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2013 | Book

Realization Problem for Descriptor Positive Fractional Continuous-Time Linear Systems Read chapter Read first chapter

Advances in the Theory and Applications of Non-integer Order Systems

5th Conference on Non-integer Order Calculus and Its Applications, Cracow, Poland

Editors: Wojciech Mitkowski, Janusz Kacprzyk, Jerzy Baranowski

Publisher: Springer International Publishing

Book Series : Lecture Notes in Electrical Engineering

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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About this book

This volume presents various aspects of non-integer order systems, also known as fractional systems, which have recently attracted an increasing attention in the scientific community of systems science, applied mathematics, control theory. Non-integer systems have become relevant for many fields of science and technology exemplified by the modeling of signal transmission, electric noise, dielectric polarization, heat transfer, electrochemical reactions, thermal processes, acoustics, etc. The content is divided into six parts, every of which considers one of the currently relevant problems. In the first part the Realization problem is discussed, with a special focus on positive systems. The second part considers stability of certain classes of non-integer order systems with and without delays. The third part is focused on such important aspects as controllability, observability and optimization especially in discrete time. The fourth part is focused on distributed systems where non-integer calculus leads to new and interesting results. The next part considers problems of solutions and approximations of non-integer order equations and systems. The final and most extensive part is devoted to applications. Problems from mechatronics, biomedical engineering, robotics and others are all analyzed and solved with tools from fractional systems. This volume came to fruition thanks to high level of talks and interesting discussions at RRNR 2013 - 5th Conference on Non-integer Order Calculus and its Applications that took place at AGH University of Science and Technology in Kraków, Poland, which was organized by the Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering.

Table of Contents

Frontmatter

Realization Problem

Frontmatter
Realization Problem for Descriptor Positive Fractional Continuous-Time Linear Systems
Abstract
The realization problem for descriptor positive fractional continuous-time linear systems with regular pencils is formulated and solved. Conditions for the existence of positive realizations of the descriptor fractional systems are established and procedures for computation of the realizations of improper transfer matrices are proposed. Effectiveness of the proposed procedures are demonstrated on numerical examples.
Tadeusz Kaczorek
Positive Stable Minimal Realization of Fractional Discrete-Time Linear Systems
Abstract
The positive stable minimal realization problem fractional discrete-time linear systems is formulated and a method for finding a positive stable minimal realization of a given proper transfer matrix is proposed. Sufficient conditions for the existence of a positive stable minimal realization of this class of linear systems are established. A procedure for computation of a positive stable realization is proposed and illustrated by a numerical example.
Łukasz Sajewski

Stability

Frontmatter
Frequency Domain Method for Stability Analysis of Linear Continuous-Time State-Space Systems with Double Fractional Orders
Abstract
The stability problem of continuous-time linear systems described by the state equation with double fractional orders has been considered. The frequency domain method for stability checking of the system with commensurate or non-commensurate orders has been given. The method proposed is based on the Argument Principle. The considerations are illustrated by numerical examples.
Mikołaj Busłowicz
Stability of Fractional Difference Systems with Two Orders
Abstract
In the paper we study the stability of nonlinear systems with the Caputo fractional difference with two orders. The Lyapunov direct method is used to analyze the stability of a system. The sufficient conditions for uniform stability and uniform asymptotic stability are presented.
Małgorzata Wyrwas, Ewa Girejko, Dorota Mozyrska, Ewa Pawłuszewicz
Stability Conditions of Fractional Discrete-Time Scalar Systems with Two Delays
Abstract
In the paper the stability problems of fractional discrete-time linear scalar systems with two delays are considered. Using the classical D-partition method boundaries of the stability regions in the parameter space are determined. Based on the stability regions new conditions for practical stability and for asymptotic stability are given.
Andrzej Ruszewski

Controllability, Observability and Optimal Control

Frontmatter
Constrained Controllability of h-Difference Linear Systems with Two Fractional Orders
Abstract
The problem of controllability in finite number of steps with control constrains of h-difference linear control systems with two fractional orders is studied. There are considered systems with the Caputo type h-difference operators and with controls which values are from a given convex and bounded subset of the control space. Necessary and sufficient conditions for constrained controllability in finite number of steps are given.
Ewa Pawłuszewicz, Dorota Mozyrska
Observability of Positive Fractional-Order Discrete-Time Systems
Abstract
In the paper the positive linear discrete-time fractional-order (non-commensurate and commensurate order) systems described in the state space are considered. Definition and necessary and sufficient conditions for the positivity, observability are given and proven. The considerations are illustrated by a numerical example.
Wojciech Trzasko
Optimal Control Problem for Fractional Dynamic Systems – Linear Quadratic Discrete-Time Case
Abstract
Dynamic optimization problems for integer (not fractional) order systems have been widely considered in literature (see e.g. [6, 13, 18, 21]). Mathematical fundamentals of the fractional calculus are given in the monographs [22-24] and the fractional differential equations and their applications have been addressed in [17, 19]. The numerical simulation of the fractional order control systems has been investigated in [7]. One of the fractional discretization method has been presented in [20]. Some optimal control problems for fractional order systems have been investigated in [1-5, 11, 12, 27]. Fractional Kalman filter and its application have been addressed in [25, 26]. Some recent interesting results in fractional systems theory and its applications to standard and positive systems can be found in [14-16].
Andrzej Dzieliński, Przemysław M. Czyronis

Distributed Parameter Systems

Frontmatter
Stabilization of Wave Equation Using Standard/Fractional Derivative in Boundary Damping
Abstract
We discuss the problem of stabilization of wave equation by means of the standard or fractional derivative in boundary damping. The problem is being reduced to a selection between the proportional or fractional integrator of order 1 − α feedback controllers. The fractional integration leads to the strong asymptotic stability only, while the proportional feedback control can ensure the exponential stability. This means that exponential stability is not robust around the value α = 1. We shall discuss mathematical and control theory aspects of this fact.
Piotr Grabowski
Fundamental Solutions to the Central Symmetric Space-Time Fractional Heat Conduction Equation and Associated Thermal Stresses
Abstract
The space-time fractional heat conduction equation with the Caputo time fractional derivative and the Riesz fractional Laplace operator is investigated. The fundamental solutions to the Cauchy and source problems as well as associated thermal stresses are found in the case of spherical symmetry. The numerical results for temperature and stresses are presented graphically for various orders of space and time derivatives.
Yuriy Povstenko
Variable Order Fractional Isoperimetric Problem of Several Variables
Abstract
In this work we study three types of partial variable order fractional operators. Using integration by parts formulas for variable order fractional integrals, we prove necessary optimality condition of Euler–Lagrange type for multi-dimensional isoperimteric problem.
Tatiana Odziehjewicz
Mittag-Leffler Pattern in Anomalous Diffusion
Abstract
Various systems described by the bi-fractional Fokker-Planck-Smoluchowski equation display some very general and universal properties. These universal characteristics originate in the underlying competition between long jumps (fractional space derivative) and long waiting times (fractional time derivative). Using a few selected model examples the universal features of anomalous diffusion will be demonstrated.
Bartłomiej Dybiec

Solutions and Approximations

Frontmatter
Piecewise Affine Representation of Discrete in Time, Non-integer Order Systems
Abstract
The multi-model approach has been often used for modeling and control of physical processes in recent years, leading to a class of so-called switched systems. Their properties, particularly the stability, observability and controllability analysis, have become one of active research topics in control theory and applications. In the paper a method of modeling nonlinear, discrete in time, non-integer order systems by means of piecewise affine multi-models is proposed, and then the special cases of such models are described. The discussion is illustrated with results of simulation tests.
Stefan Domek
Exact Solution of Two-Term Nonlinear Fractional Differential Equation with Sequential Riemann-Liouville Derivatives
Abstract
In this paper we derive a general solution for a class of nonlinear sequential fractional differential equations (SFDEs) with Riemann -Liouville (R-L) derivatives of arbitrary order. The solution of such an equation exists in arbitrary interval (0,b], provided nonlinear term obeys the respective Lipschitz condition. We prove that each pair of stationary functions of the corresponding R-L derivatives leads to a unique solution in the weighted continuous functions space.
Marek Błasik, Małgorzata Klimek
Laguerre Polynomial Approximation of Fractional Order Linear Systems
Abstract
This paper presents a finite dimensional approximation of fractional order linear systems and its connection with transport equation. The main results show, that the linear fractional order system can be approximated by a finite number of linear differential equations. Appropriate error estimate in C 0 norm is presented. Next, the solution of fractional order linear system is presented as a linear functional of the solution of transport equation. This result establishes a connection between the semi groups theory and fractional system theory. Considerations are illustrated with simple example of fractional oscillator.
Piotr Bania, Jerzy Baranowski
Solutions of Systems with Two-Terms Fractional Difference Operators
Abstract
Systems with generalized two-terms fractional difference operators are discussed. By the choice of a certain kernel, these operators can be reduced to the standard fractional integrals and derivatives. We study existence of solutions to such systems.
Ewa Girejko, Dorota Mozyrska, Małgorzata Wyrwas
Comparison of h-Difference Fractional Operators
Abstract
We compare three different types of h-difference fractional operators: Grünwald-Letnikov, Caputo, Riemann-Liouville types of operators. There is introduced the formula for fundamental matrix of solutions for linear systems of h-difference fractional equations with Grünwald-Letnikov type operator while the one with Caputo type or Riemann-Liouville type is well known. We present new formulas for linear control systems with the mentioned operators.
Dorota Mozyrska, Ewa Girejko, Małgorzata Wyrwas

Applications

Frontmatter
Reflection Symmetry in Fractional Calculus – Properties and Applications
Abstract
In this paper we define Riesz type derivatives symmetric and anti-symmetric w.r.t. the reflection mapping in finite interval [a,b]. Functions determined in [a,b] are split into parts with well determined reflection symmetry properties in a hierarchy of intervals [a m ,b m ],  m ∈ ℕ, concentrated around an arbitrary point. For these parts - called the [J]-projections of function, we prove the representation and integration formulas for the introduced fractional symmetric and anti-symmetric integrals and derivatives. It appears that they can be reduced to operators determined in arbitrarily short subintervals [a m ,b m ]. The future application in the reflection symmetric fractional variational calculus and the generalization of previous results on localization of Euler-Lagrange equations are discussed.
Małgorzata Klimek, Maria Lupa
A General Fractional-Order Thermal Model for Buildings and Its Properties
Abstract
The paper presents a general model of the temperature dynamics in buildings. The modeling approach relies on thermodynamics, in particular heat transfer, principles. The model considers heat loses by conduction and ventilation and internal heat gains. The parameters of the model can be determined uniquely from the geometry of the building and thermal properties of construction materials. The model is described by fractional-order differential equations and is presented using state space notation. The stability property of the model is analyzed. An illustrative example is provided.
Pawel Skruch
Heat Transfer Modeling in Ceramic Materials Using Fractional Order Equations
Abstract
Using classic numerical methods in modeling of heat transfer in ceramic materials causes imprecision results. This paper presents the new way of modeling using fractional order equations. Obtained numerical results were compared with registered heat transfer distribution using infrared camera. Comparison shows that presented method may have much more accuracy.
Anna Obrączka, Jakub Kowalski
A Comparative Study of PI λ D μ Controller Approximations Exemplified by Active Magnetic Levitation System
Abstract
The PI λ D μ DFOC was examined when applied to the Active Magnetic Levitation System. This research is based on the Prof. Ivo Petras Toolbox for fractional controller synhesis. The point of interest is the PID configuration applied at the simulation and experimental stages. The search for the optimal controller form is dependent on the quality measure in the transition phase when the external excitation load is activated. The digital control experiment was carried out in the MATLAB/Simulink using a USB I/O board. The controller realisations are compared and discussed.
Adam Piłat
The Application of Fractional Order Differential Calculus for the Description of Temperature Profiles in a Granular Layer
Abstract
In this article we will present the findings of an actual experiment on the flow of air through a bulkhead filled with granular material. The determined temperature profiles in the discussed bulkhead at different external and internal temperatures will be compared to a numerical description based on fractional order differential calculus. Fractional order differential calculus is being used increasingly widely in construction, as its application facilitates obtaining more precise modelling.
Ewa Szymanek
Fractional-Order P2D β Controller for Uncertain Parameter DC Motor
Abstract
In this paper an uncertain-parameter DC motor controlled with the use of non integer order P2D β controller with uncertain-parameters is considered. For this system an analysis of BIBO (Bounded Input Bounded Output) stability with respect to uncertainty of plant’s parameters was done. Results were with an example depicted.
Wojciech Mitkowski, Krzysztof Oprzędkiewicz
Synchronization of the Chaotic Ikeda Systems of Fractional Order
Abstract
The paper considers the problem of synchronization of two fractional Ikeda delay systems via master/slave configuration with linear coupling. Using numerical simulations effects of fractional order and the coupling rate on synchronization is investigated. Simulations are performed using Ninteger Fractional Control Toolbox for MatLab.
Mikołaj Busłowicz, Adam Makarewicz
Analog Modeling of Fractional Switched-Order Derivatives: Experimental Approach
Abstract
The article presents experimental results of modeling switched-order integrators based on domino-ladder approximations of order 0.5 and 0.25. Results were obtained for increasing and decreasing the fractional order. As fractional order impedances, a half-order domino ladder impedance, and quarter-order domino ladder structure were used. The quarter-order impedance was implemented with using over 5000 discrete elements. The experimental circuits are based on switching scheme that is numerically identical to the second order type of fractional variable order derivative. Experimental results were analyzed and compared with numerical results.
Dominik Sierociuk, Michal Macias, Wiktor Malesza
Fractional-Order Models of the Ultracapacitors
Abstract
In the paper, dynamic behavior of the ultracapacitors is investigated and analyzed. The ultracapacitors are represented by equivalent electrical circuit models and mathematically described by fractional-order differential equations. The identification procedure is proposed to identify the parameters of the models. The results of numerical simulations are compared with the results measured experimentally in the physical system.
Pawel Skruch, Wojciech Mitkowski
Non-integer Order PI α D μ Control ICU-MM
Abstract
The article presents dynamical system model that describes glycemia. It is based on four differential equations that simulates glucose dynamics of traumatised patient’s blood (at ICU). Authors present description of basic model and method of tuning PI α D μ controller parameters based on the integrated absolute error as the performance index.
Waldemar Bauer, Jerzy Baranowski, Wojciech Mitowski
Comparison of Fractional- and Integer-Order Filters in Filtration of Myoelectric Activity Acquired from Biceps Brachii
Abstract
This study assesses the viability of filtration of myoelectric signal using fractional-order filters. We acquired raw EMG signal from m. biceps brachii during isometric maximal voluntary contraction from ten test subjects; tested conventional and fractional Butterworth filters of two order groups; and compared the results in terms of offline filtration.
Tomasz Moszkowski, Elzbieta Pociask
Variable-, Fractional-Order Dead-Beat Control of a Robot Arm
Abstract
In the paper a synthesis method of the variable-, fractional – order dead – beat controller is proposed. It is applied to control of a robot arm described as a simple integrating element. The system structure is presented. The transient characteristic of a closed – loop system with the proposed controller are measured and compared with computer simulations performed for classical controllers.
Piotr Duch, Maciej Łaski, Sylwester Błaszczyk, Piotr Ostalczyk
Backmatter
Metadata
Title
Advances in the Theory and Applications of Non-integer Order Systems
Editors
Wojciech Mitkowski
Janusz Kacprzyk
Jerzy Baranowski
Copyright Year
2013
Publisher
Springer International Publishing
Electronic ISBN
978-3-319-00933-9
Print ISBN
978-3-319-00932-2
DOI
https://doi.org/10.1007/978-3-319-00933-9