identification of systems containing linear dynamic and static nonlinear elements identification of nonlinear systems which can be represented by combinations of linear dynamic and static nonlinear elements are considered previous results by the authors based on correlation analysis are combined to provide a unified treatment for this class of systems it is shown that systems composed of cascade feedforward feedback and multiplicative connections of linear dynamic and zero memory nonlinear elements can be identified in terms of the individual component subsystems from measurements of the system input and output only

fuzzy piecewise multilinear and piecewise linear systems as universal approximators in sobolev norms a piecewise multilinear and a piecewise linear model for fuzzy systems are introduced and their approximation capabilities investigated classical results on the approximation of regular functions are improved showing that fuzzy systems may keep their semantic structure while approximating to any degree of accuracy not only sufficiently regular functions but also their derivatives

brief paper identification of dynamic systems using piecewise affine basis function models piecewise affine pwa basis function autoregressive exogenous bpwarx models are proposed in this paper for nonlinear black box identification a bpwarx model is a weighted sum of pwa basis bpwa functions which are the minimum or maximum of n 1 affine functions in n dimensions since the bpwa functions have a universal representation capability for continuous pwa functions the bpwarx models provide better accuracy than the hinging hyperplane arx hharx models with the same number of parameters and the same order of computational complexity when using a modified gauss newton algorithm to build the models from input output data

paper zeros of sampled systems the zeros of the discrete time system obtained when sampling a continuous time system are explored theorems for the limiting zeros for large and small sampling periods are given conditions which guarantee that the sampled system has stable zeros are also presented

strong consistence of recursive identification for wiener systems the paper concerns identification of the wiener system consisting of a linear subsystem followed by a static nonlinearity f with no invertibility and structure assumption recursive estimates are given for coefficients of the linear subsystem and for the value f v at any fixed v the main contribution of the paper consists in establishing convergence with probability one of the proposed algorithms to the true values this probably is the first strong consistency result for this kind of wiener systems a numerical example is given which justifies the theoretical analysis

recursive prediction error identification using the nonlinear wiener model the nonlinear wiener model consisting of a linear dynamic block in cascade with a static nonlinearity is considered a recursive prediction error identification algorithm based on the wiener model is derived the linear dynamic block is modelled as a siso transfer function operator and the static nonlinearity is approximated with a piecewise linear function a theoretical analysis of the method is carried out and conditions for local convergence to the true parameter vector are given in particular the analysis shows that the input signal should be such that there is signal energy in the whole range of the piecewise linear approximation a numerical example illustrates the performance of the algorithm further practical guidelines on how to apply the algorithm are also included in the paper

convergence analysis of recursive identification algorithms based on the nonlinear wiener model recursive identification algorithms based on the nonlinear wiener model are presented a recursive identification algorithm is first derived from a general parameterization of the wiener model using a stochastic approximation framework local and global convergence of this algorithm can be tied to the stability properties of an associated differential equation since inversion is not utilized noninvertible static nonlinearities can be handled which allows a treatment of for example saturating sensors and blind adaptation problems gauss newton and stochastic gradient algorithms for the situation where the static nonlinearity is known are then suggested in the single input single output case the proposed methods can outperform conventional linearizing inversion of the nonlinearity when measurement disturbances affect the output signal for fir finite impulse response models it is also proved that global convergence of the schemes is tied to sector conditions on the static nonlinearity in particular global convergence of the stochastic gradient method is obtained provided that the nonlinearity is strictly monotone the local analysis performed for iir infinite impulse response models illustrates the importance of the amplitude contents of the exciting signals

a multilayer neural network with piecewise linear structure and back propagation learning a multilayer neural network which is given a two layer piecewise linear structure for every cascaded section is proposed the neural networks have nonlinear elements that are neither sigmoidal nor of a signum type each nonlinear element is an absolute value operator it is almost everywhere differentiable which makes back propagation feasible in a digital setting both the feedforward signal propagation and the backward coefficient update rules belong to the class of regular iterative algorithms this form of neural network specializes in functional approximation and is anticipated to have applications in control communications and pattern recognition

statistical analysis of shape through triangulation of landmarks a study of sexual dimorphism in hominids two objects with homologous landmarks are said to be of the same shape if the configuration of landmarks of one object can be exactly matched with that of the other by translation rotation reflection and scaling in an earlier paper the authors proposed statistical analysis of shape by considering logarithmic differences of all possible euclidean distances between landmarks tests of significance for differences in the shape of objects and methods of discrimination between populations were developed with such data in the present paper the corresponding statistical methodology is developed by triangulation of the landmarks and by considering the angles as natural measurements of shape this method is applied to the study of sexual dimorphism in hominids

generalization of hinging hyperplanes the model of hinging hyperplanes hh can approximate a large class of nonlinear functions to arbitrary precision but represent only a small part of continuous piecewise linear cpwl functions in two or more dimensions in this correspondence the influence of this drawback for black box modeling is first illustrated by a simple example then it is shown that the above shortcoming can be amended by adding a sufficient number of linear functions to current hinges it is proven that any cpwl function of n variables can be represented by a sum of hinges containing at most n 1 linear functions hence the model of a sum of such expanded hinges is a general representation for all cpwl functions the structure of the novel general representation is much simpler than the existing generalized canonical representation that consists of nested absolute value functions this characteristic is very useful for black box modeling based on the new general representation an upper bound on the number of nestings of nested absolute value functions of a generalized canonical representation is established which is much smaller than the known result

on the nice behaviour of the gaussian projection filter with small observation noise when projecting on the manifold of gaussian densities the projection filter has been shown to be equal to a mcshane fisk stratonovich mfs derivation of the gaussian assumed density filter starting from this point we study the asymptotic behaviour of the gaussian projection filter when the covariance of the observation noise tends to zero we prove that the mean square difference between the true state of the system and the estimate given by the projection filter is bounded by a constant which is proportional to the magnitude of the observation noise

maximum likelihood identification of wiener models the wiener model is a block oriented model having a linear dynamic system followed by a static nonlinearity the dominating approach to estimate the components of this model has been to minimize the error between the simulated and the measured outputs we show that this will in general lead to biased estimates if there are other disturbances present than measurement noise the implications of bussgang s theorem in this context are also discussed for the case with general disturbances we derive the maximum likelihood method and show how it can be efficiently implemented comparisons between this new algorithm and the traditional approach confirm that the new method is unbiased and also has superior accuracy

parameter identification of wiener systems with discontinuous nonlinearities abstract the paper deals with modeling and parameter identification of wiener type nonlinear dynamic systems having discontinuous asymmetric piecewise linear characteristics with preloads and dead zones multiple application of a decomposition technique provides a special form of static nonlinearity description its incorporation into the wiener model leads to an input output equation where all the model parameters are separated and can be estimated iteratively using input output data and internal variable estimates more illustrative examples are included to demonstrate the feasibility of the proposed approach

a recursive local linear estimator for identification of nonlinear arx systems asymptotical convergence and applications in this paper we propose a recursive local linear estimator rlle for nonparametric identification of nonlinear autoregressive systems with exogenous inputs narx first the rlle is introduced next the strong consistency as well as the asymptotical mean square error properties of the rlle are established and then an application of the rlle to an additive nonlinear system is discussed the rlle provides recursive estimates not only for the function values but also their gradients at fixed points a simulation example is provided to confirm the theoretical analysis

adaptive control of weiner type nonlinear systems abstract this study is concerned with the problem of adaptive control of a wiener type nonlinear process it is assumed that the linear dynamic part of the process can be represented by a pulse transfer function of known order with stable inverse followed by a known time delay the memoryless nonlinearity is one to one on a compact set and belongs to one of two classes 1 piecewise linear with known breakpoints or 2 smooth function in which case an approximation is made it is represented by linear splines the output signal from the linear part is not available for measurement the global stability of the proposed model reference adaptive control scheme is established subject to the assumption that the nonlinearity can be represented exactly by the linear spline function with a given set of breakpoints otherwise there will be a residual tracking error which depends on the nonlinearity approximation error but the boundedness of all signals can be assured the proposed scheme is used for adaptive control of the ph process in a continuous flow reactor

limit set dichotomy and multistability for a class of cooperative neural networks with delays recent papers have pointed out the interest to study convergence in the presence of multiple equilibrium points eps multistability for neural networks nns with nonsymmetric cooperative nonnegative interconnections and neuron activations modeled by piecewise linear pl functions one basic difficulty is that the semiflows generated by such nns are monotone but due to the horizontal segments in the pl functions are not eventually strongly monotone esm this notwithstanding it has been shown that there are subclasses of irreducible interconnection matrices for which the semiflows although they are not esm enjoy convergence properties similar to those of esm semiflows the results obtained so far concern the case of cooperative nns without delays the goal of this paper is to extend some of the existing results to the relevant case of nns with delays more specifically this paper considers a class of nns with pl neuron activations concentrated delays and a nonsymmetric cooperative interconnection matrix a and delay interconnection matrix a the main result is that when a a satisfies a full interconnection condition then the generated semiflows which are monotone but not esm satisfy a limit set dichotomy analogous to that valid for esm semiflows it follows that there is an open and dense set of initial conditions in the state space of continuous functions on a compact interval for which the solutions converge toward an ep the result holds in the general case where the nns possess multiple eps i e is a result on multistability and is valid for any constant value of the delays

configuration of continuous piecewise linear neural networks the problem of constructing a general continuous piecewise linear neural network is considered in this paper it is shown that every projection domain of an arbitrary continuous piecewise linear function can be partitioned into convex polyhedra by using difference functions of its local linear functions based on these convex polyhedra a group of continuous piecewise linear basis functions are formulated it is proven that a linear combination of these basis functions plus a constant which we call a standard continuous piecewise linear neural network can represent all continuous piecewise linear functions in addition the proposed standard continuous piecewise linear neural network is applied to solve some function approximation problems a number of numerical experiments are presented to illustrate that the standard continuous piecewise linear neural network can be a promising tool for function approximation

hard frequency domain model error bounds from least squares like identification techniques the problem of deriving so called hard error bounds for estimated transfer functions is addressed a hard bound is one that is sure to be satisfied i e the true system nyquist plot will be confined with certainty to a given region provided that the underlying assumptions are satisfied by blending a priori knowledge and information obtained from measured data it is shown how the uncertainty of transfer function estimates can be quantified the emphasis is on errors due to model mismatch the effects of unmodeled dynamics can be considered as bounded disturbances hence techniques from set membership identification can be applied to this problem the approach taken corresponds to weighted least squares estimation and provides hard frequency domain transfer function error bounds the main assumptions used in the current contribution are that the measurement errors are bounded that the true system is indeed linear with a certain degree of stability and that there is some knowledge about the shape of the true frequency response

a constructive method for multivariate function approximation by multilayer perceptrons mathematical theorems establish the existence of feedforward multilayered neural networks based on neurons with sigmoidal transfer functions that approximate arbitrarily well any continuous multivariate function however these theorems do not provide any hint on how to find the network parameters in practice it is shown how to construct a perceptron with two hidden layers for multivariate function approximation such a network can perform function approximation in the same manner as networks based on gaussian potential functions by linear combination of local functions

recursive nonlinear estimation geometry of a space of posterior densities the paper establishes a geometric formulation for nonlinear parameter estimation using reduced statistics if a reduced rather than sufficient statistic is used in estimation an equivalence class of densities p t rather than the true posterior density p t is determined two questions arise in this connection 1 what kind of a reduced statistic allows recursive computations 2 what is the appropriate representative p t of the equivalence class p t typically the first question is not posed at all and the second one is resolved by heuristic considerations the present paper attempts to cast the problem into a solid mathematical structure it closely follows the differential geometric approach suggested in kulhav automatica 26 545 555 1990 but goes into more detail roughly admissible statistics are characterized here as homomorphisms of a group containing all possible likelihoods a representative of the pertinent equivalence class is constructed by a projection along this class onto an orthogonal submanifold going through a prespecified density p and imbedded in the manifold of possible posteriors the obtained projection p t has attractive extremal properties it minimizes the kullback leibler distance from the reference point p and at the same time the dual kullback leibler distance from the true point p t

some extensions of a new method to analyze complete stability of neural networks in a recent work a new method has been introduced to analyze complete stability of the standard symmetric cellular neural networks cnns which are characterized by local interconnections and neuron activations modeled by a three segment piecewise linear pwl function by complete stability it is meant that each trajectory of the neural network converges toward an equilibrium point the goal of this paper is to extend that method in order to address complete stability of the much wider class of symmetric neural networks with an additive interconnecting structure where the neuron activations are general pwl functions with an arbitrary number of straight segments the main result obtained is that complete stability holds for any choice of the parameters within the class of symmetric additive neural networks with pwl neuron activations i e such a class of neural networks enjoys the important property of absolute stability of global pattern formation it is worth pointing out that complete stability is proved for generic situations where the neural network has finitely many isolated equilibrium points as well as for degenerate situations where there are infinite nonisolated equilibrium points the extension in this paper is of practical importance since it includes neural networks useful to solve significant signal processing tasks e g neural networks with multilevel neuron activations it is of theoretical interest too due to the possibility of approximating any continuous function e g a sigmoidal function using pwl functions the results in this paper confirm the advantages of the method of forti and tesi with respect to lasalle approach to address complete stability of pwl neural networks

set membership identification of systems with parametric and nonparametric uncertainty a method for parameter set estimation in which the system model is assumed to contain both parametric and nonparametric uncertainty is presented in the disturbance free case the parameter set estimate is guaranteed to contain the parameter set of the true plant in the presence of stochastic disturbances the parameter set estimate obtained from finite data records is shown to have the property that it contains the true plant parameter set with probability one as the data length tends to infinity

pattern recognition based on scale invariant discriminant functions some probability models for classifying individuals as belonging to one of two or more populations using scale invariant discriminant functions are considered the investigation is motivated by practical situations where the observed data on an individual are in the form of ratios of some basic measurements or measurements scaled by an unknown nonnegative number the probability models are obtained by considering a p vector random variable x with a known distribution and deriving the distribution of the random vector y g x 1 x where g x is a nonnegative measure of size such that g x g x for 0 explicit expressions are obtained for the densities of what are called angular gaussian compositional gaussian type 1 and compositional gaussian type 2 distributions

instrumental variable methods for identification of hammerstein systems abstract consistency conditions for instrumental variable methods applied to hammerstein models are derived and analysed it is necessary to have a model that is not over parametrized and an input that is strongly persistently exciting i e the inputs and their powers are jointly persistently exciting some specific choices of the instruments are proved to give consistency those instruments are formed as filtered inputs and powers of the input consistency can be guaranteed if either the input is white noise or if a certain transfer function is positive real the vector of instruments is in general of larger dimension than the parameter vector it is shown then that a non singular linear transformation of the instruments gives a new and still consistent estimate some simulations using the iv variants proposed in this paper are also included

an analytic geometry approach to wiener system frequency identification this paper addresses the problem of wiener system identification the underlying linear subsystem is stable but not necessarily parametric the nonlinear element in turn is allowed to be nonparametric noninvertible and nonsmooth as wiener models are uniquely defined up to an uncertain multiplicative factor it makes sense to start the frequency identification process estimating the system phase which is common to all models to this end a consistent estimator is designed using analytic geometry tools accordingly the system frequency behavior is characterized by a family of lissajous curves interestingly all these curves are candidates to modelling the system nonlinearity but the most convenient one is the less spread of them finally the frequency gain is in turn consistently estimated optimizing an appropriate cost function involving the obtained phase and nonlinearity estimates

canonical piecewise linear networks in this paper mapping networks will be considered from the viewpoint of the piecewise linear pwl approximation the so called canonical representation plays a kernel role in the pwl representation theory while this theory has been researched intensively in the contents of mathematics and circuit simulations little has been seen in the research area about the theoretical aspect of neural networks this paper modifies this theory and applies it as a mathematical support for mapping networks the main modification is a higher level generalization of the canonical representation with proofs of its availability in the set of pwl functions the modified theory will first be used to study the canonical pwl feature of the popular multilayer perceptron like mlpl networks second it will be seen that the generalized canonical representation is itself suitable for a network implementation which is called the standard canonical pwl network more generally the family of generalized canonical pwl networks is defined as those which may take the canonical pwl representation as a mathematical model this family is large and practically meaningful the standard canonical pwl networks may be taken as representatives in the family the modification of the pwl representation theory as well as the introduction of this theory in the theoretical study of mapping networks which provide a new concept of mapping networks i e the canonical pwl network family may be regarded as the main contributions of the paper

quantifying the error in estimated transfer functions with application to model order selection previous results on estimating errors or error bounds on identified transfer functions have relied on prior assumptions about the noise and the unmodeled dynamics this prior information took the form of parameterized bounding functions or parameterized probability density functions in the time or frequency domain with known parameters it is shown that the parameters that quantify this prior information can themselves be estimated from the data using a maximum likelihood technique this significantly reduces the prior information required to estimate transfer function error bounds the authors illustrate the usefulness of the method with a number of simulation examples how the obtained error bounds can be used for intelligent model order selection that takes into account both measurement noise and under modeling is shown another simulation study compares the method to akaike s well known fpe and aic criteria

gaussian approximation in recursive estimation of multiple state of nonlinear wiener systems abstract the paper concerns recursive state estimation in wiener systems where the output is a scalar weighted sum of multiple states a gaussian approximation to the computed conditional probability distribution for the scalar sum of states is shown to lead to a practicable numerical algorithm for estimating the multiple states this algorithm mimics the kalman filter simulations show the method giving good performance when the nonlinearity is severe quantization

h control of a wiener type system a wiener system is a system which can be modelled as a linear dynamic followed by a static gain the goal of this paper is to develop a robust h compensator for controlling an siso wiener system the controller also takes the form of a wiener model the design approach consists of the approximation of the non linear gain using a piecewise linear pwl function and in using a linear controller for each sector obtained from this approximation therefore the general controller structure can be stated as a linear dynamic compensator in series with a pwl static gain as an illustrative case a neutralization ph reaction between a strong acid and a strong base in the presence of a buffer agent is dealt with computer simulations are developed for showing the performance of the proposed controller

morphometric tools for landmark data geometry and biology 1 introduction 2 preliminaries 3 landmarks 4 distance measures 5 shape coordinates 6 principal axes of shape change for triangles 7 features of shape comparison 8 retrospect and prospect

towards identification of wiener systems with the least amount of a priori information iir cases in this paper we investigate what constitutes the least amount of a priori information on the nonlinearity so that the linear part is identifiable in the non gaussian input case under the white noise input three types of a priori information are considered quadrant information point information and monotonic information in all three cases identifiability has been established and the corresponding nonparametric identification algorithms are developed along with their convergence proofs

design variables for bias distribution in transfer function estimation estimation of transfer functions of linear systems is one of the most common system identification problems several different design variables chosen by the user for the identification procedure affect the properties of the resulting estimate in this paper it is investigated how the choices of prefilters noise models sampling interval and prediction horizon i e the use of k step ahead prediction methods influence the estimate an important aspect is thai the true system is not assumed to be exactly represented within the chosen model set the estimate will thus be biased it is shown how the distribution of bias in the frequency domain is governed by a weighting function which emphasizes different frequency bands the weighting function in turn is a result of the previously listed design variables it is shown e g thai the common least squares method has a tendency to emphasize high frequencies and that this can be counteracted by prefiltering it is also shown that asymptotically it is only the prediction horizon itself and not how it is split up into sampling interval times number of predicted sampling instants that affects this weighting function

identification of block oriented nonlinear systems starting from linear approximations a survey abstract block oriented nonlinear models are popular in nonlinear system identification because of their advantages of being simple to understand and easy to use many different identification approaches were developed over the years to estimate the parameters of a wide range of block oriented nonlinear models one class of these approaches uses linear approximations to initialize the identification algorithm the best linear approximation framework and the approximation framework or equivalent frameworks allow the user to extract important information about the system guide the user in selecting good candidate model structures and orders and prove to be a good starting point for nonlinear system identification algorithms this paper gives an overview of the different block oriented nonlinear models that can be identified using linear approximations and of the identification algorithms that have been developed in the past a non exhaustive overview of the most important other block oriented nonlinear system identification approaches is also provided throughout this paper

nonparametric identification of wiener systems a wiener system i e a system in which a linear dynamic part is followed by a nonlinear and memoryless one is identified no parametric restriction is imposed on the functional form of the nonlinear characteristic of the memoryless subsystem and a nonparametric algorithm recovering the characteristic from input output observations of the whole system is proposed its consistency is shown and the rate of convergence is given an idea for identification of the impulse response of the linear subsystem is proposed results of numerical simulation are also presented

asymptotic variance expressions for identified black box transfer function models identification of black box transfer function models is considered it is assumed that the transfer function models possess a certain shift property which is satisfied for example by all polynomial type models expressions for the variances of the transfer function estimates are derived that are asymptotic both in the number of observed data and in the model orders the result is that the joint covariance matrix of the transfer functions from input to output and from driving white noise source to the additive output disturbance respectively is proportional to the inverse of the joint spectrum matrix for the input and driving noise multiplied by the spectrum of the additive output noise the factor of proportionality is the ratio of model order to number of data this result is independent of the particular model structure used the result is applied to evaluate the performance degradation due to variance for a number of typical model uses some consequences for input design are also drawn

parameter identification of wiener systems with multisegment piecewise linear nonlinearities the paper deals with the identification of wiener systems having multisegment piecewise linear nonlinearities a special form of nonlinearity representation is used in the wiener system description the resulting output equation contains the least possible number of parameters required for the blocks of given system the proposed iterative method enables simultaneous estimation of both the linear block parameters and all the parameters characterizing the nonlinearity i e the slopes of linear segments and the constants determining the partition of domain limits for the constants are the only a priori knowledge required