poles and zeros at infinity of linear time varying systems the notions of poles and zeros at infinity and their relations are extended to the case of linear continuous time varying systems this study is based on the notion of a newborn systems which is in a mathematical point of view a graded module extension over the noncommutative ring of differential operators it is proved to be a relevant generalization to the time varying case of the equivalence class for the so called restricted equivalence of rosenbrock s polynomial matrix descriptions the authors approach is intrinsic and unifies the definitions previously given in the literature in the time invariant case

an intrinsic algebraic setting for poles and zeros of linear time varying systems in this paper poles and zeros are defined for linear time varying systems using suitable ground field extensions the definitions of the system poles transmission poles invariant zeros hidden modes etc are given in an intrinsic module based framework and are consistent in the sense that the poles are connected to the stability of the system and the zeros to the zeroing of the output for non zero inputs in particular it is proved that the necessary and sufficient condition for a continuous time system to be exponentially stable is similar to the well known condition in the time invariant case

some basic structural properties of generalized linear systems abstract this paper shows how to give a kalman state variable realization to generalized linear systems simple characterizations of controllability observability of their duality and of a minimal realization are proposed our techniques based on elementary module theory are applicable to time varying systems as well as constant systems the note ends with a brief discussion of asymptotic observers in the constant case

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

the moore penrose inverse of a morphism with factorization abstract given an m by n matrix a of rank r over a field with an involutory automorphism it is well known that a has a moore penrose inverse if and only if rank a a r rank aa by use of the full rank factorization theorem this result may be restated in the category of finite matrices as follows if a 1 r a 2 is an epic monic factorization of a m n through r then a has a moore penrose inverse if and only if a a 1 r a 2 and a 1 r a 2 a are respectively epic monic factorizations of a a n n and aa m m through r this characterization of the existence of moore penrose inverses is extended to arbitrary morphisms with epic monic factorizations

notes on the definition of behavioural controllability we take another look at the behavioural definition of controllability for discrete 1d systems and its extension to multidimensional nd systems defined on z n or n n we suggest that the current definition for nd systems is inappropriate for systems defined on the domain n n and that care has to be taken even with the 1d definition we propose a new definition which applies to all standard classes of discrete nd systems

algebraic characterization of approximate controllability of behaviours of spatially invariant systems we give algebraic characterizations of the properties of autonomy and of controllability of behaviours of spatially invariant dynamical systems consisting of distributional solutions w that are periodic in the spatial variables to a system of partial differential equations corresponding to a polynomial matrix m g c 1 d mxn

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 behavioral approach to control of distributed systems this paper develops a theory of control for distributed systems i e those defined by systems of constant coefficient partial differential operators via the behavioral approach of willems the study here is algebraic in the sense that it relates behaviors of distributed systems to submodules of free modules over the polynomial ring in several indeterminates as in the lumped case behaviors of distributed arma systems can be reduced to ar behaviors this paper first studies the notion of ar controllable distributed systems following the corresponding definition for lumped systems due to willems it shows that as in the lumped case the class of controllable ar systems is precisely the class of ma systems it then shows that controllable 2 d distributed systems are necessarily given by free submodules whereas this is not the case for n d distributed systems n ge 3 this therefore points out an important difference between these two cases this paper then defines two notions of autonomous distributed systems which mimic different properties of lumped autonomous systems control is the process of restricting a behavior to a specific desirable autonomous subbehavior a notion of stability generalizing bounded input bounded output stability of lumped systems is proposed and the pole placement problem is defined for distributed systems this paper then solves this problem for a class of distributed behaviors

relationship between poles and zeros of input output and chain scattering systems abstract a system is frequently represented by transfer functions in an input output characterization however such a system under mild assumptions can also be represented by transfer functions in a port characterization frequently referred to as a chain scattering representation due to its cascade properties the chain scattering representation is used throughout many fields of engineering this paper studies the relationship between poles and zeros of input output and chain scattering representations of the same system

composite wavelet bases for operator equations this paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit n cube these bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains this covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo differential equations with special emphasis on boundary integral equations in this case it is natural to model the boundary surface as a union of parametric images of the unit cube it will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch here the relevant properties are the validity of norm equivalences in certain ranges of sobolev scales as well as appropriate moment conditions

quasi poles of linear time varying systems in an intrinsic algebraic approach abstract in a previous piece of work it has been shown that the exponential stability of a linear time varying ltv system can be evaluated using new definitions of the poles of such a system the latter are given by a fundamental set of roots of the skew polynomial p which defines the autonomous part of the system such a set may not exist over the initial field k of definition of the coefficients of the system but can exist over a suitable field extension k k it is shown here that conditions for stability can also be obtained using linear factors of the polynomial p over another field extension k which may be smaller k k k the roots of these factors are called the quasi poles of the system the necessary condition for system stability expressed in function of these quasi poles is more restrictive than the one involving a fundamental set of roots

wavelet bases on a manifold using the decomposition method we present in this paper constructions of multiresolution analyses on a compact riemannian manifold m of dimension n n n these analyses are generated by a finite number of basic functions and are adapted to the study of the sobolev spaces h1 m and h01 m

periodic polynomial interpretation for structural properties of linear periodic discrete time systems the structural properties of linear periodic discrete time systems are analyzed in the periodic polynomial representation it is shown that the classical polynomial approach for linear time invariant systems can be extended to periodic systems new definitions and properties are given in terms of skew polynomial rings and periodic difference algebra necessary and sufficient conditions for the characterization of reachability controllability observability and reconstructibility are given in this framework

wavelet bases on the l shaped domain abstract we present in this paper two elementary constructions of multiresolution analyses on the l shaped domain d in the first one we shall describe a direct method to define an orthonormal multiresolution analysis in the second one we use the decomposition method for constructing a biorthogonal multiresolution analysis these analyses are adapted for the study of the sobolev spaces h s d s n

biorthogonal multiresolution analysis on a triangle and applications we present in this paper new constructions of biorthogonal multiresolution analysis on the triangle we use direct method based on the tensor product to construct dual scaling spaces on next we construct the associated wavelet spaces and we prove that the associated wavelets have compact support and preserve the original regularity finally we describe some regular results which are very useful to establish the norm equivalences as applications we prove that the wavelet bases constructed in this paper are adapted for the study of the sobolev spaces h 0 s and h s s n and are easy to implement wavelets have unfolded their full computation efficiently in numerical and applied analysis the properties of wavelet bases provide a rigorous analysis for dynamical systems we construct in this work biorthogonal wavelet bases on a triangle these bases are adapted to the study of the sobolev spaces the bases allow many concrete numerical examples as numerical simulation for elliptic problems or image processing

the moore idempotents of a matrix abstract let r be a commutative ring with 1 and with an involution and let mr be the category of finite matrices over r with the involution aij a ij a ji if a is in mr then there exists a unique list e0 es of pairwise orthogonal symmetric idempotents of r that are characterized by several properties involving the rank and squared volume of the eia in particular a has a moore penrose inverse in mr iff esa 0

symmetric morphisms and the existence of moore penrose inverses abstract let x y be a morphism of an additive category with an involution then has a moore penrose inverse with respect to iff is invertible for some annihilator n x of iff there exists an object z and morphisms x z z z z y such that 2 and 1 z and 1 z are invertible in this case 1 1 z 1 1 z 1

a behavioral approach to delay differential systems we will study linear time invariant delay differential systems from the behavioral point of view as it was introduced for dynamical systems by willems dynam report 2 1989 pp 171 269 a ring cal h which lies between bbb r s z z 1 and bbb r s z z 1 will be presented whose elements can be interpreted as a generalized version of delay differential operators on cal c infty bbb r bbb r in this framework a behavior is the kernel of such an operator using the ring cal h an algebraic characterization of inclusion respectively equality of the behaviors under consideration is given finally controllability of the behaviors is characterized in terms of the rank of the associated matrices in the case of time delay state space systems this criterion becomes the known hautus criterion for spectral controllability

rosenbrock models and their homotopy equivalence as is known the notion of homotopy equivalence is a fundamental notion of mathematics and was introduced in order to formalize a relation that is weaker than isomorphism in this note we define a homotopy equivalence of rosenbrock systems and show that it coincides with the classical equivalences of rosenbrock and fuhrmann next we show that the homotopy equivalence does preserve the important properties of a system including the properties at infinity when these are properly understood finally we define in a simple manner the states and motions of a system and claim that they are homotopy invariants

continuous time linear predictive control and flatness a module theoretic setting with examples a general flatness based framework for linear continuous time predictive control is presented the mathematical setting which is valid for multivariable systems is provided by the algebraic theory of modules where a controllable system corresponds to a finitely generated free module over a principal ideal ring any basis of this free module is a flat output which yields an easy calculation of the predicted trajectory this formalism permits one to handle non minimum phase systems system constraints and to deal with additive perturbations three concrete case studies namely a dc motor a flexible system and a cement mill are analysed and simulations are given these examples are written in such a way that any reader who is not familiar with module theory may nevertheless grasp the proposed control strategy

duality for differential difference systems over lie groups in modern mathematical systems theory there exist two consistent ways of defining and describing a linear system i in the behavioral approach a linear system is the kernel mathfrak b of a matrix valued operator r in a power of a signal space w ii in the module theoretic setting a linear system is the cokernel m of the above matrix r these two formulations have connections the minimal conditions under which they are equivalent are investigated in this paper the general theory is applied to differential difference systems over lie groups

finite poles and zeros of linear systems an intrinsic approach several kinds of finite poles and zeros are defined and their relationships are studied in an intrinsic manner some of these definitions are more precise than the usual ones all kinds of multiplicities of an input output decoupling zero are defined here in addition we believe that our approach clarifies those notions and it makes it possible to very concisely establish fundamental relationships between various poles and zeros some new relationships are derived the notions of input output stability and transfer stability are also introduced several examples illustrate the theory

a volume associated with m x n matrices abstract let a mxnr with nonzero singular values 1 2 r the volume of a vol a is defined as zero if r 0 and otherwise vola ri 1 i or equivalently vola det2aij summing over all r x r nonsingularsubmatrices aij the matrix volume vol a generalizes the absolute value of the determinant from nonsingular to arbitrary matrices any r dimensional unit cube in r at is mapped under a into a parallelepiped of volume vol a this paper gives properties and applications of vol a in particular the moore penrose inverse of a is a convex combination of ordinary inverses of its maximal nonsingular submatrices a i j ij or is a convex combination of moore penrose inverses of its maximal full rank submatrices a i i j j where the convex weights are proport ional to the squares of volumes of the corresponding submatrices since a a the matrix a is a convex combination of inverses of submatrices of a interestingly the weights of these convex combinations are the above weights in transposed position

multidimensional behaviours an algebraic approach to control theory for pde this paper gives an overview of the algebraic theory of multidimensional behaviours and it contains some new aspects as well some known results are restated with a new proof after a short introduction to behavioural systems theory we briefly describe the mathematical background of the algebraic approach then we study structural properties such as autonomy and over determined systems on the one hand and controllability and flat systems on the other

on invariant zeros of linear systems of pdes in this paper we provide a behavioral framework in which to describe and extend the concept of linear dynamics introduced by fliess from the one dimensional 1d to the multidimensional nd framework we provide an alternative description of the invariant zeros of a system equivalent to the smith zero description in the 1d case and use this to generalize the concept and characterization of invariant zeros to the nd case in particular we show that the definitions are equivalent in the 1d case to those in the classical literature we provide new results on the structural relations of nd invariant and transmission zeros

on the extension of solutions of linear constant coefficient pde abstract we solve the following over determined boundary value problem the extension problem let r be a matrix whose entries are linear partial differential operators with constant coefficients let be a non empty open bounded convex set we consider the homogeneous system r f 0 in a neighborhood of subject to the boundary condition f g in a neighborhood of for a given g we give a criterion for the unique existence of a smooth solution f to this problem there are two obvious necessary conditions g is smooth and r g 0 in a neighborhood of we characterize the class of differential operators r for which the problem is solvable for any g satisfying the necessary conditions finally in the case where the solution is non unique we consider the possibility of obtaining uniqueness by fixing several components of the desired solution

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

a behavioral approach to the pole structure of one dimensional and multidimensional linear systems we use the tools of behavioral theory and commutative algebra to produce a new definition of a finite pole of a linear system this definition agrees with the classical one and allows a direct dynamical interpretation it also generalizes immediately to the case of a multidimensional nd system we make a natural division of the poles into controllable and uncontrollable poles when the behavior in question has latent variables we make a further division into observable and unobservable poles in the case of a one dimensional 1d state space model the uncontrollable and unobservable poles correspond respectively to the input and output decoupling zeros whereas the observable controllable poles are the transmission poles most of these definitions can be interpreted dynamically in both the 1d and nd cases and some can be connected to properties of kernel representations we also examine the connections between poles transfer matrices and their left and right matrix fraction descriptions mfds we find behavioral results which correspond to the concepts that a controllable system is precisely one with no input decoupling zeros and an observable system is precisely one with no output decoupling zeros we produce a decomposition of a behavior as the sum of subbehaviors associated with various poles this is related to the integral representation theorem which describes every system trajectory as a sum of integrals of polynomial exponential trajectories

brief paper output feedback pole placement for linear time varying systems with application to the control of nonlinear systems the output feedback pole placement problem is solved in an input output algebraic formalism for linear time varying ltv systems the recent extensions of the notions of transfer matrices and poles of the system to the case of ltv systems are exploited here to provide constructive solutions based as in the linear time invariant lti case on the solutions of diophantine equations also differences with the results known in the lti case are pointed out especially concerning the possibilities to assign specific dynamics to the closed loop system and the conditions for tracking and disturbance rejection this approach is applied to the control of nonlinear systems by linearization around a given trajectory several examples are treated in detail to show the computation and implementation issues

key problems in the extension of module behaviour duality the duality for linear constant coefficient partial differential equations between behaviours and finitely generated modules over the operator ring is a very powerful tool linking equation structure to dynamic behaviour this duality is critically dependent on the choice of signal space in this paper we discuss two key algebraic problems which form an obstacle to the extension of this theory to general signal spaces the first of these is the so called willems closure problem which limits the ability of system equations to directly describe the system the second is the elimination problem the general solution of which depends upon an algebraic property injectivity of the signal space we demonstrate the importance of these problems in the module behaviour framework and some of the useful consequences of a full or partial solution the issues here are of particular relevance to the extension of the current duality theory for behaviours defined by linear partial differential equations from the case of constant to non constant coefficients

wavelet bases on a triangle abstract in the present paper we find new constructions of orthonormal multiresolution analyses on the triangle in the first one we describe a direct method to define an orthonormal multiresolution analysis which is adapted for the study of the sobolev spaces h 0 s s n in the second one we add boundary conditions for constructing an orthonormal multiresolution analysis which is adapted for the study of the sobolev spaces hs s n the associated wavelets preserve the original regularity and are easy to implement

on interconnections control and feedback the purpose of this paper is to study interconnections and control of dynamical systems in a behavioral context we start with an extensive physical example which serves to illustrate that the familiar input output feedback loop structure is not as universal as we have been taught to believe this leads to a formulation of control problems in terms of interconnections subsequently we study interconnections of linear time invariant systems from this vantage point let us mention two of the results obtained the first one states that any polynomial can be achieved as the characteristic polynomial of the interconnection with a given plant provided the plant is not autonomous the second result states that any subsystem of a controllable system can be implemented by means of a singular feedback control law these results yield pole placement and stabilization of controllable plants as a special case these ideas are finally applied to the stabilization of a nonlinear system around an operating point

direct methods for primary decomposition leti be an ideal in a polynomial ring over a perfect field we given new methods for computing the equidimensional parts and radical ofi for localizingi with respect to another ideal and thus for finding the primary decomposition ofi our methods rest on modern ideas from commutative algebra and are direct in the sense that they avoid the generic projections used by hermann 1926 and all others until now

the contact problem for linear continuous time dynamical systems a geometric approach in this paper linear time invariant dynamical systems described by a combination of differential equalities and static inequalities in state space formulation are investigated of special interest is the contact problem the effect of the boundary of the constraint set on the behavior of the system this effect is studied by dividing the state space in a number of disjunct subsets it is shown that these subsets are invariant under linear state feedback in our framework a specific place is reserved for modeling the laws of collision i e physical modeling which are regarded as external factors our main results are a system theoretical framework in which we describe what happens upon contact and a definition of the constrained state space system in terms of its restricted behavior the results presented here can be considered as an extension for restricted linear systems of the classic positive invariance theory for linear systems

the determinal rank idempotents of a matrix abstract let m r be the category of finite matrices over a commutative ring r with 1 for a m r of determinantal rank r let i r a be the ideal of r generated by the r r minors of a there exists a unique list e 1 e t of pairwise orthogonal idempotents of r that sum to 1 such that if r i rank e i a then rank a r 1 r 2 r t e i is the identity element of i r i e i a for 1 i t and either e t a 0 or i r t e t a does not possess an identity element characterizations are given for various generalized inverses of a e 1 a e t a in particular a is regular iff e t a 0

a remark on willems trajectory characterization of linear controllability abstract willems trajectory characterization of linear controllability is related to some module theoretic property

generalized inverses over integral domains in an earlier paper one of the authors showed that a matrix of rank r over an integral domain has a generalized inverse if and only if a linear combination of all the r r minors of the matrix is one in the same paper a procedure for constructing a generalized inverse from such a linear combination was also given in the present paper we show that any reflexive generalized inverse can be obtained by that procedure we also show that if the integral domain is a principal ideal domain every generalized inverse can be obtained by that procedure it is also shown that a matrix a of rank r over an integral domain has moore penrose inverse if and only if the sum of squares of all r r minors of a is an invertible element of the integral domain

categories of matrices with only obvious moore penrose inverses abstract let r be an associative ring with 1 and with an involution a and let mr be the category of finite matrices over r with the involution aij aij ji then the following two statements are equivalent i if a in mr has a moore penrose inverse with respect to then a is permutationally equivalent to a matrix of the form b 0 0 0 with b invertible ii if 1 a in r then at most one of the a s is not zero

controllability of 2 d systems the concept of controllability of 2 d systems is introduced in a behavioral framework this property is defined as a systems concept and is characterized in terms of system representations state space realizations for controllable 2 d systems are presented the structure of state space systems is analyzed and the notions of trimness reachability and observability are introduced and characterized these realizations do not depend on any assumptions concerning causality

the moore penrose inverse over a commutative ring abstract let r be a commutative ring with 1 and with an involution a and let mr be the category of finite matrices over r with the involution a ij a ij a ji a matrix a m n in mr of determinantal rank r such that u a q r m q r m det a det a has a moore penrose inverse u a in r is said to be moore invertible with moore idempotent u a u a if u a u a a a for every matrix a of mr a has a moore penrose inverse with respect to if and only if a is the sum of moore invertible matrices whose moore idempotents are pairwise orthogonal

primeness of multivariate polynomial matrices abstract different characterizations of primeness of polynomial matrices that are mutually equivalent in the one dimensional case of univiriate polynomials lead to various primeness concepts in the multivariate situation applied to systems theory this fact gives rise to a more refined view to observability and controllability of multidimensional systems algorithms for testing primeness properties are given based on computer algebraic techniques