Studies in Systems, Decision and Control. Series editor Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland

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1 Studies in Systems, Decision and Control Volume 13 Series editor Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland

2 About this Series The series "Studies in Systems, Decision and Control" (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control- quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. More information about this series at

3 Tadeusz Kaczorek Krzysztof Rogowski Fractional Linear Systems and Electrical Circuits ABC

4 Tadeusz Kaczorek Faculty of Electrical Engineering Białystok University of Technology Białystok Poland Krzysztof Rogowski Faculty of Electrical Engineering Białystok University of Technology Białystok Poland ISSN ISSN (electronic) ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer Cham Heidelberg New York Dordrecht London c Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Preface This monograph covers some selected problems of positive and fractional electrical circuits composed of resistors, coils, capacitors and voltage (current) sources. The monograph consists of 8 chapters, 4 appendices and a list of references. Chapter 1 is devoted to fractional standard and positive continuous-time and discrete-time linear systems without and with delays. The state equations and their solutions of linear continuous-time and discrete-time linear systems are presented. Necessary and sufficient conditions for the internal and external positivity of the linear systems are given. Solutions of the descriptor standard and fractional linear systems using the Weierstrass Kronecker decomposition and Drazin inverse matrix method are also presented. In chapter 2 the standard and positive fractional electrical circuits are considered. The fractional electrical circuits in transient states are analyzed. The reciprocity theorem, equivalent voltage source theorem and equivalent current source theorem are presented. Descriptor linear electrical circuits and their properties are investigated in chapter 3. The Weierstrass Kronecker decomposition method and the shuffle algorithm method are discussed. The regularity, pointwise completeness and pointwise degeneracy of descriptor electrical circuits is analyzed. The descriptor fractional standard and positive electrical circuits are also investigated. Chapter 4 is devoted to the stability of fractional standard and positive linear electrical circuits. It is shown that the electrical circuits with resistances can be also unstable. The reachability, observability and recontsructability of fractional positive electrical circuits and their decoupling zeros are analyzed in chapter 5. Necessary and sufficient conditions for the reachability, observability and reconstructability are established and illustrated by examples of electrical circuits. The decompositions of the pairs (A, B) and(a, C) of the electrical circuits are given and the decoupling zeros of the positive electrical circuits are proposed. The fractional linear electrical circuits with feedbacks are considered in chapter 6. The zeroing of the state vector of electrical circuits by state and output-feedbacks is discussed. In chapter 7 the problem of minimum

6 VI Preface energy control for standard and fractional systems with and without bounded inputs has been solved. In chapter 8 the fractional continuous-time 2D linear systems described by the Roesser type models are investigated. The fractional derivatives and integrals of 2D functions are introduced. The descriptor fractional 2D model is proposed and its solution is derived. The standard fractional 2D Roesser type models are also investigated. In Appendix A some basic definitions and theorems on Laplace transforms and Z-transforms are given. The elementary row and column operations on matrices are recalled in Appendix B. In Appendix C some properties of the nilpotent matrices are given. Definition of Drazin inverse of matrices and its properties are presented in Appendix D. The monograph contains some original results of the authors, most of which have already been published. It is dedicated to scientists and Ph.D. students from the field of electrical circuits theory and control systems theory. We would like to express our gratitude to Professors Mikołaj Busłowicz, Krzysztof Latawiec and Wojciech Mitkowski for their invaluable remarks, comments and suggestions which helped to improve the monograph. Białystok, June 2014 Tadeusz Kaczorek Krzysztof Rogowski

7 Contents Preface... V 1 Fractional Differential Equations Definition of Euler Gamma Function and Its Properties Mittag-LefflerFunction DefinitionsofFractionalDerivative-Integral Riemann-Liouville Definition CaputoDefinition Solutions of the Fractional State Equation ofcontinuous-timelinearsystem PositivityoftheFractionalSystems External Positivity of the Fractional Systems Positive Continuous-Time Linear Systems with Delays Positive Linear Systems Consisting of n Subsystems withdifferentfractionalorders Linear Differential Equations with Different FractionalOrders Positive Fractional Systems with Different FractionalOrders Descriptor Fractional Continuous-Time Linear Systems Solution of the Descriptor Fractional Systems Drazin Inverse Method for the Solution of Fractional Descriptor Continuous-Time Linear Systems Definition of n-orderdifference State Equations of the Discrete-Time Fractional Linear Systems FractionalSystemswithoutDelays FractionalSystemswithDelays... 31

8 VIII Contents 1.12 Solution of the State Equations of the Fractional Discrete-TimeLinearSystem FractionalSystemswithDelays Fractional Systems with Delays in State Vector FractionalSystemswithoutDelays PositiveFractionalLinearSystems ExternallyPositiveFractionalSystems Fractional Different Orders Discrete-Time Linear Systems Positive Fractional Different Orders Discrete-Time Linear Systems Descriptor Fractional Discrete-Time Linear Systems Solutionto thestateequation Positive Fractional Electrical Circuits FractionalElectricalCircuits PositiveFractionalElectricalCircuits Fractional R, C, e Type Electrical Circuits Fractional R, L, e Type Electrical Circuits Fractional R, L, C Type Electrical Circuits Analysis of the Fractional Electrical Circuits intransientstates ReciprocityTheoremforFractionalCircuits Equivalent Voltage Source Theorem and Equivalent CurrentSourceTheorem Descriptor Linear Electrical Circuits and Their Properties DescriptorLinearElectricalCircuits Regularity of Descriptor Electrical Circuits Pointwise Completeness of Descriptor Electrical Circuits Pointwise Degeneracy of Descriptor Electrical Circuits Descriptor FractionalLinearElectricalCircuits Polynomial Approach to Fractional Descriptor Electrical Circuits Positive Descriptor Fractional ElectricalCircuits Pointwise Completeness and Pointwise Degeneracy of Positive Fractional Descriptor Electrical Circuits Stability of Positive Standard Linear Electrical Circuits Stability of Positive Electrical Circuits Positive Unstable R, L, e ElectricalCircuits

9 Contents IX 4.3 Positive Unstable G, C, i s ElectricalCircuit Positive Unstable R, L, C, e Type Electrical Circuits Reachability, Observability and Reconstructability of Fractional Positive Electrical Circuits and Their Decoupling Zeros Decomposition of the Pairs (A, B) and(a, C) oflinear Circuits Reachability of Positive Electrical Circuits Observability of Positive Electrical Circuits Constructability of Positive Electrical Circuits Decomposition of the Positive Pair (A, B) Decomposition of the Positive Pair (A, C) Decoupling Zeros of the Positive Electrical Circuits Reachability of Positive Fractional Electrical Circuits Observability of Positive Fractional Electrical Circuits Standard and Fractional Linear Circuits with Feedbacks Linear Dependence on Time of State Variable in Standard ElectricalCircuitswithStateFeedbacks Zeroing of the State Vector of Standard Circuits by State-Feedbacks Zeroing of the State Vector of Standard Circuits by Output-Feedbacks Zeroing of the State Vector of Fractional Electrical CircuitsbyState-Feedbacks Minimum Energy Control of Electrical Circuits Minimum Energy Control of Positive Standard Electrical Circuits Minimum Energy Control of Fractional Positive Electrical Circuits Minimum Energy Control of Fractional Positive Electrical Circuits with Bounded Inputs Fractional 2D Linear Systems Described by the Standard and Descriptor Roesser Model with Applications Fractional Derivatives and Integrals of 2D Functions Descriptor Fractional 2D Roesser Model and Its Solution Fractional-Order Model of the Long Transmission Line Standard Fractional 2D Roesser Model and Its Solution GeneralizationofCayley-HamiltonTheorem

10 X Contents A Laplace Transforms of Continuous-Time Functions and Z-Transforms of Discrete-Time Functions A.1 Convolutions of Continuous-Time and Discrete-Time FunctionsandTheirTransforms A.2 LaplaceTransformsofDerivative-Integrals A.3 Laplace Transforms of Two-Dimensional Fractional Differintegrals A.4 Z-TransformsofDiscrete-TimeFunctions B Elementary Operations on Matrices C Nilpotent Matrices D Drazin Inverse Matrix D.1 Definition and Properties of Drazin Inverse Matrix D.2 Procedure for Computation of Drazin Inverse Matrices Bibliography Index

11 List of Symbols Γ (x) R R + C N Z Z + R n m the Euler gamma function the set of real numbers the set of real nonnegative numbers the set of complex numbers the set of natural numbers the set of integers the set of nonnegative integers the set of n m real matrices R n = R n 1 the set of n-components real vectors R n m + the set of n m real matrices with nonnegative entries R n + = Rn 1 + the set of n-rows real vectors with nonnegative components I n the n n identity matrix M n the set of n n Metzler matrices a A a is an element of the set A deta determinant of the matrix A ranka rank of the matrix A AdjA adjoint matrix degp (s) degree of polynomial (polynomial matrix) P (s)

12 XII List of Symbols inverse matrix Drazin inverse matrix transpose matrix ImA image of the matrix A kera kernel of the matrix A n! the factorial of natural number n diag[a 1,...,a n ] diagonal matrix with a 1,...,a n on diagonal R(x) real part of complex number x I(x) imaginary part of complex number x E α (z) one parameter Mittag-Leffler function of z E α,β (z) two parameters Mittag-Leffler function of z 0It n f(t) n-multiple integral of the function f(t) on the interval (0,t) 0It α f(t) Riemann-Liouville fractional (α-order) integral RL 0 Dt αf(t) Riemmann-Liouville fractional (α-order) derivative-integral C 0 Dα t f(t) Caputo fractional (α-order) derivative-integral 0Δ n k x i n-order backward difference of x i on the interval [0,k] L [f(t)] = F (s) Laplace transform of the function f(t) Z [f k ]=F(z) Z-transform of the function f k f(t) = df(t) dt first order derivative of continuous-time function f(t) A 1 A D A T f (n) (t) = dn f(t) dt n n-order derivative of continuous-time function f(t) f(t) g(t) the convolution of functions f(t) and g(t) δ(t) Dirac impulse function 1(t) unit step function