on system equivalence this note establishes the relationship between strong system equivalence 3 and complete system equivalence 6 and deduces two sets of compact conditions for strong system equivalence of polynomial realizations

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

on spectral clustering of hss preconditioner for generalized saddle point matrices abstract for the nonsingular generalized saddle point matrix of a hermitian positive definite or semidefinite leading block we rigorously analyze clustering property for the eigenvalues of the corresponding preconditioned matrix with respect to the hermitian and skew hermitian splitting preconditioner the result shows that these eigenvalues are clustered around 0 2 and a few points located on the unit circle centered at 1 as the iteration parameter is close to 0

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

transformations of matrix pencils and implications in linear systems theory a new transformation of matrix pencils is proposed and its theory formally developed its restriction to generalized state space system matrix pencils is shown to be an appropriate generalization of the usual state space transformation and to represent a closed form expression in system matrix terms of verghese s concept of strong equivalence of such pencils

restricted invariant factor assignment under state feedback abstract for a given matrix pair a b a square necessary and sufficient conditions are established for the existence of a state feedback matrix f such that a bf is in the same similarity class as a partially prescribed square matrix

generalized proper inverse of polynomial matrices and the existence of infinite decoupling zeros a necessary and sufficient condition is presented for the existence of a generalized proper rational inverse for nonsquare polynomial matrices the condition is then proved to be equivalent to the absence of infinite decoupling zeros this condition provides a simple test for the absence of infinite decoupling zeros in the polynomial fraction form it can also be used to remove the infinite decoupling zeros in order to achieve a strongly irreducible system and to provide a new look of the unimodular matrix operation effect at infinity for the left and right polynomial fraction forms

infinite frequency structure and a certain matrix laurent expansion abstract a method for determining the smith mcmillan form at infinity of a rational matrix is derived by considering the laurent expansion at infinity of the matrix this method is used to provide a new test for the absence of infinite zeros in a rational matrix and a new formula for calculating the highest degree among the largest minors of a polynomial matrix

symmetries of linear systems of second order ordinary differential equations in this paper several problems concerning the lie algebra structure of symmetries and variational symmetries of a general linear system of second order ordinary differential equations are studied in particular a necessary and sufficient condition is obtained in terms of the coefficients of the system for the system s symmetry algebra to be of maximal dimension i e n2 4n 3 and isomorphic to sl n 2 r the well known symmetry algebra of the free particle equation x 0 when this condition is satisfied it is proved that the system is lagrangian and that its variational symmetry algebra is isomorphic to a fixed n2 3n 6 2 dimensional lie algebra whose structure levi mal cev decomposition and realization by means of a matrix algebra is determined for the particular case of isotropic systems which includes as far as is known all the examples treated in the literature explicit formulas for the generators of both the symmetry algebra and the variational symmetry algebra are obtained

a relaxed deteriorated pss preconditioner for nonsymmetric saddle point problems from the steady navier stokes equation for nonsymmetric saddle point problems arising from the steady navier stokes equations pan ng and bai presented a deteriorated positive definite and skew hermitian splitting dpss preconditioner pan et al 2006 to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the unconditional convergence property of the dpss iteration method is proved and a relaxed dpss rdpss preconditioner is proposed the rdpss preconditioner is much closer to the coefficient matrix than the dpss preconditioner in certain norm which straightforwardly results in an rdpss iteration method the convergence conditions of the rdpss iteration are analyzed and the optimal parameter which minimizes the spectral radius of the rdpss iteration matrix is derived using the rdpss preconditioner to accelerate some krylov subspace methods like gmres is also studied the eigenproperty of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the rdpss preconditioner

on the analysis of discrete linear time invariant singular systems the discrete singular equation over an interval can represent a two point boundary value problem or it can be considered as a dynamical relation developing forward in time a theory is provided by giving analytic solutions and discussing system properties in both cases that encompasses both interpretations the singular system fundamental matrix is used to provide analytic solutions to a time invariant discrete singular equation defined over an interval the two distinct cases in which the singular relation is interpreted as a two point boundary value problem or as a forward dynamical system on the interval are both considered also considered is the case in which the singular relation is considered as a backward dynamical system and the relationship between the index of nilpotence and the laurent series coefficients resulting from the solutions of certain state space equations is shown reachability and observability are discussed and the point is made that these properties are different depending on how the singular relation is interpreted

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

on the solution space of discrete time ar representations over a finite time horizon abstract the main purpose of this work is to determine the forward and backward solution space of a nonregular discrete time ar representation i e a k 0 in a finite time horizon where a is a polynomial matrix and is the forward shift operator the construction of the behavior is based on the structural invariants of the polynomial matrix that describes the ar representation i e the finite and infinite elementary divisors and the right and left minimal indices of a

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

a note on pss preconditioners for generalized saddle point problems abstract in this note the positive semidefinite and skew hermitian splitting pss preconditioners introduced by pan et al 2006 10 are further extended to the generalized saddle point problems with nonzero 2 2 blocks the model problem of oseen equations is used to illustrate the presented spectral properties as well as the performance of the pss preconditioner

natural preconditioning and iterative methods for saddle point systems the solution of quadratic or locally quadratic extremum problems subject to linear ized constraints gives rise to linear systems in saddle point form this is true whether in the continuous or the discrete setting so saddle point systems arising from the discretization of partial differential equation problems such as those describing electromagnetic problems or incompressible flow lead to equations with this structure as do for example interior point methods and the sequential quadratic programming approach to nonlinear optimization this survey concerns iterative solution methods for these problems and in particular shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods these preconditioners are related to the original extremum problem and their effectiveness in terms of rapidity of convergence is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matri

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

on conditions guaranteeing two polynomial matrices possess identical zero structures some sufficient conditions for two polynomial matrices to simultaneously possess identical finite and infinite zero structures are developed a special case of this problem that of obtaining conditions under which a polynomial matrix has no infinite zeros is also considered and certain connections with previous work are established

a new notion of equivalence for discrete time ar representations we present a new equivalence transformation termed divisor equivalence that has the property of preserving both the finite and the infinite elementary divisor structures of a square non singular polynomial matrix this equivalence relation extends the known notion of strict equivalence which dealt only with matrix pencils to the general polynomial matrix case it is proved that divisor equivalence characterizes in a closed form relation the equivalence classes of polynomial matrices that give rise to fundamentally equivalent discrete time auto regressive representations

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

sl 3 r realisations and the damped harmonic oscillator the point and contact lie groups for the damped harmonic oscillator are found and several realisations of the sl 3 r lie group are analysed the non critical and critical cases are studied separately and a sort of limiting procedure is defined which leaves the lie algebra structure unchanged some physical applications of these results are also pointed out

eigenvalue estimates for saddle point matrices of hermitian and indefinite leading blocks we study the eigenvalue bounds for the nonsingular saddle point matrices of hermitian and indefinite 1 1 and 2 2 blocks without imposing the restrictions that the 1 1 blocks are positive definite on the kernels of the 2 1 blocks

boundary recursion for descriptor variable systems it is shown that if an n dimensional descriptor variable system satisfies the properties of solvability and controllability the family of solutions is n dimensional and the set of two point boundary values that are consistent with the system can be expressed as the solutions to a linear system of equations involving only the boundary values this equation is termed a boundary mapping equation and it provides a compact representation of the system the boundary mapping generalizes the state transition matrix associated with state variable systems the author presents the boundary mapping theory describes the recursive processes required and applies both to some special cases including general linear time invariant systems and adjoint systems

structure smith macmillan form and coprime mfds of a rational matrix inside a region p ω abstract the structure of the smith macmillan form of a rational matrix t s inside a region p where is asymmetric with respect to the real axis subset of the finite complex plane c is determined algorithmic procedures based on elementary row and column operations over the euclidean ring r p s consisting of all rational functions with no poles in p are given coprimeness in p of a pair of rational matrices is studied in detail these results lead to constructive procedures for determining the coprime in p matrix fraction descriptions of t a

interpretation of symmetry transformations a general interpretation of symmetry transformations for an arbitrary regular multidimensional differential system is given in terms of its constants of motion the results indicate those of recent works in the literature as special cases and a query posed in one of them is solved

when nonlinear differential equations are equivalent to linear differential equations a necessary and sufficient condition is established for the existence of a 1 1 transformation of a system of nonlinear differential equations to a system of linear equations the obtained theorems enable one to construct such transformations from the invariance groups of differential equations the hodograph transformation the legendre transformation and lie s transformation of the monge ampsre equation are shown to be special cases noninvertible transformations are also considered examples include burgers equation a nonlinear diffusion equation and the liouville equation introduction in this work we study transformations mapping nonlinear differen tial equations to linear differential equations in a 1 1 manner based upon the group analysis of differential equations we obtain necessary and sufficient conditions for the existence of such transformations the established theorems not only allow us to determine the existence of the transformations but also enable us to actually construct these transformations from invariance groups of the nonlinear equations in the following analysis two types of transformations are considered 1 the invariance groups of differential equations and 2 the mappings which transform nonlinear differential equations to linear differential equations theorems will be proved based upon the following observations clearly if there exists a 1 1 mapping between any two differential equations it must inject properties of one equation into the other including their invariance properties for this reason the often ignored fact that any linear differential equation admits an invariance group related to the superpo sition principle becomes very significant the generator of this particular group depends upon an arbitrary solution of the linear equation it follows then that any nonlinear equation transformable to a linear equation by a 1 1 mappingmust admit an invariance group whose generator depends upon an arbitrary solution of some linear differential equation the idea of comparing invariance groups of differential equations in the search of mappings connecting the equations was first used by bluman in his study of burgers equation 1 and it was applied to the study of the mappings of one dimensional linear parabolic equations to the heat equation 2 in the first section of this paper some basic properties of various transformations are summarized the theorems are presented in the second section the examples in the third section include the hodograph transformation the legendre transformation lie s transformation of themonge ampre equation and the equation ux uxx uyy 0 in the last section we examine properties of noninvertible mappings of burgers equation the diffusion equation u 2ux uy 0 and the liouville equation u y e u respectively into linear equations

a modified dimensional split preconditioner for generalized saddle point problems abstract for large and sparse saddle point linear systems arising from 2d linearized navier stokes equations benzi and guo recently studied a dimensional split ds preconditioner appl numer math 61 2011 66 76 by further applying it to generalized saddle point problems in this paper we present a modified dimensional split mds preconditioner this new preconditioner is based on a splitting of the generalized saddle point matrix resulting in an unconditional convergent fixed point iteration the basic iteration is accelerated by a krylov subspace method like restarted gmres the implementation of the mds preconditioner is discussed and a similar case is also analyzed finally numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the mds preconditioner

equivalence transformations of rational matrices and applications the known theories of transformations between polynomial matrices are extended to the case of rational matrices specifically equivalence between rational matrices having possibly different dimensions is defined and this has the property of preserving the zero structure of rational matrices in the region c some implications of this new equivalence transformation for linear system theory are also provided

the spectral properties of the hermitian and skew hermitian splitting preconditioner for generalized saddle point problems in this paper we consider the hermitian and skew hermitian splitting hss preconditioner for generalized saddle point problems with nonzero 2 2 blocks the spectral property of the preconditioned matrix is studied in detail under certain conditions all eigenvalues of the preconditioned matrix with the original system being non hermitian will form two tight clusters one is near 0 0 and the other is near 2 0 as the iteration parameter approaches to zero from above so do all eigenvalues of the preconditioned matrix with the original system being hermitian numerical experiments are given to demonstrate the results

relations between strict equivalence invariants and structure at infinity of matrix pencils the structure at infinity of matrix pencils is investigated via their smith macmillan forms at infinity 1 relations between some strict equivalence invariants and pole zero structure at infinity are established

on the lie backlund symmetry of linear ordinary differential equations the symmetry of a second order linear ordinary differential equation and a system of coupled linear ordinary differential equations is investigated the equations for characteristic functions which determine the coordinates of the infinitesimal generators of the lie backlund transformations are found general solutions of equations for characteristic functions are constructed and the form of infinitesimal generators is obtained

a jordan control canonical form for singular systems a canonical form for controllable singular systems modulo restricted system equivalence is presented the canonical form respects the decomposition of the singular system into a regular and an impulsive part the construction is based on popov s control canonical form and on the jordan canonical form recently developed by hinrichsen and pratzel wolters continuity properties are analysed and the cellular partition of the orbit space induced by the canonical form is briefly discussed

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

regularized dpss preconditioners for generalized saddle point linear systems abstract by introducing a regularization matrix and an additional iteration parameter a new class of regularized deteriorated positive definite and skew hermitian splitting rdpss preconditioners are proposed for generalized saddle point linear systems compared with the well known hermitian and skew hermitian splitting hss preconditioner and the regularized hss preconditioner bai 2019 studied recently the new rdpss preconditioners have much better computing efficiency especially when the 1 1 block matrix is non hermitian it is proved that the corresponding rdpss stationary iteration method is unconditionally convergent in addition clustering property of the eigenvalues of the rdpss preconditioned matrix is studied in detail two numerical experiments arising from the meshfree discretization of a static piezoelectric equation and the finite element discretization of the navier stokes equation show the effectiveness of the new proposed preconditioners

on coprime rational function matrices abstract the concept of coprimeness of matrices with elements in a field of fractions is introduced we focus on the field of rational functions and define when two rational matrices are coprime with respect to different rings the definition of coprimeness at infinity of polynomial matrices is obtained as a particular case moreover the fact that coprimeness is a local property is proved and it is shown how coprimeness changes by performing mobius transformations

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

a fundamental notion of equivalence for linear multivariable systems a fundamental form of equivalence of polynomial matrix descriptions of linear multivariable systems is defined based on the existence of a bijective map between the finite and infinite solution sets of the differential equations describing the two systems the connection with the system matrix relationship of full system equivalence is established

infinite frequency structure preserving transformations for general polynomial system matrices a recent transformation proposed for polynomial matrices is extended to the systems theory case of general polynomial system matrices two such extensions are proposed depending on whether the system matrix or its normalized form is used the relationship between both these transformations is established and under them all finite and infinite frequency system zero and pole structure is shown to be invariant

contact symmetries of the harmonic oscillator the symmetry group of contact transformations for the one dimensional harmonic oscillator is determined its generators contain two arbitrary functions i e it is an infinite parameter lie group

on the semi convergence of regularized hss iteration methods for singular saddle point problems abstract recently bai and benzi proposed a class of regularized hermitian and skew hermitian splitting rhss iteration methods for solving the nonsingular saddle point problem in this paper we apply this method to solve the singular saddle point problem in the process of the semi convergence analysis we get that the rhss method and the hss method are unconditionally semi convergent which has improved the previous results then some spectral properties of the corresponding preconditioned matrices and a class of improved preconditioned matrices are analyzed finally some numerical experiments on linear systems arising from the discretization of the stokes equation are presented to illustrate the feasibility and effectiveness of this method and the corresponding preconditioners

convergence properties of preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite matrices for the non hermitian and positive semidefinite systems of linear equations we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned hermitian and skew hermitian splitting iteration methods we then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned hermitian and skew hermitian splitting iteration methods

a canonical parameterization of the kronecker form of a matrix pencil abstract the kronecker form is the classical canonical form for matrix pencils under strict equivalence transformation consider a matrix pencil whose entries are smooth functions of a parameter vector the kronecker form of the parameterized pencil will in general be a discontinuous function of the parameters for a linear time invariant control system these discontinuities correspond to a change in the finite and infinite invariant zero structure since many control strategies require knowledge of or place restrictions on the zero structure these points of discontinuity are of considerable interest in this paper a general approach to the study of such points is developed in the framework of singularity theory we derive a miniversal parameterization of a given pencil that is a parameterized family of pencils that i includes the given pencil ii is locally equivalent to any other family up to a change of parameters and iii uses the fewest number of parameters to achieve this property all nearby zero structures can be obtained by varying parameter values in the miniversal parameterization from all miniversal parameterizations one having a particularly uncluttered representation is selected as canonical however some restrictions on the finite elementary divisors are then required

structure and smith macmillan form of a rational matrix at infinity the use of special row and column operations in the reduction of a rational matrix to its smith macmillan form at infinity is investigated the connections between this reduction procedure and the valuation approach are established a graphical method for finding the smith macmillan form of a rational matrix at infinity from its bode magnitude array and some new results on realization theory for polynomial matrices are presented

contact symmetries of general linear second order ordinary differential equations using 1 1 mappings the complete symmetry groups of contact transformations of general linear second order ordinary differential equations are determined from two independent solutions of those equations and applied to the harmonic oscillator with and without damping

matrix pencil equivalents of a general 2 d polynomial matrix a two level algorithm which reduces a general 2 d polynomial matrix to an equivalent matrix pencil form is presented the exact nature of the equivalence transformation connecting the matrix with its associated pencil is established

a variant of the hss preconditioner for complex symmetric indefinite linear systems abstract using the equivalent block two by two real linear systems we establish a new variant of the hermitian and skew hermitian splitting hss preconditioner for a class of complex symmetric indefinite linear systems the new preconditioner is not only a better approximation to the block two by two real coefficient matrix than the well known hss preconditioner but also resulting in an unconditional convergent fixed point iteration the quasi optimal parameter which minimizes an upper bound of the spectral radius of the iteration matrix is analyzed eigen properties and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed finally two numerical examples are provided to show the efficiency of the new preconditioner

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

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

some new estimates on the complex eigenvalues of the hss preconditioned matrix in this paper we derive some new bounds on the complex eigenvalues of the hermitian and skew hermitian splitting hss preconditioner for saddle point problems and improve those stated in previously published works

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

matrix fractions and strict system equivalence unimodular equivalence ue is defined for matrix fraction decompositions mfds it is shown that if an mfd is regarded as a quadruple d s n s i then uni modular equivalence and fuhrmann strict system equivalence coincide it is thus established that ue for mfds and system similarity for state space representations are exact counterparts a brief discussion on extended unimodular equivalence eue of polynomial matrices is included and a short polynomial matrix proof that the invariant factors of the smith form are preserved under eue is given

an algebraic theory for design of controllers for linear multivariable systems part i structure matrices and feedforward design in this two part paper a theory is presented in which several control problems can be solved the theory is applicable to a general class of linear multivariable systems where the measured output does not have to be equal to the controlled output or the state the system may be affected by nonmeasurable disturbances only controllers which stabilize the system and have proper transfer functions are allowed i e the controllers have to be physically realizable it is shown that the solutions to control problems of servo type e g problems of model matching decoupling and invertibility are special cases of the solution to a more general problem analogously the solutions to problems of regulator type e g disturbance decoupling output regulation and pole placement are also speclal cases of the solution to a more general problem it is shown that problems of servo type and regulator type can be solved independently of each other in part i generalized polynomials are introduced as a mathematical framework structue matrices which describe how well a system can be controlled are defined finally problems of servo type are solved part ii mainly deals with problems of regulator type

infinite elementary divisors of a matrix polynomial and implications a new characterization of infinite poles and zeros in terms of a natural extension of the idea of infinite elementary divisors of a matrix pencil to matrix polynomials is developed it is used to show that two previous descriptions of infinite frequency structure are equivalent subsequently a polynomial matrix transformation which simultaneously preserves the finite and infinite zero structure is presented

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

algebraic and dynamic characterizations of poles and zeros at infinity abstract the presence of infinite frequency poles and zeros in a rational transfer function matrix is considered and certain algebraic characterizations for their occurrence are given one such characterization is shown to enable the transmission effects due to infinite poles and zeros to be ascertained in an extremely simple and direct way

fundamental equivalence of discrete time ar representations we examine the problem of equivalence of discrete time auto regressive representations dtarrs over a finite time interval two dtarrs are defined as fundamentally equivalent fe over a finite time interval 0 n if their solution spaces or behaviours are isomorphic we generalize the concept of strict equivalence se of matrix pencils to the case of general polynomial matrices and in turn we show that fe of dtarrs implies se of the underlying polynomial matrices

two new variants of the hss preconditioner for regularized saddle point problems two new preconditioners which can be viewed as improved variants of the hermitian and skew hermitian splitting hss preconditioner are presented for regularized saddle point problems the unconditionally convergent properties of the corresponding iteration methods are deduced and the selection of optimal iteration parameters resulting in fast convergence for the two iteration methods are discussed moreover eigenvalue bounds of the preconditioned matrices and upper bounds of the degree of their minimal polynomials are obtained compared with the hss preconditioner they are much better approximations to the coefficient matrix of the saddle point problem and they have better convergence properties and spectrum distributions numerical experiments from the stokes problem are presented which illustrates the advantages of the two preconditioners over the hss preconditioner and some other preconditioners to accelerate the convergence rate of gmres

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

on matrix polynomials with the same finite and infinite elementary divisors abstract a criterion is presented that characterizes when two matrix polynomials of any size rank and degree have the same finite and infinite elementary divisors this characterization inherits a coprimeness condition of the extended unimodular equivalence defined by pugh and shelton 17 in the set of real or complex matrix polynomials satisfying the constraint that the difference between the number of rows and columns is constant this extended unimodular equivalence is first generalized to matrices of any size with elements in any principal ideal domain

descriptor systems decomposition into forward and backward subsystems it is shown that a decomposition of the descriptor space with certain properties induces a decomposition of the descriptor system into forward and backward subsystems different decompositions correspond to different apportionments of the required boundary conditions between initial and final times applications include several different but equivalent methods for solving the descriptor system and a clarification of the relations between some previously known methods of solution

spectral equivalence of matrix polynomials and the index sum theorem abstract the concept of linearization is fundamental for theory applications and spectral computations related to matrix polynomials however recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive because some structured polynomials do not have any linearization with the same structure this phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations motivated by this fact we introduce equivalence relations that allow the possibility of matrix polynomials with coefficients in an arbitrary field to be equivalent with the same spectral structure but have different sizes and degrees these equivalence relations are directly modeled on the notion of linearization and consequently inherit the simplicity applicability and most relevant properties of linearizations simultaneously though they are much more flexible in the possible degrees of equivalent polynomials this flexibility allows us to define in a unified way the notions of quadratification and l ification to introduce the concept of companion form of arbitrary degree and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second frobenius companion forms the properties of l ifications are studied in depth in this process a fundamental result on matrix polynomials the index sum theorem is recovered and extended to arbitrary fields although this result is known in the systems theory literature for real matrix polynomials it has remained unnoticed by many researchers it establishes that the sum of the finite and infinite partial multiplicities together with the left and right minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial the index sum theorem turns out to be a key tool for obtaining a number of significant results on the possible sizes and degrees of l ifications and companion forms on the minimal index preservation properties of companion forms of arbitrary degree as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree this paper presents many new results blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields therefore we have written the paper in an expository and self contained style that makes it accessible to a wide variety of readers

an extension of wolovich s definition of equivalence of linear systems wolovich s classical definition of equivalence for linear systems 1974 is extended to the generalized study of linear systems it is shown that the resulting equivalence is an alternative characterization of the notion of full system equivalence underlying its fundamental role in the generalized study of linear systems

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

realizations in generalized state space form for polynomial system matrices and the definitions of poles zeros and decoupling zeros at infinity this paper contains three main results firstly a realization procedure is presented that brings a polynomial system matrix of a non proper multivariable system to generalized state space form such that all relevant properties including phenomena of redundancy associated with finite and infinite decoupling zeros are retained secondly new definitions are proposed for poles zeros and decoupling zeros at infinity of a general polynomial system matrix as a third result a theorem of rosenbrock 1974 e on the redundancy of lcr multiports is generalized to include the decoupling zeros at infinity

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