from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

minimality of descriptor representations under external equivalence abstract necessary and sufficient conditions are derived for the minimality of descriptor representations under external equivalence these conditions are stated completely in terms of the matrices e a b c and d use is made of the close connection between the descriptor representation and the so called pencil representation this connection is further exploited to derive the transformation group for minimal descriptor representations it is shown that the transformations coincide with the operations of strong equivalence as introduced by verghese et al ieee trans aut control ac 26 811 831 1981

on the spectral and combinatorial structure of 2d positive systems abstract the dynamics of a 2d positive system depends on the pair of nonnegative square matrices that provide the updating of its local states in this paper several spectral properties such as finite memory separability and property l which depend on the characteristic polynomial of the pair are investigated under the nonnegativity constraint and in connection with the combinatorial structure of the matrices some aspects of the perron frobenius theory are extended to the 2d case in particular conditions are provided guaranteeing the existence of a common maximal eigenvector for two nonnegative matrices with irreducible sum finally some results on 2d positive realizations are presented

solvability conditions consistency and weak consistency for linear differential algebraic equations and time invariant singular systems the general case we present several solvability concepts for linear differential algebraic equations daes with constant coefficients on the positive time axis as well as for the associated singular systems and investigate under which conditions these concepts are met next we derive necessary and sufficient conditions for global consistency of initial conditions for the dae as well as for the system and generalize these conditions with respect to our concept of weak consistency our distributional approach enables us to generalize results in an earlier paper where singular systems are assumed to have a regular pencil in the sense of gantmacher in particular we establish that global weak consistency in the system sense is equivalent to impulse controllability

input output and state space representations of finite dimensional linear time invariant systems in this paper we develop various representations for systems described by aset of high order differential equations of the form r0w r1w rsw s 0 with r0 r1 rs not necessarily square matrices the variables w are the external variables particular attention is paid to the problem of obtaining minimal state space realizations and input output or input state output representations of such systems

common minimal multiples and divisors for rational matrix functions abstract we extend to rational matrix functions the theory of common multiples and common divisors of regular matrix polynomials developed in 12 attention is focused mainly on problems of existence uniqueness and construction from local data

continuous varying linear systems abstract we discuss implicit systems of ordinary linear differential equations with time variable coefficients their solutions in the signal space of hyperfunctions according to sato and their solution spaces called time varying linear systems or behaviours from the system theoretic point of view the basic result inspired by an analogous one for multidimensional constant linear systems is a duality theorem which establishes a categorical one one correspondence between time varying linear systems or behaviours and finitely generated modules over a suitable skew polynomial ring of differential operators this theorem is false for the signal spaces of infinitely often differentiable functions or of meromorphic hyper functions or of distributions on r it is used to obtain various results on key notions of linear system theory several new algorithms for modules over rings of differential operators and in particular new grobner basis algorithms due to insa and pauer make the system theoretic results effective

p class groups of certain extensions of degree p kkt p be an odd prime number in this article we study the distribution of p class groups of cyclic number fields of degree p and of cyclic extensions of degree p of an imaginary quadratic field whose class number is coprime to p we formulate a heuristic principle predicting the distribution of the p class groups as galois modules which is analogous to the cohen lenstra heuristics concerning the prime to p part of the class group although in our case we have to fix the number of primes that ramify in the extensions considered using results of gerth we are able to prove part of this conjecture furthermore we present some numerical evidence for the conjecture

unknown input observers for 2d state space models in this paper dead beat unknown input observers uios for two dimensional 2d state space models are investigated dead beat uios are observers which produce an exact estimate of the original system state trajectory after a finite number of evolution steps independently of the system and observer initial conditions and of the unknown disturbances that affect the system functioning necessary and sufficient conditions for the existence of dead beat uios are provided as well as a parameterization of all dead beat uio transfer matrices comparisons with luenberger type uios are also carried out and the extension of the paper results to the case of asymptotic uios with a given rate of convergence and to roesser models are finally discussed

explicit computation of galois p groups unramified at p abstract we introduce a new method for finding galois groups by modifying the p group generation algorithm using number theoretical information thus by exhaustive search we calculate the galois groups of some p extensions that were previously beyond reach

descriptor variable systems and optimal state regulation linear systems of the form e dot x ax bu with e singular are treated it is desired to find a control which drives the system asymptotically to the origin minimizing a quadratic cost functional no restrictions are placed on initial conditions the cost associated with the impulsive behavior of the system is examined as well as existence and uniqueness of the optimal control through a sequence of coordinate transformations it is proven that the optimal control can be found by solving a reduced order riccati equation

hautus type conditions for controllability of implicit linear discrete time systems revisited new simple proofs of validity of hautus type conditions for controllability of not necessarily regular implicit linear discrete time systems are given the new proofs do not require any previous knowledge of implicit systems theory

heuristics for p class towers of imaginary quadratic fields cohen and lenstra have given a heuristic which for a fixed odd prime p leads to many interesting predictions about the distribution of p class groups of imaginary quadratic fields we extend the cohen lenstra heuristic to a non abelian setting by considering for each imaginary quadratic field k the galois group of the p class tower of k i e g k mathrm gal k infty k where k infty is the maximal unramified p extension of k by class field theory the maximal abelian quotient of g k is isomorphic to the p class group of k for integers c ge 1 we give a heuristic of cohen lenstra type for the maximal p class c quotient of g k and thereby give a conjectural formula for how frequently a given p group of p class c occurs in this manner in particular we predict that every finite schur sigma group occurs as g k for infinitely many fields k we present numerical data in support of these conjectures

invariant subspaces and invertibility properties for singular systems the general case abstract open loop definitions and properties of several subspaces for general singular systems are characterized by means of a fully algebraic distributional framework simple recursive algorithms for producing these spaces as well as related duality aspects turn out to follow directly from these definitions next we provide definitions and conditions for two notions of left right invertibility of a general singular system in terms of our distributions subspaces and rosenbrock s system matrix and we show which conditions represent the gap between our invertibility concepts finally we prove that in many cases left right invertibility is equivalent to left right invertibility of the system matrix

on behavior homomorphisms and system equivalence in this paper we introduce the concept of a behavior homomorphisms and isomorphisms and use it to present a uni ed approach to the study of equivalence of di erent behavior representations concentrating on a special representation we call normalized arma narma representation c 2001 elsevier science b v all rights reserved

a tutorial on the geometric analysis of linear time invariant implicit systems abstract a background of recent work in nonregular and regular implicit systems of the form e x ax bu is provided then the output nulling unknown input and composite subspaces are defined for singular systems with recursions given for their computation inner subspaces that is those in the domain of e and a are distinguished from outer subspaces those in the codomain of e and a the notions of composite preimage and composite image are introduced the result is a framework that unifies some of the recent work in the geometric theory of implicit systems system properties are discussed for both regular and nonregular systems and duality is investigated

rational spectral factorization using state space methods abstract we describe a state space based algorithm to find the minimum phase spectral factor of any rational spectral density whether it be proper improper or polynomial

a note on the state space realization of 2d fir transfer functions abstract some properties of minimal state space realizations of 2d fir transfer functions are investigated it is shown that hidden modes are allowed in minimal realization and consequently there exist internally unstable minimal realizations of fir transfer functions

realization and canonicity for implicit systems this paper studies realization theory and canonicity conditions for implicit systems here an implicit system is any system defined by an input output finite dimensional linear time invariant relation in discrete or continuous time they are also called generalized or descriptor systems in the literature the familiar definitions of canonicity and minimality are extended to such systems and a so called reduced form proves that as for classical systems minimality and canonicity are two equivalent concepts we replace the notion of transfer function by a notion of external form and prove that minimal realization can always be achieved

structural matrix minimization algorithm for implicit descriptions in this paper we propose a matrix minimization algorithm for linear time invariant implicit e a b c descriptions of the type ex t ax t bu t y t cx t given the initial matrices e a b c we obtain an externally equivalent minimal realization em am bm cm after applying a matricial procedure based on maximal row and column compressions

directed graphs 2d state models and characteristic polynomials of irreducible matrix pairs abstract the definition and main properties of a 2d digraph namely a directed graph with two kinds of arcs are introduced under the assumption of strong connectedness the analysis of its paths and cycles is performed based on an integer matrix whose rows represent the compositions of all circuits and on the corresponding row module natural constraints on the composition of the paths connecting each pair of vertices lead to the definition of a 2d strongly connected digraph for a 2d digraph of this kind the set of vertices can be partitioned into disjoint 2d imprimitivity classes whose number and composition are strictly related to the structure of the row module irreducible matrix pairs i e pairs endowed with a 2d strongly connected digraph are subsequently discussed equivalent descriptions of irreducibility naturally extending those available for a single irreducible matrix are obtained these refer to the free evolution of the 2d state models described by the pairs and to their characteristic polynomials finally primitivity is viewed as a special case of irreducibility and completely characterized in terms of 2d digraphs characteristic polynomials and 2d system dynamics

invariants and canonical forms for systems structural and parametric identification the basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled with reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set then the set of polynomial input output models of the type p z y t q z u t and the equivalence relation induced by the premultiplication of p and q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced since the two sets of canonical forms share common sets of complete independent invariants very simple algebraical links between state space and input output canonical forms can be deduced the previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input output sequences generated by a linear discrete time invariant multivariable system the results obtained in the identification of a real process are then reported

when is a storage function a state function in discrete time the purpose of this paper is to investigate when a storage function is a state function in discrete time as shown by trentelman and willems systems control lett 32 1997 pp 249 259 siam j control optim 36 1998 pp 1703 1749 every storage function is a state function in continuous time at first glance the same claim seems to hold also in discrete time contrary to this expectation this is not true in general in fact the discrete time counterpart involves not only some different but also some more difficult issues compared with the continuous time case this paper addresses these issues exactly and shows that every nonnegative storage function is a state function of a supply rate with a linear time invariant dynamical system in discrete time

survey paper structured low rank approximation and its applications fitting data by a bounded complexity linear model is equivalent to low rank approximation of a matrix constructed from the data the data matrix being hankel structured is equivalent to the existence of a linear time invariant system that fits the data and the rank constraint is related to a bound on the model complexity in the special case of fitting by a static model the data matrix and its low rank approximation are unstructured we outline applications in system theory approximate realization model reduction output error and errors in variables identification signal processing harmonic retrieval sum of damped exponentials and finite impulse response modeling and computer algebra approximate common divisor algorithms based on heuristics and local optimization methods are presented generalizations of the low rank approximation problem result from different approximation criteria e g weighted norm and constraints on the data matrix e g nonnegativity related problems are rank minimization and structured pseudospectra

duality in polynomial models with some applications to geometric control theory duality is studied in the context of polynomial models for linear systems the output injection group the dual of the feedback group is studied and a polynomial characterization of c a invariant subspaces as well as of the maximal reachability subspaces contained in ker c is given

matrix pairs in two dimensional systems an approach based on trace series and hankel matrices two dimensional system dynamics depends on matrix pairs that represent the shift operators along coordinate axes the structure of a matrix pair is analysed according to its characteristic polynomial and to the traces of suitable matrices in the algebra generated by the elements of the pair necessary and sufficient conditions for properties l and p are provided by resorting to hankel matrix theory finite memory and separable systems as well as two dimensional systems whose characteristic polynomials exhibit one dimensional structures are finally characterized in terms of spectral properties and traces

discrete time average positivity and spectral factorization in a behavioral framework abstract quite recently willems and trentelman siam j control opt 36 1998 1703 1749 have proposed quadratic differential forms for system analysis and synthesis from the behavioral point of view in this paper we develop their discrete time version called quadratic difference forms and some related fundamental concepts with respect to dissipativeness for a dynamical system which interacts with its environment along square summable trajectories moreover we provide an algorithm for spectral factorization of polynomial matrices as one example of quadratic difference forms a numerical example will be given to show the validity of our result

the equivalence structure of descriptor representations of systems with possibly inconsistent initial conditions abstract necessary and sufficient conditions for minimality of descriptor representations of impulsive smooth behaviors are derived we obtain a complete set of transformations by which minimal descriptor representations that give rise to the same behavior can be transformed into each other in particular this leads to a jump behavioral interpretation of the notion of strong equivalence of descriptor representations

realization and partial fractions abstract we discuss the relation between two intrinsically different proposals that have been made in the literature concerning the representation by constant matrices of rational matrices given in fractional form it turns out that the relation is most naturally studied in the framework of partial fraction decompositions we develop the realization theory for decompositions with respect to arbitrary complementary parts of the extended complex plane which may for instance correspond to stability and instability an isomorphism is obtained which connects the spaces used in the two methods and several identities relating to the mcmillan degree are derived in a direct way finally a new computational procedure is given to obtain the partial fraction decomposition of a rational matrix given in fractional form

autonomous subbehaviours and output nulling subspaces the aim of the present paper is to point out some connections between geometric control theory and the theory of behaviours specifically given a behaviour we analyse the set of all autonomous subbehaviours and relate those to some polynomial matrix completion problems we use this to analyse the different module structures on some vectorial polynomial spaces subsequently we apply this analysis to the study and the polynomial characterization of output nulling subspaces controlled invariant subspaces and the natural feedback induced module structures in these spaces toeplitz operators and wiener hopf factorizations play an important role