input output stabilization of linear systems on spl zopf a formal framework is set up for the discussion of generalized autoregressive with external input models of the form ay bu where a and b are linear operators with the main emphasis being on signal spaces consisting of bounded sequences parametrized by the integers different notions of stability are explored and topological notions such as the idea of a closed system are linked with questions of stabilizability in this very general context various problems inherent in using spl zopf as the time axis are analyzed in this operatorial framework

binet cauchy kernels on dynamical systems and its application to the analysis of dynamic scenes we propose a family of kernels based on the binet cauchy theorem and its extension to fredholm operators our derivation provides a unifying framework for all kernels on dynamical systems currently used in machine learning including kernels derived from the behavioral framework diffusion processes marginalized kernels kernels on graphs and the kernels on sets arising from the subspace angle approach in the case of linear time invariant systems we derive explicit formulae for computing the proposed binet cauchy kernels by solving sylvester equations and relate the proposed kernels to existing kernels based on cepstrum coefficients and subspace angles we show efficient methods for computing our kernels which make them viable for the practitioner besides their theoretical appeal these kernels can be used efficiently in the comparison of video sequences of dynamic scenes that can be modeled as the output of a linear time invariant dynamical system one advantage of our kernels is that they take the initial conditions of the dynamical systems into account as a first example we use our kernels to compare video sequences of dynamic textures as a second example we apply our kernels to the problem of clustering short clips of a movie experimental evidence shows superior performance of our kernels

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

worst case input output identification we consider worst case l1 identification of causal linear shift invariant systems from time series many results are given on general aspects of identification algorithm performance existence of optimal algorithms robust convergence and input experiment design the identification methodology studied here is compatible with the modelling requirements of modern robust control design

nonnegativity of discrete quadratic functionals corresponding to symplectic difference systems we study the nonnegativity of quadratic functionals with separable endpoints which are related to the discrete symplectic system s in particular we characterize the nonnegativity of these functionals in terms of i the focal points of the natural conjoined basis of s and ii the solvability of an implicit riccati equation associated with s this result is closely related to the kernel condition for the natural conjoined basis of s we treat the situation when this kernel condition is possibly violated at a certain index to accomplish this goal we derive a new characterization of the set of admissible pairs sequences that does not require the validity of the above mentioned kernel condition finally we generalize our results to the variable stepsize case

a behavioral approach to linear exact modeling the behavioral approach to system theory provides a parameter free framework for the study of the general problem of linear exact modeling and recursive modeling the authors present the solution of the continuous time polynomial exponential time series modeling problem both recursive and nonrecursive solutions are provided and classified according to properties like complexity and controllability it is shown in particular that recursive modeling corresponds to updating by means of a cascade inter connection of systems as a special case the solution of several other problems such as rational interpolation realization and modeling of arbitrary discrete time time series is obtained

reduced class groups grafting relative invariants let x t be a regular stable conical action of an algebraic torus on an affine normal conical variety x defined over an algebraically closed field of characteristic zero we define a certain subgroup of cl x t and characterize its finiteness in terms of a finite t equivariant galois descent x of x consequently we show that the action x t is equidimensional if and only if there exists a t equivariant finite galois covering x x such that x t is cofree moreover the order of gal x x is controlled by a certain subgroup of cl x the present result extends thoroughly the equivalence of equidimensionality and cofreeness of x t for a factorial x the purpose of this paper is to evaluate orders of divisor classes associated to modules of relative invariants for a krull domain with a group action this is useful in studying on equidimensional torus actions as above the generalization of r p stanley s criterion for freeness of modules of relative invariants plays an important role in showing key assertions

structural identification of muitivariabie systems by the eigenvector method abstract the structural identification of linear muitivariabie systems can be performed on the basis of the predicted percentage of reconstruction errors ppcre associated to a sequence of increasing complexity models this paper extends some previous results describing the properties of the ppcre for canonical models whose parameters are estimated by means of the eigenvector method

eigenvalue theory for time scale symplectic systems depending nonlinearly on spectral parameter abstract in this paper we develop the eigenvalue theory for general time scale symplectic systems in which the dependence on the spectral parameter is allowed to be nonlinear at the same time we do not impose any controllability or strict normality assumptions we prove the oscillation theorems for eigenvalue problems with dirichlet separated and jointly varying endpoints including the periodic boundary conditions we also allow the boundary conditions depending on the spectral parameter our new theory generalizes and unifies recently published results on continuous time linear hamiltonian systems and discrete symplectic systems with nonlinear dependence on and on time scale symplectic systems with linear dependence on the results of this paper are also new in the special case of linear hamiltonian systems with variable endpoints as well as for sturm liouville dynamic equations

more about continuity of dynamical systems abstract in this note we continue the investigations started in 1 we give sufficient conditions for the convergence of the external behavior of discrete time time invariant finite dimensional linear input state output systems we also state some interesting conjectures

picone type identities and definiteness of quadratic functionals on time scales in this paper we derive a new sufficient condition for the nonnegativity of time scale quadratic functionals associated to time scale symplectic systems to establish this result a new global picone formula is derived another proof of a special case of the result is shown to be obtained via a sturmian comparison technique furthermore we derive several new picone type identities which in particular do not impose a certain delta differentiability assumption and we survey known ones from the literature the results in this paper complete our earlier work on the definiteness of a time scale quadratic functional in terms of its corresponding time scale symplectic system

principal solutions at infinity for time scale symplectic systems without controllability condition abstract in this paper we introduce a new concept of a principal solution at infinity for nonoscillatory symplectic dynamic systems on time scales the main ingredient is that we avoid the controllability or normality condition which is traditionally assumed in this theory in the current literature we show that the principal solutions at infinity can be classified according to the eventual rank of their first component and that the principal solutions exist for all values of the rank between explicitly given minimal and maximal values the minimal value of the rank is connected with the eventual order of abnormality of the system and it gives rise to the so called minimal principal solution at infinity we show that the uniqueness property of the principal solutions at infinity is satisfied only by the minimal principal solution in this study we unify and extend to arbitrary time scales the recently introduced theory of principal and recessive solutions at infinity for possibly abnormal continuous time linear hamiltonian differential systems and discrete time symplectic systems moreover the new theory on time scales also shows that in some results from the continuous time theory the needed assumptions can be simplified

riccati equations for abnormal time scale quadratic functionals abstract this paper focuses on developing new riccati type conditions for an abnormal time scale symplectic system s these conditions provide characterizations of the nonnegativity with and without a certain image condition and positivity of the quadratic functionals associated with such a system the novelty of these conditions rely on the natural conjoined basis x a u a of s in which x a t is not necessarily invertible and thus the system s could be abnormal these results are new even in the special case of continuous time as are some of them in the discrete time setting

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

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

control oriented system identification a worst case deterministic approach in h sub infinity the authors formulate and solve two related control oriented system identification problems for stable linear shift invariant distributed parameter plants in each of these problems the assumed a priori information is minimal consisting only of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the first of these problems involves identification of a point sample of the plant frequency response from a noisy finite output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest this problem leads naturally to the second problem which involves identification of the plant transfer function in h sub infinity from a finite number of noisy point samples of the plant frequency response concrete plans for identification algorithms are provided for each of these two problems

guaranteed stability with subspace methods abstract we show how stability of models can be guaranteed when using the class of identification algorithms which have become known as subspace methods in many of these methods the a matrix is obtained or can be obtained as the product of a shifted matrix with a pseudo inverse we show that whenever the shifted matrix is formed by introducing one block of zeros in the appropriate position then a stable model results the cost of this is some possibly large distortion of the results but in some applications that is outweighed by the advantage of guaranteed stability

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

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

hamilton jacobi theory over time scales and applications to linear quadratic problems abstract in this paper we first derive the verification theorem for nonlinear optimal control problems over time scales that is we show that the value function is the only solution of the hamilton jacobi equation in which the minimum is attained at an optimal feedback controller applications to the linear quadratic regulator problem lqr problem gives a feedback optimal controller form in terms of the solution of a generalized time scale riccati equation and that every optimal solution of the lqr problem must take that form a connection of the newly obtained riccati equation with the traditional one is established problems with shift in the state variable are also considered as an important tool for the latter theory we obtain a new formula for the chain rule on time scales finally the corresponding lqr problem with shift in the state variable is analyzed and the results are related to previous ones

multidimensional realisation theory and polynomial system solving abstractmultidimensional systems are becoming increasingly important as they provide a promising tool for estimation simulation and control while going beyond the traditional setting of one dimen

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

the predictable degree property and row reducedness for systems over a finite ring motivated by applications in communications we consider linear discrete time systems over the finite ring z p r we solve the open problem of deriving a theory of row reduced representations for these systems we introduce a less restrictive form of representation than the adapted form introduced by fagnani and zampieri we call this form the composed form we define the concept of p predictable degree property and p row reduced we demonstrate that these concepts coupled with the composed form provide a natural setting that extends several classical results from the field case to the ring case in particular the classical rank test on the leading row coefficient matrix is generalized we show that any annihilator of a behavior mathfrak b of a pre specified degree is uniquely parametrized with finitely many coefficients in terms of a kernel representation in p row reduced composed form the underlying theory is the theory of reduced p basis for submodules of z q p r xi that is developed in this paper we show how to construct a p row reduced kernel representation in composed form by constructing a reduced p basis for the module mathfrak b perp

identification of stabilizable systems closed loop approximation approximate modelling and identification of linear shift invariant possibly unstable discrete time systems or plants is studied in a framework compatible with the so called robust stability concept for feedback systems this unified framework is based on approximate modelling of the plant in the gap and graph metrics which is achieved here through approximation of certain closed loop transfer functions by finite dimensional systems properties of this approximate inverse modelling approach are studied and concrete rate of approximation results are given furthermore a consistency result in the gap and graph metrics is given for a certain experimental estimate of the plant constructed from closed loop input output data in a stationary noise set up under mild conditions on the unknown plant

time scale symplectic systems without normality abstract we present a theory of the definiteness nonnegativity and positivity of a quadratic functional f over a bounded time scale the results are given in terms of a time scale symplectic system s which is a time scale linear system that generalizes and unifies the linear hamiltonian differential system and discrete symplectic system the novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of f and in establishing equivalent conditions for the nonnegativity of f without any normality assumption to reach this goal a new notion of generalized focal points for conjoined bases x u of s is introduced results on the piecewise constant kernel of x t are obtained and various picone type identities are derived under the piecewise constant kernel condition the results of this paper generalize and unify recent ones in each of the discrete and continuous time setting and constitute a keystone for further development in this theory

a behavioral approach to estimation and dead beat observer design with applications to state space models the observer design problem is investigated in the context of linear left shift invariant discrete behaviors whose trajectories have supports on z necessary and sufficient conditions for the existence of a dead beat observer of some relevant variables from some measured ones in the presence of some unmeasured and irrelevant variables are introduced and a complete parametrization of all dead beat observers is given equivalent conditions for the existence of causal dead beat observers are then derived finally several classical problems addressed for state space models like state estimation the design of unknown input observers or the design of fault detectors and identifiers possibly in the presence of disturbances are cast in this general framework and the aforementioned equivalent conditions and parametrizations are specialized to these cases

on the choice of inputs in identification for robust control the thesis that noisy identification has close ties to the study of the singular value decomposition of perturbed matrices is investigated in particular by assuming an upper bound on the norm of the perturbation one can obtain a convex parametrization of an uncertain family of systems which contains the system generating the data in this approach the second smallest singular value s of an appropriately defined data matrix becomes a quantity of importance as it provides an upper bound for the size of the uncertain family this yields a new tool leading to the design of input functions which are optimal or persistently exciting from the point of view of identification for robust control

continuity of latent variable models the authors study the continuity of the behavior of dynamical systems as a function of the parameters in their behavioral equations a system is defined in terms of its behavior and continuity requires that this behavior be continuous in the limit the problem is demonstrated by means of an example involving an rlc circuit whose port behavior exhibits a surprising discontinuity as a function of the numerical values of the elements in the circuit the main result states that a system described by means of difference equations involving manifest external and latent internal variables will have a continuous behavior in the limit if the limit system is observable

the most powerful unfalsified model for data with missing values abstract the notion of the most powerful unfalsified model plays a key role in system identification since its introduction in the mid 80s many methods have been developed for its numerical computation all currently existing methods however assume that the given data is a complete trajectory of the system motivated by the practical issues of data corruption due to failing sensors transmission lines or storage devices we study the problem of computing the most powerful unfalsified model from data with missing values we do not make assumptions about the nature or pattern of the missing values apart from the basic one that they are a part of a trajectory of a linear time invariant system the identification problem with missing data is equivalent to a hankel structured low rank matrix completion problem the method proposed selects rank deficient complete submatrices of the incomplete hankel matrix under specified conditions the kernels of the submatrices form a nonminimal kernel representation of the data generating system the final step of the algorithm is reduction of the nonminimal kernel representation to a minimal one apart from its practical relevance in identification missing data is a useful concept in systems and control classic problems such as simulation filtering and tracking control can be viewed as missing data estimation problems for a given system the corresponding identification problems with missing data are data driven equivalents of the classical simulation filtering and tracking control problems

new second order optimality conditions for variational problems with c 2 hamiltonians the generalized problem of bolza with a c2 hamiltonian is considered necessary and sufficient conditions are obtained respectively in terms of the accessory problem the existence of conjoined basis and the existence of a solution to a riccati equation with boundary conditions no strengthened legendre clebsch condition is required when applied to a general optimal control problem these results include and generalize known results

a general hankel norm approximation scheme for linear recursive filtering let x t and y t be stochastic processes which are weakly stationary and stationarily correlated we consider the problem of finding an approximate recursive low dimensional filter of x t based on the observation of the past of y t using hankel norm techniques several estimation problems have been investigated in the past using these techniques we present here a general framework which includes many of these approaches as special cases we also discuss some new applications the approximate filter so constructed allows for an a priori bound on the estimation error

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

bias correction in least squares identification abstract the least squares method in system identification leads generally to biased parameter estimates a conceptually simple modification is to estimate the bias and to compute compensated parameter estimates when white output noise is the only disturbance this principle compensated least squares cls can readily be used to obtain consistent estimates the main purpose of the paper is to investigate the accuracy of the parameter estimates obtained when the cls method is used it is proved that the estimates are asymptotically gaussian distributed an explicit expression for the covariance matrix is given it is also shown that a commonly used instrumental variable method gives asymptotically better accuracy than the compensated least squares method

the range error test in the structural identification of linear multivariable systems among the approaches that have been proposed in recent years for the identification of discrete time linear multivariable systems very efficient procedures are based on the introduction of a set of input output canonical models and on the separate determination of the process structure structural identification and parameters parametric identification this paper introduces a new simple test which allows us to perform the structural identification without the drawbacks usually associated with singularity tests the proposed test gives an a priori prediction of the model performance in order to avoid trial and error procedures in the determination of a suitable structure for the process to be identified

linear hamiltonian difference systems disconjugacy and jacobi type conditions abstract we consider a linear hamiltonian difference system for the so called singular case so that discrete sturm liouville equations of higher order are included in our theory we introduce the concepts of focal points for matrix valued and generalized zeros for vector valued solutions of the system and define disconjugacy for linear hamiltonian difference systems we prove a reid roundabout theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional among them the strengthened jacobi s condition and a condition on a certain riccati difference equation the key to this theorem is a discrete version of picone s identity furthermore for the sake of generalization of our theorem we introduce controllability for linear hamiltonian difference systems and prove a reid roundabout theorem for a more general functional and more general boundary conditions

divisorial free modules of relative invariants on krull domains an effective construction of relative invariants plays an important role in the study of finite reflection groups e g j e humphreys reflection groups and coxeter groups cambridge stud adv math vol 29 cambridge univ press cambridge 1990 using a combinatorial method r p stanley cf r p stanley relative invariants of finite groups generated by pseudo reflections j algebra 49 1977 134 148 invariants of finite groups and their applications to combinatorics bull amer math soc n s 1 1979 475 511 generalized such classical result to a criterion for a module sym v of invariants of g relative to a character to be sym v g free of rank one in the case where g is any finite complex subgroup of gl v his criterion is useful in invariant theory of finite groups and in combinatorics in this paper we will study on relative invariants of a group g consisting of automorphisms of a krull domain r from the view point of a generalized partial result on ramifications in number theory mentioned in h nakajima reduced ramification indices of quotient morphisms under torus actions j algebra 242 2001 536 549 we will give a criterion for r to be a free rg module of rank one for a 1 cocycle of g in the unit group u r and consequently establish a criterion for module of relative invariants of a finite central extension of algebraic tori to be free in terms of local characters which is similar to one in h nakajima relative invariants of finite groups j algebra 79 1982 218 234 r p stanley relative invariants of finite groups generated by pseudo reflections j algebra 49 1977 134 148

approximation of linear operators in the 2 norm abstract the problems of approximating linear operators in the 2 induced norm which are 1 finite dimensional unstructured and 2 infinite dimensional structured hankel have been solved the solutions of these two problems exhibit striking similarities these similarities suggest the search of a unifying framework for the approximation of linear operators in the 2 induced norm

driving variable realizations and the non negative realization problem for controllable behaviours non negative linear systems traditionally investigated within the state space framework have been recently analysed within the behavioural setting in a couple of recent papers j w nieuwenhuis when to call a linear system nonnegative linear algebra its appl 281 1998 pp 43 58 m e valcher nonnegative linear systems in the behavioural approach the autonomous case linear algebra and its appl 2000 319 pp 147 162 several definitions and results about non negative behaviours and in particular about non negative autonomous behaviours have been derived moreover the non negative realization problem for autonomous behaviours has been fully explored in m e valcher non negative realization of autonomous systems in the behavioral approach siam j on control and opt 2001 40 pp 540 556 thus deriving an extended set of necessary and sufficient conditions for an autonomous behaviour to admit a non negative realization here we focus our attention on the non negative re

system theory for system identification abstract prediction and control problems require for their solution mathematical models in the form of dynamic systems in system theory the approach to this problem is to investigate realization problems for system identification novel in this paper is the emphasis on the realization problem for observation based realizations and on the realization problem for control a major aspect of these realization problems is computation and complexity algebraic approaches to realization problems may lead to the distinction of subclasses of systems with required finiteness properties to decompositions and to abstractions for hierarchical systems

reduced ramification indices of quotient morphisms under torus actions abstract according to hilbert s theory on ramifications in number theory the p part of the order of the inertia group of a prime divisor of a normal domain of characteristic p with a finite galois group is equal to the p part of its reduced ramification index over its restriction to the fixed subring under the action of the galois group this classical correspondence plays a fundamental role in the study of relative invariants of finite groups on an affine factorial variety in arbitrary characteristics the purpose of this paper is to extend the relationship stated above to one in the case of regular actions of affine algebraic groups and especially in the torus actions our generalization is also regarded as a criterion for finite extensions of tori to be central in terms of invariant theory and seems to be useful in extending the studies on relative invariants of finite groups and related materials and on torus invariants

representations of stochastic processes the purpose of this paper is to extend the deterministic behavioural theory of j c willems to a stochastic setting this extension consists of characterizations of discrete time gaussian processes with rational spectral density

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

matrix lie theory and measure chains in this paper we investigate the relation between lie matrix groups mg mg n k of n n matrices with k entries and their so called lie h algebras lh mg a lie h algebra for a matrix group mg mg n k is defined as the set a mat n k ah i mg if h 0 and a mat n k exp at mg for all t r if h 0 where mat n k is the set of n n matrices with k entries and i mat n k is the identity matrix we investigate the structural relations between the matrix group mg and its lie h algebras there are results concerning the group and topological isomorphy of the group and its lie h algebra for h 0 the subsequent h 0 limiting behaviour hereby we can reveal the exact relation between the commutator in mg and the lie bracket in l0 mg in a final result we give a condition on the linear operator function in x a t x x i which keeps the solution trajectory in mg

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

robust input output stabilization on z for persistent signals robustness of feedback stabilization is studied for causal linear time invariant input output systems defined on the full time axis z the signal space setting used includes also certain unbounded signals this study has been possible due to the use of two operator plant models ay bu this avoids the intrinsic difficulties of unbounded causal input output plant mappings y gu on z the full time axis robust stabilization results obtained here are analogous to those on the semi infinite time axis n in the finite dimensional case however there are certain important differences between the full time axis results and the semi infinite time axis results for infinite dimensional systems even when two operator plant models are used

a structure theory for linear dynamic errors in variables models we deal with problems connected with the identification of linear dynamic systems in situations when inputs and outputs may be contaminated by noise the case of uncorrelated noise components and the bounded noise case is considered if also the inputs may be contaminated by noise a number of additional complications in identification arise in particular the underlying system is not uniquely determined from the population second moments of the observations a description of classes of observationally equivalent systems is given continuity properties of mappings relating classes of observationally equivalent systems to the spectral densities of the observations are derived and the classes of spectral densities corresponding to a given maximum number of outputs are studied

a behavioral approach to time varying linear systems part 1 general theory we develop a behavioral approach to linear time varying differential algebraic systems the analysis is almost everywhere in the sense that the statements hold on mathbb r setminus mathbb t where mathbb t is a discrete set controllability observability and autonomy are introduced and related to the behavior of the system classical results on the behavior of time invariant systems are studied in the context of time varying systems

brief paper axiomatic characterization of linear differential systems and operators in his ieee trans aut contr paper in 1991 willems posed the following question given a set of smooth trajectories when does there exist a linear constant coefficient differential operator whose kernel is precisely the given set we show that the properties that are necessary and sufficient for a set of smooth trajectories to admit a kernel representation are linearity time invariance jet closedness and jet determinedness it is interesting to note that the properties of jet closedness and jet determinedness were introduced by willems himself in his automatica paper in 1986

optimal hankel norm identification of dynamical systems abstract the problem of optimal approximate system identification is addressed with a newly defined measure of misfit between observed time series and linear time invariant models the behavioral framework is used as a suitable axiomatic setting for a non parametric introduction of system complexity and allows for a notion of misfit of dynamical systems that is independent of system representations the misfit function introduced here is characterized in terms of the induced norm of a hankel operator associated with the data and a co inner kernel representation of a model sets of pareto optimal models are defined as feasible trade offs between complexity and misfit of models and it is shown how all pareto optimal models are characterized as exact models of compressed data sets obtained from hankel norm approximations of data matrices this leads to new conceptual algorithms for optimal approximate identification of time series 1997 elsevier science ltd

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

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

symplectic difference systems variable stepsize discretization and discrete quadratic functionals discrete quadratic functionals with variable endpoints for variable stepsize symplectic difference systems are considered a comprehensive study is presented for characterizing the positivity of such functionals in terms of conjugate intervals conjoined bases and implicit and explicit riccati equations with various forms of boundary conditions moreover necessary conditions for the nonnegativity of these functionals are obtained in terms of the above notions furthermore we show that a variable stepsize discretization of a continuous time nonlinear control problem leads to a discrete linear quadratic problem and a hamiltonian difference system which are special cases of their symplectic counterparts

criterion based equivalence for equation error models the problem of modeling an input output data sequence is considered when applying equation error least squares identification methods to matrix fraction description mfd models in the case that a set of models has been parametrized using the notion of i o input output equivalence it is shown that the identified model will be essentially dependent on the specific parameterization chosen as an alternative a notion of model equivalence is introduced that is related to the identification criterion the pseudocanonical or overlapping parametrization of mfd models appears to constitute a set of canonical forms for this least squares based model equivalence

dipolynomial minimal bases and linear systems in ar representation abstract this paper deals with a module theoretic approach to the dipolynomial matrix parametrization of discrete time behavior systems as introduced by willems canonical minimal lag representations for these behaviors are constructed which are tighter than the corresponding polynomial representations

state space approach to behavioral systems theory the dirac bergmann algorithm this paper develops an approach to behavioral systems theory in which a state space representation of behaviors is utilised this representation is a first order hybrid representation of behaviors called pencil representation an algorithm well known after dirac and bergmann db is shown to be central in obtaining a constraint free and observable cfo state space representation of a behavior results and criteria for asymptotic stability controllability inclusions and markovianity of behaviors are derived in terms of the matrices of this representation which involve linear algebraic processes in their computation

brief optimal errors in variables filtering this paper deals with optimal minimal variance filtering in an errors in variables framework differently from many other contexts errors in variables models treat all variables in a symmetric way no partition of the variables into inputs and outputs is required and assume additive noise on all the variables the filtering technique described in this paper can be easily implemented in a recursive way and does not require the use of a riccati equation at every update the results of monte carlo simulations have shown the effectiveness and consistency of the approach

applications of time scale symplectic systems without normality recently the authors obtained new characterizations of the positivity and nonnegativity of a time scale quadratic functional f with separable endpoints related to a time scale symplectic system s in these results the assumption of normality is absent in this paper we present applications of such results namely without assuming normality we derive sturmian comparison theorems results for general jointly varying endpoints and characterizations of the positivity of f via the corresponding time scale riccati equation a certain perturbed quadratic functional and a time scale riccati inequality these results generalize and unify many recent as well as classical ones

the rank of reduced dispersion matrices psychometricians working in factor analysis and econometricians working in regression with measurement error in all variables are both interested in the rank of dispersion matrices under variation of the diagonal elements psychometricians concentrate on cases in which low rank can be attained preferably rank one the spearman case econometricians cocentrate on cases in which the rank cannot be reduced below the number of variables minus one the frisch case in this paper we give an extensive historial discussion of both fields we prove the two key results in a more satisfactory and uniform way we point out various small errors and misunderstandings and we present a methodological comparison of factor analysis and regression on the basis of our results

consistency of system identification by global total least squares global total least squares gtls is a method for the identification of linear systems where no distinction between input and output variables is required this method has been developed within the deterministic behavioural approach to systems in this paper we analyse statistical properties of this method when the observations are generated by a multivariable stationary stochastic process in particular sufficient conditions for the consistency of gtls are derived this means that when the number of observations tends to infinity the identified deterministic system converges to the system that provides an optimal appoximation of the data generating process the two main results are the following gtls is consistent if a guaranteed stability bound can be given a priori if this information is not available then consistency is obtained if gtls is applied to the observed data extended with zero values in past and future

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

linear quadratic optimization for systems in the behavioral approach in this paper the following formulation of the linear quadratic optimal control problem for dynamical systems in the behavioral setting is proposed given a linear time invariant and complete system find the set of trajectories of the system that optimize a quadratic type cost function and that satisfy some linear static constraints this formulation provides a unifying framework where several generalized versions of the classical lq optimal control can be stated and solved the existence of solutions is first discussed it is shown that a necessary and sufficient condition for the existence of solutions may be obtained as a by product of a reduction procedure translating the problem into an equivalent one of minimum complexity such a procedure is based on the theory of ell 2 systems in the behavioral setting once the complexity is reduced a parametrization of the set of optimal solutions is obtained by employing a behavioral realization technique and a two step optimization procedure

oscillation theorems and rayleigh principle for linear hamiltonian and symplectic systems with general boundary conditions the aim of this paper is to establish the oscillation theorems rayleigh principle and coercivity results for linear hamiltonian and symplectic systems with general boundary conditions i e for the case of separated and jointly varying endpoints and with no controllability normality and strong observability assumptions our method is to consider the time interval as a time scale and apply suitable time scales techniques to reduce the problem with separated endpoints into a problem with dirichlet boundary conditions and the problem with jointly varying endpoints into a problem with separated endpoints these more general results on time scales then provide new results for the continuous time linear hamiltonian systems as well as for the discrete symplectic systems this paper also solves an open problem of deriving the oscillation theorem for problems with periodic boundary conditions furthermore the present work demonstrates the utility and power of the analysis on time scales in obtaining new results especially in the classical continuous and discrete time theories

when to call a linear system nonnegative abstract in this paper we will consider discrete time invariant linear systems that allow for an input state output representation with a finite dimensional state space and that have a finite number of inputs and outputs the basic issue in this paper is when to call these systems nonnegative an important concept in this respect is that of the most powerful unfalsified model

quivers geometric invariant theory and moduli of linear dynamical systems we use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems a general approach to this problem is presented and applied to two well known cases we show how both lomadze s and helmke s compactification arises naturally as a geometric invariant theory quotient both moduli spaces are proven to be smooth projective manifolds furthermore a description of lomadze s compactification as a quot scheme is given whereas helmke s compactification is shown to be an algebraic grassmann bundle over a quot scheme this gives an algebro geometric description of both compactifications as an application we determine the cohomology ring of helmke s compactification and prove that the two compactifications are not isomorphic when the number of outputs is positive