linear and nonlinear algorithms for identification in h sub infinity with error bounds a linear algorithm and a nonlinear algorithm for the problem of system identification in h sub infinity posed by helmicki et al 1990 for discrete time systems are presented the authors derive some error bounds for the linear algorithm which indicate that it is not robustly convergent however the worst case identification error is shown to grow as log n where n is the model order a robustly convergent nonlinear algorithm is derived and bounds on the worst case identification error in the h sub infinity norm are obtained

schur recursions error formulas and convergence of rational estimators for stationary stochastic sequences an exact and approximate realization theory for estimation and model filters of second order stationary stochastic sequences is presented the properties of j lossless matrices as a unifying framework are used necessary and sufficient conditions for the exact realization of an estimation filter and a model filter as a submatrix of a j lossless system are deduced an extension of the so called schur algorithm yields an approximate j lossless realization based on partial past information about the process the geometric properties of such partial realizations and their convergence are studied finally connections with the nevanlinna pick problem are made and how the techniques presented constitute a generalization of many aspects of the levinson szego theory of partial realizations is shown as a consequence generalized recursive formulas for reproducing kernels and christoffel darboux formulas are obtained in this paper the scalar case is considered the matrix case will be considered in a separate publication

on the estimation of transfer functions this paper treats the close conceptual relationships between basic approaches to the estimation of transfer functions of linear systems the classical methods of frequency and spectral analysis are shown to be related to the well known time domain methods of prediction error type via a common empirical transfer function estimate asymptotic properties of the estimates obtained by the respective methods are also described and discussed an important feature that is displayed by this treatment is a frequency domain weighting function that determines the distribution of bias in case the true system cannot be exactly described within the chosen model set the choice of this weighting function is made in terms of noise models for time domain methods the noise model thus has a dual character from the system approximation point of view

on approximation of stable linear dynamical systems using laguerre and kautz functions approximation of stable linear dynamical systems by means of so called laguerre and kautz functions which are the laplace transforms of a class of orthonormal exponentials is studied since the impulse response of a stable finite dimensional linear dynamical system can be represented by a sum of exponentials times polynomials it seems reasonable to use basis functions of the same type assuming that the transfer function of a system is bounded and analytic outside a given disc it is shown that laguerre basis functions are optimal in a mini max sense this result is extended to the two parameter kautz functions which can have complex poles while the poles of laguerre functions are restricted to the real axis by conformai mapping techniques the two parameter kautz approximation problem is recast as two laguerre approximation problems thus the well developed theory of laguerre functions can be applied to analyze kautz approximations unilateral shifts are used to further develop the connection between laguerre functions and kautz functions results on 2 and approximation using kautz models are given furthermore the weighted l2 kautz approximation problem is shown to be equivalent to solving a block toeplitz matrix equation

robust identification in h abstract we consider system identification in the banach space h in the framework proposed by helmicki jacobson and nett it is shown that there is no robustly convergent linear algorithm for identifying exponentially stable systems in the presence of noise which is not tuned to prior information about the unknown system or noise various nonlinear algorithms some closely related to one of gu and khargonekar are analysed and results on trigonometric interpolation used to provide new error bounds an application of these techniques to approximation is given and finally some numerical results are provided for illustration

asymptotically optimal orthonormal basis functions for lpv system identification a global model structure is developed for parametrization and identification of a general class of linear parameter varying lpv systems by using a fixed orthonormal basis function obf structure a linearly parametrized model structure follows for which the coefficients are dependent on a scheduling signal an optimal set of obfs for this model structure is selected on the basis of local linear dynamic properties of the lpv system system poles that occur for different constant scheduling signals the selected obf set guarantees in an asymptotic sense the least worst case modeling error for any local model of the lpv system through the fusion of the kolmogorov n width theory and fuzzy c means clustering an approach is developed to solve the obf selection problem for discrete time lpv systems based on the clustering of observed sample system poles

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

orthogonal rational functions for system identification numerical aspects recently there has been a growing interest in the use of orthogonal rational functions orfs in system identification there are many advantages over more classical techniques probably due to a known explicit expression for the basis functions when the orthogonality weight is uniformly equal to 1 the so called malmquist basis the attention has been on the development of methods using this basis however for some discrete identification problems this choice of the orthogonality weight may still lead to serious numerical problems due to the ill conditioning of the linear system of equations to be solved in this note we give an algorithm based on a more general system of orf to overcome the numerical problem and which allows for a fast order update of the estimate

variable neighborhood search principles and applications abstract systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization called variable neighborhood search vns we present a basic scheme for this purpose which can easily be implemented using any local search algorithm as a subroutine its effectiveness is illustrated by solving several classical combinatorial or global optimization problems moreover several extensions are proposed for solving large problem instances using vns within the successive approximation method yields a two level vns called variable neighborhood decomposition search vnds modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed vns svns heuristic finally we show how to stabilize column generation algorithms with help of vns and discuss various ways to use vns in graph theory i e to suggest disprove or give hints on how to prove conjectures an area where metaheuristics do not appear to have been applied before

approximation of infinite dimensional systems a fourier series based method for approximation of stable infinite dimensional linear time invariant system models is discussed the basic idea is to compute the fourier series coefficients of the associated transfer function t sub d z and then take a high order partial sum two results on h sup infinity convergence and associated error bounds of the partial sum approximation are established it is shown that the fourier coefficients can be replaced by the discrete fourier transform coefficients while maintaining h sup infinity convergence thus a fast fourier transform algorithm can be used to compute the high order approximation this high order finite dimensional approximation can then be reduced using balanced truncation or optimal hankel approximation leading to the final finite dimensional approximation to the original infinite dimensional model this model has been tested on several transfer functions of the time delay type with promising results

the heterogeneous p median problem for categorization based clustering the p median offers an alternative to centroid based clustering algorithms for identifying unobserved categories however existing p median formulations typically require data aggregation into a single proximity matrix resulting in masked respondent heterogeneity a proposed three way formulation of the p median problem explicitly considers heterogeneity by identifying groups of individual respondents that perceive similar category structures three proposed heuristics for the heterogeneous p median hpm are developed and then illustrated in a consumer psychology context using a sample of undergraduate students who performed a sorting task of major u s retailers as well as a through monte carlo analysis

robust identification and interpolation in h abstract we consider system identification in h in the framework proposed by helmicki jacobson and nett an algorithm using the jackson polynomials is proposed that achieves an exponential convergence rate for exponentially stable systems it is shown that this and similar identification algorithms can be successfully combined with a model reduction procedure to produce low order models connections with the nevanlinna pick interpolation problem are explored and an algorithm is given in which the identified model interpolates the given noisy data some numerical results are provided for illustration finally the case of unbounded random noise is discussed and it is shown that one can still obtain convergence with probability 1 under natural assumptions

paper on the estimation of transfer functions this paper treats the close conceptual relationships between basic approaches to the estimation of transfer functions of linear systems the classical methods of frequency and spectral analysis are shown to be related to the well known time domain methods of prediction error type via a common empirical transfer function estimate asymptotic properties of the estimates obtained by the respective methods are also described and discussed an important feature that is displayed by this treatment is a frequency domain weighting function that determines the distribution of bias in case the true system cannot be exactly described within the chosen model set the choice of this weighting function is made in terms of noise models for time domain methods the noise model thus has a dual character from the system approximation point of view

silhouettes a graphical aid to the interpretation and validation of cluster analysis a new graphical display is proposed for partitioning techniques each cluster is represented by a so called silhouette which is based on the comparison of its tightness and separation this silhouette shows which objects lie well within their cluster and which ones are merely somewhere in between clusters the entire clustering is displayed by combining the silhouettes into a single plot allowing an appreciation of the relative quality of the clusters and an overview of the data configuration the average silhouette width provides an evaluation of clustering validity and might be used to select an appropriate number of clusters

a frequency domain iterative identification algorithm using general orthonormal basis functions in this paper a two step frequency domain identification algorithm is proposed in which the identified model is parameterized in terms of general orthonormal basis functions furthermore an update scheme is presented where the poles of the basis functions are adjusted iteratively and the convergence properties of this scheme with respect to unknown but bounded noise are studied the algorithm is applied to experimental frequency response data obtained from a compact disk player

robust identification in the disc algebra using rational wavelets and orthonormal basis functions worst case identification using corrupted frequency response measurements is considered in a framework which allows one to consider a variety of rational basis functions these include laguerre models kautz models and rational wavelets conditions are given for model sets to be complete in the disc algebra and examples are given explicit error bounds are provided for approximate modelling using specific basis functions

identification and application of bounded parameter models abstract the place of parameter bounding algorithms in identification methodology is discussed such algorithms provide a radical alternative to the computation of parameter point estimates and covariances instead of p d f or mean and covariance for the noise and prior parameter estimates they require bounds in an essentially deterministic model formulation the paper is a preliminary review of some standard tasks namely experiment design testing for outliers toleranced prediction and worst case control design in the context of parameter bounding identification computational requirements of these tasks are examined and the limitations of existing parameter bounding algorithms are noted

a class of algorithms for identification in h abs traet in this paper a class of algorithms for the problem of system identification in are investigated these algorithms are characterized by a two stage structure and involve a class of window functions some conditions in terms of properties of the window functions are derived which guarantee robust convergence of the algorithms identification errors are analyzed for several common window functions this leads to some insights into the trade off between the error induced by approximation and that due to noise

a stochastic multidimensional scaling procedure for the spatial representation of three mode three way pick any j data this paper presents a new stochastic multidimensional scaling procedure for the analysis of three mode three way pick any j data the method provides either a vector or ideal point model to represent the structure in such data as well as floating model specifications e g different vectors or ideal points for different choice settings and various reparameterization options that allow the coordinates of ideal points vectors or stimuli to be functions of specified background variables a maximum likelihood procedure is utilized to estimate a joint space of row and column objects as well as a set of weights depicting the third mode of the data an algorithm using a conjugate gradient method with automatic restarts is developed to estimate the parameters of the models a series of monte carlo analyses are carried out to investigate the performance of the algorithm under diverse data and model specification conditions examine the statistical properties of the associated test statistic and test the robustness of the procedure to departures from the independence assumptions finally a consumer psychology application assessing the impact of situational influences on consumers choice behavior is discussed

convex fuzzy k medoids clustering abstract k medoids clustering is among the most popular methods for cluster analysis despite its use requiring several assumptions about the nature of the latent clusters in this paper we introduce the convex fuzzy k medoids cfkm model which not only relaxes the assumption that objects must be assigned entirely to one and only one medoid but also that medoids must be assigned entirely to one and only one cluster the resulting model is convex thus its resolution is completely robust to initialization to illustrate the usefulness of the cfkm model we compare it with two fuzzy k medoids clustering models the fuzzy k medoids fkm and the fuzzy clustering with multi medoids fmmdd both solved approximately by heuristics because of their hard computational complexity our experiments with both synthetic and real world data as well as a user survey reveal that the model is not only more robust to the choice hyperparameters of the fuzzy clustering task but also that it can uniquely discover important aspects of data inherently fuzzy in nature

periodic solutions of abel differential equations for a class of polynomial non autonomous differential equations of degree n we use phase plane analysis to show that each equation in this class has n periodic solutions the result implies that certain rigid two dimensional systems have at most one limit cycle which appears through multiple hopf bifurcation

extended ho kalman algorithm for systems represented in generalized orthonormal bases this paper considers the construction of minimal state space models of linear time invariant systems on the basis of system representations in terms of generalized orthogonal basis function expansions starting from the classical ho kalman algorithm that solves the problem using markov parameter expansions a generalization is obtained by analysing the matrix representations of the hankel operators in generalized orthonormal bases using the so called hambo domain techniques an efficient algorithm is given to implement the proposed method

worst case system identification in h sub spl infin validation of a priori information essentially optimal algorithms and error bounds resolves several important open issues pertaining to a worst case control oriented system identification problem known as identification in h sub spl infin first a method is presented for developing confidence that certain a priori information available for identification is not invalid this method requires the solution of a certain nondifferentiable convex program second an essentially optimal identification algorithm is constructed this algorithm is worst case strongly optimal to within a factor of two finally new upper and lower bounds on the optimal identification error are derived and used to estimate the identification error associated with the given algorithm interestingly the development of each of these results draws heavily upon the classical nevanlinna pick interpolation theory as such the authors results establish a clear link between the areas of system identification and optimal interpolation theory both the formulation and techniques in this paper are applicable to problems where the frequency data available for identification may essentially be arbitrarily distributed

a construction of rational wavelets and frames in hardy sobolev spaces with applications to system modeling using the daubechies wavelet theory we establish rational wavelet decompositions of the hardy sobolev classes on the half plane the decay of wavelet coefficients is analyzed and error bounds for approximation are given we give applications to the modeling of linear systems and to the model reduction of infinite dimensional systems

the p median problem for cluster analysis a comparative test using the mixture model approach recently mulvey and crowder mulvey j h crowder 1979 cluster analysis an application of lagrangian relaxation management sci 25 329 340 suggested that the p median problem might be useful for cluster analysis problems where the goal is to group objects described by a vector of characteristics in such a way that objects in the same group are somehow more alike than objects in different groups the intent of this paper is to test mulvey and crowder s proposal using the mixture model approach i e by applying a number of algorithms including one for the p median problem to a set of objects randomly sampled from a number of known multivariate populations and comparing the ability of each algorithm to detect the original populations in order to evaluate the results a generalized partition comparison measure and its distribution are developed using this measure results from various algorithms are compared

rational basis functions for robust identification from frequency and time domain measurements this paper investigates the use of general bases with fixed poles for the purposes of robust estimation these bases which generalise the common fir laguerre and two parameter kautz ones are shown to be fundamental in the disc algebra provided a very mild condition on the choice of poles is satisfied it is also shown that by using a min max criterion these bases lead to robust estimators for which error bounds in different norms can be explicitly quantified the key idea facilitating this analysis is to re parameterise the chosen model structures into a new one with equivalent fixed poles but for which the basis functions are orthonormal in h 2 d

the p median problem a survey of metaheuristic approaches the p median problem is one of the basic models in discrete location theory as with most location problems it is classified as np hard and so heuristic methods are usually used to solve it metaheuristics are frameworks for building heuristics in this survey we examine the p median with the aim of providing an overview on advances in solving it using recent procedures based on metaheuristic rules

shift operator induced approximations of delay systems certain shift operator induced finite dimensional approximations of an important class of delay systems are studied h infty h2 and l1 error bounds are given for stable delay systems furthermore a multiple shift formula generated by a second order pade shift is analyzed further and an exact asymptotic h infty error formula is derived the studied class of shift operator induced approximations provides an appealing and most transparent approach for determining good finite dimensional models of delay systems

realisation and approximation of linear infinite dimensional systems with error bounds the class of linear infinite dimensional systems with finite dimensional inputs and outputs whose impulse response h satisfies h in l 1 cap l 2 0 infty mathbb c p times m and induces a nuclear hankel operator is said to be of nuclear type for this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies furthermore explicit l infty bounds on the transfer function errors l 1 and l 2 bounds on the impulse response errors and hilbert schmidt and nuclear bounds on the hankel operator errors are obtained these truncations also generate an approximating sequence to the optimal hankel norm approximations to the original system and various error bounds of these approximants are deduced

cubic systems and abel equations abstract we investigate cubic systems which can be transformed to an equation of abel form the conditions for the origin to be a centre and in particular an isochronous centre are obtained the maximum number of limit cycles which can bifurcate from a fine focus is determined and some information is obtained about the global phase portrait

three way metric unfolding via alternating weighted least squares three way unfolding was developed by desarbo 1978 and reported in desarbo and carroll 1980 1981 as a new model to accommodate the analysis of two mode three way data e g nonsymmetric proximities for stimulus objects collected over time and three mode three way data e g subjects rendering preference judgments for various stimuli in different usage occasions or situations this paper presents a revised objective function and new algorithm which attempt to prevent the common type of degenerate solutions encountered in typical unfolding analysis we begin with an introduction of the problem and a review of three way unfolding the three way unfolding model weighted objective function and new algorithm are presented monte carlo work via a fractional factorial experimental design is described investigating the effect of several data and model factors on overall algorithm performance finally three applications of the methodology are reported illustrating the flexibility and robustness of the procedure

brief robustness in h identification we consider identification in both the time domain for suitable choices of inputs and the frequency domain using a deterministic formulation that is less conservative than the usual worst case approach in that it allows for a certain number of outliers two new identification methods the orlicz and the top k method respectively are proposed and analysed theoretical h error bounds for the identified models are provided and the proposed methods are illustrated by a numerical example

optimal asymptotic identification under bounded disturbances the intrinsic limitation of worst case identification of linear time invariant systems using data corrupted by bounded disturbances when the unknown plant is known to belong to a given model set is studied this is done by analyzing the optimal worst case asymptotic error achievable by performing experiments using any bounded input and estimating the plant using any identification algorithm it is shown that under some topological conditions on the model set there is an identification algorithm which is asymptotically optimal for any input and the optimal asymptotic error is characterized as a function of the inputs these results which hold for any error metric and disturbance norm are applied to three specific identification problems identification of stable systems in the l sub 1 norm identification of stable rational systems in the h sub infinity norm and identification of unstable rational systems in the gap metric for each of these problems the general characterization of optimal asymptotic error is used to find near optimal inputs to minimize the error

a simple solution of some composition conjectures for abel equations abstract trigonometric abel differential equations appear in the study of the number of limit cycles and the center focus problem for certain families of planar polynomial systems the composition centers are a class of centers for trigonometric abel equations which have been widely studied during last years we characterize this type of centers as the ones given by couples of trigonometric polynomials for which all the generalized moments vanish they also coincide with the strongly and the highly persistent centers our result gives a simple and self contained proof of the so called composition conjecture for trigonometric abel differential equations we also prove a similar version of this result for abel equations with polynomial coefficients

modeling of linear fading memory systems motivated by questions of approximate modeling and identification we consider various classes of linear time varying bounded input bounded output bibo stable fading memory systems and the characterizations are proved these include fading memory systems in general almost periodic systems and asymptotically periodic systems we also show that the norm and strong convergence coincide for bibo stable causal fading memory systems

discrete time system modelling in with orthonormal basis functions abstract in this paper model sets for linear time invariant systems spanned by fixed pole orthonormal bases are investigated the obtained model sets are shown to be complete in l p t 1 the lebesque spaces of functions on the unit circle t and in c t the space of periodic continuous functions on t the l p norm error bounds for estimating systems in l p t by the partial sums of the fourier series formed by the orthonormal functions are computed for the case 1 some inequalities on the mean growth of the fourier series are also derived these results have application in estimation and model reduction

center conditions at infinity for abel differential equations an abel differential equation y p x y 2 q x y 3 is said to have a center at a set a a 1 a r of complex numbers if y a 1 y a 2 y a r for any solution y x with the initial value y a 1 small enough the polynomials p q are said to satisfy the polynomial composition condition on a if there exist polynomials p q and w such that p p and q q are representable as p x p w x q x q w x and w a 1 w a 2 w a r we show that for wide ranges of degrees of p and q restricted only by certain assumptions on the common divisors of these degrees the composition condition provides a very accurate approximation of the center one up to a finite number of configurations not accounted for to our best knowledge this is the first general i e not restricted to small degrees of p and q or to a very special form of these polynomials result in the center problem for abel equations as an important intermediate result we show that at infinity according to an appropriate projectivization of the parameter space the center conditions are given by a system of the moment equations of the form as a1 p k q 0 s 2 r k 0 1

approximation and identification of continuous time systems an approximation approach to the identification of stable continuous time possibly infinite dimensional systems is studied for this purpose concrete exponential leg endre series methods are proposed and analysed here for the l1 and l2 approximations of continuous time systems rate of approximation results are given for the proposed methods for an important class of delay systems the proposed exponential legendre series method for the l2 approximation has a certain optimal rate of approximation property for this class of delay systems these results show that exponential legendre series techniques have promising properties in connection with the l1 and l2 identifications of stable continuous time systems the proposed l1 identification method has a direct application to the robust design of control systems

an explicit bound of the number of vanishing double moments forcing composition we give two new characterizations of pairs of polynomials or trigonometric polynomials that form a composition pair one of them proves that the cancellation of a given number of double moments implies that they form a composition pair this number only depends on the maximum degree of both polynomials this is the first time that composition is characterized in terms of the cancellation of an explicit number of double moments our results allow to recognize the composition centers for polynomial and trigonometric abel differential equations

parameter identification in the presence of non parametric dynamic uncertainty abstract this paper provides identification laws which yield monotonic parameter error reduction in the absence of external noise inputs but in the presence of non parametric dynamic uncertainty unmodelled dynamics this paper considers two non parametric uncertainty sets motivated by robust control the weighted ball in h and the weighted ball in the induced l2 norm the non parametric uncertainty produces signals which corrupt the signals used in identification this paper shows that the corruption is non conservatively bounded in an l2 sense this l2 bound is then compared with the signal energy with the result used to enable disable the parameter adjustment of a gradient least squares adjustment mechanism it is proved that the resulting parameter adjustment yields non increasing parameter errors with strictly decreasing parameter errors whenever the parameter errors are distinguishable from zero given the system input and output histories

system identification using laguerre models the traditional approach of expanding transfer functions and noise models in the delay operator to obtain linear in the parameters predictor models leads to approximations of very high order in cases of rapid sampling and or dispersion in time constants by using prior information about the time constants of the system more appropriate expansions related to laguerre networks are introduced and analyzed it is shown that the model order can be reduced compared to arx fir ar modeling by using laguerre models furthermore the numerical accuracy of the corresponding linear regression estimation problem is improved by a suitable choice of the laguerre parameter consistency error bounds persistence of excitation conditions and asymptotic statistical properties are investigated this analysis is based on the result that the covariance matrix of the regression vector of a laguerre model has a toeplitz structure

optimal partitioning of a data set based on the p median model although the k means algorithm for minimizing the within cluster sums of squared deviations from cluster centroids is perhaps the most common method for applied cluster analyses a variety of other criteria are available the p median model is an especially well studied clustering problem that requires the selection of p objects to serve as cluster centers the objective is to choose the cluster centers such that the sum of the euclidean distances or some other dissimilarity measure of objects assigned to each center is minimized using 12 data sets from the literature we demonstrate that a three stage procedure consisting of a greedy heuristic lagrangian relaxation and a branch and bound algorithm can produce globally optimal solutions for p median problems of nontrivial size several hundred objects five or more variables and up to 10 clusters we also report the results of an application of the p median model to an empirical data set from the telecommunications industry

gennclus new models for general nonhierarchical clustering analysis a general class of nonhierarchical clustering models and associated algorithms for fitting them are presented these metric clustering models generalize the shepard arabie additive clusters model in allowing for 1 either overlapping or nonoverlapping clusters 2 either symmetric one way clustering or nonsymmetric two way clustering proximities input data and 3 either symmetric or diagonal weights the gennclus algorithms utilize alternating least squares methods combining ordinary and constrained least squares nonlinear constrained mathematical programming and combinatorial optimization techniques in estimating model parameters in addition to developing the mathematical bases of these models a comprehensive set of monte carlo simulations of the different models is reported two applications concerning brand switching data and celebrity brand proximities are discussed finally extensions to three way models nonmetric analyses and other model specifications are provided

system identification by discrete rational atoms a novel adaptive frequency domain system identification method for linear time invariant systems is proposed in this paper it finds poles for discrete rational atoms with discrete frequency responses the theoretical foundation including adaptive decomposition principle and decomposition convergence rate is established the algorithm of the adaptive pursuit is also provided in this paper

some new results on abel equations abstract abel equations of the form r a t r2 b t r3 t t0 t1 are of interest because of their close relation to planar vector fields in this paper three new results on abel equations are presented 1 an asymptotic expansion of the solutions of abel equations in terms of the coefficient functions a and b 2 a simple recurrence relation for bautin quantities of abel equations 3 a center condition for a class of polynomial abel equations

the fundamental role of general orthonormal bases in system identification the purpose of this paper is threefold firstly it is to establish that contrary to what might be expected the accuracy of well known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure secondly it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros this is achieved by employing certain new results on generalized fourier series and the asymptotic properties of toeplitz like matrices in such a way that the new variance expressions presented here encompass pre existing ones as special cases via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification namely that orthonormal bases are much more than an implementational option offering improved numerical properties in fact they are an intrinsic part of estimation since as shown here orthonormal bases quantify the asymptotic variability of the estimates whether or not they are actually employed in calculating them

model quality evaluation in set membership identification abstract identification from corrupted input output measurements of systems that do not necessarily belong to the model class used is investigated this leads to a nonstandard set membership sm identification problem the goodness of different model classes is measured by the conditional radius of information a generalization of the radius in standard sm identification theory giving a measure of the minimal worst case modeling error upper and lower bounds on the radius are derived for linearly parameterized model classes specific formulas for the upper and lower bounds are given for the case of h 2 identification of exponentially stable systems in the presence of powerbounded noise the bounds are shown to coincide with the conditional radius when the model space dimension is equal to the number of output measurements an almost optimal identification algorithm is derived giving identification error within the range of the derived bounds

estimation of model quality this paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems this is a very large field we therefore concentrate on approaches that have been motivated by the need for reliable models for control system design this will involve a discussion of efforts that go under the titles of estimation in th worst case estimation estimation in l1 and stochastic embedding of undermodelling a central theme of this survey is to examine these new methods with reference to the classic bias variance tradeoff in model structure selection

center problem for trigonometric lienard systems we give a complete algebraic characterization of the non degenerated centers for planar trigonometric lienard systems the main tools used in our proof are the classical results of cherkas on planar analytic lienard systems and the characterization of some subfields of the quotient field of the ring of trigonometric polynomials our results are also applied to some particular subfamilies of planar trigonometric lienard systems the results obtained are reminiscent of the ones for planar polynomial lienard systems but the proofs are different

on the uniform approximation of discrete time systems by generalized fourier series in this paper model sets for linear time invariant discrete time systems spanned by fixed orthonormal bases are studied it is shown that the fourier series of the system transfer function with respect to these bases converges uniformly on the unit circle if the frequency response of the system is dini lipschitz continuous

non autonomous equations related to polynomial two dimensional systems periodic solutions of certain one dimensional non autonomous differential equations are investigated equation 1 4 the independent variable is complex the motivation which is explained in the introductory section is the connection with certain polynomial two dimensional systems several classes of coefficients are considered in each case the aim is to estimate the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients in particular we need to know when there is a full neighbourhood of periodic solutions we give a number of sufficient conditions and investigate the implications for the corresponding two dimensional systems

identification in h using pick s interpolation the problem of system identification in h is studied using the pick interpolation theorem a tuned nonlinear interpolation algorithm is proposed and the resulting error bound in h norm is derived an example is used to illustrate the use of the interpolation algorithm

universal centres and composition conditions in this paper we characterize the universal centers of the ordinary differential equations d d i 1 ai i 1 where ai are trigonometric polynomials in terms of the composition conditions these centers are closely related with the classical poincare center problem for planar analytic differential systems additionally we show that the notion of universal center is not invariant under changes of variables and we also provide different families of universal centers finally we characterize all the universal centers for the quadratic polynomial differential systems

the p median problem under uncertainty abstract consider the need to currently locate p facilities but it is possible that up to q additional facilities will have to be located in the future there are known probabilities that 0 r q facilities will need to be located the p median problem under uncertainty is to find the location of p facilities such that the expected value of the objective function in the future is minimized the problem is formulated on a graph properties of it are proven an integer programming formulation is constructed and heuristic algorithms are suggested for its solution the heuristic algorithms are modified to reduce the run time by about two orders of magnitude with minimal effect on the quality of the solution optimal solutions for many problems are found effectively by cplex computational results using the heuristic algorithms are presented

robust approximate modelling of stable linear systems robust approximation and worst case approximate modelling of stable shift invariant systems from corrupted transfer function estimates are studied in the h sense connections between the problem formulations of the present work and certain problems of worst case system identification notably the helmicki jacobson nett problem formulation for identification in h are established issues of model set selection are addressed using the n width concept a concrete result establishes a priori knowledge for which a certain rational model set is optimal in then width sense a notion of robust convergence is defined so that any untuned approximation method satisfying it has a generic well posedness property for systems in the disk algebra the existence of robustly convergent approximation methods based on any complete model set in the disk algebra is shown in a constructive way a framework is given in which approximate models can be obtained as stable perturbations of the true system these can be combined

the number of polynomial solutions of polynomial riccati equations consider real or complex polynomial riccati differential equations a x y b 0 x b 1 x y b 2 x y 2 with all the involved functions being polynomials of degree at most we prove that the maximum number of polynomial solutions is 1 resp 2 when 1 resp 0 and that these bounds are sharp for real trigonometric polynomial riccati differential equations with all the functions being trigonometric polynomials of degree at most 1 we prove a similar result in this case the maximum number of trigonometric polynomial solutions is 2 resp 3 when 2 resp 1 and again these bounds are sharp although the proof of both results has the same starting point the classical result that asserts that the cross ratio of four different solutions of a riccati differential equation is constant the trigonometric case is much more involved the main reason is that the ring of trigonometric polynomials is not a unique factorization domain