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 new approach to the derivation of binary non markovian kinetic equations abstract a universal method of derivation of infinite hierarchies for partial distribution functions and correlation forms in the thermodynamic limit has been developed it is based on the consideration of reacting systems in the fock space hierarchy closure methods available in the literature are shown to give incorrect binary kinetic equations of the reaction a b b in some critical cases a new approach to hierarchy closure has been proposed it consists in neglecting contributions from four particle correlations and in adapting the faddeev method of the three body theory to the extraction of a binary part of three particle evolution for the model of the reaction a b b the proposed method gives correct kinetic equations obtained earlier on the basis of diagram summation it gives the theoretical basis for derivation of binary kinetic equations for realistic reacting systems

radon nikodym theorem and vitali hahn saks theorem on fuzzy number measures in banach spacers abstract radon nikodym theorem is established on fuzzy number measures in banach spaces vitali hahn saks theorem is generalized to set valued measures and fuzzy number measures

optimal robust stabilization and dissipativity synthesis by behavioral interconnection given a nominal plant together with a fixed neighborhood of this plant the problem of robust stabilization is to find a controller that stabilizes all plants in that neighborhood in an appropriate sense if a controller achieves this design objective we say that it robustly stabilizes the nominal plant in this paper we formulate the robust stabilization problem in a behavioral framework with control as interconnection we also formulate a relevant behavioral mathcal h infty synthesis problem which will be instrumental in solving the robust stabilization problem we use both rational and polynomial representations for the behaviors under consideration necessary and sufficient conditions for the existence of robustly stabilizing controllers are obtained using the theory of dissipative systems we will also find the optimal stability radius i e the smallest upper bound on the radii of the neighborhoods for which there exists a robustly stabilizing controller this smallest upper bound is expressed in terms of certain storage functions associated with nominal control system

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

synthesis of dissipative systems using quadratic differential forms part ii the problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative our formulation and solution of this problem is completely representation free the system dynamics are specified by a differential behavior and the performance is specified through a quadratic differential form we view control as interconnection a controller constrains a distinguished set of system variables the control variables the resulting behavior of the to be controlled variables is called the controlled behavior the constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior obtained by setting the control variables equal to zero and the plant behavior obtained by leaving the control variables unconstrained the main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications the essential requirement is a coupling condition an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior

state maps for linear systems modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential algebraic equations as such state space models are not a natural starting point for modeling while they have utmost importance in the simulation and control phase the paper addresses the problem of computing state variables for systems of linear differential algebraic equations of various forms the point of view from which the problem is considered is the behavioral one as put forward in j c willems automatica j ifac 22 1986 pp 561 580 dynamics reported 2 1989 pp 171 269 ieee trans automat control 36 1991 pp 259 294

every storage function is a state function it is shown that for linear dynamical systems with quadratic supply rates a storage function can always be written as a quadratic function of the state of an associated linear dynamical system this dynamical system is obtained by combining the dynamics of the original system with the dynamics of the supply rate

behavioral controllability of delay differential systems in this paper we will prove that the system described by the delay differential equation r d dt delta w 0 with delta the unit delay operator is controllable if and only if the rank of r lambda e lambda is constant for all lambda in bkc this condition is compared with the existing results obtained both by the analytic approach and by the algebraic approach to delay differential systems

on polynomial matrix spectral factorization by symmetric extraction we revise the 1963 davis algorithm 2 for the spectral factorization of a para hermitian nonnegative polynomial matrix phi by symmetric factor extraction this algorithm is careless about zeros at infinity by introducing the notion of diagonal reducedness of phi we obtain an easy sufficient test for the absence of zeros at infinity we show then how to get phi diagonally reduced by diagonal excess reduction steps similar to the oono and yasuura steps removing all zeros at infinity and then how to remove synunetrically finite zeros while keeping el diagonally reduced hence free of zeros at infinity this results in a revised symmetric extraction spectral factorization algorithm with monotone degree control an example shows the didactical conceptual simplicity of the method appropriate symmetric extraction is discovered by revising and discovering important particular one sided factor extraction properties of polynomial matrices

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

a many particle derivation of non markovian kinetic equations of reversible reaction a b c in solutions based on the effective pairs approach abstract the non markovian kinetic equations for local concentrations have been derived in the effective pairs approximation on the basis of the hierarchy for correlation patterns of the reversible reaction a b c the equations are shown to agree with the kinetic and thermodynamic detailed balancing principles as compared to the integral encounter theory equations obtained earlier the above equations have a wider time range of applicability and may form the basis for the development of a binary approximation for the reversible reaction a b c in solutions

lq control of behavior systems in kernel representation abstract in this paper we consider linear systems given through a kernel representation p together with a quadratic cost functional h we will see that the solvability of the associated infinite horizon optimal control problem is equivalent to cyclo dissipativity and that the solution of the optimal control problem can be obtained from the kernel representation of a para hermitian matrix polynomial which can be easily constructed from p and h in contrast to previous work on this topic we avoid image representations as far as possible the results can be applied to descriptor systems see section 4 to obtain results which so far were only known for regular descriptor systems with index 1

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

stability of systems with the duhem hysteresis operator in this paper we discuss the dissipativity property of the counterclockwise duhem operator sufficient conditions on the functions which define the duhem operator are given such that the duhem operator has counterclockwise input output dynamics in particular an explicit construction of the storage functions satisfying the counterclockwise dissipation inequality is given the constructed storage function is also related to the underlying anhysteresis function which is commonly used to describe hysteresis in magnetic materials the results can thus facilitate analysis of systems with the counterclockwise duhem operator via the dissipativity approach

t matrix representation and long time behavior of observables in the theory of migration influenced irreversible reactions in liquid solutions abstract the theory of migration influenced irreversible reactions upon binary encounters of reactants in liquid solutions is formulated in terms similar to those of quantum scattering theory free motion of a reacting pair in the configuration space is a random walk in three dimensional infinite space and arbitrary markovian motion over internal degrees of freedom in the case of mixing by free motion general asymptotic properties of the free resolvent describing this motion have been established general long time kinetic law of the attainment of steady state values by the observables in bulk and geminate reactions has been deduced thermodynamically not only the universality of their long time dependence is important but also the fact that the rate of attaining the steady state values is completely determined by macroscopic quasi equilibrium observables

the kalman yakubovich popov lemma in a behavioural framework the classical kalman yakubovich popov lemma provides a link between dissipativity of a system in state space form and the solution to a linear matrix inequality in this paper we derive the kyp lemma for linear systems described by higher order differential equations the result is an lmi in terms of the original coefficients in which the dissipativity problem is posed subsequently we study the connection between dissipativity and spectral factorization of polynomial matrices this enables us to derive a new algorithm for polynomial spectral factorization in terms of an lmi in the coefficients of a polynomial matrix

closed form solutions of singular kyp lemma strongly passive systems and fast lossless trajectories in this paper we deal with a special class of passive systems which possess the characteristic property of having no finite spectral zeros we call these systems strongly passive it is well know

from time series to linear system part ii exact modelling abstract in the second part of this paper the problem of finding an exact model for a q dimensional infinite time series is considered first a mathematical vocabulary for discussing exact modelling is developed it is then shown how the results of part i guarantee the existence of a most powerful ar model for an observed time series two algorithms for obtaining such an ar model are subsequently derived one of these algorithms gives a shortest lag input output model the problem of obtaining a minimal state space realization of the observed time series is also considered in order to do that realization theory based on the truncated behaviour is developed as an extensive example the classical situation with impulse response measurements is discussed

dissipativity of pseudorational behaviors this paper studies dissipativity for a class of infinite dimensional systems called pseudorational in the behavioral context extending the finite dimensional counterpart we show that a pseudorational behavior is dissipative if and only if it admits a storage function or a dissipation function for its proof we derive a new necessary and sufficient condition for entire functions of exponential type in the so called paley wiener class to allow a symmetric factorization characterization of dissipative behaviors and linear quadratic optimal behaviors are also given

the strict dissipativity synthesis problem and the rank of the coupling qdf the problem of existence of a controlled behavior that is strictly dissipative with respect to a quadratic supply rate is studied the relation between strictness and the rank of a suitable coupling condition that combines the dissipativity properties of the hidden behavior and the orthogonal complement of the plant behavior is analyzed

systems with sector bound nonlinearities a behavioral approach the class of nonlinear dynamical systems obtained by interconnection of a linear time invariant dynamical system and a memoryless time invariant sector bound nonlinearity has been widely studied in literature in this paper it is shown that many important results for this class of systems can be elegantly unified using behavioral theory of dynamical systems systems with slope restricted nonlinearities are also considered

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

a characterization of solutions of the discrete time algebraic riccati equation based on quadratic difference forms this paper is concerned with a characterization of all symmetric solutions to the discrete time algebraic riccati equation dare dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper along the line of the continuous time results due to trentelman and rapisarda h l trentelman p rapisarda pick matrix conditions for sign definite solutions of the algebraic riccati equation siam j contr optim 40 3 2001 969 991 we show that the solvability of the dare is equivalent to a certain dissipativity of the associated discrete time state space system as a main result we characterize all unmixed solutions of the dare using the pick matrix obtained from the quadratic difference forms this characterization leads to a necessary and sufficient condition for the existence of a non negative definite solution it should be noted that when we study the dare and the dissipativity of the discrete time system there exist two difficulties which are not seen in the continuous time case one is the existence of a storage function which is not a quadratic function of state another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the dare due to the infinite generalized eigenvalues of the associated hamiltonian pencil one of the main contributions of this paper is to demonstrate how to resolve these difficulties

dissipative switched linear differential systems the authors develop a dissipativity theory for switched systems whose dynamical modes are described by systems of higher order linear differential equations they give necessary and sufficient conditions for dissipativity based on systems of linear matrix inequalities lmis constructed from the coefficient matrices of the differential equations describing the modes the relationship between dissipativity and stability is also discussed and an application to the stabilization of power converters is provided

a theory of passive linear systems with no assumptions we present two linked theorems on passivity the passive behavior theorem parts 1 and 2 part 1 provides necessary and sufficient conditions for a general linear system described by a set of high order differential equations to be passive part 2 extends the positive real lemma to include uncontrollable and unobservable state space systems

totally dissipative systems in a totally dissipative behavior all non trivial trajectories dissipate energy a characterization of such behaviors is given in terms of properties of the one and two polynomial matrices associated with the supply rate and with their kernel and image representation

fuzzy valued measure abstract the main purpose of this paper is to introduce and develop the notion of a fuzzy valued measure in separable banach space this definition of fuzzy valued measure is a natural generalization of the set valued measure radon nikod m theorems for fuzzy valued measure are established

a many particle derivation of the integral encounter theory non markovian kinetic equations of the reversible reaction a b c in solutions the non markovian binary kinetic equations for a rather realistic model of association dissociation reaction a b c in liquid solutions are derived using a newly developed method based on a many particle consideration the method leads to an infinite hierarchy for correlation patterns that can be truncated at the first step on a two particle level to give the integral encounter theory result it is shown that although being restricted in time the theory predicts true equilibrium concentrations of reactants and the equilibrium constant corresponding to the thermodynamic detailed balancing principle

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

on the extension of the fuzzy number measures in banach spaces part i representation of the fuzzy number measures abstract in this paper the definition of the fuzzy number measures which extends the concepts of the set valued measures in banach spaces is introduced the relations between the fuzzy number measures and the set valued measures are studied the definition of the indefinite integral of the random fuzzy numbers is also introduced and its properties are discussed

dissipation inequalities for the analysis of a class of pdes in this paper we develop dissipation inequalities for a class of well posed systems described by partial differential equations pdes we study passivity reachability induced input output norm boundedness and input to state stability iss we consider both cases of in domain and boundary inputs and outputs we study the interconnection of pde pde systems and formulate small gain conditions for stability for pdes polynomial in dependent and independent variables we demonstrate that sum of squares sos programming can be used to compute certificates for each property therefore the solution to the proposed dissipation inequalities can be obtained via semi definite programming the results are illustrated with examples

radon nikody m theorem and conditional expectation of fuzzy valued measures and variables abstract the notions of fuzzy valued measure and conditional expectation of a fuzzy valued variable are defined the main result of the paper is a proof of the existence of the radon nikod m derivative of the fuzzy valued measure a convergence theorem for conditional expectations is proved as well

deterministic kalman filtering in a behavioral framework the purpose of this paper is to obtain a deterministic version of the kalman filtering equations we will use a behavioral description of the plant specifically an image representation the resulting algorithm requires a matrix spectral factorization we also show that the filter can be implemented recursively

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

generalizing the algebraic riccati equation to higher order behavioral systems the well known kalman yakubovich popov lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality see r a b van der geest and h l trentelman systems control lett 32 1997 pp 283 290 i masubuchi systems control lett 55 2006 pp 158 164 in this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of the so called lur e equation t reis linear algebra appl 434 2011 pp 152 173 which mainly is a linear matrix inequality with a rank minimizing condition while the results about the solvability of the algebraic riccati equation j c willems ieee trans automat control 16 1971 pp 621 634 cannot be obtained from the kalman yakubovich popov lemma we will see that with the lur e equation this is possible

random fuzzy number integrals in banach spaces abstract in this paper a new kind of random fuzzy number integral is introduced by the pettis aumann integral of the set valued functions in banach spaces the existence of the random fuzzy integral is also shown levi s convergence theorem of the integral is given a special mention

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

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

a minimal state map for strictly lossless positive real behaviors in this paper behaviors with equal input and output cardinalities and with lossless positive real transfer functions corresponding to certain input output partitions are considered based on cauer synthesis method an algorithm to obtain a minimal state map of such behaviors starting from an observable image representation is given this algorithm makes use of polynomial algebraic methods to arrive at a minimal state map it is shown that the algorithm has application in synthesis and analysis of lossless electrical networks

on the characterization of the duhem hysteresis operator with clockwise input output dynamics abstract in this paper we investigate the dissipativity property of a certain class of duhem hysteresis operator which has clockwise cw input output i o behavior in particular we provide sufficient conditions on the duhem operator such that it is cw and propose an explicit construction of the corresponding function satisfying dissipation inequality of cw systems the result is used to analyze the stability of a second order system with hysteretic friction which is described by a dahl model

dissipativity of uncontrollable systems storage functions and lyapunov functions dissipative systems have played an important role in the analysis and synthesis of dynamical systems the commonly used definition of dissipativity often requires an assumption on the controllability of the system in this paper we use a definition of dissipativity that is slightly different and less often used in the literature to study a linear time invariant possibly uncontrollable dynamical system we provide a necessary and sufficient condition for an uncontrollable system to be strictly dissipative with respect to a supply rate under the assumption that the uncontrollable poles are not mixed i e no pair of uncontrollable poles is symmetric about the imaginary axis this condition is known to be related to the solvability of a lyapunov equation we link lyapunov functions for autonomous systems to storage functions of an uncontrollable system the set of storage functions for a controllable system has been shown to be a convex bounded polytope in the literature we show that for an uncontrollable system the set of storage functions is unbounded and that the unboundedness arises precisely due to the set of lyapunov functions for an autonomous linear system being unbounded further we show that stabilizability of a system results in this unbounded set becoming bounded from below positivity of storage functions is known to be very important for stability considerations because the maximum stored energy that can be drawn out is bounded when the storage function is positive in this paper we establish the link between stabilizability of an uncontrollable system and existence of positive definite storage functions in most of the results in this paper we assume that no pair of the uncontrollable poles of the system is symmetric about the imaginary axis we explore the extent of necessity of this assumption and also prove some results for the case of single output systems regarding this necessity

stabilization lyapunov functions and dissipation for linear time invariant systems any stabilizing controller for a given plant can be associated with a supply rate with respect to which the plant in open loop is half line dissipative we also prove the equivalence between the stability of the interconnection of two systems and their half line dissipativity with respect to a supply rate and its negative

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

realization of lossless systems via constant matrix factorizations we study the realization problem for linear time invariant systems described by higher order differential equations which are j lossless with constant j our approach is based on the factorization of a constant matrix obtained in a straightforward way from the storage function of the system state equations are obtained directly from this factorization and from the original system representation

kalman yakubovich lemma in the behavioural setting in this paper the kalman yakubovich lemma is formulated and proved in the behavioural theoretic setting the behavioural framework is used to generalize the lemma for behaviours with two manifest variables with respect to supply rates that are quadratic differential forms qdfs

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem