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

a quasi reversibility regularization method for an inverse heat conduction problem without initial data in this paper we study an inverse heat conduction problem without initial data in a bounded domain in which the cauchy data at x 0 are given and the solution in 0

numerical solution to a sideways parabolic equation the inverse heat conduction problem can be considered to be a sideways parabolic equation in the quarter plane this is a model of a problem where one wants to determine the temperature on both sides of a thick wall but one side is inaccessible to measurements a numerical procedure for this severely ill posed problem is suggested which consists of two steps namely a mollification of the data and a marching difference scheme the numerical method is proved to be stable several computational results are presented and discussed copyright 2001 john wiley sons ltd

determining surface temperatures from interior observations we consider the inverse heat transfer problem for the case of the one dimensional heat equation in a quarter plane and present a new solution algorithm together with error bounds of logarithmic convexity type our method is based on regularizing the initial value problem for the heat equation run sideways we describe a numerical experiment with rapidly varying temperature profiles and explain how the method can be extended to problems with temperature dependent thermal properties

algebraic aspects of controllability for ar systems this paper examines some aspects of controllability for behaviour systems in ar representation the approach is based on the theory of dipolynomial modules and dipolynomial matrices explicit criteria for controllability in terms of the representing matrices are derived and an effective test algorithm is given furthermore controllability indices are introduced for ar systems the new theory is compared with existing index concepts in the literature on linear systems

l infty error bounds in partial deconvolution of the inverse gaussian pulse when a c infty approximation to the dirac 5 function in the form of an inverse gaussian pulse is used as input into a linear time invariant system the output waveform is an approximation to that system s green s function in which the singularities have been smoothed out the ill posed deconvolution problem for the output signal aims at reconstructing these singularities by exploiting the smoothing properties of the inverse gaussian kernel we prove that partial deconvolution of the output waveform given l 2 a priori bounds on the data noise and the unknown green s function results in l infty error bounds for the regularized solution and its derivatives consequently when the l 2 norm of the output noise is sufficiently small partial deconvolution is a pointwise reliable c infty function which in turn approximates the desired green s function in many applications

from time series to linear system part iii approximate modelling abstract in the third part of this paper the problem of finding a linear time invariant complete system which models an observed time series will be continued however in this part it will be assumed that the model is required to explain the observations only approximately this raises the question of what is meant by an approximate model and how one should judge a model against data which in a strict sense falsify the model the initial sections of this paper introduce the methodology either the maximal admissible complexity is fixed and the best fitting model in the model class is sought or the maximally tolerated misfit is fixed and the least complex model in the model class is sought this is illustrated by means of finite dimensional models where natural complexity and misfit functions suggest themselves subsequently what is meant by the complexity of a linear time invariant complete system is discussed a concrete complexity function is proposed which is defined in terms of the richness of the behaviour on finite time intervals and is in one to one correspondence with the number of inputs the number of states and the observability indices of the underlying minimal i s o system the next definition is that of the misfit between an observed time series and a system a concrete misfit function is proposed which takes into consideration the lag structure of the system and in how far the observations fail to corroborate the ar equations which define the system s behaviour how the misfit may be computed by means of the correlation function of the data is then shown these concepts are subsequently used in order to set up two algorithms for approximate modelling of an observed q dimensional time series the first algorithm fixes the maximal complexity in a sense this comes down to fixing the number and the lags of the required ar equations the second algorithm fixes the tolerated misfit it is adaptive in nature in the sense that the number and the structure of the resulting ar equations will depend on the observations these ideas and algorithms are illustrated by means of two simulated examples

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

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

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

infinitely divisible pulses continuous deconvolution and the characterization of linear time invariant systems this paper addresses the problem of determining the impulse response of a linear time invariant system by probing the system with a causal c infty approximation to the dirac delta function we analyze the ill posed deconvolution problem which results from a wide choice of possibly multimodal infinitely divisible probe pulses the notion of infinite divisibility is shown to play a key role when the system s response is suspected of having nondifferentiable singularities we reformulate the volterra integral equation as a cauchy problem for a linear partial differential equation in two independent variables and introduce the concepts of partial and continuous deconvolution we then show that partial deconvolution of the output waveform results in l infty error bounds for the regularized solution and its derivatives under l 2 a priori bounds on the data noise and the unknown system response using the poisson summation formula and fft algorithms we construct an efficient computational alg

separation of year and site effects by generalized linear models in regionalization of annual floods this paper explores the utility of generalized linear models glms and generalized linear mixed models glmms for regionalization and record augmentation of flood data because both models allow the separation of site and year effects each gives an estimate of a site s mean annual flood free from effects of the particular years in which floods were recorded in addition a glm gives a measure of the extent to which a short flood record can be regarded as representative of a longer period the way in which glms and glmms are formulated effectively unites two problems that have largely been regarded as separate namely regional regression and data augmentation glmm model structure implicitly includes correlation between flood records at neighboring sites and both models contain facilities for testing whether time trends exist in flood data for testing whether such trends are regionally uniform when they are found to exist and for testing which climatological and or physiographic variables are useful for information transfer by appropriate selection of probability distribution and link function biases are avoided that arise where log transformed data are back transformed to the scale on which flood flows are recorded the paper illustrates glms and glmms by fitting them to flood records of variable length from 12 sites in the ibicui drainage basin in southern brazil while the models are extremely flexible and are easily fitted for gamma distributed data and to data following certain other probability distributions the assumptions on which they are based are arguably more complex than in more familiar methods requiring careful checking procedures

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

identification of dynamic green s functions in structural networks this paper presents the mathematical and computational basis of a method for experimentally identifying the dynamic behavior of linear structural systems the method consists of exciting the structure with specific pulses and reconstructing the dynamic green s functions by deconvolution of the measured response the reconstruction procedure involves the solution of an ill posed integral equation in the presence of noise inverse gaussian functions which have mathematical properties that render the deconvolution problem tractable are used as probing pulses two examples are presented for which dynamic green s functions are reconstructed from synthetic response data 1 a nondispersive medium consisting of a mechanical slider with free body motion and 2 a network that may be representative of a large orbiting structure with rigidly connected torsional members and timoshenko beams it is shown that residual effects of noise that may be detectable in the reconstructed green s functions are effectively smoothed out when these functions are used to predict the response to forces such as may be induced by control systems finally an example is presented of deconvolution implemented in slow motion

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

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

une interpretation algebrique de la transformation de laplace et des matrices de transfert abstract the tensor product of the module of a linear system with the quotient field of the ring of linear differential operators is a vector space where even in the time varying case a formal laplace transform and the transfer matrix are most naturally defined several classic problems are examined in this algebraic setting the relationship between left right coprime matrix decomposition and controllability observability the state variable canonical realization the transfer algebra with respect to parallel and series connections the input output inversion and model matching

algebraic geometric aspects of feedback stabilization this paper develops a theory of feedback stabilization for siso transfer functions over a general integral domain which extends the well known coprime factorization approach to stabilization necessary and sufficient conditions for stabilizability of a transfer function in this general setting are obtained these conditions are then refined in the special cases of unique factorization domains ufds noetherian rings and rings of fractions it is shown that these conditions can be naturally interpreted geometrically in terms of the prime spectrum of the ring this interpretation provides a natural generalization to the classical notions of the poles and zeros of a plant the set of transfer functions is topologized so as to restrict to the graph topology of vidyasagar ieee trans automatic control ac 29 1984 pp 403 418 when the ring is a bezout domain it is shown that stability of a feedback system is robust in this topology when the ring is a ufd this theory is then applied to the problem of sta

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

stability and stabilization of multidimensional input output systems in this paper we discuss stability and stabilization of continuous and discrete multidimensional input output io behaviors of dimension r which are described by linear systems of complex partial differential resp difference equations with constant coefficients where the signals are taken from various function spaces in particular from those of polynomial exponential functions stability is defined with respect to a disjoint decomposition of the r dimensional complex space into a stable and an unstable region with the standard stable region in the one dimensional continuous case being the set of complex numbers with negative real part a rational function is called stable if it has no poles in the unstable region an io behavior is called stable if the characteristic variety of its autonomous part has no points in the unstable region this is equivalent to the stability of its transfer matrix and an additional condition the system is called stabilizable if there is a compensator io system such that the output feedback system is well posed and stable we characterize stability and stabilizability and construct all stabilizing compensators of a stabilizable io system parametrization the theorems and proofs are new but essentially inspired and influenced by and related to the stabilization theorems concerning multidimensional io maps as developed for instance by bose guiver shankar sule xu lin ying zerz and quadrat and of course the seminal papers of vidyasagar youla and others in the one dimensional case in contrast to the existing literature the theorems and proofs of this paper do not need or employ the so called fractional representation approach i e various matrix fraction descriptions of the transfer matrix thus avoiding the often lengthy matrix computations and seeming to be of interest even for one dimensional systems at least to the author an important mathematical tool new in systems theory is gabriel s localization theory which only in the case of ideal convex shankar sule unstable regions coincides with the usual one algorithmic tests for stability stabilizability and ideal convexity and the algorithmic construction of stabilizing compensators are addressed but still encounter many difficulties see in particular the open problems listed by xu e

the construction of noetherian operators abstract we give an elementary and constructive purely algebraic proof for the existence of noetherian differential operators for primary submodules of finite dimensional free modules over polynomial algebras by means of these operators the submodule can be described by differential conditions on the associated characteristic variety this important result and the terminology are due to v p palamodov however his l ehrenpreis and later j e bjork s proofs of the existence theorem use complicated algebraic and analytic techniques and are not constructive as far as we see the idea to characterize primary ideals by their associated differential operators is due to w grobner but m noether s fundamentalsatz is based on similar ideas and is obviously the origin of palamodov s terminology

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

regional hydrologic analysis ordinary and generalized least squares revisited generalized least squares gls regional regression procedures have been developed for estimating river flow quantiles a widely used gls procedure employs a simplified model error structure and average covariances when constructing an approximate residual error covariance matrix this paper compares that gls estimator an idealized gls estimator based on the simplifying assumptions of with true underlying statistics in a region the best possible gls estimator obtained using the true residual error covariarice matrix and the ordinary least squares estimator useful analytic expressions are developed for the variance of and for previously examined cases the average sampling mean square error mses of was the same as the mses of and the mses of usually was larger than the mses of both and the loss in efficiency of was mostly due to estimating streamflow statistics employed in the construction of the residual error covariance matrix rather than the simplifying assumptions in presently employed gls estimators the new analytic expressions were used to compare the performance of the ols and gls estimators for new cases representing greater model variability across sites as well as the effect return period has on the estimators relative performance for a more heteroscedastic model error variance and larger return periods some increase in the mses of relative to the mses of was observed

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

zero structures of n d systems a detailed definition of the zero structure of an n d polynomial matrix is given the nature of the invariance of this zero structure is determined in both the polynomial matrix and polynomial system matrix contexts some connections with previous work are made

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

a mollification method for a noncharacteristic cauchy problem for a parabolic equation abstract in this paper the noncharacteristic cauchy problem ncp formula is considered with lp r p 1 it is proved that a solution of ncp exists if and only if is infinitely differentiable and n lp r c 2n s2n n n for certain constantscands ncp is well known to be severely ill posed a small perturbation in the cauchy data may cause a dramatically large error in the solution the following mollification method is suggested for solving ncp in a stable way if lp r is given inexactly by lp r then we mollify by convolutions with the dirichlet kernel and the de la vallee poussin kernel the exact solution of ncp is approximated by the solution of the mollified problem with a reasonable choice of mollification parameters which yields error estimates of the holder type by the method we can work with the data inlp r p 1 and obtain several sharp stability estimates inlp andl norms of the holder type for the solution of the problem the method can easily be implemented numerically using the fast fourier transform a stable marching difference scheme based on this method is suggested several numerical examples are given which show that the method is effective

simplified tikhonov and fourier regularization methods on a general sideways parabolic equation the inverse heat conduction problem ihcp can be considered to be a sideways parabolic equation in the quarter plane and now the results available in the literature on ihcp mainly devoted to the standard sideways heat equation numerical methods have been developed also for more general equations but in most cases the stability theory and convergence proofs have not been generalized accordingly this paper remedies this by a simplified tikhonov and a new fourier regularization methods on a general sideways parabolic equation some known results for sideways heat equation are only the special case of the conclusions in this paper the numerical example shows that the computation effect is satisfactory

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

the input output and controllable behavioural structures of a state space representation given the matrices of a linear state space representation we find an expression for a universal left annihilator of the matrix ctwovec zi a c and hence derive kernel representations for the input output behaviour and by duality the controllable part more generally in the discrete case we derive representations for the l completion for different values of l and the sub behaviours of trajectories reachable in a given time interval the representations are in certain trim canonical forms which are intimately connected with structure indices as a by product of the state elimination procedure we obtain a minimal state space representation for a given behaviour in terms of an arbitrary one

determining surface temperature and heat flux by a wavelet dual least squares method we apply a wavelet dual least squares method to a general sideways parabolic equation for determining surface temperature and surface heat flux connecting meyer wavelet bases with a special project method dual least squares method we can obtain a regularized solution meanwhile order optimal error estimates between the approximate solution and exact solution are proved

development of regional regression relationships with censored data when no discharge record is available for a site a regional regression relationship can be used to estimate low flow quantiles problems arise in the derivation of such models when some at site quantile estimates are reported as zero one concern is that quantile estimates reported as zero may be in the range from zero to the measurement threshold a second concern is that a logarithmic transformation cannot be used with zero quantile estimates so traditional log linear least squares estimators cannot be computed this paper uses visual examples and monte carlo simulation to compare the performance of techniques for estimating the parameters of a regional regression model when some at site quantile estimates are zero ordinary least squares ols techniques employed in practice include adding a small constant to all at site quantile estimates denoted olsc or neglecting all observation reported as zero denoted olsd olsc and olsd performed poorly compared to the use of a tobit model which is a maximum likelihood estimator mle procedure that represents the below threshold estimates as a range from zero to the threshold level for a small amount of censoring the olsd method can be acceptable a weighted tobit model that accounts for the heteroscedasticity of the residuals in the regression model provided relatively little gain over the ordinary tobit model

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

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

module theoretic zero structures for system matrices the foundation for a coordinate free theory of the poles of a linear dynamical system was laid in 1965 by the module theoretic work of kalman however although the theoretical and application importance of transfer function zeros for feedback system design was widely known for single input single output systems by 1955 it remained for rosenbrock in 1970 to propose the ideas of transmission zeros input decoupling zeros and output decoupling zeros for multi input multi output systems by means of matrix theoretic methods a coordinate free module theoretic treatment of transmission zeros for a multi input mufti output transfer function was given in 1981 by wyman and sain this paper extends these coordinate free module theoretic studies to include systems which need not be controllable or observable interpretation of the rosenbrock system matrix is given on three levels rational finitely generated free modular and torsion divisible on the second level an omega zero module z omega is de

a mollification method for ill posed problems a mollification method for a class of ill posed problems is suggested the idea of the method is very simple and natural if the data are given inexactly then we try to find a sequence of mollification operators which map the improper data into well posedness classes of the problem mollify the improper data within these mollified data our problem becomes well posed and when these facts are in hand we try to obtain error estimates and optimal or quasi optimal mollification parameters the method is working not only for problems in hilbert spaces but also for problems in banach spaces applications of the method to concrete problems like numerical differentiation parabolic equations backwards in time the cauchy problem for the laplace equation one and multidimensional non characteristic cauchy problems for parabolic equations in infinite or finite domains give us very sharp stability estimates of h older continuous type in these cases the method is optimal in the sense that it gives the same order of h older continuous dependence on the data as for the regularized problems furthermore the method may be implemented numerically using fast fourier transforms for the first time a uniform stability estimate of h older continuous type of the solution of the heat equation backwards in time in the space l p bbb r for all p in 1 infty could be established by our mollification method a new simple sharp pointwise estimate of h older type for the weak solution of a non characteristic cauchy problem for parabolic equations in a finite domain is established

probe waveforms and deconvolution in the experimental determination of elastic green s functions we propose a new time domain method for the experimental determination of the impulse response of linear systems the technique centers around the use of specifically designed probe waveforms these waveforms are particular c infty approximations to the dirac delta function and the heaviside unit step function and lead to a subsequent time domain deconvolution problem which can be implemented as a cauchy initial value problem this approach allows for continuous deconvolution a powerful option in the presence of noise we orient the discussion to the context of acoustic emission and elastic green s functions and present several numerical reconstructions of sharp signals from smooth synthetic data

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

a spectral regularization method for solving surface heat flux on a general sideways parabolic we consider a general sideways heat equation which is a well known ill posed problem a small noise in the exact data may cause dramatically large errors in the solution the literature available are mainly devoted to determining the temperature with constant coefficients equation in this paper a fourier regularization method for determining surface heat flux is given an error estimate for a general sideways parabolic equation is established

a modified method for a non standard inverse heat conduction problem abstract a non standard inverse heat conduction problem is considered data are given along the line x 1 and the solution at x 0 is sought the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data in order to solve the problem numerically it is necessary to employ some regularization method in this paper we study a modification of the equation where a fourth order mixed derivative term is added error estimates for this equation are given which show that the solution of the modified equation is an approximation of the heat equation a numerical implementation is considered and a simple example is given some numerical results show the usefulness of the modified method

state space realization of 2 d finite dimensional behaviours this paper deals with the state space realization of autonomous autoregressive two dimensional 2 d systems in the context of the behavioural approach an arbitrary autoregressive 2 d system sig

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

causal and stable input output structures on multidimensional behaviors in this work we study multidimensional nd linear differential behaviors with a distinguished independent variable called time we define in a natural way causality and stability of input output structures with respect to this distinguished direction we make an extension of some results in the theory of partial differential equations demonstrating that causality is equivalent to a property of the transfer matrix which is essentially hyperbolicity of the p c operator defining the behavior cal b c 0 y we also quote results which in effect characterize time autonomy for the general systems case stability is likewise characterized by a property of the transfer matrix we prove this result for the 2d case and for the case of a single equation for the general case it requires solution of an open problem concerning the geometry of a particular set in mathbb c n in order to characterize input output stability we also develop new results on inclusions of kernels freeness of variables and closure with respect to cal s cal s prime and associated spaces which are of independent interest we also discuss stability of autonomous behaviors which we believe to be governed by a corresponding condition

strong controlled invariance of behavioural nd systems abstractin this paper we give a stronger version of the notion of behavioral controlled invariance introduced in pereira rocha 2017 in the context of regular partial interconnections in such interconnections the variables are divided into two sets the variables to be controlled and the variables on which it is allowed to enforce restrictions control variables moreover regularity means that the restrictions of the controller do not overlap with the ones already implied by the laws of the original behavior a complete characterization of strong controlled invariance for nd behaviors is derived making use of a special controller behavior known as the canonical controller

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

control systems analysis for the fornasini marchesini 2d systems model progress after four decades the development of a control systems theory for multidimensional linear systems i e systems with more than one independent variable is an area where a very significant volume of research is based

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

central difference schemes in time and error estimate on a non standard inverse heat conduction problem inverse heat conduction problems ihcp are severely ill posed in the sense that the solution if it exists does not depend continuously on the data but now the results available in the literature on ihcp are mainly devoted to the standard sideways heat equation this paper remedies this by a central difference schemes in time which itself has a regularizing effect for a non standard ihcp which appears in some applied subjects an error estimate is obtained and the error estimate also gives information about how to choose the step length in the time discretization a numerical example shows that the computational effect of this method is satisfactory

strong practical stability and stabilization of differential linear repetitive processes abstract differential linear repetitive processes evolve over a subset of the upper right quadrant of the 2d plane where the unique feature is a series of sweeps or passes through a set of dynamics governed by the solution of a linear matrix differential equation over a finite duration t 0 where is termed the pass length or duration on each pass an output termed the pass profile is produced which acts as a forcing function on and hence contributes to the dynamics of the next pass profile the result can be oscillations in the pass to pass direction that cannot be controlled by direct application of standard or 1d linear systems theory the existing stability theory demands a bounded input bounded output property uniformly which in the case of the along the pass dynamics means for t 0 and for k t 0 0 0 0 where the integer k 0 denotes the pass number or index the pass length is always finite however and hence this stability theory could well be too strong in many cases and in particular impose very strong conditions in terms of control law design this paper develops an alternative in such cases by relaxing the requirement for the bounded input bounded output property to hold when k and t simultaneously provides an explanation of the implications of this in the frequency domain and then develops control law design algorithms

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

research on system zeros a survey this report provides a review of notable publications regarding the topic of zeros of multivariable systems in the period from 1970 into 1989 particular attention is paid to relationships between definitions of such zeros and their theoretical development transmission invariant system module infinite and finite zeros are addressed the latter subdivided into a hierarchical structure with respect to information considered in the definition process

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

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