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

stability of switched linear differential systems we study the stability of switched systems whose dynamic modes are described by systems of higher order linear differential equations not necessarily sharing the same state space concatenability of trajectories at the switching instants is specified by gluing conditions i e algebraic conditions on the trajectories and their derivatives at the switching instants we provide sufficient conditions for stability based on lmis for systems with general gluing conditions we also analyse the role of positive realness in providing sufficient polynomial algebraic conditions for stability of two modes switched systems with special gluing conditions

generalised autoregressive models and their trajectories the set of classical ar models is not compact so that a sequence of ar models may fail to converge to anything reasonable a compactification leads to ar models with arbitrary singularity at infinity these limit ar models are represented by pairs of polynomial and proper rational matrices satisfying certain compatibility condition it is shown that they produce trajectories in a very natural way then the classical equivalence theorem is extended to them stating that two ar models having the same behaviour are equivalent

testing nonlinear regression parameters under heteroscedastic normally distributed errors summary likelihood ratio tests for parameters estimated assuming normally distributed errors are examined under a variety of homoscedastic and heteroscedastic variance assumptions it is assumed that yij the jth observation from the ith population is distributed as n u xij wbi ai a xij oi 2 by varying a and oi this model contains the most obvious examples of homoscedastic and heteroscedastic variability under this model the null hypothesis of equal jbi s can easily be tested using the likelihood ratio criterion also considered is the two sample chi square statistic x u s u where u is the difference vector of nonlinear least squares parameter estimates and s is an estimate of the covariance matrix of u monte carlo simulation using the von bertalanffy growth curve as an example is used to evaluate several test statistics for type i error rates under different sampling assumptions

realization by inspection we investigate which first order representations can be obtained from high order representations of linear systems by inspection that is just by rearrangement of the data under quite weak conditions it is possible to obtain minimal realizations in the so called pencil form under stronger conditions one can obtain minimal realizations in standard state space form by inspection the development is based on a reformulation of the realization problem as a problem of finding a complete set of basis vectors for the nullspace of a given constant matrix since no numerical computation is needed the realization method in particular is suitable for situations in which some of the coefficients are symbolic rather than numerical

a controllability test for general first order representations we derive a new controllability rank test for general first order representations the criterion generalizes the well known controllability rank test for linear input state systems as well as a controllability rank test by mertzios et al for descriptor systems

system structure and singular control the general continuous time linear quadratic control problem is considered it is shown that recently developed linear system theoretic properties and algorithms play an important role in solving this singular control problem

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

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

a general realization theory for higher order linear differential equations in this note we show that the geometric quotient under a natural group action of the generalized state space systems recently considered by schumacher kuijper and geerts is algebraically isomorphic to the space of a homogeneous autoregressive system this result essentially follows from work of stromme published earlier in the algebraic geometry literature in particular these generalized state space systems represent a realization of the space of homogeneous autoregressive systems

a comparison of several exact and approximate tests for structural shift under heteroscedasticity abstract it is known that the chow test for equality of regression coefficients is not robust to heteroscedasticity much recent attention has been paid to this problem by way of new tests the bulk of the new procedures are justified only asymptotically and little has been done to examine finite sample properties even less attention has been focused on comparisons of tests under either the null or alternative hypotheses in this paper we redress these shortcomings and also contribute by suggesting additional tests the results indicate that there are tests available with good sampling characteristics some of which are computationally easy to carry out from both the standpoint of the researcher s time and computation time

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

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

transformations of linear systems under external equivalence abstract we consider systems of linear differential and algebraic equations in which some of the variables are distinguished as external variables two systems are called equivalent if the set of solutions for the external variables is the same for both systems we give an operational form for this definition of equivalence i e we describe a set of system transformations having the property that two systems are equivalent if and only if they can be taken into each other by transformations from that set next an algorithm is described to transform a given system in general form to a system in minimal state space form this algorithm differs from existing methods in that it first takes the equations to first order form so that each subsequent step can be formulated and interpreted in state space terms we also compute the structure indices in terms of a state space description in nonminimal form and use this to prove the minimality of the end result of the algorithm finally an application is shown to the problem of ill posedness of feedback connections

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

a test for behavioral equivalence external equivalence of behavioral representations is investigated within an impulsive smooth framework as an attendant problem initial value consistency is also considered explicit necessary and sufficient conditions are derived applying the external equivalence result to conventional system representations reveals some interesting interpretations

brief nonlinear robust stabilization by conicity and qft techniques a technique is proposed to address the robust stability of a certain class of nonlinear systems it is based on conic confinement of the nonlinear part of the dynamics the tight nonconservative bounding cones are characterized and an algorithm is provided after that the nonlinearity is modelled as a complex uncertainty lying inside a small gain cone stability under additional arbitrary uncertainties in the linear part given as frequencial qft templates is addressed by robust qft techniques deriving conditions in the form of frequencial boundaries for the controller

smooth impulsive linear systems interconnections smooth impulsive sm imp linear systems are the smallest class containing all classical linear systems and stable with respect to limits in this article we show that they behave well under the operation of interconnection namely it turns out that the mcmillan degree of a regular interconnection is always equal to the sum of the mcmillan degrees of the components in other words all regular interconnections of sm imp linear systems automatically are regular feedback interconnections for this more general class of linear systems we also study the regular interconnection problem

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

welch s approximate solution for the behrens fisher problem the wald likelihood ratio and score statistics have been calculated for the behrens fisher problem these statistics provide asymptotically equivalent and weakly optimal tests the test based on the wald statistic was found to be very similar to the v test discussed by welch 1937 hence the v test was compared with the likelihood ratio and score tests in a power study since the powers were very similar and the v test was more convenient to use from several points of view we recommend this test sizes of the v test as well as a power comparison with the usual pooled t test were also obtained

smooth impulsive linear systems axiomatic description abstract paper concerns with linear systems having arbitrary singularity at infinity these systems deserve to study since they are limits of familiar classical linear systems an axiomatic description of such systems is obtained

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

box type approximations in nonparametric factorial designs abstract linear rank statistics in nonparametric factorial designs are asymptotically normal and in general heteroscedastic in a comprehensive simulation study the asymptotic chi squared law of the corresponding quadratic forms is shown to be a rather poor approximation of the finite sample distribution motivated by this problem we propose simple finite sample size approximations for the distribution of quadratic forms in factorial designs under a normal heteroscedastic error structure these approximations are based on an f distribution with estimated degrees of freedom that generalizes ideas of patnaik and box simulation studies show that the nominal level is maintained with high accuracy and in most cases the power is comparable to the asymptotic maximin wald test data driven guidelines are given to select the most appropriate test procedure these ideas are finally transferred to nonparametric factorial designs where the same quadratic forms as in the parametric case are applied to the vector

realization of autoregressive equations in pencil and descriptor form a linear system described by autoregressive equations with a given input output structure cannot be transformed to standard state space form if the implied input output relation is nonproper instead a realization in descriptor form must be used in this paper it is shown how to obtain minimal descriptor realizations from autoregressive equations without separating finite and infinite frequencies and without going through a reduction process external equivalence is used so that even situations in which there is no transfer matrix can be considered the approach is based on the so called pencil representation of linear systems and it is shown that there is a natural realization of autoregressive equations in pencil form in this way the link between the realization theories of willems and fuhrmann can also be clarified

brief paper switched nonlinear differential algebraic equations solution theory lyapunov functions and stability we study switched nonlinear differential algebraic equations daes with respect to existence and nature of solutions as well as stability we utilize piecewise smooth distributions introduced in earlier work for linear switched daes to establish a solution framework for switched nonlinear daes in particular we allow induced jumps in the solutions to study stability we first generalize lyapunov s direct method to non switched daes and afterwards obtain lyapunov criteria for asymptotic stability of switched daes developing appropriate generalizations of the concepts of a common lyapunov function and multiple lyapunov functions for daes we derive sufficient conditions for asymptotic stability under arbitrary switching and under sufficiently slow average dwell time switching respectively

representability of impulsive input output behaviors in descriptor form abstract it is known that the descriptor representation of linear smooth behaviors imposes no constraints on the partitioning of external variables into inputs and outputs the issue is reconsidered here in the setting of impulsive behaviors in this context there are two different descriptor representations that may be used the conventional form and the unconventional one it is shown that the conventional descriptor form again does not constrain the choice of inputs and outputs but the unconventional form does the latter representation only allows input output partitionings in which inputs are dirac free this notion is newly defined here

hypothesis testing in linear models when the error covariance matrix is nonscalar a well developed exact statistical theory exists for hypothesis testing in the normal linear regression model when the errors are independent and homoscedastic in the more general case where the error covariance matrix is nonscalar and depends on a set of unknown parameters exact analysis is difficult and reliance is usually placed on asymptotic approximations for large sample size n in this paper higher order asymptotic expansions are developed for comparing the size and power of some common procedures for testing linear hypotheses on the regression coefficients in a class of generalized normal linear models the class investigated is essentially the same as in breusch 4 and magnus 6 and includes many of the examples of heteroscedasticity and autocorrelation discussed in the literature we assume simply that the regressors are nonrandom and that the error covariance matrix is a smooth function of a few parameters that can be efficiently estimated by maximum likelihood tests based on the wald likelihood ratio and lagrange multiplier principles are considered these principles lead to three tests which though distinct in finite samples are locally asymptotically equivalent and share certain asymptotic optimality properties of course there are infinitely many other tests that are asymptotically equivalent to the ones examined here although the techniques of this paper can be applied to any of them our results concern only the tests arising from the three traditional principles we show that to a second order of approximation under local alternatives the likelihood ratio test statistic is a simple average of the wald statistic and the lagrange multiplier statistic when the null hypothesis is one dimensional the three tests are to second order equally powerful that is after the critical regions are adjusted so that the tests have to order n i the same size the local power functions differ by terms of smaller order than n when the null hypothesis contains more than one

seminormality and local cohomology of toric face rings abstract we characterize the toric face rings that are normal respectively seminormal extending results about local cohomology of brun bruns ichim li and romer of seminormal monoid rings and stanley toric face rings we prove the vanishing of certain graded parts of local cohomology of seminormal toric face rings the combinatorial formula we obtain generalizes hochster s formula we also characterize all necessarily seminormal toric face rings that are f pure or f split over a field of characteristic p 0 an example is given to show that f injectivity does not behave well with respect to face projections of toric face rings finally it is shown that weakly f regular toric face rings are normal affine monoid rings

free resolutions of simplicial posets abstract a simplicial poset a poset with a minimal element and whose every interval is a boolean algebra is a generalization of a simplicial complex stanley defined a ring a p associated with a simplicial poset p that generalizes the face ring of a simplicial complex if v is the set of vertices of p then a p is a k v module we find the betti polynomials of a free resolution of a p and the local cohomology modules of a p generalizing hochster s corresponding results for simplicial complexes the proofs involve splitting certain chain or cochain complexes more finely than in the simplicial complex case corollaries are that the depth of a p is a topological invariant and that the depth may be computed in terms of the cohen macaulayness of skeleta of p generalizing results of munkres and hibi

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

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

impulsive smooth behavior in multimode systems part ii minimality and equivalence this is the second part of a two part study of linear multimode systems in the first part it was argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant and both first order and polynomial representations were introduced that satisfy this requirement here we determine the conditions under which first order representations are minimal we also show how two minimal representations of the same behavior are related this leads in particular to an appropriate state space isomorphism theorem the minimality conditions are given a dynamic interpretation

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

impulsive smooth behavior in multimode systems part i state space and polynomial representations abstract a switched or multimode system is one that can switch between various modes of operation we consider here switched systems in which the modes of operation are characterized as linear finite dimensional systems not necessarily all of the same mcmillan degree when a switch occurs from one of the modes to another of lower mcmillan degree the state space collapses and an impulse may result followed by a smooth evolution under the new regime this paper is concerned with the description of such impulsive smooth behavior on a typical interval we propose an algebraic framework modeled on the class of impulsive smooth distributions as defined by hautus both state space and polynomial representations are considered and we discuss transformations between the two forms

tests for equality between sets of coefficients in two linear regressions under heteroscedasticity abstract structural shift is a common problem in a relationship dealing with time series data chow 1960 developed a test to detect such a shift under the assumption that observations both before and after the shift have the same variance structural shifts however often accompany changes in variance as well and the chow test is not robust to such changes two relatively robust tests are proposed and are found to be highly powerful

quotient algebras of stanley reisner rings and local cohomology we study certain subcomplexes of an arbitrary simplicial complex such that hmi k hmi k for any 0 i dim k here hmi k is the ith local cohomology module of the stanley reisner ring k of over a field k our technique is an elegant approach to one of the most generalized versions of the rank selection theorems of j munkres 1984 michigan math j 31 113 128 theorem 6 4 and r stanley 1979 trans amer math soc 249 139 157 theorem 4 3

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

output feedback pole placement with dynamic compensators derives a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most q by using this rank condition the authors establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system the authors proofs also come with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed loop poles

an approximate test for comparing heteroscedastic regression models abstract this article addresses the problem of testing whether the vectors of regression coefficients are equal for two independent regression models when the error variances are unequal the usual chow statistic appropriate when equality of variances can be assumed is modified by replacing the pooled residual variance in the denominator with a weighted average of the residual variances from each data set the weights are functions of the mean of the eigenvalues of w x 1 x 1 x 1 x 1 x 2 x 2 1 both numerator and denominator are then approximated by scalar multiples of chi squared distributions the parameters of these approximated distributions are chosen to equate their first two moments to those of the exact distribution the resulting approximation for the modified chow statistic c is an f distribution with degrees of freedom that depend on the two sample sizes the number of regressor variables the average eigenvalue of w and the true ratio of error variances since the latter is unknow