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

descriptor representations without direct feedthrough term abstract descriptor representations are considered that are given by e a b c d with d 0 minimality under external equivalence is characterized in terms of the matrices e a b and c also transformations are given by which minimal e a b c representations are related under external equivalence the transformations turn out to be more simple than in the d 0 case algorithms for rewriting an e a b c d representation in e a b c form are also given finally a realization procedure is presented for obtaining a minimal e a b c representation for a system that is given in polynomial matrix fractional form

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

feedback and pole placement in descriptor variable systems the effects and uses of applying linear feedback to continuous time descriptor systems are studied structural changes resulting from feeding back the slow and fast parts of the trajectory separately are characterized it is shown that under certain conditions related to controllability the poles of the slow subsystem may be shifted arbitrarily and the impulsive behaviour of the fast subsystem may be eliminated

generalized state space realizations of non proper rational transfer functions in this paper we propose an algebraic realization theory for polynomial transfer functions based on classical module theory realizations are given by linear systems whose generalized state space has a natural module structure over the ring of formal power series k z 1 existence and uniqueness of minimal realizations are proved and their properties are described the dimension of the minimal realization is called generalized degree of the polynomial transfer function and it is proved to be equal to the rank of an associated generalized hankel matrix finally a correspondence between realizations and fractional representations over k z 1 of polynomial matrices is established and some properties of such representations are investigated

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

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

bifurcations in a delayed fractional complex valued neural network complex valued neural networks cvnns with integer order have attracted much attention and which have been well discussed fractional complex valued neural networks fcvnns are more suitable to describe the dynamical properties of neural networks but have rarely been studied it is the first time that the stability and bifurcation of a class of delayed fcvnn is investigated in this paper the activation function can be expressed by separating into its real and imaginary parts by using time delay as the bifurcation parameter the dynamical behaviors that including local asymptotical stability and hopf bifurcation are discussed the conditions of emergence of bifurcation are obtained furthermore it reveals that the onset of the bifurcation point can be delayed as the order increases finally an illustrative example is provided to verify the correctness of the obtained theoretical results

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

first order representations of fliess models abstract fuhrmann s state space construction in its generalized form is used to obtain a general theory of first order representations of fliess models defined over an arbitrary noetherian commutative ring the case of arbitrary linear delay differential equations is involved

boundary value descriptor systems well posedness reachability and observability in this paper we introduce the class of two point boundary value descriptor systems tpbvds discrete time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system state at the two endpoints of the system s interval of definition by introducing a standard form for regular pencils we obtain a new and simple generalized cayley hamilton theorem that simplifies our investigation of well posedness green s function solutions and reachability and observability for tpbvds there are two distinct notions of reachability and observability that one can define for tpbvds associated with processes that propagate inward from and outward toward the boundaries we investigate each of these in detail obtaining among other things far simpler forms for the reachability and observability results than found previously in the literature in addition we describe several methods for the efficient solution of tpbvds one involving recursions from each end of t

multistability and multiperiodicity of delayed bidirectional associative memory neural networks with discontinuous activation functions abstract in this paper the multistability and multiperiodicity issues are discussed for 2 n dimensional delayed bidirectional associative memory neural networks with r level discontinuous activation functions sufficient conditions are established to ensure the existence of r n locally exponentially stable equilibria as an extension of multistability conditions are established to ensure the existence of r n locally exponentially stable periodic solutions when external inputs are periodic finally two examples are given to illustrate the effectiveness of the obtained results

canonical forms for descriptor systems under restricted system equivalence abstract the problem of canonical forms of implicit descriptor systems under restricted system equivalence is considered it is shown that the canonical form is directly related to the echelon form of the autoregressive matrix inducing the external behavior of the system the proposed canonical form is a generalization of the popov form and is also valid for singular systems under transfer equivalence

poles and zeros of matrices of rational functions abstract this expository paper considers the problem of defining poles and zeros with multiplicity including those at infinity for a matrix of rational functions over a field the underlying theme question for the paper is does the number of poles of a matrix equal the number of zeros and what structural meaning underlies such an assertion our approach is motivated by ideas from linear system theory and control engineering control theoretic ideas are not a prerequisite and we hope to encourage specialists in classical linear and commutative algebra to investigate this promising source of new algebra problems

simultaneous state and input reachability for linear time invariant systems abstract in this paper we give an explicit solution to the behavioral reachability problem for linear time invariant systems which amounts to finding an explicit control law that reaches a given final input state pair u 1 x 1 in a given finite time t 1 we first tackle the case of state space realizations and we then extend the obtained results to the case of implicit realizations for this we use the geometric approach and some results of the viability theory some complements are given about the existing relationships between reachability and pole placement as well as some notions of unicity and existence of solution

homological identities for closed orbits the classical lefschetz formula expresses a relationship between the fixed points of certain transformations and homological invariants when applied to continuous systems one obtains the hopf formula relating the singularities of a flow to the euler characteristic of the underlying manifold the main goal of this paper is to prove an analogous formula that relates the closed orbits of a stable flow to a homological invariant of the manifold there are serious obstruction to finding such a formula for flows the first is that there are flows on any closed manifold of dimension 4 of zero euler characteristic with no periodic orbits whatsoever 27 these flows however are unstable by pugh s c a closing lemma 19 so it is natural to restrict attention to flows that satisfy axiom a and the no cycles property since smale has shown that these flows are o stable 23 secondly there are gradient flows on any closed manifold that are axiom a no cycles and have no closed orbits 24 so it is natural to study only nonsingular flows the possibility that a lefschetz type formula for such flows existed was suggested to us by r bowen even nonsingular morse smale flows however are difficult to deal with since asimov s geometric index is not an invariant of the nonsingular homotopy class 2 we must further restrict the class of flows to assume that the closed orbits relate in some manner to the homology of the underlying manifold we define in sect 2 the class of homology proper flows such a flow has the property that closed orbits of large period lie in large homology classes in sect 7 we show that a homology proper axiom a no cycles flow does satisfy a lefschetz type formula that relates invariants of the closed orbits to a manifold invariant this formula is closely related to franks study of smale flows on s 3 in which the linking numbers of closed orbits are used to compute the alexander polynomial of a certain link indeed the manifold invariant used in sect 7 is a close analogue of an alexander polynomial that we call the alexander

mittag leffler synchronization of delayed fractional order bidirectional associative memory neural networks with discontinuous activations state feedback control and impulsive control schemes this paper is concerned with the drive response synchronization for a class of fractional order bidirectional associative memory neural networks with time delays as well as in the presence of disc

rationality for isolated expansive sets abstract an expansive isolated set of a diffeomorphism with nondegenerate periodic points has a rational zeta function

generalized state space realizations from matrix fraction descriptions the problem of the realization of a matrix fraction description mfd of a nonproper transfer function is considered an algorithm for the realization is developed and it is shown that it is a natural generalization of wolovich s method 1973 for the case of proper systems the resulting generalized state space system is shown to be reachable and the conditions for its minimality in terms of the coprimeness and column reducedness of the mfd are derived the relation between the controllability indexes of the realization and the forney order 1975 of the mfd is established

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 generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

geometric minimization under external equivalence for implicit descriptions we consider minimality under external equivalence for linear time invariant implicit e a b c descriptions of the type ex t ax t bu t y t cx t necessary and sufficient conditions for external minimality have been previously given from an algebraic point of view by kuijper we propose here alternative geometric characterizations as well as a geometric reduction procedure that extracts from a given e a b c description a minimal model of the same type we also geometrically characterize in terms of the geometry of e a b c the corresponding minimal dimensions for the state space and for the state equation space

global exponential stability and global convergence in finite time of delayed neural networks with infinite gain this paper introduces a general class of neural networks with arbitrary constant delays in the neuron interconnections and neuron activations belonging to the set of discontinuous monotone increasing and possibly unbounded functions the discontinuities in the activations are an ideal model of the situation where the gain of the neuron amplifiers is very high and tends to infinity while the delay accounts for the finite switching speed of the neuron amplifiers or the finite signal propagation speed it is known that the delay in combination with high gain nonlinearities is a particularly harmful source of potential instability the goal of this paper is to single out a subclass of the considered discontinuous neural networks for which stability is instead insensitive to the presence of a delay more precisely conditions are given under which there is a unique equilibrium point of the neural network which is globally exponentially stable for the states with a known convergence rate the conditions are easily testable and independent of the delay moreover global convergence in finite time of the state and output is investigated in doing so new interesting dynamical phenomena are highlighted with respect to the case without delay which make the study of convergence in finite time significantly more difficult the obtained results extend previous work on global stability of delayed neural networks with lipschitz continuous neuron activations and neural networks with discontinuous neuron activations but without delays

homomorphisms of modules associated with polynomial matrices with infinite elementary divisors abstract if the inverse of a nonsingular polynomial matrix l has a polynomial part then one can associate with l a module over the ring of proper rational functions which is related to the structure of l at infinity in this paper we characterize homomorphisms of such modules

torsion and closed geodesics on complex hyperbolic manifolds letx be a compact complex manifold covered by complex hyperbolicn space with the induced metric each stable horocycle has a cocomplex structure preserved by the geodesic flow to a closed geodesic one can thus associate a piece of the poincare map with a holomorphic fixed point the resulting atiyah bott fixed point indices together with the length and multiplicity of as a periodic orbit determine the contribution of to certain zeta functionsrp z 0 p n from the leading coefficient ofr p atz 0 and the hodge numbersh ij x we calculate the ray singer bar partial torsiont p x this indicates that the known connections between torsion and the dynamical features of closed orbits continue to hold in the holomorphic category

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

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

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

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

the fried conjecture in small dimensions we study the twisted ruelle zeta function x s for smooth anosov vector fields x acting on flat vector bundles over smooth compact manifolds in dimension 3 we prove fried conjecture relating reidemeister torsion and x 0 in higher dimensions we show more generally that x 0 is locally constant with respect to the vector field x under a spectral condition as a consequence we also show fried conjecture for anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic 3 manifolds this gives the first examples of non analytic anosov flows and geodesic flows in variable negative curvature where fried conjecture holds true

a cell structure for the set of autoregressive systems the set of autoregressive systems generalizes the set of transfer functionsin a natural way we describe a topology for the set of all autoregressive systems of fixed size and bounded mcmillan degree we show that this topological space has the structure of a finite cw complex

controllability and observability indices for linear time invariant systems a new approach abstract this paper is devoted to an investigation of the controllability and observability indices for linear time invariant discrete time systems these indices are here introduced at an intrinsic level in the context of the behavioral approach to systems theory for the class of the complete controllable systems it is proven that the controllability indices can be interpreted as a minimal and complete set of invariants with respect to a very natural systems equivalence autonomous phenomena are also discussed

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

existence of solutions of semilinear stochastic delay evolution inclusions in a hilbert space in this paper the existence of mild solutions of a class of semilinear stochastic evolution inclusions with delays in a hilbert space is studied the results are obtained by using a fixed point theorem for condensing map due to martelli the result is illustrated with an example

alternative solution of discrete time kalman filter abstract this paper concerns the closed form solutions of the minimum variance and the optimal filter gain of the discrete time kalman filter we directly construct the closed form solution of riccati equations and the optimal gain of the filter from the system parameters the approach is based on the time series treatment of the dual system of the stochastic systems the closed form solution of the minimum variance yields the riccati equation

analytic torsion and closed geodesics on hyperbolic manifolds pour une variete hyperbolique orientee fermee x et une representation orthogonale de 1 x on definit une fonction zeta de type ruelle r s pour re s grand comme le produit sur les geodesiques closes premieres des facteurs det i exp sl ou l est la longueur de on continue analytiquement r s et on calcule son terme principal en s 0

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

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

left invertibility and duality for linear systems left and right invertibilities as well as inversion procedures and algorithms have been widely studied for years we consider here for linear time invariant lti systems such types of one side invertibilities but in the time domain which is more demanding than just in the frequency domain i e for the transfer function matrix tfm new geometric conditions are given for time domain td left invertibility as well as structural conditions which require not only full column rank but also the absence of some types of finite zeros namely the observable ones for general monic full column rank systems having observable zeros we enhance a cascade decomposition which allows for the td left inversion of the pole factor and for the tfm left inversion of the zero factor duality in the classical tfm setting just relies on transposition in the present td context this is not relevant we give necessary and sufficient conditions for td right invertibility that show that the underlying duality is indeed the one between zeros and poles which is a posteriori natural full row rank lti systems can be td right inverted if and only if they have no finite observable poles

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

a simple procedure for the stabilization of a class of uncontrollable generalized systems this paper gives a simple procedure for the stabilization with a prescribed rate of convergence of the free motions of a class of possibly single input uncontrollable generalized continuous time time invariant linear systems preserving the impulse free property if any

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

trajectory control and interconnection of 1d and nd systems in this paper we examine the relationship between control viewed as concatenation of trajectories and control viewed as interconnection of systems we show that for one dimensional linear time invariant systems the ability to obtain a given subsystem by regular interconnection a prerequisite for any feedback type structure is equivalent to the ability to drive any trajectory into that subsystem however in the case of multidimensional systems the former is a stronger property than the latter trajectory controllability can however be expressed as a regular interconnection of behaviors in an extended variable space by introducing latent or auxiliary variables this leads as a by product to the notion of controlling a system by means of latent variables