ebook img

Polynomial and Rational Matrices: Applications in Dynamical Systems Theory PDF

513 Pages·2007·3.922 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Polynomial and Rational Matrices: Applications in Dynamical Systems Theory

Communications and Control Engineering Publishedtitlesinclude: StabilityandStabilizationofInfinite IdentificationandControlUsingVolterraModels DimensionalSystemswithApplications FrancisJ.DoyleIII,RonaldK.Pearson Zheng-HuaLuo,Bao-ZhuGuo andBobatundeA.Ogunnaike andOmerMorgul Non-linearControlforUnderactuated NonsmoothMechanics(Secondedition) MechanicalSystems BernardBrogliato IsabelleFantoniandRogelioLozano NonlinearControlSystemsII RobustControl(Secondedition) AlbertoIsidori JürgenAckermann L2-GainandPassivityTechniques FlowControlbyFeedback inNonlinearControl OleMortenAamoandMiroslavKrsti´c ArjanvanderSchaft LearningandGeneralization ControlofLinearSystemswithRegulation (Secondedition) andInputConstraints MathukumalliVidyasagar AliSaberi,AntonA.Stoorvogeland ConstrainedControlandEstimation PeddapullaiahSannuti GrahamC.Goodwin,MaríaM.Seron RobustandH∞Control andJoséA.DeDoná BenM.Chen RandomizedAlgorithmsforAnalysis ComputerControlledSystems andControlofUncertainSystems EfimN.RosenwasserandBernhardP. RobertoTempo,GiuseppeCalafiore Lampe andFabrizioDabbene ControlofComplexandUncertainSystems SwitchedLinearSystems StanislavV.EmelyanovandSergeyK. ZhendongSunandShuzhiS.Ge Korovin SubspaceMethodsforSystemIdentification RobustControlDesignUsingH∞Methods TohruKatayama IanR.Petersen,ValeryA.Ugrinovski DigitalControlSystems andAndreyV.Savkin IoanD.LandauandGianlucaZito ModelReductionforControlSystemDesign MultivariableComputer-controlledSystems GoroObinataandBrianD.O.Anderson EfimN.RosenwasserandBernhardP.Lampe ControlTheoryforLinearSystems DissipativeSystemsAnalysisandControl HarryL.Trentelman,AntonStoorvogel (Secondedition) andMaloHautus BernardBrogliato,RogelioLozano, FunctionalAdaptiveControl BernhardMaschkeandOlavEgeland SimonG.FabriandVisakanKadirkamanathan AlgebraicMethodsforNonlinearControlSystems Positive1Dand2DSystems (Secondedition) TadeuszKaczorek GiuseppeConte,ClaudeH.Moogand AnnaMariaPerdon Tadeusz Kaczorek Polynomial and Rational Matrices Applications in Dynamical Systems Theory 123 TadeuszKaczorek,Prof.drhab.inz˙. InstituteofControlandIndustrialElectronics FacultyofElectricalEngineering WarsawUniversityofTechnology 00-662Warsaw ul.Koszykowa75m.19 Poland SeriesEditors E.D.Sontag·M.Thoma·A.Isidori·J.H.vanSchuppen BritishLibraryCataloguinginPublicationData Kaczorek,T.(Tadeusz),1932- Polynomialandrationalmatrices:applicationsin dynamicalsystemstheory.-(Communicationsandcontrol engineering) 1.Automaticcontrol-Mathematics2.Electrical engineering-Mathematics3.Matrices4.Linearsystems 5.Polynomials I.Title 629.8’312 ISBN-13:ISBN-13:9781846286049 ISBN-10:ISBN-10:1846286042 LibraryofCongressControlNumber:2006936878 CommunicationsandControlEngineeringSeriesISSN0178-5354 ISBN 978-1-84628-604-9 e-ISBN 1-84628-605-0 Printedonacid-freepaper ©Springer-VerlagLondonLimited2007 Apartfromanyfairdealingforthepurposesofresearchorprivatestudy,orcriticismorreview,as permittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicencesissued bytheCopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbe senttothepublishers. Theuseofregisterednames,trademarks,etc.inthispublicationdoesnotimply,evenintheabsenceof aspecificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandtherefore freeforgeneraluse. Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinfor- mationcontainedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsor omissionsthatmaybemade. 987654321 SpringerScience+BusinessMedia springer.com Preface This monograph covers the selected applications of polynomial and rational matrices to the theory of both continuous-time and discrete-time linear systems. It is an extended English version of its preceding Polish edition, which was based on the lectures delivered by the author to the Ph.D. students of the Faculty of Electrical Engineering at Warsaw University of Technology during the academic year 2003/2004. The monograph consists of eight chapters, an appendix and a list of references. Chapter 1 is devoted to polynomial matrices. It covers the following topics: basic operations on polynomial matrices, the generalised Bézoute theorem, the Cayley(cid:16)Hamilton theorem, elementary operations on polynomial matrices, the choosing of a basis for a space of polynomial matrices, equivalent polynomial matrices, reduced row matrices and reduced column matrices, the Smith canonical form of polynomial matrices, elementary divisors and zeros of polynomial matrices, similarity of polynomial matrices, the Frobenius and Jordan canonical forms, cyclic matrices, pairs of polynomial matrices, the greatest common divisors and the smallest common multiplicities of matrices, the generalised Bezoute identity, regular and singular matrix pencil decompositions, and the Weierstrass(cid:16)Kronecker canonical form of a matrix pencil. Rational functions and matrices are discussed in Chap. 2. With the basic definitions and operations on rational functions introduced at the beginning, the following issues are subsequently addressed: decomposition into the sum of rational functions, operations on rational matrices, the decomposition of a matrix into the sum of rational matrices, the inverse matrix of a polynomial matrix and its reducibility, the McMillan canonical form of rational matrices, the first factorization of rational matrices and the application of rational matrices in the synthesis of control systems. Chapter 3 addresses normal matrices and systems. A rational matrix is called normal if every non-zero minor of size 2 of the polynomial matrix of the denominator is divisible by the minimal polynomial of this matrix. It has been proved that a rational matrix is normal if and only if its McMillan polynomial is equal to the smallest common denominator of all the elements of the rational matrix. Further, the following issues are discussed: the fractional forms of normal vi Preface matrices, the sum and product of normal matrices, the inverse matrix of a normal matrix, the decomposition of normal matrices into the sum of normal matrices, the structural decomposition of normal matrices, the normalisation of matrices via feedback and electrical circuits as examples of normal systems. The problem of the realisation of normal matrices is addressed in Chap. 4. The problem formulation is provided; further the following issues are discussed: necessary and sufficient conditions for the existence of minimal and cyclic realisations, methods of computing the realisation with the state matrix in both the Frobenius and Jordan canonical forms, structural stability and the computation of the normal transfer function matrix Chapter 5 is devoted to normal singular systems. In particular it focuses on discrete singular systems, cyclic pairs of matrices, the normal inverse matrices of cyclic pairs, normal transfer matrices, reachability and cyclicity of singular systems, cyclicity of feedback systems, computation of equivalent standard systems for singular systems. It is shown that electrical circuits consisting of resistances and inductances or resistances and capacities, together with ideal voltage (current) sources, constitute examples of singular continuous-time systems. Both the Kalman decomposition and the structural decomposition of the transfer matrix are generalised to the case of singular systems. Polynomial matrix equations, both rational and algebraic, are discussed in Chap. 6. The chapter begins with unilateral polynomial equations with two unknown matrices. Subsequently the following issues are addressed: the computation of minimal degree solutions to matrix equations, bilateral polynomial equations, the computation of rational solutions to polynomial equations, matrix equations of the m-th order, the Kronecker product of matrices and its applications, and the methods for computing solutions to Sylvester and Lapunov matrix equations. Chapter 7, the last one, is devoted to the problem of realisation and perfect observers for linear systems. A new method for computing minimal realisation for a given improper transfer matrix is provided together with the existence conditions; subsequently the methods for computing full and reduced order observers, as well as functional perfect observers, for 1D and 2D systems are given. In Chap. 8 some new results (published and unpublished) are presented on positive linear discrete-time and continuous-time systems with delays: asymptotic and robust stability, reachability, minimum energy control and positive realisation problem. The Appendix contains some basic definitions and theorems pertaining to the controllability and observability of linear systems. The monograph contains some original results of the author, most of which have already been published. It is haped that this monograph will be of value to Ph.D. students and researchers from the field of control theory and circuit theory. It can be also recommended for undergraduates in electrical engineering, electronics, mechatronics and computer engineering. I would like to express my gratitude to Professors M. Bus(cid:225)owicz and J. Klamka, the reviewers of the Polish version of the book, for their valuable comments and Preface vii suggestions, which helped to improve this monograph. I also wish to thank my Ph.D. students, the first readers of the manuscript, for their remarks. I wish to extend my special thanks to my Ph.D. students Maciej Twardy, Konrad Markowski and Stefan Krzemi(cid:276)ski for their valuable help in the preparation of this English edition. T. Kaczorek Contents Notation.................................................................................................................xv 1PolynomialMatrices............................................................................................1 1.1 Polynomials...................................................................................................1 1.2 Basic Notions and Basic Operations on Polynomial Matrices.......................5 1.3 Division of Polynomial Matrices...................................................................9 1.4 Generalized Bezoute Theorem and the Cayley–Hamilton Theorem...........16 1.5 Elementary Operations on Polynomial Matrices.........................................20 1.6 Linear Independence, Space Basis and Rank of Polynomial Matrices........23 1.7. Equivalents of Polynomial Matrices...........................................................27 1.7.1 Left and Right Equivalent Matrices...................................................27 1.7.2 Row and Column Reduced Matrices..................................................30 1.8 Reduction of Polynomial Matrices to the Smith Canonical Form...............32 1.9 Elementary Divisors and Zeros of Polynomial Matrices.............................37 1.9.1 Elementary Divisors...........................................................................37 1.9.2 Zeros of Polynomial Matrices............................................................39 1.10 Similarity and Equivalence of First Degree Polynomial Matrices.............42 1.11 Computation of the Frobenius and Jordan Canonical Forms of Matrices..45 1.11.1 Computationof theFrobenius Canonical Formof a Square Matrix..............................................................................................45 1.11.2 Computation of the Jordan Canonical Form of a Square Matrix......47 1.12 Computation of Similarity Transformation Matrices.................................49 1.12.1 Matrix Pair Method..........................................................................49 1.12.2 Elementary Operations Method........................................................54 1.12.3 Eigenvectors Method........................................................................57 1.13 Matrices of Simple Structure and Diagonalisation of Matrices.................59 1.13.1 Matrices of Simple Structure............................................................59 1.13.2 Diagonalisation of Matrices of Simple Structure.............................61 1.13.3 Diagonalisation of anArbitrarySquare Matrix by the Useofa Matrix with Variable Elements........................................................65 1.14 Simple Matrices and Cyclic Matrices........................................................67 1.14.1 Simple Polynomial Matrices............................................................67 x Contents 1.14.2 Cyclic Matrices................................................................................69 1.15 Pairs of Polynomial Matrices.....................................................................75 1.15.1 Greatest Common DivisorsandLowest Common Multiplicities of Polynomial Matrices........................................................................75 1.15.2 Computationof Greatest CommonDivisorsof a Polynomial Matrix..............................................................................................77 1.15.3 Computation of Greatest Common Divisors and Smallest Common Multiplicities of Polynomial Matrices.............................................78 1.15.4 Relatively Prime Polynomial Matrices and the Generalised Bezoute Identity.............................................................................................84 1.15.5 Generalised Bezoute Identity...........................................................86 1.16 Decomposition of Regular Pencils of Matrices.........................................87 1.16.1 Strictly Equivalent Pencils...............................................................87 1.16.2WeierstrassDecomposition of Regular Pencils................................92 1.17 Decomposition of Singular Pencils of Matrices........................................95 1.17.1Weierstrass–KroneckerTheorem.....................................................95 1.17.2 Kronecker IndicesofSingularPencils and Strict Equivalenceof Singular Pencils.............................................................................102 2 RationalFunctions andMatrices...................................................................107 2.1 Basic Definitions and Operations on Rational Functions..........................107 2.2 Decomposition of a Rational Function into a Sum of Rational Functions.116 2.3 Basic Definitions and Operations on Rational Matrices............................124 2.4 Decomposition of Rational Matrices into a Sum of Rational Matrices.....128 2.5 The Inverse Matrix of a Polynomial Matrix and Its Reducibility..............132 2.6 Fraction Description of Rational Matrices and the McMillan Canonical Form..........................................................................................................136 2.6.1 Fractional Forms of Rational Matrices.............................................136 2.6.2 Relatively Prime Factorization of Rational Matrices.......................146 2.6.3Conversion of aRational Matrix intothe McMillan Canonical Form...............................................................................................152 2.7 Synthesis of Regulators.............................................................................155 2.7.1 SystemMatrices and theGeneralProblemof Synthesis of Regulators......................................................................................155 2.7.2 Setof RegulatorsGuaranteeing Given Characteristic Polynomials of a Closed-loop System ...............................................................159 3 Normal MatricesandSystems........................................................................163 3.1 Normal Matrices........................................................................................163 3.1.1 Definition of the Normal Matrix......................................................163 3.1.2 Normality of the Matrix [Is – A]-1 for a Cyclic Matrix....................164 3.1.3 Rational Normal Matrices................................................................168 3.2 Fraction Description of Normal Matrices..................................................170 3.3 Sum and Product of Normal Matrices and Normal Inverse Matrices........175 3.3.1 Sum and Product of Normal Matrices..............................................175 3.3.2 The Normal Inverse Matrix..............................................................180 3.4 Decomposition of Normal Matrices...........................................................182 Contents xi 3.4.1Decomposition of Normal Matrices into theSumofNormal Matrices.........................................................................................182 3.4.2 Structural Decomposition of Normal Matrices................................185 3.5 Normalisation of Matrices Using Feedback...............................................191 3.5.1 State-feedback .................................................................................191 3.5.2 Output-feedback...............................................................................197 3.6 Electrical Circuits as Examples of Normal Systems..................................200 3.6.1 Circuits of the Second Order............................................................200 3.6.2 Circuits of the Third Order...............................................................203 3.6.3 Circuits of the Fourth Order and the General Case..........................210 4 The Problem ofRealization ...........................................................................219 4.1 Basic Notions and Problem Formulation...................................................219 4.2 Existence of Minimal and Cyclic Realisations..........................................220 4.2.1 Existence of Minimal Realisations...................................................220 4.2.2 Existence of Cyclic Realisations......................................................224 4.3 Computation of Cyclic Realisations..........................................................226 4.3.1Computationofa RealisationwiththeMatrixAinthe Frobenius CanonicalForm..............................................................................226 4.3.2Computationofa Cyclic Realisationwith MatrixAinthe Jordan CanonicalForm..............................................................................232 4.4 Structural Stability and Computation of the Normal Transfer Matrix.......244 4.4.1 Structural Controllability of Cyclic Matrices...................................244 4.4.2 Structural Stability of Cyclic Realisation.........................................245 4.4.3Impact of the Coefficients oftheTransferFunction onthe System Description.....................................................................................247 4.4.4Computationof theNormal TransferMatrix on theBasis of Its Approximation...............................................................................249 5 Singularand Cyclic NormalSystems.............................................................255 5.1 Singular Discrete Systems and Cyclic Pairs..............................................255 5.1.1 Normal Inverse Matrix of a Cyclic Pair...........................................257 5.1.2 Normal Transfer Matrix...................................................................260 5.2 Reachability and Cyclicity.........................................................................264 5.2.1 Reachability of Singular Systems.....................................................264 5.2.2 Cyclicity of Feedback Systems........................................................267 5.3Computation of Equivalent Standard Systems for Linear Singular Systems.....................................................................................................272 5.3.1Discrete-time Systemsand Basic Notions ......................................272 5.3.2 Computation of Fundamental Matrices............................................276 5.3.3 Equivalent Standard Systems...........................................................279 5.3.4Continuous-time Systems ...............................................................282 5.4 Electrical Circuits as Examples of Singular Systems................................285 5.4.1 RL Circuits.......................................................................................285 5.4.2 RC Circuits.......................................................................................288 5.5 Kalman Decomposition.............................................................................291 5.5.1 Basic Theorems and a Procedure for System Decomposition..........291

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.