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

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

the asymptotics of a bessel kernel determinant which arises in random matrix theory in random matrix theory the local correlations of the laguerre and jacobi unitary ensemble in the hard edge scaling limit can be described in terms of the bessel kernel b x y xyj x yj y j y xj x x2 y2 x y 0 1 in particular the so called hard edge gap probabilities p r can be expressed as the fredholm determinants of the corresponding integral operator b restricted to the finite interval 0 r using operator theoretic methods we are going to compute their asymptotics as r i e we show that p r det i b l2 0 r exp r24 r 22logr g 1 2 2 where g stands for the barnes g function in fact this asymptotic formula will be proved for all complex parameters satisfying re 1

universal parametric correlations of eigenvalues of random matrix ensembles eigenvalue correlations of random matrix ensembles as a function of an external parametric perturbation are investigated via the dyson brownian motion model in the situation where the level density has a hard edge singularity by solving a linearized hydrodynamical equation a universal dependence of the density density correlator on the external field is found as an application we obtain a formula for the variance of linear statistics with the parametric dependence exhibited as a laplace transform

skew prime polynomial matrices the polynomial model approach abstract we examine the concept of skew primeness of polynomial matrices in terms of the associated polynomial model it is shown that skew primeness can be characterized in terms of the property of decomposition of a vector space relative to an endomorphism this basic result is then applied to the special case of nonsingular polynomial matrices we investigate the nonuniqueness of skew complements of a skew prime pair it is shown that the space of equivalence classes of skew complements is in bijective correspondence with a finite dimensional linear space finally the equivalence of the solutions to the problem of output regulation with internal stability obtained via geometric methods and via polynomial matrix techniques is shown explicitly

new results on the algebraic theory of linear systems the solution of the cover problems abstract we give the complete solution of a large class of problems in linear system theory the so called cover problems these problems are formulated and solved both in the state space and in the input output frameworks the key concept which allows the effective parametrization of all solutions of the cover problems is that of the partial realizations of a sequence of matrices it is shown that the solutions of the state space cover problems can be expressed as state spaces of the partial realizations of appropriately defined sequences of matrices and the solution of the input output cover problems can be expressed as a simple function of the functions of the partial realizations of the sequences of mentioned above

towards a compactification of the set of conditioned invariant subspaces a compactification of the set of conditioned invariant subspaces of fixed dimension for an observable pair c a is proposed it contains the almost conditioned invariant subspaces of the same dimension in certain cases the compactification is shown to be smooth and a complete geometric description is given in the case of a single output system

a behavioral approach to classic polynomial matrix descriptions abstract in this note we aim to show how the well known definitions and results obtained for the classic polynomial matrix descriptions pmds of a dynamic system kailath linear systems 1980 rosenbrock state space and multivariable theory 1970 wolovich linear multivariable systems 1974 can be seen as particular instances of the analogous definitions and results obtained within the more general behavioral setting more precisely pmds of dynamic systems correspond to a class of linear time invariant dynamic systems that we will call extended state space models for these models internal stability as well as the structural properties can be defined and fully characterized by applying the definitions and criteria introduced within the behavioral approach as a result we will obtain the main results of the pmd s approach also the control problem and the observer design problem are here stated in behavioral terms and solved finally the introduction of the dual system of an extended state space model completes the analysis

on observability subspaces the paper presents an in depth study of topics in geometric control pertaining to observer theory from a functional point of view we give characterizations of several classes of subspaces including observability almost observability and reconstructibility subspaces we solve completely the problem of spectral assignability for observer dynamics by generalizing rosenbrock s pole placement theorem these results are then applied to observer theory

tensored polynomial models the theme of the present paper is the study of two different versions of tensor products of functional models and present some applications to various problems related to system theory among those are stability of higher order systems tangent spaces spaces of intertwining maps invariant factors of tensor products of linear transformations and the solvability of sylvester equations

a polynomial characterization of mathcal a mathcal b invariant and reachability subspaces based on the state space model of p fuhrmann a link is laid between the geometric approach to linear system theory as developed by w m wonham and a s morse and the approach based on polynomial matrices in particular polynomial characterizations of a b invariant and reachability subspaces are given these characterizations are used to prove the equivalence of the disturbance decoupling problem and the exact model matching problem and also to connect the polynomial matrix and the geometric approach to the construction of observers finally constructive procedures and conditions are given for computing the supremal a b invariant subspace and reachability space and for checking the solvability of the exact model matching problem

indefinite inner product spaces associated with hermitian polynomial matrices abstract nonsingular hermitian polynomial matrices are completely characterized by bases and shift operators of suitably defined inner product spaces

on the state of behaviors the theme of the present paper is the study of the concept of state and the corresponding state maps in the context of willems behavioral theory we concentrate on markovian system and their representation in terms of first order difference or differential systems we follow by a full analysis of the special case of state systems the embedding of a linear system in a state system via the use of state maps arriving at state representations or equivalently to a realization theory for behaviors minimality is defined and characterized and a state space isomorphism theorem is established realization procedures based on the shift realization are developed as well as a rigorous analysis of the construction of state maps the paper owes much to rapisarda and willems p rapisarda j c willems state maps for linear systems siam j contr optim 35 1997 1053 1091

gaussian fluctuation for the number of particles in airy bessel sine and other determinantal random point fields we prove the central limit theorem clt for the number of eigenvalues near the spectrum edge for certain hermitian ensembles of random matrices to derive our results we use a general theorem essentially due to costin and lebowitz concerning the gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions as another corollary of the costin lebowitz theorem we prove the clt for the empirical distribution function of the eigenvalues of random matrices from classical compact groups

on the causal factorization problem in this note the formalism of the infinite zero module and of the infinite pole module is used to analyze the causal factorization problem this allows us to express its solutions in terms of zeros and poles at infinity and to have a better understanding of both the problem and the techniques previously proposed by various authors to solve it moreover we obtain a characterization in terms of the existence of certain causal factorizations of the transfer functions having no infinite zeros

distribution functions for random variables for ensembles of positive hermitian matrices distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive hermitian matrices the inverse fourier transform of the distribution is shown to be a fredholm determinant of a certain operator that is an analogue of a wiener hopf operator the asymptotic formula shows that up to the terms of order o 1 the distributions are gaussian

functional models in linear algebra abstract this paper examines the role of functional models in linear algebra and some related areas of application notably linear systems theory this approach adopts an intermidiate level of abstraction somewhere between abstract module theory on the one hand and matrix theory on the other it is a powerful method which simplifies many proofs provides insights and illuminates the less obvious connections between linear algebra and operator theory approximation theory moment problems interpolation and control theory

duality in time varying linear systems a module theoretic approach abstract using elementary module theory an intrinsic definition of the dual or adjoint of a generalized time varying linear system is given with this the duality of controllability and observability is recovered from their intrinsic module theoretical definitions the duality of state feedback and output injection is discussed both in the static case and for its quasistatic generalization related brunovsky type canonical forms are derived in the quasistatic case the corresponding controllability indices and their duals the observability indices are defined intrinsically

parametrization of c a invariant subspaces in this paper we classify and parametrize the c a invariant subspaces of a given observable pair c a for that purpose we associate with every c a invariant subspace v a k s module m v of laurent series in s 1 and parametrize these modules via their uniquely determined kronecker hermite bases the discrete and continuous parameters thus obtained are interpreted in slate space terms finally we illustrate the results by presenting a complete overview over the set of all c a invariant subspaces for a given pair c a 6 3 6 in dual brunovsky canonical form

the distribution and moments of the smallest eigenvalue of a random matrix of wishart type abstract given a random rectangular m n matrix with elements from a normal distribution what is the distribution of the smallest singular value to pose an equivalent question in the language of multivariate statistics what is the distribution of the smallest eigenvalue of a matrix from the central wishart distribution in the null case we give new results giving the distribution as a simple recursion this includes the more difficult case when n m is an even integer without resorting to zonal polynomials and hypergeometric functions of matrix arguments with the recursion one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials exponentials and at worst ordinary hypergeometric functions we further elaborate on the special cases when n m 0 1 2 and 3 and give a numerical table of the expected values for 2 m 25 and 0 n m 25

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

regulator synthesis using c a b pairs the concept of a c a b pair of subspaces was introduced in an earlier paper by the author to study problems of compensator design here it is shown that the same concept plays a basic role in the theory of tracking and regulation from a few fundamental properties the main results of regulator theory are derived in a relatively simple and straightforward manner another new feature is that we are able to allow the presence of an unknown input thereby unifying the fields of tracking and regulation and disturbance decoupling all results are fully constructive and an example is given to illustrate the theory

a new approach to fast polynomial interpolation and multipoint evaluation abstract the fastest known algorithms for the problems of polynomial evaluation and multipoint interpolation are devastatingly unstable numerically because of their recursive use of polynomial divisions we apply a completely distinct approach to compute approximate solutions to both problems equally fast but with improved numerical stability our approach relies on new techniques so far not used in this area we reduce the problems to vandermonde matrix computations and then exploit some recent methods for improving computations with structured matrices

on hamiltonian rational transfer functions abstract a sequel to a previous paper fuhrmann 1983 this paper contains a detailed study of rational transfer functions having hamiltonian symmetry a detailed realization procedure is described that reflects this symmetry as a corollary the williamson canonical form for hamiltonian maps in symplectic spaces is derived

on bezoutians van der monde matrices and the lienard chipart stability criterion we show how the bezoutian can be reduced by congruence to diagonal form with the congruence given by a van der monde matrix this result is applied to obtain new proofs of some classical stability criteria and in particular of the lienardchipart theorem

asymptotics of level spacing distributions for random matrices asymptotic formulas for the probability of finding exactly n eigenvalues in an interval of length s for large s and fixed n are given for random matrices taken from the gaussian ensembles 1 2 4 these exact results are compared with the predictions of a continuum coulomb gas model due to dyson

on a hardy space approach to the analysis of spectral factors the paper presents a study of several problems related to spectral factorizations we assume only a very weak form of coercivity for the p x p spectral function phi and look at the set wm of all rectangular p x m spectral factors the main object is the arithmetization of the geometry of the set of minimal stable spectral factors by employing hardy space techniques and the arithmetic of inner functions particular attention is paid to the study of various partial orders associated with the set of spectral factors

disturbance decoupling by measurement feedback with stability or pole placement in this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system the problem is attacked using the geometric approach through the concepts of a bmod mathcal b invariant and controllability subspaces and their duals a mid k invariant and complementary observability subspaces the solution of this problem has an interesting structure consisting of a feedback processor which decomposes into i a disturbance decoupling loop ii a disturbance input stabilization or pole placement loop and iii a controlled output stabilization or pole placement loop

on the lyapunov equation coinvariant subspaces and some problems related to spectral factorizations a geometric approach to stochastic realization theory and hence to spectral factorization problems has been developed by lindquist and picci 1985 1991 and lindquist et al 1995 most of this work was done abstractly fuhrmann and gombani 1998 adopted an entirely hardy space approach to this set of problems studying the set of rectangular spectral factors of given size for a weakly coercive spectral function the parametrization of spectral factors in terms of factorizations of related inner functions as developed in fuhrmann 1995 had to be generalized this led to a further understanding of the partial order introduced by lindquist and picci in the set of stable spectral factors in the present paper we study the geometry of finite dimensional coinvariant subspaces of a vectorial hardy space h 2 via realization theory emphasizing the role of the lyapunov equation in lifting the hardy space metric to the statespace domain we follow this by deriving state space formulas for rectangular spect

on the zeros of a minimal realization in an earlier work the authors have introduced a coordinate free module theoretic definition of zeros for the transfer function g s of a linear multivariable system a b c the first contribution of this paper is the construction of an explicit k z module isomorphism from that zero module z g to v r where v is the supremal a b invariant subspace contained in kerc and r is the supremal a b controllable subspace contained in kerc and where a b c constitutes a minimal realization of g s the isomorphism is developed from an exact commutative diagram of k vector spaces the second contribution is the introduction of a zero signal generator and the establishment of a relation between this generator and the classic notion of blocked signal transmissions

a study of behaviors abstract the paper presents the results of the study of behavior theory as developed by j c willems from the point of view of polynomial and rational models considering behaviors in the discrete time case to be generalizations of rational models a natural focal point becomes the concept of a behavior homomorphism we give a characterization of behavior homomorphisms and analyze their invertibility properties in terms of embeddings in unimodular polynomial matrices these results which are of intrinsic interest are then applied to the uniform derivation of a large number of results for equivalence in different classes of behavior representations to a certain extent these are generalizations of the strict system equivalence concept for the class of polynomial matrix description of systems in the style of rosenbrock a study of behavioral controllability is undertaken and gives some new insights into connections with geometric control theory

remarks on the inversion of hankel matrices the paper continues the investigation into the links between algebraic system theory more specifically the partial realization problem and the problem of hankel matrix inversion the representation of the inverse of a hankel matrix as a bezoutiant and the bezout equation play a principal role

on a connection between spectral factorization and geometric control theory we investigate here how the geometric control theory of basile marro and wonham can be obtained in a hilbert space context as the byproduct of the factorization of a spectral density with no zeros on the imaginary axis we show how controlled invariant subspaces can be obtained as images of orthogonal projections of co invariant subspaces onto a semi invariant markovian subspace of the hardy space of square integrable functions analytic in the right half plane output nulling subspaces are then related to a particular spectral factorization problem a similar construction is presented for controllability subspaces and a new algorithm for the computation of these subspaces is presented

a functional approach to the stein equation abstract this paper applies the theory of tensor products of functional models to the study of the stein equation we analyse the connections between the sylvester and stein equations as well as those between the anderson jury bezoutian associated with the polynomial sylvester equation and the t bezoutian associated with the homogeneous polynomial stein equation the functional setting allows us to extend the theory of finite section toeplitz matrices and their inversion

quasi symmetries of determinantal point processes the main result of this paper is that determinantal point processes on mathbb r corresponding to projection operators with integrable kernels are quasi invariant in the continuous case under the group of diffeomorphisms with compact support theorem 1 4 in the discrete case under the group of all finite permutations of the phase space theorem 1 6 the radon nikodym derivative is computed explicitly and is given by a regularized multiplicative functional theorem 1 4 applies in particular to the sine process as well as to determinantal point processes with the bessel and the airy kernels theorem 1 6 to the discrete sine process and the gamma kernel process the paper answers a question of grigori olshanski

generalized polynomial bezoutian with respect to a jacobson chain basis over an arbitrary field abstract we introduce a so called generalized polynomial bezoutian with respect to a jacobson chain basis over an arbitrary field some characterization of this kind of matrix such as the barnett type factorization and the intertwining relation with generalized hypercompanion matrix are obtained the diagonal reduction formula via the generalized confluent vandermonde matrix similar to that of classical bezoutian is presented the method used is based on polynomial module and operator representation

tangent spaces of rational matrix functions we consider the task of determining tangent spaces for classes of rational matrix valued functions our analysis is based on methods from control theory and in particular the theory of polynomial models explicit descriptions of tangent spaces of rational transfer functions stable rational transfer functions rational inner functions and symmetric rational transfer functions are obtained moreover a new proof of delchamps s decomposition formula for the tangent bundle of rational transfer functions is given a riemannian metric as well as a symplectic structure is defined 0 1998 elsevier science inc

an algebraic approach to the partial realization problem abstract the partial realization problem is treated algebraically using polynomial models the main idea is the kernels of certain projections of the hankel map which give rise to orthogonal direct sum decomposition of the state space minimality of partial realizations is based on minimal elements of these subspaces which arise as a natural consequence of the structural properties

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

polynomial characterizations of h f invariant subspaces with applications abstract a direct study of h f invariant subspaces associated with the polynomial fractional system model z pq 1r w is undertaken the earlier results on the polynomial characterization of these subspaces are extended to cover the general case for full rank transfer matrices the smallest unobservability subspace containing im g and the largest reachability subspace in ker h are described in terms of factors of the polynomial system matrix a new polynomial model and its superspaces play the central role in all these characterizations the results obtained are applied to two measurement feedback problems in linear control theory yielding geometric interpretations for two rational matrix equations

a homeomorphism between observable pairs and conditioned invariant subspaces abstract a bijective correspondence between similarity classes of observable systems c a and n codimensional conditioned invariant subspaces of a pair c a is constructed that leads to a homeomorphism of the spaces this is applied to the parametrization of inner functions of fixed mcmillan degree proofs using state space methods as well as using polynomial models are given

on the parametrization of conditioned invariant subspaces and observer theory the present paper is an in depth analysis of the set of conditioned invariant subspaces of a given observable pair c a we do this analysis in two different ways one based on polynomial models starting with a characterization obtained in p a fuhrmann linear operators and systems in hilbert space 1981 ieee trans automat control ac 26 1981 284 the other being a state space approach toeplitz operators projections in polynomial and rational models wiener hopf factorizations and factorization indices all appear and are tools in the characterizations we single out an important subclass of conditioned invariant subspaces namely the tight ones which already made an appearance in p a fuhrmann u helmke systems control lett 30 1997 217 a precursor of the present paper of particular importance for the study of the parametrization of the set of conditioned invariant subspaces of an observable pair c a is the structural map that associates with any reachable pair with only the input dimension constrained a uniquely determined conditioned invariant subspace the construction of this map uses polynomial models and the shift realization new objects the partial observability and reachability matrices are introduced which are needed for the state space characterizations kernel and image representations for conditioned invariant subspaces are derived uniqueness of a kernel representation of a conditioned invariant subspace is shown to be equivalent to tightness we pass on to an analysis and derivation of the kronecker hermite canonical form for full column rank rectangular polynomial matrices this extends the work of a e eckberg a characterization of linear systems via polynomial matrices and module theory ph d thesis mit cambridge ma 1974 g d forney siam j control optim 13 1973 493 and d hinrichsen h f munzner and d pratzel wolters systems

a polynomial approach to hankel norm and balanced approximations abstract a new unified approach to problems of hankel norm and balanced approximations is presented which is based on a combination of polynomial algebra and the geometry of invariant subspaces contrary to state space methods where contact with external properties of systems is indirect the approach presented yields new insights into basic properties of hankel norm approximation and balanced realizations several approximation results are interpreted geometrically in terms of projections also duality results in this area are printed out

generalized polynomial bases and the bezoutian abstract a foundation polynomial is used to induce polynomial bases for fn 1 x the vector space of polynomials of degree less than n over an arbitrary field f the associated bases are then used to block diagonalize the bezout matrix of two polynomials under congruence

minimal factorizations of rational matrix functions in terms of polynomial models the problem of minimal factorization of rational bicausal matrices is considered using polynomial models the factors are given explicitly in terms of polynomial matrices

the algebraic riccati equation a polynomial approach abstract polynomial models are used to give a unified approach to the problem of classifying the set of all real symmetric solutions of the algebraic riccati equation

hankel operators that commute with second order differential operators abstract suppose that is a continuous and self adjoint hankel operator on l 2 0 with kernel x y and that l f d d x a x d f d x b x f x with a 0 0 if a and b are both quadratic hyperbolic or trigonometric functions and satisfies a suitable form of gauss s hypergeometric differential equation or the confluent hypergeometric equation then l l the paper catalogues the commuting pairs and l including important cases in random matrix theory there are also results proving rapid decay of the singular numbers of hankel integral operators with kernels that are analytic and of exponential decay in the right half plane

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

integrable operators and the squares of hankel operators integrable operators arise in random matrix theory where they describe the asymptotic eigenvalue distribution of large self adjoint random matrices from the generalized unitary ensembles this paper gives sufficient conditions for an integrable operator to be the square of a hankel operator and applies the condition to the airy associated laguerre modified bessel and whittaker functions

a note on continuous behavior homomorphisms the purpose of this note is the characterization of the class of behavior homomorphisms that have nice polynomial representations the key to this is a judicious application of duality theory together with additional continuity requirements

observers for linear time varying systems we give characterizations and necessary and sufficient existence conditions for tracking and asymptotic observers for linear functions of the state of a linear finite dimensional time varying state space system we specialize the results to affine parameter varying systems and bilinear control systems

a new realization theory a realization theory that is based on the kalman system theory and motivated by the fuhrmann realization theory is presented in the fuhrmann realization theory the realization state space is defined by a natural polynomial module k sub q which is related to a nonsingular polynomial matrix q the module k sub q is formed of all f where q sup 1 f is strictly proper in the realization theory proposed the realization state space is defined by a different module m sub q which is formed of properly truncated strictly proper formal power series of q sup 1 f the realization theory can be used to prove the dual realization scheme induced by the fuhrmann realization theory