the vinnicombe metric for nonlinear operators describes an extension of the vinnicombe metric on linear operators to a pseudometric on nonlinear operators a metric for finite dimensional time varying operators is shown to be capable of guaranteeing stability and performance robustness and reduces to the standard vinnicombe metric for the time invariant operator case which is known to be less conservative than the gap metric the analysis exploits the time varying operator equivalents of unstable poles and normalized coprime fractional descriptions in addition a time varying operator equivalent of the winding number is defined

local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

nonparametric estimates of regression quantiles and their local bahadur representation let x y be a random vector such that x is d dimensional y is real valued and y theta x varepsilon where x and varepsilon are independent and the alpha th quantile of varepsilon is 0 alpha is fixed such that 0 0 and set r p m 2p d where m is a nonnegative integer smaller than p let t theta denote a derivative of theta of order m it is proved that there exists a pointwise estimate hat t n of t theta based on a set of i i d observations x 1 y 1 cdots s n y n that achieves the optimal nonparametric rate of convergence n r under appropriate regularity conditions further a local bahadur type representation is shown to hold for the estimate hat t n and this is used to obtain some useful asymptotic results

on the radius of convergence of interconnected analytic nonlinear input output systems a complete analysis is presented of the radii of convergence of the parallel product cascade and feedback interconnections of analytic nonlinear input output systems represented as fliess operators such operators are described by convergent functional series which are indexed by words over a noncommutative alphabet their generating series are therefore specified in terms of noncommutative formal power series given growth conditions for the coefficients of the generating series for the subsystems the radius of convergence of each interconnected system is computed assuming the subsystems are either all locally convergent or all globally convergent in the process of deriving the radius of convergence for the feedback connection it is shown definitively that local convergence is preserved under feedback this had been an open problem in the literature until recently

sensitivity and robust stability of general input output systems the author considers a general model of an input output system that is governed by nonlinear operator equations which relate the input the state and the output of the system this model encompasses feedback systems as a special case assuming that the governing equations depend on a parameter a which is allowed to vary in a neighborhood of a nominal value a sub 0 in a linear space the author studies the dependence of the system behavior on a a system is considered insensitive if for any fixed input the output depends continuously on a similarly the system is robust if it is stable for each a in a neighborhood of a sub 0 stability is defined as an appropriate continuity of the input output operator the results give various sufficient conditions for insensitivity and robustness applications of the theory are discussed including the estimation of the difference of operator inverses and the insensitivity and robust stability of a hilbert network a feedback feedforward system a traditional feedback system and a time varying dynamical system described by a linear vector differential equation on 0 infinity

reference values obtained by kernel based estimation of quantile regressions in this paper we study the problem of estimating non parametrically a quantile regression curve as it applies to computing reference values we propose an automatic procedure that uses a symmetrized nearest neighbor kernel estimator of conditional distributions we also discuss ways of measuring the dispersion of quantile regression estimator one is based on the asymptotic distribution of such quantiles while the other relies on the bootstrap method the results of a small simulation study show that the methods of the paper perform rather well even in a situation where a good parametric solution is available as an example we analyze a small part of a data set that was collected to establish reference values for blood velocity in different parts of the umbilical cord of human fetuse as they grow toward birth

construction of the left coprime fractional representation for a class of nonlinear control systems abstract a constructive approach to the left coprime fractional representations for a class of stable and unstable nonlinear control systems described by certain multi input multi output nonlinear ordinary differential equations is proposed the method works for the coprimeness defined by verma 15

stability analysis for time varying systems via quadratic constraints abstract in this paper we introduce the concept of quadratic constraint qc to deal with various stability problems for infinite dimensional linear discrete time varying ltv systems first we derive a necessary and sufficient condition for the close loop stability based on quadratic constraints qcs an equivalent condition of this stability criterion presents the relationship between the stabilization of each finite dimensional truncation system and that of the whole system moreover applying this stability criterion and the complete finiteness of the nest algebra we show that the plant is stabilizable if and only if it has a single strong representation finally we use qcs to give a necessary and sufficient condition for the robust stabilization of connected uncertainty set defined by gap metric

characterization of all nonlinear stabilizing controllers via observer based kernel representations this paper deals with the characterization of all nonlinear stabilizing controllers observer based kernel representations are introduced to avoid the difficulty which occurs in constructing the state space realizations of the parametrization of input output operators the parametrization of all stabilizing controllers for a given plant and a class of stabilizing plant controller pairs are derived by employing observer based kernel representations furthermore the relationship between the results via kernel representation approach and the existing results via state space approach is clarified

a parametrization for closed loop identification of nonlinear systems based on differentially coprime kernel representations in this paper we use the notion of a differentially coprime kernel representation to parametrize the set of all nonlinear plants stabilized by a given nonlinear controller using a so called youla parameter and to unify understanding of some stability concepts for nonlinear systems by utilizing the differential kernel representation concept we are able to convert a closed loop identification problem into one of open loop identification the main advantage of our approach using kernel representations over fractional descriptions is that we address a larger class of nonlinear systems the idea of a differential kernel representation allows us also to clarify the relationship between three different notions of internal stability available in the literature the results in the paper thus provide new insights to the stability of nonlinear feedback systems

further results on stability of xd t ax t bx t t abstract our aim in this communication is to improve the existing results concerning the stability of x dot t ax t bx t it is shown that the unstable characteristic roots could be more accurately located in a bounded and restrictive region of the complex plane an iterative scheme is presented for numerical implementation of the results numerical computations are presented for illustration and comparison with previous results

robust right coprime factorization and robust stabilization of nonlinear feedback control systems robust right coprime factorization and robust stabilization of nonlinear feedback control systems are studied the concept of robust right coprime factorization of nonlinear operators for feedback control systems is introduced some conditions for the robustness of a right coprime factorization of a nonlinear plant under unknown but bounded perturbations are derived an example with a closed form solution is included to illustrate the general theory and a step by step construction of the robust factorization and the robust stabilization

the bandwidth minimization problem for cyclic caterpillars with hair length 1 is np complete in this paper we show that the bandwidth minimization problem remains np complete for cyclic caterpillars with hair length 1 cyclic caterpillars with hair length 1 are graphs in which the removal of all pendant vertices results in a simple cycle

non linear systems stabilization and coprimeness a theory of coprimeness is developed for non linear systems with the intention of constructing analytic tools for the solution of the problem of stabilizing a non linear system through the application of additive non linear feedback the qualitative features of the resulting theory are strikingly similar to the situation in the linear case

internally stable nonlinear systems with disturbances a parameterization this paper deals with the control of a nonlinear system whose output is subjected to an additive disturbance the main result is a simple parameterization of the set of all system responses that can be obtained through internally stable control of the given system the parameterization provides a clear indication of the effects of the disturbance on the response of the stabilized closed loop system the class of achievable responses is determined by the numerator of a right coprime fraction representation of the system being controlled

a quantitative condition for the well posedness of interconnected dynamical systems a sufficient condition for the well posedness of interconnected dynamical systems is reported the condition is stated in terms of the uniform instantaneous gains of subsystems and interconnections and includes vidyasagar s condition 8 as a special case the result is useful for testing the validity of the dynamical model of a large scale system which is obtained by combining the models of subsystems

conditional empirical processes consider a sequence i xi yi i 1 of independent d 1 random vectors with common d f h text h left text x y right mathbb p left text x x y y right text left text x y right in mathbb r text d 1

sensitivity and robust stability of some control systems containing a nonlinear plant sufficient conditions for insensitivity and robust stability of some frequently used control systems which contain a nonlinear plant and are not of a feedback type are presented these conditions are derived from some results of the author 1991 on general input output systems they are stated in terms of the transmission operators that describe the blocks constituting the system

on robustness in control and lti identification near linearity and non conic uncertainty robustness issues are studied in the context of linear models in linear controller design and in system identification when the true system is nonlinear the notion of nearly linear system is generalized to include time varying and open loop unstable systems this class of systems although it presents the simplest possible global generalization of linear systems is in many ways nontrivial for the purpose of linear controller design and linear model identification from input output data this is mainly due to the presence of non conic uncertainty which is not included in standard treatments of robust control theory and stochastic system identification theory signal distribution theory a realistic non stochastic signal analysis tool is used to study the limiting least squares estimates of linear finite impulse response fir and autoregressive with external input arx model parametrizations for two classes of nonlinear systems some connections to worst case analysis of linear model identification are also discussed

on the parametrization of nonlinear finite gain over set stabilizing controllers the concepts of gain and incremental gain over set have been introduced to alleviate some of the limitations of the conventional notions of gain and incremental gain in this paper the class of all finite gain over set stabilizing controllers is parametrized in terms of a finite gain over set stable map q provided that the plant p is finite incremental gain over set stable an alternate theorem for finite gain over ball stability of the closed loop is also developed which provides a superset and a subset of the set of all finite gain over set stabilizing compensators for the feedback system

identification of dynamic models in complex networks with prediction error methods basic methods for consistent module estimates abstract the problem of identifying dynamical models on the basis of measurement data is usually considered in a classical open loop or closed loop setting in this paper this problem is generalized to dynamical systems that operate in a complex interconnection structure and the objective is to consistently identify the dynamics of a particular module in the network for a known interconnection structure it is shown that the classical prediction error methods for closed loop identification can be generalized to provide consistent model estimates under specified experimental circumstances two classes of methods considered in this paper are the direct method and the joint io method that rely on consistent noise models and indirect methods that rely on external excitation signals like two stage and iv methods graph theoretical tools are presented to verify the topological conditions under which the several methods lead to consistent module estimates

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

percentile smoothing using piecewise polynomials with covariates a nonparametric procedure for the joint smoothing of a series of percentile curves is presented it allows separate curves to be fitted over contiguous age ranges constraining them to join smoothly covariates can be incorporated thus allowing efficient estimation simultaneously for subgroups

the gap metric for time varying systems abstract a formula is given relating the gap metric to a 2 block uniform optimization problem georgiou s formula for the gap between time invariant systems is obtained as a special case

finding the minimum bandwidth of an interval graph abstract an assignment of unique integers to the vertices of a graph is called a linear layout the bandwidth minimization problem bandwidth is the following given a graph g v e and an integer k determine whether there exists a linear layout of g such that the maximum difference between adjacent vertices is bounded by k interval graphs are the intersection graphs of a family of intervals of the real line bandwidth remains np complete even when restricted to special subclasses of trees we show that bandwidth can be solved in time o n2 for interval graphs moreover for a given interval graph a linear layout with minimum bandwidth can be constructed in time o n2logn as a by product we get that this construction can be done for proper interval graphs in time o nlog n m

reference curves based on non parametric quantile regression reference curves which take time into account such as those for age are often required in medicine but simple systematic and efficient statistical methods for constructing them are lacking classical methods are based on parametric fitting polynomial curves semi parametric methods are also widely used especially in europe here we propose a new methodology for the estimation of reference intervals for data sets based on non parametric estimation of conditional quantiles the derived methods should be applicable to all clinical or more generally biological variables that are measured on a continuous quantitative scale as an example we analyse a data set collected to establish reference curves for biophysical properties of the skin of healthy french women the results are compared to those obtained with royston s polynomial parametric method and the semi parametric lms approach

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

graph topologies gap metrics and robust stability for nonlinear systems graph topologies for nonlinear operators which admit coprime factorizations are defined w r t a gain function notion of stability in a general normed signal space setting several metrics are also defined and their relationship to the graph topologies are examined in particular relationships between nonlinear generalizations of the gap and graph metrics georgiou type formulae and the graph topologies are established closed loop robustness results are given w r t the graph topology where the role of a coercivity condition on the nominal plant is emphasized

singular perturbation margin and generalised gain margin for linear time invariant systems in this paper we propose a singular perturbation margin spm and a generalised gain margin ggm as stability metrics for single input single output siso linear time invariant lti systems from the view of singular perturbations and regular perturbations which have bijective correspondences with the classical phase margin pm and the gain margin gm respectively both of the numerical and analytical time domain spm and ggm assessment methods are provided and relationships between the singular perturbation parameter pm of the perturbed system pm and spm of the nominal system and the monotonically increasing phase of the fast system are also revealed these results make it possible to assess the pm of the nominal system in the time domain for siso lti systems using the spm with a standardised testing system called pm gauge as demonstrated by examples the concepts of spm and ggm can be used as metrics of stability margins for linear time varying systems and nonlinear systems

technical communique recursive identification of nonlinear plants operating in closed loop using kernel representations in this paper we extend the family of algorithms presented in algorithms for identification of continuous time nonlinear systems a passivity approach in a isidori f ramnabhi lagarrigue w respondek eds nonlinear control in the year 2000 vol 2 pp 13 44 berlin springer automatica 37 2000 469 for the identification of continuous time nonlinear plants operating in closed loop the new algorithms presuppose that one can construct a stable kernel representation for the to be identified model structure the new theory results in a less restrictive passivity condition the main novelty is that the identification of unstable plants can be tackled by an appropriate choice of the kernel representation i e there is an additional degree of freedom when constructing the kernel representation the implicit stability of the controller is still required by the new passivity condition

a flexible and fast method for automatic smoothing abstract the choice of smoothing parameter or bandwidth is crucial when applying nonparametric regression estimators such as kernel estimators the optimal choice depends on the data at hand a data driven bandwidth selection close to the optimal one would make these curve estimators objective more reliable and easier to use minimizing residual mean squared error criteria such as cross validation have been frequently proposed to estimate the optimal smoothing parameter empirical and theoretical evidence indicates that cross validation rules and related methods lead to rather variable estimated optimal bandwidths the method presented here builds on estimating the asymptotically optimal bandwidth from the data since estimators for the residual variance and for an asymptotic expression for the bias are plugged into the asymptotic formula such selection rules are called plug in estimators the functional that quantifies bias is approximated by the integrated squared second derivative of the regre

generating series for interconnected analytic nonlinear systems given two analytic nonlinear input output systems represented as fliess operators four system interconnections are considered in a unified setting the parallel connection product connection cascade connection and feedback connection in each case the corresponding generating series is produced and conditions for the convergence of the corresponding fliess operator are given in the process an existing notion of a composition product for formal power series has its set of known properties significantly expanded in addition the notion of a feedback product for formal power series is shown to be well defined in a broad context and its basic properties are characterized

estimation of heteroscedasticity in regression analysis consider the regression model y i g t i varepsilon i 1 leq i leq n with nonrandom design variables t i and measurements y i for the unknown regression function g cdot we assume that the data are heteroscedastic i e e varepsilon 2 i sigma 2 i not equiv operatorname const and investigate how to estimate sigma 2 i if sigma 2 i sigma 2 t i with a smooth function sigma 2 cdot initial estimators tilde sigma 2 i can be improved by kernel smoothers and the resulting class of estimators is shown to be uniformly consistent these estimates can be used to improve the estimation of the regression function g itself in parametric and nonparametric models further applications are suggested

nonlinear unity feedback systems and q parametrization this paper concerns nonlinear systems defines a new concept of stability and extends to unity feedback systems the technique of q parametrization introduced by zames and developed by desoer chen and gustafson we specify 1 a global parametrization of all controllers that s stabilize a given s stable plant 2 a parametrization of a class of controllers that stabilize an unstable plant 3 necessary and sufficient conditions for a nonlinear controller to simultaneously stabilize two nonlinear plants

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

variance function estimation in regression the effect of estimating the mean abstract the authors consider estimation of a variance function g in regression problems such estimation requires simultaneous estimation of the mean function f we obtain sharp results on the extent to which the smoothness of f influences best rates of convergence for estimating g for example in nonparametric regression with two derivatives on g classical rates of convergence are possible if and only if the unknown f satisfies a lipschitz condition of order 1 3 or more if a parametric model is known for g then g may be estimated n 1 2 consistently if and only if f is lipschitz of order 1 2 or more optimal rates of convergence are attained by kernel estimators

linear functions of concomitants of order statistics with application to nonparametric estimation of a regression function abstract let xi yi i 1 2 n be independent identically distributed as x y then the rth ordered x variate is denoted by xr n and the associated y variate the concomitant of the rth order statistic by y r n this paper considers statistics of the form and more generally of the form where j is a bounded smooth function and may depend on n under certain regularity conditions the asymptotic normality of these statistics is established these statistics are used to construct consistent estimators of various conditional quantities for example e y x x p y a x x and var y x x

on the stabilizing conic sectors obtained using small gain techniques abstract this paper is concerned with the analysis of the scope and properties of the stabilizing conic sectors obtained using small gain and loop transformation sglt techniques first it is shown how sglt techniques are able to generate the set of all nonlinear stabilizing controllers by using transformations based on dynamical 2 2 block systems configurations then stabilizing cones obtained from static configurations are considered they induce only subsets of all stabilizing controllers however providing simplification and tractability for these cones maximality is characterized in terms of mu values and frequency responses finally an algorithm for deriving maximal cones is presented

computing kernel smoothed conditional quantiles from many observations abstract quantiles of a variable y conditional on another variable x when plotted against x can be a useful descriptive tool these plots give a quick impression of the functional form of the relation between x and the location spread and shape of the conditional distribution of y if several y are observed for each x then sample quantiles could be calculated for each x the resulting quantile plot may be quite noisy however and smoothing across x may be desired this article presents an algorithm that calculates kernel smoothed conditional quantiles with a cross validation choice of bandwidth for x it is computationally feasible for large data sets when x assumes a small number of values since it requires only one pass through the full data set the cross validation does not require a pass through the data because of simplifications arising from the l 1 loss function being based on absolute values the technique is illustrated by plotting the conditional quantiles of the net wealth of a sample of

design adaptive nonparametric regression abstract in this article we study the method of nonparametric regression based on a weighted local linear regression this method has advantages over other popular kernel methods moreover such a regression procedure has the ability of design adaptation it adapts to both random and fixed designs to both highly clustered and nearly uniform designs and even to both interior and boundary points it is shown that the local linear regression smoothers have high asymptotic efficiency i e can be 100 with a suitable choice of kernel and bandwidth among all possible linear smoothers including those produced by kernel orthogonal series and spline methods the finite sample property of the local linear regression smoother is illustrated via simulation studies nonparametric regression is frequently used to explore the association between covariates and responses there are many versions of kernel regression smoothers some estimators are not good for random designs such as in observational studies and

on the robust stabilizability of uncertain linear time invariant plants using nonlinear time varying controllers abstract we consider general families of linear time invariant plants described by a nominal plant model with additive or multiplicative unstructured modeling uncertainty we show that such families of plants are robustly uniformly stabilizable by nonlinear time varying controllers if and only if they are robustly stabilizable by linear time invariant ones some auxiliary results on hankel and l norm approximation are also obtained

remarks on computing coprime factors for distributed parameter systems presents an indirect method for computing coprime factors for a class of linear time invariant distributed parameter systems the method overcomes the numerical instability of an existing direct decomposition method it is shown that coprime factorization can be reduced to constructing simultaneously stabilizing finite dimensional controllers for which an efficient iterative procedure is developed

conditions for abstract nonlinear regulation this paper studies the general problem of the existence of output dynamic stabilizers for a control system the controller is not assumed to have any special structure necessary and sufficient conditions are given in terms of new notions which generalize the usual ones of stabilizability and detectability in the particular cases of analytic and bilinear systems these conditions are considerably simplified

a note on the coprime fractional representation of nonlinear feedback systems abstract a very large infinite dimensional banach space of bounded nonlinear operators is suggested as an underlying framework for studies of nonlinear systems control meanwhile it is pointed out that all results obtained in verma 6 can be extended to this much larger family of nonlinear control systems

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

the gap metric robustness of stabilization of feedback systems in this paper we introduce the gap metric to study the robustness of the stability of feedback systems which may employ not necessarily stable open loop systems we elaborate on the computational aspects of the gap metric and provide upper bounds to the gap in cases where the exact formulas do not apply by admissible uncertainties we mean those which preserve closed loop stability and a specified small tolerance on the i o behavior of a feedback system we show that admissible uncertainties are precisely those which are constrained in the gap finally we conclude that any metric which preserves a continuous relationship between open loop systems and the corresponding stable feedback interconnections must have the topology of the gap metric

a comparative study of input output stability results at present many similar but disparate input output i o stability criteria exist without means for comparison it is unclear which result is best used in any given application this paper proposes a means for comparison between i o stability results involving norms and inner products of inputs and outputs the extended conic sector theorem provides a framework for determining which results are least conservative and most broadly applicable in so doing numerous existing stability results are unified and revealed as more powerful than previously thought

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

a new factorization of special nonlinear discrete systems and their applications in this paper a new factorization of discrete nonlinear systems is presented this factorization is based on the star product an operation corresponding to the linear connection of systems a relevant algorithm is developed based on a theoretical background finally some applications to system analysis bibo stability and feedback design are provided

right coprime factorizations and stabilization for nonlinear systems the problem of constructing right coprime factorizations which are based on the graph of an input output map instead of the bezout identity of nonlinear input output maps is considered the map is assumed to arise from a state variable realization with a fixed initial state the main result is that the existence of a stabilizing state feedback implies the existence of a right coprime factorization for the map the technique is illustrated by application to nonlinear systems which are affine in the control and have a controllable linear past and nonlinear systems which are feedback linearizable a notion of input output stability that requires a bound on the magnitude of the input signals is introduced methods of constructing such bounds are developed for a locally feedback linearizable system the problem of input bounds is transferred to the equivalent linear system this leads to a technique that allows state and input constraints for a feedback linearizable system to be mapped to the equivalent linear system

a new algorithm for the solution of a polynomial matrix diophantine equation the main contribution of this work is to provide a new algorithm for the general solution of a polynomial matrix diophantine equation the theory is implemented via illustrative examples

fraction representations of non linear systems a simplified approach abstract a non linear system has a right fraction representation if it can be represented as pq 1 where p and q are stable systems and it has a left fraction representation if it can be represented as g 1 t where g and t are stable systems we develop here a theory of right and of left fraction representations for discrete time non linear systems with bounded input sequences we indicate the connection between fraction representations and the stabilization problem for non linear systems

on finding the minimum bandwidth of interval graphs abstract let g v e be an interval graph with v n and e m an o n 2 log n algorithm was proposed in kratsch inform comput 74 140 158 1987 to find the bandwidth of g we show that this algorithm is wrong and provide a corrected version of the same also it was observed in 4 that the bandwidth of a proper interval graph can be computed in o n log n m time we show how this idea can be modified slightly to yield an o n m algorithm

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

diophantine equations in control a survey abstract this survey is also a tutorial whose aim is to explain the role of diophantine equations in the synthesis of feedback control systems these are linear equations in a ring and result from a fractional representation of the systems involved the cornerstone of the exposition is a simple parametrization of all stabilizing controllers for a given plant one can then choose in principle the best controllers for various applications these ideas evolved from early attempts to use polynomial equations in the design of discrete time linear systems by now they have been extended to continuous time infinite dimensional time varying and non linear systems

an exponential time 2 approximation algorithm for bandwidth the bandwidth of a graph g on n vertices is the minimum b such that the vertices of g can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b in this paper we present a 2 approximation algorithm for the bandwidth problem that takes worst case o 1 9797 n o 3 0 6 2 1 7 n time and uses polynomial space this improves both the previous best 2 and 3 approximation algorithms of cygan et al which have o 3 n and o 2 n worst case running time bounds respectively our algorithm is based on constructing bucket decompositions of the input graph a bucket decomposition partitions the vertex set of a graph into ordered sets called buckets of almost equal sizes such that all edges are either incident to vertices in the same bucket or to vertices in two consecutive buckets the idea is to find the smallest bucket size for which there exists a bucket decomposition the algorithm uses a divide and conquer strategy along with dynamic programming to achieve the improved time bound

robustness analysis of nonlinear feedback systems an input output approach this paper presents an approach to robustness analysis for nonlinear feedback systems we pursue a notion of model uncertainty based on the closeness of input output trajectories which is not tied to a particular uncertainty representation such as additive parametric structured etc the basic viewpoint is to regard systems as operators on signal spaces we present two versions of a global theory where stability is captured by induced norms or by gain functions we also develop local approaches over bounded signal sets and give a treatment for systems with potential for finite time escape we compute the relevant stability margin for several examples and demonstrate robustness of stability for some specific perturbations e g small time delays we also present examples of nonlinear control systems which have zero robustness margin and are destabilized by arbitrarily small gap perturbations the paper considers the case where uncertainty is present in the controller as well as the plant and the generalization of the approach to the case where uncertainty occurs in several subsystems in an arbitrary interconnection

optimal hankel norm identification of dynamical systems abstract the problem of optimal approximate system identification is addressed with a newly defined measure of misfit between observed time series and linear time invariant models the behavioral framework is used as a suitable axiomatic setting for a non parametric introduction of system complexity and allows for a notion of misfit of dynamical systems that is independent of system representations the misfit function introduced here is characterized in terms of the induced norm of a hankel operator associated with the data and a co inner kernel representation of a model sets of pareto optimal models are defined as feasible trade offs between complexity and misfit of models and it is shown how all pareto optimal models are characterized as exact models of compressed data sets obtained from hankel norm approximations of data matrices this leads to new conceptual algorithms for optimal approximate identification of time series 1997 elsevier science ltd

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

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

a state space approach to parameterization of stabilizing controllers for nonlinear systems a state space approach to the youla parameterization of stabilizing controllers for linear and nonlinear systems is suggested the stabilizing controllers or a class of stabilizing controllers for nonlinear systems are characterized as fractional transformations of stable parameters the main idea behind this approach is to decompose the output feedback stabilization problem into state feedback and state estimation problems the parameterized output feedback controllers have separation structures this machinery allows the parameterization of stabilizing controllers to be conducted directly in state space without using coprime factorization

method for the global analysis of non linear parametrized systems a method for the global analysis of non linear systems is presented which provides a framework for the analysis of both continuous and discrete time non linear systems for the diverse range of model types by extending a numerical approach to enable the analysis of non linear parametrized non linear systems a global perspective is achieved that is difficult to obtain using analytical methods alone simple examples are used to illustrate how the method characterizes the system stability at the same time as detecting typical qualitative bifurcation phenomena a non linear feedback system is analysed in order to show how the approach provides both qualitative and quantitative information over a defined region of the system parameter space the method proves to be a powerful tool which can be used to probe the non linear characteristics of a system

operator based robust control design for nonlinear plants with perturbation in this article operator based robust control design for nonlinear plants with perturbation is considered by using robust right coprime factorisation approach in detail the property of unimodular operator is studied and a control design structure is proposed based on the proposed design scheme the designed system is robustly stable also output tracking performance can be realised simultaneously finally the effectiveness of the proposed design scheme is demonstrated by a simulation example

detectability and stabilizability of time varying discrete time linear systems the concepts of detectability and stabilizability are explored for time varying systems we study duality invariance under feedback an extended version of the lemma of lyapunov existence of stabilizing feedback laws linear quadratic filtering and control and the existence of approximate canonical forms

kernel and nearest neighbor estimation of a conditional quantile let x 1 z 1 x 2 z 2 x n z n be iid as x z and for 0 p 1 let p x denote the conditional p quantile of z given x x i e p z p x x x p kernel and near est neighbor estimators of p x are proposed in order to study the asymptotics of these estimates bahadur type representations of the sample conditional quantiles are obtained these representations are used to examine the important issue of choosing the smoothing parameter by a local approach for a fixed x based on weak convergence of these estimators the results lead to asymptotic linear models which motivate certain estimators

normalized coprime factorizations for linear time varying systems abstract in this paper we show that a finite dimensional linear time varying continuous time system admits normalized coprime factorizations if and only if it admits a stabilizable and detectable realization we construct state space formulas for these factorizations using the stabilizing solutions to standard riccati differential equations in the process we give a simple proof that stabilizability and detectability are sufficient to ensure the existence of such solutions based on these results and on recent advances in the theory of h optimization we present an algorithm to compute the distance between two systems in the gap metric