Linear Systems over Finite Fields - Startseite TU Ilmenau · Linear Systems over Finite Fields —...

Linear Systems over Finite Fields —

Modeling, Analysis, and Synthesis

Der Technischen Fakultät der

Universität Erlangen-Nürnberg

zur Erlangung des Grades

DOKTOR-INGENIEUR

vorgelegt von

Johann Reger

Erlangen 2004

Als Dissertation genehmigt vonder Technischen Fakultät derUniversität Erlangen-Nürnberg

Tag der Einreichung: 01. 06. 2004Tag der Promotion: 15. 07. 2004Dekan: Prof. Dr. rer. nat. A. WinnackerBerichterstatter: Prof. Dr.-Ing. T. Moor

Prof. Dr.-Ing. D. Abel

Vorwort

Die vorliegende Arbeit entstand während meiner Tätigkeit als Promotionsstipendiat und wissen-schaftlicher Assistent am Lehrstuhl für Regelungstechnikder Friedrich-Alexander-UniversitätErlangen-Nürnberg. Die Inspiration zur Arbeit gaben im wesentlichen die Vorarbeiten von HerrnProf. Dr.-Ing. Dieter Franke, dem das Verdienst gebührt, die Steuerungstechnik wieder in der Re-gelungstechnik verankert zu haben. Auf Herrn Prof. Arthur Gill geht ein reichhaltiger Fundus anErgebnissen zur Theorie linearer Schieberegister zurück,auf die ein Gutteil des Kapitels zur Ana-lyse gründet. Zahlreiche Anregungen zog ich auch aus den Arbeiten von Herrn Prof. Dr. DieterBochmann zum Booleschen Differentialkalkül und von Herrn Prof. William Wolovich zur Poly-nommatrixmethode; sie finden sich in den Kapiteln zur Modellbildung und Synthese wieder.

An erster Stelle bedanke ich mich bei Herrn Prof. Dr.-Ing. Thomas Moor für die Übernahme desReferats, ebenso herzlich bei Herrn Prof. Dr.-Ing. Dirk Abel für die Übernahme des Korreferats.Herrn Prof. Dr.-Ing. Günter Roppenecker gilt mein Dank für die Übernahme des Prüfungsvorsitzes,meine besondere Wertschätzung aber für die Gewährung des nötigen Freiraums wie auch für diegroßzügige Bereitstellung von Mitteln zur Förderung des wissenschaftlichen Austauschs. HerrnProf. Dr. Hans Kurzweil danke ich für seine Hilfestellung inFragen der Algebra, nicht zuletzt aberfür sein Mitwirken im Prüfungskollegium. Hervorheben möchte ich auch die gute Zusammenarbeitmit meinen Kollegen am Lehrstuhl, dabei insbesondere mit den Herren Dipl.-Ing. Klaus Schmidt,Dr.-Ing. Joachim Deutscher und Dr.-Ing. Armin Schleußinger. Ferner danke ich allen Studenten,die mit ihren Arbeiten einen Beitrag zum Gelingen dieser Arbeit geleistet haben. Herrn Dipl.-Ing.Klaus Schmidt danke ich für die sorgfältige und kritische Durchsicht der Arbeit. Not at least, warmthanks to Dr. Jonathan Magee in Galway for polishing what wassupposed to be English.

Ein Dankeschön geht auch an die Studienstiftung des deutschen Volkes und die Deutsche For-schungsgemeinschaft, ohne deren Unterstützung, ideell wie finanziell, die Arbeit in dieser Formnicht möglich gewesen wäre.

Erlangen, im Juni 2004 Johann Reger

You’re a well paid scientistYou only talk in factsYou know you’re always right’Cause you know how to prove itStep by stepA PhD to show you’re smartWith textbook formulasBut you’re used upJust like a factory hand

The Dead Kennedys “Well Paid Scientist”

Table of Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1

1.2 Contribution of this Dissertation . . . . . . . . . . . . . . . . . .. . . . . . . . . 3

1.3 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 4

2 Mathematical Preliminaries 5

2.1 Fundamental Concepts of Group Theory . . . . . . . . . . . . . . . .. . . . . . . 6

2.2 Polynomials over Finite Fields . . . . . . . . . . . . . . . . . . . . .. . . . . . . 9

2.3 Linear Transformations and Matrices . . . . . . . . . . . . . . . .. . . . . . . . . 12

2.3.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Invariants of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . .. . 20

2.3.4 The Rational Canonical Form . . . . . . . . . . . . . . . . . . . . . .. . 20

2.4 An Image Domain for Finite Fields . . . . . . . . . . . . . . . . . . . .. . . . . . 22

2.4.1 TheA-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2 Table of Correspondences . . . . . . . . . . . . . . . . . . . . . . . .. . 23

3 Finite State Automata in the State Space 25

3.1 The Relation of Boolean Algebra with the Finite FieldF2 . . . . . . . . . . . . . . 28

3.2 Methods for Determining the State Space Model . . . . . . . . .. . . . . . . . . . 30

3.2.1 The Disjunctive Normal Form Method . . . . . . . . . . . . . . . .. . . . 31

3.2.2 The Reed-Muller Generator Matrix Method . . . . . . . . . . .. . . . . . 32

3.2.3 Deterministic State Space Model . . . . . . . . . . . . . . . . . .. . . . . 33

VIII TABLE OF CONTENTS

4 Analysis of Linear Systems over Finite Fields 37

4.1 Linear Modular Systems (LMS) . . . . . . . . . . . . . . . . . . . . . . .. . . . 39

4.2 Homogeneous LMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

4.2.1 Cyclic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.2 Nilpotent Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62

4.2.3 Arbitrary Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68

4.3 Inhomogeneous LMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71

4.3.1 Linearization by Translation . . . . . . . . . . . . . . . . . . . .. . . . . 71

4.3.2 Non-linearizable Parts . . . . . . . . . . . . . . . . . . . . . . . . .. . . 74

4.3.3 General Inhomogeneous LMS . . . . . . . . . . . . . . . . . . . . . . .. 78

4.3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Synthesis of Linear Systems over Finite Fields 83

5.1 Controllability of an LMS . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 84

5.1.1 Controllability Matrix and Controllability Indices. . . . . . . . . . . . . . 85

5.1.2 The Controllability Companion Form . . . . . . . . . . . . . . .. . . . . 86

5.2 Synthesis in the Image Domain . . . . . . . . . . . . . . . . . . . . . . .. . . . . 88

5.2.1 Linear State Feedback and its Structural Constraints. . . . . . . . . . . . 88

5.2.2 Controller Design in the Image Domain — why? . . . . . . . . .. . . . . 89

5.2.3 The Polynomial Matrix Fraction of the Transfer Matrix. . . . . . . . . . . 90

5.2.4 Synthesis Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96

5.2.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.6 Non-controllable Parts . . . . . . . . . . . . . . . . . . . . . . . . .. . . 102

5.2.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Conclusions and Future Work 115

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117

IX

Appendix 119

A Permutations of a Block Matrix 119

B The Transformation Matrix on Rational Canonical Form 121

C The Jordan Normal Form over an Extension Field ofFq 123

D General Solution of Linear Systems using Singular Inverses 131

E Rank Deficiency of a Matrix-Valued Polynomial Function 135

F Solving the Linear State Equation in the Image Domain 141

G List of Publications 143

References 145

X TABLE OF CONTENTS

Zusammenfassung

Im Mittelpunkt der Untersuchungen stehen dynamische Systeme, welche sich in einem diskretenZustandsraummodell über einem endlichen Körper abbilden lassen. Dazu werden Methoden zuralgebraischen Modellierung und, speziell für den linearenFall, strukturelle Verfahren zur Analyseund Synthese herausgearbeitet.

Ausgangspunkt der Modellbildung ist eine Tabelle, in der inAbhängigkeit jedes Zustands undEingangs entsprechende Nachfolgezustände verzeichnet sind. Auf dieser Grundlage werden zweiVerfahren aus der Booleschen Algebra bzw. Kodierungstheorie vorgestellt, welche die Berechnungeiner Zustandsübergangsfunktion gestatten, die ausschließlich auf den Operationen Konjunktionund exklusive Disjunktion beruht. Wie in der Arbeit gezeigt, ist diese Darstellung einer Darstel-lung über einem endlichen Körper mit zwei Elementen äquivalent, so daß damit auf einfache Weiseeine Zustandsübergangsfunktion über einem endlichen Körper gewonnen wird. Diese schließt auchden nicht-deterministischen Fall mehrerer Folgezuständeein und ermöglicht die algebraische Dar-stellung jedes endlichen Automaten. Als besonders einfacherweist sich dabei die Berechnung mitder Methode, welche auf der Verwendung einer sogenannten Reed-Muller-Generator-Matrix fußt.

Der Analyseteil der Arbeit stellt eine detaillierte Untersuchung des Zustandsübergangsverhaltensautonomer linearer Systeme über einem beliebigen endlichen Körper dar. Dabei wird auf die reich-haltige Theorie linearer Schaltkreise der sechziger Jahrezurückgegriffen, im Gegensatz dazu aberwird die Herleitung nicht auf tiefergehende Erkenntnisse über endliche Ringe gegründet, sondernauf fortgeschrittenen Kapiteln der linearen Algebra. Auf diesem Weg wird gezeigt, wie strukturel-le algebraische Eigenschaften der Systemdynamikmatrix, genauer die Perioden ihrer Elementar-teilerpolynome, das Übergangsverhalten von Zuständen bestimmen, womit ein notwendiges wiehinreichendes Kriterium zur Zerfällung des Zustandsraumsin zyklische und nicht-zyklische Un-terräume hergeleitet wird. Ist ein endlicher Automat als ein solches lineares System darstellbar,so wird damit zum ersten Mal ein Kriterium vorgestellt, welches alle Automatenzyklen in Längeund Anzahl liefert sowie das nicht-zyklische Übergangsverhalten beschreibt. Es ergibt sich alsoeine Methode, mit Hilfe derer sich der zugehörige Automatengraph alleine unter Verwendung derSystemdynamikmatrix eindeutig bestimmen läßt. Wie am Endedes Analyseteils ausgeführt, er-hält man ein dem entsprechendes Ergebnis auch für affin-lineare autonome Systeme über einemendlichen Körper.

XII ZUSAMMENFASSUNG

Mit dem Wissen um den Einfluß der Elementarteilerpolynome auf das zyklische Verhalten einesautonomen linearen Systems über einem endlichen Körper gelingt im letzten Teil der Arbeit erst-malig die Herleitung eines Verfahrens zur gezielten Vorgabe des zyklischen Verhaltens eines linea-ren Systems über einem endlichen Körper. Dazu eignen sich statische lineare Zustandsrückführun-gen, die aber, will man im Mehrgrößenfall alle Elementarteilerpolynome vorgeben und nicht nurdas charakteristische Polynom im geschlossenen Regelkreis, nicht mit klassischen Zeitbereichsver-fahren wie z. B. der vollständigen modalen Synthese ermittelt werden können. Ein demZ-Bereichentsprechender Bildbereich über einem endlichen Körper,A-Bereich genannt, erweist sich hier-bei als Schlüssel zur Lösung des Problems: die imA-Bereich erklärte Polynommatrixmethodegestattet gerade die Lösung dieses Vorgabeproblems. Die Lösung erfolgt in zwei Schritten: MitHilfe des Rosenbrockschen Kontrollstrukturtheorems wirddie Frage der Existenz einer Zustands-rückführung, welche die gewünschten Elementarteilerpolynome im geschlossenen Kreis realisiert,geklärt. Die Synthese der Rückführung selbst geschieht durch Umformung der Nennermatrix ei-ner rechtsprimen Polynommatrixzerlegung der Übertragungsmatrix bzgl. einer Zustandsdarstel-lung in Steuerbarkeitsnormalform. Die dazu nötigen Schritte beschreibt ein Algorithmus, der einederartige Nennermatrix für den geschlossenen Regelkreis liefert. Eine einfache Rechnung ergibtdann die gesuchte Matrix der Zustandsrückführung. Eine Diophantische Gleichung muß hierzunicht gelöst werden. Der zweite Abschnitt erweitert das Verfahren auf lineare Systeme mit nicht-steuerbarem Anteil. Hierzu wird eine an die Steuerbarkeitsnormalform angelehnte Darstellung ab-geleitet, die den steuerbaren und nicht-steuerbaren Systemanteil offenlegt. Zur Unterbindung desEinflusses seitens des Anfangszustands des nicht-steuerbaren Systemanteils werden steuerbarerund nicht-steuerbarer Systemanteil voneinander entkoppelt, was sich für beliebige lineare Syste-me mit nicht-steuerbarem Anteil als immer möglich herausstellt. Des weiteren wird ein neuartigesKriterium vorgestellt, mit Hilfe dessen sich genau entscheiden läßt, wann eine solche Entkoppe-lung die Elementarteilerpolynome der Systemdynamikmatrix beeinflußt — dies in Erweiterungdes bekannten Ergebnisses, daß eine derartige Entkoppelung das charakteristische Polynom derSystemdynamikmatrix des geschlossenen Kreises nicht zu verändern vermag. Am Ende dieserUntersuchungen steht eine Methode, welche die Anwendung des zuvor für steuerbare Systemeermittelten Algorithmus auf das steuerbare Teilsystem erlaubt, dabei aber das charakteristischePolynom des nicht-steuerbaren Teilsystems unberührt läßt.

Chapter 1

Introduction

Discrete event systems are characterized by a discrete state transition behavior which is driven byan asynchronous occurrence of discrete events rather than by the propagation of continuous time.To some extent, this behavior can be reconciled with the state transition behavior of a discrete timecontinuous system if one is content with giving up equidistant time ticks in favor of just counterinstants that indicate the occurrence of an event. In this respect, it seems promising to formulate astate space model for discrete event systems.

State space models are the dominant and successful paradigmfor representing continuous dynamicsystems. For the most part, this is due to the profound knowledge about linear algebra that capturesmany real world system properties in appropriate algebraicproperties. This may explain whyconsiderable effort has been made to setup a link between linear algebra and discrete event systems.

1.1 Motivation

Discrete space models using so-calledarithmetical polynomialshave been introduced for repre-senting a class of discrete event systems with a finite numberof states, i. e. finite state automata[Fra94, Fra96]. An other method employsWalsh functionsfor modeling deterministic finite stateautomata as autonomous linear systems [Son99, Son00]. There are major drawbacks in both ap-proaches. In order to point out some of them, consider an example system in the framework ofarithmetical polynomials1 with, for simplicity, an autonomous state equation ofn-th order

x(k+1) = Ax(k) . (1.1)

In this regard,A ∈Zn×n is the dynamics matrix,k∈N0 is a counter, and fork fixed,x(k)∈ Bn is astate vector with boolean entries, 1 and 0 only. The algebraic operations addition and multiplicationare understood in the usual sense.

1Similar arguments apply for the Walsh-function approach.

2 CHAPTER 1 — INTRODUCTION

Problem 1 Assume that the task is to calculate a statext ∈Bn which “returns” aftert ∈N counterinstances, i. e. a so-called periodic state withAt xt = xt shall be calculated. Consequently, theequation

(I −At)xt = 0, (1.2)

has to be solved, which as long as the matrix(I −At)∈Zn×n shows some rank deficiency has a so-lution xt ∈ Qn. This solution obviously does not need to be boolean. Conversely, letxt,1,xt,2 ∈ Bn

be two boolean solutions of equation (1.2). Then in the presupposed sense of addition, generally,

xt,1+xt,2 /∈ Bn,

that is, superposition does not apply for these systems. Consequently, though on the face of it theexample system appears to be linear, it is non-linear in nature.

Problem 2 Let r ≤ n be the rank deficiency of the matrixI −At . Then the general solution ofequation (1.2) takes the form

xt =r∑

i=1

ci xit (1.3)

with xit ∈ Qn specific, but coefficientsci ∈ Q arbitrary. If solutionsxt are admissible only if

xt ∈ Bn then the coefficientsci have to be suitably selected for renderingxt boolean. The entailedcalculations can be very cumbersome. The reason is that the general problem of solving

Ax = b

with A ∈Zn×n andb∈Zn for x∈Bn is a non-polynomial complete (NP-complete) problem, whichfor the purposes here, shall mean that this problem cannot besolved in less thannm calculations,whateverm∈ N may be chosen [CLR90]. Hence, the calculations for solving such problems canbe considered tractable for small numbersn only, which is strongly opposed to the rational casext ∈ Qn in which one were already done with the solution in (1.3), having an effort of less thann2

calculations with the Gauß-algorithm for example.

These outlined shortcomings obviously originate from the model which admits integral numbersfor the parameters in the state equation but claims to keep the states and inputs boolean.

In the author’s opinion, the solution of both problems demands for a change in the algebraic setting:

• Operations should map numbers into the same set of numbers,

• The set of numbers should include the inverse elements for addition and multiplication,2

• The cardinality of the set of numbers should be finite.

An algebraic system which takes into account all these issues is the concept of a finite field. It isthis notion that governs the development of theory to be exposed in the following chapters.

2Except for the zero element.

SECTION 1.2 — CONTRIBUTION OF THIS DISSERTATION 3

An other motivation of this work was to obtain a system model over a finite field that is generalenough for encompassing the description of any deterministic or non-deterministic finite state au-tomaton. In the approaches from above, basically, models for deterministic automata (arithmeticalpolynomials) or even without inputs (Walsh-functions) were derived. That is why, in this work,the modeling was started from the scratch by employing standard methods from boolean algebraand coding theory.

1.2 Contribution of this Dissertation

Based on a discrete state space model over a finite field, the first main contribution of this workis to provide a sufficient and necessary criterion for a complete algebraic locating of the periodicand aperiodic state transition behavior of linearly modeled finite state automata. This criterion isan essential enhancement compared to former results withinthe approaches of arithmetical poly-nomials and Walsh-functions, which both offered necessarycriteria only [Fra94, Son00]. For anapplication of the criterion, the set of elementary divisorpolynomials with respect to the systemdynamics matrix has to be calculated. It turns out that this set can be divided into a set of periodicand aperiodic elementary divisor polynomials, where the former part is related to cycles in thestate graph representation, and the latter part reflects thetree-like structures. Finally, the theory isextended to cover the case of affine-linear systems over finite fields as well.

A further important novelty presented in this work is a method for synthesizing linear state feed-back which, under the assumption of controllability, imposes a desired state transition structureon these linear systems in the closed-loop, i. e. a set of desired periodic and aperiodic elementarydivisor polynomials. To this end, an image domain for functions over finite fields is presented andconnected to methods from the polynomial approach, which isa well-established image domainmethod for the controller synthesis of linear continuous systems [Wol74, Ant98]. The outcome isan algorithm, which first checks whether the requirements burdened by the structural constraintsfrom Rosenbrock’s control structure theorem are met. If theanswer is positive, a desired set of ele-mentary divisor polynomials can be realized by static statefeedback. In a second step, an appropri-ate feedback matrix is derived by manipulating a denominator matrix of a right-prime polynomialmatrix fraction, which results from the transfer matrix of the system representation in controllabil-ity companion form. This approach has the advantage that thesolution of a Diophantine equationis not required, and that, opposed to continuous systems where controllability matrices tend to beill-conditioned, linear systems over finite fields are not subject to this numerical problem. Thus,for linear systems over finite fields this approach of feedback design allows to fully benefit from itsadvantages without incurring its drawbacks. In the closingpart of the work, the methods formerlyderived for a controllable system are extended to the case with an uncontrollable subsystem. Thisis done by decoupling the uncontrollable subsystem from thecontrollable subsystem, for the latterof which an appropriate feedback is designed by resorting tothe methods from before. In this

4 CHAPTER 1 — INTRODUCTION

regard, a criterion of when a decoupling has an influence on the closed-loop elementary divisorpolynomials is derived, which completes the known result that such a decoupling does not alterthe characteristic polynomial of the system dynamics matrix. Moreover, it is worth mentioningthat any linear system of equations occurring in this context can be solved with algorithms of justpolynomial complexity.

1.3 Dissertation Overview

The algebraic foundation of this work, consisting particularly of basics from group theory, poly-nomials over finite fields, and linear algebra, is given in Chapter 2. This chapter also presents animage domain for functions over finite fields. In an exemplarymanner, Chapter 3 exposes twomethods from boolean algebra and coding theory for derivingthe general non-linear state spacemodel over a finite field with characteristic 2, for deterministic and non-deterministic systems. Theanalysis of linear and affine-linear systems over finite fields is dealt with in Chapter 4. It comprisesthe development of a criterion for a complete state space decompositon into periodic and aperiodicsubspaces. In Chapter 5, the cycle sum synthesis problem is solved for controllable and partiallyuncontrollable linear systems over finite fields by adaptingimage domain feedback design methodson an image domain over a finite field. In Chapter 6 the main points are summarized and directionsfor future research are suggested.

Chapter 2

Mathematical Preliminaries

In engineering sciences, and particularly in control engineering, it is common practice to take thefield on which the calculations are carried out for granted. Usually this field is the field of real (orcomplex) numbers, and since this field is a well-establishedparadigm, all the calculation rules andtheorems, e. g. for the zeroes of polynomials, are used and understood in a rather all-embracingmanner. Nevertheless, an infinite set of numbers is always assumed, at least implicitly.

On the contrary, an awareness of the algebraic fundamentalsof discrete mathematics is of crucialimportance when state space representations are used to model finite state automata. As the numberof states is finite for finite state automata, this limitationnaturally demands a finite state spacebecause the automaton states have to be mapped somehow into acorresponding algebraic domain.Conversely, all states in this state space have to have theircounterpart in the automaton, whichmeans that a bijection must exist between each other. When automaton states are related by a nextstate function, an appropriate function must exist on the algebraic domain as well — that is forconstructing those functions, on the algebraic domain somealgebraic operations have to be given,as for instance addition and multiplication. All this indicates that the algebraic domain needs to beclosed under these algebraic operations. Thus, the nature of the problem can be characterized bythe notion of a group, in this case a finite group, and under stronger assumptions, the character ofthe problem is close to the concept of a finite field. Having established the field property withinthe mathematical model of description, the typical standard propositions for (general) fields can beapplied. Next to them, those propositions must be taken intoconsideration which originate fromthe finiteness of the field. The latter propositions cause many differences in everyday calculations.

For all these reasons, the indispensable terminology from finite field theory which is the prerequi-site for an easier understanding of the automaton model in Chapter 3 shall be prearranged. Yet, thisreview can only be far from complete. In the main, it refers tothe comprehensive and thoroughintroduction to finite fields from Lidl and Niederreiter [LN94]. The train of thought, especiallythe introduction to vector spaces and matrices, follows thestyle presented in [DH78]. Reference

6 CHAPTER 2 — MATHEMATICAL PRELIMINARIES

is made as well to [McE87, Sel66] where polynomials over finite fields are the focus of attention.Additionally, the reader may refer to [Lan84, BM77] whereinmost of the algebraic fundamentalsthat are introduced here can be found in detail. All these stress the development and expositionof the mathematical background, whereas [Boo67] serves automata theory from a more practicalpoint of view.

The chapter is organized as follows: in Section 2.1 fundamental concepts and basic terminologyof group theory are recalled. Using the result of Fermat’s little theorem some important propertiesof polynomials over finite fields, in particular the conceptsof irreducibility and periodicity, areintroduced in Section 2.2. Provided with this knowledge, vector spaces, linear transformations andmatrices are introduced in Section 2.3. Special attention is directed to matrices in standardized(canonical) forms because the properties of a matrix becomevisible right away by inspecting thecorresponding matrix in certain canonical forms. Since difference equations prospectively becomealgebraic in an image domain, an operational calculus for finite fields is presented in Section 2.4.This operational calculus prepares the way for an algebraicexamination and synthesis of linearmodular systems in Chapter 5. Some remarks demonstrate the differences between finite andinfinite fields.

2.1 Fundamental Concepts of Group Theory

In algebraic systems, elements of a set are connected by applying operations on them. If an oper-ation relates two elements of the set, then the operation is called a binary operation. The first andmost general algebraic system is a semigroup.

Definition 2.1 (Semigroup)A semigroup(S,∗) is a nonempty setS together with a binary operation∗ such that

1. For alla,b∈ S, a∗b∈ S.

2. The operation∗ is associative, i. e.a∗ (b∗c) = (a∗b)∗c for anya,b,c∈ S. �

An example of a semigroup is(N,+), the set of positive integersN together with the binary oper-ation+ defined as addition. Note that in this case there is no positive integere∈ N such that forsome positive integera ∈ N, a+ e= a. That is an identity element is missed here, which wouldbee= 0 if the set of positive integers were enlarged by the number 0. On that account one moreassumption can be added.

Definition 2.2 (Monoid)A semigroup(M,∗) with setM and binary operation∗ is a monoid if an identity elemente∈ M

exists such that for alla∈ M, a∗e= e∗a = a. �

SECTION 2.1 — FUNDAMENTAL CONCEPTS OFGROUP THEORY 7

In other words: a monoid is a set that is closed under an operation and has an identity element bymeans of which the operation maps elements onto itself. If the task is to solve for an element inthe operation, monoids have to be enhanced by one more property. An inverse element must existfor each element of the set, i. e. the more powerful concept ofa group is necessary.

Definition 2.3 (Group)A group(G,∗) is a setG together with a binary operation∗ such that

1. For alla,b∈ G, a∗b∈ G.

2. The operation∗ is associative, i. e.a∗ (b∗c) = (a∗b)∗c for anya,b,c∈ G.

3. An identity element,e∈ G, exists such that for alla∈ G, a∗e= e∗a = a.

4. For alla∈ G exists an inverse elementa−1 ∈ G such thata∗a−1 = a−1∗a = e.

Moreover, a group is commutative (or Abelian) if for alla,b∈ G, a∗b = b∗a. A group is calledfinite if the setG contains finitely many elements. �

By virtue of the last part of this definition, systems of equations can be solved for variables.1 Anexample for a group is(R,+), the set of real numbersR with respect to addition; the identityelement ise= 0, the inverse element is the corresponding negative element. The set of rationalnumbersQ\{0} together with multiplication is another simple example of agroup; the identityelement of this group,(Q\{0}, ·), is e= 1, the inverse element is the corresponding quotient of theelement.

The next step is to extend the basic theory by adding a second operation.

Definition 2.4 (Ring)A ring (R,+, ·) is a setR together with two binary operations, addition+, and multiplication·,such that

1. R is a commutative group with respect to addition.

2. R is a semigroup with respect to multiplication.

3. R is distributive with respect to these operations, that isa · (b+c) = a ·b+a ·c and(b+c) ·a= b ·a+c·a for all a,b,c∈ R.

1This is the main obstacle when using dioids like the (max,+)-algebra, where inverses do not exist in general.


A ring is called commutative if its multiplication is commutative. In a ring(R,+, ·) the identityelement with respect to+ is denoted by 0, the identity with respect to· is denoted by 1.2 �

It is essential for a ring that an inverse operation for the multiplication need not exist. If, how-ever, a multiplicative inverse is required – for example when solving for multiplicatively boundindeterminates — the concept of a field becomes important.

Definition 2.5 (Field)A ring (F,+, ·) is a fieldF = (F,+, ·) if the subsetF\{0} is a commutative group with respect tothe multiplication·. A field F with q elements, denoted byFq, is called finite ifq is finite.3 �

According to common notational practice the symbol “·”, indicating multiplication, will be omit-ted, unless it is necessary. In the further chapters a special type of finite field is utilized that isbased on the division remainder operationmodulo. The following theorem is shown by simplychecking the field property.

Theorem 2.1 (Galois-Field)The set of integral numbers{0,1, . . . ,q−1}, whereq is a prime number, together with the binaryoperations addition and multiplication moduloq, is a finite field, called a Galois-FieldFq. �

To spot the need for the primality ofqconsider the nonempty set{0,1, . . . ,q−1}with cardinalityq,endowed with the operations addition and multiplication modulo q, respectively. The interestingcase is whenq > 1. Assume thatq > 1 were not a prime such that a factorization isq = p1 p2

with pi 6= 1 and obtainp1 p2 = 0 modq. If a field Fq existed then an inverse elementp−11 such

that p−11 p−1

1= 1 modq must also exist since 1 is the identity element of multiplication. As aconsequence, the conclusion would bep2 = 0 modq or equivalently thatq| p2, a contradiction.Therefore, such a field does not exist.

Remark 2.1The property thatab= 0 iff a = 0∨ b = 0 for a,b∈ F often is referred to that fields are devoid ofzero divisors. �

It can be shown that any finite field is equivalent to some Galois-Field or at least to one of itsextension fields4; for a proof see [LN94]. Hence, as the field property is required in finite automata

2Note that in this definition addition and multiplication, denoted by the symbols “+” and “·”, represent a gen-eralization of the ordinary concept of addition and multiplication. They are not to be confused with the ordinaryconcept, since the above-given definition of a ring applies to arbitrary operations, which have the stated properties ofcommutativity, associativity and distributivity.

3The symbolF is used to emphasize the field character. For exampleF denotes a field, whereasF would mark aset, which would not necessarily be equipped with any operations. Subscripts indicate the cardinality of the underlying(finite) set. Without a subscript no further assumptions on the field, whether finite or infinite, are made.

4An example which bases on the notion of extension fields is considered in Appendix C.

SECTION 2.2 — POLYNOMIALS OVER FINITE FIELDS 9

models over a finite set, any of those models can be described by means of some suitable Galois-Field. For this reason, the main attention is paid to Galois-Fields, and for simplicity, if a finite fieldis in question then always a respective Galois-Field is assumed.

Theorem 2.2 (Fermat’s Little Theorem)Let an integral numberq be a prime number. Then for all integersλ, q dividesλq−λ. In caseswhere the integerλ is not divisible by the primeq, q dividesλq−1−1. �

Proof In order to explain this, the first part of the theorem to be proven is rephrased byλq ≡ λmodq, expressing the equivalence moduloq of λq andλ. An induction argument will be used.To start with, the first integerλ = 1 is checked, which obviously verifies this part of the theorem.From the binomial theorem(a+b)n =

∑ni=0

(ni

)an−i bi it follows

(λ+1)q =

q∑

i=0

(qi

)

λq−i = λq+qλq−1 +q(q−1)

2λq−2+ · · ·+qλ+1

= λq+1+q(λq−1+q−1

2λq−2 + · · ·+λ)

=⇒ (λ+1)q ≡ λq+1 modq.

Applying the induction hypothesisλq ≡ λ modq yields

(λ+1)q ≡ λ+1 modq,

and since the statement already holds forλ = 1 the first part of the theorem has been shown.Concerning the second part, it is clear that with

λq ≡ λ modq ⇐⇒ λ(λq−1−1) ≡ 0 modq,

the first part of the theorem implies that eitherλ or λq−1−1 is divisible byq. Due to the assumptionthatq does not divideλ, finally the second part of the theorem is obtained. �

Remark 2.2Fermat’s little theorem is important whenever polynomialsover a finite fieldFq are concerned. Thetheorem implies that many polynomials over a finite fieldFq can be identical to zero for arbitraryλ ∈ Fq, because these polynomials may contain polynomial factorsλq−λ, which are zero moduloq. In contrast, a polynomial over the infinite field of real numbersR is identical to zero iff allcoefficients are zero. �

2.2 Polynomials over Finite Fields

A fundamental property of polynomials, which is according to Gauß’ fundamental theorem ofalgebra, is that all polynomials over the field of real numbers R can be factored (reduced) in


quadratical factors, or over the extension fieldC in linear factors. As will be demonstrated in thissection, for finite fieldsFq in general this is not the case.

It can easily be verified that the set of polynomials togetherwith customary polynomial additionand polynomial multiplication is a ring.

Theorem 2.3 (Ring of Polynomials)The set of all polynomialsp(λ) =

∑

i ai λi with indeterminateλ and i = 0,1,2, . . . coefficientsai

in a fieldF together with polynomial addition and polynomial multiplication is a ring, called thering of polynomials over the fieldF. It is denoted byF[λ]. �

Remark 2.3In the ring of polynomials, the 0-element (identity elementwrt. polynomial addition) is the poly-nomial 0 in which all coefficientsai are zero. The respective 1-element (identity element wrt.polynomial multiplication) is the polynomial 1 in which thecoefficienta0 = 1 and all other coef-ficients are zero. �

For convenience some fundamentals are recalled.

Definition 2.6 (Monic Polynomial)A polynomialp(λ) =

∑di=0ai λi with degreed is called monic ifad = 1. �

Definition 2.7 (Irreducible Polynomial)A non-constant polynomialp ∈ F[λ] is called irreducible overF, wheneverp(λ) = g(λ)h(λ) inF[λ], then eitherg(λ) or h(λ) is a constant. �

In view of irreducibility, Gauß’ fundamental theorem of algebra can be rephrased.

Theorem 2.4 (Unique Factorization Theorem)Any polynomialp∈ F[λ] can be written in the form

p = a p1e1 · · · pk

ek , (2.1)

wherea∈ F, p1, . . . , pk are distinct monic irreducible polynomials inF[λ], ande1, . . . ,ek are posi-tive integers. Moreover, this factorization is unique apart from the sequence of the factors. �

For the infinite fieldR it is a well-known fact that all factor polynomials are at most of seconddegree overR, that isei ≤ 2 in Theorem 2.4. This does not apply for finite fieldsFq. For example:p(λ) = λ5 + λ2 + λ + 1 = (λ3 + λ + 1)(λ + 1)2 for p ∈ F2[λ], becauseλ3 + λ + 1 andλ + 1 areirreducible overF2. However, withq∈ F3[λ] the case is a different one, as showsq(λ) = λ3+λ+

1 = (λ2+λ+2)(λ+2). Consequently, reducibility of a polynomial depends on thefield.

The latter concepts apply for fields in general. The following concepts apply only if the fieldsunder concern are finite. Besides the degree of a polynomial,it can be shown that for any non-zeropolynomial over a finite field, another characteristic integer exists.

SECTION 2.2 — POLYNOMIALS OVER FINITE FIELDS 11

Definition 2.8 (Period of a Polynomial)Let p∈ Fq[λ] be a non-zero polynomial over the finite fieldFq. If p(0) 6= 0, then the least positiveintegerτ for which p(λ) dividesλτ − 1 is called the period (or order) of the polynomialp. Ifp(0) = 0, thenp(λ) = λhg(λ), whereh∈ N andg∈ Fq[λ] with g(0) 6= 0, and the periodτ of thepolynomialp is defined as the period ofg. �

As a consequence, all polynomials with periodτ represent factors of the polynomialλτ −1. Forpolynomials which are powers of irreducible polynomials, so-called powered polynomials, thefollowing theorem is taken from [LN94].

Theorem 2.5 (Period of a Powered Polynomial)Let p∈ Fq[λ] be an irreducible polynomial over the finite fieldFq with p(0) 6= 0 and periodτ. Letf ∈ Fq[λ] be f = pe with e∈ N. Let l be the least integer such thatql ≥ e. Then the poweredpolynomial f has the periodql τ. �

Example 2.1The period of the polynomialf (λ) = λ4+λ2+1∈ F2[λ] is to be calculated. From the sequence

λ4+λ2+1 → λ6+λ4 +λ2 → λ6+1,

that is from,λ2 f (λ)+ f (λ) = (λ2+1) f (λ) = λ6+1 ⇒ f (λ)|λ6+1

it follows τ f = 6. Using the factorizationf = p2 with p(λ) = λ2+λ+1∈ F2[λ] the sequence

λ2+λ+1 → λ3+λ2+λ → λ3+1

or in other words,

λ p(λ)+ p(λ) = (λ+1)p(λ) = λ3+1 ⇒ p(λ)|λ3+1

shows thatτp = 3. Thus, observinge= 2 and Theorem 2.5 results inl = 1, in the first place, andtherefore withτp = 3 finally the periodτ f = 21 ·3 = 6 is obtained.5 �

In practice, periods of polynomials need not be calculated manually. For finite fieldsFq startingfrom characteristicq = 2 up toq = 7 the respective periods of irreducible polynomials can befound in tabulars such as in [LN94], in [PW72] for polynomials overF2 up to degree 34, or areinternally tabulated and calculated in computer algebra software.6

Remark 2.4Nilpotent polynomialsp∈ Fq[λ], i. e. polynomials of the formp = λk for some positive integerk,are not periodic by definition (see Definition 2.8). With the fact that in a factorization of a polyno-mial apart from nilpotent polynomial factors only irreducible polynomials and their powers mayoccur, polynomials over finite fields are either periodic or nilpotent. This implies some importantconsequences to be discussed in Chapter 4. �

5The introductory proof of Theorem 4.4 in Chapter 4 gives moreinsight to the above-used algorithm.6Typical such software are for instance the packages MapleR© or MathematicaR©.


2.3 Linear Transformations and Matrices

2.3.1 Vector Spaces

Having developed the basis of algebraic systems as groups and rings of polynomials over finitesets one may then define a vector space.

Definition 2.9 (Linear Space, Vector Space)A (linear vector) space over a fieldF, denoted byV = (V,+,F), is a setV which is a commutativegroup with respect to addition+ together with a fieldF such that

1. The operations on the fieldF are addition and multiplication.

2. For anyv ∈ V and anya∈ F, av = va∈ V. (closedness)

3. For arbitraryu,v ∈ V and arbitrarya,b∈ F,

(a) a(u+v) = au+av (distributivity 1)

(b) (a+b)u = au+bu (distributivity 2)

(c) (ab)u = a(bu) = b(au) = (ba)u (associativity)

(d) 1F u = u (1F is the identity element wrt. multiplication inF)

The elements ofV are called vectors, the elements ofF are called scalars. The identity elementof the group(V,+) is given by the zero vector0, the inverse element of an elementv ∈ V is itscorresponding negative element−v. �

Definition 2.10 (Subspaces)Let V = (V,+,F) and V = (V,+,F) with V ⊆ V be two vector spaces associated to the sameoperations defined over the same fieldF. ThenV is called a subspace ofV. �

In a vector spaceV = (V,+,F), vectorsvi ∈ V, i = 1, . . . ,n, can be combined such that withai ∈ F

the vectorv = a1v1+a2v2+ · · ·+anvn

is a so-called linear combination of the vectorsvi .

Definition 2.11 (Linear Independence)Vectorsvi ∈ V, i = 1, . . . ,n, of a vector spaceV = (V,+,F) are said to be linearly independent ifffor ai ∈ F, i = 1, . . . ,n

a1v1+a2v2+ · · ·+anvn = 0

implies thatai = 0 for all i = 1, . . . ,n. Otherwise the vectors are called linearly dependent.�

SECTION 2.3 — LINEAR TRANSFORMATIONS AND MATRICES 13

With the notion of linear independence a set of vectors may bedefined which can be used torepresent a vector space.

Definition 2.12 (Basis of a Vector Space)A set of vectorsB = {b1,b2, . . .} is a basis of the vector spaceV = (V,+,F) if

1. B ⊆ V ,

2. B spansV, i. e. allv∈V are linear combinations of the vectorsbi with respective coordinatesvi ∈ F, hence

v =∑

i

vi bi .

3. The vectors inB are linearly independent.

The cardinality ofB is called the dimension dim(V) of the vector spaceV.7 Any v ∈ V can berepresented by a tuple(v1,v2, . . .), a so-called vector of coordinates, with respect to a basisB. �

Remark 2.5Finite dimensional vector spaces over finite fields are pointspaces consisting of a finite numberof vectors — in a sense, the vector space is void between any two points. Moreover, the commonnotion of a scalar product does not yield an expressive notion of distance. For these reasons, adefinition of convergence is not straight-forward for finitefields.8 �

Remark 2.6According to common practice,Fn denotes then-dimensional column vector space overF. Anyvectorv ∈ Fn can be termed by a column vector of coordinates

v =

v1

v2...

vn

,

instead of which, for convenience, often the transposevT = (v1,v2, . . . ,vn) will be used. �

Example 2.2The setV = {v1,v2,v3} of column vectorsvT

1 = (1,1,2,0), vT2 = (0,1,1,1) andvT

3 = (1,0,1,2)

over the finite fieldF3 with addition and multiplication modulo 3, respectively, generates a sub-spaceV ∈ F4

3 . Observing thatv3 = v1+2v2 ,

7If B contains infinitely many elements thenV is called a vector space of infinite dimension.8This imposes strong restrictions on the existence of approximation models.


the vectorv3 is a linear combination of the linearly independent vectorsv1 andv2. Then a basisof V is {v1,v2} andv3, the column vector of coordinates forv3 with respect to that basis ofV, isvT

3 = (1,2). The dimension of this space is dim(V) = 2. �

Now, one is in the position to define transformation maps on vector spaces.

Definition 2.13 (Linear Transformations)Let V = (V,+,F) andV = (V,+,F) be vector spaces over the fieldF. The mappingt : V → V iscalled a linear transformation (mapping) or homomorphism if for all v1,v2 ∈ V and for alla,b∈ F

t(av1+bv2) = at(v1)+bt(v2) = av1+bv2

wherevi = t(vi) ∈ V is the image ofvi ∈ V in the vector spaceV under the mappingt. If the map-ping t is a one-to-one mapping ofV onto V then the linear transformationt is called nonsingularandV andV are termed isomorphic. Otherwise the linear transformation t is called singular. �

As per Definition 2.12, any vectorv of ann-dimensional vector spaceV can be expressed by useof a basisB = {b1, . . . ,bn}. If t : V → V, that isv is subject to a linear transformation into ann-dimensional vector spaceV, then first by linearity

t(v) = t

(n∑

j=1

a j b j

)

=

n∑

j=1

a j t(b j) (2.2)

with the result that linear transformations can be expressed by transformations of the basis. Sec-ondly, all t(b j), j = 1, . . . ,n, and t(v) are vectors of the vector spaceV with a base, sayB =

{b1, . . . , bn}, hence

t(v) =

n∑

j=1

a j

n∑

i=1

ti j bi , (2.3)

t(v) =

n∑

i=1

ai bi . (2.4)

Swapping the sums in equation (2.3) and equating it with (2.4) yields

n∑

i=1

bi

(

ai −n∑

j=1

a j ti j

)

= 0. (2.5)

SinceB = {b1, . . . , bn} is a basis, all vectorsbi are linearly independent, thus, the expression inparentheses is zero, that is

ai =

n∑

i=0

ti j a j . (2.6)


Using a matrixT the entries of which areti j , i = 1, . . . , n, j = 1, . . . ,n, and column vectors for thecoordinatesaT = (a1, . . . ,an) andaT = (a1, . . . , an) linear transformations can be computed by thematrix-vector product

a = Ta, a∈ Fn, a∈ F n, T ∈ F n×n (2.7)

employing the coordinates with respect to the corresponding bases only. Moreover, in equation(2.7) the vector spaces are indicated by making use of the typical symbolic notation.

2.3.2 Matrices

The subsequent concepts are intensively used and understood for infinite fields. At a first glance,however, it might be not so clear how to calculate, for instance, the rank or nullspace of a matrixover a finite field. Therefore, some well-known terminology shall be re-exposed and exemplified.

Definition 2.14 (Elementary Column and Row Operations on Matrices)Let coli(A) be thei-th column vector (rowi(A) the i-th row vector) of a matrixA ∈ Fm×n anda 6= 0∈ F. Then elementary column (row) operations onA are:

1. replacing coli(A) by a ·coli(A)(rowi(A) by a · rowi(A)

),

2. interchanging coli(A) and colj(A)(rowi(A) and rowj(A)

),

3. replacing coli(A) by coli(A)+a ·colj(A)(rowi(A) by rowi(A)+a · rowj(A)

), j 6= i. �

Elementary column (row) operations are performed by right and left multiplication with so-calledelementary matrices, which assure scaling, interchangingand replacing of rows (columns).

Example 2.3By elementary row operations the following system of equationsAx = b with

A =

1 0 11 1 00 1 1

, b =

100

can be solved forx. First, assume the case of a field of rational numbersQ, hence,A ∈ Q3×3 andb ∈ Q3, and it shall be solved forx ∈ Q3. By elementary row operations

1 0 1 11 1 0 00 1 1 0

→

1 0−1 −10 1−1 −10 1−1 −0

→

1 0−1 −10 1−1 −10 0−2 −1

→

1 0−1 −10 1−1 −10 0−1 1/2

a system of equations is obtained which easily is solved recursively. The unique solution isxT =

(1/2,−1/2,1/2).


If the case of a finite fieldF2 is assumed thenA ∈ F3×32 andb ∈ F3

2 and the task is to solve forx ∈ F3

2 , now all operations taken modulo 2. Hence it follows,

1 0 1 11 1 0 00 1 1 0

→

1 0 1 10 1 1 10 1 1 0

→

1 0 1 10 1 1 10 0 0 1

and from the last row it is obvious that there is no solutionx ∈ F32 . �

Definition 2.15 (Rank of a Matrix)The column (row) rank of a matrixA ∈ Fm×n is the number of linearly independent column (row)vectors ofA. If the column (row) rank isn (m) then the matrixA is said to have full column (row)rank. �

It can be seen that for square matrices column and row rank arethe same. In this regard theysimply havea rank.

Remark 2.7 (Column and Row Space)The column (row) vectors of a matrixA ∈ Fm×n generate or span the so-called column (row)space of the matrixA. The associated dimension of the column (row) space of a matrix A equalsits column (row) rank. Moreover, the column space of a matrixis a subspace ofFm. �

Since scaling and interchanging of vectors in a basis does not vary the vector space, elementaryrow operations on matrices do not change the row space of a matrix. This will be made use of byrecourse to the example above. The rank of the square matrixA ∈ Q3×3 is to be determined. Thelast transformation comprises three linearly independentrow vectors,(1,0,1), (0,1,−1), (0,0,1),hence, the dimension of the associated row vector space is three and asA is square, rank(A) = 3 as-sumingA ∈Q3×3. ForA ∈ F3×3

2 there are two linearly independent row vectors,(1,0,1), (0,1,1),alternatively, there is only a choice of two linearly independent column vectors, which results inrank(A) = 2 in the caseA ∈ F3×3

2 .

Linear equations asAx = b with A square are always uniquely solvable ifA is of full (maximal)rank. In these cases a linear left transformation to be applied onA which mapsA to the identitymatrix I exists. This leads to the following theorem.

Theorem 2.6 (Inverse of a Matrix)Iff a matrix A ∈ Fn×n is of full rank then a unique inverse matrix denoted byA−1 ∈ Fn×n existssuch that

A−1A = I .

In this caseA is termed invertible. �


With the inverse matrixA−1 the unique solution of a system of equationsAx = b readsx = A−1b.

Remark 2.8If a linear transformationt on a vector spaceV = (V,+,F) mapsV ontoV, that ist : V → V, thenthe corresponding matrixT is square and has full rank, thus the matrixT is invertible. In additionto that, such linear transformations are called nonsingular, see Definition 2.13. For this reasoninvertible matrices are said to be nonsingular. �

Many major properties of a matrix are invariant by its structure and are preserved under elementaryrow and column operations, so-called similarity transformations.

Definition 2.16 (Similarity of Matrices)MatricesA1, A2 ∈ Fn×n are similar if

A2 = TA1T−1 (2.8)

for some invertible matrixT ∈ Fn×n. �

An interpretation of similarity is given by a change of coordinates. Consider a linear transformationa : V → V on a vector spaceV = (V,+,F) represented byx′ = Ax with a square matrixA thatis not necessarily nonsingular. According to equation (2.7) let the coordinatesx be subject to acoordinate transformationt : V → V with x = Tx and a nonsingular square matrixT. Then thequestion arises: how can the linear transformationa be translated into the new coordinates? Somesimple calculation steps give the answer

x′ = TAT−1 x . (2.9)

Consequently,A = TAT−1 represents the matrix of the linear transformationa with regard to thenew coordinates.

Definition 2.17 (Kernel and Image of a Matrix)Let the matrixA ∈ Fm×n represent the mappinga : Fn → Fm. Then the set

Ker(A) := {x ∈ Fn |Ax = 0}

is termed kernel or nullspace of the matrixA and the set

Im(A) := {b ∈ Fm|∃x ∈ Fn : Ax = b}

is referred to as the image of the matrixA. �

In order to simplify the terminology, any vectors will implicitly mean column vectors unless ex-plicitly specified in a different way.


Example 2.4As an example recall the linear transformation with the matrix A ∈ F3×3

2 from above. Then thekernel and image ofA can be calculated by

1 0 1 b1

1 1 0 b2

0 1 1 b3

→

1 0 1 b1

0 1 1 b1 +b2

0 1 1 b3

→

1 0 1 b1

0 1 1 b1+b2

0 0 0 b1 +b2+b3

Therefore, the kernel is

Ker(A) =

x3

x3

x3

,x3 ∈ F2

=

x3

111

,x3 ∈ F2

and the image

Im(A)={

b ∈ F3 |b1+b2+b3 = 0}

=

b2+b3

b2

b3

,b2,b3 ∈ F2

=

b2

110

+b3

101

,b2,b3 ∈ F2

. �

Remark 2.9The complement to the concept of rank is the concept of nullity. The nullity of a matrixA ∈ Fm×n,or equivalently, of its linear mappinga : Fn → Fm is the dimension of the nullspace of the matrixA, nullity(A) = dim(Ker(A)), whereas the rank is the dimension of the image concerningA,rank(A) = dim(Im(A)). Connecting both, an important theorem states that

nullity(A)+ rank(A) = n = dim(Fn) . �

The subsequent definitions prepare the introduction of the Smith canonical form of a matrix. Tothis end, matrices with variant coefficients need to be considered.

Definition 2.18 (Rational and Polynomial Matrix)A matrix R(λ), the elements of which are fractions of polynomials inF[λ] is called a rationalmatrix. If the denominator polynomial of each element ofR(λ) is equal to one then the matrix isa polynomial matrix.9 �

9Let F[λ]n×m denote the set ofn×m polynomial matrices with entries that are polynomials in the ringF[λ], i. e.with coefficients in the fieldF. Accordingly,Fq[λ]n×m denotes the respective finite field version.


Elementary column and row operations can be extended to row and column multiplications withpolynomial factors using matrices, which are a generalization to elementary matrices: unimodularmatrices.

Definition 2.19 (Unimodular Matrix)A polynomial matrix whose inverse matrix is a polynomial matrix is called unimodular. �

It is easy to show that the determinant of an unimodular (polynomial) matrix is a nonzero scalarin the underlying fieldF. Unimodular matrices are the main ingredient for stating the followingimportant theorem.

Theorem 2.7 (Smith Form of the Characteristic Matrix)For any matrixA ∈ Fn×n unimodular matricesU(λ),V(λ) ∈ F[λ]n×n exist such that

U(λ)(λI −A)V(λ) = S(λ) (2.10)

with the characteristic matrix(λI −A) corresponding toA and the unique polynomial matrix

S(λ) =

c1(λ) 0 · · · 00 c2(λ)

....... . . 0

0 · · · 0 cn(λ)

, (2.11)

in which the monic polynomialsci+1 |ci , i = 1, . . . ,n−1. The polynomial matrixS(λ) ∈ F[λ]n×n iscalled the Smith canonical form of (the characteristic matrix wrt.) A.10 �

The calculation of the unimodular matricesU(λ) andV(λ) is most efficiently carried out by em-ploying symbolic procedures which are offered by computer algebra packages such as MapleR© andMathematicaR©. For an algorithm see [Boo67, p. 268 ff.], [Gil69, p. 222 ff.]or [LT85].

By equating the Smith canonical forms of two matricesA1 andA2 and employing Definition 2.19for unimodular matrices finally leads the following important result11

Theorem 2.8 (Smith Form Similarity Criterion)Two square matricesA1 andA2 are similar iff they have the same Smith form. �

Since the polynomialsci(λ), i = 1, . . . ,n, in the Smith canonical form are preserved under similar-ity transformations this gives rise to define invariants.

10In general, respective Smith canonical forms exists for arbitrary non-square polynomial matrices.11Showing that the corresponding transformation matrices are constant matrices is more involved [Kai80].


2.3.3 Invariants of Matrices

Definition 2.20 (Invariant Polynomials)The unique (non-constant) monic polynomialsci(λ), i = 1, . . . ,n, referring to the Smith formS(λ) ∈ F[λ]n×n of a matrixA ∈ Fn×n are the invariant polynomials (similarity invariants) ofA. �

Note that the product of all invariant polynomials equals the characteristic polynomial, that iscpA(λ) = det(λ I −A) =

∏

i ci(λ). The uppermost polynomialc1(λ) in the Smith form can beidentified with the minimal polynomial mpA(λ) of the matrixA, which is the polynomial of leastdegree such that the associated matrix polynomial holds identically zero, that is mpA(A)≡ 0. Fromthe divisibility property of the invariant polynomials it is clear that mpA|cpA.

With Theorem 2.4 the invariant polynomials can be decomposed into factors.

Definition 2.21 (Elementary Divisor Polynomials, Elementary Divisors)Let ci ∈ F[λ], i = 1, . . . ,n, be the invariant polynomials of a matrixA ∈ Fn×n andci = p

ei,1i,1 · · · pei,Ni

i,Ni

be the unique factorization ofci into Ni factors due to Theorem 2.4. Then, theN =∑n

i=1Ni

non-constant monic factor polynomialspei, ji, j , i = 1, . . . ,n and j = 1, . . . ,Ni, are termed elementary

divisor polynomials ofA. The set of (integral) powersej , j = 1, . . . ,N, is referred to as the set ofelementary divisors ofA. �

2.3.4 The Rational Canonical Form

In addition to the Smith form (2.11), another canonical formreferring to the elementary divisorpolynomials will be used. This involves the notion of a companion matrix.

Definition 2.22 (Companion Matrix)Let pC(λ) = λd +

∑d−1i=0 ai λi ∈ F[λ] be a monic polynomial of degreed. Then the(d×d)-matrix12

C =

0 0 0 · · · 0 −a0

1 0 0 · · · 0 −a1

0 1 0 · · · 0 −a2......

.... . .

......

0 0 0 · · · 0 −ad−20 0 0 · · · 1 −ad−1

(2.12)

overF is called the companion matrix with respect to the polynomial pC(λ). �

By transforming the characteristic matrix of a companion matrix on its Smith form [LT85] thefollowing useful property of a companion matrix can be shown[BM77, p. 339].

12in literature, sometimes the transpose of this matrix


Theorem 2.9 (Characteristic and Minimal Polynomial of a Companion Matrix)Let pC(λ) be the defining polynomial, cpC(λ) the characteristic polynomial, and mpC(λ) the min-imal polynomial, all with respect to a companion matrixC. Then the following property holds:

cpC(λ) ≡ mpC(λ) ≡ pC(λ) , (2.13)

that is, companion matrices are so-called non-derogatory matrices. �

Definition 2.23 (Rational Canonical Form)The block diagonal matrixArat ∈ Fn×n with

Arat = diag(C1, . . . ,CN) , (2.14)

whereCi , i = 1, . . . ,N are companion matrices, is called a rational canonical form. �

Theorem 2.10 (Uniqueness of the (Classical) Rational Canonical Form of a Matrix)For any matrixA ∈ Fn×n exists a similarity transformation

Arat = TAT−1 (2.15)

by virtue of an invertible constant matrixT such thatArat is a rational canonical form, which com-prises thei = 1, . . . ,N companion matricesCi with respect to the elementary divisor polynomialspi of the matrixA.

Apart from the ordering of the companion matricesCi the matrixArat is unique and the numberNis maximal regardingA. �

The matrixArat often will be referred to, simply, as the rational canonicalform of A. An algorithmfor computing the transformation matrixT is given in [Gil69, p. 225 ff.]. An alternative is presentedin Appendix B.

Example 2.5The following Smith form of a matrixA ∈ F6×6

2

S(λ) =

λ5 +λ4+λ2+λ 0 0 0 0 00 λ+1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

=

(λ2+λ+1)(λ+1)2λ 0 0 0 0 00 λ+1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

apparently has the elementary divisor polynomials

p1(λ) = λ2+λ+1, p2(λ) = (λ+1)2, p3(λ) = λ, p4(λ) = λ+1.


Then the corresponding companion matrices and the rationalcanonical form ofA are

C1 =

(0 11 1

)

, C3 =(0)

,

C2 =

(0 11 0

)

, C4 =(1)

,

−→ Arat =

0 1 0 0 0 01 1 0 0 0 00 0 0 1 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 0 1

. �

Remark 2.10The Jordan canonical form of a matrix, which would follow from diagonalizing the rational canon-ical form, is omitted here because the Jordan canonical formis accompanied by the notion of anextension fieldFqk, k = 1,2, . . ., associated toFq. As will be shown in Appendix C, for finite fieldsthe calculation of roots in this extension fieldFqk is much more cumbersome than it is in the exten-sion field associated to the field of real numbersR, which isC, the field of complex numbers.�

2.4 An Image Domain for Finite Fields

In this section discrete functionsf : N0 → Fq are dealt with. These functions can be interpreted asan infinite sequence of function valuesf (0), f (1), f (2), . . . ∈ Fq. Similar to discrete functions overthe infinite field of real numbers, for which the so-calledZ-transform applies, a counterpart existsfor the finite fieldFq. This will be referred to as theA-transform. The basic idea of this transform,an operational calculus for finite fields, was introduced forfeedback shift register synthesis at theend of the fifties [Huf56, Fri59] and widely used for linear coding, linear sequential switchingcircuits and networks in the sixties of the past century; see[Kau65] which is a compilation ofthe most important contributions in that field. Though it wasmade use of frequently, a formalderivation of the calculus was performed quite late [Boo62,Ric65, Boo67]. After a decade thistransform was rediscovered for the use within the analysis of linear automata [Wun75, Gös91].Recently, it was applied for establishing a binary system theory [Wen00].

2.4.1 TheAAA-Transform

The application of theA-transform in Chapter 5 will have the very narrow focus of transforming afirst order linear recurring sequence into an algebraic expression with respect to an image domain.Therefore, the mathematical theory shall be exposed only briefly.13

13During the years the image domain representation was definedand redefined in several other ways. Huffmann[Huf56] originally defined theD-transform byD{ f (k)} = F(D) =

∑∞k=0 f (k)Dk, thusD = 1/a. In the same manner

SECTION 2.4 — AN IMAGE DOMAIN FOR FINITE FIELDS 23

Definition 2.24 (AAA-Transform)TheA-transform for a discrete functionf (k) overFq with f (k) = 0,∀k < 0 is

A{ f (k)} = F(a) :=∞∑

k=0

f (k)a−k ,

in which addition and multiplication are taken moduloq. �

In this sense, the original domain is the domain on which the discrete functionf (k) resides andthe image domain is the domain whereF(a), theA-transform of f (k), is defined.14 To any valueof f (k) in thek-domain corresponds exactly one monomial in theA-domain (it can be shown thatthe converse is true as well). Both domains are related by a bijection of transform and inversetransform. As a consequence, any calculation can be carriedout, alternatively, on either of theboth domains.

In view of Definition 2.24 the inverse transform can be determined according to the followingtheorem.

Theorem 2.11 (Inverse of theAAA-Transform)The inverse transform of theA-transform is given by

A−1{F(a)} = f (0), f (1), f (2), . . .

f (k) = [akF(a)]ind , (2.16)

in which the operation[akF(a)]ind provides the addend of the rational expressionak F(a) which isindependent ofa.15 �

The inverse operationA−1 precisely transforms image domain functions into corresponding infi-nite sequences of function values in the original domain.

2.4.2 Table of Correspondences

If the transform of linear systems of equations is taken intoaccount then only a few rather obvioustransformation rules are relevant (see Table 2.1).

Wunsch [Wun75] and Gössel [Gös91] used the symbolsζ andd, respectively, instead of the parameterD whereasRichalet [Ric65] defined an operational calculus as perF(p) =

∑∞k=0 f (k) p−(k+1) with parameterp, as a consequence

F(p) = F(a)/p. In general, causality off (k), i. e. f (k) = 0,∀k < 0, is an assumption common to all the definitionsused. The definition presented here is taken from [Wen00] andits laws are in wide accordance with those from theZ-transform. This clue, at least to the author, is reason enough to use theA-transform. Whatever may be preferred,the transform tables are easily rephrased.

14Clearly,F(a) is an operator, but for the purposes here it can simply be considered as a quotient of polynomials.15As an alternative to equation (2.16) the rational functionF(a) can be written in partial fraction form (after a

polynomial division if necessary) and then some correspondences may be used for the backward transform by meansof which sequences of function values are obtained, typically. To obtain functional expressions forf (k) involves morecomputational effort. The reader may refer to [Boo62] for the details.


original domain (function ofk) image domain(function ofa)

αi ∈ Fq : g(k) :=n∑

i=1

αi fi(k) αi ∈ Fq : G(a) =n∑

i=1

αi Fi(a)

g(k) :=k∑

i=0

f (i) G(a) = aa−1 F(a)

g(k) :=k∑

i=0

f1(i) f2(k− i) G(a) = F1(a)F2(a)

g(k) := f (k+1) G(a) = aF(a)+a f(0)

Table 2.1: A-transform for functionsf (k) with f (k) = 0,∀k < 0

Applications are part of Chapter 5.

Chapter 3

Finite State Automata in the State Space

The study of dynamic systems presupposes a system model. There are many models for discreteevent systems, and each model has pros and cons depending itspurpose. Established models aregraphical representations like Petri nets or automaton graphs, set-oriented representations invokingformal languages, or algebraic models as the (max,+)-algebra. This chapter presents two methodsfor determining a particular algebraic model with respect to a subclass of discrete event systems,finite state automata. The result is a discrete state space model over a finite field.

In order not to loose oneself in too much technicalities, consider the following illustration (Fig-ure 3.1) which reflects a deterministic automaton model of a simple conveyor belt and its interpre-tation as a discrete state space model over a finite field — Table 3.1 depicts the meaning of theused symbols, Figure 3.2 illustrates the conveyor belt states.

1 2 3 4

(00

) (01

) (10

) (11

)

εε

off

off on

on

wpl

wpa

u = 0

u = 0

u = 0u = 0

u = 1

u = 1

u = 1

u = 1

Figure 3.1: Simple conveyor belt — automaton graph (top), state space model interpretation (bottom)

26 CHAPTER 3 — FINITE STATE AUTOMATA IN THE STATE SPACE

Symbol Type Meaning

on Event motor of the conveyor belt switched on

off Event motor of the conveyor belt switched off

wpa Event workpiece arrives at sensor of conveyor belt

wpl Event workpiece leaves sensor of conveyor belt

ε Event empty string (no event taking place)

1 State (initial) conveyor belt at rest, without workpiece

2 State moving conveyor belt, without workpiece

3 State moving conveyor belt, workpiece detected at sensor

4 State conveyor belt at rest, workpiece detected at sensor

Table 3.1: Meaning of the symbols in the automaton graph used in Figure 3.1

Figure 3.2: Conveyor belt states in the automaton graph of Figure 3.1

In the state space model,u is to be understood as an input variable that takes on valuesu∈ {0,1}.Settingu = 1 results in switching the conveyor belt motor into the corresponding other mode —from on to off and from off to on — andu = 0 means to stay in the respective mode. For practicalpurposes,u = 1 is the actual control action,u = 0 corresponds to doing nothing. The statesx ofthe automaton graph are coded in binary vectors with values in {0,1}× {0,1}. In light of themathematical preliminaries from Chapter 2, for a fixed counter instantk ∈ N0 the respective statevectorx(k) and input vectoru(k) can be interpreted as elements of a vector space over a finite fieldwith characteristic 2, i. e. fork∈ N0 fixedx(k) ∈ F2

2 andu(k) ∈ F2.

Additionally, the inputu(k) can be considered as a necessary condition that has to be met in instantk for enabling a state transition from statex(k) to statex(k+ 1). For instance in the example

27

problem from above, if the inputu(k) = 0 for all k∈ N0 then beginning in some instant, two of thestates show a 2-periodic transient behavior

· · ·(

01

)

→(

10

)

→(

01

)

· · ·

and the remaining two states are mapped onto itself.

Considerations like those lead to the central question of this chapter:

How to determine an algebraic relation that represents the state evolution dependingon the input?

On the basis of an algebraic relation, a behavior as in the above-sketched example problem can berelated to a corresponding algebraic property, which for the case of linear models will be examinedin detail in Chapter 4 and Chapter 5 where the focus is on the analysis and synthesis of linearsystems over finite fields, whereas this chapter is to demonstrate how such discrete models canbe derived by referring to methods that base on an elementarylevel of boolean algebra — seeRemark 3.1. From this boolean algebra point of view, discrete models over a finite fieldFq arenatural just forq = 2. For this reason, the development of the discrete state space models willconcentrate on the case ofF2 as well as the examples in the subsequent Chapters 4 and 5, thoughthe examinations in these later chapters encompass the general caseFq for q an arbitrary prime.

The chapter continues with a review of the minimum necessarybasics from boolean algebra inSection 3.1. Besides the standard boolean operations, in particular the isomorphism of booleanalgebra with the finite fieldF2 is stressed. Utilizing the disjunctive normal form and the Reed-Muller form, two methods for deriving a finite field model overF2 out of a state transition table oran automaton graph are presented in Section 3.2. Final remarks comment on how these methodscan be applied for modeling deterministic systems.

Remark 3.1From a more general point of view, discrete state space modelcan be defined as polynomial dy-namic systems over an arbitrary finite fieldFq, for which it is essential that the respective transitionfunction is a partial, polynomial function overFq. In order to develop methods for such systemsthat are rich in content, one is bound to make considerable enlargements within “algebraic tool-box”, that means, it is necessary to introduce basics from algebraic geometry and eliminationtheory [CLO98], for example ideals, varieties and Gröbner-bases. For the scope of analysis andsynthesis of only linear systems as it is the goal in the Chapters 4 and 5 this would much meanoverdoing it. The reader may refer to the excellent works from Hervé Marchand and Michel LeBorgne at the Institut National de Recherche en Informatique et Automatique (INRIA) and theInstitut de Recherche en Informatique et Système Aléatoires (IRISA), both in Rennes (France) —for a short overview see [LBL89, LBL91, BLL91], deeper insight offers the thesis [Mar97] andthe technical reports [ML97, ML99, PML99]. �


3.1 The Relation of Boolean Algebra with the Finite FieldF2

Some basics from boolean algebra are required for determining the discrete system model over afinite field [BP81, Tha88].

Definition 3.1 (Boolean Operations)Given the setB = {0,1}. The boolean operations AND “∧” (conjunction), OR “∨” (disjunction),XOR “⊕” (exclusive disjunction) and NOT “¯” (negation) are definedonB as follows:

x1 x2 x1∧x2

0 0 00 1 01 0 01 1 1

x1 x2 x1∨x2

0 0 00 1 11 0 11 1 1

x1 x2 x1⊕x2

0 0 00 1 11 0 11 1 0

x x

0 11 0 �

Boolean operations allow to construct boolean functions, which usually are expressed in normalform representations. In practice, special types of normalforms help to reduce the logic complex-ity by diminishing the number of logical devices and admit aneasier decomposition into logicalsubfunctions in order to improve modularity. The next normal form is standard.

Definition 3.2 (Disjunctive Normal Form (DNF))A boolean functionf : Bn → B with indeterminatesxT = (x1, . . . ,xn) ∈ Bn is given in disjunctivenormal form (DNF) if withcT = (c1, . . . ,cn) ∈ Bn

f (x) =∨

c∈Bn

(

f (c)∧n∧

i=1

(xi ⊕ci))

. �

As already suggested by the denotation “normal form”, the following holds true.

Theorem 3.1 (Uniqueness of the DNF of a Boolean Function)For any boolean functionf : Bn → B a representation in a disjunctive normal form exists. Exceptfor ordering this representation is unique. �

Example 3.1The disjunctive normal form of the boolean functionf : B2 → B with f (x1,x2) = x1⊕x2 is

f (x1,x2) =(

f (0,0)∧ (x1⊕0)∧ (x2⊕0))∨(

f (0,1)∧ (x1⊕0)∧ (x2⊕1))∨

(f (1,0)∧ (x1⊕1)∧ (x2⊕0)

)∨(

f (1,1)∧ (x1⊕1)∧ (x2⊕1))

=((x1∧ (x2⊕1)

)∨((x1⊕1)∧x2

)= (x1∧ x2)∨ (x1∧x2), x1,x2 ∈ B .

Hence, the XOR-operation can be eliminated by means of the operations AND, OR and NOT.�

SECTION 3.1 — THE RELATION OF BOOLEAN ALGEBRA WITH THE FINITE FIELD F2 29

An other important normal form is based on XOR and AND [Zhe27,Mul54, Ree54, HHL+00].1

Definition 3.3 (Zhegalkin Form (Reed-Muller Form))A boolean functionf : Bn → B with indeterminatesxT = (x1, . . . ,xn) ∈ Bn is given in Zhegalkinform (Reed-Muller form) if

f (x) =⊕

S∈2I

(

δS ∧∧

i∈S

xi

)

in which 2I is the power set of the index setI = {1,2, . . . ,n} andδS ∈ B are constants. �

Theorem 3.2 (Uniqueness of the Zhegalkin Form of a Boolean Function)For any boolean functionf : Bn → B a representation in a Zhegalkin form (Reed-Muller form)exists. Except for ordering this representation is unique. �

Example 3.2The Zhegalkin form of an arbitrary boolean functionf : B2 → B with indeterminatesx1, x2 reads

f (x1,x2) = δ /0 ⊕ (δ1∧x1)⊕ (δ2∧x2)⊕ (δ1,2∧x1∧x2)

in whichδ /0,δ1,δ2,δ1,2 ∈ B are constants. �

Any boolean operation in Definition 3.1 can be expressed by XOR and AND only. To this end,observe that

x = 1⊕x, x∈ B, (3.1)

which applying DeMorgan’s Law results in

x1∨x2 = x1∧ x2 = 1⊕ ((1⊕x1)∧ (1⊕x2)) = x1⊕x2⊕ (x1∧x2), x1,x2 ∈ B . (3.2)

The underlying algebraic system with the operations XOR andAND is a ring with respect to(B,⊕,∧), the so-called boolean ring, and by deeper inspection it canbe inferred thatB\{0} is acommutative group with respect to∧, see the Definitions 2.3 and 2.5. As a consequence,(B,⊕,∧)

is a field. Even further, the operations XOR and AND on the setB = {0,1} can be identified withaddition modulo 2 and multiplication modulo 2 on the fieldF2, that is the following importanttheorem can be stated.

Theorem 3.3 (Isomorphism ofF2 and B)The setB = {0,1} together with the operations+ := ⊕ and· := ∧ is a finite field. The finite fieldB is isomorphic to the Galois-FieldF2. �

1In the year 1927, Zhegalkin was the first to discover that sucha normal form exists for any boolean function.Reed and Muller, to whom these forms now are attributed, 1954reinvented this normal form and extended the notionto arbitrary rings over finite fieldsFq. Currently, many different Reed-Muller forms circulate inthe literature. Exactlyspeaking, the Reed-Muller form presented here is the so-called positive polarity Reed-Muller expansion, a denotationthat refers to the positive exponents regarding the indeterminates (all equal 1 for the caseF2).


Since any boolean functionf can be manipulated so as to obtain its Zhegalkin form which com-prises the operations⊕ and∧ only, the calculation of the finite field representation off overF2

amounts to a simple interchange of⊕ by +mod2 and∧ by ·mod2, respectively (starting from hereaddition and multiplication understood modulo 2). Then forx1,x2 ∈ B the following equivalencesto F2 apply:

x1∧x2 ⇐⇒ x1x2 (3.3)

x1∨x2 ⇐⇒ x1 +x2 +x1x2 (3.4)

x1⊕x2 ⇐⇒ x1 +x2 (3.5)

x ⇐⇒ 1+x (3.6)

These equivalences indicate how to transform an arbitrary boolean function into its respectivecounterpart over the finite fieldF2.

3.2 Methods for Determining the State Space Model

Considering the concepts from boolean algebra presented above, this section exposes two methodsfor determining a discrete system model over the finite fieldF2. The outcome of these methods isan implicit multilinear transition functionf : Fn

2 ×Fn2 ×Fm

2 → F2 according to

f (x(k+1),x(k),u(k)) = 0 =∑

S1∈2In

∑

S2∈2In

∑

S3∈2Im

δS1,S2,S3

(∏

i∈S1

xi(k+1))(∏

j∈S2

x j(k))(∏

l∈S3

ul (k))

, (3.7)

in whichk∈ N0 is a counter,x(k) ∈ Fn2 andu(k) ∈ Fm

2 are the state and input vector of the systemat instantk, respectively. The setsIn = {1,2, . . . ,n}, Im = {1,2, . . . ,m} are index sets, 2In denotesthe (possibly empty) power set ofIn andδS1,S2,S3 ∈ F2 are constants. The functionf takes onthe value 0 at a fixed instantk if x(k), x(k+ 1), u(k) represents an admissible evolution of statex(k) into x(k+ 1) under the inputu(k) in the underlying system, otherwise the value off is 1.2

Inspecting the next statex(k+1) for fixedk, multiple successorsx(k+1) may be observed sincefis an implicit function for the next statex(k+1) — strictly speaking a relation. This indicates thatthe finite field representation (3.7) is capable of modeling non-deterministic finite state automataas well.

2It is a peculiarity of discrete models over finite fieldsFq that one single functionf is sufficient for representingthe state transition behavior even forn > 1 states. For the finite fieldF2 this can easily be seen when consideringi = 1, . . . ,n equationsfi(x) = 0 in which only thosex = x⋆ are considered admissible that solve alln equations. As aconsequence,f (x) := 1+

∏ni=1(1+ fi(x)) = 0. A similar result can be shown for the general caseFq.

SECTION 3.2 — METHODS FORDETERMINING THE STATE SPACE MODEL 31

(

00

) (

01

)

(

10

) (

11

)

u = 0 u = 1

Figure 3.3: Graph of an example automaton (above) andits state table (to the right). The column markedfc signifieswhether a transition from(x1,x2)

T to (x′1,x

′2)

T under inputu is admissible and vice versa.

u x′2 x′1 x′2 x′1 fc

0 0 0 0 0 10 0 0 0 1 00 0 0 1 0 10 0 0 1 1 00 0 1 0 0 10 0 1 0 1 10 0 1 1 0 00 0 1 1 1 00 1 0 0 0 10 1 0 0 1 00 1 0 1 0 00 1 0 1 1 00 1 1 0 0 00 1 1 0 1 10 1 1 1 0 00 1 1 1 1 11 0 0 0 0 11 0 0 0 1 01 0 0 1 0 11 0 0 1 1 01 0 1 0 0 01 0 1 0 1 11 0 1 1 0 01 0 1 1 1 01 1 0 0 0 11 1 0 0 1 01 1 0 1 0 01 1 0 1 1 01 1 1 0 0 11 1 1 0 1 11 1 1 1 0 01 1 1 1 1 1

3.2.1 The Disjunctive Normal Form Method

In what follows, a single input example is used to introduce the main steps for obtaining the discretetransition function over the finite fieldF2 for a non-deterministic automaton. The underlyingalgorithm can be generalized easily and is omitted for clearness.

Consider the automaton depicted in Figure 3.3. The nodes arecoded by binary vectors, whichrepresent the statesxT = (x1,x2) ∈ F2

2 . Arcs connect the states and indicate admissible transitionsbetween the states. Marked arcs denote that the transition is admissible only if a certain conditionimposed on the input variablesu is satisfied, that is ifu = 1. If no marking is specified on an arcthen a transition is admissible under any choice of inputs. Obviously, there are arcs with the samemarking that lead to different successor states, i. e. the automaton is non-deterministic.

For abbreviation, let the symbolk be omitted within the denotation ofxi(k) and u(k) and letxi(k+1) be abbreviated byx′i instead. In order to work out the state transition function,the logicalinterconnection of the statex and inputu and successor statex ′ is translated into a state table (right


hand side of Figure 3.3). Clearly, each row in the state tablecorresponds to a function valuefc = 1if a transition in the automaton graph is admissible,fc = 0 if it is not.

Therefore, in view of the DNF, Definition 3.2, and along the lines of Example 3.1 the DNF of thefunction f represented in the state table of Figure 3.3 reads

f (x′1,x′2,x1,x2,u) = ux′2x′1x2x1∨ ux′2x′1x2x1∨ ux′2x′1x2x1∨ ux′2x′1x2x1∨ ux′2x′1x2x1∨

ux′2x′1x2x1∨ ux′2x′1x2x1∨ux′2x′1x2x1∨ux′2x′1x2x1∨ux′2x′1x2x1∨ux′2x′1x2x1∨ux′2x′1x2x1∨ux′2x′1x2x1∨ux′2x′1x2x1 = 1, (3.8)

⇐⇒ f (x′1,x′2,x1,x2,u) = x′2x′1x2x1∨ x′2x′1x2x1∨ ux′2x′1x2x1∨ x′2x′1x2x1∨x′2x′1x2x1∨

x′2x′1x2x1∨x′2x′1x2x1∨ux′2x′1x2x1 = 1, (3.9)

in which the abbreviationa∧b = ab is used. Observe that by applying DeMorgan’s law

a1∨a2∨ . . .∨ak = 1 ⇐⇒ (1⊕a1)∧ (1⊕a2)∧ . . .∧ (1⊕ak) = 0, (3.10)

all disjunctions can be eliminated. The remaining negations in (3.9) vanish by setting ¯a = a⊕1,thus, a function that comprises∧ and⊕ only is obtained. Therefore, with (3.3) and (3.5), therepresentation of the transition function in the finite fieldF2 is

f (x′1x′2,x1,x2,u) =(1+(1+x′2)(1+x′1)(1+x2)(1+x1)

)· · ·

(1+x′2x′1x2x1

)(1+ux′2x′1(1+x2)(1+x1)

)= 0, (3.11)

⇐⇒ f (x′1x′2,x1,x2,u) = x1 +x1x′1 +x2x′1+x2x′2 +x1x2x′2 +x′1x′2 +x1x′1x′2 +

x1x2x′1x′2+x′1u+x1x′1u+x2x′1u+x1x2x′1u = 0. (3.12)

3.2.2 The Reed-Muller Generator Matrix Method

The regular tabulation of the state table in Figure 3.3 — which originates from binary counting,row by row — gives a hint for determining the transition function f more efficiently. So-calledReed-Muller codes, well-known in linear coding theory, exploit this property [HHL+00].

Consider the Reed-Muller generator matrixGi ∈ F2i×2i

2 , i ∈N0, which is defined recursively as per

Gi :=

(Gi−1 0Gi−1 Gi−1

)

, G0 := 1. (3.13)

Using this Reed-Muller generator matrix the demanded transition function reads [Ree54, HHL+00]

f (x′1,x′2, . . . ,x

′n,x1,x2, . . . ,xn,u1,u2, . . . ,um) = (G2n+mfc)

TϕϕϕT2n+m+1 = 0. (3.14)


In equation (3.14), the vectorϕϕϕ2n+m with dim(ϕϕϕ2n+m) = 22n+m is a vector of those monomials thatcan be combined from the variable sets{x1, . . . ,xn}, {u1, . . . ,um}, and{x′1, . . . ,x

′n}, which are the

entries of the state vectorx, the input vectoru and the next state vectorx ′, respectively. The vectorfc with dim(fc) = 22n+m is made up of the rightmost column entries in the state table of Figure 3.3.

It remains to explain how to tabulate the monomials inϕϕϕ2n+m. This will become clear by resortingto the example of Figure 3.3, in which

G5 =

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0

1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0

1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0

1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0

1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0

1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0

1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

, ϕϕϕ5 =

1

x1

x2

x1x2

x′1x1x′1x2x′1x1x2x′1x′2x1x′2x2x′2x1x2x′2x′1x′2x1x′1x′2x2x′1x′2x1x2x′1x′2u

x1u

x2u

x1x2u

x′1u

x1x′1u

x2x′1u

x1x2x′1u

x′2u

x1x′2u

x2x′2u

x1x2x′2u

x′1x′2u

x1x′1x′2u

x2x′1x′2u

x1x2x′1x′2u

(3.15)

are the Reed-Muller generator matrix and the respective vector of monomials. The tabulationregarding the elements ofϕϕϕ5 obeys a recursion: if starting the inspection from the top ofthe vectorthen any added new variable becomes a right factor of a copy ofthe former part of the vector, andso on. A substitution ofG5 andϕϕϕ5 from (3.15) in (3.14) verifies the result from (3.12).

3.2.3 Deterministic State Space Model

The following is to comment on how an explicit deterministicmodel can be generated for a de-terministic finite state automaton. In this case, the state tables can be reshaped as illustrated inTable 3.2. Thus, by employing the methods from Section 3.2.1and 3.2.2 a transition function canbe determined in which the next state becomes explicit, thatis

x(k+1) = f(x(k),u(k)) (3.16)


um · · · u2 u1 x′′

n · · · x2 x1 x′n · · · x′2 x′10 · · · 0 0 0 · · · 0 0 fn,1 · · · f2,1 f1,1

0 · · · 0 0 0 · · · 0 1 fn,2 · · · f2,2 f1,2

0 · · · 0 0 0 · · · 1 0 fn,3 · · · f2,3 f1,3... · · · ......

... · · · ......

... · · · ......

1 · · · 1 1 1 · · · 1 1 fn,2n+m · · · f2,2n+m f1,2n+m

Table 3.2: Typical shape of a state table regarding a deterministic system

with k∈ N0, x(k) ∈ Fn2 , andu(k) ∈ Fm

2 , which except for the field reminds of the standard form ofa non-linear discrete time continuous system.

Since the further examinations in the next chapters are restricted to the deterministic case, the ex-ample from the introductory part of this chapter shall be reconsidered, for convenience. Concern-ing this example, the respective state table, Reed-Muller generator matrix and appendant vector ofmonomials are

u x′′

2 x1 x′2 x′10 0 0 0 00 0 1 1 00 1 0 0 10 1 1 1 11 0 0 1 01 0 1 1 11 1 0 0 01 1 1 0 1

G3 =

1 0 0 0 0 0 0 01 1 0 0 0 0 0 01 0 1 0 0 0 0 01 1 1 1 0 0 0 01 0 0 0 1 0 0 01 1 0 0 1 1 0 01 0 1 0 1 0 1 01 1 1 1 1 1 1 1

, ϕϕϕ3 =

1x1

x2

x1x2

ux1ux2u

x1x2u

.

Denoting the columns regarding the next state entriesx′2 andx′1 in the state table byfc,2 andfc,1,respectively, the application of equation (3.14) results in the state equations

x1(k+1) = (G3 fc,1)TϕϕϕT

3(k)+1 = x2(k)+(x1(k)+x2(k)

)u(k) ,

x2(k+1) = (G3 fc,2)TϕϕϕT

3(k)+1 = x1(k)+u(k)+(x1(k)+x2(k)

)u(k) ,

which are non-linear in this case. Observe that by applying astate feedback

u(k) = 1+x1(k)+x2(k)

which avoids that the conveyor belt gets at rest, an affine linear system

x(k+1) =

(0 10 1

)

x(k)+

(01

)

can be obtained that can readily be analyzed with the methodsto be derived in the next chapter.


Conclusion

In this chapter, an algebraic model for the transient behavior of finite state automata is exposed.On the face of it, this algebraic model appears to be similar to a discrete time system in the worldof continuous systems. However, the discrete state space model established here resides on a finitefield. For simplicity, it is shown in an exemplary manner thatthe determination of a state spacemodel over a finite fieldF2 involves basics from boolean algebra only. By invoking the disjunctivenormal form and the Reed-Muller form of a boolean function, two methods are proposed that allowto derive non-deterministic discrete state space models over the finite fieldF2 for such systems. Inboth methods it is assumed that any state transitions are tabulated in a state table, which does notincur much a restriction as such tables can easily be obtained from other system representations,as for example automaton graphs.

The first procedure is the common boolean algebra method for determining a boolean function outof a set of its function values. The respective disjunctive normal form (DNF) of such a functioncan be simply read off the state table. In order to obtain a function that depends on the operationsXOR and AND only, all disjunctions and negations are eliminated in the DNF, which results in theReed-Muller form of this function. Employing the isomorphism of boolean algebra and of finitefields overF2, the outcome is an implicit function that typically represents a non-deterministicstate transition behavior, as demonstrated in an example.

In the second procedure the Reed-Muller form of the transition function is determined directly.This methods turns out to entail much less calculation effort as just a matrix multiplication with ageneric matrix, the so-called Reed-Muller generator matrix, is required for calculating the Reed-Muller form. For this reason, this method is pursued in the remark on deterministic systems, inwhich the impressive simplicity is illustrated by means of deriving the discrete state equation overF2 for the deterministic example problem in the introductory part of this chapter.

Chapter 4

Analysis of Linear Systems over FiniteFields

The main goal of the study in Chapter 3 was to obtain a state space model over finite fields for arather broad class of discrete event systems. Throughout the next two chapters these models shallbe restricted to the deterministic linear case; the technical term linear modular systembecameestablished for systems, in which the expressionmodularindicates the premise of a finite field.

Even though the scope of linearity seems far from any practical interest, the consideration of linearstate space models for finite state automata is worthwhile since their study reveals important insightinto the state transition behavior of automata in general. In addition to that, the class of lineardiscrete systems over finite fields naturally entails a multitude of far-reaching propositions. Not atleast, the resulting algorithms are mainly of tractable computational complexity.

When a vector space is endowed with a relation, the type of this relation imposes a particularstructure on the underlying vector space. Accordingly, linear state equations impose a somewhatparticular structure, a linear structure on the respectivestate space. Thus, the notion itself suggeststhat this linear structure of the state space can be exploited for an examination of the interconnec-tion structure of the states in the corresponding automaton. This is the basic idea behind the linearstate space models for automata recalled in Section 1.1 of the introductory chapter. Common toall of them is the attempt to determine periodicity properties of states in terms of specific systeminvariants; for instance eigenvalues of the system dynamics. Yet, using eigenvalues only necessarycriteria for periodicity properties have been stated, and without calling for a periodicity test whichutilizes the system equations with respect to each considered state [Son00], that is enumeratingthe state space, no sufficient statements have been derived so far. This lack of sufficient crite-ria is due to the absence of the field property in these state space models, and as a consequencefield-dependent concepts as are for example eigenvalues cannot be employed, which highlights thedecisive constructional flaw within these approaches.

38 CHAPTER 4 — ANALYSIS OF L INEAR SYSTEMS OVERFINITE FIELDS

Assuming a linear state space model over finite fields, it is the main objective of this chap-ter to develop and present a necessary and sufficient criterion for determining the entire struc-tural state space decomposition into periodic/aperiodic subspaces comprising the associated pe-riodic/aperiodic states. As a result, the periodic and aperiodic properties of any automaton statecan be determined by an effective method which is dependent on a similarity transformation anddivisions of polynomials only, whilst at the same time refraining from any state space enumera-tion technique (exhaustive search algorithms or testing),known to be intractable for large systemdimensions. The key concept is the notion of a linear modularsystem (LMS).

Primarily, the theory of LMS was developed by Huffmann underthe technical termlinear sequen-tial coding networksto the end of generating quasi-random binary digital numbers by employingshift registers with feedback logic [Huf56, Huf59]. The major application was in the field of error-correcting codes [PW72]. Many of the results to be presentedin this chapter can be traced backto Elspas and Friedland [Els59, Fri59], who worked out the essential properties oflinear sequen-tial (switching) circuitsunder an algebraic perspective using finite fields. A collection of thesecontributions1 is given with [Kau65]. Later developments focused on how periodic states couldbe related to invariants of the characteristic matrix of a linear sequential circuit; the main part ofthe contributions originate to Gill [Gil64, Gil66b, Gil66a, Gil69] whose results are particularlyimportant within the analysis of LMS.

The first objective of this chapter is to present the basic concept of a linear modular system. Onthis basis, autonomous and the subclasses of homogeneous and inhomogeneous linear modularsystems are introduced in Section 4.1. At the beginning of Section 4.2 the period of a state isdefined. The subsequent paragraphs of this section are concerned with the structure of the statespace of homogeneous linear modular systems in order to develop a method by means of which abrute force calculation of periodic states is rendered unnecessary. To this end, the block structureof the rational canonical form of the system dynamics matrixis shown to be in strong relation to thestructure of the state space. Since these blocks consist of cyclic and nilpotent dynamics matricesonly, their influence on the state space is considered, and bylinearity these results are superposedamounting to a statement for the case of arbitrary linear dynamics. The main objective is to statea necessary and sufficient criterion of how to decompose the entire state space into subspaces ofcertain period and cardinality. Resorting to this decomposition the actual states with specifiedperiodic property are calculated in an efficient manner. Section 4.3 presents a generalization of theresults to those inhomogeneous linear modular systems for which a translation of state allows torender the state equations linear. For the case in which sucha linearization is not possible a solutionis given as well. The section closes with the composition of the afore-obtained results. The finalpart of this chapter summarizes the main steps of the analysis developed so far (Section 4.3.3) andexposes an illustrative example (Section 4.3.4).

1the journals of which are rarely available in libraries

SECTION 4.1 — LINEAR MODULAR SYSTEMS (LMS) 39

4.1 Linear Modular Systems

A Linear Modular System (LMS) is a discrete state space system over a finite field for whichthe deterministic state transition functionf given in relation (3.16) is linear. Furthermore, LMSinclude a linear output function relating state and input with an output. An LMS represents adiscrete dynamic system over finite fields which, owing to thefield property, shows analogies tolinear discrete time systems over the field of real numbersR. Due to the fact that the state spaceof an LMS is finite and the transition functions are discrete,LMS are an adequate model for analgebraic examination of simple automata. In view of equation (3.16) a possible definition of anLMS in terms of matrices is as follows

Definition 4.1 (Linear Modular System)A linear modular (k-invariant) system overFq, denoted by LMS(q), is given by an evolution equa-tion of the form

x(k+1) = Ax(k)+Bu(k), k∈ N0 , (4.1)

called state equation, in which addition and multiplication are taken moduloq. Furthermore,

• the vectorx(k) is called state vector or state, the vectoru(k) is called input. The setX = Fnq

of states and the setU = Fmq of inputs are vector spaces, termed state space and input space,

respectively.2

• the dimensionn of the state space, that isn = dim(X), is the order or dimension of an LMS,

• the matrixA ∈ Fn×nq is called the (system) dynamics matrix, the matrixB ∈ Fn×m

q is theinput matrix (all matrices independent fromk). �

For simplicity, the term LMS will be used any time when it is clear that the primeq is not yetspecified. A system is said to be autonomous if the input has noinfluence on the evolution of state.For an LMS according to equation (4.1) this amounts toB = 0. Moreover, such an autonomousLMS is called homogeneous. If otherwise for all instantsk the vectorBu(k) = const. =: b isnon-zero then the autonomous LMS is called inhomogeneous.

Obviously, LMS are based on all the concepts from algebra that have been developed so far inChapter 2 and Chapter 3. For LMS(2) the latter chapter, Chapter 3, introduced methods fromBoolean algebra for deriving the respective state space model over the finite fieldF2. That is whymost of the examples will be confined to the case ofF2 though the results are formulated forFq

with q an arbitrary prime.

2More precisely,x(k) is the vector-valued mappingx : N→ Fnq andu(k) is the vector-valued mappingu : N→ Fm

q ,both mapping an instantk to a column vector inX andU, respectively. In order to keep the denotation simple this willnot be stressed unless necessary.


4.2 Homogeneous LMS

Firstly, linear homogeneous LMS according to the simplest version of the state space representa-tion in equation (4.1), that is equations of the form

x(k+1) = Ax(k) (4.2)

with x(k) ∈ Fnq andu(k) ∈ Fm

q , k∈ N0, shall be dealt with. In the autonomous case, evidently, anyinformation must be included in the structure of the dynamics matrixA. The transition of statex(k)to x(k+1) is determined by the linear mapping which corresponds to thematrix A. By applyingthis mappingκ times the statex(k) is mapped to the statex(k+κ). Consider the sequence of statesx(k), x(k+1), . . . ,x(k+κ) which originates from aκ-fold sequence of these mappings. It is easyto see that beginning from an instanti, when some statex(i) occurs again at instanti + t, t ∈ N, allt −1 states in between these identical states show the same characteristict-periodic behavior forincreasing instancesκ > i + t. This idea is characterized in the following definition.3

Definition 4.2 (Period of a State)A statext of an LMS is calledt-periodic if

xt ∈ Xt , Xt :={

ξξξ ∈ Fnq |∃ t ∈ N, ξξξ = At ξξξ ∧ ∀ i ∈ N, i < t, ξξξ 6= A i ξξξ

}

.

Xt is the set oft-periodic states. �

A particular periodic state is the zero statext = 0 (null state). In fact, any autonomous LMScomprises a zero state with periodt = 1. Apart from the zero state there need not exist anotherperiodic state in the state space of an LMS. If only the zero state is periodic all state transitionsterminate finally in the zero state. If there is at least one other periodic state (besides the zerostate) then some of the state transitions terminate in the periodic structure/s associated to thisother periodic state/s with some period. Such periodic structures will be referred to as cycles, adenotation which becomes perspicuous in the state graph, see Figure 4.1.

Apparently, for a linear modular system (4.2) thet-periodic statesxt ∈ Fnq can be determined from

the linear system of equations(I −At)xt = 0, (4.3)

which for example by employing Gauß’ algorithm can be solvedfor xt concerning all conceivableperiodst; see the previous Example 2.3.

However, doing that way is by no means an efficient procedure for determining all periodic statesin the state spaceFn

q with respect to an autonomous LMS. The most crucial problem is the typically

3A similar argument is valid for autonomous non-linear deterministic discrete state space systems over finite fields.

SECTION 4.2 — HOMOGENEOUSLMS 41

(000

)

(001

)

(110

)

(010

)

(011

) (101

)

(111

) (100

)

Figure 4.1: Typical example for a state graph with cycles in an LMS of order 3

huge number ofqn states such that after subtracting the 1-periodic zero state, periodic states up tothe maximal possible period oftmax= qn−1 (all but one state in the same cycle) may occur.

In the worst case the calculation of all periodic statesxt , t = 1,2, . . .qn−1, entails the effort of

• (qn−2)-fold multiplication of the matrixA for calculating the matrixAt ,

• qn−1 times solving the linear system of equation (4.3).

A remaining problem is that these calculations still do not yield the sets of states sorted by itsperiod. The reason is that all solutionsxtd of

(I −Atd)xtd = 0

for any integertd such thattd|t, that are any divisorstd which dividet, are solutions of (4.3) as well.Hence, ordinary evaluations of (4.3) as the method proposedin [Son99] bear the risk of enormouscomputational expenses.

Here on the contrary, the proposal is to benefit from algebraic properties within the structure ofthe dynamics matrixA. The interconnection of the states is given by the linear modular system(4.2), which relates the statex(k) in instantk with the next statex(k+ 1). It is obvious that a(bijective) change of coordinates on the states, which justamounts to a relabeling of states, doesnot affect the state graph, hence the state interconnectionstructure. Concerning linear systems (4.2)a linear change of coordinates on the statex results in a similarity transformation of the dynamicsmatrix A. Among others, the most revealing similarity transformation of A is the transformationinto the rational canonical formArat = TAT−1, introduced in equation (2.14). The matrixArat istransformed into a maximal number of diagonal blocks, each of which is entirely specified by oneof the elementary divisor polynomials ofA, which are invariant under similarity transformations.As these invariants ofA andArat, respectively, cannot be altered by a coordinate change, theseinvariants must express the interconnection structure of the states. For this reason, the elementarydivisor polynomials have to express the periodicity properties of all states in the respective statespace.


As a consequence, when analyzing for periodic states, one isjustified in examining the periodicityproperties by means of the structurally simpler rational canonical formArat = diag(C1,C2, . . . ,CN)

of the dynamics matrixA. Thus, by transformingx = Tx with the invertible matrixT ∈ Fn×nq ,

which only renumbers the state vectors, equation (4.3) can be turned into

(I −diag(C t1,C

t2, . . . ,C

tN)) xt = 0. (4.4)

Recall that the companion matricesCi , i = 1, . . . ,N are determined by the elementary divisorpolynomials ofA each of which is a power of an irreducible polynomial. Since the matrix in (4.4)is of block-diagonal type the system of equations can be partitioned intoN subproblems to besuperposed finally. With regard to the solution of equation (4.4) two disjoint sets with cardinalityNc andNn yieldingN = Nc +Nn can be distinguished due to the finiteness of the underlying field4

1. the set of allNc cyclic companion matrices, i. e. allCi such thatC tii = I for someti ∈ N,

2. the set of allNn nilpotent companion matrices, i. e. allCi such thatC tii = 0 for someti ∈ N.

Remark 4.1If the dynamics matrixA is not singular then all companion matricesCi in Arat are cyclic. Fur-thermore, the rightmost column entries of nilpotent companion matrices are all zero. Therefore,for finding out whether a companion matrix is nilpotent it is not necessary to calculate any matrixpower. �

Remark 4.2In Section 4.2.2 nilpotent companion matrices referring toelementary divisor polynomials of theform pC(λ) = λh with h ∈ N will be concerned separately as these are not related to periodicsubspaces. �

It remains to introduce some notation in order to facilitatereferences in the subsequent sections.

As the rational canonical formArat is determined except for the ordering of its block-diagonalcompanion matrices, its companion matrices can be reordered such that

Arat = diag(Ac,An) , (4.5)

in which Ac andAn collect the cyclic and nilpotent companion matrices withinArat, respectively.Thus, the system is decomposable into a cyclic and a nilpotent subsystem

xc(k+1) = Ac xc(k) , (4.6)

xn(k+1) = An xn(k) (4.7)

4Note that companion matrices over the infinite field of real numbers can be neither cyclic nor nilpotent.


with k ∈ N0 andxT(k) =: (xTc (k), xT

n(k)) such thatxc(k) ∈ Fncq , xn(k) ∈ Fnn

q andFncq ×Fnn

q = Fnq ,

that isn = nc +nn. Furthermore,Nc cyclic andNn nilpotent companion matrices composeAc andAn, respectively. Results regarding any of these subsystems can be superposed due to linearity.The associated original statesx are given by the inverse coordinate transformation5

x = T−1 x . (4.8)

In what follows, if not specified any dynamics matrixA of an LMS shall be assumed in its re-ordered rational canonical formArat — since resorting to the appropriate transformation matrixT any dynamics matrix can be transformed into its appropriaterational canonical form and viceversa, which therefore incurs no loss in generality.

4.2.1 Cyclic Dynamics

First of all, by referring to equation (4.6) the cyclic system partAc and its respective state spaceFncq

shall be examined. This amounts to the case as if a cyclic dynamics matrixA were under concern.An adaption of condition (4.4) to the cyclic system part (4.6) allows to calculate at-periodic statexc,t ∈ Fnc

q by means of the followingNc equations

(I −C ti ) x(i)

c,t = 0, i = 1, . . . ,Nc , (4.9)

to be fulfilled all at once. In the latter relation theNc-fold composition

xc,t =

x(1)c,t

x(2)c,t...

x(Nc)c,t

, x(i)c,t ∈ Fdi

q , i = 1, . . . ,Nc (4.10)

indicates theNc disjoint subspaces satisfyingFd1q ×·· ·×F

dNcq = Fnc

q .

On the other hand, each of thesei = 1, . . . ,Nc subspaces has its ownti-periodic vectorsx(i)c,ti which

are determined by(I −C ti

i ) x(i)c,ti = 0. (4.11)

Consequently, each of thesei = 1, . . . ,Nc subspaces has its own periodic properties, to be examinedwith equation (4.11). As the problem is linear, the periodicproperties of the entire state spaceFnc

q = Fd1q ×·· ·×F

dNcq are determined by the periodic properties of its parts (Section 4.2.1.1). In

the next step, the superposition yields all periodic statesxc,t with regard to equation (4.10), thatis for the cyclic partAc, which is block-diagonally composed of cyclic companion matrices only(Section 4.2.1.2).

5The transformation matrixT transformingA into Arat = TA T−1 results inT = ΠΠΠT, whereT is the transformationmatrix such thatArat = TAT −1 and the matrixΠΠΠ is the orthogonal permutation matrix that provides the exchanges ofcompanion matrices inArat according toArat = ΠΠΠAratΠΠΠT. For details see Chapter A of the Appendix.


4.2.1.1 Periodic States of an LMS with Cyclic Companion Matrix as Dynamics

Thus, the first task is to determine the periodti which corresponds to a statex(i)c,ti in the subsystem

x(i)c (k+1) = Ci x

(i)c (k).6 The key to the solution of the problem is to take advantage from the notion

of annihilating polynomials of matrices and their properties [LT85].

Definition 4.3 (Annihilating Polynomial)Let M ∈ Fn×n be a matrix. Then any polynomialp∈ F[λ] for which

p(M) ≡ 0

is called annihilating polynomial ofM . �

The standard examples for annihilating polynomials of a matrix M are its minimal polynomialmpM (λ) and its characteristic polynomial cpM (λ), the latter of which is characterized in the well-known theorem of Cayleigh-Hamilton.

Theorem 4.1 (Cayleigh-Hamilton Theorem)Let M ∈ Fn×n be a matrix. Then the respective characteristic polynomialcpM (λ) := det(λ I −M)

is one of its annihilating polynomials:

cpM (M) ≡ 0. �

A simple consequence on companion matrices is due to Theorem2.9.

Corollary 4.1 (Cayleigh-Hamilton Theorem for Companion Matrices)Let pC ∈ F[λ] be the defining polynomial of the companion matrixC. Then

pC(C) ≡ cpC(C) ≡ mpC(C) ≡ 0. �

The following theorems provides a link between divisibility of polynomials and singularity ofmatrices, which will be exploited in the proofs of the remaining part of this section.

Theorem 4.2 (Minimal Polynomials Divide Annihilating Polynomials)Let M ∈ Fn×n be a matrix, mpM (λ) its minimal polynomial andf ∈ F[λ]. Then f is an annihilatingpolynomial ofM iff mpM divides f, that is

f (M) ≡ 0 ⇐⇒ mpM (λ)| f (λ) . �

6Throughout Section 4.2.1.1 the notationx andC will be used instead of the more involvedx(i)c,ti andCi , respectively.


Theorem 4.3 (Singularity of a Matrix Polynomial)Let M ∈ Fn×n be a matrix, mpM (λ) its minimal polynomial andf ∈ F[λ]. Then the matrixf (M)

is singular iff mpM (λ) and f (λ) have at least one common factor, that is7

det( f (M)) = 0 ⇐⇒ gcd(mpM (λ), f (λ)

)6= 1. �

Recall, that each companion matrixCi , i = 1, . . . ,Nc is already of particular type —Ci correspondsto thei-th elementary divisor polynomial ofAc, each of which is a power of one single irreduciblepolynomial only (and not of more of them). Thus, a promising guideline for the subsequent exam-ination is the following: Firstly, the perspective is confined to a cyclic companion matrix definedby an irreducible polynomial. Secondly, a cyclic companionmatrix that is defined by a power ofan irreducible polynomial is dealt with, resorting to the results of stage one.

Companion Matrices of an Irreducible Polynomial

In this part of the section the periodic properties of the state space with respect to a cyclic compan-ion matrix of an irreducible polynomial shall be derived. Tothis end, the following lemma will behelpful.

Lemma 4.1 (Least Exponent of Unipotency of a Matrix)Let mpM ∈ Fq[λ] be the minimal polynomial with respect to a cyclic matrixM ∈ Fd×d

q . Let τdenote the period of mpM (λ). Then the leastt ∈ N such that

M t − I ≡ 0

is t = τ. �

Proof According to Definition 2.8 and since for cyclicM , pM (0) 6= 0, a periodτ of mpM (λ)

exists, this period is the leastτ such that mpM (λ)|λτ−1, or equivalently

g(λ)mpM (λ) = λτ −1

for some polynomialg(λ); where the degree ofpM is minimal (minimal polynomial), the degreeof λτ −1 is minimal (period of a polynomial), hence the degree ofg(λ) is minimal. But then with

g(M)mpM (M) = M τ − I ,

and with the annihilating polynomial of least degree, that is mpM (M) ≡ 0, immediately followsthe result. �

Equipped with this result the central theorem of this paragraph can be proven.

7The expression gcd(a,b) is the greatest common divisor ofa andb.


Theorem 4.4 (Periodic State Space wrt. a Companion Matrix ofan Irreducible Polynomial)Given an LMS whose dynamics matrix is a cyclic companion matrix C ∈ Fd×d

q that correspondsto an irreducible polynomialpC ∈ Fq[λ] of periodτ. Then in the LMS

• all non-zero state vectorsx ∈ Fdq areτ-periodic,

• the states formν = (qd−1)/τ disjoint cycles with periodτ (called cycles of lengthτ),

• the remaining zero-vector has period 1. �

Proof By Corollary 4.1 the defining polynomialpC of the companion matrixC is its annihilatingpolynomial of least degree. Then sinceC is cyclic Lemma 4.1 implies that the periodτ of pC isthe least number such that

Cτ − I ≡ 0.

Therefore, for arbitraryx ∈ Fdq

(Cτ − I)x ≡ 0,

which expresses that any non-zero statex is periodic of period at mostt = τ.

It remains to show that any of these statesx cannot be periodic witht < τ. Assumet ∈ N is theperiod of a statex. Then

(Ct − I)x = 0,

wheret is the least such number. Excluding the trivial casex = 0, which is 1-periodic, the matrixCt − I has to be singular. Consequently, by Theorem 4.3 the minimalpolynomial mpC(λ) andthe polynomialλt − 1 must have a common factor. Since by assumptionpC(λ) = mpC(λ) isirreducible8, the polynomialλt −1 can only be power of mpC(λ), which by Theorem 4.2 entailsthat

(Ct − I)x ≡ 0.

Hence, by Lemma 4.1 the conclusion ist = τ and any non-zero state vectorx in the correspondingstate spaceFd

q is τ-periodic. Leaving aside the zero state, the remainingqd −1 states inFdq form

ν = (qd−1)/τ cycles of lengthτ. �

Remark 4.3The minimally possible period of ad-th degree polynomial overFq is its degreed, its maximallypossible period isqd−1, which yields the inequality

d ≤ τ ≤ qd−1.

The left inequality is clear from the divisibility ofλτ −1 by the respective polynomial. The rightinequality concerns the period of a so-called primitive polynomial. The result is in accordancewith the fact that in any LMS maximally all but the zero state of the qd states inFd

q can make uptogether one single cycle. �

8It is here where the distinction into irreducible polynomials gets its justification.


Example 4.1Given a companion matrixC ∈ F3×3

2 together with its irreducible (characteristic and minimal)polynomialpC ∈ F[λ],

C =

0 0 11 0 10 1 0

, pC(λ) = λ3+λ+1 = 0,

the coefficients of which represent the third column ofC. Multiplying pC(C) by C yields thefollowing

C5 +C3+C2 ≡ 0 | + C3+C+ I ≡ 0

C5+C2 +C+ I ≡ 0

C6 +C3+C2 +C ≡ 0 | + C3+C+ I ≡ 0

C6+C2 + I ≡ 0

C7+C3 +C ≡ 0 | + C3+C+ I ≡ 0

C7+ I ≡ 0

where by construction the minimal numbert = 7 is obtained (see Lemma 4.1). Consequently, theperiod of pC(λ) is τ = 7, which implies thatλ3 + λ + 1|λ7 + 1, i. e. pC(λ) dividesλ7 + 1.9 ByTheorem 4.4 all non-zero state vectorsx ∈ F3

2 are 7-periodic and withν = (23−1)/7 = 1 there isexactly one cycle of lengthτ = 7. �

Companion Matrices of Powered Irreducible Polynomials

It remains to examine the periodic property of the state space regarding a cyclic companion matrixC ∈ Fd×d

q of a powered irreducible polynomial

pC(λ) = mpC(λ) = (pirr,C(λ))e.

This examination is held separate from the past paragraph because in this case, in contrast tothe former, the state spaceFd

q can be decomposed further. It will be organized in three steps:firstly, it will be shown that the kernel of the matrixpC(C) can be decomposed intoe nestedlinear subspaces. Its dimensions shall be determined in a second step. With the knowledge of thedimensions, the number of states in the respective space is clear and the period of these states willbe derived, leading to the main theorem of this section.

First, consider thej = 0,1. . . ,esets of vectors

N j := Ker(

(pirr,C(C)) j)

, N0 := {0} (4.12)

9In light of Remark 4.3 this period is maximal and the irreducible polynomialλ3+λ+1 is a primitive polynomial.


which are linear spaces and thej = 1, . . . ,esets of vectors

D j :={

x ∈ Fdq |(pirr,C(C)) j x = 0∧ (pirr,C(C)) j−1x 6= 0

}

, (4.13)

for which obviously holdsD j = N j −N j−1, j = 1, . . . ,e. (4.14)

Furthermore, note that with this notation and with

(pirr,C(C)) j x 6= 0 =⇒ (pirr,C(C))κ x 6= 0, ∀κ < j (4.15)

and for the minimal polynomial

(pirr,C(C))e≡ 0 =⇒ (pirr,C(C))ex = 0, ∀x ∈ Fdq (4.16)

one obtains the following property.

Lemma 4.2 (Nesting Property of Nullspaces)Let pC(λ) = (pirr,C(λ))e, e∈ N, be thed-th degree defining polynomial inFq[λ] of a companionmatrixC ∈ Fd×d

q . Assume the basis polynomialpC,irr(λ) to be irreducible overFq. Then the strictinclusion property (nesting) applies10

N0 ⊂ N1 ⊂ N2 ⊂ . . . ⊂ Ne = Fdq , (4.17)

where theN j are nullspaces,N0 := {0} andN j := Ker((pirr,C(C)) j

), j = 1, . . . ,e. �

Remark 4.4With equation (4.13) Lemma 4.2 can be expressed equivalently by

Fdq =

e⋃

j=0

D j , D0 := {0} , (4.18)

in whichDi ∩D j = /0 for i 6= j. �

For deriving the dimension of the spacesN j , j = 1, . . . ,e the following lemma will be employed.

10On account of the intention to derive the periodicity properties of linear systems over finite fields, the statementof the theorem is restricted to finite fields, though the nesting property of course holds as well for infinite fields forwhich an analog is given, for example, by generalized eigenvectorsvk of k-th order defined by

(λ I −A)k vk = 0 ∧ (λ I −A)k−1vk 6= 0.

Here the nesting property shows that thek-th order generalized eigenvector space, spanned by the generalized eigen-vectors ofk-th ordervk, contains the(k−1)-th order generalized eigenvector space spanned byvk−1 and so forth.


Lemma 4.3 (Invariance of Nullspaces)Let (pirr,C) j ∈ Fq[λ] be the defining polynomial of the cyclic companion matrixC∈ Fd×d

q in whichthe polynomialpirr,C(λ) is irreducible overFq. Then thej = 1, . . . ,e nullspaces of the matrices(pirr,C(C)) j are invariant under the transformationC, i. e. on anyx j ∈ Ker

((pirr,C(C)) j

). �

Proof By virtue of the non-singularity of cyclic companion matricesC and by commutativity ofmatrix multiplication with regard to matrix polynomials ofthe same matrix, it follows

(pirr,C(C)) j x j = 0 ⇐⇒ C(pirr,C(C)) j x j = 0 ⇐⇒ (pirr,C(C)) j Cx j = 0

for anyx j ∈Ker((pirr,C(C)) j

), j = 1, . . . ,e. This implies the invariance ofN j := Ker

((pirr,C(C)) j

)

under the mappingC. �

In view of Lemma 4.3, forj = 1, . . . ,edefine the mapC|N j : N j → N j such that

C|N j x j = Cx j , ∀x j ∈ N j (4.19)

describes the portion of the linear transformC acting on the subspaceN j only. As a consequence,

(pirr,C(C|N j ))j x j = (pirr,C(C)) j x j , ∀x j ∈ N j

which with equation (4.12) reads

(pirr,C(C|N j ))j x j = 0, ∀x j ∈ N j .

Thus, the polynomial(pirr,C(λ)) j is an annihilating polynomial ofC|N j in the spaceN j , hence, itis a multiple of the minimal polynomial mpC|N j

(λ) — see Theorem 4.2 — and since(pirr,C(λ)) j

is power of an irreducible polynomial the minimal polynomial that corresponds to matrixC|N j canonly comply with

mpC|N j(λ) = (pirr,C(λ))κ

for someκ = 1, . . . , j.

For a contradiction argument assumeκ < j. Using the fact that the minimal polynomial is the leastdegree annihilating polynomial of matrixC|N j in the respective space, that is mpC|N j

(C|N j ) ≡ 0,

it follows(pirr,C(C|N j ))

κ x j = 0, ∀x j ∈ N j .

But on the contrary, forκ < j vectorsx⋆j ∈ N j exist such that

(pirr,C(C|N j ))j x⋆

j = 0 ∧ (pirr,C(C|N j ))κ x⋆

j 6= 0

as stated by the strict inclusion within the nesting property — see Lemma 4.2. By contradiction,the minimal polynomial ofC|N j in N j is

mpC|N j(λ) = (pirr,C(λ)) j (4.20)

and since to any minimal polynomial corresponds a companionmatrix the dimension of which isthe degree of its minimal polynomial, the subsequent lemma has been proven.


Lemma 4.4 (Dimension of Nullspaces)Let (pirr,C) j ∈ Fq[λ] be the defining polynomial of the cyclic companion matrixC ∈ Fd×d

q withpolynomialpirr,C of degreeδ and irreducible overFq. Let the j = 1, . . . ,enullspacesN j be definedby N j := Ker

((pirr,C(C)) j

). Then the dimension of each nullspaceN j is

dim(N j) = deg((pirr,C) j)= δ j . �

An other consequence of(pirr,C(λ)) j being minimal polynomial of the matrixC|N j in the spaceN j is that by Lemma 4.1 the least numbert such that

(C|N j )t − I ≡ 0

is t = τ j , with τ j denoting the period of the (powered) polynomial(pirr,C(λ)) j — see Definition 2.5for the period of a powered polynomial overFq. Hence allx j ∈ N j are at mostτ j -periodic.

In order to see whether there are states with periodt ∈ N with t < τ j , assume thatx j ∈ N j ist-periodic, i. e.

((C|N j )

t − I)

x j = 0.

Then, excluding the trivial casex j = 0, the matrix(C|N j )t − I is singular which per Theorem 4.3

implies that the minimal polynomial mpC|N j(λ) has a factor in common withλt −1. For reason

that the only factors of this minimal polynomial are powers of pirr,C(λ) this means thatλt −1 isdivided by(pirr,C(λ))κ for someκ ≤ j, analogously

g(λ)(pirr,C(λ))κ = λt −1

for some polynomialg(λ). Therefore,

g(C|N j )(pirr,C(C|N j ))κ = (C|N j )

t − I (4.21)

In light of Remark 4.4 right-multiplication by an arbitrarystatex j ∈ N j yields

g(C|N j )(pirr,C(C|N j ))κ x j = ((C|N j )

t − I)x j (4.22)

which implies thatx j is t-periodic if

(pirr,C(C|N j ))κ x j = 0. (4.23)

According to the definitions ofN j andD j and under assumption ofκ < j this equation can onlybe solved ifx j ∈ Nκ. But then the states inD j are exactly those which areτ j -periodic.

Lemma 4.5 (Period of the States inD j)Let the dynamics matrix of an LMS be given by a cyclic companion matrixC ∈ Fd×d

q the definingpolynomial of which is(pirr,C)e ∈ Fq[λ]. Moreover, let thej = 1, . . . ,e setsD j be defined as inequation (4.13). Then any state vector in the setD j is τ j -periodic, whereτ j is the period of thepolynomial(pirr,C(λ)) j . �


Collecting the past results, referring to a cyclic companion matrixC and for j = 0, . . . ,e the j-thnested subspaceN j := Ker

((pirr,C(C)) j

)contains exactlyq jδ states of the spaceFd

q . To sum it up,this leads to

• 1 state inN0 := {0}, the zero state,

• qδ −1 states in the difference setD1 = N1−N0,

• q2δ −qδ states in the difference setD2 = N2−N1,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• qeδ −q(e−1)δ states in the difference setDe = Ne−Ne−1 with Ne = Fdq andd = eδ .

All q jδ−q( j−1)δ states inD j have periodτ j such thatν j =(q jδ−q( j−1)δ)/τ j cycles ofτ j-periodic

states lie in the spaceD j . Adding up the number of all states inD j from j = 1, . . . ,eplus the zerostate results in

1+

e∑

j=1

q jδ−q( j−1)δ = qeδ = qd

which shows that the entire spaceFdq is composed of those cycles. Accordingly, the following

important theorem has been deduced.

Theorem 4.5 (Periodic Nullspace Decomposition of a Companion Matrix)Given a cyclic companion matrixC ∈ Fd×d

q with respect to thed-th degree polynomialpC(λ) =

(pirr,C(λ))e, wherepirr,C(λ) is an irreducible polynomial overFq of degreeδ such thatd = eδ.Then

1. the associated state spaceFdq is entirely composed of periodic states as per

ν0 = 1 cycles of length τ0 = 1ν1 =

(qδ −1

)/τ1 ” τ1 = τ

ν2 =(q2δ −qδ)/τ2 ” τ2 = qτ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .ν j =

(q jδ−q( j−1)δ)/τ j ” τ j = ql j τ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .νe =

(qeδ −q(e−1)δ)/τe ” τe = qle τ

where eachl j , j = 1, . . . ,e, is the least integer such thatql j ≥ j,

2. theτ j -periodic statesxτ j ∈ Fdq that build these cycles follow from the system of equations

(pirr,C(C)) j xτ j = 0 ∧ (pirr,C(C)) j−1xτ j 6= 0, j = 1, . . . ,e

wherexτ0 = 0 and(pirr,C(C))0 = I . �


The periodic decomposition can be written in a more convenient form by applying

Definition 4.4 (Cycle Sum)The cycle sumΣ is the formal sum of cycle terms

Σ = ν1[τ1]∔ν2[τ2]∔ . . .∔νκ[τNΣ] , (4.24)

in which the cycle termνi[τi ] denotesνi cycles of lengthτi and the binary operation∔ satisfiesνi [τ]∔ν j [τ] = (νi +ν j)[τ]. The number of different-length cycles inΣ is denoted byNΣ. �

Using this definition the result of Theorem 4.5 can be rewritten as

Σ = 1[1]∔qδ −1

τ1[τ1]∔

q2δ −qδ

τ2[τ2]∔ . . .∔

qeδ −q(e−1)δ

τe[τe] , (4.25)

in which τ j , j = 1, . . . ,e, marks the periods of the polynomial(pirr,C(λ)) j to be computed mostsimply via Theorem 2.5 by means of the periodτ of its respective basis polynomialpirr,C(λ).

Some examples for cycle sums with regard to powers of irreducible polynomials overF2 can betaken from Tabular 4.1.

4.2.1.2 Periodic States of an LMS Composed of Cyclic Companion Matrices

The knowledge of how the periodic states of an LMS with a dynamics matrix which is a cycliccompanion matrixCi of an elementary divisor polynomial can be calculated allows the generaliza-tion of the obtained result by benefiting from the linearity of LMS. Linearity grants the possibilityof superposing the results for alli = 1, . . . ,Nc subspaces referring to theNc elementary divisorpolynomials ofAc.

Recall from equation (4.9) that the followingi = 1, . . . ,Nc equations11

(I −C ti ) x(i)

c,t = 0

have to be fulfilled all at once in order to yield at-periodic statexTc,t =

(x(1)

c,tT, . . . , x(Nc)

c,tT)

in the

overall Nc-fold composed cyclic part of the state spaceFd1q × ·· · × F

dNcq = Fnc

q . Each of thesei = 1, . . . ,Nc subspaces can be decomposed completely into cycles with an associated cycle sumaccording to equation (4.25).

The subsequent lemma provides a means for determining the period of a composed state.

11Beginning from here, the exact notationx(i)c,ti andCi instead of, loosely,x andC becomes important again.


Polynomial e CycleSumνi[τi ]

(x+1)e 1 1[1]

2 2[1]∔1[2]

3 2[1]∔1[2]∔1[4]

4 2[1]∔1[2]∔3[4]

5 2[1]∔1[2]∔3[4]∔2[8]

(x2+x+1)e 1 1[1]∔1[3]

2 1[1]∔1[3]∔2[6]

3 1[1]∔1[3]∔2[6]∔4[12]

(x3+x+1)e, (x3+x2 +1)e 1 1[1]∔1[7]

2 1[1]∔1[7]∔4[14]

3 1[1]∔1[7]∔4[14]∔16[28]

(x4+x+1)e, (x4+x3 +1)e 1 1[1]∔1[15]

2 1[1]∔1[15]∔8[30]

3 1[1]∔1[15]∔8[30]∔64[60]

(x4+x3 +x2 +x+1)e 1 1[1]∔3[5]

2 1[1]∔3[5]∔25[10]

(x5+x2 +1)e, . . . 1 1[1]∔1[31]

2 1[1]∔1[31]∔16[62]

Table 4.1: Cycle sums of irreducible polynomials overF2 risen to the power ofe— taken from [Els59]

Lemma 4.6 (Period of Two Composed States)Let x(1) be at1-periodic substate inFd1

q andx(2) be at2-periodic substate inFd2q with respect to

the linear modular systems induced by the matricesA1 andA2, respectively. Then the composedstatexT = (x(1)T,x(2)T) in F

d1+d2q = F

d1q ×F

d2q with respect to the composed linear modular system

induced by the matrixA := diag(A1,A2) is periodic with periodt = lcm(t1, t2), denoting the leastcommon multiple oft1 andt2. �

Proof The period ofx is the least integert such thatx = At x. This impliesx(1) = A t1x(1) and

x(2) = A t2x(2), hence,t is a common multiple oft1 andt2, which are the periods ofx(1) andx(2),

respectively. Sincet1, t2 andt are the least such integers,t has to be the least product of all factorsof t1 andt2, that is the least common multiple oft1 andt2, t = lcm(t1, t2). �

An immediate generalization of this lemma owing to induction is the following


Theorem 4.6 (Period of Multiple Composed States)Let T = {τ1, . . . ,τNc} denote a set of periods regarding the periodic substatesx(i)

c in thei = 1, . . . ,Nc

linear modular subsystemsx(i)c (k+ 1) = Ac,i x

(i)c (k) with cyclic dynamics matricesAc,i. Let the

overall LMS be composed according toAc = diag(Ac,1, . . . ,Ac,Nc) such thatxc(k+1) = Ac xc(k)

with xTc = (x(1)

cT, . . . , x(Nc)

cT). Let P = {P1,P2, . . . ,P2Nc} denote the power set ofT. Then the

respective state spaceFdq , d = dim(xc), of the composed cyclic LMS is made up of states with

periodsti = lcm(Pi), i = 1,2, . . . ,2Nc. �

Remark 4.5In general the periodsti, i = 1,2, . . . ,2Nc of the states in the composed system are not distinct andcan be summed up as described in Definition 4.4. �

Theorem 4.6 gives evidence about the periods, i. e. about thecycle lengths, that occur in an LMS.The next step is to deal with the numbers of theses cycles. Forclearness, consider again an LMSthe dynamicsAc of which consists of two cyclic companion matricesC1 andC2, one for each ofthe two elementary divisor polynomials,

Ac =

(C1 00 C2

)

, C1 ∈ Fd1×d1q ,C2 ∈ Fd2×d2

q (4.26)

with corresponding cycle sums

Σ1 = 1[1]∔ν(1)1 [τ(1)

1 ]∔ν(1)2 [τ(1)

2 ]∔ . . .∔ν(1)e1 [τ(1)

e1 ] , (4.27)

Σ2 = 1[1]∔ν(2)1 [τ(2)

1 ]∔ν(2)2 [τ(2)

2 ]∔ . . .∔ν(2)e2 [τ(2)

e2 ] . (4.28)

Henceqdi , the number of states in the respective subsystemsi = 1,2, is

qdi = 1+

ei∑

j=1

ν(i)j τ(i)

j ,

which furthermore implies the number of states in the overall system

qd1+d2 =(

1+

e1∑

j=1

ν(1)j τ(1)

j

)(

1+

e2∑

k=1

ν(2)k τ(2)

k

)

=

= 1+

e1∑

j=1

ν(1)j τ(1)

j︸︷︷︸

a j

+

e2∑

k=1

ν(2)kj τ(2)

k︸︷︷︸

ak

+

e1∑

j=1

e2∑

k=1

ν(1)j ν(2)

k τ(1)j τ(2)

k︸︷︷︸

a jk

. (4.29)

The addends 1,a j , ak, a jk in the overall summation of state numbers in equation (4.29)are due tothe following subsystem state combinations:


1: zero state referring to subsystem 1 combined with the zerostate referring to subsystem 2

⇒ There results 1 state of period 1 inFd1+d2q .

a j : zero state referring to subsystem 2 combined with eachj = 1, . . . ,e1 of theν(1)j states of period

τ(1)j in subsystem 1

⇒ For eachj = 1, . . . ,e1 resultν(1)j τ(1)

j states of periodτ(1)j in F

d1+d2q .

ak: zero state referring to subsystem 1 combined with eachk= 1, . . . ,e2 of theν(2)k states of period

τ(2)k in subsystem 2

⇒ For eachk = 1, . . . ,e2 resultν(2)k τ(2)

k states of periodτ(2)k in F

d1+d2q .

a jk: combination of eachj = 1, . . . ,e1 of theν(1)j states of periodτ(1)

j referring to subsystem 1 with

eachk = 1, . . . ,e2 of theν(2)k states of periodτ(2)

k referring to subsystem 2

⇒ For each pairj = 1, . . . ,e1 and k = 1, . . . ,e2 result ν(1)j ν(2)

k τ(1)j τ(2)

k states of period

lcm(τ(1)

j ,τ(2)k

)in F

d1+d2q .

As a consequence of the last item, the number

ν jk =ν(1)

j ν(2)k τ(1)

j τ(2)k

lcm(τ(1)

j ,τ(2)k

) = ν(1)j ν(2)

k gcd(τ(1)

j ,τ(2)k

)

is the number of cycles consisting of states the period of which is τ jk = lcm(τ(1)

j ,τ(2)k

)and the

expression gcd(τ(1)

j ,τ(2)k

)denotes the greatest common divisor ofτ(1)

j andτ(2)k . Hence, a product

of cycle terms may be defined.

Definition 4.5 (Product of Cycle Terms)The product

ν1[τ1]ν2[τ2] = ν1ν2gcd(τ1,τ2)[lcm(τ1,τ2)] (4.30)

is called the cycle term product. �

By means of the denotation of sum and product of cycle terms, according to Definition 4.4 and 4.5,the setting can be extended to the superposition of cycle sums. Reverting to the equations (4.27)


and (4.28) the product

Σ = Σ1Σ2 =

=(

1[1]∔ν(1)1 [τ(1)

1 ]∔ν(1)2 [τ(1)

2 ]∔ . . .∔ν(1)e1 [τ(1)

e1 ])(

1[1]∔ν(2)1 [τ(2)

1 ]∔ν(2)2 [τ(2)

2 ]∔ . . .∔ν(2)e2 [τ(2)

e2 ])

= 1[1]

e1

∔j=1

ν(1)j [τ(1)

j ]

e2

∔k=1

ν(2)k [τ(2)

k ]

e1

∔j=1

e2

∔k=1

ν(1)j ν(2)

k gcd(τ(1)

j ,τ(2)k

)[lcm(τ(1)

j ,τ(2)k

)](4.31)

represents the cycle sumΣ of the composed LMS with respect to matrixAc in equation (4.26). Thenext theorem is an obvious generalization of this result.

Theorem 4.7 (Superposition of Cycle Sums)The cycle sumsΣi , i = 1, . . . ,Nc corresponding toNc disjoint cyclic linear subspaces of an LMSsuperpose distributively in accordance with the product

Σ = Σ1Σ2 · · ·ΣNc . �

Remark 4.6In order to calculate the product in Theorem 4.7 in a simple fashion, the result from equation (4.31)can be extended to addends consisting of more than two factors by allowing for the definitions

ν(i)j := ν(i1)

j1· · · ν(iη)

jη gcd(

τ(i1)j1

, . . . ,τ(iη)jη

)

, (4.32)

τ(i)j := lcm

(

τ(i1)j1

, . . . ,τ(iη)jη

)

(4.33)

with the abbreviationsi = (i1, . . . , iη), j = ( j1, . . . , jη) andη≤Nc such that the cycle sumΣ consists

of addends of the formν(i)j

[τ(i)

j

]only. �

All these results from above are combined in the main theoremof this section, Section 4.2.1.

Theorem 4.8 (Cycle Sum of an Autonomous LMS with Cyclic Dynamics)Let the dynamics matrixAc = diag(C1, . . . ,CNc) ∈ Fnc×nc

q of an autonomous LMS(q) be block-diagonally composed ofi = 1, . . . ,Nc cyclic companion matricesCi, each with respect to one ofthe i = 1, . . . ,Nc elementary divisor polynomialspCi ∈ Fq[λ] of degreedi . Let each elementarydivisor polynomialpCi be given in fully factored formpCi = (pirr,Ci)

ei subject to its irreduciblefactor polynomialpirr,Ci of degreeδi such thatdi = ei δi . Then

1. each elementary divisor polynomialpCi contributes the cycle sum

Σi = 1[1]∔qδi −1

τ(i)1

[τ(i)

1

]∔

q2δi −qδi

τ(i)2

[τ(i)

2

]∔ . . .∔

qeiδi −q(ei−1)δi

τ(i)ei

[τ(i)

ei

], (4.34)


whereτ(i)j denotes the period of the polynomial(pirr,Ci)

j . For the entire autonomous LMS(q)the cycle sumΣ follows by superposition of all cycle sumsΣi as perΣ = Σ1Σ2 · · ·ΣNc.

2. Theτ(i)j -periodic statesx(i)

c, j ∈ Fdiq that build thei-th cycle sumΣi are the solutions of the

system of equations

(pirr,Ci(Ci))j x(i)

c, j = 0 ∧ (pirr,Ci(Ci))j−1 x(i)

c, j 6= 0, j = 1, . . . ,ei , (4.35)

wherex(i)c,0 = 0 and(pirr,Ci(Ci))

0 = I . Abbreviatei = (i1, . . . , iη), j = ( j1, . . . , jη) for which

holds i1 ≤ . . . ≤ iη, j1 ≤ . . . ≤ jη with η ≤ Nc. With a slight abuse of notation, letx(i)c,j

T =(

x(i1)c, j1

T, . . . , x(iη)c, jη

T)

denote a vectorx(i)c,j ∈ Fnc

q that results from the composition ofη subvec-

tors x(i1)c, j1

, . . . , x(iη)c, jη, where non-specified vectors are zero vectors of corresponding dimen-

sion.12 Then any statex(i)c,j ∈ Fnc

q which is composable of arbitrary solutionsx(i1)c, j1

, . . . , x(iη)c, jη

of equation (4.35) for some fixedi andj has periodτ(i)j = lcm

(

τ(i1)j1

, . . . ,τ(iη)jη

)

. �

Remark 4.7With respect to an elementary divisor polynomialpCi (λ) of degreedi that is irreducible, obviously

there areδi = di and j = 1, hence, from equation (4.34) it followsΣi = 1[1]∔(qdi −1)/τ(i)1 [τ(i)

1 ] and

all non-zero substate vectors inFdiq areτ(i)

1 -periodic, that is all of them are solutions of equation(4.35). �

Recapitulating the past results, the determination of the cycle sum of an autonomous LMS(q) withdynamics matrixA takes the following steps:

i. Calculate thei = 1, . . . ,Nc periodic elementary divisor polynomialspCi (λ) of the dynamicsmatrixA by dint of deriving either

• the factor polynomialspCi (λ) of the invariant polynomials using the Smith normalform S(λ) of A ,

• or the rational canonical formArat of A. The polynomials with respect to each com-panion matrix in the rational canonical form are the elementary divisor polynomials.

The nilpotent elementary divisor polynomialspCi(λ) = λhi , i = 1, . . . ,Nn, do not contributehere due to Remark 4.2 — see Section 4.2.2 for the corresponding discussion.

12For an illustration, letAc = diag(C1,C2,C3,C4) be a cyclic dynamics matrix inF9×92 with pC1(λ) = pC2(λ) =

λ3 + λ + 1, pC3(λ) = λ2 + λ + 1, pC4(λ) = 1 irreducible defining polynomials of the companion matrices C1 = C2,

C3, C4 such that the periods of the polynomials areτ(1)1 = τ(2)

1 = 7, τ(3)1 = 3 andτ(4)

1 = 1. Then the state vector

x(i)c,j

T = (x(i1)c, j1

T, x(i2)c, j2

T) with i = (i1, i2) = (1,3) andj = ( j1, j2) = (1,1) is a short cut denotation for any 21-periodic

state vector(x(1)c,1

T,0T, x(3)c,1

T,0) one of which is for example((1,0,1),(0,0,0),(1,0),0).


ii. Determine the irreducible factor polynomialspirr,Ci(λ) for eachi = 1, . . . ,Nc periodic ele-mentary divisor polynomial such thatpCi = (pirr,Ci )

ei .

iii. To each periodic polynomial(pirr,Ci)j , i = 1, . . . ,Nc, j = 1, . . . ,ei, assign the periodsτ(i)

j .

iv. Compute the cycle sumΣi regarding eachi = 1, . . . ,Nc periodic elementary divisor polyno-mial pCi (λ).

v. The cycle sumΣ of the entire autonomous LMS(q) then follows by distributively superposingall cycle setsΣi , i = 1, . . . ,Nc.

vi. Simplify the cycle sumΣ by collecting cycle terms of the same period using Definition4.4.

Remark 4.8 (Comment on the Complexity of the Method)The first three steps within the above-stated method, the calculation of the Smith form of the dy-namics matrix, the factorization of the invariant polynomials into elementary divisor polynomials,and the determination of its basis polynomial periods deserve some deeper examination of its com-putational complexity.13,14

For determining the Smith normal form of ann by n matrix Iliopoulus [Ili89] derived a polynomialcomplexity bound ofO(n4), which was decreased down toO(n3) by Storjohann [Sto00].

The best known algorithm for factoring polynomials over finite fields is the algorithm of Berlekampand von zur Gathen [Ber70, Gat87], for a review see [Sho90]. According to [Sho90], factoring and-th degree polynomial over a finite fieldFq entails a complexity less thanO(M(d) + qd2+ε),whereM(d) denotes the complexity of multiplying twod by d matrices — the least known up-per bound for this is aboutO(d2.4). The expressionO(dε) denotes a polynomial of finite degreein logd. Consequently, forq fixed, factoring of polynomials over finite fields is of polynomialcomplexity.

The periods of the irreducible basis polynomials of the elementary divisor polynomials can befound in numerous tables, for example in [LN94]. For polynomial periods which are not tabulated,say due to the magnitude of its degree, there is a very severe obstacle: Though on the face of it,determining periods seems to be a simple task it turns out that it is as complex as determining theorder of an element of a group or a discrete logarithm [BJT97]. This problem has been addressedby Meijer in [Mei96] and shown to be at least as complex as factoring integers. Unfortunately, atthe moment, there is no polynomial time algorithm for factoring integers into primes.

Thus, an important consequence is that the presented methodfor solving the problem of deter-mining the cycle set of an LMS allows to computationally benefit only in the case of rather small

13It is sufficient here to consider just the computational timecomplexity.14Besides multiplication, step v comprises the calculation of greatest common divisors and least common multiples.

For this task algorithms of polynomial complexity exist, for example the Euclidian Algorithm.


degrees of the basis polynomials; tabulars are available about up to degreeδ = 100. In this regard,it can be expected that the greater the number of invariant polynomials of the dynamics matrixand the more these polynomials factor into elementary divisor polynomials the less likely the latterpractical bound may be exceeded. �

To the structural information expressed in the cycle sum corresponds the calculation of the respec-tive periodic states of the autonomous LMS(q). Provided that all information after item v fromabove is at hand the following procedure will serve this purpose:

1. Calculate theτ(i)j -periodic substatesx(i)

c, j for all cycle sumsΣi , i = 1, . . . ,Nc by making use ofequation (4.35) in view of Remark 4.7.

2. For all periodsτ(i)j which occur in the unsimplified cycle sumΣ of the superposition (step v

from above) determine theτ(i)j -periodic statesx(i)

c,j ∈ Fncq by part 2 of Theorem 4.8.

3. In the overall state spaceFnq = Fnc+nn

q theτ(i)j -periodic statesx(i)

j ∈ Fnq follow by composing

x(i)c,j with the 1-periodic zero state ofFnn

q such thatx(i)j

T =(

x(i)c,j

T,0T)

is τ(i)j -periodic.

4. Theτ(i)j -periodic state vectors are given in original coordinates by x(i)

j = T−1x(i)j =ΠΠΠT−1x(i)

jas per equation (4.8).

4.2.1.3 Example: an LMS with Cyclic Dynamics Matrix

Consider a dynamics matrixA ∈ F5×52 of an LMS(2) and its appendant Smith normal formS(λ) ∈

F2[λ]5×5 with S(λ) = U(λ)(λ I +A)V(λ),

A =

1 0 0 1 11 1 0 0 10 0 1 0 10 0 0 0 11 0 0 0 1

, S(λ) =

(λ2+λ+1)(λ+1)2 0 0 0 00 λ+1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

with unimodular matrices

U(λ) =

λ λ3+1 λ4+λ3+λ+1 1 λ3+λ2+λ+10 1 λ 0 10 0 1 0 00 0 0 0 11 0 0 0 0


and

V(λ) =

λ2+1 λ2+1 λ+1 1 0λ λ+1 0 0 01 1 0 0 0

λ3+λ2 λ3+λ2 λ2 λ+1 1λ+1 λ+1 1 0 0

.

Here the only invariant polynomials6= 1 of the matrixA are

c1(λ) = (λ2+λ+1)(λ+1)2, c2(λ) = λ+1

as indicated by the Smith normal form. Thus,A has the elementary divisor polynomials

pC1(λ) = λ2+λ+1, pC2(λ) = (λ+1)2, pC3(λ) = λ+1 ,

none of which are of the formλh for some integerh, hence, all elementary divisor polynomials areperiodic in accordance with the assumption of a cyclic dynamics matrixA. The corresponding basepolynomial degrees areδ1 = 2, δ2 = 1 andδ3 = 1, respectively. Consequently, the correspondingrational canonical formArat = TAT−1 together with its transformation matrixT reads15

Arat = diag(C1,C2,C3) =

0 1 0 0 01 1 0 0 00 0 0 1 00 0 1 0 00 0 0 0 1

,T =

1 0 0 1 10 0 0 1 11 0 1 0 10 0 1 1 10 1 1 0 1

,T−1 =

1 1 0 0 01 1 1 0 10 1 0 1 01 1 1 1 01 0 1 1 0

.

In view of Definition 2.8 and Theorem 2.5 the corresponding periods are

pirr,C1(λ) = λ2+λ+1|λ3+1 =⇒ τ(1)1 =3

pirr,C2(λ) = λ+1 =⇒ τ(2)1 =1

(pirr,C2(λ)

)2= (λ+1)2 = λ2+1 =⇒ τ(2)

2 =2

pirr,C3(λ) = λ+1 =⇒ τ(3)1 =1

Theorem 4.8 yieldsΣ1 = 1[1]∔1[3], Σ2 = 2[1]∔1[2], Σ3 = 2[1]

and by superposition according to Theorem 4.7 it follows

Σ = Σ1Σ2Σ3 =(1[1]∔1[3]

)(2[1]∔1[2]

)(2[1])=(2[1]∔1[2]∔2[3]∔1[6]

)(2[1])=

= 4[1]∔2[2]∔4[3]∔2[6] .

15A simple method for obtaining the transformation matrixT which transformsA into Arat employing MapleR© ispresented in Chapter B of the Appendix.


Alternatively,Σ can be calculated in view of Remark 4.6 via

Σ = Σ1Σ2Σ3 =(1[1]∔1[3]

)(2[1]∔1[2]

)(2[1])

= 1 ·2 ·2 gdc(1,1,1) [lcm(1,1,1)]∔1 ·1 ·2 gdc(1,2,1) [lcm(1,2,1)]∔

1 ·2 ·2 gdc(3,1,1) [lcm(3,1,1)]∔1 ·1 ·2 gdc(3,2,1) [lcm(3,2,1)]

= 4[1]∔2[2]∔4[3]∔2[6] .

Therefore, the LMS(2) represented by the dynamics matrixA comprises 4 cycles of length 1, 2cycles of length 2, 4 cycles of length 3 and 2 cycles of length 6.

The respective periodic statesx(1)c,1 and x(3)

c,1 for the irreducible elementary divisor polynomialspC1(λ) andpC3(λ) immediately result from Remark 4.7, that is

x(1)c,1 ∈

{(10

)

,

(01

)

,

(11

)}

, x(3)c,1 ∈ {(1)}

whereas for the reducible elementary divisor polynomialpC2(λ) equation (4.35) yields

(pirr,C2(C2))1 x(2)

c,1 = 0 ∧ (pirr,C2(C2))0 x(2)

c,1 6= 0

⇐⇒ (C2+ I) x(2)c,1 = 0 ∧ I x(2)

c,1 6= 0

⇐⇒(

1 11 1

)

x(2)c,1 = 0 ∧ x(2)

c,1 6= 0 =⇒ x(2)c,1 ∈

{(11

)}

(pirr,C2(C2))2 x(2)

c,2 = 0 ∧ (pirr,C2(C2))1 x(2)

c,2 6= 0

⇐⇒ (C2+ I)2 x(2)c,2 = 0 ∧ (C2+ I) x(2)

c,2 6= 0

⇐⇒(

0 00 0

)

x(2)c,2 = 0 ∧

(1 11 1

)

x(2)c,2 6= 0 =⇒ x(2)

c,2 ∈{(

10

)

,

(01

)}

Now the overall periodsτ(i)j of the associated statesx(i)

c,j can be determined from all possible com-positions which are vectors in the transformed coordinates(Figure 4.2). For instance, take thevector

xT1 = (0,0,1,0,0)

which was determined to be 2-periodic. Its counterpart in the original coordinates is given by theinverse transformationx1 = T−1 x1 with

xT1 = (0,1,0,1,1)

The simple calculation

A2x1 = x1


τ = 1 τ = 2 τ = 3 τ = 6

x∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

00000

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

x(2)c,(1)∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

00110

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

x(3)c,(1)∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

00001

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

x(2,3)c,(1,1)∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

00111

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

x(2)c,(2)∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

00100

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

00010

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

x(2,3)c,(2,1)∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

00101

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

00011

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

x(1)c,(1)∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

10000

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

01000

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

11000

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

x(1,2)c,(1,1)∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

10110

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

01110

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

11110

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

x(1,3)c,(1,1)∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

10001

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

01001

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

11001

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

x(1,2,3)c,(1,1,1)∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

10111

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

01111

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

11111

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

x(1,2)c,(1,2)∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

10100

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

01100

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

11100

1

C

C

C

C

C

C

C

C

A

,

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

0

B

B

B

B

B

B

B

B

@

10010

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

01010

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

11010

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

,

x(1,2,3)c,(1,2,1)∈

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

0

B

B

B

B

B

B

B

B

@

10101

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

01101

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

11101

1

C

C

C

C

C

C

C

C

A

,

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

0

B

B

B

B

B

B

B

B

@

10011

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

01011

1

C

C

C

C

C

C

C

C

A

,

0

B

B

B

B

B

B

B

B

@

11011

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

,

Figure 4.2: Periodic states in transformed coordinates referring to Example 4.2.1.3

verifies thatx1 is 2-periodic. Furthermore, deriving

Ax1 = x2, xT2 = (0,0,1,1,1)

is in accordance with the result of

x2 = T−1 x2, xT2 = (0,0,0,1,0),

in whichx2 is the remaining 2-periodic state vectorx2 expressed in original coordinates.

The entire state graph with states represented in transformed coordinates is depicted in Figure 4.3.

4.2.2 Nilpotent Dynamics

This section presents an investigation of the nilpotent part An within the rational canonical formArat of the dynamics matrixA in the decomposition according to equation (4.5). The focuswilllie again on the interconnection structure of the states, this time concerning the substatesxn ∈ Fnn

q .With this regard reconsider equation (4.7)

xn(k+1) = An xn(k) , k∈ N0 , An = diag(C1, . . . ,CNn) (4.36)


0

B

B

@

00000

1

C

C

A

0

B

B

@

00110

1

C

C

A

0

B

B

@

00001

1

C

C

A

0

B

B

@

00111

1

C

C

A

0

B

B

@

00100

1

C

C

A

0

B

B

@

00010

1

C

C

A

0

B

B

@

00101

1

C

C

A

0

B

B

@

00011

1

C

C

A

0

B

B

@

10000

1

C

C

A

0

B

B

@

01000

1

C

C

A

0

B

B

@

11000

1

C

C

A

0

B

B

@

10110

1

C

C

A

0

B

B

@

01110

1

C

C

A

0

B

B

@

11110

1

C

C

A

0

B

B

@

10001

1

C

C

A

0

B

B

@

01001

1

C

C

A

0

B

B

@

11001

1

C

C

A

0

B

B

@

10111

1

C

C

A

0

B

B

@

01111

1

C

C

A

0

B

B

@

11111

1

C

C

A

0

B

B

@

10100

1

C

C

A

0

B

B

@

01010

1

C

C

A

0

B

B

@

11100

1

C

C

A

0

B

B

@

10010

1

C

C

A

0

B

B

@

01100

1

C

C

A

0

B

B

@

11010

1

C

C

A

0

B

B

@

10101

1

C

C

A

0

B

B

@

01011

1

C

C

A

0

B

B

@

11101

1

C

C

A

0

B

B

@

10011

1

C

C

A

0

B

B

@

01101

1

C

C

A

0

B

B

@

11011

1

C

C

A

Figure 4.3: State graph of the LMS discussed in Example 4.2.1.3; states given in transformed coordinates

wherexn(k) ∈ Fnnq and alli = 1, . . . ,Nn matricesCi are nilpotent companion matrices with respect

to the elementary divisor polynomialspCi (λ) = λhi , hi ∈ N, implying Chii ≡ 0. Note that with

Remark 4.1 anyh×h nilpotent companion matrixC uniquely reads

C =

0 0 · · · 0 01 0 · · · 0 00 1 · · · 0 0...

......

......

0 0 · · · 1 0

, dim(C) = h. (4.37)

Collecting all nilpotent companion matrices inAn yieldsnn =∑Nn

i=1hi.


Definition 4.6 (Levels of a State Graph)Let an LMS be given by a nilpotent dynamics matrixAn. Then the associated states which reachthe zero statexn = 0 in l steps are called states of levell (in the state graph). The set of states inlevel l is termed levell . �

4.2.2.1 State Graph of an LMS with a Nilpotent Companion Matrix as Dynamics Matrix

In the special case when the dynamics matrix of an LMS is a nilpotent companion matrix thefollowing applies.

Theorem 4.9 (State Graph of an LMS with Minimal Polynomial λh)In an LMS the dynamics matrix of which is a nilpotent companion matrixC ∈ Fh×h

q the followingstatements hold:

• the state graph consists ofh levels,

• the levell = 1,2. . . ,h comprises(q−1)ql−1 states,q0 := 1,

• any non-zero, non-terminal state hasq confluent states,

• the zero state hasq−1 confluent states (in level 1). �

Proof Any statexTh = (x1,x2, . . . ,xh) ∈ Fh

q with x1 ∈ Fq−{0} and arbitraryxi 6=1 ∈ Fq has nopredecessors sincexh /∈ Im(C). The number of states in this partition is(q−1)qh−1. The mappingof xh ∈ Fq by means of matrixC results inxT

h−1 = (0,x1,x2, . . . ,xh−1). Thus, the next partition ismade up of all such statesxh−1. These all haveq confluent states, which is due to the number ofvalues corresponding to the “omitted”xh in the last shift. The number of states in this partitionis (q− 1)qh−2. Proceeding in the same manner, finally, leeds toxT

1 = (0, . . . ,0,x1) which withx1 ∈ Fq − {0} are q− 1 states in this partition all of which again haveq confluent states andterminate in the remaining zero state in a last step. Counting the states of all partitions yields

(q−1)qh−1+(q−1)qh−2+ · · ·+(q−1)+1 = 1+(q−1)h−1∑

i=0

qi = 1+(q−1)qh−1q−1

= qh ,

which is the total number of states inFhq . As a consequence, all these states formh levels in the

state graph. �

Example 4.2For an LMS with nilpotent dynamicsA = An = C ∈ F3×3

2 with

C =

0 0 01 0 00 1 0


(000

)

(001

)

(010

) (011

)

(100

) (101

) (111

) (110

)

Figure 4.4: State graph of a 3-rd order LMS with nilpotent companion matrix as dynamics (Example 4.2)

Theorem 4.9 implies a graph as shown in Figure 4.4. �

4.2.2.2 State Graph of an LMS with Arbitrary Nilpotent Dynam ics Matrix

The structural examination of the state graph of an LMS with arbitrary nilpotent dynamics matrixAn entails more effort. Let a companion matrix (of dimensionhi) referring to an elementary divisorpolynomialλhi be denoted byCλhi , for simplicity, and consider the ordered recollection

diag(Cλ, . . . ,Cλ︸︷︷︸

µ1 blocks

,Cλ2, . . . ,Cλ2︸︷︷︸

µ2 blocks

, . . . ,Cλhmax, . . . ,Cλhmax︸︷︷︸

µhmax blocks

) ,

hmax∑

j=1

µj = Nn , (4.38)

of the diagonal blocks of matrixAn, which is a recollection by ascending block dimensionshi .In the sequel, since this form can be obtained fromAn by simple permutations16, without loss ofgeneralityAn will be assumed in the form (4.38).

With (4.37) an immediate result is the following.

Theorem 4.10 (Termination into the Zero State)Any statexn ∈ Fnn

q of an LMS with nilpotent dynamics matrixAn ∈ Fnn×nnq , which consists of

i = 1, . . . ,Nn nilpotent companion matricesCi of dimensionhi , terminates in at mosthmax =

max(h1, . . . ,hNn) steps into the zero vector. �

16See Chapter A of the Appendix for the details.


Proof The mapping of an arbitrary vector by virtue of a nilpotent companion matrixCi shifts thevector entries one digit down and sets the first entry to zero.Hence, afterhi such mappings withCi

the zero vector is obtained. In the overall system with respect toAn, consequently, max(h1, . . . ,hNn)

mappings ofxn by means ofAn yield the zero vector for the first time which then persists for anyfurther mappings usingAn. �

Thus, the subsequent consequence of Theorem 4.10 becomes obvious.

Corollary 4.2 (Number of Levels)The state graph of an LMS with nilpotent dynamics matrixAn consisting ofi = 1, . . . ,Nn nilpotentcompanion matricesCi of dimensionhi consists ofhmax = max(h1, . . . ,hNn) levels. �

In contrast to cyclic dynamics matrices, which in its rational canonical form comprise specificcoefficients of its elementary divisor polynomials still (rightmost columns in the respective com-panion matrices), nilpotent dynamics matrices depend on the numbershi only. For this reason, it isworthwhile to interpret equation (4.7) as a linear system ofequations forxn(k). To this end, con-sider theκ-fold mapping fromxn(k) to xn(k+κ) by means ofAn, amounting to the linear systemof equations

Aκn xn(k) = xn(k+κ) . (4.39)

Similar to (4.9), equation (4.39) comprises all information about the interconnection structure ofthe states.

Since the matrixAn is singular a more general concept of an inverse is requestedfor solving (4.39).So-called singular inverses (g-Inverses) grant this facility. Taking this into account, a solution ofequation (4.39) exists iffxn(k+ κ) ∈ Im(Aκ

n), which, referring to Chapter D of the Appendix, isequivalent to

(Aκn (AT

n)κ − I) xn(k+κ) = 0. (4.40)

Given existence, a general solution of equation (4.39) is

xn(k) = (ATn)κ xn(k+κ)+

(I − (AT

n)κAκn

)z, ∀z∈ Fnn

q (4.41)

where the first addend marks the particular solution and the second addend the homogeneoussolution, respectively.

Under the assumption that the statexn(k+κ) meets the solvability condition (4.40) the number ofdifferent solutions is expressed by

(I − (AT

n)κAκn

)z.

The matrix(ATn)κAκ

n is a diagonal matrix and its zero rows incur components fromz, which is anarbitrary vector inFnn

q . Thus, counting these zero rows allows to calculate the number of statesthat reach an arbitrary given state, if possible, in at mostκ steps.


In light of the fact thatAn is a block-diagonal matrix and that for nilpotent companionmatrices

(CT)κ Cκ =

{0h, 1 ≤ h ≤ κ

diag(Ih−κ,0κ), κ < h ≤ hmax(4.42)

the number of zero rowsr(κ) in matrix (ATn)κAκ

n is

r(κ) =κ∑

j=1

µj j +hmax∑

j=κ+1

µj κ , κ = 1, . . . ,hmax (4.43)

in which the denotation introduced in equation (4.38) is applied. Finally, simple combinatoricsimply the following theorem.

Theorem 4.11 (Number of States Reaching an Arbitrary State)Let An be a nilpotent dynamics matrix of an LMS in recollected form as per equation (4.38) andlet xn ∈ Fnn

q be an arbitrary state with(Aκn (AT

n)κ− I) xn = 0 andκ = 1, . . . ,hmax. Then the number

of states which reachxn in κ steps isqr(κ), r(κ) =∑κ

j=1µj j +∑hmax

j=κ+1µj κ. �

Remark 4.9The condition(Aκ

n (ATn)κ− I) xn = 0 in Theorem 4.11 is fulfilled trivially ifxn = 0. Consequently,

the zero statexn = 0 is reachable from arbitrary states. �

On this account the number of states in levell can be calculated.

Corollary 4.3 (Number of States in each Level)Let An be a nilpotent dynamics matrix of an LMS in recollected form as per equation (4.38). Thenthe number of states in levell is given by

η(l) = qr(l)−qr(l−1), r(l) =

l∑

j=1

µj j +hmax∑

j=l+1

µj l , r(0) := 0. �

Theorem 4.11 and Corollary 4.3 are sufficient to construct the state graph of a nilpotent LMS. Ifthe task is to determine particular states then equation (4.41) serves this purpose.

4.2.2.3 Example: an LMS with Nilpotent Dynamics Matrix

For simplicity the following nilpotent dynamics matrixAn ∈ F7×72 of an LMS shall be given in the

recollected rational canonical form according to equation(4.38), that is

An = diag(Cλ2,Cλ2,Cλ3)


with Nn = 3 nilpotent companion matrices. From a comparison with equation (4.38) it follows thatµ1 = 0, µ2 = 2, µ3 = 1 andhmax = 3, where the latter with Definition 4.6 indicates a state graphwith 3 levels. Corollary 4.3 yields

r(0) := 0, r(1) =1∑

j=1µj j +

3∑

j=2µj = 3,

r(2) =2∑

j=1µj j +

3∑

j=3µj 2 = 6, r(3) =

3∑

j=1µj j = 7

by means of which follow the number of states per level

η(1) = 2r(1)−2r(0) = 7,

η(2) = 2r(2)−2r(1) = 56,

η(3) = 2r(3)−2r(2) = 64.

Hence, by Theorem 4.11 the number of states that reach any given state

• in 1 step is either 0 or 2r(1) = 8,

• in 2 steps is either 0 or 2r(2) = 64,

• in 3 steps is either 0 or 2r(3) = 128.

In this regard, 0 means that this given state cannot be reached because the solvability condition inequation (4.40) is not met. Figure 4.5 depicts the resultingstate graph.

4.2.3 Arbitrary Dynamics

The general statement follows by superposition of the cyclic and nilpotent subsystem. As hasbeen shown in equation (4.5) a similarity transformation can be used to decompose the dynamicsmatrix A into Arat = diag(Ac,An) which comprises a cyclic matrixAc and a nilpotent matrixAn,provided that either of both parts, cyclic and nilpotent, exists. Collecting equations (4.6) and (4.7)together in one equation and definingxT = (xT

c , xTn) ∈ Fnc+nn

q = Fnq the t-fold mapping ofx with

Arat obviously results in

Atrat

(xc

xn

)

=

(At

c xc

Atn xn

)

which makes clear how the results of the cyclic and nilpotentsubsystem can be superposed: com-posed statesxT = (xT

c , xTn) of the formxc 6= 0 andxn = 0 constitute a cyclic state graph according

to Theorem 4.8, whereas states of the formxc = 0 and xn 6= 0 constitute a so-called null tree, a


Figure 4.5: State graph of the 7-th order LMS with nilpotent dynamics matrix (Example 4.2.2.3)

state graph, all states of which incrementally approach thezero vector (null state) according toTheorem 4.11 and Corollary 4.3. For the remaining composed states withxc 6= 0 and xn 6= 0 —due to invertibility of matrixAc the cyclic subsystem is characterized by unique predecessor andsuccessor states — the overall state interconnection structure is governed by the nilpotent matrixAn only. Hence, these structures are made up of trees with the shape of the null tree and all thesetrees incrementally approach and terminate in a periodic state xn = 0. For this reason, the con-struction of the overall state graph amounts to simply attaching a null tree (to be determined byAn) to each periodic state (following fromAc).

The following theorem completes the results that have been derived so far such that the entire stategraph of the overall LMS can be determined completely.

Theorem 4.12 (State Graph of an LMS with Arbitrary Dynamics)Let Arat = diag(Ac,An) be the (reordered) rational canonical form of the dynamics matrix A ofan LMS, whereAc andAn are cyclic and nilpotent matrices, given with its associated cycle sumand null tree, respectively. Let the corresponding composed state be expressed in transformedcoordinates, i. e.xT = (xT

c , xTn). Then any periodic state of the formxc 6= 0 andxn = 0 represents

the root of a tree which has the structure of the null tree associated toAn and the remaining stateswith xc = 0 are the respective non-periodic tree states. �


4.2.3.1 Example: an LMS with Arbitrary Dynamics Matrix

For an illustration of how to construct the state graph of an LMS the dynamics matrix of whichcomprises both, a cyclic and a nilpotent subsystem, consider the dynamics matrix

A = diag(Ac,An), Ac =

(0 11 1

)

∈ F2×22 , An =

(0 01 0

)

∈ F2×22

which is given already in (reordered) rational canonical form. To matrixAc corresponds the singleelementary divisor polynomialλ2+λ+1 which is irreducible and of periodτ = 3. By Theorem 4.8the cycle sum of the respective LMS results inΣ = 1[1]∔1[3].

The single nilpotent submatrixAn is associated to the the single elementary divisor (and minimal)polynomialλ2, by Theorem 4.9 the corresponding null tree consists ofh = 2 levels and the levelnumberl = 0,1,2 comprises 1, 1 and 2 states, respectively. The single statein level l = 1 has 2confluent states, whereas the zero state has only one confluent state. Due to Theorem 4.12 thismeans to attach this null tree to any of the 4 cyclic states (see Figure 4.6).

replacements

1000

1101

0110

0111

0100

1001

1111

1110

1100

0101

1011

1010

0000

0001

0010

0011

Figure 4.6: State graph of the 4-th order LMS discussed in Example 4.2.3.1

SECTION 4.3 — INHOMOGENEOUSLMS 71

4.3 Inhomogeneous LMS

In this section, systems are in the focus which comply with

x(k+1) = Ax(k)+b , (4.44)

wherex(k) ∈ Fnq is the state at instantk∈ N0 and the additional constant vectorb ∈ Fn

q , the affinepart, enlarges the former homogeneous LMS by an inhomogeneous expression. Hence, thesesystems are termed inhomogeneous LMS. Based on the theory derived so far, it is promisingto make efforts in reducing these systems to the homogenous case. Hence, one is left with thequestion under which conditions such a transform is possible, for example by a suitable changeof coordinates. If these conditions cannot be met then an exact borderline should drawn, whichrenders the investigation of such an inhomogeneous LMS easier. This will be the guideline in whatfollows.

4.3.1 Linearization by Translation

The bijective mappingx = x−x0 represents a translation of the state vectorx by the shiftx0. Thefollowing Lemma points out when this translation can be usedfor transforming an inhomogeneousLMS according to (4.44) into a linear one.

Lemma 4.7 (First Condition for Transformability into a Homo geneous LMS)An inhomogeneous LMS given in accordance with equation (4.44) can be transformed into anhomogeneous LMS with dynamics matrixA iff b ∈ Im(I −A). �

Proof Assume an LMS withx′ = A x, which expresses the linear mapping of a statex to itssuccessorx′ in some transformed coordinates. Then, the respective (bijective) coordinate transformx = φ(x) assures the existence of a vector0 = φ(x0) for one uniquex0 in the original coordinates.Sincex = 0 is a fixpoint under the mappingA, the vectorx0 is this fixpoint expressed in the originalcoordinates, hence in the inhomogeneous LMS holds

x0 = Ax0+b

⇐⇒ (I −A)x0 = b (4.45)

and to require solvability forx0 is to requireb ∈ Im(I −A).

Conversely, ifb ∈ Im(I −A) then ax0 exists for which(I −A)x0 = b applies. Specify a coordinatetransformx = φ(x) such thatφ(x0) = 0. For example, choose the bijective mappingx = x− x0

and letx′ = x′− x0 be the successor state ofx. Then by use of the inverse transform in the state


equation, i. e.

x′+x0 = A (x+x0)+b = A x+Ax0 +(I −A)x0

= A x+x0

⇐⇒ x′ = A x ,

an LMS in the transformed coordinates is obtained, as was theother part of the claim. �

Hence the question is: which are the requirements onA and onb such thatb ∈ Im(I −A)?

To this end, the statex is transformed using equation (4.8), which transforms the dynamics matrixA of an LMS into its respective rational canonical formArat. Consequently, the transformed formof equation (4.45) is

(I −Arat) x0 = b

with the denotationb = Tb. In more detail, this equation can be written by resorting tothecompanion matrices with respect to theN elementary divisor polynomials ofA, accordingly

(I −Ci) x(i)0 = bi , i = 1, . . . ,N . (4.46)

Equation (4.46) is always solvable ifA does not contain elementary divisor polynomialspCi(λ) =

(λ−1)l i , for somel i ∈ N. In this case all matrices(I −Ci) are non-singular, hence invertible, asimplied by Theorem 4.3.

On the contrary, if for somei the defining polynomial of the companion matrixCi ∈ Fdi×diq is

pCi (λ) = (λ−1)l i , l i ∈ N, then

I −Ci =

1 0 0 · · · 0 a0,i

−1 1 0 · · · 0 a1,i

0 −1 1 · · · 0 a2,i......

.... . .

......

0 0 0 · · · −1 1+adi−1,i

and with the non-singular transformation matrix

L =

1 0 0 · · · 01 1 0 · · · 01 1 1 · · · 0...

......

. . ....

1 1 1 · · · 1

(4.47)

equation (4.46) becomes

L (I −Ci) =

1 0 0 · · · a0,i

0 1 0 · · · a0,i +a1,i

0 0 1 · · · a0,i +a1,i +a2,i......

.... . .

...0 0 0 · · · 1+

∑di−1j=0 a j ,i

x(i)0 =

b1,i

b1,i + b2,i

b1,i + b2,i + b3,i...∑di

j=1 b j ,i

(4.48)


Note that

1+

di−1∑

j=0

a j ,i = pCi (1) (4.49)

with regard to the polynomialpCi (λ) = (λ−1)l i . But pCi (1) = 0 which in consequence means thatfor assuringbi ∈ Im(I −Ci) the relation

di∑

j=1

b j ,i = 0. (4.50)

has to be satisfied with respect to all numbersi which refer to elementary divisor polynomials ofthe formpCi(λ) = (λ−1)l i . With Lemma 4.7 this yields a proof of

Theorem 4.13 (General Condition for Transformability into a Homogeneous LMS)Let an inhomogeneous LMS be given by its state equation in rational canonical form

x(k+1) = Aratx(k)+ b = diag(C1, . . . ,CN)

x(1)(k)...

x(N)(k)

+

b1...

bN

with i = 1, . . . ,N companion matricesCi associated to the elementary divisor polynomials of therespective dynamics matrixA. Then the inhomogeneous LMS can be transformed into an equiv-alent LMS by a change of coordinates iff the sum of vector entries is zero for those vectorsbi

which correspond to elementary divisor polynomials of the form pCi (λ) = (λ−1)l i , if any. If noelementary divisor polynomial is of this form then the inhomogeneous LMS is equivalent to anLMS without any further restriction. �

If this transformability condition is satisfied then the respective shift vectorx(i)0 concerning the

partial coordinate transformx(i) = x(i) − x(i)0 can be determined. To this end, observe that with

equation (4.48), with the relation 1+∑di−1

j=0 a j ,i = 0 and with the kernel

Ker(I −Ci) = Ker(L (I −Ci)

)

= span{(−α0,i ,−α0,i −α1,i , . . . ,−α0,i −α1,i . . .−αdi−2,i ,1)T}

= span{(1+α1,i + · · ·+αdi−1,i ,1+α2,i + · · ·+αdi−1,i , . . . ,1+αdi−1,i ,1)T} (4.51)

the homogeneous solution can be obtained. A particular solution of equation (4.48) is given bysetting the last entry of the solution vector equal to zero. Both ideas are summarized within thefollowing theorem.

Theorem 4.14 (Translation into a Homogeneous LMS)Assume that for an inhomogeneous LMS in rational canonical form the transformability conditionof Theorem 4.13 is fulfilled. Then each subsystemi = 1, . . . ,N of the inhomogeneous LMS can betransformed into an LMS by use of a translation of the substate x(i), that isx(i) = x(i)− x(i)

0 , where


1. for elementary divisor polynomialspCi(λ) 6= (λ−1)l i ,

x(i)0 = (I −Ci)

−1 bi ,

2. for elementary divisor polynomialspCi(λ) = (λ−1)l i ,

x(i)0 =

b1,i

b1,i +b2,i...∑di−1

j=1 b j ,i

0

+

1+adi−1,i + · · ·+a1,i

1+adi−1,i + · · ·+a2,i...1+adi−1,i

1

zi , ∀zi ∈ Fq .

With these results, in the overall system the respective translation of state is

x = x−(

x(1)0

T, . . . , x(N)0

T)T

that transforms the inhomogeneous LMS into a linear one. �

Remark 4.10All free parameterszi in part 2 of Theorem 4.14 can be chosenzi = 0, for simplicity. �

4.3.2 Non-linearizable Parts

In order to complete the result, subsystems

x(i)(k+1) = Ci x(i)(k)+ bi (4.52)

of an inhomogeneous LMS have to be concerned which do not suffice the transformability condi-tion of Theorem 4.13.

Theorem 4.15 (Cycle Sum of the Non-linearizable Subsystem of an Inhomogeneous LMS)Assume that for an inhomogeneous LMS in rational canonical form the transformability conditionof Theorem 4.13 is not fulfilled byNu subsystems associated toi = 1, . . . ,Nu elementary divisorpolynomialspCi (λ) of degreedi . Then allqdi states of subsystemi areti-periodic with

ti =

{ql i if ∃l i ∈ N : ql i > di > ql i−1

ql i+1 otherwise (∃l i ∈ N : ql i = di)(4.53)

and the subsystemi contributes the cycle sum

Σi =qdi

ti[ti] . (4.54)

to the cycle sum

Σu =qd1+···+dNu

lcm(t1, . . . , tNu)[lcm(t1, . . . , tNu)] (4.55)

of all Nu non-linearizable subsystems. �


Proof The following statements are to be proven:

1) any state in thei-th subspace has a unique predecessor, thus is periodic,

2) all states have periodti as in equation (4.53),

3) the cycle sum ofNu non-linearizable subsystems is given by equation (4.55).

In order to facilitate the notation, the system indexi will be omitted.

ad 1) Rewrite equation (4.52) as per

Cx(k) = x(k+1)− b

Since with pC = (λ− 1)d the companion matrixC is invertible any statex(k+ 1) has aunique predecessor statex(k) = C−1(x(k+1)− b), hence, the state graph cannot show treestructure, and due to the assumption that a vectorx0 solving(I −C) x0 = b does not existthere is no statex(k+1) = x(k). As a consequence, all states have to be periodic witht > 1.

ad 2) Starting with equation (4.52), the existence of at-periodic statex(k+ t) = x(k) =: xt meansthat

(I −Ct) xt =

(t−1∑

i=0

Ci

)

b (4.56)

holds for a least integert. The left hand side of this equation comprises a geometric series,which results in

(t−1∑

i=0

Ci

)

(I −C) xt =

(t−1∑

i=0

Ci

)

b

⇐⇒(

t−1∑

i=0

Ci

)(

(I −C) xt − b)

= 0 (4.57)

There are three conceivable ways of solving equation (4.57):

a) a trivial solution(I −C) xt − b = 0 exists,

b) there are vectorsx⋆ ∈ Ker(∑t−1

i=0 Ci) with x⋆ := (I −C) xt − b,

c)t−1∑

i=0Ci ≡ 0.

ad a) By assumption the inhomogeneous LMS is not linearizable, therefore, such a trivialsolution does not exist as per Lemma 4.7.


ad b) Equation (4.57) can be solved non-trivially iff the determinant det(∑t−1

i=0 Ci) = 0.Then somex⋆ might be found in Ker(

∑t−1i=0 Ci). As by Theorem 4.3 the polynomials

pC(λ) = (λ−1)l and∑t−1

i=0 λi must have a common factor, it results that the matrix(I −C) has a rank deficiency and the vectorx⋆ is a solution of

(I −C) x⋆ = 0 (4.58)

by means of whichxt can be determined with

(I −C) xt = x⋆ + b . (4.59)

By use of the kernel in equation (4.51) the solution of equation (4.58) is

(x⋆)T = z(1+ad−1+ · · ·+a1,1+ad−1+ · · ·+a2, . . . ,1+ad−1,1) ∀z∈ Fq . (4.60)

With this result and by applying the solvability condition (4.50), equation (4.59) issolvable iff

d∑

i=1

x⋆i + bi = 0. (4.61)

For testing this, first, calculate the coefficients in (4.60)by recalling that

(λ−1)d =

d∑

i=0

(di

)

(−1)d−i λi , (4.62)

hence,

ai =

(di

)

(−1)d−i , i = 0, . . . ,d (4.63)

and by equation (4.60) the sum of entries in vectorx⋆ reads

d∑

i=1

x⋆i = z(d+(d−1)ad−1+(d−2)ad−2+ · · ·+a1) = z

d∑

i=1

iai

= zd∑

i=1

i

(di

)

(−1)d−i . (4.64)

The only interesting cases are vector-valued, thus, assumed > 1 in the evaluation of

d∑

i=1

x⋆i = z

d∑

i=1

i

(di

)

(−1)d−i = zd∑

i=1

d!(d− i +1−1)!(i −1)!

(−1)d−i

= zdd∑

i=1

(d−1)!((d−1)− (i −1))!(i −1)!

(−1)d−i = zdd∑

i=1

(d−1i −1

)

(−1)d−i

= zdd−1∑

j=0

(d−1

j

)

(−1)d−1− j = zd(λ−1)d−1∣∣∣λ=1

= 0 (4.65)


and consequentlyd∑

i=1

x⋆i + bi =

d∑

i=1

bi . (4.66)

But then by condition (4.61) equation (4.59) is solvable iff

d∑

i=1

bi = 0 (4.67)

which contradicts the assumption that the inhomogeneous LMS is non-linearizable;see Theorem 4.13.

ad c) Since there are periodic states it remains only that

t−1∑

i=0

Ci ≡ 0 (4.68)

is satisfied for some least integert. With pC(λ) = mpC(λ) = (λ−1)d Theorem 4.2implies that

g(λ)(λ−1)d =t−1∑

i=0

λi (4.69)

for some polynomialg(λ). Multiplication by(λ−1) yields

g(λ)(λ−1)d+1 = λt −1 (4.70)

for a least integert. But then equation (4.70) implies thatt is the period of the poly-nomial(λ−1)d+1. This periodt results from Theorem 2.5, accordingly

t = τ(λ−1)d+1 =

{

τ(λ−1)d = ql if ql > d > ql−1, l ∈ N

q τ(λ−1)d = ql+1 if ql = d, l ∈ N(4.71)

In part 1 it has been shown that all states in the respective state space are periodic.Observe, that equation (4.68) is state-independent. Hence, all states are periodic ofthe same periodt and altogether they constituteqd/t cycles of lengtht

ad 3) Using Theorem 4.7 for the superposition ofi = 1, . . . ,Nu such subsystems yields

ΣNu =

(qd1

t1[t1]

)

· · ·(

qdNu

tNu

[tNu]

)

=qd1 · · ·qdNu

t1 · · ·tNu

gcd(t1, . . . , tNu) [lcm(t1, . . . , tNu)]

=qd1+···+dNu

lcm(t1, . . . , tNu)[lcm(t1, . . . , tNu)]

in accordance with equation (4.55); which completes the proof. �


Remark 4.11In the subspace with regard to an elementary divisor polynomial (λ−1)d of a homogeneous LMS,the maximal possible period of the subspace states isτmax= ql with ql ≥ d > ql−1. If this subsys-tem is a non-linearizable inhomogeneous LMS with dimensiond = ql then the maximal possibleperiod of the subspace states isτmax= ql+1, that isq-times the maximal period of an homogenousLMS. �

4.3.3 General Inhomogeneous LMS

Summing up the results of the past sections, an inhomogeneous LMS as in equation (4.44) can besplit into three uncoupled subsystems

xc,u(k+1) = Ac,u xc,u(k)+ bu (4.72)

xc,s(k+1) = Ac,sxc,s(k) (4.73)

xn(k+1) = An xn(k) (4.74)

First, the system with cyclic dynamics matrixAc,u represents the non-linearizable subsystem re-gardingAc; all elementary divisor polynomials ofAc,u are of the formpCi (λ) = (λ−1)l i and thecorresponding sum of coefficients is

∑

i bi 6= 0. The second system with cyclic dynamics matrixAc,s represents the linearizable subsystem regardingAc; any elementary divisor polynomial ofAc,s that is of the formpCi(λ) = (λ−1)l i is associated to

∑

i bi = 0. Finally the third system, thenilpotent subsystem with dynamics matrixAn, is always linearizable by a shift of state.

Finally, a possible combination of the main results developed in Section 4.2 and in Section 4.3 forcalculating the cycle sum and the state graph of an inhomogeneous LMS is the following:

1. Transform the coordinates of the states in the inhomogeneous LMS such that with

x(k+1) = Ax(k)+b ⇐⇒ x(k+1) = Aratx(k)+ b

the dynamics matrixArat is in ordered rational canonical formArat = diag(Ac,An), whichcontains the dynamics matrices of the cyclic and nilpotent subsystems.

2. Check the linearizability condition of Theorem 4.13 for any cyclic subsystem (4.52) con-cerningAc and split this inhomogeneous LMS into a linearizable subsystem with dynamicsmatrixAc,s and a non-linearizable subsystem with dynamics matrixAc,u.

3. Determine the cycle sumΣs of the subsystem concerningAc,s by use of Theorem 4.8.

4. Determine the cycle sumΣu of the subsystem concerningAc,u by use Theorem 4.15.

5. Superpose the cycle sums referring to Theorem 4.7 according toΣ = ΣsΣu to obtain the cyclesumΣ of the overall inhomogeneous LMS.


6. Calculate the null tree with respect to the nilpotent subsystemAn by means of Theorem 4.11and Corollary 4.3.

7. For obtaining the state graph of the entire inhomogeneousLMS attach the above-derivednull tree to any periodic state with regard to the cycle sumΣ; as per Theorem 4.12.

4.3.4 Example

For a demonstration of the results consider an inhomogeneous LMS(2) of ordern = 6 with adynamics matrixA that is given in ordered rational canonical form according to

x(k+1) = Ax(k)+b, A = diag(C(λ+1)2,C(λ+1)3,Cλ), bT = (bT1 ,bT

2 ,bT3),

C(λ+1)2 =

(0 11 0

)

, C(λ+1)3 =

0 0 11 0 10 1 1

, Cλ = 0, b1 =

(11

)

, b2 =

111

, b3 = 1,

hence, the dynamics matrixA = diag(Ac,An) is (already) decomposed into a cyclic block-diagonalmatrix Ac = diag(C(λ+1)2,C(λ+1)3) and a nilpotent block-diagonal matrixAn = Cλ.

If the task is to determine the cycle sum of this inhomogeneous LMS, one is left with two compan-ion matrices in the blocks ofA, these areC(λ+1)2 andC(λ+1)3 with defining polynomial(λ−1)2

and(λ−1)3, respectively, which both are candidates for representingnon-linearizable subsystems.Thus, examine the mod2-sum of coefficients ofb1

d1∑

j=1

b j ,1 = 1+1 = 0

and ofb2d2∑

j=1

b j ,2 = 1+1+1 = 1 6= 0.

These signify that the subsystem corresponding toAc,s = C(λ+1)2 can be linearized as per

xc,s(k+1) = Ac,sxc,s(k) (4.75)

using the translated statexc,s = x1−x(1)0 with a shift vectorx(1)

0 , which by Theorem 4.13 is guaran-

teed to exist and via Theorem 4.14 together with Remark 4.10 can be chosen asx(1)0 = (1,0)T. In

addition to that, the subsystem corresponding toAc,u = C(λ+1)3 andbu = b2 cannot be linearized,hence

xc,u(k+1) = Ac,uxc,u(k)+bu . (4.76)


Moreover, it is clear that the nilpotent subsystem associated toAn is linearizable by the shift ofstatexn = xn−1, that is

xn(k+1) = An xn(k) .

Then with Theorem 4.7, the cycle sumΣ of the entire inhomogeneous LMS can be determined bythe superposition of the cycle sumΣs andΣu of the systems (4.75) and (4.76), respectively.

ConcerningΣs, recall Theorem 4.8, which withAc,s = C(λ+1)2 means to calculate the cycle sumwith respect toC(λ+1)2 only, i. e.

Σs = 1[1]∔21−1

1[1]∔

22−21

2[2] = 1[1]∔1[1]∔1[2] = 2[1]∔1[2] .

RegardingΣu, recall Theorem 4.15, which withAc,u = C(λ+1)3 amounts to determine the cyclesum regardingC(λ+1)3 only. As with the degreed2 = 3 and for some leastl ∈ N

2l > d2 > 2l−1 =⇒ 4 > 3 > 2

it follows that the period of(λ+1)3 is t3 = 4 and

Σu =23

4[4] = 2[4] .

Superposition of the cycle sumsΣs andΣu yields the cycle sum of the overall inhomogeneous LMS

Σ = ΣuΣs = 2[4](2[1]∔1[2]

)=(2[4]2[1]∔2[4]1[2]

)

= 2 ·2 gcd(1,4) [lcm(1,4)]∔1 ·2 gcd(2,4) [lcm(2,4)] = 4[4]∔4[4] = 8[4] .

Finally, the nilpotent subsystemAn = 0 implies a null tree, which attached to each state in these 8cycles of length 4, results in the state graph given in Figure4.7.

· · · 8 times · · ·

Figure 4.7: State graph of the inhomogeneous LMS of order 4 analyzed in Example 4.3.4

Conclusion

In this chapter, a method for analyzing the state interconnection structure within state spaces ofautonomous linear dynamic systems over finite fields, so-called linear modular system (LMS), is


developed. The analysis method is based on the inspection ofthe elementary divisor polynomials,which are invariant under similarity transforms. When examining how the state space decomposesinto subspaces, in particular into subspaces with periodicstates, the period of these polynomials,which is a characteristic integral number common to all polynomials over a finite field, is shownto be decisive. Refraining from finite ring theory and setting up the theoretical development ona fundamental level of linear algebra and simple combinatorics only, the main theorem on thedecomposition of the state space of a linear dynamic system into periodic subspaces is derived.The main outcome is a new criterion which allows to determineall cycles in length and number,and further, its respective periodic states, a result that can be applied in a straight-forward manneron automata modeled as LMS. It turns out that periodic transition behavior occurs already in thecase of linear automata, hence, it is a linear phenomenon andis not to be confused with limit cyclesas are encountered within non-linear continuous time systems.

A further result refers to the non-periodic elementary divisor polynomials, which correspond tonilpotent companion matrices. Again, these polynomials comprise the information necessary fordetermining the state interconnection structure within the associated non-periodic subspaces thestate graph of which is a tree. Employing particular singular inverses a constructive procedure forderiving this state graph is presented, which is as before based on fundamental linear algebra only.By connecting the former results the structural analysis ofautonomous LMS is rendered possibleand verified in examples.

In an extension of the theory derived so far, affine-linear autonomous dynamic systems over fi-nite fields (called inhomogeneous LMS) are concerned. Two conditions are deduced which statewhether an inhomogeneous LMS can be reduced to a linear one bya coordinate transform in formof a state translation. If these conditions are fulfilled then the state graph can be constructed byresorting to the methods from above. If these conditions cannot be met, however, a theorem isproven that connects the periodicity of the states in the respective subspace with the period ofthe corresponding elementary divisor polynomial of the dynamics matrix of the inhomogeneousLMS. The combination of all results grants efficient means for a structural analysis of autonomousinhomogeneous LMS in general, as is illustrated in the closing part of this chapter.

Chapter 5

Synthesis of Linear Systems over FiniteFields

In the last chapter, for state space models that are autonomous linear dynamic systems over a finitefield, conditions were presented by means of which the state space can be decomposed into sub-spaces whose states, typically, are periodic or correspondto tree states; both indicating a respectivestructure in the state graph.

Regarding non-autonomous systems, inputs are at one’s disposal which can be used for control-ling the state evolution. Further equipped with knowledge about the current state — provided bymeasurement, for example — the input can be related to an appropriate function of the state, aso-called (static) control law, in order to synthesize desired system properties in a feedback con-trol loop. By virtue of a control law the closed-loop system is rendered autonomous. As a linearsystem remains linear under linear state feedback, consequently, the resulting autonomous closed-loop system again can be analyzed with the methods presentedin Chapter 4. If the initial pointis a certain closed-loop behavior then a usual task is to design a control that ensures this desiredbehavior. Thus naturally, the notion of controllability comes to the fore.

If the purpose of control is to guarantee certain closed-loop properties then a natural question is toask for criteria about the existence of a suitable state feedback, and subsequently, how this suitablecontrol law can be chosen. For linear continuous systems Rosenbrock’s control structure theoremaddresses the existence problem by relating the (desired) closed-loop invariant polynomials withthe (control-invariant) controllability indices [Wol74,Kuc91]. Thus, Rosenbrock’s control struc-ture theorem is of particular value when the goal of feedbacksynthesis is to fit the closed-loop sys-tem with desired elementary divisor polynomials, which hasalready been shown sufficient for im-posing a specific cycle sum on an LMS in Chapter 4. Since the outcome is an existence statement,the actual feedback law still has to be derived. For specifying invariant polynomials, image domainmethods are well-known to be the suitable tool in the continuous world [Kuc91, Ant98, DH01]. To

84 CHAPTER 5 — SYNTHESIS OFL INEAR SYSTEMS OVERFINITE FIELDS

this end, theA-transform introduced in Chapter 2 is worked out for an adaption of the polynomialapproach onto systems over finite fields for the first time. Theresult is an algorithm that allowsto synthesize a state linear feedback for imposing a set of desired elementary divisor polynomialson the closed-loop system so as to obtain a closed-loop stategraph comprising desired structuralelements (i. e. states of certain periods, particular null tree). Expressed in a more formal way, thisalgorithm solves the following control problem:

Definition 5.1 (Cycle Sum Synthesis Problem (CSSP))Given an LMS as in Definition 4.1 with measurable state. Design a linear state feedback such thatthe closed-loop LMS shows a specified cycle sum. �

The chapter is organized as follows: Basic notions as controllability of an LMS, controllabilityindices, and the controllability companion form (CCF) are presented in Section 5.1. The maincontribution of this chapter is represented by Section 5.2 which develops a method for synthe-sizing a state feedback controller in an image domain. By invoking basics from the polynomialapproach, a polynomial matrix fraction representation of the closed-loop transfer matrix is derivedthat ensures a specified periodic behavior. The result is a synthesis algorithm yielding the respec-tive state feedback matrix. The chapter ends with an outlookof how to integrate a non-controllablepart of the system and gives an illustration of the presentedmethod in an example.

5.1 Controllability of an LMS

The property of controllability is a key prerequisite when considering control systems.

Definition 5.2 (Controllability)An LMS of order n is l -controllable iff for all ordered pairs of states(x1,x2), x1,x2 ∈ Fn

q , thesystem can be driven (steered) from statex1 to statex2 in l steps. An LMS is controllable iff it isl -controllable for somel . �

In order to establish an easy criterion for checking controllability, resolve the recursion within thestate equation (4.1) of an LMS according to Definition 4.1 so as to obtain its solution

x(k) = Ak x(0)+

k−1∑

i=0

Ak−1−i Bu(i) . (5.1)

In light of Definition 5.2 rewrite equation (5.1) in the form

(B,AB, . . . ,Ak−1B

)(uT(0), . . . ,uT(k−1))T = x(k)−Akx(0) (5.2)

SECTION 5.1 — CONTROLLABILITY OF AN LMS 85

which indicates that an input sequenceu(0), . . . ,u(k−1) that drives an arbitrary statex(0) to another arbitrary statex(k) in k steps (steering problem) can only be found if1

Im(B,AB, . . . ,Ak−1B) = Fnq .

Furthermore, applying the theorem of Cayleigh-Hamilton, Theorem 4.1, onA yields that any col-umn vector of some matrixAk for k ≥ n can be expressed as some linear combination of corre-sponding column vectors of matricesA i with i < n. Hence, any state can be reached in at mostnsteps, and a controllability criterion can be stated which is in full accordance with the well-knownresult in the continuous case.

Theorem 5.1 (Controllability Criterion)An LMS of ordern is l -controllable iff the matrix(B,AB, . . . ,A l−1B) ∈ Fn×lm

q has (full) rankn. �

Definition 5.3 (Controllability Matrix)Given an LMS of ordern. The matrix(B,AB, . . . ,An−1B) ∈ Fn×nm

q is called controllability matrixof the LMS. �

Under assumption of controllability and by choosing new coordinates, the state equation (4.1) ofan LMS can be transformed into the so-called controllability companion form.

5.1.1 Controllability Matrix and Controllability Indices

Given controllability, as per Theorem 5.1, a reduced controllability matrix L ∈ Fn×nq of an LMS

can be determined by choosingn linearly independent column vectors from the controllabilitymatrix in Definition 5.3. These linearly independent columnvectors are chosen in a way such thatthe appendant powers ofA are minimal, see [Wol74]. This procedure yields the invertible matrix

L =(b1, . . . ,Ac1−1b1,b2, . . . ,Ac2−1b2, . . . ,bm, . . . Acm−1bm

), (5.3)

where thei = 1, . . . ,m vectorsbi are the respective column vectors of the input matrixB.2

Definition 5.4 (Controllability Indices)Let the reduced controllability matrixL ∈ Fn×n

q of an LMS be given as in equation (5.3). Thei = 1, . . . ,m numbersci ∈ N are called controllability indices. �

1Given solvability, an appropriate generalized inverse matrix according to Appendix D is given by a matrix whosecolumn space is the orthogonal complement space of(B,AB, . . . ,Ak−1B) — see the discussion in Section 5.2.6. For amethod that employs a deadbeat-like feedback for solving the steering problem refer to the example in Appendix E.

2Without loss of generality, the column vectors of the input matrix B can be assumed linearly independent since,otherwise, the inputs would depend on each other which contradicts liberality in input choice.


Referring to [Wol74] again, some important properties of controllability indices shall be listed inthe following theorem.

Theorem 5.2 (Properties of Controllability Indices)Let ci , i = 1, . . . ,m, be the controllability indices with respect to an LMS of order n. Then thefollowing properties hold:

• the set of controllability indices is unique,

• the set of controllability indices is invariant with respect to a change of state coordinates,

• the LMS is controllable iff∑m

i=1ci = n. �

5.1.2 The Controllability Companion Form

For controllable LMS, the state equation (4.1) can be transformed into a particular form by achange of state coordinatesx using other characteristic coordinatesxc = Qx. The calculation ofthe transformation matrixQ ∈ Fn×n

q is based on the reduced controllability matrixL from (5.3)and its associatedi = 1, . . . ,m controllability indicesci . The resulting state equation is calledcontrollability companion form (CCF) [Wol74, Ant98].

Definition 5.5 (Controllability Companion Form (CCF))An LMS of ordern with m inputs is represented in controllability companion form (CCF) iff itsstate equation reads

xc(k+1) = Acxc(k)+Bcu(k), Ac ∈ Fn×nq , Bc ∈ Fn×m

q , (5.4)

Ac =

Ac11 Ac

12 · · · Ac1m

Ac21 Ac

22 · · · Ac2m

......

. . ....

Acm1 Ac

m2 · · · Acmm

, Acii =

0 1 0 · · · 00 0 1 · · · 0...

......

. . ....

0 0 0 · · · 1x x x · · · x

, Aci j ,i 6= j =

0 0 0 · · · 00 0 0 · · · 0...

......

. . ....

0 0 0 · · · 0x x x · · · x

,

Bc =

Bc1

Bc2...

Bcm

, Bci =

0 · · · 0 0 · · · 0...

......

.... . .

...0 · · · 0 0 · · · 00 · · · bc

ci i x · · · x

, bcci i = 1

whereAci j ∈ F

ci×c jq , Bc

i ∈ Fci×mq andbc

ci i is the element in theci-th row andi-th column of eachmatrix Bc

i , i, j = 1, . . . ,m. Moreover, any symbol “x” represents an arbitrary number inFq. �

SECTION 5.1 — CONTROLLABILITY OF AN LMS 87

The non-determined rows within the system in CCF can be concentrated in two matrices,

Acσ :=

rowσ1(Ac)

rowσ2(Ac)

...rowσm(Ac)

, Bcσ :=

rowσ1(Bc)

rowσ2(Bc)

...rowσm(Bc)

=

1 x x · · · x0 1 x · · · x0 0 1 · · · x...

.... . . . . .

...0 0 0 · · · 1

, (5.5)

in which σi :=∑i

j=1c j , i = 1, . . . ,m and the expression rowi(.) denotes thei-th row of a matrix.Both matrices will be needed in Section 5.2.3.

The question of which matrix to choose for transforming an arbitrary controllable system into therepresentation in CCF is answered in the next theorem.

Theorem 5.3 (CCF Transformation Matrix)Let an LMS of ordern with m inputs be controllable withi = 1, . . . ,m controllability indicesci

andσi :=∑i

j=1c j , i = 1, . . . ,m. Furthermore, letL−1 ∈ Fn×nq denote the inverse of the respective

reduced controllability matrix. Then a change of state coordinatesxc = Qx by virtue of the matrix

Q =

qT1

qT1 A...

qT1 Ac1−1

qT2

qT2 A...

qT2 Ac2−1

...qT

mAcm−1

, qTi = rowσi (L

−1), i = 1, . . . ,m (5.6)

transforms the state equation into CCF with dynamics matrixAc and input matrixBc as per

Ac = QAQ−1, Bc = QB, (5.7)

respectively.3 �

Remark 5.1 (Ordered Controllability Companion Form)For facilitating the algorithm in Section 5.2.4 that solvesthe cycle sum synthesis problem in Def-inition 5.1 it is advisable to have the controllability indices in decreasing order. Without loss of

3For continuous systems, the matrixL often is ill-conditioned. Due to the numerical problems incurred, thecalculation ofL−1 usually is avoided. For matrices over finite fields such problems cannot arise.


generality, this property can easily be obtained by renaming the input vector entries in a mannersuch that the powers ofA in the reduced controllability matrix (5.3) obeyc1 ≥ . . . ≥ cm.

Without renumbering of the inputs, the block matricesAci j ∈ F

ci×c jq in Ac can be reordered with

respect to decreasing controllability indices by a respective permutation matrixΠΠΠ, as described inAppendix A. Note that a consequence of this procedure is thatthe input matrix transforms as well,that is, the new input matrix becomesΠΠΠBc. Hence, the matrix that corresponds toBc

σ is in generalnot upper triangular anymore. �

In light of this conceptual framework, a synthesis method for imposing a specific cycle sum on anLMS can be developed.

5.2 Synthesis in the Image Domain

Changing the elementary divisor polynomials, which is equivalent to changing the invariant poly-nomials of an LMS, is closely related to changing the eigenvalues of the system dynamics. Fromthe theory of linear discrete time systems over the field of real numbers it is well-known that achange of eigenvalues of the system dynamics can be achievedby introducing a (static) linear statefeedback.

5.2.1 Linear State Feedback and its Structural Constraints

In view of the controllability companion form and by linearity of the system to be controlled a firstsimple state feedback is a linear form4

u(k) = Kx(k) . (5.8)

This leeds to the closed-loop state representation

x(k+1) = (A +BK)x(k) (5.9)

in which the matrixA +BK is the closed-loop dynamics.

Remark 5.2The influence of state feedback can be studied easily by considering the state space representationin CCF because any controllable LMS can be transformed into CCF. For this purpose, note thatBc

σ is invertible, thusBc generally can be altered by right-multiplication with(Bcσ)−1 such that the

resulting product matrix chooses thei-th rows,i = 1, . . . ,m, from any feedback matrix to the right.Thus, the correspondingσi-th rows of the closed-loop dynamics matrix can be changed completelyat choice. �

4which can be extended by an additional new input, of course

SECTION 5.2 — SYNTHESIS IN THE IMAGE DOMAIN 89

Therefore, the question arises whether or at least to which extent the closed-loop invariant poly-nomials can be chosen freely. This major question is answered by Rosenbrock’s control structuretheorem [Kai80, Ros70], which can be shown to apply to systems over finite fields as well.5

Theorem 5.4 (Rosenbrock’s Control Structure Theorem)Given a controllable LMS of ordern with controllability indicesc1 ≥ . . . ≥ cm and desired monicinvariant polynomialsci,K ∈ Fq[λ] with ci+1,K |ci,K , i = 1, . . . ,m−1, and

∑mi=1deg(ci,K ) = n. Then

a matrix K ∈ Fm×nq exists such thatA + BK has the desired invariant polynomialsci,K iff the

inequalitiesk∑

i=1

deg(ci,K) ≥k∑

i=1

ci , k = 1,2, . . . ,m (5.10)

are satisfied. �

Rosenbrock’s control structure theorem is of particular importance for the controller problem onhand because it entails a limit when focusing on maximal liberality in the choice of closed-loopinvariant polynomials, that is, in the choice of a desired cycle sum. If the inequalities in (5.10)can be verified for a desired set of closed-loop invariant polynomials then an appropriate feedbackmatrix exists, which can be determined by a method, one is still free to choose.

5.2.2 Controller Design in the Image Domain — why?

On the face of it, pole placing methods cannot be applied since specifying invariant polynomi-als, generally, is a stronger requirement than specifying eigenvalues. Consequently, standard poleplacing methods do not meet the requirements and some specific method for synthesizing a statefeedback for MIMO-systems is necessary. A well-established method for systems over the fieldof real numbers is the parametric approach [Rop86, DH01]. Inthis approach, besides the closed-loop eigenvalues, the remaining degrees of freedom are usedfor specifying a linear combinationof closed-loop eigenvectors, the so-called parameter vectors, with the purpose of achieving a par-ticular closed-loop behavior.

However, parametric approach techniques turn out to be inapplicable if the control objective is todesign a state feedback that fits an LMS with a set of invariantpolynomials. This is due to thefollowing peculiar reasons:

• In the parametric approach it is fundamental that the open-loop and closed-loop eigenvaluesare distinct. Applying this approach to LMS would incur thatthe invariant polynomials inthe open and the closed loop have to be different, which is a restrictive assumption in theframework of LMS.

5See remark in [Kai80, p. 517].


• The assignment of multiple eigenvalues, which would be indispensable for the realizationof rather standard cycle sums (e. g. cycles of even length fora model overF2), proves to becumbersome because the computation of chains of generalized eigenvectors is required.

• As eigenvalues of matrices over a finite fieldFq are roots of some polynomial the notion ofzeroes becomes important. These zeroes typically lie in some extension field ofFq, the sizeof which deeply depends on the degree of the polynomial and ofits factors. Moreover, theseextension fields have no defining element in common [LN94]. This is a severe differenceto the field of real numbers in which any polynomial inR[λ] can be factored into quadraticirreducible polynomials overR (see Definition 2.7). Hence, any zero of a polynomial inR[λ] lies in the corresponding extension field, which is the field of complex numbersC withunique defining elementi =

√−1. Conversely, such a uniform factorization is not possible

for polynomials inFq[λ] with the consequence that the computation of eigenvalues intheextension field ofFq entails enormous symbolical computation effort.6

• The structural theorem imposes realizability constraintson the invariant polynomials in theclosed-loop system. Thus, observing these constraints viasome suitable set of closed-loopinvariant polynomials immediately yields the respective Smith form of the closed-loop dy-namics. Once given the Smith form of the closed-loop dynamics, an appropriate imagedomain framework provides simple straight-forward methods for determining the suitablestate feedback matrix.7

In view of these issues, image domain design techniques as inparticular the polynomial approach,will be adapted for the state feedback synthesis of LMS.

5.2.3 The Polynomial Matrix Fraction of the Transfer Matrix

An image domain representation of an LMS can be obtained by resorting to theA-transform in-troduced in Definition 2.24. In the image domain, the counterpart of the state equation (4.1) reads

aX(a) = AX(a)+BU(a)+ax(0) (5.11)

in which capital letters in bold face with argument denote the respectiveA-transformed variables.This representation directly leads to theA-transform of the system state8

X(a) = (aI −A)−1(BU(a)+ax(0)). (5.12)

6Refer to Appendix C for an example.7In the same manner, this statement holds true for the frequency domain with respect to continuous systems.8Appendix F exposes how the inverseA-transform can be used for determining the solution of the state equation.


The rational matrixF(a) = (aI −A)−1B in equation (5.12) can be identified with the customaryform of a transfer matrix since

X(a)∣∣∣x(0)=0

= F(a)U(a) = (aI −A)−1BU(a) . (5.13)

Nevertheless, the interpretation ofF(a) as a transfer matrix is of minor importance here. In thefollowing, just a particular decomposition of the transfermatrix will be used since it prepares theground for a simple method of setting the invariant polynomials ofA +BK , i. e. in the closed loop.

5.2.3.1 Fundamentals

The polynomial matrix method is based on a particular decomposition of the transfer matrix in(5.13). To this end, those notions and concepts of the polynomial matrix approach which aresignificant for the purpose here are recalled from [Wol74, Kuc91, Ant98], for convenience.

First the concepts of factors and primeness of polynomial matrices shall be introduced.

Definition 5.6 (Right Divisor and Left Divisor of a Polynomial Matrix)Let P(a), L(a) andR(a) be polynomial matrices. If

P(a) = L(a)R(a)

thenR(a) andL(a) are called right divisor and left divisor ofP(a), respectively. �

Definition 5.7 (Greatest Common Right (Left) Divisor of Polynomial Matrices)Let P(a) andQ(a) be polynomial matrices. A greatest common right (left) divisor of the poly-nomial matricesP(a) andQ(a) is a common right (left) divisor which is a left (right) multiple ofevery common right (left) divisor ofP(a) andQ(a). �

Definition 5.8 (Right-prime and Left-prime Polynomial Matr ices)Polynomial matrices with the same number of columns (rows) are termed right-prime (left-prime)if their greatest common right divisors (greatest common left divisors) are unimodular matrices.�

In order to state a less involved criterion for testing primality of polynomial matrices, a Bézoutdentity can be derived for polynomial matrices over finite fields as well. The proof can be kept tothe lines in [Kai80, p. 379].

Lemma 5.1 (Bézout Identity)Let P(a) andQ(a) be polynomial matrices. ThenP(a) andQ(a) are right-prime (left-prime) iffpolynomial matricesX(a), Y(a)

(X(a), Y(a)

)exist such that the equations

X(a)P(a)+Y(a)Q(a) = I(P(a) X(a)+Q(a) Y(a) = I

)(5.14)

hold. �


Lemma 5.1 is the basis for deriving a controllability criterion that is based on the primality ofpolynomial matrices only, and renders the calculation of eigenvalues in some field extension un-necessary — the proof in [Ros70] applies also to the finite field case.

Theorem 5.5 (Controllability Criterion)Let A be the dynamics matrix andB be the input matrix of an LMS. The LMS is controllable iffthe polynomial matrices(aI −A) andB are left-prime. �

Definition 5.9 (Polynomial Matrix Fraction)A right (left) polynomial matrix fraction RPMF (LPMF) of a rational matrixR(a) is an expressionof the following form

R(a) = N(a)D−1(a)(R(a) = D−1(a) N(a)

)(5.15)

in which the denominator matricesD(a), D(a) and the numerator matricesN(a), N(a) are polyno-mial matrices. �

Lemma 5.2 (Existence of a Polynomial Matrix Fraction)For any rational matrixR(a) there exists a right-prime RPMF (left-prime LPMF). �

Consequently, any transfer matrix can be represented by some polynomial matrix fraction. Inparticular, the transfer matrix representation in (5.13) is a LPMF. Moreover, it is a left-primeLPMF iff the LMS is controllable. The proof of the following theorem, taken from [Kai80, p. 410],applies right away to the finite field case.

Lemma 5.3 (Generalized Bézout Identity)Let a rational matrixR(a) be given in a right-prime RPMF and left-prime LPMF as per

R(a) = N(a)D−1(a) = D−1(a) N(a) .

Then polynomial matricesX(a), Y(a), X(a), andY(a) exist such that

(D(a) N(a)

−X(a) Y(a)

)(X(a) −N(a)

Y(a) D(a)

)

=

(I 00 I

)

. (5.16)

Moreover, all block matrices in equation (5.16) are unimodular. �

Under the assumptions in Lemma 5.3 it follows that

(D(a) N(a)

0 I

)(X(a) −N(a)

Y(a) D(a)

)

=

(I 0

Y(a) D(a)

)

(5.17)


which, with unimodularity of the matrix in the middle, implies similarity of

(D(a) N(a)

0 I

)

and

(I 0

Y(a) D(a)

)

.

In view of the identity matrices comprised, the matrices from above again can be converted uni-modularly. Hence, the matrices

(D(a) 0

0 I

)

and

(I 00 D(a)

)

are similar and the following theorem has been shown.

Theorem 5.6 (Invariant Polynomials of Denominator Matrices)Let a rational matrixR(a) be given in right-prime RPMF and a left-prime LPMF. Then bothdenominator matrices have the same nonunity invariant polynomials. �

As a result, if the LPMF in equation (5.13) is left-prime thenthe corresponding right-prime RPMFhave the same nonunity invariant polynomials. The next section presents a suchlike right-primeRPMF.

5.2.3.2 Polynomial Matrix Fraction for Systems in CCF

For LMS in controllability companion form (CCF), see equations (5.4) and (5.5), a right-primeRPMF can be determined in a closed analytical expression9.

Theorem 5.7 (RPMF of the Transfer Matrix for a System in CCF)Let the state equation of a controllable LMS of ordern with m inputs and controllability indicesc1, . . . ,cm be given in controllability companion form according to equations (5.4) and (5.5). Thena right-prime RPMF of the transfer matrixF(a) = (aI −A)−1B ∈ Fq(a)n×m is

F(a) = P(a)D−1(a) (5.18)

with the denominator matrixD(a) ∈ Fq[a]m×m as per

D(a) = (Bcσ)−1(ΛΛΛ(a)−Ac

σ P(a)), (5.19)

9referring to thestructure theoremin [Wol74, p. 196],[Ant98, p. 291]


the numerator matrixP(a) ∈ Fq[a]n×m and the diagonal matrixΛΛΛ(a) ∈ Fq[a]m×m given as

P(a) =

1 0 · · · 0a 0 · · · 0...

......

...ac1−1 0 · · · 0

0 1 · · · 0...

......

...0 ac2−1 · · · 0...

......

...0 0 · · · 1...

......

...0 0 · · · acm−1

, ΛΛΛ(a) =

ac1 0 · · · 00 ac2 · · · 0...

......

...0 0 · · · acm

, (5.20)

all operations to be understood overFq. �

Proof The correctness of equality (5.19) can be checked easily by multiplying with the respectivedenominator matrices and appealing to the shifting property of the nilpotent blocks inAc

σ — therest is straight-forward. The matricesP(a) andD(a) are right-prime since Theorem 5.1 is fulfilledwith the choice of

X(a) =

1 0 . . . . . . . . . . . . . . . .0 . . . 0 1 0 . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .0 . . . . . . . . . 0 1 0. . .

, Y(a) = 0,

where thei = 1, . . . ,m unity entries in the matrixX(a) are in thei-th row andρi-th column,ρi := 1+

∑i−1j=0c j . �

In light of Remark 5.2, for an LMS in CCF which is subject to an extended linear state feedbackof the form

u(k) = Kx(k)+w(k) (5.21)

it is obvious that the CCF-structure is preserved. This leads to

Corollary 5.1 (RPMF of the Closed-Loop Transfer Matrix for a System in CCF)Let the transfer matrix of ann-th order controllable LMS withm inputs regarding the controllabilitycompanion form be given as per Theorem 5.7. Furthermore, assume that this LMS is subject to an(extended) state feedback law in CCF

u(k) = K cxc(k)+w(k) , (5.22)

whereK c = KQ−1∈Fm×nq is the feedback matrix with respect to the transformxc = Qx into CCF.

Then with the denotation in (5.20) a right-prime RPMF of the closed-loop transfer matrix reads

FK (a) = P(a)D−1K (a) (5.23)


with denominator matrixDK (a) ∈ Fm×mq according to

DK (a) = (Bcσ)−1(ΛΛΛ(a)−Ac

σ,KP(a)), Ac

σ,K = Acσ +Bc

σ K c , (5.24)

in whichAcσ,K ∈ Fm×n

q can be altered arbitrarily by the feedback matrixK c. �

5.2.3.3 Properties

The main property of the right-prime RPMF for the system in the closed loop as given in Corol-lary 5.1 follows from Theorem 5.6 and is summarized in the following theorem.

Theorem 5.8 (Nonunity Invariant Polynomials ofaI − (A +BK) and DK (a) Coincide)Let the state equation of a controllable LMS be given in controllability companion form accordingto equations (5.4) and (5.5). Moreover, letDK (a) denote the denominator matrix of the closed-loop transfer matrix in the right-prime RPMF as per Corollary 5.1. Then the polynomial matricesaI − (A +BK) andDK (a) have the same nonunity invariant polynomials. �

As a result, desired closed-loop invariant polynomials canbe synthesized by means ofDK (a)

which determines the feedback matrixK uniquely. Thus, by finding an adequateDK (a) the cyclesum synthesis problem (CSSP) as stated in Definition 5.1 can be solved.

In order to be in accordance with the notation in the literature, the column degree and the highestcolumn degree coefficient matrix shall be defined [Wol74, Ant98].

Definition 5.10 (Column Degree of a Polynomial Matrix)Let M(a) ∈ Fq[a]n×m be an arbitrary polynomial matrix. The degree of the highestdegree mono-mial in the indeterminatea regarding thei-th column vector ofM(a) is termed thei-th columndegree ofM(a). Thei = 1, . . . ,mcolumn degrees are denoted by∂c,i(M).10 �

Definition 5.11 (Highest Column Degree Coefficient Matrix)Let M(a) ∈ Fq[a]n×m be an arbitrary polynomial matrix. The highest column degree coefficientmatrix ΓΓΓc(M) ∈ Fn×m

q is the matrix made up of coefficients with respect to the highest degreeaterms in each column ofM . �

In equation (5.23) the structure of the denominator matrixDK (a) of the closed-loop system transfermatrixF(a)= P(a)D−1

K (a) in RPMF reveals that the following properties are invariantunder linearstate feedback [Wol74].

10The subscript “c” is to emphasize column degrees, in contrast to row degrees.


Theorem 5.9 (Invariants under Linear State Feedback)Let an LMS of ordern with m inputs be subject to linear state feedback according to (5.21).Resorting to the denotation introduced in Corollary 5.1 andequation (5.24) the following termsare invariant under linear state feedback:

1. the numerator matrixP(a),

2. thei = 1, . . . ,mcolumn degrees ofD(a), i. e. ∂c,i(DK ) = ∂c,i(D),

3. the highest column degree coefficient matrix ofD(a), i. e.ΓΓΓc(DK ) = ΓΓΓc(D). �

Remark 5.3By simple inspection, thei = 1, . . . ,m controllability indicesci can be identified with the controlinvariant column degrees∂c,i(D), accordingly

∂c,i(D) = ci , i = 1, . . . ,m (5.25)

and for an LMS in CCF the matrix

ΓΓΓc(D) = (Bcσ)−1 (5.26)

is the invariant highest column degree coefficient matrix.11 �

5.2.4 Synthesis Algorithm

Solving the CSSP, see Definition 5.1, means to find an adequatestate feedback matrixK for fittingthe respective LMS withi = 1, . . . ,mdesired invariant polynomialsci,K (a) in the closed loop. Thisamounts to determine a denominator matrixDK (a) that meets the following conditions:

C1) Thei = 1, . . . ,m invariant polynomials ofDK (a) coincide with the desired closed-loop in-variant polynomialsci,K (a).

C2) The column degrees ofDK (a) equal the controllability indicesci of the LMS.

C3) The highest column degree coefficient matrix regardingDK (a) equals(Bcσ)−1.

11The reader who is familiar with the polynomial approach recognizes by invertibility ofΓΓΓc(D) that the denominatormatrixD(a) is column reduced.


5.2.4.1 Comments on the Algorithm

The following algorithm extends an algorithm presented in [Kuc91, p. 123]. However, the fol-lowing algorithm does not depend on the solution of a Diophantine equation. Instead, it employsCorollary 5.1, that is a transform of the state equation intoCCF.12 In order to keep the algorithmsimple, an LMS in ordered controllability companion form according to Remark 5.1 will be as-sumed, i. e. the controllability indices are arranged in decreasing orderc1 ≥ . . . ≥ cm.

The algorithm begins with checking realizability first, which means that the inequalities (5.10)in Rosenbrock’s control structure theorem, Theorem 5.4, have to be verified for thei = 1, . . . ,mdesired closed-loop invariant polynomialsci,K (a). As these are invariant polynomials, the choiceof polynomials is restricted toci+1,K (a)|ci,K(a) for i = 1, . . . ,m−1 and

∑mi=1deg(ci,K (a)) = n.

Given realizability, a denominator matrix of the form

D⋆(a) = diag(c1,K (a), . . . ,cm,K (a)) (5.27)

appears to be a promising start because this matrix already covers the conditions C1 and C3 —obviously, condition C1 is satisfied. Condition C3 is fulfilled since the polynomialsci,K (a) aremonic which impliesΓΓΓc(D⋆) = Im. The latter property turns out sufficient for covering conditionC3 as with the unimodular matrix(Bc

σ)−1 the inspection of the polynomial matrix

D⋆K (a) := ΛΛΛ(a)− (Ac

σ +Bcσ K c)P(a) (5.28)

in lieu of DK (a) is equivalent becauseD⋆K (a) andDK (a) = (Bc

σ)−1D⋆K (a) have the same invariant

polynomials. Consequently, the task with respect to condition C3 turns into generatingD⋆K (a) in

place ofDK (a), and now the goal is to achieve a highest column degree coefficient matrix

ΓΓΓc(D⋆K ) = Im, (5.29)

which is in accordance with the matrix from the start asΓΓΓc(D⋆) = Im.

In order to fulfill condition C2, the column degrees inD⋆(a) are adapted by performing suitableelementary row and column operations onD⋆(a); being unimodular operations they do not changethe invariant polynomials. By construction ofD⋆(a) in virtue of decreasing polynomial degrees itis clear that column degrees∂i(D⋆(a)) > ci for some increasing numberi = 1,2, . . . can be reducedby raising the column degrees∂ j(D⋆(a)) < c j for some decreasing numberj = m,m−1, . . ., thepossibility of which is ensured by Rosenbrock’s control structure theorem.

In the course of these unimodular transforms the resulting polynomial matrixD++(a) may finallyhave a highest column degree coefficient matrixΓΓΓc(D++) 6= Im. Therefore, a multiplication with(ΓΓΓc(D++))−1 may be necessary so as to be conform to equation (5.29), expressing the adaptedversion of condition C3.

12The calculation of the transformation matrixQ ∈ Fn×nq by inversion ofL ∈ Fn×n

q , see Theorem 5.3, does not incurnumerical problems — unlike in the real caseL ∈ Rn×n, whereL often emerges to be ill-conditioned.


5.2.4.2 The Algorithm

The last section accounted for the following algorithm for solving the CSSP [RS03, Sch02]

Theorem 5.10 (Synthesis Algorithm for Solving CSSP)Let a controllable LMS be given in ordered CCF with controllability indicesci , i = 1, . . . ,m. Letci,K ∈ Fq[a], i = 1, . . . ,m, be desired invariant polynomials withc j+1,K |c j ,K , j = 1, . . . ,m−1 and∑m

i=1deg(ci,K ) = n.

If possible, the following steps yield a denominator matrixDK (a) for solving the CSSP:

1. Check Rosenbrock’s control structure theorem, Theorem 5.4, forci andci,K (a).

• if the inequalities (5.10) are fulfilledthengoto step 2,

• else

– return “A suitable denominator matrixDK (a) does not exist.”

2. DefineD⋆(a) := diag(c1,K , . . . ,cm,K ).

3. Examine the column degrees ofD⋆(a).

• if the column degrees ofD⋆(a) equal the ordered list of controllability indicesthengoto step 6.

• else

– Detect the first column ofD⋆(a) which differs from the ordered list of controlla-bility indices, starting with column 1. Denote this column colu(D⋆).(deg(colu(D⋆)) > cu)

– Detect the first column ofD⋆(a) which differs from the ordered list of controlla-bility indices, starting with columnm. Denote this column cold(D⋆).(deg(cold(D⋆)) < cd)

4. Adapt the column degrees ofD⋆(a) by unimodular transformations.

• Multiply rowd(D⋆) by a and add the result to rowu(D⋆) in D⋆(a) → D+(a) .

• if deg(colu(D+)) = deg(colu(D⋆))−1 then

– D+(a) → D++(a) andgotostep 5.

• else

– Definer := deg(colu(D⋆))−deg(cold(D⋆))−1


– Multiply cold(D+) by ar and subtract the result from colu(D+) in D+(a) →D++(a) .

5. GenerateΓΓΓc(D++) and setD⋆(a) = (ΓΓΓc(D++))−1D++(a) andgoto step 3

6. D⋆K (a) := D⋆(a) andreturn D ⋆

K (a)

If a closed-loop denominator matrixD⋆K (a) is returned then a linear state feedback matrixK exists

that solves the CSSP. �

It only remains to show that the feedback matrixK can be computed from the closed-loop denom-inator matrixD⋆

K (a), uniquely. To this end, recall equation (5.24), that is

Bcσ DK (a) = ΛΛΛ(a)−Ac

σ,KP(a) = D⋆K (a)

which leads toAc

σ,K P(a) = D⋆K (a)−ΛΛΛ(a) . (5.30)

By comparison of coefficients the matrixAcσ,K is obtained which with the right hand part of equa-

tion (5.24) results inK c = (Bc

σ)−1(Acσ,K −Ac

σ) (5.31)

Finally, the respective coordinate transform yields

K = K cQ , (5.32)

which, unless assured by renamed inputs, has to allow for thepermutation matrix that could havebeen necessary for transforming the CCF into its ordered form according to Remark 5.1.

Remark 5.4There is some flexibility in choosing the unimodular transforms onD⋆(a), for instance, the additionof some row to an other row may not affect any column degree. Due to this flexibility, besides thedenominator matrixD⋆

K (a) that is returned by the algorithm, generally, other polynomial matricesmay meet the conditions C1–C3. �

5.2.5 Example

This section is to illustrate the latter notions by an example taken from [RS03, Sch02]. Note thatappendant calculations were carried out by making use of standard calculations with the packageLinearAlgebra in MapleR©. Given an LMS overF2 of ordern = 5 with m= 2 inputs,

x(k+1) = Ax(k)+Bu(k), A =

1 0 1 0 10 1 1 0 11 1 0 1 01 0 1 1 11 1 0 0 1

, B =

1 00 10 10 10 1

.


The corresponding reduced controllability matrix according to equation (5.3) results in

L = (b1,Ab1,b2,Ab2,A2b2) =

1 1 0 0 00 0 1 1 10 1 1 0 00 1 1 1 10 1 1 0 1

,

which shows that them= 2 controllability indicesc1 = 2, c2 = 3 are not arranged decreasingly. Inlight of Remark 5.1 this entails the need of renaming the inputs, hence

u1(k) = u2(k), u2(k) = u1(k), =⇒ b1 = b2, b2 = b1 .

In what follows, let all variables that are affected by this renaming be marked by a hat. Now thecorresponding reduced controllability matrix becomes

L = (b1,A b1,A2 b1, b2,A b2) =

0 0 0 1 11 1 1 0 01 0 0 0 11 1 1 0 11 0 1 0 1

.

which has the required ordering property since ˆc1 = 3, c2 = 2. Its inverse matrix reads

L−1 =

0 1 1 1 00 0 0 1 10 0 1 0 11 1 0 1 00 1 0 1 0

and by Theorem 5.3 with

σ1 = c1 = 3, σ2 = c1 + c2 = 5

and

qT1 = rowσ1(L

−1) = (0,0,1,0,1), qT2 = rowσ2(L

−1) = (0,1,0,1,0)

the transformation matrix

Q =

qT1

qT1 A

qT1 A2

qT2

qT2 A

=

0 0 1 0 10 0 0 1 10 1 1 1 00 1 0 1 01 1 0 1 0


for the state transformxc = Qx is obtained. Consequently, the LMS can be represented in theCCF

xc(k+1) = Acxc(k)+Bc u(k), Ac =

0 1 0 0 00 0 1 0 00 0 0 0 00 0 0 0 11 0 0 1 0

, Bc

0 01 00 00 00 1

from which the matrices

Acσ =

(0 0 0 0 01 0 0 1 0

)

, Bcσ =

(1 00 1

)

can be extracted for use within the image domain state feedback design.

As a control objective, assume that this LMS shall have 4 cycles of length 1, 2 cycles of length2, 4 cycles of length 3 and 2 cycles of length 6 in the closed-loop. This is a CSSP with desiredinvariant polynomialsc1,K (a) = (a2 +a+1)(a+1)2 andc2,K (a) = a+1, see Example 4.2.1.3 inChapter 4.

An appropriate state feedback matrixK can be determined by using the algorithm proposed inTheorem 5.2.4, accordingly

i−→ 1∑

i=1deg(ci,K(a)) = 4≥

1∑

i=1ci = 3

√

2∑

i=1deg(ci,K(a)) = 5≥

2∑

i=1ci = 5

√

ii−→ D⋆(a) =

((a2+a+1)(a+1)2 0

0 a+1

)

=

(a4+a3 +a+1 0

0 a+1

)

iii , iv−→ D+(a) =

(a4+a3+a+1 a2+a

0 a+1

)

=⇒ D++(a) =

(a+1 a2+a

a3 +a2 a+1

)

v−→ ΓΓΓc(D++) =

(0 11 0

)

=⇒ D⋆(a) = (ΓΓΓc(D++))−1D++(a) =

(a3+a2 a+1a+1 a2 +a

)

iii ,vi−→ D⋆K (a) =

(a3+a2 a+1a+1 a2 +a

)

With D⋆K (a) the feedback matrixK can be computed. First, employing equation (5.30) yields


Acσ,K

1 0a 0a2 00 10 a

=

(a3 00 a2

)

︸︷︷︸

ΛΛΛ(a)

+

(a3+a2 a+1a+1 a2+a

)

︸︷︷︸

D⋆K (a)

=

(a2 a+1

a+1 a

)

and by comparison of coefficients results

Acσ,K =

(0 0 1 1 11 1 0 0 1

)

.

which, secondly, as per (5.31) implies

K c =

(1 00 1

)-1

︸︷︷︸

Bcσ

((0 0 1 1 11 1 0 0 1

)

︸︷︷︸

Acσ,K

+

(0 0 0 0 01 0 0 1 0

)

︸︷︷︸

Acσ

)

=

(0 0 1 1 10 1 0 1 1

)

.

Thus, the feedback matrixK with respect to the renamed inputs follows from (5.32), thatis

K = K cQ =

(1 1 1 1 01 0 0 1 1

)

which by swapping rows results in the desired state feedbackmatrix

K =

(1 0 0 1 11 1 1 1 0

)

.

5.2.6 Non-controllable Parts

Controllable LMS ofn-th order are characterized by the existence of a set ofn linearly inde-pendent column vectors in the controllability matrixL , see Definition 5.3. If the LMS containsnon-controllable parts then it consists of an uncontrollable subsystem of order ¯n≥ 1 and of a con-trollable subsystem of corresponding ordern− n, and the controllability matrix only comprisesn− n linearly independent vectors, which span the controllablevector space. Nevertheless it ispossible to adapt the synthesis method of the last section.

To this end, consider the linearly independent column vectors in the non-square reduced controlla-bility matrix L ∈ F

n×(n−n)q in equation (5.3), i. e.

L =(b1, . . . ,Ac1−1b1,b2, . . . ,Ac2−1b2, . . . ,bm, . . . Acm−1bm

),

m∑

i=1

ci = n− n < n, (5.33)


which can be completed by a set of ¯n linearly independent column vectors that span the orthogonalcomplement space of the controllable subspace with respectto the entire spaceFn

q , see[Wol74].Let these vectors be the column vectors of the matrixL n ∈ Fn×n

q . Therefore, with

LTnL = 0 (5.34)

the adapted reduced controllability matrix

L = (L ,L n) , L ∈ Fn×nq (5.35)

becomes an invertible matrix.

This extension byL n acts as if the number of inputs were extended by ¯nand the input matrixB wereaugmented to the right by the ¯n linearly independent columns ofL n. Then by linear independence,each of these vectors can be interpreted as corresponding toa controllability index equal to one,which implies controllability of this extended system, andaccordingly, the extended system can betransformed into CCF as per Definition 5.5. For this reason, proceeding in the same manner as forconstructing the transformation matrixQ in Theorem 5.3, that is, by calculating the inverse matrixL−1 and again usingσi =

∑ij=1c j , i = 1, . . . ,m by defining

qTi := rowσi(L

−1), (5.36)

the matrix

Q =

qT1

qT1 A...

qT1 Ac1−1

qT2

qT2 A...

qT2 Ac2−1

...qT

mAcm−1

(5.37)

is obtained, which here withQ ∈ F(n−n)×nq is non-square. In view of the interpretation of controlla-

bility index one from above, the missing ¯n rows in order to quadratically supplementQ are simplythe respective last rows ofL−1. These can be collected in

Qn =

rown−n+1(L−1)

rown−n+2(L−1)...

rown(L−1)

(5.38)


with Qn ∈ F n×nq . Consequently, an adapted version of the transformation matrix13 reads

Q =

(QQn

)

. (5.39)

whose partitioning induces a partitioning of the statex(k), the dynamics matrixA, and the inputmatrix B, i. e.

x(k) =

(xc(k)x c(k)

)

=

(QQn

)

x(k) , (5.40)

A =

(Ac Ac c

A cc A c

)

=

(QQn

)

A(

QQn

)−1

, (5.41)

B =

(Bc

B c

)

=

(QQn

)

B (5.42)

such that the state equation is in a CCF-like form

x(k+1) = A x(k)+ Bu(k) . (5.43)

“CCF-like” is to stress that the special choice ofL n as an orthogonal complement — see equation(5.34) — implies even more structure than the representation in CCF. First, since the vectorsA i b j ,i ≥ c j depend linearly on those vectors withi < c j note that

QnA i b j = 0, i = 0,1,2, . . . , j = 1, . . . ,m (5.44)

by means of which an obvious consequence in (5.42) is that

B c= 0. (5.45)

Moreover, the firstn− n columns inQ−1 are orthogonal to the ¯n rows inQn, by definition. Butas the orthogonal complement space with respect to the row space ofQn is spanned by then− ncolumn vectors ofL an other immediate implication of (5.44) in (5.41) is

A cc= 0. (5.46)

Now it is clear that the lower system part with dynamics matrix A c cannot be influenced, neitherby some input nor by the statesxc(k) from the upper system, hence, represents an autonomousuncontrollable subsystem. Even though by virtue ofAc c the uncontrollable subsystem can takeinfluence on the upper system, however, the upper system represents a controllable subsystem.This is due to the fact that by choosing a suitable input — feedback of the uncontrollable statesx c(k) — any element of the non-zero rows inAc ccan be specified arbitrarily, for example turned

13which is invertible by construction


into zero, see Section 5.2.6.1. For this reason, a feedback of the remaining controllable statesxc(k) allows of designing a customary controller for the controllable subsystem embodied by thedynamics matrixAc.

Summarizing the latter, the following theorem has been shown.

Theorem 5.11 (Adapted CCF for Uncontrollable Systems)Let an uncontrollable LMS of ordern with m inputs havei = 1, . . . ,mcontrollability indicesci with

σ j :=∑ j

i=1ci andn := n−∑m

i=1ci . Then the transformations in (5.40)-(5.42) transform the stateequation of the LMS into

x(k+1) =

(Ac Ac c

0 A c

)

x(k)+

(Bc

0

)

u(k) . (5.47)

• The matricesAc∈F(n−n)×(n−n)q andBc∈F

(n−n)×mq represent an(n− n)-th order controllable

subsystem in CCF, in complete accordance with Definition 5.5.

• The matrixA c∈ F n×nq represents an ¯n-th order (autonomous) uncontrollable subsystem.

• The matrixAc c∈ F(n−n)×nq represents a coupling of controllable and uncontrollable subsys-

tem. Moreover, itsσ j -th rows, j = 1, . . . ,m, are the only non-zero rows inAc c. �

5.2.6.1 Decoupling the Uncontrollable from the Controllable Subsystem

The synthesis goal of specifying a set of desired elementarydivisor polynomials (CSSP) in theoverall closed-loop dynamics matrix of an uncontrollable LMS can be combined with a decouplingthat breaks the influence of the uncontrollable subsystem onthe controllable subsystem. Referringto the solution of the state equation (5.1) it turns out that this influence is incurred by non-zeroinitial states with respect to the uncontrollable subsystem only, and not by the input-sided portion.An additional benefit of decoupling is that the block-diagonal structure of the decoupled dynamicsmatrix allows to simplify the synthesis.

Theorem 5.11 indicates an easy way of how to decouple the uncontrollable subsystem from thecontrollable subsystem by an appropriate feedback. Obviously, decoupling requires that

Ac cx c(k)+ Bcu(k) = 0 (5.48)

which can be achieved by a state feedback in the simple formu(k) = Kx(k). In transformedcoordinates the feedback matrix becomes

K = K Q−1 (5.49)


which in a partitioned form

K = (K c, K c c) (5.50)

means that

u(k) = K x(k) = K c xc(k)+ K c cx c(k) . (5.51)

Thus, recalling equation (5.48) the influence of the coupling matrixAc cvanishes if

Bc K c c= −Ac c (5.52)

or with the denotation from (5.5) with (5.31), equivalently, if

K c c= −(Bcσ)−1Ac c

σ (5.53)

in which Bcσ andAc c

σ comprise the alterable entries in the matricesBc andAc c, respectively. There-fore, the following theorem has been established.

Theorem 5.12 (Decoupling by State Feedback)Let an LMS of ordern with m inputs be uncontrollable. Let the uncontrollable subsystem be oforder n < n. Then the uncontrollable subsystem can always be decoupledfrom the controllablesubsystem by linear state feedback with respect to the uncontrollable subsystem states. �

The Influence of Decoupling on the Closed-Loop Invariants

In light of the benefits from decoupling it is important to discuss its influence on the invariantpolynomials of the dynamics matrix. On account of this, the following issues are addressed in thesequel:

• development of a sufficient criterion of when the dynamics matrix of a decoupled uncontrol-lable LMS shows the same invariant polynomials as the respective coupled LMS,

• present a particular case in which the invariant polynomials remain unaltered by decouplingwhatever the coupling matrix in the dynamics matrix is,

• show that the characteristic polynomial of the dynamics matrix is invariant under decoupling,

• determine the elementary divisor polynomials of a decoupled dynamics out of the elementarydivisor polynomials of its parts.

First, recall the following theorem, taken from literaturefor simplicity [GLR82, p. 342 ff.].


Theorem 5.13 (Similarity Criterion)Let M11 ∈ Fn1×n1, M22 ∈ Fn2×n2, andM12 ∈ Fn1×n2. The matrices

(M11 M12

0 M22

)

and

(M11 0

0 M22

)

are similar iff the linear matrix equation

M11X −X M 22 = M12 (5.54)

has a solution. �

A solution which is valid in the majority of the cases is due to[Gan58, p. 208 ff.] and [New74].14

Theorem 5.14Using the notation of Theorem 5.13 the linear matrix equation (5.54) has a solution for arbitrarycoupling matricesM12 ∈ Fn1×n2 if

gcd(cpM11(λ),cpM22

(λ)) = 1.

Moreover, this solution is unique. �

The far-reaching interpretation for the purpose of this work is

Corollary 5.2 (Invariant Polynomials under Decoupling)Let an uncontrollable LMS of ordern with m inputs be given in the notation of Theorem 5.11.Then decoupling the uncontrollable subsystem from the controllable subsystem by use of a statefeedback (5.53) does not change the invariant polynomials of the dynamics matrix iff the linearmatrix equation

AcX −X A c= Ac c (5.55)

has a solution. A unique such solution exists if

gcd(cpAc(λ),cpA c(λ)

)= 1.

In this case the invariant polynomials remain unchanged under decoupling whateverAc c. �

14From linearity of equation (5.54) it is clear that the general statement can be derived as well; the general solutionis attainable pursuing the lines in Appendix D. Since in addition to that, more notational effort is incurred by theassociated introduction of theKronecker-product of matrices for transforming the equation into a standard linearequation, the generalization becomes quite involved and distracts too much from the purposes here. Hence, keepingtrack of this way shall be left to the reader.


Thus, if the conditions stated in Corollary 5.2 are met then decoupling cannot change the invariantpolynomials of the dynamics matrix; under these assumptions this can only be done by an adequatefeedback with respect to the controllable part. Conversely, if the conditions stated in Corollary 5.2are not met then the influence of decoupling on the invariant polynomials is still subject to a limit,but a less restrictive one.

Theorem 5.15 (Invariance of Characteristic Polynomials under Decoupling)Let an uncontrollable LMS of ordern with m inputs be given in the notation of Theorem 5.11.Then Decoupling the uncontrollable subsystem from the controllable subsystem by use of a statefeedback (5.53) does not change the characteristic polynomial of the dynamics matrix. �

Proof Using unimodular matricesU(λ) = diag(Uc(λ),U c(λ)) andV(λ) = diag(Vc(λ),V c(λ))

the characteristic matrixλ I − A can be transformed into

U(λ)(λ I − A)V(λ) = U(λ)

(λ I − Ac Acc

0 λ I − A c

)

V(λ) =

(Sc(λ) Uc(λ)AccV c(λ)

0 Sc(λ)

)

, (5.56)

which clearly has the same invariant polynomials and elementary divisor polynomials, respec-tively, asA. Hence,

cpA(λ) = det(U(λ)(λ I − A)V(λ)

)= det

(Sc(λ)

)det(Sc(λ)

)= cpAc(λ)cpA c(λ)

which does not show any dependency onAcc. As a result, decoupling has no influence on thecharacteristic polynomial of the dynamics matrix. �

SettingAcc equal to zero in equation (5.56), the same argument can be used for showing an impor-tant result [LT85], which applied to this work is

Theorem 5.16 (Elementary Divisor Polynomials of a Decoupled Dynamics Matrix)Let an uncontrollable LMS of ordern with m inputs be given in the notation of Theorem 5.11 andlet the uncontrollable subsystem be decoupled from the controllable subsystem by use of a statefeedback (5.53). Then the set of elementary divisor polynomials regarding the matrix diag(Ac, A c)

is the union of the sets of elementary divisor polynomials with respect toAc, A c. �

As the respective elementary divisor polynomials divide each other this statement is sufficientfor constructing the Smith normal form of diag(Ac, A c) out of the Smith normal forms regardingAc andA c — the formal description is rather cumbersome and omitted for clearness. A simplerformal description can be achieved in cases for which cpAc(λ) and cpA c(λ) have no common factor[New72, New74].

Theorem 5.17 (Smith Normal Form of a Compound Matrix)Let an uncontrollable LMS of ordern with m inputs be given in the notation of Theorem 5.11 witharbitrary coupling matrixAc c∈ F (n−n)×n. Without loss of generality, assume ¯n≤ n− n. Moreover,let gcd

(cpAc(λ),cpA c(λ)

)= 1. Denote the Smith normal form with respect toAc andA cby

Sc(λ) = diag(c1(λ),c2(λ), . . . ,cn−n(λ)), Sc(λ) = diag(c1(λ), c2(λ), . . . , cn(λ)) ,


respectively. Then the Smith normal formS(λ) with respect to

A(λ) =

(Ac Ac c

0 A c

)

isS(λ) = diag(c1c1, c2c2, . . . ,cncn , cn+1,cn+2, . . . ,cn−n , 1,1, . . . ,1) . �

In the preceding lines, the focus was on how to decouple the uncontrollable subsystem from thecontrollable subsystem of an LMS without changing the elementary divisor polynomials of theLMS. The subsequent remark comments on the remaining case.

Remark 5.5 (Conservation of Elementary Divisor Polynomials of the Uncontrollable Part)Even though decoupling does not change the characteristic polynomial of the dynamics matrix,however, decoupling can change the elementary divisor polynomials of the controllable subsystemif the condition in Corollary 5.2 is not met. This change can only affect the distribution of thefactors in the characteristic polynomial, for instance by joining elementary divisor polynomialsinto a new one of higher degree. But note that in any case the elementary divisor polynomials ofA care preserved in the overall system, which can be concluded from the respective Smith normalform Sc in equation (5.56) and the fact that the uncontrollable system is an autonomous LMS.�

Consequently, if striving to solve the CSSP for a set of desired elementary divisor polynomialsin the overall closed-loop dynamics matrix of an uncontrollable LMS then this set has to includethe set of elementary divisor polynomials with respect to the autonomous dynamicsA c of theuncontrollable subsystem. This condition has to be fulfilled in addition to the condition imposedby Rosenbrock’s control structure theorem.

5.2.6.2 Solving the Cycle Sum Synthesis Problem (CSSP) for Uncontrollable LMS

In what follows, the CSSP is solved assuming the decoupling of the uncontrollable from the con-trollable subsystem, for simplicity. After decoupling theremaining unspecified parameters residein the feedback matrixK c. This matrix can be determined by the algorithm presented inTheo-rem 5.2.4 since the corresponding subsystem of ordern− n is controllable. For specifying the setof closed-loop elementary divisor polynomials it is first necessary to enclose the set of elemen-tary divisor polynomials regarding the uncontrollable dynamicsA c. Then, if in accordance withRosenbrock’s control structure theorem, the algorithm returns a denominator matrix, hereD⋆

Kc(a),which by a comparison of coefficients according to equation (5.30) yields a matrixAc

σ,Kc that can

finally be used for determining the feedback matrixK c, see equation (5.31). The resulting feedbackmatrix is

K c = (Bcσ)−1(Ac

σ,Kc − Acσ) . (5.57)


With K c andK c call entries in the overall feedback matrixK are fixed. Summing up, a possibleprocedure that solves the CSSP for an LMS with uncontrollable subsystem and decoupling is thefollowing:

1. Transform the state equation into an adapted CCF as in equations (5.41) and (5.42).

2. Calculate the elementary divisor polynomials of the uncontrollable dynamics matrixA c

3. Check whether the desired set of closed-loop elementary divisor polynomials includes theset of elementary divisor polynomials ofA c.

4. Check whether the desired closed-loop invariant polynomials with respect to the controllablepart meet Rosenbrock’s control structure theorem, Theorem5.4.

5. If both checks are positive then

• use the synthesis algorithm in Theorem 5.2.4 in order to determine a matrixK c forfeeding back the controllable states (5.31) and

• with (5.53) calculate the decoupling state feedback matrixK c cwhich feeds back theuncontrollable states.

• A feedback matrix that solves the CSSP isK = K Q with K = (K c, K c c).

Remark 5.6As already indicated, there are cases in which the set of desired closed-loop invariant polynomialscannot be realized under decoupling. Then a particular change of the coupling matrix might helpto solve the problem. Notwithstanding, a coupling conserves that the state evolution dependson the initial state of the uncontrollable subsystem. Moreover, a coupling can just be used fora reassembly of the elementary divisor polynomials concerning the controllable part. For thesereasons, this is not commented on in a deeper fashion. �

5.2.7 Example

Consider an LMS overF2 of ordern = 5 with m= 1 input and

A =

0 1 1 0 11 0 1 1 11 0 0 0 10 1 0 1 11 1 0 1 1

, B =

00010

.


The corresponding reduced controllability matrix according to equation (5.3) results in

L = (B,AB,A2B) =

0 0 00 1 00 0 11 1 10 1 1

which reveals that the sole controllability index isc1 = σ1 = 3 and implies that the LMS has anuncontrollable subsystem of order ¯n = 2. In view of equation (5.35) a simple choice of linearlyindependent column vectors yields

L n =

1 00 10 00 00 0

,

thus, the adapted reduced controllability matrix and its inverse reads

L = (L ,L n) =

0 0 0 1 00 1 0 0 10 0 1 0 01 1 1 0 00 1 1 0 0

, L−1 =

0 0 0 1 10 0 1 0 10 0 1 0 01 0 0 0 00 1 1 0 1

.

Theorem 5.3 and equation (5.39) allows to calculate the transformation matrix, i. e. from

qT1 = rowσ1(L

−1) = (0,0,1,0,0), Qn =

(1 0 0 0 00 1 1 0 1

)

follows the transformation matrix

Q =

qT1

qT1 A

qT1 A2

Qn

=

0 0 1 0 01 0 0 0 11 0 1 1 01 0 0 0 00 1 1 0 1

which via (5.41) and (5.42) transforms the LMS in adapted CCFwith

A =

0 1 0 0 00 0 1 0 00 1 1 1 00 0 0 0 10 0 0 1 1

, B =

00100

.


By inspection, the matrices

Ac =

0 1 00 0 10 1 1

, Ac c=

0 00 01 0

, A c=

(0 11 1

)

, Bc =

001

are obtained in the same way as the matrices

Acσ = (0,1,1), Ac c

σ = (1,0), Bcσ = 1.

Since the matricesAc andA c are companion matrices, each of them has only one non-unity in-variant polynomial,cAc(a) andcA c(a), the coefficients of which are are just the elements of thematricesAc

σ andA cσ, respectively. Hence,

cAc(a) = a3 +a2+a = a(a2+a+1), cA c(a) = a2 +a+1,

which in view of Theorem 5.16 reveals the Smith normal form ofthe dynamicsA after decou-pling.15 Note thatcA c(a) is an invariant polynomial with regard to the uncontrollable subsystem.Therefore, it has to be enclosed in the set of desired closed-loop elementary divisor polynomials.

Let the desired (single) closed-loop invariant polynomialwith respect to the controllable subsystembe cAc,K (a) = (a+ 1)(a2 + a+ 1). This choice satisfies Rosenbrock’s control structure theorem,Theorem 5.4, since deg(cAc,K (a)) = c1 = 3 and additionallycA c(a)|cAc,K (a). Consequently, theCSSP with desired closed-loop elementary divisor polynomials

p1(λ) = a+1, p2(λ) = a2+a+1, p3(λ) = a2+a+1

is solvable.

Obviously, the algorithm in Theorem 5.2.4 returns the denominator matrix

D⋆Kc(a) = (a+1)(a2+a+1)

which according to equation (5.30) yields

Acσ,Kc

1aa2

= a3+(a+1)(a2+a+1) = 1.

By comparing coefficients this means that

Acσ,Kc = (1,0,0) .

15This Smith normal form is different from the Smith normal form S(λ) = diag(a(a2 + a+ 1)2,1, . . . ,1) of thecoupled dynamics matrixA which is in full accordance with Corollary 5.2 because equation (5.55) has no solution.


With respect to the controllable part, using equation (5.31) results in the feedback matrix

K c = (Bcσ)−1(Ac

σ,Kc +Acσ) = (1,1,1) .

Simple inspection, or reference to equation (5.53), yieldsthe decoupling state feedback matrix

K c c= (1,0) .

With these results, the overall feedback matrix finally reads

K = (K c, K c c)Q = (1,0,0,1,1) .

Conclusion

In this chapter, a method for solving the cycle sum synthesisproblem (CSSP) is presented for thefirst time. This problem is solved by an algorithm that fits a given multiple-input LMS, for instancerepresenting a linear automaton model, with a specific cycleand/or tree structure in its state graph.In view of the analysis part, Chapter 4, this clearly means toalter the set of invariant polynomialsof the dynamics matrix into a set of desired ones by determining an adequate input sequence. Asthis problem is strongly related to controllability, first,the notion of controllability is recalled fromlinear continuous system theory and reinterpreted into thefinite field case. It turns out that thisconcept applies as it stands due to the field property.

Rosenbrock’s control structure theorem is known to be the appropriate means for answering thequestion of when a linear state feedback exists that sets desired closed-loop invariant polynomialsby linear state feedback. Fortunately, this theorem is valid for systems over finite fields as well,and as a consequence, an answer to the question whether a CSSPcan be solved by linear statefeedback can always be given.

For synthesizing a linear state feedback, however, original domain methods for multiple-inputsystem synthesis as for example the parametric approach show to be insufficient for solving theCSSP — occurring problems are discussed extensively. In this context, the crucial problem isthat an extension field is required for expressing eigenvalues and respective chains of generalizedeigenvectors. For finite fields the concept of an extension field is more involved because generallypolynomials do not factor in a common extension field. Thus, employing the parametric approachbecomes very cumbersome — see Appendix C for the symbolic calculation effort. These prob-lems naturally lead to synthesis methods in an image domain where polynomial factorizations inextension fields can be avoided.

The method employed here is the polynomial matrix method. Itallows a complete specificationof the invariant polynomials in the closed-loop dynamics, hence, solving the CSSP. Its main in-gredients are polynomial matrix fractions of transfer matrices in an image domain. For LMS in


a particular state space representation, in the so-called controller companion form (CCF), a right-prime right polynomial matrix fraction in the image domain is obtained as an analytic expressionwithout refactorization of the transfer matrix. Two other properties associated to representationsin CCF proof to be of major value: feedback leaves the CCF-structure invariant, a property that ispreserved in the image domain as well. Additionally, for systems over finite fields the similaritytransform into CCF over finite fields is not subject to numerical problems by ill-conditioned matri-ces, consequently, the inversion of the (reduced) controllability matrix is unproblematic. By virtueof this factorization it can easily be seen that the closed-loop denominator (polynomial) matrix ofthe right-prime right polynomial matrix fraction for the LMS in CCF-representation can be speci-fied to exact the extent which is imposed by Rosenbrock’s control structure theorem — it is just thisdenominator matrix that has to comprise the desired closed-loop nonunity invariant polynomials.Therefore, by referring to such a closed-loop denominator matrix, if in accordance with the controlstructure theorem, a feedback matrix is determined withoutinvolving the solution of a diophantineequation. The main result is a synthesis algorithm which fora controllable LMS receives a set ofdesired closed-loop invariant polynomials as an input and returns a suitable denominator matrix— as long as the CSSP is solvable.

The second part of the chapter extends the results to LMS withan uncontrollable subsystem. Tothis end, the state equation of the LMS is transformed into anadapted CCF which reveals theautonomous uncontrollable subsystem. In light of the afore-presented synthesis algorithm which isbased on controllability of the LMS, a separation into controllable and uncontrollable subsystemsis ensured by a decoupling. This further breaks the influenceof the initial state with respect tothe uncontrollable subsystem on the evolution of state regarding the controllable subsystem. It isshown that such a decoupling exists for arbitrary LMS with uncontrollable part. As decoupling canalter the invariant polynomials of the dynamics matrix a novel criterion is developed which drawsan exact line between when a decoupling takes an influence andwhen it entails no influence on theinvariant polynomials. This criterion extends the well-known fact that such a decoupling cannotchange the characteristic polynomial of the closed-loop dynamics. As a result, for uncontrollableLMS under decoupling, again, the afore-developed synthesis algorithm can be used for specifyingthe invariant polynomials, this time with respect to the controllable subsystem.

Chapter 6

Conclusions and Future Work

In this dissertation, a discrete state space model over a finite field is presented. For linear systems ofthis kind, methods are developed that allow a structural analysis and feedback controller synthesis.The general philosophy throughout this work is not to invokedeeper knowledge about finite ringtheory but instead to ground the development and the proofs on a fundamental level of matrixtheory. This way, a less involved but still consistent theory of linear systems over a finite field isgiven.

6.1 Summary

The mathematical preliminaries in Chapter 2 recall the algebraic fundamentals which are indis-pensable for a theoretical treatment of discrete dynamic systems over finite fields. In particular,the basic concepts of finite groups and finite fields are emphasized since these concepts implysome peculiarities for polynomials over finite fields, e. g. their periodicity and reducibility proper-ties. In view of the subsequent analysis and synthesis methods, a review of an elementary level oflinear algebra is given so as to motivate structural invariants of linear systems like invariant poly-nomials and elementary divisor polynomials. The closing part of the chapter briefly establishes aZ-domain-like image domain for functions over finite fields, which due to its similarity to theZ-domain formulae provides the benefit of using a variety of image domain methods for continuoussystems.

For an illustration of the underlying idea of discrete dynamic systems over finite fields, first, asimple conveyor belt example is discussed at the beginning of Chapter 3. This example underlinesthe need for formal methods for deriving a model that describes the input-dependent evolutionof state for such systems. For this purpose, a coding scheme that permits to determine a purelyalgebraic transition relation, similar to a state space model in the continuous world, is introduced.

116 CHAPTER 6 — CONCLUSIONS AND FUTURE WORK

Its construction is confined to the case of the finite field withcharacteristic 2 which is shown tobe isomorphic to a boolean ring with the consequence that basics from boolean algebra can beemployed for the model construction. Two ways for deriving this model have been pointed out:the first method invokes the calculation of the disjunctive normal form, elimination of the negationsand using the law of DeMorgan. The second method is based on Reed-Muller generator matriceswhich prove to be tailored for the problem as less computation effort is required for deriving thecoefficients of the transition function in view. In a generalprospect, both methods yield a scalarimplicit multilinear transition relation over the finite field with characteristic 2. If the systemsunder consideration are deterministic then this transition relation becomes a transition function.For depicting the ease of use of the Reed-Muller generator matrix method, it is applied to theconveyor belt example.

In Chapter 4 autonomous linear systems over finite fields are considered. The main result is that forthe class of finite state automata which can be represented asautonomous linear modular systems,for the first time, a necessary and sufficient criterion is deduced that allows to determine all au-tomaton cycles in length and number, the so-called cycle sum. For using this criterion, the periodsof the periodic elementary divisor polynomials regarding the system dynamics matrix have to bedetermined. These periods are in strong relation with a complete decomposition of the state spaceinto periodic subspaces, hence, are in strong relation witha respective criterion for the automatoncycles. It is worth mentioning that this criterion does not resort to a state space enumeration pro-cedure, what, apart from more elegance, promises less calculation effort as long as the requiredperiods can be found in tabulars (true for polynomial degrees of about 100). In contrast, the por-tion of non-periodic elementary divisor polynomials does not contribute to the cyclic behavior,but instead it entirely constitutes the state transition behavior which takes the form of a tree ina state graph. The results are superposed in a general statement on autonomous linear modularsystems. The final part of this chapter gives answers to the afore-posed questions for systems thatare extended by an affine constant term.

The final contribution of this dissertation, in Chapter 5, isto present the first method for solving thecycle sum synthesis problem for non-autonomous linear modular systems. In light of the resultsfrom Chapter 4 this task is associated with specifying the elementary divisor polynomials of thesystem dynamics. Assuming controllability it is shown thatthe cycle sum synthesis problem canbe solved for non-autonomous systems by closing the loop; naturally referring to the notion of statefeedback. As standard strategies for the computation of feedback for multiple-input systems likethe parametric approach turn out to be inadequate for solving the problem, methods which encom-pass the synthesis of invariant polynomials have to be used.In this regard, image domain methodsare found to be suitable. The proposed solution of the cycle sum synthesis problem is twofold.In a first step, Rosenbrock’s control structure theorem is recalled in order to answer the questionwhether a state feedback exists that fits the closed-loop dynamics matrix with a desired set of el-ementary divisor polynomials. If this answer is positive, in a second step, such a feedback matrix

SECTION 6.2 — FUTURE WORK 117

is determined by modifying the denominator matrix of a right-prime right polynomial matrix frac-tion of the transfer matrix with respect to the system in controllability companion form. The resultis an algorithm that computes the denominator matrix of the right-prime right polynomial matrixfraction corresponding to the desired closed-loop transfer matrix, which after simple calculationsyields a desired state feedback matrix. An advantage of thiscontrollability companion form basedapproach is that the solution of a Diophantine equation for obtaining the state feedback matrix isnot necessary. The second part of this chapter enhances the setting on systems with uncontrol-lable subsystem. To this end, an adapted form of the controllability companion form is introducedwhich reveals the controllable and uncontrollable subsystem. In order to break the influence of theinitial state with regard to the uncontrollable subsystem,both subsystems are decoupled, which isshown to be feasible for arbitrary linear modular systems with uncontrollable part. A further resultprovides a criterion of when a decoupling has an influence on the elementary divisor polynomialsof the system dynamics, which extends the well-known fact that such a decoupling cannot changethe characteristic polynomial of the closed-loop dynamics. Finally, a procedure is proposed whichallows to apply the afore-presented algorithm for the controllable subsystem, leaving the charac-teristic polynomial of the uncontrollable subsystem unchanged.

6.2 Future Work

Many practically relevant systems are non-linear in nature, which for systems over finite fieldsmeans polynomially non-linear. For facing this problem, the development of appropriate lineariza-tion techniques may be a first conceivable step of future work. In a next step, further research maykeep track of the non-linear case, for which then real non-linear methods for analysis and controlhave to be established. Potentially fruitful work could be based on using ideal theoretic methodslike Gröbner-bases [CLO98] for effective transformationsof non-linear system models [NMGJ01],as proposed in the advanced approaches worked out in Rennes [Mar97, ML97, ML99, PML99]and Linköping [Ger95, Gun97]. In this regard, some attention could be directed to the structuralanalysis of the non-linear transition equations.

118 CHAPTER 6 — CONCLUSIONS AND FUTURE WORK

Appendix A

Permutations of a Block Matrix

Let a matrixA ∈ Fn×n be partitioned in blocks as per

A =

A11 A12 · · · A1p

A21 A22 · · · A2p...

.... . .

...Ap1 Ap2 · · · App

, A i j ∈ Fdi×d j , i, j = 1, . . . , p (A.1)

with n =∑p

i=1di . An other composition out of the same submatricesA i j from A that shows thesame structure is the block matrix

A =

Ak1k1 Ak1k2 · · · Ak1kp

Ak2k1 Ak2k2 · · · Ak2kp...

.... . .

...Akpk1 Akpk2 · · · Akpkp

, Akik j ∈ Fdki

×dkj , ki ,k j = 1, . . . , p, (A.2)

which results from a permutation of the submatrices inA with respect to the permutation of di-mension numbers from(d1, . . . ,dp) into (dk1, . . . ,dkp).

The following theorem is a straight-forward extension of the notion of an elementary matrix withscalar entries to an elementary matrix with matrix entries,a collection of which provides the nec-essary row and column permutations for transformingA into A.

Theorem A.1 (Permutation Matrix of a Block Matrix)Let A ∈ Fn×n be a block matrix as denoted in equation (A.1) with the respective ordered set ofdimension numbers(d1, . . . ,dp). Let A ∈ Fn×n be a block matrix that is a an other composition ofthe submatrices with respect toA as per equation (A.2) with the ordered set of dimension numbers

120 APPENDIX A. PERMUTATIONS OF A BLOCK MATRIX

(dk1, . . . ,dkp). Then the matrix

ΠΠΠ =

ΠΠΠ11 ΠΠΠ12 · · · ΠΠΠ1p

ΠΠΠ21 ΠΠΠ22 · · · ΠΠΠ2p...

.... . .

...ΠΠΠp1 ΠΠΠp2 · · · ΠΠΠpp

, ΠΠΠi j =

{

Idki, j = ki

0dkid j , j 6= ki

(A.3)

is orthogonal and transforms the block matrixA into the block matrixA according to

A = ΠΠΠAΠΠΠT . (A.4)

�

Example A.1The block matrix

A =

A11 A12 A13

A21 A22 A23

A31 A32 A33

, A i j ∈ Fdi×d j , i, j = 1, . . . ,3

with the ordered set of dimension numbers(d1,d2,d3) is to be transformed into

A =

A33 A31 A32

A13 A11 A12

A23 A21 A22

.

The ordered set of dimension numbers associated toA is (d3,d1,d2), hence,k1 = 3, k2 = 1, k3 = 2and the permutation matrix becomes

ΠΠΠ =

0d3d1 0d3d2 Id3

Id1 0d1d2 0d1d3

0d2d1 Id2 0d2d3

with the respective square identity matrices and generallynon-square zero matrices. �

Appendix B

The Transformation Matrix on RationalCanonical Form

For any matrixA over a finite fieldFq the commandfrobenius in theShare-Libraryof the com-puter algebra package MapleR© admits to calculate the so-called frobenius normal formAF, whichis a rational canonical form with respect to the invariant polynomials in non-factored form; thecorresponding procedure returns the transformation matrix TF as well. This procedure can be usedto determine the transformation matrixT which transformsA into the rational canonical formArat.To this end, it is taken advantage from the following relations.

By definitionArat = TAT−1 (B.1)

andAF = TFAT−1

F . (B.2)

SinceArat andAF are similar matrices it is clear that

AF = QAratQ−1 (B.3)

being equivalent toArat = Q−1AFQ (B.4)

which employing equation (B.2) can be transformed into

Arat = Q−1TFA (Q−1TF)−1 . (B.5)

A comparison with equation (B.1) yields the required

T = Q−1TF , (B.6)

because the matricesQ andTF are at one’s disposal after transformingA andArat both intoAF viaequations (B.2) and (B.3), using the MapleR© commandfrobenius.

122 APPENDIX B. THE TRANSFORMATION MATRIX ON RATIONAL CANONICAL FORM

Appendix C

The Jordan Normal Form over anExtension Field ofFq

For completeness, the Jordan normal form of a matrix over a finite field Fq shall be derived in anexemplary, tutorial manner.1 For simplicity, let a matrix inF8×8

2 be given as

A :=

0 0 0 0 0 1 0 01 0 0 0 0 0 0 00 1 0 0 0 1 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 1

,

in rational canonical form. The respective Smith normal form S(λ) = U(λ)(λ I −A)V(λ) reads

S(λ) = diag(c(λ), 1, 1, 1, 1, 1, 1, 1

),

c(λ) = λ8+λ7+λ6+λ4 +λ3+λ+1 = (λ3+λ+1)2︸︷︷︸

=: p1(λ)

(λ2+λ+1)︸︷︷︸

=: p2(λ)

,

wherep1(λ) andp2(λ) are the elementary divisor polynomials inF2[λ]. The corresponding trans-formation matrices are

U(λ) =

0

B

B

B

B

B

B

B

B

B

B

B

@

λ3 +λ2 +λ λ4 +λ3 +λ2 λ5 +λ4 +λ3 λ6 +λ5 +λ4 λ7 +λ6 +λ5 λ4 +λ3 +λ+1 λ8 +λ4 +λ2 λ6 +λ2 +1λ2 λ3 λ4 λ5 λ6 λ3 +λ λ7 +λ6 +λ4 +λ2 +1 λ5 +λ4 +λ2 +10 0 0 0 0 0 1 01 λ λ2 +1 λ3 +λ λ4 +λ2 λ 0 00 0 0 0 1 0 0 00 0 0 1 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 0

1

C

C

C

C

C

C

C

C

C

C

C

A

1It is assumed that the reader is familiar with the derivationof the (complex) Jordan normal form for matrices withcoefficients over the field of real numbersR. For the details from finite field algebra refer to [LN94].

124 APPENDIX C. THE JORDAN NORMAL FORM OVER AN EXTENSION FIELD OF Fq

and

V(λ) =

λ13+ λ11+ λ6+ λ4+ λ3+ λ2 λ6 + λ4+ λ2 λ6 + λ4+ λ2 λ 0 0 1 0λ12+ λ10+ λ5+ λ3+ λ2+ λ λ5 + λ3+ λ λ5 + λ3+ λ 1 0 0 0 0

λ9 + λ5+ λ3+ λ2 λ2 λ2 0 λ 1 0 0λ8 + λ4+ λ2+ λ λ λ 0 1 0 0 0λ7 + λ3+ λ +1 1 1 0 0 0 0 0

λ12+ λ6+ λ5+ λ4+ λ2+ λ λ5 + λ λ5 + λ 1 λ2 λ 0 1λ7 + λ3+ λ 1 0 0 0 0 0 0λ8 + λ4+ λ2 λ 1 0 0 0 0 0

,

all operations to be taken modulo 2.

The Jordan-form of a matrix is based on a linear factor representation of the elementary divisorpolynomials over some splitting field, i. e. an extension field in which such a factorization ispossible. For a finite fieldFq the respective extension fields are defined upon the same operationsasFq and have the same characteristicq. In contrast toFq, the extension fields containqn elements— that is why those fields are denoted byFqn. Any field element ofFqn can be represented asa polynomial inFq[γ] with a field-specific elementγ that is a symbolic zero2 of an n-th degreeirreducible polynomialpirr ∈ Fq[λ] and calculations can be carried out in the associated residueclass ringFq[λ]/pirr(λ). Consequently, any polynomialf ∈ Fq[λ] is reduced withpirr(λ) by takingthe remainder polynomialr(λ) from

f (λ) = pirr(λ)q(λ)+ r(λ) ,

obtained after possibly successive polynomial divisions until the degree ofr(λ) is less than thedegree ofpirr(λ). An irreducible polynomial of degreen over a finite fieldFq splits into linearfactors in any extension fieldFqm for which n|m. Hence,Fqn is the least such splitting extensionfield because it contains the fewest field elements.

Therefore in the example problem, the irreducible basis polynomials with respect to the elementarydivisor polynomialsp1(λ) andp2(λ),

pirr,1(λ) = λ3+λ+1, pirr,2(λ) = λ2+λ+1,

can be factored according to3

pirr,1(λ) = (λ+λ1,1)(λ+λ1,2)(λ+λ1,3), pirr,2(λ) = (λ+λ2,1)(λ+λ2,2) ,

whereλ1,1,λ1,2,λ1,3 ∈ F23 andλ2,1,λ2,2 ∈ F22 are the zeroes in the extension fieldsF23 andF22.

2A customary analog is the symbolic zero “i” of the polynomialλ2 + 1 over the field of real numbersR. Thissymbolic zero is sufficient for describing any complex number in a polynomial of degree one.

3In the finite fieldF2 addition and subtraction coincide.

125

Now assume thatα is a zero (in a splitting extension field ofFq) of ann-th degree polynomial thatis irreducible overFq. Then from [LN94] recall that alln zeroes of this polynomial can be writtenas

α, αq, αq2, . . . , αqn−1

.

In the example, ifα ∈ F22 is a zero ofpirr,2(λ) = λ2 + λ + 1 then substituting this zero in theidentityλ2 = (λ2+λ+1)+λ+1 yieldsα2 = α+1, accordingly,

pirr,2(λ) = λ2+λ+1 = (λ+α)(λ+α2) = (λ+α)(λ+α+1) .

In general, the elements ofFqn can be interpreted as the zeroes of the polynomial

λqn −λ = λ(λ−1) · · ·(λ− (q−1))ρ(λ) ,

in which ρ(λ) is a polynomial that contains exactly those polynomials irreducible overFq thedegree of which is a proper divisor ofn.4 It is clear that the elements ofFq are always included inits extension fields. Additionally, in the particular caseF2 it follows that

λ2n+λ = λ(λ+1)

2n−2∑

i=0

λi .

Resorting to the example, the elements ofF23 are the zeroes of

λ23+λ = λ8+λ = λ(λ+1)

6∑

i=0

λi = λ(λ+1)(λ3+λ2 +1)(λ3+λ+1) ,

a polynomial that comprises all third degree irreducible polynomials overF2, and analogously, theelements ofF22 are the zeroes of

λ22+λ = λ4+λ = λ(λ+1)(λ2+λ+1) .

For ann-th degree polynomialp∈ Fq[λ] that consists ofi = 1, . . . ,N irreducible factor polynomialspirr,i of degreeδi to the power ofei , i. e.

p(λ) = pe1irr,1(λ) · · · peN

irr,N(λ) ,

4In the ringFq[λ] the numberNq(n) of monic irreducible polynomials of degreen is given by

Nq(n) =1n

∑

d|nµ(n

d

)

qd

in which the summation is carried out over all divisorsd of n.The functionµ(d) is the Möbius-function

µ(d) =

1 if d = 1(−1)k if d is the product ofk distinct primes0 if d is divisible by the square of a prime


the least extension field which contains allN fieldsFqδi is the fieldFqδ with δ = lcm(δi , . . . ,δN).

Consequently, if the degreen=∑N

i=1 δi ei of polynomialp is divisible by the square of some primenumber thenδ < n.

Returning to the example, an extension field for factoring the invariant polynomialc(λ) into linearfactors has to contain both finite fields,F23 andF22. In consequence,δ = 6 andF26 is the leastappropriate field.

Furthermore, an irreducible polynomial of degree 6 is needed for describing the elements ofF26,e. g. λ6 +λ3 +1. Let γ denote a zero of this polynomial inF26. With the arguments from above,usingγ any field element ofF26 can be represented as a polynomial in the ringF2[γ] of maximaldegree 5. It turns out thatc(λ) splits into linear factors according to

c(λ) = (λ+ γ3)(λ+ γ3+1)︸︷︷︸

= λ2+λ+1

(λ+ γ4+ γ2 + γ)2(λ+ γ5+ γ4)2(λ+ γ5+ γ2 + γ)2︸︷︷︸

= (λ3+λ+1)2

with the respective zeroes

λ1 = γ3, λ2 = γ3 +1, λ3 = γ4+ γ2 + γ, λ4 = γ5+ γ4, λ5 = γ5+ γ2 + γ .

Note that given the zeroesλ1 andλ3 the other zeroes are implied by

(γ3)2 = (γ6+ γ3+1)+ γ3 +1 = γ3 +1

(γ4+ γ2 + γ)2 = (γ6+ γ3+1)γ2 + γ5+ γ4 = γ5 + γ4

(γ5+ γ4)4 = (γ6+ γ3+1)(γ4+ γ2 + γ)+ γ5+ γ2+ γ = γ5 + γ2+ γ

which is in accordance with above since

λ2 = λ21, λ4 = λ2

3, λ5 = λ43 .

Note thatλ1, . . . ,λ5 are the eigenvalues ofA rendering the respective characteristic matrixλi I −Asingular. MatrixA has 5 elementary divisor polynomials over the splitting field F26, thus, there are5 Jordan-chains in the Jordan matrix: 3 of length 2 and 2 of length 1. As a result, the Jordan-formreads

J =

λ1 0 0 0 0 0 0 00 λ2 0 0 0 0 0 00 0 λ3 1 0 0 0 00 0 0 λ3 0 0 0 00 0 0 0 λ4 1 0 00 0 0 0 0 λ4 0 00 0 0 0 0 0 λ5 10 0 0 0 0 0 0 λ5

=

γ3 0 0 0 0 0 0 00 γ3 +1 0 0 0 0 0 00 0 γ4 + γ2+ γ 1 0 0 0 00 0 0 γ4 + γ2 + γ 0 0 0 00 0 0 0 γ5 + γ4 1 0 00 0 0 0 0 γ5 + γ4 0 00 0 0 0 0 0 γ5 + γ2 + γ 10 0 0 0 0 0 0 γ5 + γ2+ γ

.

127

The associated transformation matrixT which transforms the matrixA into its similar Jordan-matrix J as perA = TJT−1 is given by the respective Jordan-chains of generalized eigenvectorsv j

i for the i-th elementary divisor polynomials. These comply with the recursion

(λi I −A)v j+1i = v j

i , (λi I −A)v0i = 0,

for any eigenvalueλi of A. Here, the Jordan-chains for thei = 1, . . . ,5 eigenvaluesλi result in

v01, v0

2, v03 → v1

3, v04 → v1

4, v05 → v1

5

with

v01 =

0000001γ3

, v02 =

0000001

γ3 +1

, v03 =

γ5 + γ2 + γγ4 + γ2+ γ+1

γ4 + γ2 + γ1

γ5 + γ4+1γ5 + γ2+ γ+1

00

, v13 =

γ4 + γ2+ γ+1010

γ5 + γ2+ γ+1000

v04 =

1γ5 + γ2+ γ+1

γ4 + γ2 + γγ5 + γ4+1γ5 + γ2 + γ

γ5 + γ4

00

, v14 =

0γ4 + γ2+ γ+1

0γ5 + γ2 + γ

0100

, v05 =

γ5 + γ2 + γ1

γ5 + γ4 +1γ5 + γ4

γ5 + γ2+ γ+1γ4 + γ2 + γ

00

, v15 =

0γ4 + γ2+ γ+1

0γ5 + γ2+ γ+1

0γ5 + γ2 + γ

00

showing the geometric multiplicity of one with respect to each eigenvalue. Hence, the respectivetransformation matrix reads

T =(

v01,v

02,v

03,v

13,v

04,v

14,v

05,v

15

)

.

Given for convenience, the respective inverse matrix is

T−1 =

0 0 0 0 0 0 γ3 +1 10 0 0 0 0 0 γ3 10 γ5 + γ2+ γ 0 γ5 + γ4 +1 0 γ4 + γ2 + γ 0 0

γ5 + γ2 + γ γ5 + γ2 + γ+1 γ5 + γ4 +1 1 γ4 + γ2 + γ γ5 + γ4 0 01 0 γ5 + γ2+ γ 0 γ4 + γ2 + γ 0 0 0

γ5 + γ4 γ5 + γ2+ γ γ5 + γ4 +1 γ4 + γ2+ γ γ4 + γ2+ γ+1 γ5 + γ2+ γ+1 0 0γ4 + γ2+ γ+1 0 γ5 + γ2+ γ 0 γ5 + γ2+ γ+1 0 0 0

1 γ5 + γ2+ γ γ4 + γ2+ γ γ5 + γ2 + γ+1 γ5 + γ4 γ5 + γ4 +1 0 0

.


Remark C.1The above-employed irreducible polynomialλ6+λ3 +1 has the period 96= 26−1, hence, it is nota primitive polynomial. If choosing the irreducible polynomial λ6 + λ + 1 instead, thenc(λ) isfactorable as

c(λ) = (λ2+λ+1)(λ3+λ+1)2 = (λ+ ε5+ ε4+ ε3 + ε+1)(λ+ ε5+ ε4+ ε3+ ε)(λ+ ε3+ ε2+ ε)2(λ+ ε4+ ε2+ ε+1)2(λ+ ε4+ ε3+1)2

in which ε ∈ F26 is a zero ofλ6 +λ +1. Converse to the polynomialλ6 +λ3 +1, the polynomialλ6 +λ+1 is an irreducible polynomial of maximal period, hence 26−1 = 63 is the period.

An important theorem in Galois-theory states that any non-zero element in an extension fieldFqn

can be represented as some power of a zero of ann-th degree irreducible polynomial overFq iffthis polynomial is primitive. Simple calculations show that

ε21 = ε5+ ε4+ ε3+ ε+1

ε42 = (ε21)2 = ε5+ ε4+ ε3+ εε27 = ε3+ ε2+ εε54 = (ε27)2 = ε4+ ε2+ ε+1

ε45 = (ε54)2 = ε4+ ε3+1,

which implies thatc(λ) can be represented as

c(λ) = (λ+ ε21)(λ+ ε42)(λ+ ε27)2(λ+ ε54)2(λ+ ε45)2 .

Employing primitive polynomials renders polynomial calculations much easier as such expressedelements ofFqn are periodic with periodqn−1. For instance

ε108 = ε45ε63 = ε45

since withλ6+λ+1|λτ −1 with periodτ = 63 followsε63 = 1.

Conversely, no zeroγ of the non-primitive irreducible polynomialλ6 + λ3 + 1 over F2 can beused for representing the field elements inF26 as powers ofγ. Nevertheless, there is a reason fortaking a zero of some non-primitiven-th degree irreducible polynomial for defining the elementsof Fqn in a more complicated polynomial representation: primitive polynomials of arbitrary degreen cannot be calculated in an efficient straight-forward manner. If enumerations in tabulars are notsufficient any more there is fairly no way out from taking zeroes of just irreducible polynomialsthe determination of which is complicated enough for arbitrary degreesn. �

Remark C.2The task of polynomial synthesis by starting with some desired zeroes in an extension fieldFqδ

turns out to be tricky. The problem is that any desired zeroγ in the extension fieldFqδ has to

129

be accompanied by a specification of corresponding conjugatesγq, γq2, . . . and so on, in order to

obtain a polynomial inFq[λ] whose coefficients are elements fromFq only. Moreover, the numberof necessary conjugates depends on the degree of the polynomial of which it is root of. Similarly,if specifying an eigenvector inFn

qn then its conjugate eigenvectors have to be specified as well.

An unpleasant consequence is that a controller synthesis bymeans of eigenvalue placement andparameter vector specification in some extension field, as proposed in the parametric approach[Rop86, DH01], becomes very cumbersome as many distinct cases entail unwieldy parameteriza-tion formulae. This is the main reason why the presented workis not based on the notion of anextension field. �

Appendix D

General Solution of Linear Systems usingSingular Inverses

The solvability condition and the general solution of the linear system of equations

Ax = b (D.1)

shall be determined, whereA ∈ Fm×n is a possibly singular and non-square matrix andx ∈ Fn,b ∈ Fm are column vectors. Refering to [LT85, CW94] the basic relation for an inverse matrix isthe following.

Definition D.1 (Generalized Inverse)Let A ∈ Fm×n andG ∈ Fn×m with

AGA = A . (D.2)

Such a matrixG is called generalized inverse (g-inverse, singular inverse). �

This definition generalizes many concepts of inverse matrices, for example it is in accordance withthe customary definition of an inverseG = A−1 if the matrix A is square and invertible, or inaccordance with the left (right) inverse matrixG = (ATA)−1AT (G = AT(AAT)−1) with respectto a non-square matrixA whose column (row) rank is its column (row) dimension.

Without loss of generality, any matrixA ∈ Fm×n of rankr can be written as1

A =

(Ar×r Ar×(n−r)

A(m−r)×r A(m−r)×(n−r)

)

and a construction scheme [CW94] for a corresponding generalized inverse matrixG ∈ Fn×m is

G =

(A−1

r×r −A−1r×r

(Ar×(n−r)G21Ar×r −Ar×rG12A(m−r)×r −Ar×(n−r)G22A(m−r)×r

)A−1

r×r G12

G21 G22

)

1Some elementary row and column operations may be necessary.

132 APPENDIX D. GENERAL SOLUTION OF L INEAR SYSTEMS USINGSINGULAR INVERSES

in which the matricesG12, G21 andG22 can be chosen arbitrarily, thus,g-inverses are not uniquein general. A simple form of ag-inverse is

G =

(A−1

r×r 00 0

)

, G ∈ Fn×m. (D.3)

Employing a generalized inverseG, equation (D.1) is solvable if a vectorx exists such that

b = Ax = AGAx = AGb ⇐⇒ (I −AG)b = 0.

Conversely, if(I −AG)b = 0 thenb = AGb and a (particular) solutionxp = Gb exists.

Theorem D.1 (Solvability Condition)Let A ∈ Fm×n andb ∈ Fm. Then the linear system of equationsAx = b is solvable iff

(I −AG)b = 0 (D.4)

for some generalized inverse matrixG ∈ Fn×m with respect toA. �

Given solvability the general solution ofAx = b has the form

x = xh+xp (D.5)

in whichxh is the solution of the homogeneous equationAxh = 0 andxp is a particular solution ofAxp = b. As has already been shown above such a particular solution is

xp = Gb . (D.6)

For the purpose of deriving the homogeneous solutionxh multiply A by an arbitrary vectorz∈ Fn

and use the general inverse relation in Definition D.1 to obtain

Az = AGAz ⇐⇒ A(I −GA)z= 0.

Hence, the homogeneous solution is

xh = (I −GA)z (D.7)

for z∈ Fn arbitrary.

Theorem D.2 (General Solution of a Linear System of Equations)Let A ∈ Fm×n, b ∈ Fm. Let G ∈ Fn×m be a respective general inverse matrix that satisfies thesolvability condition in Theorem D.1. Then the general solution of the linear systemAx = b is

x = (I −GA)z+Gb (D.8)

for arbitraryz∈ Fn. �

133

Example D.1If C ∈ Fn×n is a nilpotent, hence singular, companion matrix an exampleof a generalized inversematrix G simply follows from

CCTC = C ,

that isG = CT. Note that(CT)κ Cκ = diag(In−κ,0κ), κ ∈ N ,

which implies that forκ ≥ 2(CT)κ Cκ 6= (CTC)κ . �

134 APPENDIX D. GENERAL SOLUTION OF L INEAR SYSTEMS USINGSINGULAR INVERSES

Appendix E

Rank Deficiency of a Matrix-ValuedPolynomial Function

When investigating the dimension of a linear subspace, for example when generalized eigenspacesare concerned, a criterion for the rank deficiency of a matrixf (A) as a value of a polynomialfunction f is advantageous. Particularly, in cases when the elementary divisor polynomials of thematrix A and the multiplicities of the zeroes regarding the polynomial f (λ) are known, a simpleformula for the rank off (A) can be derived [Gan58].

First, observe that for any matrixA ∈ Fn×n a Jordan-formJ ∈ Fn×ns with A = TJT−1 exists.1 A

polynomial functionf : Fn×n → Fn×n applied onA yields

f (A) = T f (J)T−1 . (E.1)

Given thei = 1, . . . ,Ns elementary divisor polynomialspi ∈ Fs[λ] of A

p1(λ) = (λ−λ1)e1, p2(λ) = (λ−λ2)

e2, . . . , pNs(λ) = (λ−λNs)eNs

made up of the (not necessarily distinct) eigenvaluesλi ∈ Fs with respect toA, the Jordan-formreads

J = diag(J1, . . . ,JNs) , (E.2)

in which each matrixJi corresponds to an elementary divisor polynomial. These matrices showthe form

Ji = λi Iei +Nei (E.3)

1The fieldFs is a splitting field ofF. In other wordsF is an extension field ofF in which any polynomial inF[λ]

can be factored in linear factors with coefficients inFs. Example: the field of complex numbersC is a splitting fieldwith respect to the field of real numbersR.

136 APPENDIX E. RANK DEFICIENCY OF A MATRIX -VALUED POLYNOMIAL FUNCTION

whereNei is aei ×ei nilpotent matrix

Nei =

0 1 0 · · · 00 0 1 · · · 0...

......

. . . 00 0 0 · · · 10 0 0 · · · 0

(E.4)

andIei is the respective identity.

As J consists of diagonal blocks only, it follows

f (J) = diag( f (J1), . . . , f (JNs)) (E.5)

and the rank deficiency off (J) can be obtained by summing up the rank deficiencies of the matricesf (J1), . . . , f (JNs).

To this end, let the polynomialf (λ) be represented by its Lagrange-Sylvester interpolation poly-nomial regardingλi, accordingly2

f (λ) = f (λi)+f ′(λi)

1!(λ−λi)+

f (2)(λi)

2!(λ−λi)

2+ . . . (E.6)

by means of which fori = 1, . . . ,Ns

f (Ji) = f (λi) Iei +f ′(λi)

1!(Ji −λi Iei)+

f (2)(λi)

2!(Ji −λi Iei)

2+ . . .

= f (λi) Iei +f ′(λi)

1!Nei +

f (2)(λi)

2!N2

ei+ · · ·+ f (ei−1)(λi)

(ei −1)!Nei−1

ei

since withNei = Ji −λi Iei from (E.3) it turns out that the series truncates due to the nilpotency ofNei . Hence, the result is

f (Ji) =

f (λi)f ′(λi)

1! · · · f (ei−1)(λi)(ei−1)!

0 f (λi).. .

......

..... . f ′(λi)

1!0 0 · · · f (λi)

(E.7)

On the one hand, ifλi is no zero of the polynomialf (λ) then f (Ji) is of full rank. On the otherhand, ifki is the multiplicity of a zeroλi with regard tof (λ) then for reason of

f (λi) = f ′(λi) = . . . = f (ki−1)(λi) = 0, f (ki)(λi) 6= 0, i = 1, . . . ,Ns (E.8)

the rank deficiency off (Ji) is ki , unless the dimension of the matrixNei is less thanki . The resultis fixed in the following theorem.

2Derivatives used are only formal derivatives and do not imply continuity.

137

Theorem E.1 (Rank Deficiency of a Matrix-Valued Polynomial Function)Let A ∈ Fn×n be a matrix and let thei = 1, . . . ,Ns elementary divisor polynomialspi(λ) over thesplitting fieldFs be given as

p1(λ) = (λ−λ1)e1, p2(λ) = (λ−λ2)

e2, . . . , pNs(λ) = (λ−λNs)eNs .

Furthermore, letf ∈ F[λ] be a polynomial. Then the rank deficiency∆ of the matrix f (A) is

∆ =

Ns∑

i=1

min(ki ,ei)

whereki is the multiplicity of the eigenvalueλi as a zero off . �

Example E.1Consider a companion matrixC ∈ Fd×d

q whosed-th degree defining polynomial

pC(λ) =(pirr,C(λ)

)e

is thee-th power of an irreducible polynomialpirr,C(λ) of degreeδ. Hence,pC(λ) is the onlyelementary divisor polynomial overFq.

The dimensions of the nullspaces concerning the matrices

f j(C) =(pirr,C(C)

) j, j = 1, . . . ,e

are to be determined by means of Theorem E.1.

From Appendix C recall that an irreducible polynomial of degreeδ overFq has exactlyδ distinctzeroes in the extension fieldFqδ, which represents a corresponding splitting field. Thus,

(λ−λ1)e, (λ−λ2)

e, . . . , (λ−λδ)e.

are theδ elementary divisor polynomials ofC over the splitting fieldFqδ.

As the dimension of the nullspaces off j(C) equals the rank deficiency∆ j of f j(C), consequently,Theorem E.1 can be applied. This results in the nullspace dimensions

∆ j =

δ∑

i=1

min( j,e) =

δ∑

i=1

j = j δ, j = 1, . . . ,e

with respect to the matricesf j(C). �

Example E.2Given a controllable LMS with dynamics matrixA ∈ Fn×n

q and input matrixB ∈ Fn×mq , respec-

tively. Let K ∈ Fm×nq denote a feedback matrix associated to a static state feedback of the form


u(k) = Kx(k). As the closed-loop system is autonomous it is easy to see that the solution of theclosed-loop state equation reads

x(k) = (A +BK)kx(0) = (AK )k x(0)

with the closed-loop dynamics matrixAK . Assume that the feedback matrixK is to be determinedsuch that a maximal set of initial statesx(0) can be steered to some desired statexd in a minimalnumber of stepsk. Then by transforming the state as perx = x−xd, solving the problem amountsto determine a feedback matrixK complying with

0 = (AK )k x(0)

which renders(AK )k = 0 with a full rank deficiency∆ = n in a minimal number of stepsk. Thismeans thatAK must be nilpotent, thus, its characteristic polynomial hasto be

cpAK(λ) = λn .

This control objective reminds of standard deadbeat-control [Kuc91] where all eigenvalues of theclosed-loop dynamics matrix are equal to zero and the closed-loop dynamics matrixAK in CCFshows simple companion form. However, Theorem E.1 gives a hint for more refinement. The rea-son is that in the MIMO-case there is some liberty in choosingthe elementary divisor polynomials,which are exactly the factors of the characteristic polynomial and coincide with the invariant poly-nomials, in this case. In view of this fact and Theorem E.1 denote f (AK ) = (AK)k and set theNelementary divisor polynomials

pi(λ) = λei , i = 1, . . . ,N,N∑

i=1

ei = n.

Then by Theorem E.1 the rank deficiency of(AK )k is

∆ =N∑

i=1

min(k,ei) .

Increasing the rank deficiency∆ by increasing the numberN of elementary divisor polynomialsis not possible abovem. This originates from the fact that by Theorem 5.3,m is the maximumnumber of achievable nilpotent diagonal blocks in the closed-loop dynamics matrix represented inCCF, see Definition 5.5. Thus,N = m.

Again in light of Definition 5.5, the step numberk that is necessary for obtaining(AK )k = 0 can bebounded from above by the dimension of the largest nilpotentdiagonal block matrix. This resultsin

∆ =

m∑

i=1

min(k,ei) =

m∑

i=1

k = km

139

and, consequently, thei = 1, . . . ,m elementary divisor exponentsei have to be chosen equal to theordered controllability indicesci of the LMS, hence

pi(λ) = λci , i = 1, . . . ,m c1 ≥ . . . ≥ cm.

Since these elementary divisor polynomialspi(λ) coincide with the invariant polynomials, it isobvious that the inequalities in Rosenbrock’s control structure theorem, Theorem 5.4, are satisfied.Following the lines in Chapter 5.2.4, the polynomial matrix

D⋆K (a) = diag(ac1, . . . ,acm)

can be used for determining a feedback matrixK that guarantees to drive any statex(0) into somedesired statexd in maximalk = c1 steps, which as compared to standard deadbeat-control takesalways less thann steps. �

Appendix F

Solving the Linear State Equation in theImage Domain

The image domain representation of the state equation (4.1)of an LMS directly leads to theA-transform of the system state

X(a) = (aI −A)−1(BU(a)+ax(0)),

recalling (5.12) for convenience. Employing the inverseA-transform, Theorem 2.11, allows todetermine the well-known solution of the state equation. With the geometric series formula appliedon the expression

(aI −A)−1 =1a

(

I − Aa

)−1=

1a

∞∑

i=0

(Aa

)i

the original domain function results in

x(k) =[

akX(a)]

ind=[

ak (aI −A)−1(BU(a)+ax(0))]

ind

=[

ak1a

∞∑

i=0

(Aa

)i(

B∞∑

j=0

u( j)a− j +ax(0))]

ind

=[( ∞∑

i=0

A i ak−i)

x(0)]

ind+[( ∞∑

i=0

A i ak−i−1)

B( ∞∑

j=0

u( j)a− j)]

ind

=[ ∞∑

i=0

A i ak−i x(0)]

ind+

∞∑

i=0

A i Bu(k− i −1)

= Ak x(0)+Bu(k−1)+ABu(k−2)+ · · ·+Ak−1Bu(0) ,

which equals the expression for the solution of the state equation, given in (5.1).

142 APPENDIX F. SOLVING THE L INEAR STATE EQUATION IN THE IMAGE DOMAIN

Appendix G

List of Publications

[1] J. Reger, “Deadlock Analysis for Deterministic Finite State Automata using Affine LinearModels”, in: Proc. of 2001 European Control Conference, (Porto, Portugal), 2001.

[2] J. Reger, “Cycle Analysis for Deterministic Finite State Automata”, in:IFAC Proc. of 15thWorld Congress, (Barcelona, Spain), 2002.

[3] J. Reger and K. Schmidt, “Modeling and Analyzing Finite State Automata in the Finite FieldGF(2)”, in: Proc. of 4th MATHMOD, (Vienna, Austria), Argesim, 2003.

[4] K. Schmidt and J. Reger, “Synthesis of State Feedback forLinear Automata in the FiniteField GF(2)”, in:Proc. of 4th MATHMOD, (Vienna, Austria), Argesim, 2003.

[5] J. Reger, “Analysis of Multilinear Systems using Gröbner-bases over the Finite Field GF(2)”,in: Proc. of 4th MATHMOD, (Vienna, Austria), Argesim, 2003. best conference posteraward

[6] J. Reger and K. Schmidt, “Aspects on Analysis and Synthesis of Linear Discrete Systemsover the Finite FieldFq”, in: Proc. of 2003 European Control Conference, (Cambridge,United Kingdom), 2003.

[7] J. Reger and K. Schmidt, “Modeling and Analyzing Finite State Automata in the Finite FieldGF(2)”, Mathematics and Computers in Simulation (MATCOM), 66(1–2):193–206, 2004.

[8] J. Reger and K. Schmidt, “A Finite Field Framework for Modelling, Analysis and Controlof Finite State Automata”,Mathematical and Computer Modelling of Dynamical Systems(MCMDS), 2004. accepted for publication.

[9] K. Schmidt, J. Reger, and T. Moor, “Hierarchical Controlfor Structural Decentralized DES”,in: Proc. of 7th Workshop on Discrete Event Systems (WODES), (Reims, France), 2004.accepted for publication.

144 APPENDIX G. LIST OF PUBLICATIONS

References

[Ant98] P. ANTSAKLIS. Linear Systems. McGraw-Hill, New York, 1998.

[Ber70] E. BERLEKAMP. Factoring polynomials over large finite fields.Mathematics of Com-putation, 24(111):713–735, 1970.

[BJT97] J. BUCHMANN , M. JACOBSEN, AND E. TESKE. On some computational problemsin finite abelian groups.Mathematics of Computation, 66(220):1663–1687, 1997.

[BLL91] A. B ENVENISTE, P. LE GUERNIC, AND M. LE BORGNE. Dynamical systems overgalois fields and DEDS control problems. InProc. of 30th Conf. Decision and Control,pages 1505–1509. IEEE Publications, 1991. Brighton, UK.

[BM77] G. BIRKHOFF AND S. MACLANE. A Survey of Modern Algebra. MacMillan, NewYork, 1977.

[Boo62] T. L. BOOTH. An Analytical Representation of Signals in Sequential Networks. InProc. of Symposium on The Mathematical Theory of Automata, New York, 1962. Poly-technic Press and J. Wiley and Sons.

[Boo67] T. L. BOOTH. Sequential Machines and Automata Theory. Wiley, New York, 1967.

[BP81] D. BOCHMANN AND C. POSTHOFF. Binäre Dynamische Systeme. R. Oldenbourg,München, 1981.

[CLO98] D. COX, J. LITTLE , AND D. O’SHEA. Using Algebraic Geometry. Springer, NewYork, 1998.

[CLR90] T. H. CORMEN, C. E. LEISERSON, AND R. L. RIVEST. Introduction to Algorithms.MIT Press, Cambridge, MA, 1990.

[CW94] W. CASPARY AND K. W ICHMANN . Lineare Modelle. Oldenbourg, München, 1994.

[DH78] L. L. DORNHOFF ANDF. E. HOHN. Applied Modern Algebra. Macmillan, New York,1978.

146 REFERENCES

[DH01] J. DEUTSCHER ANDP. HIPPE. Parametric compensator design in the frequency do-main. Int. J. Control, (74):1467–1480, 2001.

[Els59] B. ELSPAS. The theory of autonomous linear sequential networks.IEEE Trans. CircuitTheory, 6:45–60, 1959.

[Fra94] D. FRANKE. Sequentielle Systeme — Binäre und Fuzzy Automatisierung mit arith-metischen Polynomen. Vieweg, Braunschweig, 1994.

[Fra96] D. FRANKE. Arithmetische Logik — Ein Brückenschlag zwischen diskreten Steue-rungen und klassischen Regelungen.Automatisierungstechnik, 44:553–563, 1996.

[Fri59] B. FRIEDLAND. Linear modular sequential circuits.IEEE Trans. Circuit Theory,6:71–68, 1959.

[Gan58] F. R. GANTMACHER. Matrizenrechnung, volume 1. VEB, Berlin, 1958.

[Gat87] J. VON ZUR GATHEN. Factoring polynomials and primitive elements for specialprimes.Theoretical Computer Science, 52:77–89, 1987.

[Ger95] R. GERMUNDSSON. Symbolic Systems — Theory, Computation and Applications.PhD thesis, Linköping, 1995.

[Gil64] A. GILL . Analysis of Linear Sequential Circuits by Confluence Sets.IEEE Trans.Computers, 30:226–231, 1964.

[Gil66a] A. GILL . Graphs of Affine Transformations, with Applications to Sequential Circuits.In Proc. of the 7th IEEE International Symposium on Switching and Automata Theory,pages 127–135. IEEE Publications, 1966. Berkeley, California, USA.

[Gil66b] A. GILL . Linear Sequential Circuits: Analysis, Synthesis, and Applications. McGraw-Hill, New York, 1966.

[Gil69] A. GILL . Linear Modular Systems. In L. A. ZADEH AND E. POLAK , editors,SystemTheory. McGraw-Hill, New York, 1969.

[GLR82] I. GOHBERG, P. LANCASTER, AND L. RODMAN. Matrix Polynomials. AcademicPress, New York, 1982.

[Gös91] M. GÖSSEL. Automatentheorie für Ingenieure. Akademie Verlag, Berlin, 1991.

[Gun97] J. GUNNARSSON. Symbolic Methods and Tools for Discrete Event Dynamic Systems.PhD thesis, Linköping, 1997.

147

[HHL+00] D. HANKERSON, D. G. HOFFMANN, D. A. LEONARD, C. C. LINDNER, K. T.PHELPS, C. A. RODGER, AND J. R. WALL . Coding Theory and Cryptography —The Essentials. Marcel Dekker Inc., New York, 2nd edition, 2000.

[Huf56] D. A. HUFFMANN. Information Theory, chapter The Synthesis of Linear SequentialCoding Networks. Academic Press, New York, 1956. Paper readat a Symposium onInformation Theory held at the Royal Institution, London, September 1955.

[Huf59] D. A. HUFFMANN. A linear circuit viewpoint on error-correcting codes.IEEE Trans.Circuit Theory, 6:45–60, 1959.

[Ili89] C. S. ILIOPOULUS. Worst-case complexity bounds on algorithms for computingthecanonical structure of finite abelian groups and the Hermiteand Smith normal formsof an integer matrix.SIAM J. Comput., 18(4):658–669, 1989.

[Kai80] T. KAILATH . Linear Systems. Prentice Hall, Englewood Cliffs, 1980.

[Kau65] W. H. KAUTZ, editor. Linear Sequential Switching Circuits - Selected TechnicalPa-pers. Holden-Day, San Francisco, 1965.

[Kuc91] V. KUCERA. Analysis and design of discrete linear control systems. Prentice Hall,New York, 1991.

[Lan84] S. LANG, editor. Algebra. Addison-Wesley, Yale University, New Haven, Connecti-cut, 1984.

[LBL89] M. L E BORGNE, A. BENVENISTE, AND P. LE GUERNIC. Polynomial ideal theoreticmethods in discrete event, and hybrid dynamical systems. InProc. 1989 IEEE Work.CACSD, December 1989. Tampa, USA.

[LBL91] M. L E BORGNE, A. BENVENISTE, AND P. LE GUERNIC. Polynomial DynamicalSystems over Finite Fields. In G. JACOB AND F. LAMNABHI -LAGARRIGUE, editors,Lecture Notes in Computer Science, volume 165, pages 212–222. Springer, Berlin,1991.

[LN94] R. LIDL AND H. NIEDERREITER. Introduction to finite fields and their applications.Cambridge Univ. Press, New York, 1994.

[LT85] P. LANCASTER AND M. T ISMENETSKY. The Theory of Matrices. Academic Press,San Diego, 2nd edition, 1985.

[Mar97] H. MARCHAND. Méthodes de synthèse d’automatismes d’écrits par des systèmes àévénements discrets finis. PhD thesis, Université de Rennes, October 1997.

148 REFERENCES

[McE87] R. J. MCELIECE. Finite fields for computer scientists and engineers. Kluwer Aca-demic Publishers, Dordrecht, 1987.

[Mei96] A. M EIJER. Groups, Factoring and Cryptography.Mathematics Magazine, 69(2):103–109, 1996.

[ML97] H. M ARCHAND AND M. LE BORGNE. Partial Order Control and Optimal Controlof Discrete Event Systems modeled as Polynomial Dynamical Systems over Galoisfields. Rapport de recherche IRISA 1125, IRISA, October 1997.

[ML99] H. M ARCHAND AND M. LE BORGNE. The Supervisory Control Problem of DiscreteEvent Systems using Polynomial Methods. Rapport de recherche IRISA 1271, IRISA,October 1999.

[Mul54] D. E. MULLER. Application of boolean algebra to switching circuit design and toerror detection.IEEE Transactions on Computers, (3):6–12, 1954.

[New72] M. NEWMAN. Integral Matrices. Academic Press, New York, 1972.

[New74] M. NEWMAN. The Smith Normal Form of a Partitioned Matrix.Journal of Researchof the National Bureau of Standards - B, Mathematical Sciences, 778(1), 1974.

[NMGJ01] D. NEŠIC, I. M. Y. M AREELS, T. GLAD , AND M. JIRSTRAND. Software for Con-trol System Analysis and Design, Symbol Manipulation. In J.WEBSTER, editor,Wiley Encyclopedia of Electrical and Electronics Engineering Online. Wiley, 2001.http://www.interscience.wiley.com:83/eeee/.

[PML99] S. PINCHINAT, H. MARCHAND, AND M. LE BORGNE. Symbolic Abstractions of Au-tomata and their application to the Supervisory Control Problem. Rapport de rechercheIRISA 1279, IRISA, November 1999.

[PW72] W. W. PETERSON ANDE. J. WELDON. Error-Correcting Codes. MIT Press, Cam-bridge, MA, 2nd edition, 1972.

[Ree54] I. S. REED. A class of multiple-error-correcting codes and the decoding scheme.IEEETransactions on Information Theory, (4):38–49, 1954.

[Ric65] J. RICHALET. Operational Calculus for Finite Rings.IEEE Trans. Circuit Theory,12:558–570, 1965.

[Rop86] G. ROPPENECKER. On Parametric State Feedback Design.Int. J. Control, (43):793–804, 1986.

[Ros70] H. H. ROSENBROCK. State-space and Multivariable Theory. Thomas Nelson Ltd.,London, 1970.

149

[RS03] J. REGER AND K. SCHMIDT. Aspects on Analysis and Synthesis of Linear DiscreteSystems over the Finite FieldFq. In Proc. of 2003 European Control Conference,Cambridge, United Kingdom, 2003.

[Sch02] K. SCHMIDT. Entwurf von Zustandsrückführungen für lineare diskrete Systemeüber GF(2) mit Hilfe der Polynommatrixmethode. Master’s thesis, Lehrstuhl fürRegelungstechnik, Friedrich-Alexander-Universität Erlangen-Nürnberg, March 2002.

[Sel66] E. S. SELMER. Linear recurrence relations over finite fields. Technical report, De-partment of Mathematics, University of Bergen, 1966.

[Sho90] V. SHOUP. On the deterministic complexity of factoring polynomialsover finite fields.Information Processing Letters, 33(5):261–267, 1990.

[Son99] J. SONNENBERG. A New Method for Describing and Analyzing Finite DeterminedAutomata by Walsh Functions. InProc. of 1999 European Control Conference, Karls-ruhe, Germany, 1999.

[Son00] J. SONNENBERG. Verfahren zur linearen Modellierung dynamischer ereignisdiskreterSysteme mittels Walsh-Funktionen. VDI-Verlag, Düsseldorf, 2000.

[Sto00] A. STORJOHANN. Algorithms for Matrix Canonical Forms. PhD thesis, Departmentof Computer Science, Swiss Federal Institute of Technology– ETH, Zürich, 2000.

[Tha88] A. THAYSE. Boolean calculus and differences. InLecture Notes in Computer Science,volume 101. Springer, Berlin, 1988.

[Wen00] W.-M. WENDLER. Elements of Linear Binary System Theory. InProc. of 4th Inter-national Workshop on Boolean Problems, pages 37–46, Freiberg (Germany), 2000.

[Wol74] W. A. WOLOVICH. Linear Multivariable Systems. Springer, New York, 1974.

[Wun75] G. WUNSCH. Systemtheorie. Akademische Verlagsgesellschaft Geest & Portig K.-G.,Leipzig, 1975.

[Zhe27] I. I. ZHEGALKIN . Über eine Technik zur Berechnung von Sätzen in der symbolischenLogik. Mat. sbornik, (34):9–28, 1927. transl. from russian: Prof. Dr. D. Bochmann.

150 REFERENCES

Lebenslauf

Zur Person:

Johann Regergeboren am 18. 06. 1971 in Erbendorf in der Oberpfalzverheiratet, ein Kind

Schulbildung:

1977–1981 Grundschule in Waldeck1981–1990 Gymnasium in Eschenbach i. d. Opf.Juni 1990 Abschluss mit Abitur

Wehrdienst:

1990–1992 2jähriger Wehrdienst als Soldat auf Zeit in der Stabskompanie derPanzergrenadierbrigade 10 in Weiden i. d. Opf.

Studium:

1992–1998 Studium der Fertigungstechnik an der Friedrich-Alexander-UniversitätErlangen-Nürnberg und an der University of Liverpool

1993–1995 Tutor am Lehrstuhl für Technische Mechanik1995–1999 Tutor am Lehrstuhl für Angewandte Mathematik II1997 Praktikum bei der Siemens AG, Medical Solutions in Erlangen,

Forchheim und Nürnberg1994 Preis der FAG Kugelfischer-Stiftung für erzielte Examensergebnisseseit 1995 Stipendiat der Studienstiftung des deutschen Volkes und Mitglied

des internationalen Studentenprogramms SSP der Siemens AGDez. 1998 Studienabschluss Dipl.-Ing.

Hochschultätigkeit:

1999–2001 Promotionsstipendiat der Studienstiftung des deutschen Volkesseit 2002 Wissenschaftlicher Assistent am Lehrstuhl für Regelungstechnik

der Universität Erlangen-Nürnberg

Linear Systems over Finite Fields - Startseite TU Ilmenau · Linear Systems over Finite Fields —...

Documents

Transcript of Linear Systems over Finite Fields - Startseite TU Ilmenau · Linear Systems over Finite Fields —...