root n consistent semiparametric regression on considere une variable aleatoire x z dans 7b r p 7b r q on construit un estimateur generalisant l estimateur des moindres carres ordinaires en inserant des estimateurs non parametriques de la regression dans la projection orthogonale non lineaire sur z

erratic behavior of invariant circles in standard like mappings abstract we study mappings like the standard map but with the sine replaced by another function by varying a parameter in this function it changes smoothly from the sine to a piecewise linear pwl function the function is always analytic except when it is pwl taking the function close enough to the pwl function it is found that kam tori can reappear after they disappeared when the nonlinearity is increased when plotting the boundary around the area where a kam torus with noble winding number exists in the plane of the non linearity versus the distance of the function to the pwl function self similarity is observed

a kam theory for conformally symplectic systems efficient algorithms and their validation abstract we present a kam theory for some dissipative systems geometrically these are conformally symplectic systems i e systems that transform a symplectic form into a multiple of itself for systems with n degrees of freedom depending on n parameters we show that it is possible to find solutions with a fixed n dimensional diophantine frequency by adjusting the parameters we do not assume that the system is close to integrable but we present the results in an a posteriori format our unknowns are a parameterization of the quasi periodic solution and some parameters in the system we formulate an invariance equation that expresses that the system with the parameters leaves invariant the solution given by the embedding we show that if there is a sufficiently approximate solution of the invariance equation which also satisfies some non degeneracy conditions then there is a true solution nearby the smallness assumptions above can be understood either in sobolev or in analytic norms the a posteriori format has several consequences a smooth dependence on the parameters including the singular limit of zero dissipation b estimates on the measure of parameters covered by quasi periodic solutions c convergence of perturbative expansions in dissipative analytic systems d bootstrap of regularity i e that all tori which are smooth enough are analytic if the map is analytic e a numerically efficient criterion for the breakdown of the quasi periodic solutions the proof is based on an iterative quadratically convergent method the iterative step takes advantage of some geometric identities these identities also lead to an efficient algorithm if we discretize the parameterization with n terms a modified newton step requires o n storage and o n log n operations the a posteriori theorems allow one to be confident on the numerical results even very close to breakdown the algorithm does not require that the system is close to integrable so that a continuation algorithm is guaranteed to continue the tori till the breakdown in practice only limitation being the total memory and the precision the algorithms have been implemented and run in other papers

the existence and asymptotic properties of a backfitting projection algorithm under weak conditions we derive the asymptotic distribution of a new backfitting procedure for estimating the closest additive approximation to a nonparametric regression function the procedure employs a recent projection interpretation of popular kernel estimators provided by mammen marron turlach and wand 1997 and the asymptotic theory of our estimators is derived using the theory of additive projections reviewed in bickel klaassen ritov and wellner 1993 our procedure achieves the same bias and variance as the oracle estimator based on knowing the other components and in this sense improves on the method analysed in opsomer and ruppert 1997 we provide high level conditions independent of the sampling scheme we then verify that these conditions are satisfied in a regression and a time series autoregression under weak conditions

comments on robust fragile or optimal with reply the issue of fragile controllers produced by using popular robust and optimal control synthesis methods was raised by keel bhattacharyya in the above mentioned paper ibid vol 42 1997 the commenter points out that the controller fragility problem described in the paper is 1 well known in the literature 2 due to the poor choice of examples 3 can easily be fixed by changing the optimization criteria and 4 can be fixed by changing the implementation in reply keel bhattacharyya disagree with him on all counts and explain their argument

gap metric on the subgraphs of systems and the robustness problem the gap metric between the shift invariant subspaces of the graphs or subgraphs of systems is investigated under certain index conditions it is shown that the gap metric on the subgraphs shares the same fundamental property of robust stability as those well known metrics such as the gap metric and the spl nu gap metric it is also shown that the spl nu gap metric between two systems is the distance measured by the gap metric between their respective sets of all subgraphs under certain index conditions

one step local quasi likelihood estimation summary local quasi likelihood estimation is a useful extension of local least squares methods but its computational cost and algorithmic convergence problems make the procedure less appealing particularly when it is iteratively used in methods such as the back fitting algorithm cross validation and bootstrapping a one step local quasi likelihood estimator is introduced to overcome the computational drawbacks of the local quasi likelihood method we demonstrate that as long as the initial estimators are reasonably good the one step estimator has the same asymptotic behaviour as the local quasi likelihood method our simulation shows that the one step estimator performs at least as well as the local quasi likelihood method for a wide range of choices of bandwidths a data driven bandwidth selector is proposed for the one step estimator based on the pre asymptotic substitution method of fan and gijbels it is then demonstrated that the datadriven one step local quasi likelihood estimator performs as well as the maximum local quasilikelihood estimator by using the ideal optimal bandwidth

variable bandwidth kernel estimators of regression curves on montre qu en termes de l erreur en moyenne quadratique integree asymptotique les estimateurs du noyau avec un certain choix de largeur de bande local sont superieurs aux estimateurs ordinaires du noyau avec un choix de largeur de bande global si on utilise des largeurs de bandes optimales

pointwise gap metrics on transfer matrices a family of metrics pointwise gap metrics in the space of real rational matrices of fixed size is developed and used to study open loop and closed loop stability robustness of linear time invariant finite dimensional continuous time systems it is shown that pointwise gap metrics have the desired qualitative properties for the study of stability robustness necessary and sufficient conditions on the robustness are obtained in terms of the radii of the pointwise gap metric balls centered at the nominal plant and or the nominal controller comparison with the gap metric and the graph metric is made all of these metrics induce the same topology many of the quantitative properties of pointwise gap metrics are the same as those of the gap metric although they differ in value pointwise gap metrics in the scalar case have a very simple expression which is potentially useful for accessing the relationship between the uncertainty of physical parameters and uncertainty measured by pointwise gap metrics

on slater s criterion for the breakup of invariant curves abstract we numerically explore slater s theorem in the context of dynamical systems to study the breakup of invariant curves slater s theorem states that an irrational translation over a circle returns to an arbitrary interval in at most three different recurrence times expressible by the continued fraction expansion of the related irrational number the hypothesis considered in this paper is that slater s theorem can be also verified in the dynamics of invariant curves hence we use slater s theorem to develop a qualitative and quantitative numerical approach to determine the breakup of invariant curves in the phase space of area preserving maps

fractal boundary for the existence of invariant circles for area preserving maps observations and renormalisation explanation abstract breakup of a golden invariant circle is studied in an extended version of the standard map with two parameters the critical line in parameter space turns out to have a cantor set of cusps which can be organized in a self similar tree a theoretical explanation is conjectured in terms of a horseshoe for a renormalisation operator the results of some other researchers on similar systems are shown to fit this explanation as well

singularities of periodic orbits near invariant curves abstract we show numerically for standard like maps how the singularities in the complex parameter of the function which conjugates the map to a rotation of rational period behave when the period goes to an irrational number furthermore we propose a numerical method to extrapolate the radius of convergence of the series parametrizing the solution of periodic orbits the results are compared with analyses performed by pade approximants greene s method root criterion and the prediction by renormalization theory

superefficiency in nonparametric function estimation fixed parameter asymptotic statements are often used in the context of nonparametric curve estimation problems e g nonparametric density or regression estimation in this context several forms of superefficiency can occur in contrast to what can happen in regular parametric problems here every parameter point e g unknown density or regression function can be a point of superefficiency we begin with an example which shows how fixed parameter asymptotic statements have often appeared in the study of adaptive kernel estimators and how superefficiency can occur in this context we then carry out a more systematic study of such fixed parameter statements it is shown in four general settings how the degree of superefficiency attainable depends on the structural assumptions in each case

nonparametric and distribution free estimation of the binary threshold crossing and the binary choice models in this paper it is shown that it is possible to identify binary threshold crossing models and binary choice models without imposing any parametric structure either on the systematic function of observable exogenous variables or on the distribution of the random term this identification result is employed to develop a fully nonparametric maximum likelihood estimator for both the function of observable exogenous variables and the distribution of the random term the estimator is shown to be strongly consistent and a two step procedure for its calculation is developed the paper also includes examples of economic models that satisfy the conditions that are necessary to apply the results

sufficient conditions for robust bibo stabilization given by the gap metric a relation between coprime fractions and the gap metric is presented using this result we provide some sufficient conditions for robust bibo stabilization for a wide class of systems these conditions allow the plant and the compensator to be disturbed simultaneously

a semiparametric maximum likelihood estimator this paper presents a procedure for analyzing a model in which the parameter vector has two parts a finite dimensional component 8 and a nonparametric component a the procedure does not require parametric modeling of but assumes that the true density of the data satisfies an index restriction the idea is to construct a parametric model passing through the true model and to estimate by setting the score for the parametric model to zero the score is estimated nonparametrically and the estimator is shown to be n consistent and asymptotically normal the estimator is then shown to attain the semiparametric efficiency bound characterized in begun et al 1983 for multivariate nonlinear regression simultaneous equations partially specified regression index regression censored regression switching regression and disequilibrium models in which the error densities are unknown

the rank of demand systems theory and nonparametric estimation w m gorman s 1981 concept of engel curve rank is extended to apply to any demand system rank is shown to have implications for specification separability and aggregation of demands a simple nonparametric test of rank using engel curve data is described and applied to u s and u k consumer survey data the test employs a new general method for testing the rank of estimated matrices the results are used to assess theoretical and empirical aggregation error in representative consumer models and to explain a representative consumer paradox copyright 1991 by the econometric society

graph topology and gap topology for unstable systems a reformation is provided of the graph topology and the gap topology for a general setting including lumped linear time invariant systems and distributed linear time invariant systems in the frequency domain some essential properties and their comparisons are clearly presented in the reformulation it is shown that the gap topology is suitable for general systems rather than square systems with unity feedback it is shown that whenever an unstable plant can be stabilized by feedback it is a closed operator mapping a subspace of the input space to the output space hence the gap topology can always be applied whenever the unstable plant can be stabilized the graph topology and the gap topology are suitable for different subsets of systems and have many similar characteristics if one confines them to the same subset they will be identical the definitions of the graph metric and the gap metric are discussed

non integrability and continuation of fixed points of 2 n dimensional perturbed twist maps abstract in this paper a simple criterion to prove non integrability of symplectic perturbed twist mappings in 2 n dimensions is developed for sufficiently small perturbations in addition an upper bound for the number of isolating integrals the system can possess is provided a criterion for the analytic continuation of isolated periodic orbits in case of a small nonzero perturbation of twist maps is found the evaluation of their linear stability character by obtaining a simplified expression of the eigenvalues of the jacobian matrix concludes the theoretical part the theory is finally applied to a twist map perturbed by the suris potential and also a four dimensional map with physical interest

robust stabilization in the gap metric controller design for distributed plants the problem of robustness in the gap metric for infinite dimensional systems is considered the problem of computing the optimal controller and the optimal robustness radius for a class of systems whose normalized coprime factors have elements which are h sub infinity functions with continuous boundary values is studied the underlying hankel and related operators which are important in the gap optimization problem are studied and relations between their singular values and vectors are established a computational approach to the optimal robustness problem is developed for single input single output systems whose transfer function is an inner function in h sub infinity times a rational function the procedure is applied to a general first order delay system and a closed form formula is obtained for the optimal controller the frequency response plots of the compensated system for various values of time delay are examined

natural boundaries for area preserving twist maps we consider kam invariant curves for generalizations of the standard map of the form x y x y y f x wheref x is an odd trigonometric polynomial we study numerically their analytic properties by a pade approximant method applied to the function which conjugates the dynamics to a rotation in the complex plane natural boundaries of different shapes are found in the complex plane the analyticity region appears to be a strip bounded by a natural boundary whose width tends linearly to 0 as tends to the critical value

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

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

computation of domains of analyticity for some perturbative expansions of mechanics abstract we compute the domain of analyticity of some perturbative expansions for invariant circles appearing in mechanics we use pade approximants for the perturbative expansions and introduce methods to ascertain their domain of convergence we also use non perturbative methods based on direct computation of the invariant circles and in analogy with greene s criterion approximation by circles with rational rotation we find that the domains computed by all the methods agree within the limits of accuracy we also study rigorously the nature of the singularities when the frequency is rational

nonparametric two stage estimation of conditional choice probabilities in a binary choice model under uncertainty abstract in this paper estimation of conditional choice probabilities in a binary choice model under uncertainty is considered under weak nonparametric restrictions on expectations and the error distribution the estimation method follows a two stage strategy the first stage estimates expectations using realizations of agents future and the second stage estimates conditional choice probabilities using choice data and the expectations estimates this paper gives conditions under which the two stage nonparametric kernel estimator is uniformly strongly consistent and asymptotically normal

continuity of singular perturbations in the graph topology abstract for a certain model for singular perturbations in control systems which we motivate by a simple example we show that under weak assumptions continuity in the graph topology holds as the perturbation parameter tends to zero this may be contrasted with a result by cobb who considered a different model for singular perturbations and who found a strong condition to be necessary for continuity in that model our proof techniques are based on the characterization due to qiu and davison of the graph topology as a topology of uniform convergence

on the gap metric and coprime factor perturbations abstract new conditions are derived for when the distance between two linear systems in the gap metric is less than one by including a coprimeness assumption in the coprime factor uncertainty description it is shown that an open ball in the gap metric is equivalent to an open ball phrased in terms of coprime factor perturbations a new criterion is given for robust stabilization

the robustness of feedback systems with bounded complexity controllers h sub spl infin methods for the analysis and design of robust feedback control systems have sometimes been criticized for an apparent conservatism however recent results have shown that they can provide the least conservative results possible under a particular set of assumptions for example a ball in the v gap metric is the largest set of plants that can be guaranteed to be stabilized a priori by a controller known only to satisfy a bound on the induced norm of a particular closed loop operator nevertheless there are examples of uncertainty which whilst large when measured by the v gap metric would be regarded as relatively benign by an experienced designer of control systems the present paper examines the possibility that in arriving at this judgment such a designer is implicitly using the knowledge that he will always choose the least complex controller necessary to do the job it is shown that given an appropriate bound on the complexity of the controller significantly stronger a priori robustness results can be obtained

brief controller tuning freedom under plant identification uncertainty double youla beats gap in robust stability in iterative schemes of identification and control one of the particular and important choices to make is the choice for a model uncertainty structure capturing the uncertainty concerning the estimated plant model structures that are used in the recent literature encompass e g gap metric uncertainty coprime factor uncertainty and the vinnicombe gap metric uncertainty in this paper we study the effect of these choices by comparing the sets of controllers that guarantee robust stability for the different model uncertainty bounds in general these controller sets intersect however in particular cases the controller sets are embedded leading to uncertainty structures that are favourable over others in particular when restricting the controller set to be constructed as metric bounded perturbations around the present controller the so called double youla parametrization provides a set of robustly stabilizing controllers that is larger than corresponding sets that are achieved by using any of the other uncertainty structures this is particularly of interest in controller tuning problems

nonparametric estimation of a generalized additive model with an unknown link function mean of a random scalar y conditional on a random vector x e rd let h x e yix x denote the conditional mean function this paper is concerned with estimating h under weak assumptions about its functional form when x is a multidimensional continuous random variable the paper describes a model that nests single index additive and multiplicative models and that can be estimated with a faster rate of convergence in probability than can a fully nonparametric model nonparametric estimation of h is usually unattractive when x is multidimensional the rate of convergence of nonparametric estimators decreases rapidly as the dimension of x increases and nonparametric estimates can be very imprecise with samples of practical size when x is multidimensional the curse of dimensionality this problem can be overcome through the use of dimension reduction methods they reduce the effective dimension of the estimation problem by making assumptions about the functional form of h that are stronger than those of a fully nonparametric estimator but weaker than those of a finite dimensional parametric model one method of dimension reduction is semiparametric single index modeling the best known and most easily estimated semiparametric single index model assumes that 1 1 h x g 3 x where 3 is an unknown d x 1 vector and g is an unknown function the vector 3 can be estimated with an n 1 2 rate of convergence in probability which is the rate that would be obtained if g were known han 1987 hardle and stoker 1989 horowitz and hardle 1996 ichimura 1993 klein and spady 1993 newey and stoker 1993 powell stock and stoker 1989 sherman 1993 stoker 1991 g can be estimated with the rate of convergence n r 2r 1 where r is an integer that depends on the smoothness of g but not on d dimension reduction can also be achieved by using a nonparametric generalized additive model in such a model

connection of multiplicative relative perturbation in coprime factors and gap metric uncertainty in this paper it is shown that a linear uncertain system described by a certain l multiplicative or relative perturbation in its coprime factors that are not necessarily normalized is the same as the one described by a gap or l gap metric ball hence all the stability robustness results for gap or l gap metric uncertainty carry over to this type of coprime factor perturbation uncertain systems described by h multi plicative or relative perturbations in coprime factors are also studied in this paper necessary and su cient conditions for robust stability of a feedback system with coprime factors of both the plant and the controller subject to simultaneous h multiplicative or relative perturbations are obtained

on the computation of the gap metric abstract this paper deals with the computation of the gap metric introduced by zames and el sakkary 17 it is shown that the gap between two systems p1 p2 is precisely the maximum of the two expressions inf q h d i n i d j n j q for i j equal to 1 2 and 2 1 and ni di being normalized right coprine factorizations of pi i 1 2 in the sense of vidyasagar 12 this expression is computable using well known techniques from interpolation theory

greene s residue criterion for the breakup of invariant tori of volume preserving maps abstract invariant tori play a fundamental role in the dynamics of symplectic and volume preserving maps codimension one tori are particularly important as they form barriers to transport such tori foliate the phase space of integrable volume preserving maps with one action and d angles for the area preserving case greene s residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits even though kam theory applies to the three dimensional case the robustness of tori in such systems is still poorly understood we study a three dimensional reversible volume preserving analogue of chirikov s standard map with one action and two angles we investigate the preservation and destruction of tori under perturbation by computing the residue of nearby periodic orbits we find tori with diophantine rotation vectors in the spiral mean cubic algebraic field the residue is used to generate the critical function of the map and find a candidate for the most robust torus

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

a robust control framework for linear time invariant spatially distributed systems a robust control framework for linear time invariant lti spatially distributed systems is outlined in this paper we adopt an input output approach which takes account of the spatially distributed nature of the input and output signals for such systems the approach is a generalization of h infty xspace control in the sense that the 2 norm in both time and space is used to quantify the size of signals it is shown that a frequency domain representation in the form of a graph symbol exists for every lti spatially distributed system under very mild assumptions the graph symbol gives rise to left and right coprime representations if the system is also stabilizable we investigate fundamental issues of feedback control such as feedback stability and robust stability to plant and or controller uncertainty quantified in the gap metric this includes a generalization of the sefton ober gap formula to the infinite dimensional operator case a design example in which an electrostatically destabilized membrane is feedback stabilized concludes the paper

frequency domain uncertainty and the graph topology a new metric on linear time invariant systems is defined this metric is no greater than the gap metric and is in fact the smallest metric for which a certain robust stabilization result holds unlike other known metrics which induce the graph topology it has a clear frequency response interpretation this allows questions regarding robustness in the face of parametric uncertainty to be considered in terms of this metric

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

the link between the shape of the irrational aubry mather sets and their lyapunov exponents we consider the irrational aubry mather sets of an exact symplectic monotone c 1 twist map of the two dimensional annulus introduce for them a notion of c 1 regularity related to the notion of bouligand paratingent cone and prove that a mather measure has zero lyapunov exponents if and only if its support is c 1 regular almost everywhere a mather measure has nonzero lyapunov exponents if and only if its support is c 1 irregular almost everywhere an aubry mather set is uniformly hyperbolic if and only if it is irregular everywhere the aubry mather sets which are close to the kam invariant curves even if they may be c 1 irregular are not too irregular i e have small paratingent cones the main tools that we use in the proofs are the so called green bundles

an obstruction method for the destruction of invariant curves abstract an important point in a conservative dynamical system which is a parameter depending perturbation of an integrable system is the destruction of the invariant curves for small perturbations many invariant curves subsist according to kam theorem an increase in the size of perturbations produces eventually the destruction of the curves the stochastic zone is then very large in this paper we present for the standard map a geometrical approach to obtain values of the parameter for which no invariant curves exist this is due to the existence of heteroclinic points as it has been observed using other methods there are several scaling properties in the phenomena which are also discussed here

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

nonparametric matching and efficient estimators of homothetically separable functions for vectors z and w and scalar v let r v z w be a function that can be non parametrically estimated consistently and asymptotically normally such as a distribution density or conditional mean regression function we provide consistent asymptotically normal nonparametric estimators for the functions g and h where r v z w h vg z w and some related models this framework encompasses homothetic and homothetically separable functions and transformed partly additive models r v z w h v g z w for unknown functions g and h such models reduce the curse of dimensionality provide a natural generalization of linear index models and are widely used in utility production and cost function applications we also provide an estimator of g that is oracle efficient achieving the same performance as an estimator based on local least squares when h is known

the parallel projection operators of a nonlinear feedback system abstract this paper defines and studies a pair of nonlinear parallel projection operators associated with a nonlinear feedback system these operators have been seen to play an important role in the robustness and design of linear systems especially in the theory of the gap metric the use of weighted gaps in control system design and glover mcfarlane loop shaping we show that the input output l 2 stability of a feedback system amounts to a coordinatization of the input and output spaces which is also equivalent to the existence of a pair of nonlinear parallel projection operators onto the graph of the plant and the inverse graph of the controller respectively these projections are shown to have equal norms whenever one of the feedback elements is linear a bound on this norm is given in the case of passive systems with unity negative feedback

estimating semiparametric arch models by kernel smoothing methods we investigate a class of semiparametric arch models that includes as a special case the partially nonparametric pnp model introduced by engle and ng 1993 and which allows for both flexible dynamics and flexible function form with regard to the news impact function we show that the functional part of the model satisfies a type ii linear integral equation and give simple conditions under which there is a unique solution we propose an estimation method that is based on kernel smoothing and profiled likelihood we establish the distribution theory of the parametric components and the pointwise distribution of the nonparametric component of the model we also discuss efficiency of both the parametric part and the nonparametric part we investigate the performance of our procedures on simulated data and on a sample of s p500 index returns we find evidence of asymmetric news impact functions consistent with the parametric analysis

model reduction with relative multiplicative error bounds and relations to controller reduction balanced model reduction with a priori relative multiplicative error bounds in l sub spl infin norm is studied inverse weighted balanced truncation iwbt is proposed that is a variant of the enns algorithm it is shown that iwbt is equivalent to the balanced stochastic truncation bst this equivalence leads to the extension of an improved relative multiplicative error bound for bst obtained by wang and safonov to general nonsquare transfer matrices natural relations between iwbt and observer based controller reduction are investigated which motivate the simultaneous reduction of the plant and controller robust stability conditions for simultaneous reduction of the plant and the controller are established for the closed loop system consisting of full order plant and reduced order controller as a by product robust stability and stabilization problems are solved for general coprime factor plant descriptions with relative multiplicative h sub spl infin norm bounded uncertainties similar results are obtained for general coprime factors of the feedback controller involving relative multiplicative h sub spl infin norm bounded uncertainties

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

feedback stability under simultaneous gap metric uncertainties in plant and controller abstract the stability robustness of a feedback system is studied in this paper by assuming that the plant and the controller are subject to independent uncertainties and that the uncertainties are measured by the gap metric a fairly complete solution is obtained by exploring the trigonometric structure of the graphs of the plant and the controller

simulation and the asymptotics of optimization estimators on demontre un theoreme de limite centrale general pour des estimateurs definis par minimisation de la longueur d une fonction de critere aleatoire a valeurs vectorielles aucune hypothese de regularite suffisante n est imposee sur la fonction de critere pour que les resultats puissent etre appliques a une classe d estimlateurs de simulation assez large des analyses completes de deux estimateurs de simulation l un introduit par pakes l autre par mcfadden illustrent l application des theoremes generaux

nonparametric identification of nonlinear time series projections abstract we study the possibility of identifying general linear and nonlinear time series models using nonparametric methods the kernel estimators of the conditional mean and variance are used as a basis and the properties of these quantities as model indicators are briefly discussed some drawbacks are pointed out and motivated by these we introduce projections as tools of identification the projections are especially useful for additive modeling expressions for the asymptotic bias and variance are obtained the projection of the conditional variance is suggested as a tool for identifying heteroscedastic time series the results are illustrated by simulations for both the estimators of the projections and the estimators of the conditional mean and variance

nonparametric estimation of nonadditive random functions we present estimators for nonparametric functions that are nonadditive in unobservable random terms the distributions of the unobservable random terms are assumed to be unknown we show that when a nonadditive nonparametric function is strictly monotone in an unobservable random term and it satisfies some other properties that may be implied by economic theory such as homogeneity of degree one or separability the function and the distribution of the unobservable random term are identified we also present convenient normalizations to use when the properties of the function other than strict monotonicity in the unobservable random term are unknown the estimators for the nonparametric function and for the distribution of the unobservable random term are shown to be consistent and asymptotically normal we extend the results to functions that depend on a multivariate random term the results of a limited simulation study are presented

error bounds in the gap metric for dissipative balanced approximations abstract we derive an error bound in the gap metric for positive real balanced truncation and positive real singular perturbation approximation we prove these results by working in the context of dissipative driving variable systems as in behavioral and state signal systems theory in such a framework no prior distinction is made between inputs and outputs dissipativity preserving balanced truncation of dissipative driving variable systems is addressed and a gap metric error bound is obtained bounded real and positive real input state output systems are manifestations of a dissipative driving variable system through particular decompositions of the signal space under such decompositions the existing bounded real and positive real balanced truncation schemes can be seen as special cases of dissipative balanced truncation and the new positive real error bounds follow

uncertainty in the weighted gap metric a geometric approach the stability of control systems is studied in the context of weighted input output signal spaces necessary and sufficient conditions for a controller to stabilize a plant are given in terms of geometric notions these geometric quantities can be calculated by solving h optimization problems maximally stabilizing controllers in a weighted signal space are introduced and characterized in terms of nehari extensions the robustness properties of maximally stabilizing controllers are analysed in terms of weighted coprime factor uncertainty necessary and sufficient conditions are established for a controller of a given plant to be the maximally stabilizing controller of the plant with respect to a weight an upper bound for the mixed sensitivity of a control system is given where the controller is the maximally stabilizing controller of the plant

invariant curves for area preserving twist maps far from integrable one parameter families of area preserving twist maps of the formf e x y x y ef x y ef x are considered various invariant curves for the maps corresponding tof x sin andf x sinx 1 50 sin 5x are rigorously constructed forlarge values of the nonlinearity parameter for larger values of close to critical some numerical experiments are briefly discussed

distance measures for uncertain linear systems a general theory in this paper we propose a generic notion of distance between systems that can be used to measure discrepancy between open loop systems in a feedback sense under several uncertainty structures when the uncertainty structure is chosen to be four block or equivalently normalized coprime factor uncertainty then this generic distance measure reduces to the well known nu gap metric associated with this generic distance notion we also define a generic stability margin notion that allows us to give the distance measure a feedback interpretation by deriving generic robust stability and robust performance results the proposed distance notion and the corresponding results exploit a powerful generalization of the small gain theorem which handles perturbations in rfrl infin rather than only in rfrh infin when the uncertainty structure is fixed to one of the standard structures e g additive multiplicative inverse multiplicative coprime factor four block or any mixtures of the above we give a step by step procedure based on model validation ideas that shows how the generic notion of distance and the correspondingly generic winding number conditions can be reduced to simple formulae this work provides a unified framework that captures and embeds previous results in this area and also completes the picture by showing how other results of a similar nature can be obtained from the same framework the techniques used involve only basic linear algebra so they also provide a simplification of previous advanced proofs furthermore the various distance measures so created can be used for non conservative model embedding into the smallest uncertain family an illustrative example is also given that demonstrates the superior qualities above the nu gap metric of a particular distance measure obtained from this work in situations where the plant is lightly damped all systems considered in this paper are linear time invariant

linear feedback systems and the graph topology it is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed loop stability and closed loop performance are robust properties in earlier works similar topological results were obtained by exploiting the existence of particular coprime factor system representations not known to exist in general by contrast the development here does not rely on any specific system representations systems are simply characterized as subspaces of norm bounded input output pairs and the analysis hinges on the underlying geometric structure of the feedback stabilization problem unlike other work developed within such a framework fundamental issues concerning the causality of feedback interconnections are discussed explicitly the key result of this paper concerns the difference between linear feedback interconnections with identical controllers in terms of a performance robustness related closed loop mapping upper and lower bounds on the induced norm of this difference are derived allowing for possibly infinite dimensional input output spaces and time varying behavior the bounds are both proportional to the gap metric distance between the plants which clearly demonstrates the gap to be an appropriate measure of the difference between open loop systems from the perspective of closed loop behavior to conclude an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting

on the angular metrics between linear subspaces it has recently been shown that the symmetric gauge functions on the canonical principal angles give a family of unitarily invariant metrics between linear subspaces of the same dimension in this short paper we extend such metrics to subspaces of possibly different dimensions this extension is necessary in addressing some perturbation analysis problems involving subspaces with different dimensions examples of such perturbation analysis problems are also studied in this paper using the extended metrics

a rigorous partial justification of greene s criterion we prove several theorems that lend support to greene s criterion for the existence or not of invariant circles in twist maps in particular we show that some of the implications of the criterion are correct when the aubry mather sets are smooth invariant circles or uniformly hyperbolic we also suggest a simple modification that can work in the case that the aubry mather sets have nonzero lyapunov exponents the latter is based on a closing lemma for sets with nonzero lyapunov exponents which may have several other applications

asymptotics for semiparametric econometric models via stochastic equicontinuity this paper provides a general framework for proving the square root of t consistency and asymptotic normality of a wide variety of semiparametric estimators the class of estimators considered consists of estimators that can be defined as the solution to a minimization problem based on a criterion function that may depend on a preliminary infinite dimensional nuisance parameter estimator the method of proof exploits results concerning the stochastic equicontinuity of stochastic processes the results are applied to the problem of semiparametric weighted least squares estimation of partially parametric regression models primitive conditions are given for square root of t consistency and asymptotic normality of this estimator copyright 1994 by the econometric society

breakdown of lindstedt expansion for chaotic maps in a previous paper of one of us europhys lett 59 330 336 2002 the validity of greene s method for determining the critical constant of the standard map sm was questioned on the basis of some numerical findings here we come back to that analysis and we provide an interpretation of the numerical results by showing that the conclusions of that paper were wrong as they relied on a plausible but untrue assumption hence no contradiction exists with respect to greene s method we show that the previous results based on the expansion in lindstedt series do correspond to the critical constant but for a different map the semi standard map ssm for such a map no greene s method analog is at disposal so that methods based on lindstedt series are essentially the only possible ones moreover we study the expansion for two simplified models obtained from the sm and ssm by suppressing the small divisors we call them the simplified sm and simplified ssm respectively the first case turns out to be re

orbits of unbounded energy in quasi periodic perturbations of geodesic flows abstract we show that certain mechanical systems including a geodesic flow in any dimension plus a quasi periodic perturbation by a potential have orbits of unbounded energy the assumptions we make in the case of geodesic flows are a the metric and the external perturbation are smooth enough b the geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection c the frequency of the external perturbation is diophantine d the external potential satisfies a generic condition depending on the periodic orbit considered in b the assumptions on the metric are c 2 open and are known to be dense on many manifolds the assumptions on the potential fail only in infinite codimension spaces of potentials the proof is based on geometric considerations of invariant manifolds and their intersections the main tools include the scattering map of normally hyperbolic invariant manifolds as well as standard perturbation theories averaging kam and melnikov techniques we do not need to assume that the metric is riemannian and we obtain results for finsler or lorentz metrics indeed there is a formulation for hamiltonian systems satisfying scaling hypotheses we do not need to make assumptions on the global topology of the manifold nor on its dimension

upper and lower bounds for approximation in the gap metric upper and lower bounds for the closest approximant of degree k n in the gap metric to a plant of degree n are obtained the bounds are expressed in terms of the singular values of two hankel operators defined in terms of the symbol of the graph of the plant the question of robust stability and performance of feedback systems is examined in the context of approximation of plant and controller in the gap metric

numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors we study numerically the complex domains of validity for kam theory in generalized standard mappings we compare methods based on pade approximants and methods based on the study of periodic orbits

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

the dimensionality reduction principle for generalized additive models soit x y une paire de variables aleatoires telles que x x 1 x j s etende sur c 0 1 j la distribution conditionnelle de y etant donne x x est supposee appartenir a une famille exponentielle appropriee ayant pour parametre r f x denote la dependance de vis a vis de x et f l approximation additive de f ayant la log vraisemblance esperee maximum possible

fractal diagrams for hamiltonian stochasticity abstract a new diagrammatic representation is presented for hamiltonian systems with three phase space dimensions these fractal diagrams permit an easy way to find kam boundaries and island in the poincare mapping and lead thereby to a complete description of the geography of the mapping in addition a qualitative insight is gained into the diffusion problem the standard mapping is used for demonstration