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

stochastic gradient identification of polynomial wiener systems analysis and application this paper presents analytical numerical and experimental results for a stochastic gradient adaptive scheme that identifies a polynomial type nonlinear system with memory for noisy output observations the analysis includes the computation of the stationary points the mean square error surface and the stability regions of the algorithm for gaussian data convergence of the mean is studied using l sub 2 and euclidian norms monte carlo simulations confirm the theoretical predictions that show a small sensitivity to the observation noise an application is presented for the identification of a nonlinear time delayed feedback system

determinization of fuzzy automata via factorization of fuzzy states fuzzy finite state automata over the algebra 0 1 max 0 1 in which the monoid 0 1 1 denotes a continuous triangular norm is not locally finite can accept fuzzy languages of infinite range for a given fuzzy finite state automaton which accepts a fuzzy language of infinite range we define the determinization of the fuzzy automaton via factorization of fuzzy states i e the computation of an equivalent deterministic fuzzy automaton whether it is finite this method of determinization is a generalization of the well known accessible subset construction our main contribution is to determine that the representable cycles property is the necessary and sufficient condition for determinization of a fuzzy finite state automaton via a maximal factorization of fuzzy states this property is more general than the twins property adapted for fuzzy automata which is the sufficient condition for weighted automata over the tropical semiring

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

least squares approximation method for the solution of hammerstein volterra delay integral equations in this paper an efficient numerical method is developed for solving the hammerstein volterra delay integral equations by least squares ls approximation method which is based on a polynomial of degree n to compute an approximation to the solution of hammerstein volterra delay integral equations the convergence analysis of the approximation solution relative to the exact solution of the integral equation is proved and its accuracy is illustrated on two numerical examples the study of this integral equation is important because it has as a particular case the variant of a mathematical model from epidemiology

fuzzy languages with infinite range accepted by fuzzy automata pumping lemma and determinization procedure abstract the formulation of fuzzy automata allows us to select a great variety of triangular norms depending on the selected triangular norm a fuzzy automaton can accept a fuzzy language fa language with infinite range these fuzzy automata are not equivalent to the so called deterministic fuzzy automata deterministic automata with a fuzzy subset of final states which only accept fuzzy languages with finite range in this paper we study fa languages with infinite range and a determinization procedure in order to obtain an equivalent fuzzy deterministic automaton for a given fuzzy automaton a fuzzy deterministic automaton is a fuzzy automaton which satisfies the deterministic condition in its state transition function the main contributions of our paper are 1 a pumping lemma of fa languages with infinite range 2 the formulation of fuzzy deterministic automata and a pumping lemma of fda languages 3 the necessary conditions for the determinization of fuzzy automata under continuous triangular norms which accept fuzzy languages of infinite range and 4 a determinization algorithm for fuzzy automata its correctness proof and performance

an optimal two stage identification algorithm for hammerstein wiener nonlinear systems abstract an optimal two stage identification algorithm is presented for hammerstein wiener systems where two static nonlinear elements surround a linear block the proposed algorithm consists of two steps the first one is the recursive least squares and the second one is the singular value decomposition of two matrices whose dimensions are fixed and do not increase as the number of the data point increases moreover the algorithm is shown to be convergent in the absence of noise and convergent with probability one in the presence of white noise

on fuzzy regular languages abstract we introduce the concept of a max min fuzzy language fl m and a min max fuzzy language fl m recognized by a type of fuzzy automaton m we show that if l1 and l2 are finite valued f regular languages then so are l1 l2 and l1 l2 we give a form of a fuzzified pumping lemma which we use to give a necessary and sufficient condition for fl m to be nonconstant

algebraic properties of la languages in this study we introduce the concepts of l valued regular substitution la substitution deterministic l valued regular substitution dla substitution l valued fuzzy homomorphism and its inverse images homomorphism and its inverse images for a lattice ordered monoid l we also study the properties of la languages and dla languages under the above mentioned algebraic operations the algebraic characterization of the l valued regular language is given

stochastic analysis of gradient adaptive identification of nonlinear systems with memory for gaussian data and noisy input and output measurements this paper investigates the statistical behavior of two gradient search adaptive algorithms for identifying an unknown nonlinear system comprised of a discrete time linear system h followed by a zero memory nonlinearity g spl middot the input and output of the unknown system are corrupted by additive independent noises gaussian models are used for all inputs two competing adaptation schemes are analyzed the first is a sequential adaptation scheme where the lms algorithm is first used to estimate the linear portion of the unknown system the lms algorithm is able to identify the linear portion of the unknown system to within a scale factor the weights are then frozen at the end of the first adaptation phase recursions are derived for the mean and fluctuation behavior of the lms algorithm which are in excellent agreement with monte carlo simulations when the nonlinearity is modeled by a scaled error function the second part of the sequential gradient identification scheme is shown to correctly learn the scale factor and the error function scale factor mean recursions for the scale factors show good agreement with monte carlo simulations for slow learning the stationary points of the gradient algorithm closely agree with the stationary points of the theoretical recursions the second adaptive scheme simultaneously learns both the linear and nonlinear portions of the unknown channel the mean recursions for the linear and nonlinear portions show good agreement with monte carlo simulations for slow learning the stationary points of the gradient algorithm also agree with the stationary points of the theoretical recursions

study of the lti relations between the outputs of two coupled wiener systems and its application to the generation of initial estimates for wiener hammerstein systems this paper consists of two parts in the first more theoretic part two wiener systems driven by the same gaussian noise excitation are considered for each of these systems the best linear approximation bla of the output in mean square sense is calculated and the residuals defined as the difference between the actual output and the linearly simulated output is considered for both outputs the paper is focused on the study of the linear relations that exist between these residuals explicit expressions are given as a function of the dynamic blocks of both systems generalizing earlier results obtained by brillinger brillinger d r 1977 the identification of a particular nonlinear time series system biometrika 64 3 509 515 and billings and fakhouri billings s a fakhouri s y 1982 identification of systems containing linear dynamic and static nonlinear elements automatica 18 1 15 26 compared to these earlier results a much wider class of static nonlinear blocks is allowed and the efficiency of the estimate of the linear approximation between the residuals is considerably improved in the second more practical part of the paper this new theoretical result is used to generate initial estimates for the transfer function of the dynamic blocks of a wiener hammerstein system this method is illustrated on experimental data

analysis of stochastic gradient identification of wiener hammerstein systems for nonlinearities with hermite polynomial expansions this paper investigates the statistical behavior of a sequential adaptive gradient search algorithm for identifying an unknown wiener hammerstein 1958 system whs with gaussian inputs the whs nonlinearity is assumed to be expandable in a series of orthogonal hermite polynomials the sequential procedure uses 1 a gradient search for the unknown coefficients of the hermite polynomials 2 an lms adaptive filter to partially identify the input and output linear filters of the whs and 3 the higher order terms in the hermite expansion to identify each of the linear filters the third step requires the iterative solution of a set of coupled nonlinear equations in the linear filter coefficients an alternative scheme is presented if the two filters are known a priori to be exponentially shaped the mean behavior of the various gradient recursions are analyzed using small step size approximations slow learning and yield very good agreement with monte carlo simulations several examples demonstrate that the scheme provides good estimates of the whs parameters for the cases studied

a stable adaptive hammerstein filter employing partial orthogonalization of the input signals this paper presents an algorithm that adapts the parameters of a hammerstein system model hammerstein systems are nonlinear systems that contain a static nonlinearity cascaded with a linear system in this paper the static nonlinearity is modeled using a polynomial system and the linear filter that follows the nonlinearity is an infinite impulse response iir system the adaptation of the nonlinear components is improved by orthogonalizing the inputs to the coefficients of the polynomial system the step sizes associated with the recursive components are constrained in such a way as to guarantee bounded input bounded output bibo stability of the overall system this paper also presents experimental results that show that the algorithm performs well in a variety of operating environments exhibiting stability and global convergence of the algorithm

on closure properties of l valued linear languages abstract in 2001 and 2002 daowen qiu has established a new theory of l valued automata which is based on complete residuated lattices moreover besides the finite automata several other classes of machines grammars and languages have been generalized using the ideas introduced by qiu this paper generalizes the concept of linear grammars and linear languages in this context moreover it presents some operations for l valued languages and some closure properties for the l valued linear languages

pumping lemma in context free grammar theory based on complete residuated lattice valued logic residuated lattices are important algebras and have close links with various important algebras automata theory based on complete residuated lattice valued logic called l valued automata has been established by the second author in 2001 and 2002 as a continuation of automata theory based on complete residuated lattice valued logic in this paper we mainly deal with the problem concerning pumping lemma in l valued context free languages l cfls as a generalization of the notion in the theory of formal grammars the definition of l valued context free grammars l cfgs is introduced we also discuss a special case of l cfgs l right or left linear grammars and show the equivalence between l linear grammars and l regular grammars this result shows that we generalize the pumping lemma in l valued regular languages l rls more recently established by the second author

stochastic analysis of adaptive gradient identification of wiener hammerstein systems for gaussian inputs this correspondence investigates the statistical behavior of two adaptive gradient search algorithms for identifying an unknown wiener hammerstein system whs with gaussian inputs the first scheme attempts to identify the whs with an lms adaptive filter the lms algorithm identifies a scaled version of the convolution of the input and output linear filters of the whs the second scheme attempts to identify the unknown whs with a gradient adaptive whs when the shape of the nonlinearity is known a priori the mean behavior of the gradient recursions are analyzed when the whs nonlinearity is modeled by an error function the mean recursions yield very good agreement with monte carlo simulations for slow learning

algebraic aspects of families of fuzzy languages we study operations on fuzzy languages such as union concatenation kleene intersection with regular fuzzy languages and several kinds of iterated fuzzy substitution then we consider families of fuzzy languages closed under a fixed collection of these operations which results in the concept of full abstract family of fuzzy languages or full affl this algebraic structure is the fuzzy counterpart of the notion of full abstract family of languages that has been encountered frequently in investigating families of crisp i e non fuzzy languages some simpler and more complicated algebraic structures such as full substitution closed affl full super affl full hyper affl will be considered as well in the second part of the paper we focus our attention to full affls closed under iterated parallel fuzzy substitution where the iterating process is prescribed by given crisp control languages proceeding inductively over the family of these control languages yields an infinite sequence of full affl structures with increasingly stronger closure properties

on nonparametric identification of wiener systems in this paper a new method for the identification of the wiener nonlinear system is proposed the system being a cascade connection of a linear dynamic subsystem and a nonlinear memoryless element is identified by a two step semiparametric approach the impulse response function of the linear part is identified via the nonlinear least squares approach with the system nonlinearity estimated by a pilot nonparametric kernel regression estimate the obtained estimate of the linear part is then used to form a nonparametric kernel estimate of the nonlinear element of the wiener system the proposed method permits recovery of a wide class of nonlinearities which need not be invertible as a result the proposed algorithm is computationally very efficient since it does not require a numerical procedure to calculate the inverse of the estimate furthermore our approach allows non gaussian input signals and the presence of additive measurement noise however only linear systems with a finite memory are admissible the conditions for the convergence of the proposed estimates are given computer simulations are included to verify the basic theory

equivalence in automata theory based on complete residuated lattice valued logic automata theory based on complete residuated lattice valued logic called l valued finite automata abbr l vfas was introduced by the second author in 2001 in this paper we deal with the problems of equivalence between l valued sequential machines abbr l vsms and l vfas we define l vsms and particularly present a method for deciding the equivalence between l vsms as well an algorithm procedure for deciding the equivalence between l vsms is constructed we analyze the complexity and efficiency of the algorithm procedure and obtain the relative results to l vfas moreover the definitions of l valued languages abbr l vls and l valued regular languages abbr l vrls recognized by l vfas are given and some related properties are also discussed we show an equivalent relation between l vrls and conventional regular languages by using l valued pumping lemma we get a necessary and sufficient condition for an l vl to be nonconstant

nondeterministic fuzzy automata to handle fuzzy uncertainty in system modeling nondeterministic finite automata have been generalized into fuzzy automata after a reexamination of the notions of fuzzy automata in the literature we ascertain that the fundamental property nondeterminism in nondeterministic finite automata has not been well embodied in the generalization in order to reflect nondeterminism in fuzzy automata we introduce nondeterministic fuzzy automata with or without e moves and fuzzy languages recognized by them like nondeterministic finite automata nondeterministic fuzzy automata provide a mathematical representation of nondeterministic dynamic fuzzy systems moreover we show that deterministic fuzzy automata nondeterministic fuzzy automata and nondeterministic fuzzy automata with e moves are all equivalent in the sense that they recognize the same class of fuzzy languages which is an extension of the well known equivalence among finite automata nondeterministic finite automata and nondeterministic finite automata with e moves

formal power series and regular operations on fuzzy languages in this paper we study formal power series over a quantale with coefficients in the algebra of all languages over a given alphabet and representation of fuzzy languages by these formal power series this representation generalizes the well known representation of fuzzy languages by their cut and kernel languages we show that regular operations on fuzzy languages can be represented by regular operations on power series which are defined by means of operations on ordinary languages we use power series in study of fuzzy languages which are recognized by fuzzy finite automata and deterministic finite automata and we study closure properties of the set of polynomials and the set of polynomials with regular coefficients under regular operations on power series

analysis of stochastic gradient tracking of time varying polynomial wiener systems this paper presents analytical and monte carlo results for a stochastic gradient adaptive scheme that tracks a time varying polynomial wiener 1958 system i e a linear time invariant lti filter with memory followed by a time varying memoryless polynomial nonlinearity the adaptive scheme consists of two phases 1 estimation of the lti memory using the lms algorithm and 2 tracking the time varying polynomial type nonlinearity using a second coupled gradient search for the polynomial coefficients the time varying polynomial nonlinearity causes a time varying scaling for the optimum wiener filter for phase 1 these time variations are removed for phase 2 using a novel coupling scheme to phase 1 the analysis for gaussian data includes recursions for the mean behavior of the lms algorithm for estimating and tracking the optimum wiener filter for phase 1 for several different time varying polynomial nonlinearities and recursions for the mean behavior of the stochastic gradient algorithm for phase 2 the polynomial coefficients are shown to be accurately tracked monte carlo simulations confirm the theoretical predictions and support the underlying statistical assumptions

orthogonal least squares methods and their application to non linear system identification abstract identification algorithms based on the well known linear least squares methods of gaussian elimination cholesky decomposition classical gram schmidt modified gram schmidt householder transformation givens method and singular value decomposition are reviewed the classical gram schmidt modified gram schmidt and householder transformation algorithms are then extended to combine structure determination or which terms to include in the model and parameter estimation in a very simple and efficient manner for a class of multivariate discrete time non linear stochastic systems which are linear in the parameters

statistical analysis of the single layer backpropagation algorithm i mean weight behavior the single layer backpropagation algorithm is a gradient descent method that adjusts the connection weights of a single layer perceptron to minimize the mean square error at the output it is similar to the standard least mean square al gorithm except the output of the linear combiner contains a differentiable nonlinearity in this paper we present a statis tical analysis of the mean weight behavior of the single layer backpropagation algorithm for gaussian input signals it is based on a nonlinear system identification model of the desired response which is capable of generating an arbitrary hyper plane decision boundary it is demonstrated that although the weights grow unbounded the algorithm on average quickly learns the correct hyperplane associated with the system iden tification model

fluctuation analysis of stochastic gradient identification of polynomial wiener systems this correspondence presents analytical results and monte carlo simulations for the fluctuation behavior of a stochastic gradient adaptive identification scheme this scheme identifies a polynomial wiener system linear fir filter followed by a static polynomial nonlinearity for noisy output observations the analysis includes 1 bounds and a recursion for the misadjustment matrix and 2 algorithm mean square stability regions a diagonal step size matrix for the adaptive coefficients is introduced to speed up convergence the theoretical predictions of the fluctuation analysis are supported by monte carlo simulations

fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids we study fuzzy finite automata in which all fuzzy sets are defined by membership functions whose codomain forms a lattice ordered monoid l for these l fuzzy finite automata l ffa for short we provide necessary and sufficient conditions for the extendability of the state transition function it is shown that nondeterministic l ffa nl ffa for short are more powerful than deterministic l ffa dl ffa for short then we give necessary and sufficient conditions for the simulation of an nl ffa by an equivalent dl ffa next we turn to the closure properties of languages defined by l ffas we establish closure under the regular operations and provide conditions for closure under intersection and reversal in particular we show that the family of fuzzy languages accepted by dl ffas is not closed under kleene closure operation and the family of fuzzy languages accepted by nl ffas is not closed under complement operation furthermore we introduce the notions of l fuzzy regular expressions and give the kleene theorem for nl ffas the description of dl ffas by l fuzzy regular expressions is also given finally we investigate the level structures of l ffas our results provide some insight as to what extend properties of l ffas and their languages depend on the algebraic properties of l

myhill nerode type theory for fuzzy languages and automata the myhill nerode theory is a branch of the algebraic theory of languages and automata in which formal languages and deterministic automata are studied through right congruences and congruences on a free monoid in this paper we develop a general myhill nerode type theory for fuzzy languages with membership values in an arbitrary set with two distinguished elements 0 and 1 which are needed to take crisp languages in consideration we establish connections between extensionality of fuzzy languages w r t right congruences and congruences on a free monoid and recognition of fuzzy languages by deterministic automata and monoids and we prove the myhill nerode type theorem for fuzzy languages we also prove that each fuzzy language possess a minimal deterministic automaton recognizing it we give a construction of this automaton using the concept of a derivative automaton of a fuzzy language and we give a method for minimization of deterministic fuzzy recognizers in the second part of the paper we introduce and study nerode s and myhill s automata assigned to a fuzzy automaton with membership values in a complete residuated lattice the obtained results establish nice relationships between fuzzy languages fuzzy automata and deterministic automata

statistical analysis of the single layer backpropagation algorithm ii mse and classification performance for pt i see ibid p 583 91 1993 the analysis of pt i is extended to the evaluation of the mean square error and the probability of correct classification it is shown that the mean square error and the corresponding performance surface are such that the single layer perceptron is prevented from correctly classifying with probability one until the weights converge at infinity

minimization of lattice finite automata and its application to the decomposition of lattice languages in this paper we study the minimization of lattice valued finite automata with membership values in a distributive lattice and its relationships to the decomposition of lattice valued regular languages first we establish the equivalence of nondeterministic lattice finite automata la and deterministic lattice finite automata dla furthermore we provide some characterization of lattice valued regular languages and regular operations on family of lattice valued regular languages in the sequel we introduce some notions that help clarify the concept of minimal dlas and present an effective algorithm to obtain a minimal dla from a given la using the construction of minimal dla we introduce some simple classes of lattice valued regular languages such as l unitary and l prefix ones we demonstrate that any lattice valued regular language can be decomposed as disjoint joins of such kinds of simple languages

brief paper initializing wiener hammerstein models based on partitioning of the best linear approximation this paper describes a new algorithm for initializing and estimating wiener hammerstein models which consist of two linear parts with a static nonlinearity in between the algorithm makes use of the best linear model of the system which is a consistent estimate of the systems dynamics for gaussian excitations the linear model is split in all possible ways into two sub models for all possible splits a wiener hammerstein model is initialized which means that a nonlinearity is introduced in between the two sub models the linear parameters of this nonlinearity can be estimated using least squares all initialized models can then be ranked depending on the fit typically one is only interested in the best one for which all parameters are fitted using prediction error minimization the paper explains the algorithm in detail and consistency of the initialization is proven computational aspects are investigated showing that in most realistic cases the number of splits of the initial linear model remains low enough to make the algorithm useful the algorithm is illustrated on an example where it is shown that the initialization is a tool to avoid many local minima

blind identification of simo wiener systems based on kernel canonical correlation analysis we consider the problem of blind identification and equalization of single input multiple output simo nonlinear channels specifically the nonlinear model consists of multiple single channel wiener systems that are excited by a common input signal the proposed approach is based on a well known blind identification technique for linear simo systems by transforming the output signals into a reproducing kernel hilbert space rkhs a linear identification problem is obtained which we propose to solve through an iterative procedure that alternates between canonical correlation analysis cca to estimate the linear parts and kernel canonical correlation kcca to estimate the memoryless nonlinearities the proposed algorithm is able to operate on systems with as few as two output channels on relatively small data sets and on colored signals simulations are included to demonstrate the effectiveness of the proposed technique

characterizations of fuzzy finite automata in this paper we establish some important concepts in fuzzy finite automata ffas with bifuzzy property and clarify their essential relationships first we present a number of basic definitions and properties in ffas we then define a bifuzzy family of subautomata bifuzzy source and successor operators as well as demonstrate the bifuzzy topological characterization of ffas also we clarify the relationships between our results and previous ones and thus derive a relation figure visualizing their connections afterwards we investigate further the bifuzzy separability retrievability and homomorphism fuzzy continuous mapping and open mapping are also incorporated in particular we prove a number of equivalent characterizations for these concepts and expound the relationships amongst them this also concludes that both the bifuzzy source and successor operators are fuzzy closure operators we discover that our investigation generalizes to some extent the algebraic fuzzy automata elaborated by malik mordeson and others finally the main results obtained are summarized the potential applications are indicated and a number of related problems for further study are addressed