a weak star convergence result for orthogonal rational functions we prove a weak star convergence result for orthogonal rational functions on the interval 1 1 which generalizes a well known result by rakhmanov math ussr sb 32 1977 199 as an application we use this result to approximate a certain integral which occurs in the computation of orthogonal rational functions

non parametric nonlinear system identification a data driven orthogonal basis function approach in this paper a data driven orthogonal basis function approach is proposed for non parametric fir nonlinear system identification the basis functions are not fixed a priori and match the structure of the unknown system automatically this eliminates the problem of blindly choosing the basis functions without a priori structural information further based on the proposed basis functions approaches are proposed for model order determination and regressor selection along with their theoretical justifications

the quadratic type of the 2 principal indecomposable modules of the double covers of alternating groups abstract the principal indecomposable modules of the double cover 2 a n of the alternating group over a field of characteristic 2 are enumerated using the partitions of n into distinct parts we determine which of these modules afford a non degenerate 2 a n invariant quadratic form our criterion depends on the alternating sum and the number of odd parts of the corresponding partition

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

variance error quantifications that are exact for finite model order this paper is concerned with the frequency domain quantification of noise induced errors in dynamic system estimates preceding and seminal work on this problem has suggested general expressions that are approximations whose accuracy increases with observed data length and model order in the interests of improved accuracy this paper provides new expressions whose accuracy depends only on data length they are therefore exact for arbitrarily small true model order other authors have recognized the importance of such expressions and have derived them for the case of finite impulse response like model structures in which denominators are fixed at true values and only numerator terms are estimated this paper progresses beyond this situation to address the more general output error and box jenkins structures in which full dynamics models both numerator and denominator terms and noise models may be estimated a key aspect of the work here is that it establishes that the variance quantification problem is equivalent to that of deriving the reproducing kernel for a subspace that depends on the model structure being employed

quadratic geometries projective modules and idempotents abstract let g be a finite group and let k be a field we say that a kg module v has a quadratic geometry or is of quadratic type if there exists a non degenerate equivalently non singular g invariant quadratic form on v if v is irreducible or projective indecomposable and k of odd characteristic the existence of a quadratic geometry can be read off from invariants of representation theory in terms of the group algebra the existence of such a geometry is characterized by the existence of an idempotent e which corresponds to v and is invariant under the antiautomorphism on kg given by g g 1 for g g in characteristic two the problem is more delicate for an irreducible module the existence of a quadratic geometry is partly a question in cohomology unfortunately a satisfactory answer does not seem to be known however if v is projective liftability of the module to characteristic zero may help to find the geometry more easily this is the subject of the paper the notation which we shall use throughout the paper is standard and may be taken from huppert and blackburn finite groups ii springer verlag berlin 1982 and isaacs character theory of finite groups academic press new york 1976

system identification with generalized orthonormal basis functions a least squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function where the expansion is in terms of recently introduced generalized basis functions the basis functions are orthogonal in h2 and generalize the pulse laguerre and kautz bases one of their important properties is that when chosen properly they can substantially increase the speed of convergence of the series expansion this leads to accurate approximate models with only a few coefficients to be estimated explicit bounds are derived for the bias and variance errors that occur in parameter estimates as well as in the resulting transfer function estimates

numerical analysis for one dimensional cauchy singular integral equations the paper presents a selection of modern results concerning the numerical analysis of one dimensional cauchy singular integral equations in particular the stability of operator sequences associated with different projection methods the aim of the paper is to show the main ideas and approaches such as the concept of transforming the question of the stability of an operator sequence into an invertibility problem in a certain banach algebra or the concept of certain scales of weighted besov spaces to prove convergence rates of the sequence of the approximate solutions moreover computational aspects in particular the construction of fast algorithms are discussed

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

lipschitz continuity of inner outer factorization abstract conditions are provided under which inner and outer factor of h functions on the unit disk can be found which depend lipschitz continuously on data

worst case conditional system identification in a general class of norms we consider optimality of identification of linear systems by reduced complexity models which belong to a subspace of possibly low dimension of a system space identification is based on the a priori knowledge that an unknown system belongs to a certain class and a posteriori information given by its nonexact output measurements measurement errors are assumed unknown but norm bounded the analysis is not restricted to specific norms in system and measurement spaces generic norms are considered we derive tight upper and lower bounds on the minimal local worst case identification error and define an algorithm whose error is within the derived bounds the results are specified for product norms which include l p norms email protected email protected in the system space the bounds are expressed in terms of the model error which measures a quality of representing systems by a model subspace and the diameter of information reflecting the size of a model uncertainty set the model error is further splitted into truncated model error and residual error which controls tail properties of the unknown system examples of model and input selection in some special cases are given and numerical experiments showing the behavior of the estimates reported

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

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

spline qualocation methods for boundary integral equations this paper further develops the qualocation method for the solution of integral equations on smooth closed curves qualocation is a petrov galerkin method in which the outer integrals are performed numerically by special quadrature rules here we allow the use of splines as trial and test functions and prove stability and convergence results for certain qualocation rules when the underlying petrov galerkin method is stable in the common case of a first kind equation with the logarithmic kernel a typical qualocation method employs piecewise linear splines and performs the outer integrals using just two points per interval this method with appropriate choice of the quadrature points and weights has ordero h 5 convergence in a suitable negative norm numerical experiments suggest that these rates of convergence are maintained for integral equations on open intervals when graded meshes are used to cope with unbounded solutions

adaptive receding horizon control for constrained mimo systems an adaptive control algorithm for open loop stable constrained linear multiple input multiple output systems is presented the proposed approach can deal with both input and output constraints as well as measurement noise and output disturbances the adaptive controller consists of an iterative set membership identification algorithm that provides a set of candidate plant models at each time step and a model predictive controller that enforces input and output constraints for all the plants inside the model set the algorithm relies only on the solution of standard convex optimization problems that are guaranteed to be recursively feasible the experimental results obtained by applying the proposed controller to a quad tank testbed are presented

frequency domain identification with generalized orthonormal basis functions a method is considered for the identification of linear parametric models based on a least squares identification criterion that is formulated in the frequency domain to this end use is made of the empirical transfer function estimate etfe identified from time domain data as a parametric model structure use is made of a finite expansion sequence in terms of recently introduced generalized basis functions being generalizations of the classical pulse and laguerre and kautz types of bases an asymptotic analysis of the estimated models is provided and conditions for consistency are formulated explicit and transparent bias and variance expressions are established the latter ones also valid in a situation of undermodeling

brief paper choice of free parameters in expansions of discrete time volterra models using kautz functions this work tackles the problem of modeling nonlinear systems using volterra models based on kautz functions the drawback of requiring a large number of parameters in the representation of these models can be circumvented by describing every kernel using an orthonormal basis of functions such as the kautz basis the resulting model so called wiener volterra model can be truncated into a few terms if the kautz functions are properly designed the underlying problem is how to select the free design complex poles that fully parameterize these functions a solution to this problem has been provided in the literature for linear systems which can be represented by first order volterra models a generalization of such strategy focusing on volterra models of any order is presented in this paper this problem is solved by minimizing an upper bound for the error resulting from the truncation of the kernel expansion into a finite number of functions the aim is to minimize the number of two parameter kautz functions associated with a given series truncation error thus reducing the complexity of the resulting finite dimensional representation the main result is the derivation of an analytical solution for a sub optimal expansion of the volterra kernels using a set of kautz functions

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

on worst case approximation of feasible system sets via orthonormal basis functions this note deals with the approximation of sets of linear time invariant systems via orthonormal basis functions this problem is relevant to conditional set membership identification where a set of feasible systems is available from observed data and a reduced complexity model must be estimated the basis of the model class is made of impulse responses of linear filters the objective of the note is to select the basis function poles according to a worst case optimality criterion suboptimal conditional identification algorithms are introduced and tight bounds are provided on the associated identification errors

continuous time identification of siso systems using laguerre functions this paper looks at the problem of estimating the coefficients of a continuous time transfer function given samples of its input and output data we first prove that any nth order continuous time transfer function can be written as a fraction of the form spl sigma sub k 0 sup n b sub k l sub k s spl sigma sub k 0 sup n a sub k l sub k s where l sub k s denotes the continuous time laguerre basis functions based on this model we derive an asymptotically consistent parameter estimation scheme that consists of the following two steps 1 filter both the input and output data by l sub k s and 2 estimate a sub k b sub k and relate them to the coefficients of the transfer function for practical implementation we require the discrete time approximation of l sub k s since only sampled data is available we propose a scheme that is based on higher order pade approximations and we prove that this scheme produces discrete time filters that are approximately orthogonal and consequently a well conditioned numerical problem some other features of this new algorithm include the possibility to implement it as either an off line or a quasi on line algorithm and the incorporation of constraints on the transfer function coefficients a simple example is given to illustrate the properties of the proposed algorithm

the cost of complexity in system identification in this paper we investigate the cost of complexity which is defined as the minimum amount of input power required to estimate the frequency response of a given linear time invariant system of order n with a prescribed degree of accuracy in particular we require that the asymptotic in the data length variance is less or equal to over a prespecified frequency range 0 b the models considered here are output error models with an emphasis on fixed denominator and laguerre models several properties of the cost are derived for instance we present an expression which shows how the pole of the laguerre model affects the cost these results quantify how the cost of the system identification experiment depends on n and on the model structure also they show the relation between the cost and the amount of information we would like to extract from the system in terms of b and for simplicity we assume that there is no undermodelling

orthogonal rational functions a transformation analysis finite impulse response fir models are among the most basic tools in control theory and signal processing and are routinely used in almost all fields of application the connections to orthogonal polynomials are well known however infinite impulse response iir models often provide much more compact descriptions and in many cases give improved performance the objective of this paper is to present a simple framework for the derivation and analysis of orthogonal iir transfer functions which are directly related to orthogonal rational functions orthogonality simplifies approximation analysis and leads to improved numerical properties the basic idea is to use a fractional transformation to map the problem to a new domain where an fir description is most appropriate this fir representation is then mapped back to the original domain to give an orthogonal iir representation it is then straightforward to extend many results for fir models to iir model structures with arbitrary stable poles i e prop

explicit expression of the parameter bias in identification of laguerre models from step responses this paper delivers an analysis of the least square estimation of the laguerre coefficients of a linear discrete time system from step response data the original contribution consists in an explicit formula for the bias error on the estimated coefficients due to the under modelling of the system the formula jointly with some a priori information on the neglected dynamics can be used to construct bounds on this error the results presented in this paper are illustrated with a simulation example c 2003 elsevier b v all rights reserved

brief a generalization of a standard inequality for hardy space h1 in this paper we generalize a standard inequality for the hardy space h 1 to involve a family of rational basis systems with cyclically repeated poles this inequality is used to derive an inequality that gives information on the fourier coefficients of functions in h p 1

corrected collocation methods for periodic pseudodifferential equations we propose an approximation method for periodic pseudodifferential equations which yields higher convergence rates in sobolev spaces with negative order than the collocation method the main idea consists in correcting the usual collocation solution in a certain way by the solution of a small galerkin system for the same equation both trigonometric and spline approximation methods are considered in most of the cases our convergence result even improves that of the qualocation method

brief paper synthesis of fractional laguerre basis for system approximation fractional differentiation systems are characterized by the presence of non exponential aperiodic multimodes although rational orthogonal bases can be used to model any l2 0 system they fail to quickly capture the aperiodic multimode behavior with a limited number of terms hence fractional orthogonal bases are expected to better approximate fractional models with fewer parameters intuitive reasoning could lead to simply extending the differentiation order of existing bases from integer to any positive real number however classical laguerre and by extension kautz and generalized orthogonal basis functions are divergent as soon as their differentiation order is non integer in this paper the first fractional orthogonal basis is synthesized extrapolating the definition of laguerre functions to any fractional order derivative completeness of the new basis is demonstrated hence a new class of fixed denominator models is provided for fractional system approximation and identification

orthonormal basis functions applied to optimal control design with closed loop pole location constraints this article proposes an optimal control design technique to ensure that the closed loop poles are assigned to prescribed locations in the complex plane a key element in this approach is the usage of orthonormal basis functions to parametrise the main design parameter this goal is achieved by using a quadratic cost based optimisation approach considering control effort penalisation as well as cheap control youla parametrisation is used subject to a set of constraints which guarantee stability zero steady error to constant references and disturbances and yielding the prescribed set of closed loop poles

generalized fourier and toeplitz results for rational orthonormal bases this paper provides a generalization of certain classical fourier convergence and asymptotic toeplitz matrix properties to the case where the underlying orthonormal basis is not the conventional trigonometric one but rather a rational generalization which encompasses the trigonometric one as a special case these generalized fourier and toeplitz results have particular application in dynamic system estimation theory specifically the results allow a unified treatment of the accuracy of least squares system estimation using a range of model structures including those that allow the inclusion of prior knowledge of system dynamics via the specification of fixed pole or zero locations

minimal partial realization from generalized orthonormal basis function expansions a solution is presented for the problem of realizing a discrete time lti state space model of minimal mcmillan degree such that its first n expansion coefficients in terms of generalized orthonormal basis match a given sequence the basis considered also known as the hambo basis can be viewed as a generalization of the more familiar laguerre and two parameter kautz constructions allowing general dynamic information to be incorporated in the basis for the solution of the problem use is made of the properties of the hambo operator transform theory that underlies the basis function expansion as corollary results compact expressions are found by which the hambo transform and its inverse can be computed efficiently the resulting realization algorithms can be applied in an approximative sense for instance for computing a low order model from a large basis function expansion that is obtained in an identification experiment

lipschitz continuity of h interpolation abstract optimal h interpolants may be infinitely sensitive to data however suboptimal interpolants of the aak central maximal entropy type are shown to satisfy a lipschitz condition with respect to data

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

an optimal expansion of volterra models using independent kautz bases for each kernel dimension a new solution for the problem of selecting poles of the two parameter kautz functions in volterra models is proposed in general a large number of parameters are required to represent the volterra kernels although this difficulty can be overcome by describing each kernel using a basis of orthonormal functions such as the kautz basis this representation has a wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a non linear static mapping represented by the volterra series the resulting wiener volterra model can be truncated into fewer terms if the kautz functions are properly designed the underlying problem is how to select the arbitrary complex poles that fully parametrize these functions this problem has been approached in previous research by minimizing an upper bound for the error resulting from the truncation of the kernel expansion the present paper goes even further in that each multidimensional kernel is decomposed into a set of independent kautz bases each of which is parametrized by an individual pair of conjugate kautz poles intended to represent the dominant dynamic of the kernel along a particular dimension an analytical solution for one of the kautz parameters valid for volterra models of any order is derived a simulated example is presented to illustrate these theoretical results the same approach is then used to model a real non linear magnetic levitation system with oscillatory behaviour

symmetric bilinear forms and vertices in characteristic 2 let g be a finite group and let k be an algebraically closed field of characteristic 2 suppose that m is an indecomposable kg module which affords a non degenerate g invariant symmetric bilinear form we assign to m a collection of 2 subgroups of g called its symmetric vertices each of which contains a green vertex of m with index at most 2 if m is irreducible then its symmetric vertices are uniquely determined up to g conjugacy if b is the real 2 block of g containing m we show that each symmetric vertex of m is contained in an extended defect group of b moreover we characterise the extended defect groups in terms of symmetric vertices in order to prove these results we develop the theory of involutary g algebras this allows us to translate questions about symmetric kg modules into questions about projective modules of quadratic type

asymmetric volterra models based on ladder structured generalized orthonormal basis functions in this paper an improved method to construct and estimate volterra models using generalized orthonormal basis functions gobf is presented the proposed method extends results obtained in previous works where an exact technique for optimizing the gobf parameters poles for symmetric volterra models of any order was presented the proposed extensions take place in two different ways i the new formulation is derived in such a way that each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases rather than a single common basis each of which is parameterized by an individual set of poles responsible for representing the dominant dynamic of the kernel along a particular dimension and ii the new formulation is based on a ladder structured gobf architecture that is characterized by having only real valued parameters to be estimated regardless of whether the gobf poles encoded by these parameters are real or complex valued the exact gradients of an error functional with respect to the parameters to be optimized are computed analytically and provide exact search directions for an optimization process that uses only input output data measured from the dynamic system to be modeled computational experiments are presented to illustrate the benefits of the proposed approach when modeling nonlinear systems

orthonormal basis functions in time and frequency domain hambo transform theory the class of finite impulse response fir laguerre and kautz functions can be generalized to a family of rational orthonormal basis functions for the hardy space h2 of stable linear dynamical systems these basis functions are useful for constructing efficient parameterizations and coding of linear systems and signals as required in e g system identification system approximation and adaptive filtering in this paper the basis functions are derived from a transfer function perspective as well as in a state space setting it is shown how this approach leads to alternative series expansions of systems and signals in time and frequency domain the generalized basis functions induce signal and system transforms hambo transforms which have proved to be useful analysis tools in various modelling problems these transforms are analyzed in detail in this paper and a large number of their properties are derived principally it is shown how minimal state space realizations of the system transform can be obtained from minimal state space realizations of the original system and vice versa

non asymptotic confidence regions for model parameters in the presence of unmodelled dynamics this paper deals with the problem of constructing confidence regions for the parameters of truncated series expansion models the models are represented using orthonormal basis functions and we extend the leave out sign dominant correlation regions lscr algorithm such that non asymptotic confidence regions for the parameters can be constructed in the presence of unmodelled dynamics the constructed regions have guaranteed probability of containing the true parameters for any finite number of data points the algorithm is first developed for fir models and then extended to models with generalized orthonormal basis functions the usefulness of the developed approach is demonstrated for fir and laguerre models in simulation examples

a unifying construction of orthonormal bases for system identification this paper develops a general and very simple construction for complete orthonormal bases for system identification this construction provides a unifying formulation of many previously studied orthonormal bases since the common fir and recently popular laguerre and two parameter kautz model structures are restrictive special cases of the construction presented here however in contrast to these special cases the basis vectors in the unifying construction of this paper can have arbitrary placement of pole position according to the prior information the user wishes to inject results characterizing the completeness of the bases and the accuracy properties of models estimated using the bases are provided

wiener system identification with generalized orthonormal basis functions many nonlinear systems can be described by a wiener schetzen model in this model the linear dynamics are formulated in terms of orthonormal basis functions obfs the nonlinearity is modeled by a multivariate polynomial in general an infinite number of obfs are needed for an exact representation of the system this paper considers the approximation of a wiener system with finite order infinite impulse response dynamics and a polynomial nonlinearity we propose to use a limited number of generalized obfs gobfs the pole locations needed to construct the gobfs are estimated via the best linear approximation of the system the coefficients of the multivariate polynomial are determined with a linear regression this paper provides a convergence analysis for the proposed identification scheme it is shown that the estimated output converges in probability to the exact output fast convergence rates in the order o p n f n r e p 2 can be achieved with n f the number of excited frequencies and n r e p the number of repetitions of the gobfs

an optimum time scale for discrete laguerre network any discrete time stable transfer function can be expressed by a discrete time laguerre series with a chosen time scale an optimum time scale such that an index is minimized is derived this index ensures that the coefficients of higher order laguerre functions go toward zero quickly the solution derived requires the knowledge of the impulse response of the discrete plant cases of first order plants second order underdamped plants and plants with multiunit delay are also discussed

recursive direct weight optimization in nonlinear system identification a minimal probability approach in this paper a direct weight optimization method is proposed for nonlinear system identification based on a minimal probability idea the approach has several quite attractive features and is very different from existing ones it is optimal for any given number of finite data points and at the same time possesses asymptotic convergence the estimator admits a closed form and no numerical optimization is needed theoretical analysis and numerical simulations show that the approach is a very competitive alternative to existing nonlinear identification methods

a quadrature based approach to improving the collocation method the collocation method is a popular method for the approximate solution of boundary integral equations but typically does not achieve the high order of convergence reached by the galerkin method in appropriate negative norms in this paper a quadrature based method for improving upon the collocation method is proposed and developed in detail for a particular case that case involves operators with even symbol such as the logarithmic potential operating on 1 periodic functions a smooth spline trial space of odd degree with constant mesh spacingh 1 n and a quadrature rule with 2n points where ann point quadrature rule would be equivalent to the collocation method in this setting it is shown that a special quadrature rule which depends on the degree of the splines and the order of the operator can yield a maximum order of convergence two powers ofh higher than the collocation method

sign perturbed sums a new system identification approach for constructing exact non asymptotic confidence regions in linear regression models we propose a new system identification method called sign perturbed sums sps for constructing non asymp totic confidence regions under mild statistical assumptions sps is introduced for linear regression models including but not limited to fir systems and we show that the sps confidence regions have exact confidence probabilities i e they contain the true parameter with a user chosen exact probability for any finite data set moreover we also prove that the sps regions are star convex with the least squares ls estimate as a star center the main assumptions of sps are that the noise terms are indepen dent and symmetrically distributed about zero but they can be nonstationary and their distributions need not be known the paper also proposes a computationally efficient ellipsoidal outer approximation algorithm for sps finally sps is demonstrated through a number of simulation experiments

l system approximation algorithms generated by 4 summations abstract this paper discusses linear l approximation of systems from noisy frequency domain data the models are discrete partial sum operators generated by summations including e g the fejer and the de la vallee poussin summations as special cases the main contribution of the paper is the derivation of upper bounds on these operators and on the approximation error in terms of the fourier transform of the window function the results can be used to design the first linear step in the two step approximate h identification method with desired approximation and convergence properties

choice of free parameters in expansions of discrete time volterra models using kautz functions this work tackles the problem of modeling nonlinear systems using volterra models based on kautz functions the drawback of requiring a large number of parameters in the representation of these models can be circumvented by describing every kernel using an orthonormal basis of functions such as the kautz basis the resulting model so called wiener volterra model can be truncated into a few terms if the kautz functions are properly designed the underlying problem is how to select the free design complex poles that fully parameterize these functions a solution to this problem has been provided in the literature for linear systems which can be represented by first order volterra models a generalization of such strategy focusing on volterra models of any order is presented in this paper this problem is solved by minimizing an upper bound for the error resulting from the truncation of the kernel expansion into a finite number of functions the aim is to minimize the number of two parameter kautz functions associated with a given series truncation error thus reducing the complexity of the resulting finite dimensional representation the main result is the derivation of an analytical solution for a sub optimal expansion of the volterra kernels using a set of kautz functions

on slowly time varying systems abstract a characterization of stabilizing controllers for slowly time varying systems based on input output descriptions of the plant and the controller is presented this approach generalizes standard results on slowly time varying systems with finite dimensional state space representation and allows both the plant and the controller to be infinite dimensional the controller design will be based on frozen time versions of the plant obtained at equally spaced instances in time

exact search directions for optimization of linear and nonlinear models based on generalized orthonormal functions a novel technique for selecting the poles of orthonormal basis functions obf in volterra models of any order is presented it is well known that the usual large number of parameters required to describe the volterra kernels can be significantly reduced by representing each kernel using an appropriate basis of orthonormal functions such a representation results in the so called obf volterra model which has a wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a nonlinear static mapping given by the volterra polynomial series aiming at optimizing the poles that fully parameterize the orthonormal bases the exact gradients of the outputs of the orthonormal filters with respect to their poles are computed analytically by using a back propagation through time technique the expressions relative to the kautz basis and to generalized orthonormal bases of functions gobf are addressed the ones related to the laguerre basis follow straightforwardly as a particular case the main innovation here is that the dynamic nature of the obf filters is fully considered in the gradient computations these gradients provide exact search directions for optimizing the poles of a given orthonormal basis such search directions can in turn be used as part of an optimization procedure to locate the minimum of a cost function that takes into account the error of estimation of the system output the levenberg marquardt algorithm is adopted here as the optimization procedure unlike previous related work the proposed approach relies solely on input output data measured from the system to be modeled i e no information about the volterra kernels is required examples are presented to illustrate the application of this approach to the modeling of dynamic systems including a real magnetic levitation system with nonlinear oscillatory behavior

identification of probabilistic system uncertainty regions by explicit evaluation of bias and variance errors a procedure is developed for identification of probabilistic system uncertainty regions for a linear time invariant system with unknown dynamics on the basis of time sequences of input and output data the classical framework is handled in which the system output is contaminated by a realization of a stationary stochastic process given minor and verifiable prior information on the system and the noise process frequency response pulse response and step response confidence regions are constructed by explicitly evaluating the bias and variance errors of a linear regression estimate in the model parametrizations use is made of general forms of basis functions conservatism of the uncertainty regions is limited by focusing on direct computational solutions rather than on closed form expressions using an instrumental variable method for identification the procedure is suitable also for input output data obtained from closed loop experiments

conditional central algorithms for worst case set membership identification and filtering this paper deals with conditional central estimators in a set membership setting the role and importance of these algorithms in identification and filtering is illustrated by showing that problems like worst case optimal identification and state filtering in contexts in which disturbances are described through norm bounds are reducible to the computation of conditional central algorithms the conditional chebyshev center problem is solved for the case when energy norm bounded disturbances are considered a closed form solution is obtained by finding the unique real root of a polynomial equation in a semi infinite interval

system identification using kautz models in this paper the problem of approximating a linear time invariant stable system by a finite weighted sum of given exponentials is considered system identification schemes using laguerre models are extended and generalized to kautz models which correspond to representations using several different possible complex exponentials in particular linear regression methods to estimate this sort of model from measured data are analyzed the advantages of the proposed approach are the simplicity of the resulting identification scheme and the capability of modeling resonant systems using few parameters the subsequent analysis is based on the result that the corresponding linear regression normal equations have a block toeplitz structure several results on transfer function estimation are extended to discrete kautz models for example asymptotic frequency domain variance expressions

a generalized orthonormal basis for linear dynamical systems in many areas of signal system and control theory orthogonal functions play an important role in issues of analysis and design in this paper it is shown that there exist orthogonal functions that in a natural way are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l sub 2sup n to this end use is made of balanced realizations of inner transfer functions the orthogonal functions can be considered as generalizations of for example the pulse functions laguerre functions and kautz functions and give rise to an alternative series expansion of rational transfer functions it is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion i e to obtain a good approximation by retaining only a finite number of expansion coefficients consequences for identification of expansion coefficients are analyzed and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients

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

spline qualocation methods for variable coefficient elliptic equations on curves here the stability and convergence results of oqualocation methods providing additional orders of convergence are extended from the special class of pseudodifferential equations with constant coefficient symbols to general classical pseudodifferential equations of strongly and of oddly elliptic type the analysis exploits localization in the form of frozen coefficients pseudohomogeneous asymptotic symbol representation of classical pseudodifferential operators a decisive commutator property of the qualocation projection and requires qualocation rules which provide sufficiently many additional degrees of precision for the numerical integration of smooth remainders numerical examples show the predicted high orders of convergence

frequency domain analysis of tracking and noise performance of adaptive algorithms in this paper an analysis of the tracking and noise performance of several adaptive algorithms is carried out for the case of model structures with fixed pole positions such structures have previously been proposed as an efficient generalization of the common fir model structure the focus of this work is to analyze the associated tradeoff between noise sensitivity and tracking ability in the frequency domain by illustrating how it is influenced by such things as input and noise spectral densities step size and in particular the choice of the fixed pole locations the latter influence is not described by preexisting analysis but is shown here to be amenable to attack by a particular class of orthonormal bases

control of slowly varying linear systems state feedback control of slowly varying linear continuous time and discrete time systems with bounded coefficient matrices is studied in terms of the frozen time approach this study centers on pointwise stabilizable systems these are systems for which there exists a state feedback gain matrix placing the frozen time closed loop eigenvalues to the left of a line re s gamma

explicit formulas for optimally robust controllers for delay systems this paper considers single input single output systems whose transfer functions take the form of a strictly proper rational function times a delay a closed form expression is presented for the controller which is optimally robust with respect to perturbations measured in the gap metric the formula allows the h sub spl infin loop shaping procedure of glover mcfarlane to be carried out explicitly for this class of systems without the need to first find a rational approximation of the plant the form of the controller involves a certain algebra of pseudo derivation operators these operators and their matrix generalizations play a central role in the derivation of the controller a discussion of the main properties of these operators is given an example is presented of a controller design to achieve disturbance attenuation and robust set point following for a plant with two lightly damped poles and a nontrivial time delay the performance is compared and shown to be superior to that of a smith predictor

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

characters bilinear forms and solvable groups abstract we prove a number of results about the ordinary and brauer characters of finite solvable groups in characteristic 2 by defining and using the concept of the extended nucleus of a real irreducible character in particular we show that the isaacs canonical lift of a real irreducible brauer character has frobenius schur indicator 1 we also show that the principal indecomposable module corresponding to a real irreducible brauer character affords a quadratic geometry if and only if each extended nucleus is a split extension of a nucleus

state space representation for arbitrary orthogonal rational functions abstract we first give a brief summary of the arguments that lead to the conclusion that the use of orthogonal rational functions in system identification should not be restricted to the takenaka malmquist basis because there are important cases where orthogonality with respect to a certain nonnegative weight function is required the main result however is a generalization of known state space descriptions for the takenaka malmquist basis to the more general case this is important for control problems

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

asymptotic properties of black box identification of transfer functions the problem of estimating the transfer function of a linear stochastic system is considered the transfer function is parametrized as a black box and no given order is chosen a priori this means that the model orders may increase to infinity when the number of observed data tends to infinity the consistency and convergence properties of the resulting transfer function estimates are investigated asymptotic expressions for the variances and distributions of these estimates are also derived for the case that the model orders increase it is shown that the variance of the transfer function estimate at a certain frequency is asymptotically given by the noise to signal ratio at that frequency mulliplied by the model order to number of data points ratio

brief control relevant experiment design for multivariable systems described by expansions in orthonormal bases the problem of control relevant experiment design is considered we first develop the problem in abstract form as an optimization of an unspecified closed loop performance index then we make the problem specific by considering systems that can be described via a finite expansion of orthonormal bases and designing the controller by specifying the closed loop system s sensitivity function although the resulting optimization problem is nonconvex and may admit local solutions simulations of high purity binary distillation columns are presented to demonstrate that significant improvements in controller performance may be obtained even by using the local solution

banded matrix fraction representation of triangular input normal pairs an input pair a b is triangular input normal if and only if a is triangular and aa bb i sub n where i sub n is the identity matrix input normal pairs generate an orthonormal basis for the impulse response every input pair may be transformed to a triangular input normal pair a new system representation is given a b is triangular normal and a is a matrix fraction a m sup 1 n where m and n are triangular matrices of low bandwidth for single input pairs m and n are bidiagonal and an explicit parameterization is given in terms of the eigenvalues of a this band fraction structure allows for fast updates of state space systems and fast system identification when a has only real eigenvalues one state advance requires 3n multiplications for the single input case

model quality evaluation in h sub 2 identification model quality evaluation in set membership identification is investigated in the recent literature two main approaches have been used to investigate this problem based on the concepts of n width and of radius of information in this paper it is shown that the n width is related to the asymptotic value of the conditional radius of information of the identification problem with noise free measurements upper and lower bounds of the conditional radius of information are derived for the h sub 2 identification of exponentially stable systems using approximating n dimensional models linear in the parameters in the presence of power bounded measurement errors the derived bounds are shown to be convergent to the radius for a large number of data and model dimensions moreover a formula for computing the worst case identification error for any linear algorithm is given in particular it is shown that the identification error of the least square algorithm may be increasing with respect to the model dimension peaking effect an almost optimal linear algorithm is presented that is not affected by this peaking effect and indeed is asymptotically optimal

computing orthogonal rational functions with poles near the boundary computing orthogonal rational functions is a far from trivial problem especially for poles close to the boundary of the support of the orthogonality measure in this paper we analyze some of the difficulties involved and present two different approaches for solving this problem

well posedness of h optimal control problems this paper considers the effect of perturbations of a nominal single input single output linear plant or a band of uncertainty on the solution to certain frequency domain optimal control problems weighted h infty sensitivity minimization mixed and robust sensitivity minimization will be considered along with problems of more general type a brief discussion of h p optimal control problems will also be given typical examples will be presented where optimal sensitivity does not depend continuously on the plant ill posedness and conditions for well posedness will be given it is demonstrated that similar discontinuities can occur for more general problems also suggested are ways of defining an optimization problem so that continuous dependence of the infimum is ensured

on input design in l conditional set membership identification this paper deals with input design in conditional set membership identification the problem is how to choose the input signal in order to minimize the global worst case identification error a characterization of the identification error is provided showing that the optimal input is the one that minimizes the radius of a suitable set moreover sufficient conditions under which the impulse input is optimal are provided

orthogonal basis functions in discrete least squares rational approximation we consider a problem that arises in the field of frequency domain system identification if a discrete time system has an input output relation y z g z u z with transter function g then the problem is to find a rational approximation n for g the data given are measurements of input and output spectra in the frequency points zk u zk y zk nk 1 together with some weight the approximation criterion is to minimize the weighted discrete least squares norm of the vector obtained by evaluating g n in the measurement points if the poles of the system are fixed then the problem reduces to a linear least squares problem in two possible ways by multiplying out the denominators and hide these in the weight which leads to the construction of orthogonal vector polynomials or the problem can be solved directly using an orthogonal basis of rational functions the orthogonality of the basis is important because if the transfer function n is represented with respect to a nonorthogonal basis then this least squares problem can be very ill conditioned even if an orthogonal basis is used but with respect to the wrong inner product e g the lebesgue measure on the unit circle numerical instability can be fatal in practice we show that both approaches lead to an inverse eigenvalue problem which forms the common framework in which fast and numerically stable algorithms can be designed for the computation of the orthonormal basis

brief paper efficient computation of l 1 uncertainty model from an impulse response set in this paper the problem of constructing the minimum 1 uncertainty model containing a finite set of assigned models is addressed this problem is relevant to 1 robust control design techniques in order to minimize the size of the uncertainty set associated to the nominal plant the problem is formulated as a conditional chebyshev center problem and an algorithm for its solution is proposed the algorithm converges in a finite number of steps showing computational efficiency for large size problems

approximate h sub spl infin identification using partial sum operators in a disc algebra basis h sub spl infin system identification using a basis in the disc algebra is presented the approximate model is represented by a partial sum with respect to this basis the identification problem is to estimate the expansion coefficients of this partial sum since the constructed basis functions cannot be represented analytically they are approximated in order to arrive at a model in a suitable form an algorithm is presented which calculates the model parameters from the frequency domain data set

representation and identification of non parametric nonlinear systems of short term memory and low degree of interaction in this paper we propose a new representation which is particularly useful for a class of non parametric nonlinear systems that have short term memory and low degree of interaction advantages and disadvantages of this representation are discussed and compared to existing methods both theoretically and numerically furthermore results regarding structural estimation based on the analysis of variance and on full scale identification are also provided

the asymptotic crlb for the spectrum of arma processes this paper addresses the issue of quantifying the frequency domain accuracy of autoregressive moving average arma spectral estimates as dictated by the cramer rao lower bound crlb classical work in this area has led to expressions that are asymptotically exact as both data length and model order tend to infinity although they are commonly used in finite model order and finite data length settings as approximations more recent work has established quantifications that for ar models are exact for finite model order by employing new analysis methods based on rational orthonormal parameterizations together with the ideas of reproducing kernel hilbert spaces this paper develops quantifications that extend this previous work by being exact for finite model order in all of the ar ma and arma system cases these quantifications via their explicit dependence on poles and zeros of the underlying spectral factor reveal certain fundamental aspects of the accuracy achievable by spectral estimates of arma processes

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

h identification of soft uncertainty models abstract the paper investigates the problem of identifying uncertainty models of causal siso lti discrete time bibo stable unknown systems using frequency domain measurements corrupted by gaussian noise of known covariance additive uncertainty models are looked for consisting of a nominal model and an additive dynamic perturbation accounting for the modeling error the nominal model is chosen within a class of affinely parametrized models with transfer function of given possibly low order an estimate of the parameters minimizing the h modeling error is obtained by minimizing an upper bound of the worst case with respect to the modeling error second moment of the estimation error then a bound in the frequency domain guaranteeing to include with probability the frequency response error between the estimated nominal model and the unknown system is derived

projective indecomposable modules scott modules and the frobenius schur indicator abstract let be a principal indecomposable character of a finite group g in characteristic 2 the frobenius schur indicator of is shown to equal the rank of a bilinear form defined on the span of the involutions in g moreover if the principal indecomposable module corresponding to affords a quadratic geometry then 0 this result is used to prove a more precise form of a theorem of benson and carlson on the existence of scott components in the endomorphism ring of an indecomposable g module in case the module affords a g invariant symmetric form

optimal rejection of persistent and bounded disturbances continuity properties and adaptation continuity properties of the problem of optimal rejection of bounded persistent disturbances otherwise known as l sup 1 optimal control are furnished for discrete time single input single output systems it is shown that for any plant with no zeros on the unit circle the l sup 1 optimal design is continuous as a map from the plant to the optimal closed loop solution if the plant has a unique optimal solution in the case of nonuniqueness a superoptimal criterion is proposed that will always lead to a unique solution however it still fails to supply continuity a characterization of a general class of adaptive controllers in the presence of bounded disturbances is given using this characterization an adaptive robust scheme based on the l sup 1 design methodology is proposed and it is shown to be globally convergent

brief paper on the asymptotic eigenvalue distribution of block toeplitz matrices related to the generalized orthonormal basis functions model in this paper we study the asymptotic distribution of the eigenvalues of a family of hermitian block toeplitz matrices generated by the internal signals of the so called generalized orthonormal basis functions models our results are a generalization of a well known theorem of szego which addresses only the case of hermitian toeplitz matrices

identification of rational approximate models in h sup spl infin using generalized orthonormal basis robust identification with fir models fails to be successful when the number of coefficients to be estimated becomes large caused by lightly damped modes of the plant or poles very close to the unit circle the paper presents a two stage algorithm to obtain a low order approximate model in frequency domain in a generalized orthonormal basis with guaranteed h sup spl infin error bound for deterministic linear time invariant stable systems the first stage being an l sup spl infin rational approximation and the second nonlinear step being an h sup spl infin rational approximation

h sub spl infin identification and model quality evaluation set membership h sub spl infin identification is investigated using time domain data and mixed parametric and nonparametric models as well as supposing power bounded measurement errors the problem of optimally estimating the unknown parameters and evaluating the minimal worst case identification error called radius of information is solved for classes of models affine in the parameters the radius of information is obtained as function of the h sub spl infin norm of the unmodeled dynamics a method is given for estimating this norm from the available data and some general a priori information on the unmodeled dynamics thus allowing the actual evaluation of the radius of information the radius represents a measure of the predictive ability of the considered class of models and it is then used for comparing the quality of different classes of models and for the order selection of their parametric part the effectiveness of the proposed procedure is tested on some numerical examples and compared with standard statistical criteria for model order selection