nonlinear damping estimation from rotor stability data using time and frequency domain techniques for estimating equivalent damping characteristics from transient response data involving nonlinear damping modifications to two stability testing methods the moving block technique and the sparse time domain method are presented and verified by the use of numerical simulations effects of large undamped response and noise on the performance of these techniques are evaluated for quadratic and coulomb dampings it is concluded that equivalent damping characteristics may be identified from sampled multimode and noisy nonlinear transient response data using modified versions of the moving block analysis and the sparse time domain method

gaudin models with irregular singularities abstract we introduce a class of quantum integrable systems generalizing the gaudin model the corresponding algebras of quantum hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine kac moody algebra at the critical level extending the construction of higher gaudin hamiltonians from b feigin et al 1994 17 to the case of non highest weight representations of affine algebras we show that these algebras are isomorphic to algebras of functions on the spaces of opers on p 1 with regular as well as irregular singularities at finitely many points we construct eigenvectors of these hamiltonians using wakimoto modules of critical level and show that their spectra on finite dimensional representations are given by opers with trivial monodromy we also comment on the connection between the generalized gaudin models and the geometric langlands correspondence with ramification

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

homological and monodromy representations of framed braid groups in this paper we introduce two new classes of representations of the framed braid groups one is the homological representation constructed as the action of a mapping class group on a certain homology group the other is the monodromy representation of the confluent kz equation which is a generalization of the kz equation to have irregular singularities we also give a conjectural equivalence between these two classes of representations

an identification algorithm for polynomial narx models based on simulation error minimization classical prediction error approaches for the identification of non linear polynomial narx narmax models often yield unsatisfactory results for long range prediction or simulation purposes mainly due to incorrect or redundant model structure selection the paper discusses some limitations of the standard approach and suggests two modifications namely a new index based on the simulation error is employed as the regressor selection criterion and a pruning mechanism is introduced in the model selection algorithm the resulting algorithm is shown to be effective in the identification of compact and robust models generally yielding model structures closer to the correct ones computational issues are also discussed finally the identification algorithm is tested on a long range prediction benchmark application

recursive identification using feedforward neural networks the paper is concerned with the identification of an unknown nonlinear dynamical system when only the inputs and outputs are accessible for measurement under certain assumptions it is shown that generically the system can be realized by a recursive input output model furthermore relying on the approximation properties of neural networks and the existence of effective training algorithms it is demonstrated how an effective identification model can be constructed simulation results are presented to complement the theoretical discussions

fock space representations of affine lie algebras and integral representations in the wess zumino witten models fock space representations of affine lie algebras are studied explicit forms of correction terms adding to the currentsf i z are determined it is proved that the sugawara energy momentum tensor on the fock spaces is quadratic in free bosons furthermore screening operators are constructed this implies the existence of generalized hypergeometric integrals satisfying the knizhnik zamolodchikov equation

two stage orthogonal least squares methods for neural network construction a number of neural networks can be formulated as the linear in the parameters models training such networks can be transformed to a model selection problem where a compact model is selected from all the candidates using subset selection algorithms forward selection methods are popular fast subset selection approaches however they may only produce suboptimal models and can be trapped into a local minimum more recently a two stage fast recursive algorithm tsfra combining forward selection and backward model refinement has been proposed to improve the compactness and generalization performance of the model this paper proposes unified two stage orthogonal least squares methods instead of the fast recursive based methods in contrast to the tsfra this paper derives a new simplified relationship between the forward and the backward stages to avoid repetitive computations using the inherent orthogonal properties of the least squares methods furthermore a new term exchanging scheme for backward model refinement is introduced to reduce computational demand finally given the error reduction ratio criterion effective and efficient forward and backward subset selection procedures are proposed extensive examples are presented to demonstrate the improved model compactness constructed by the proposed technique in comparison with some popular methods

state dependent parameter models of non linear sampled data systems a velocity based linearization approach discrete time models with state dependent parameters are derived for non linear sampled data systems velocity based linearization of the continuous time system followed by integration over the sampling interval is used to represent the sampled data system by a discrete time state space model with state dependent parameters by reconstructing the state a quasi arx model is obtained the parameters of which are functions of past inputs and outputs as the velocity based linearization procedure does not involve suppression of higher order terms the state dependent state space model and the quasi arx description give exact representations of the non linear system

kernel based approaches to local nonlinear non parametric variable selection in this paper we consider the variable selection problem for a nonlinear non parametric system two approaches are proposed one top down approach and one bottom up approach the top down algorithm selects a variable by detecting if the corresponding partial derivative is zero or not at the point of interest the algorithm is shown to have not only the parameter but also the set convergence this is critical because the variable selection problem is binary a variable is either selected or not selected the bottom up approach is based on the forward backward stepwise selection which is designed to work if the data length is limited both approaches determine the most important variables locally and allow the unknown non parametric nonlinear system to have different local dimensions at different points of interest further two potential applications along with numerical simulations are provided to illustrate the usefulness of the proposed algorithms

confluent kz equations for mathfrak sl _n with poincare rank 2 at infinity we construct confluent kz equations with poincar rank 2 at infinity for the case of and the integral representation for the solutions hamiltonians of these confluent kz equations are derived from suitable quantization of dlog constructed in the theory of monodromy preserving deformation mpd in jimbo et al 1981 physica d 2 306 52 our confluent kz equations can be viewed as a quantization of mpd with poincar rank 2 at infinity

minimal realization of nonlinear systems described by input output difference equations we consider a general class of nonlinear discrete time systems described by their input output i o relationship we first present the necessary and sufficient conditions for existence of a local observable state space realization of the system and an explicit algorithm for computing it in cases where the i o map and its resulting realization are nonminimal we formulate an algorithm for extracting its minimal realization whenever possible the developed theory and algorithms are illustrated by means of several examples

analogy between a 10d model for nonlinear wave wave interaction in a plasma and the 3d lorenz dynamics this paper investigates nonlinear wave wave interactions in a system that describes a modified decay instability and consists of three langmuir and one ion sound waves as a means to establish that the underlying dynamics exists in a 3d space and that it is of the lorenz type both continuous and discrete time multivariable global models were obtained from data these data were obtained from a 10d dynamical system that describes the modified decay instability obtained from zakharov s equations which characterise langmuir turbulence this 10d model is equivariant under a continuous rotation symmetry and a discrete order 2 rotation symmetry when the continuous rotation symmetry is modded out that is when the dynamics are represented with the continuous rotation symmetry removed under a local diffeomorphism it is shown that a 3d system may describe the underlying dynamics for certain parameter values the models obtained using global modelling techniques from three time series from the 10d dynamics with the continuous rotation symmetry modded out generate attractors which are topologically equivalent these models can be simulated easily and due to their simplicity are amenable for analysis of the original dynamics after symmetries have been modded out moreover it is shown that all of these attractors are topologically equivalent to an attractor generated by the well known lorenz system 2003 elsevier science b v all rights reserved

nonlinear system identification via direct weight optimization a general framework for estimating nonlinear functions and systems is described and analyzed in this paper identification of a system is seen as estimation of a predictor function the considered predictor function estimate at a particular point is defined to be affine in the observed outputs and the estimate is defined by the weights in this expression for each given point the maximal mean square error or an upper bound of the function estimate over a class of possible true functions is minimized with respect to the weights which is a convex optimization problem this gives different types of algorithms depending on the chosen function class it is shown how the classical linear least squares is obtained as a special case and how unknown but bounded disturbances can be handled most of the paper deals with the method applied to locally smooth predictor functions it is shown how this leads to local estimators with a finite bandwidth meaning that only observations in a neighborhood of the target point will be used in the estimate the size of this neighborhood the bandwidth is automatically computed and reflects the noise level in the data and the smoothness priors the approach is applied to a number of dynamical systems to illustrate its potential

nonlinear system identification an effective framework based on the karhunen loeve transform this paper proposes on the basis of a rigorous mathematical formulation a general framework that is able to define a large class of nonlinear system identifiers this framework exploits all those relationships that intrinsically characterize a limited set of realizations obtained by an ensemble of output signals and their parameterized inputs by means of the separation property of the karhunen loeve transform the generality and the flexibility of the approximating mappings ranging from traditional approximation techniques to multiresolution decompositions and neural networks allow the design of a large number of distinct identifiers each displaying a number of properties such as linearity with respect to the parameters noise rejection low computational complexity of the approximation procedure exhaustive experimentation on specific case studies reports high identification performance for four distinct identifiers based on polynomials splines wavelets and radial basis functions several comparisons show how these identifiers almost always have higher performance than that obtained by current best practices as well as very good accuracy optimal noise rejection and fast algorithmic elaboration as an example of a real application the identification of a voice communication channel comprising a digital enhanced cordless telecommunications dect cordless phone for wireless communications and a telephone line is reported and discussed

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

identification of nonlinear systems using narmax model abstract most systems encountered in the real world are nonlinear in nature and since linear models cannot capture the rich dynamic behavior of limit cycles bifurcations etc associated with nonlinear systems it is imperative to have identification techniques which are specific for nonlinear systems the problem considered in this work is the modeling of nonlinear discrete systems based on the set of input output data this is often the only approach to modeling as in most cases only external i e input output data are available this paper also discusses the practical aspects of identification of nonlinear systems the narmax nonlinear autoregressive moving average with exogenous input model provides a unified representation for a wide class of nonlinear systems and has obvious advantages over functional series representations such as volterra and wiener series this model is proven to provide a better parameter estimation and prediction accuracy than the linear model

parameterized convergence bounds for volterra series expansion of narx models evaluation of the convergence bound in the frequency domain for volterra series expansion of nonlinear systems described by narx models is studied this provides new convergence criteria under which the nonlinear system of interest has a convergent volterra series expansion and the new criteria are expressed explicitly in terms of the input magnitude model parameters and frequency variable the new convergence criteria are firstly developed for harmonic inputs and then extended to multi tone and general input cases based on the theoretical analysis a general procedure for calculating the convergence bound is provided the results provide a fundamental basis for nonlinear signal processing using the volterra series theory

bilinear realizability is equivalent to existence of a singular affine differential i o equation abstract for continuous time analytic input output maps existence of a singular differential equation relating derivatives of controls and outputs is shown to be equivalent to bilinear realizability a similar result holds for the problem of immersion into bilinear systems the proof is very analogous to that of the corresponding and previously known result for discrete time

stabilizing predictive control of nonlinear arx models in this paper a predictive control algorithm for nonlinear discrete time systems is presented starting from a state space model conditions for the asymptotic tracking of constant reference signals in a neighbourhood of a given equilibrium are first derived then it is shown that the system under control can be locally described in terms of a suitable narx nonlinear arx model which in practice can be identified by means of well established techniques for the narx model a receding horizon predictive control algorithm is proposed which guarantees local stability and robust asymptotic tracking in the neighbourhood of the equilibrium conditions for local stability and tracking are given in terms of reachability observability and transmission zeros of the linearization of the original state space system around the equilibrium an example is reported to illustrate the effectiveness of the proposed method

frequency response functions for nonlinear rational models a recursive algorithm which maps a general class of nonlinear rational model defined as the ratio of two polynomial functions into the frequency domain is derived using the harmonic expansion method the new algorithm provides for the first time a direct analytic map from the time domain rational model parameters to the higher order frequency response functions complex nonlinear time domain behaviours can be analysed and interpreted in the frequency domain and simulated examples are included to illustrate the concepts involved

difference equations versus differential equations a possible equivalence for the rossler system abstract when a set of nonlinear differential equations is investigated most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions in a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete in numerical investigations a fourth order runge kutta integration scheme is usually sufficient nevertheless sometimes a set of difference equations may be required and in this case standard schemes like the forward euler backward euler or central difference schemes are used the major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps in some cases it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps nonstandard schemes as introduced by mickens nonstandard finite difference models of differential equations world scientific 1994 allow to obtain more robust difference equations in this paper using such nonstandard scheme we propose some difference equations as discrete analogues of the rossler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used in particular it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the rossler system as long as the time step is less than the threshold value associated with the nyquist criterion

online identification of the bilinear model expansion on laguerre orthonormal bases this paper proposes a new representation of discrete bilinear model by developing its coefficients associated to the input to the output and to the crossed product on three independent laguerre orthonormal bases compared to classical bilinear model the resulting model entitled bilinear laguerre model ensures a significant parameter number reduction as well as simple recursive representation however this reduction is still subject to an optimal choice of the laguerre poles defining the three laguerre bases therefore we propose an analytical solution to optimise the laguerre poles which depend on fourier coefficients defining the bilinear laguerre model and that are identified using the regularised square error the identification procedures of the laguerre poles and fourier coefficients are combined and carried out on a sliding window to provide an online identification algorithm of the bilinear laguerre model the bilinear laguerre model as well as the proposed algorithm are illustrated and tested

smoothing data with local instabilities for the identification of chaotic systems chaotic systems are characterized by the local divergence of nearby orbits in state space this provokes a sensitive dependence on initial conditions and in turn drastically limits the accuracy of long term predictions this has important implications in the filtering of data generated by such systems in the present paper the use of global smoothers for chaotic data is investigated the ultimate objective is to be able to identify dynamically valid models from smoothed data when the identification from the original noisy data has completely failed the objective is to produce identified models that faithfully reproduce the dynamical invariants of the original system such as the geometry of the attractors in state space the largest lyapunov exponent fractal dimensions and poincare sections numerical examples are included which illustrate the main points of the paper

the determination of optimum structures for the state space representation of multivariate stochastic processes when identifying a model for a multivariate stationary stochastic process an important problem is that of determining the structure of the state variable model several overlapping parameterizations can usually be fitted to a given process and the question arises as to which structure leads to the most accurate parameter estimates the accuracy of parameter estimates is often measured by the determinant of the fisher information matrix we show that all admissible structures will give asymptotically the same value to this criterion for finite data some structures may still be better than others and two heuristic structure estimation methods are analyzed some simulation results are also presented

multiobjective parameter estimation for non linear systems affine information and least squares formulation this paper defines a class of system information affine information that includes both the dynamic residuals and some types of auxiliary information that can be used in system parameter estimation as special cases the types of information that can be cast under the affine information format give rise to quadratic functions that measure the extent to which a model fits such information and that can be aggregated in a single weighted quadratic cost functional this allows the definition of a multiobjective methodology for parameter estimation in non linear system identification which allows taking into account any type of affine information the results are presented in terms of a set of efficient solutions of the multiobjective estimation problem such a solution set is more meaningful than a single model since any affine information leads to a convex quadratic functional the whole set of efficient solutions is exactly accessible via the minimization of the quadratic functional with different weighti

affine kac moody algebras and semi infinite flag manifolds we study representations of affine kac moody algebras from a geometric point of view it is shown that wakimoto modules introduced in 18 which are important in conformal field theory correspond to certain sheaves on a semi infinite flag manifold with support on its schhubert cells this manifold is equipped with a remarkable semi infinite structure which is discussed in particular the semi infinite homology of this manifold is computed the cousin grothendieck resolution of an invertible sheaf on a semi infinite flag manifold gives a two sided resolution of an irreducible representation of an affine algebras consisting of wakimoto modules this is just the brst complex as a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent lie algebra

recursive hybrid algorithm for non linear system identification using radial basis function networks recursive identification of non linear systems is investigated using radial basis function networks a novel approach is adopted which employs a hybrid clustering and least squares algorithm the recursive clustering algorithm adjusts the centres of the radial basis function network while the recursive least squares algorithm estimates the connection weights of the network because these two recursive learning rules are both linear rapid convergence is guaranteed and this hybrid algorithm significantly enhances the real time or adaptive capability of radial basis function models the application to simulated real data are included to demonstrate the effectiveness of this hybrid approach

grey box modelling and identification using physical knowledge and bayesian techniques abstract advanced control design requires a model that describes process behaviour adequately such a model can be constructed using physical modelling or statistical identification techniques both have their disadvantages physical models are rigorous and thus often expensive to construct while black box model structures are not necessarily compatible with physical reality since both approaches have undeniable merits as well their combination seems to be attractive and rewarding this paper discusses an approach to statistical estimation where a best linearly parametrized dynamic regression model is identified which is also consistent with specified knowledge about process responses such characteristics are forced upon the black box models yielding a grey box model set it will be shown that crucial physical knowledge such as process stability and sign of stationary gains can be translated into linear inequality constraints on the black box model parameters in order to select best estimators in the grey box class a bayesian approach is adopted given a prior distribution associated with the physical knowledge and given the data likelihood a posterior distribution is constructed maximum and average a posteriori estimators are analysed explicit solutions are given for special cases of gaussian likelihood and a prior which is uniformly or piece wise linearly distributed on a linearly constrained grey box model class finally simulation results and an application to a distillation process show the advantage of the contrained estimates under realistic experiment conditions considerable variance reductions at the cost of a somewhat larger bias can be achieved indicating the potential for e g adaptive control applications

dynamical realizations of finite volterra series in this paper realizations of finite volterra series are viewed as nonlinear analytic input output systems with state space described by an analytic manifold for a minimal realization guaranteed by h j sussmann the state space which is unique up to diffeomorphism is shown to have the homogeneous space structure of a nilmanifold the quotient of two nilpotent lie groups the structure of nilmanifolds as described by a malcev is used to show that for these systems the state space has a vector space structure as a consequence of this result it is shown that a minimal realization of a finite volterra series can be described as a cascade of linear subsystems with polynomial link maps in which the dimension o f each linear subsystem is independent of the realization considered

modelling and identification of non linear deterministic systems in the delta domain this paper provides a formulation for using the delta operator in the modelling of non linear systems it is shown that a unique representation of a deterministic non linear auto regressive with exogenous input narx model can be obtained for polynomial basis functions using the delta operator and expressions are derived to convert between the shift and delta domain a delta narx model is applied to the identification of a test problem a van der pol oscillator a comparison is made with the standard shift operator non linear model and it is demonstrated that the delta domain approach improves the numerical properties of structure detection leads to a parsimonious description and provides a model that is closely linked to the continuous time non linear system in terms of both parameters and structure

fock representations of the affine lie algebra a 1 1 the aim of this note is to show that the affine lie algebraa1 1 has a natural family v of fock representations on the spacec xi yj i andj parametrized by v c2 by corresponding the highest weight of to each the parameter spacec2 forms a double cover of the weight spacec 0 c 1 with singularities at linear forms of level 2 this number is 1 times the dual coxeter number our results contain explicit realizations of irreducible non integrable highest wieghta1 1 modules for generic v

improved structure selection for nonlinear models based on term clustering in this paper the concepts of term clusters and cluster coefficients are defined and used in the context of system identification it is argued that if a certain type of term in a nonlinear model is spurious the respective cluster coefficient is small compared with the coefficients of the other clusters represented in the model once the spurious clusters have been detected the corresponding terms can be deleted from the set of candidate terms the consequences of doing this are i a drastic reduction in the size of the set of candidate terms and consequently a substantial gain in computation time is achieved ii the final estimated model is more likely to reproduce the dynamics of the original system and iii the final model is more robust to overparametrization numerical examples are included to illustrate the new procedure

application of a least absolute shrinkage and selection operator to aeroelastic flight test data identification of non linear systems involves estimating unknown parameters and model regressor selection selection of a subset of candidate terms that best describes the observed output model selection is an important step in black box modelling of any observed process this procedure is concerned with selecting a subset of parameters to give a parsimonious description of the system which may afford greater insight into the functionality of the system or a simpler controller design in this study a least absolute shrinkage and selection operator lasso technique is investigated for computing efficient model descriptions of non linear aeroelastic systems the lasso minimises the residual sum of squares by the addition of an l1 penalty term on the parameter vector of the traditional l2 minimisation problem its use for model selection is a natural extension of this constrained minimisation approach to pseudolinear regression problems which produces some model parameters that are exactly zero and the

brief paper structure detection and parameter estimation for narx models in a unified em framework in this paper we consider structure detection and parameter estimation of the nonlinear auto regressive with exogenous inputs narx model using the em expectation maximisation algorithm the parameter estimation step uses particle smoothing to obtain the necessary expectations in the e step and the parameters are then estimated in closed form in the m step the model structure detection is performed using an f test which makes use of the parameter information matrix inverse of the covariance matrix obtained from an augmentation of the em algorithm the steps for obtaining the information matrix are robust guaranteeing a positive semi definite information matrix to use in the structure detection step for the case of unknown model orders a method is proposed using the stochastic complexity sc information criterion for selecting between candidate models the sc is composed of the information matrix representing model complexity and a likelihood estimate representing model accuracy which are both generated as byproducts of the augmented em algorithm numerical results demonstrate that the em approach performs well in comparison to a standard alternative based on orthogonal least squares and also avoids the need to estimate a noise model for the case of measurement noise corrupted output signals

control of nonlinear dynamical systems using neural networks ii observability identification and control for pt i see ibid vol 4 1993 this paper considers the problems of regulation and tracking of a dynamical system when the state variables of the dynamical system are not accessible the existence of the nonlinear maps describing the identifier and controller are first established and the implications for neural network realizations are described simulation results are included to complement the theoretical discussions

identification of a continuous structure with a geometrical non linearity part i conditioned reverse path method particular effort has been spent in the field of identification of multi degree of freedom non linear systems the newly developed methods permit the structural analyst to consider increasingly complex systems the aim of this paper and a companion paper is to study by means of two methods a continuous non linear system consisting of an experimental cantilever beam with a geometrical non linearity in this paper part i the ability of the conditioned reverse path method which is a frequency domain technique to identify the behaviour of this structure is assessed the companion paper part ii is devoted to the application of proper orthogonal decomposition which is an updating technique to the test example

identification of non linear systems using empirical data and prior knowledge an optimization approach abstract the choice of a parametric model structure in empirical and semi empirical non linear modeling is usually viewed as an important and critical step however it is known that by augmenting the least squares identification criterion with a term that imposes a penalty on the non smoothness of the model an optimal non parametric model can be found explicitly the optimal non parametric model will depend on the particular form of the penalty which can be looked upon as a priori knowledge or the desired properties of the model in this paper these results are extended in several directions i we show how useful types of prior knowledge other than smoothness can be included as a term in the criterion or as a constraint and how this influences the optimal model ii dynamic models and a general prediction error penalty are considered iii we present a practical numerical procedure for the identification of a close to optimal semi parametric model the numerical approach is motivated by the difficulty of deriving the optimal non parametric model if there are complicated constraints or penalty terms in the criterion and finally iv we discuss determination of the appropriate model complexity through selecting weights on the different penalties on the basis of empirical data since the optimal non parametric model is identified using an augmented optimization criterion and all prior knowledge and desired model properties may be specified through the augmented optimization criterion it is the choice of this criterion that is critical in this approach this elevates the empirical and semi empirical modeling problems to a more transparent and engineering friendly level than is achieved by directly specifying a parametric model structure hence our results shed some more light on modern non linear empirical modeling approaches like radial basis functions splines and neural networks in addition to providing a fundamental insight into the role of different kinds of prior knowledge this high level formulation is also attractive from a practical point of view as it lies closer to engineering thinking and less guesswork is required the flexibility and power of this approach are illustrated with a semi realistic simulation example

confluent primary fields in the conformal field theory for any complex simple lie algebra we generalize the primary fields in the wess zumino novikov witten conformal field theory for a case with irregular singularities we refer to these generalized primary fields as confluent primary fields we present the screening currents ward identity a recursion rule for computing the expectation values of the products of confluent primary fields in the case of the expectation values of the products of confluent primary fields are integral formulas of solutions to confluent knizhnik zamolodchikov kz equations given in jimbo et al 2008 j phys a math theor 41 175205 by computing the operator product expansion of the energy momentum tensor t z and the confluent primary fields we obtain new differential operators moreover in the case of these differential operators are the same as those of the confluent kz equations jimbo et al 2008

a nonlinear control method based on anfis and multiple models for a class of siso nonlinear systems and its application this paper presents a novel nonlinear control strategy for a class of uncertain single input and single output discrete time nonlinear systems with unstable zero dynamics the proposed method combines adaptive network based fuzzy inference system anfis with multiple models where a linear robust controller an anfis based nonlinear controller and a switching mechanism are integrated using multiple models technique it has been shown that the linear controller can ensure the boundedness of the input and output signals and the nonlinear controller can improve the dynamic performance of the closed loop system moreover it has also been shown that the use of the switching mechanism can simultaneously guarantee the closed loop stability and improve its performance as a result the controller has the following three outstanding features compared with existing control strategies first this method relaxes the assumption of commonly used uniform boundedness on the unmodeled dynamics and thus enhances its applicability second since anfis is used to estimate and compensate the effect caused by the unmodeled dynamics the convergence rate of neural network learning has been increased third a one to one mapping technique is adapted to guarantee the universal approximation property of anfis the proposed controller is applied to a numerical example and a pulverizing process of an alumina sintering system respectively where its effectiveness has been justified

a globally consistent nonlinear least squares estimator for identification of nonlinear rational systems this paper considers identification of nonlinear rational systems defined as the ratio of two nonlinear functions of past inputs and outputs despite its long history a globally consistent identification algorithm remains illusive this paper proposes a globally convergent identification algorithm for such nonlinear rational systems to the best of our knowledge this is the first globally convergent algorithm for the nonlinear rational systems the technique employed is a two step estimator though two step estimators are known to produce consistent nonlinear least squares estimates if a n consistent estimate can be determined in the first step how to find such a n consistent estimate in the first step for nonlinear rational systems is nontrivial and is not answered by any two step estimators the technical contribution of the paper is to develop a globally consistent estimator for nonlinear rational systems in the first step this is achieved by involving model transformation bias analysis noise variance estimation and bias compensation in the paper two simulation examples and a practical example are provided to verify the good performance of the proposed two step estimator

generalized knizhnik zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform maxwell bloch system with pumping canonical quantization of the isomonodromy solutions of equations integrable via the inverse scattering transform leads to generalized knizhnik zamolodchikov equations one can solve these equations by the off shell bethe ansatz method provided the knizhnik zamolodchikov equations are related with the highest weight representations of the corresponding lie algebras these solutions can be written in terms of multi variable generalizations of special functions of the hypergeometric type in this work we consider a realization of the above scheme for the maxwell bloch system with pumping quantum states for this system are found in terms of the multi variable confluent hypergeometric function

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

dynamical effects of overparametrization in nonlinear models abstract this paper is concemed with dynamical reconstruction for nonlinear systems the effects of the driving function and of the complexity of a given representation on the bifurcation patter are investigated it is shown that the use of different driving functions to excite the system may yield models with different bifurcation patterns the complexity of the reconstructions considered is quantified by the embedding dimension and the number of estimated parameters in this respect it appears that models which reproduce the original bifurcation behaviour are of limited complexity and that excessively complex models tend to induce ghost bifurcations and spurious dynamical regimes moreover some results suggest that the effects of overparametrization on the global dynamical behaviour of a nonlinear model may be more deleterious than the presence of moderate noise levels in order to precisely quantify the complexity of the reconstructions global polynomials are used although the results are believed to apply to a much wider class of representations including neural networks

a two step method to identify parameters of piecewise linear systems based on the direct parameter estimation method and the legendre series approximation a two step method is presented to estimate by measured data the physical parameters i e mass damping stiffness and knot of nonlinear systems with symmetrical piecewise linear restoring force at first the piecewise linear restoring force of the system is approximated by legendre series a least squares direct parameter estimation method is adopted to identify the mass and damping parameters of the system and the corresponding coefficients of the legendre series secondly the physical parameters of the piecewise linear restoring force i e the primary and the secondary stiffness and the knots are extracted from the estimated legendre coefficients neither iteration nor search process is needed in this method the validity of the method is demonstrated by numerical simulation on a single degree of freedom and a three degree of freedom system with both noise free data and noise polluted data

recursive algorithm for computing the frequency response of a class of non linear difference equation models abstract a recursive algorithm is developed which is both computationally compact and enables the nth order frequency response functions of a large class of non linear difference equation models to be found without restriction on the order n

recovering map static nonlinearities from chaotic data using dynamical models abstract this paper is concerned with the estimation from chaotic data of maps with static nonlinearities a number of issues concerning model construction such as structure selection over parametrization and model validation are discussed in the light of the shape of the static non linearities reproduced by the estimated maps a new interpretation of term clusters and cluster coefficients of polynomial models is provided based on this approach the paper discusses model limitations and some useful principles to select the structure of nonlinear maps some of the ideas have been tested using several nonlinear systems including a boost voltage regulator map and a set of real data from a chaotic circuit

adaptive nn control for a class of discrete time non linear systems in this paper adaptive neural network nn control is investigated for a class of single input single output siso discrete time unknown non linear systems with general relative degree in the presence of bounded disturbances firstly the systems are transformed into a causal state space description adaptive state feedback nn control is presented based on lyapunov s stability theory then by converting the systems into a causal input output representation adaptive output feedback nn control is given finally adaptive nn observer design and observer based adaptive control are presented under the assumption of persistent excitation pe all the control schemes avoid the so called controller singularity problem in adaptive control by suitably choosing the design parameters the closed loop systems are proven to be semi globally uniformly ultimately bounded sguub simulation studies show the effectiveness of the newly proposed schemes

robust adaptive control of nonaffine nonlinear plants with small input signal changes assuming small input signal magnitudes arma models can approximate the narma model of nonaffine plants recently narma l1 and narma l2 approximate models were introduced to relax such input magnitude restrictions however some applications require larger input signals than allowed by arma narma l1 and narma l2 models under certain assumptions we recently developed an affine approximate model that eliminates the small input magnitude restriction and replaces it with a requirement of small input changes such a model complements existing models using this model we present an adaptive controller for discrete nonaffine plants with unknown system equations accessible input output signals but inaccessible states our approximate model is realized by a neural network that learns the unknown input output map online a deadzone is used to make the weight update algorithm robust against modeling errors a control law is developed for asymptotic tracking of slowly varying reference trajectories

a prediction error method for recursive indentification of nonlinear systems abstract nonlinear systems can be identified with nonlinear filters by combined estimation of parameters and state often these methods are very complex or have convergence problems as for example the extended kalman filter in this paper a new algorithm for recursive nonlinear system identification is presented convergence problems are eliminated by an improved calculation of the gradient as the total derivative of the prediction error by separately estimating parameters and states computation time is reduced the efficacy of the new identification algorithm is illustrated by studying its performance in a gravimetric filling system

analysis of the effects of time delay in nonlinear systems using generalised frequency response functions abstract a recursive algorithm for computing the generalised frequency response functions gfrfs of nonlinear time delay systems described by nonlinear differential difference equation models is derived using the operator e n and used to analyse the effects of time delay in nonlinear systems the algorithm shows an explicit relationship between the model parameters and the gfrfs such a relationship provides important insight into the behaviour of nonlinear systems the effect of delay on several properties of nonlinear systems such as harmonic generation gain compression expansion and desensitisation is studied by considering the example of a duffing oscillator with retarded damping results of the studies convincingly demonstrate that the system delay has a significant effect on several important properties of nonlinear systems

identification of models for chaotic systems from noisy data implications for performance and nonlinear filtering abstract this paper investigates the identification of global models from chaotic data corrupted by additive noise it is verified that noise has a strong influence on the identification of chaotic systems in particular there seems to be a critical noise level beyond which the accurate estimation of polynomial models from chaotic data becomes very difficult similarities with the estimation of the largest lyapunov exponent from noisy data suggest that part of the problem might be related to the limited ability of predicting the data records when these are chaotic a nonlinear filtering scheme is suggested in order to reduce the noise in the data and thereby enable the estimation of good models this prediction based filtering incorporates a resetting mechanism which enables the filtering of chaotic data and which is also applicable to non chaotic data

on critical stability of discrete time adaptive nonlinear control in this paper we examine the global stability and instability problems for a class of discrete time adaptive nonlinear stochastic control the systems to be controlled may exhibit chaotic behavior and are assumed to be linear in unknown parameters but nonlinear in output dynamics which are characterized by a nonlinear function say f x it is found and proved that in the scalar parameter case there is a critical stability phenomenon for least squares ls based adaptive control systems to be specific let the growth rate of f x be f x o spl par x spl par sup 6 with b spl ges 0 then it is found that b 4 is a critical value for global stability i e the closed loop adaptive system is globally stable if b 4 and is unstable in general if b spl ges 4 as a consequence we find an interesting phenomenon that the linear case does not have for some ls based certainty equivalence adaptive controls even if the ls parameter estimates are strongly consistent the closed loop systems may still be unstable this paper also indicates that adaptive nonlinear stochastic control that is designed based on e g taylor expansion or weierstrass approximation for nonlinear models may not be feasible in general

self tuning control of non linear armax models a control weighted self tuning minimum variance controller with a non linear difference equation structure is described an extended recursive least squares estimation algorithm is employed to provide the adaptiveness performance analysis of the controller is discussed in terms of a cumulative loss function and high order correlation functions of the system input output and residual sequences simulation results from an experiment using a model identified from a real system are also provided

spl theta adaptive neural networks a new approach to parameter estimation a novel use of neural networks for parameter estimation in nonlinear systems is proposed the approximating ability of the neural network is used to identify the relation between system variables and parameters of a dynamic system two different algorithms a block estimation method and a recursive estimation method are proposed the block estimation method consists of the training of a neural network to approximate the mapping between the system response and the system parameters which in turn is used to identify the parameters of the nonlinear system in the second method the neural network is used to determine a recursive algorithm to update the parameter estimate both methods are useful for parameter estimation in systems where either the structure of the nonlinearities present are unknown or when the parameters occur nonlinearly analytical conditions under which successful estimation can be carried but and several illustrative examples verifying the behavior of the algorithms through simulations are presented

input output modeling of nonlinear systems with time varying linear models time varying arma autoregressive moving average and armax autoregressive moving average with exogenous inputs models are proposed for input output modeling of nonlinear deterministic and stochastic systems the coefficients of these models are estimated by a random walk kalman filter rwkf this method requires no prior assumption on the nature of the model coefficients and is suitable for real time implementation since no off line training is needed a simulation example illustrates the method goodness of performance is judged by the quality of the residuals histograms autocorrelation functions and the kolmogorov smirnoff test

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

system identification of nonlinear state space models this paper is concerned with the parameter estimation of a general class of nonlinear dynamic systems in state space form more specifically a maximum likelihood ml framework is employed and an expectation maximisation em algorithm is derived to compute these ml estimates the expectation e step involves solving a nonlinear state estimation problem where the smoothed estimates of the states are required this problem lends itself perfectly to the particle smoother which provides arbitrarily good estimates the maximisation m step is solved using standard techniques from numerical optimisation theory simulation examples demonstrate the efficacy of our proposed solution

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

hypergeometric solutions to schrodinger equations for the quantum painleve equations we consider schrodinger equations for the quantum painleve equations we present hypergeometric solutions of the schrodinger equations for the quantum painleve equations as particular solutions we also give a representation theoretic correspondence between hamiltonians of the schrodinger equations for the quantum painleve equations and those of the knizhnik zamolodchikov kz equation or the confluent kz equations

input output parametric models for non linear systems part ii stochastic non linear systems recursive input output models for non linear multivariate discrete time systems are derived and sufficient conditions for their existence are defined the paper is divided into two parts the first part introduces and defines concepts such as nerode realization multistructural forms and results from differential geometry which are then used to derive a recursive input output model for multivariable deterministic non linear systems the second part introduces several examples compares the derived model with other representations and extends the results to create prediction error or innovation input output models for non linear stochastic systems these latter models are the generalization of the multivariable arm ax models for linear systems and are referred to as narmax or non linear autoregressive moving average models with exogenous inputs

existence and uniqueness of realizations of nonlinear systems necessary and sufficient conditions for the existence of either analytic or smooth symmetric realizations of nonlinear input output maps are given it is also shown that two minimal realizations are diffeomorphic

identification and control of a nonlinear discrete time system based on its linearization a unified framework this paper presents a unified theoretical framework for the identification and control of a nonlinear discrete time dynamical system in which the nonlinear system is represented explicitly as a sum of its linearized component and the residual nonlinear component referred to as a higher order function this representation substantially simplifies the procedure of applying the implicit function theorem to derive local properties of the nonlinear system and reveals the role played by the linearized system in a more transparent form under the assumption that the linearized system is controllable and observable it is shown that 1 the nonlinear system is also controllable and observable in a local domain 2 a feedback law exists to stabilize the nonlinear system locally and 3 the nonlinear system can exactly track a constant or a periodic sequence locally if its linearized system can do so with some additional assumptions the nonlinear system is shown to have a well defined relative degree delay and zero dynamics if the zero dynamics of the linearized system is asymptotically stable so is that of the nonlinear one and in such a case a control law exists for the nonlinear system to asymptotically track an arbitrary reference signal exactly in a neighborhood of the equilibrium state the tracking can be achieved by using the state vector for feedback or by using only the input and the output in which case the nonlinear autoregressive moving average narma model is established and utilized these results are important for understanding the use of neural networks as identifiers and controllers for general nonlinear discrete time dynamical systems

sparse model identification using a forward orthogonal regression algorithm aided by mutual information a sparse representation with satisfactory approximation accuracy is usually desirable in any nonlinear system identification and signal processing problem a new forward orthogonal regression algorithm with mutual information interference is proposed for sparse model selection and parameter estimation the new algorithm can be used to construct parsimonious linear in the parameters models

a gradient free adaptation method for nonlinear active noise control abstract active noise control anc problems are often affected by nonlinear effects such as saturation and distortion of microphones and loudspeakers nonlinear models and specific adaptation algorithms must be employed to properly account for these effects the nonlinear structure of the problem complicates the application of gradient based least mean squares lms algorithms due to the fact that exact gradient calculation requires executing nonlinear recursive filtering operations which pose computational and stability issues one favored solution to this problem consists in neglecting recursive terms in the gradient calculation an approximation which is not always without consequences on the convergence performance besides an efficient application of nonlinear models cannot avoid some form of model structure selection to avoid the well known effects of overparametrization and to reduce the computational load on line unfortunately the standard anc setting configures an indirect identification problem due to the presence of the secondary path in the control loop in the nonlinear case this destroys the linear regression structure of the problem even if the control filter is linear in the parameters thereby making it impossible to apply the many existing model selection methods for linear regression problems a simple and computationally wise low demanding approach is here proposed for parameter estimation and model structure selection that provides an answer to the mentioned issues the proposed method avoids altogether the use of the error gradient and relies on direct cost function evaluations a virtualization scheme is used to assess the accuracy improvements when the model is subject to parametric or structural modifications without directly affecting the control performance several simulation examples are discussed to show the effectiveness of the proposed algorithms

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

a suboptimal bootstrap method for structure detection of non linear output error models with application to human ankle dynamics identification of non linear systems involves estimating unknown parameters and structure detection selection of a subset of candidate terms that best describe the observed output this is a necessary procedure to compute an efficient system description which may afford greater insight into the functionality of the system or a simpler controller design for non linear systems simple output additive noise can generate multiplicative terms between the input output and noise the terms associated with noise need to be modelled to obtain unbiased parameter estimates significantly increasing the number of candidate terms to be estimated and considered in special cases it may be possible to use an output error oe model structure and the instrumental variable iv estimator to obtain unbiased parameters without modelling the noise this significantly reduces the dimensionality of the structure computation problem therefore in this paper we propose a suboptimal bootstrap structure detection sobsd algo

identification of mimo non linear systems using a forward regression orthogonal estimator abstract an orthogonal least squares estimator which was originally derived for single input single output systems is extended to multi input multi output non linear systems the estimator can provide information about the structure or which terms to include in the model and final parameter estimates in a very simple and efficient manner multivariable non linear model validity tests are also discussed and the results of applying the orthogonal estimator to both simulated and real data are included

a unified wavelet based modelling framework for non linear system identification the wanarx model structure a new unified modelling framework based on the superposition of additive submodels functional components and wavelet decompositions is proposed for non linear system identification a non linear model which is often represented using a multivariate non linear function is initially decomposed into a number of functional components via the well known analysis of variance anova expression which can be viewed as a special form of the narx non linear autoregressive with exogenous inputs model for representing dynamic input output systems by expanding each functional component using wavelet decompositions including the regular lattice frame decomposition wavelet series and multiresolution wavelet decompositions the multivariate non linear model can then be converted into a linear in the parameters problem which can be solved using least squares type methods an efficient model structure determination approach based upon a forward orthogonal least squares ols algorithm which involves a stepwise o

irregular conformal blocks with an application to the fifth and fourth painleve equations we develop the theory of irregular conformal blocks of the virasoro algebra in previous studies expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks however such expansions at irregular singular points were not clearly understood this is because precise definitions of irregular vertex operators had not been provided previously in this paper we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely then we define irregular conformal blocks with at most two irregular singular points as expectation values of given irregular vertex operators our definitions provide an understanding of expansions of irregular conformal blocks and enable us to obtain expansions at irregular singular points as an application we propose conjectural formulas of series expansions of the tau functions of the fifth and fourth painleve equations using expansions of i

hypothesis testing for nonlinearity detection based on an ma model in this correspondence we propose two hypothesis testing ht for nonlinearity detection these ht are based on an moving average ma model allowing us to model signals with null spectral density values unlike autoregressive ar models used in the usual parametric tests for nonlinearity detection these indexes are tested on simulated nonlinear and linear time series performances and drawbacks of these indexes are discussed with respect to the robustness of the indexes and to the difference between the theoretical and estimated laws under the hypothesis of linearity

structure identification of nonlinear dynamic systems a survey on input output approaches abstract in the past several methods have been elaborated for the identification of nonlinear dynamic systems most of the methods assume that the structure of the system is given a priori therefore they are in reality parameter estimation algorithms and structure identification is thus usually performed by repeated parameter estimation however in nonlinear system theory several methods are known to determine the structure of a system in this paper structure identification of block oriented especially cascade models of semi linear dynamic models with signal dependent parameters and of nonlinear dynamic models being linear in the parameters will be considered different structure selection methods are summarized based on step and impulse tests frequency response measurements correlation analysis repeated reproducible tests and normal operating data

remarks on the confluent kz equation for mathfrak sl _2 and quantum painleve equations for the lie algebra we give an explicit form of an irregular singular version of the kz equation in the simplest non trivial cases these kz equations reproduce the quantum painlev equations possessing affine weyl group symmetry found previously by one of the authors

non parametric identification of non linear oscillating systems the problem of system identification from a time series of measurements is solved by using non parametric additive models having only few structural information about the system a non parametric approach may be more appropriate than a parametric one for which detailed prior knowledge is needed based on non parametric regression the functions in the additive models are estimated by a penalized least squares approach using backfitting the optimal smoothing parameters are determined via generalized cross validation making this approach completely adaptive to the data the procedure is applied to identify the non linear restoring force of vibrationally excited helical wire rope isolators

variable selection algorithm for the construction of mimo operating point dependent neurofuzzy networks an input variable selection procedure is introduced for the identification and construction of multi input multi output mimo neurofuzzy operating point dependent models the algorithm is an extension of a forward modified gram schmidt orthogonal least squares procedure for a linear model structure which is modified to accommodate nonlinear system modeling by incorporating piecewise locally linear model fitting the proposed input nodes selection procedure effectively tackles the problem of the curse of dimensionality associated with lattice based modeling algorithms such as radial basis function neurofuzzy networks enabling the resulting neurofuzzy operating point dependent model to be widely applied in control and estimation some numerical examples are given to demonstrate the effectiveness of the proposed construction algorithm