.zgora.pl</A> website: <A href="http://www.amcs.uz.zgora.pl/">www.amcs.uz.zgora.pl</A> Publisher De Gruyter Poland Bogumiła Zuga 32A Str. 01-811 Warsaw, Poland Acknowledgement Electronic publication of Volume 26 (2016, Nos. 1-4) has been financed with the funds of the Polish Ministry of Science and Higher Education for science promoting activities within the agreement no. 904/P-DUN/2016 <A href="http://www.nauka.gov.pl/en/"><IMG alt="" src="https://www.degruyter.com/staticfiles/images/mnisw_opt.jpg"></A>

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<DIV align=justify> The International Journal of Applied Mathematics and Computer Science is a quarterly published in Poland since 1991 by the University of Zielona Góra in partnership with the Lubuskie Scientific Society and De Gruyter Poland, under the auspices of the Committee on Automatic Control and Robotics of the Polish Academy of Sciences. The journal strives to meet the demand for the presentation of interdisciplinary research in various fields related to control theory, applied mathematics, scientific computing and computer science. In particular, it publishes high quality original research results in the following areas: <UL> <LI>modern control theory and practice, </LI> <LI>artificial intelligence methods and their applications, </LI> <LI>applied mathematics and mathematical optimisation techniques, </LI> <LI>mathematical methods in engineering, computer science, and biology.</LI> </UL> We are primarily interested in presenting theoretical and application-orientated full-length research papers dealing with the following topics: <UL> <LI>control theory, including optimal control, system identification, adaptive and robust control, multivariable control, and non-linear systems, </LI> <LI>dynamical systems, including spatiotemporal processes, control problems, state and parameter estimation, and sensor networks, </LI> <LI>fault detection and diagnosis, including model-based approaches, observers, and classifiers, </LI> <LI>fault-tolerant control, including the control of continuous-variable and quantised systems, </LI> <LI>robotics, including modelling and simulation, mobile robots, and optimal trajectory planning, </LI> <LI>mathematical modelling and simulation, including numerical algorithms, </LI> <LI>optimisation, including mathematical optimisation techniques, global optimisation, and evolutionary algorithms, </LI> <LI>classification and pattern recognition, </LI> <LI>artificial intelligence, including neural networks, knowledge engineering, reasoning and learning models, expert and decision support systems, fuzzy systems, and search methods, </LI> <LI>mathematical biology, </LI> <LI>applications in engineering and medicine. </LI> </UL> The editors welcome proposals for exchange between similar journals. Also, all persons interested in bringing out special issues or sections of AMCS are encouraged to contact the Editor-in-Chief. Specials may be published on any important and timely subject within the scope of the journal. All papers proposed for special issues or sections will be refereed and have to meet the same criteria for scientific quality as articles presented in regular issues. Also, all journal rules and guidelines refer to papers published in specials without any exceptions. ISSN: 1641-876X (print version) </DIV>

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