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

orthogonal rational functions with real coefficients and semiseparable matrices when one wants to use orthogonal rational functions orfs in system identification or control theory it is important to be able to avoid complex calculations in this paper we study orfs whose numerator and denominator polynomial have real coefficients these orfs with real coefficients rorfs appear when the poles and the interpolation points appear in complex conjugate pairs which is a natural condition further we deduce that there is a strong connection between rorfs and semiseparable matrices

robust d stability for linear uncertain discrete time delay systems we propose criteria which guarantee robust pole location within a specified circular region for a linear uncertain discrete time delay system the perturbations are treated with structured and unstructured information examples are given to demonstrate the result

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

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

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

connection coefficients for laguerre sobolev orthogonal polynomials laguerre sobolev polynomials are orthogonal with respect to an inner product of the form p qs 0 p x q x x e x dx 0 p x q x d x w here 1 0 and p q p the linear space of polynomials with real coefficients if d x x e x dx the corresponding sequence of monic orthogonal polynomials q x has been studied by marcellan et al j com put appl math 71 1996 245 265 while if d x x dx the sequence of monic orthog onal polynomials l x was introduced by koekoek and meijer siam j math anal 24 1993 768 782 for each of these two families of laguerre sobolev polynomials here we give the explicit expression of the connection coefficients in their expansion as a series of standard la guerre polynomials the inverse connection problem of expanding laguerre polynomials in series of laguerre sobolev polynomials and the connection problem relating two families of laguerre sobolev polynomials with different parameters are also considered

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

hermite sobolev and closely connected orthogonal polynomials abstract in this paper a bilinear functional a g m o n m c m m g m which corresponds to the hermite sobolev and closely connected orthogonal polynomials is studied the domain of m m 1 n in which a is positive definite is given and the hermite sobolev and closely connected formal orthogonal polynomials are given in the nonnormal case

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

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

orthogonal polynomials on sobolev spaces old and new directions abstract during the last years orthogonal polynomials on sobolev spaces have attracted considerable attention algebraic properties distribution of their zeros and fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case in this paper we present an introductory survey about the subject the origin of the problems and their development show the interest and the promising future of this field

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

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

general sobolev orthogonal polynomials abstract in this paper we study orthogonal polynomials with respect to the inner product f g s n u fg m 1 n m u f m g m where m 0 for m 1 n and u is a semiclassical positive definite linear functional for these non standard orthogonal polynomials algebraic and differential properties are obtained as well as their representation in terms of the standard orthogonal polynomials associated with u

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

a sharp bound for the castelnuovo mumford regularity of subspace arrangements abstract we show that the ideal of an arrangement of d linear subspaces of projective space is d regular in the sense of castelnuovo and mumford answering a question of b sturmfels in particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d

brief paper frequency domain identification an algorithm based on an adaptive rational orthogonal system this paper presents a new adaptive algorithm for frequency domain identification the algorithm is related to the rational orthogonal system takenaka malmquist system this work is based on an adaptive decomposition algorithm previously proposed for decomposing the hardy space functions in which a greedy sequence is obtained according to the maximal selection criterion we modify the algorithm through necessary changes for system identification

continuous symmetrized sobolev inner products of order n i abstract given a symmetrized sobolev inner product of order n the corresponding sequence of monic orthogonal polynomials q n satisfies that q 2 n x p n x 2 q 2 n 1 x x r n x 2 for certain sequences of monic polynomials p n and r n in this paper we deduce the integral representation of the inner products such that p n and r n are the corresponding sequences of orthogonal polynomials moreover we state a relation between both inner products which extends the classical result for symmetric linear functionals

generalized linearization problems we prove that if pi x and pj x are two families of semi classical orthogonal polynomials all the linearization coefficients li j k occurring in the product of these two families satisfy a linear recurrence relation involving only the k index this property also extends to the linearization coefficients arising from an arbitrary number of products of semi classical orthogonal polynomials

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

a fast adaptive model reduction method based on takenaka malmquist systems abstract an adaptive model reduction method is proposed for linear time invariant systems based on the continuous time rational orthogonal basis takenaka malmquist basis the method is to find an adaptive approximation in the energy sense by selecting optimal points for the rational orthogonal basis the stability of the reduced models holds and the steady state values of step responses are kept to be equal furthermore the method automatically ensures the reduced system to be in the hardy space h 2 the existence of the best approximation in the hardy space h 2 by n blaschke forms is proved in the proposed approach the effectivity of this method is illustrated through three well known examples

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

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

a generalized eigenvalue problem for quasi orthogonal rational functions in general the zeros of an orthogonal rational function orf on a subset of the real line with poles among alpha 1 ldots alpha n subset mathbb c 0 cup infty are not all real unless alpha n is real and hence they are not suitable to construct a rational gaussian quadrature rule rgq for this reason the zeros of a so called quasi orf or a so called para orf are used instead these zeros depend on one single parameter tau in mathbb c cup infty which can always be chosen in such a way that the zeros are all real and simple in this paper we provide a generalized eigenvalue problem to compute the zeros of a quasi orf and the corresponding weights in the rgq first we study the connection between quasi orfs para orfs and orfs next a condition is given for the parameter so that the zeros are all real and simple finally some illustrative and numerical examples are given

some remarks on a paper by l carlitz we study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by l carlitz we show that they are a special case of the sheffer polynomials and point out some interesting connections with certain sobolev orthogonal polynomials