extended stochastic gradient identification algorithms for hammerstein wiener armax systems an extended stochastic gradient algorithm is developed to estimate the parameters of hammerstein wiener armax models the basic idea is to replace the unmeasurable noise terms in the information vector of the pseudo linear regression identification model with the corresponding noise estimates which are computed by the obtained parameter estimates the obtained parameter estimates of the identification model include the product terms of the parameters of the original systems two methods of separating the parameter estimates of the original parameters from the product terms are discussed the average method and the singular value decomposition method to improve the identification accuracy an extended stochastic gradient algorithm with a forgetting factor is presented the simulation results indicate that the parameter estimation errors become small by introducing the forgetting factor

exceptional points of an endomorphism of the projective plane let f an endomorphism of of degree 1 we show how to obtain a bound depending only on n on the number of codimension two subspaces in which are completely invariant for f where l is completely invariant for f means that f 1 l l set theoretically

adaptive digital control of hammerstein nonlinear systems with limited output sampling this paper is motivated by the practical control considerations that nonlinearity is abundant in industrial processes and output sampling rates are often limited due to hardware constraints in particular for a hammerstein nonlinear sampled data system in which the output sampling period is an integer multiple of the input updating period we derive by using a polynomial transformation technique a mathematical model which is suitable for parameter estimation with dual rate measurement data further we present an adaptive control scheme for such a dual rate nonlinear system the parameter estimation based adaptive algorithm can achieve virtually asymptotically optimal control and ensure that the closed loop system is stable and globally convergent the simulation results are included

optimal estimation theory for dynamic systems with set membership uncertainty an overview in many problems such as linear and nonlinear regressions parameter and state estimation of dynamic systems state space and time series prediction interpolation smoothing and functions approximation one has to evaluate some unknown variable using available data the data are always associated with some uncertainty and it is necessary to evaluate how this uncertainty affects the estimated variables typically the problem is approached assuming a probabilistic description of uncertainty and applying statistical estimation theory an interesting alternative approach referred to as set membership or unknown but bounded ubb error description has been investigated since the late 1960s in this approach uncertainty is described by an additive noise which is known only to have given integral typically l2 of l1 or componentwise l bounds in this paper we review the main results of this theory with special attention to the most recent advances obtained in the case of componentwise bounds

parameter estimation error bounds for hammerstein nonlinear finite impulsive response models abstract this paper presents a parameter estimation algorithm for a class of hammerstein nonlinear systems input nonlinear fir finite impulse response models and studies in detail the convergence properties of the proposed identification algorithm in the stochastic framework and derives the upper and lower bounds of the parameter estimation errors pee from the available input output data the analysis indicates that the mean square pee upper and lower bounds of the algorithm approach zero as the data length increases a simulation example is given

piecewise homotopy perturbation method for solving linear and nonlinear weakly singular vie of second kind the purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular volterra integral equation of the second kind especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem for this purpose we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like through modifying homotopy perturbation method an algorithm is successfully established to solve the linear and nonlinear weakly singular volterra integral equation of the second kind and the convergence of the algorithm is proved strictly comparisons are made between our method and other methods and the results reveal that the modified homotopy perturbation is effective

a nonparametric polynomial identification algorithm for the hammerstein system almost all existing hammerstein system nonparametric identification algorithms can recover the unknown system nonlinear element up to an additive constant and one functional value of the nonlinearity is usually assumed to be known to make the constant solvable to overcome this defect in this paper a new nonparametric polynomial identification algorithm for the hammerstein system is proposed which extends the idea in the author s previous work 1993 1994 on the hammerstein system identification to a more general and practical case where no functional value of the system nonlinearity is known a priori convergence and convergence rates in both uniform and global senses are established and simulation studies demonstrate the effectiveness and advantage of the new algorithm

a block by block method for volterra integro differential equations with weakly singular kernel the theory of a block by block method for solving volterra integro differential equations with continuous kernels see makroglou 14 5d is adapted to volterra integrodifferential equations with weakly singular kernels and a rate of convergence is given

parametric identification of structured nonlinear systems in this paper identification of structured nonlinear systems is considered using linear fractional transformations lft the a priori information regarding the structural interconnection is systematically exploited a parametric approach to the identification problem is investigated where it is assumed that the linear part of the interconnection is given and the input to the nonlinear part is measurable an algorithm for the identification of the nonlinear part is proposed the uniqueness properties of the estimate provided by the algorithm are examined it is shown that the estimate converges asymptotically to its true value under a certain persistence of excitation condition two simulated examples and a real data example are presented to show the effectiveness of the proposed algorithm

decoupling the linear and nonlinear parts in hammerstein model identification the main result of this paper is to show that the linear part can be made decoupled from the nonlinear part in hammerstein model identification therefore identification of the linear part for a hammerstein model becomes a linear problem and accordingly enjoys the same convergence and consistency results as if the unknown nonlinearity is absent

stability analysis of product t methods for abel integral equations of the second kind stability analysis of product methods for abel integral equations of the second kind based on the test equation y t 1 lambda int limits 0 1 frac y s t s alpha ds t geqq 0 0 0 and we investigate whether this property is inherited by numerical approximations toy

asymptotic properties of black box identification of transfer functions the problem of estimating the transfer function of a linear stochastic system is considered the transfer function is parametrized as a black box and no given order is chosen a priori this means that the model orders may increase to infinity when the number of observed data tends to infinity the consistency and convergence properties of the resulting transfer function estimates are investigated asymptotic expressions for the variances and distributions of these estimates are also derived for the case that the model orders increase it is shown that the variance of the transfer function estimate at a certain frequency is asymptotically given by the noise to signal ratio at that frequency mulliplied by the model order to number of data points ratio

variable transformations in the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels extensions to fredholm integral equations abstract a number of techniques that use variable transformations in numerical integration have been developed recently cf sidi numerical integration iv h brass g hammerlin eds birkhauser basel 1993 pp 359 373 laurie j comput appl math 66 1996 337 344 the use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule in this paper the application of variable transformation techniques of sidi and laurie type to the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels is considered since the transformations are such that the end points of integration need not be used as mesh points the methods introduced can be used for vie with both continuous and weakly singular kernel in a uniform way the methods have also the advantages of simplicity of application and of achieving high order of convergence the application of the idea to fredholm integral equations with continuous and weakly singular equations is also considered numerical results are included and they verify the expected increased order of convergence they were obtained by using the trapezoidal formula for the evaluation of the transformed integrals

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method

mathematical programming methods in the numerical solution of volterra integral and integro differential equations with weakly singular kernel abstract nonlinear volterra integral and integro differential equations with weakly singular kernel are considered and solved numerically using nonlinear mathematical programming methods based on minimax approximations in both cases polynomial and multiquadric approximation are used

solution of a class of volterra integral equations with singular and weakly singular kernels a class of volterra integral equations are solved in this paper where the involved kernel may be weakly singular singular and hypersingular for various cases existence and uniqueness of such volterra integral equations are established furthermore by reducing such equations to ordinary differential equations analytic solutions have been determined explicitly when restricting the solutions to certain special function classes the number of suitable solutions is given

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

resolvents and solutions of weakly singular linear volterra integral equations abstract the structure of the resolvent r t s for a weakly singular matrix function b t s is determined where b t s is the kernel of the linear volterra vector integral equation e x t a t 0 t b t s x s d s and a t is a given continuous vector function using contraction mappings in a banach space of continuous vector functions with an exponentially weighted norm we show that when b t s satisfies certain integral conditions r t s has the form r t s b t s r 1 t s where r 1 t s is the unique continuous solution of the integral equation r 1 t s b 1 t s s t b t u r 1 u s d u and b 1 t s is defined by b 1 t s s t b t u b u s d u as examples the formulas of resolvents for a couple of weakly singular kernels of practical interest are derived we also obtain conditions under which a weakly singular integral equation e has a unique continuous solution x t and show that it can be expressed in terms of r t s by x t a t 0 t r t s a s d s finally we show that there are parallel results for an alternative resolvent r t s and examine when it and r t s are equivalent

brief identification of linear systems with hard input nonlinearities of known structure this paper studies identification of systems with input nonlinearities of known structure for input nonlinearities parameterized by one parameter a deterministic approach is proposed based on the idea of separable least squares the identification problem is shown to be equivalent to an one dimensional minimization problem the method is very effective for several common static and nonstatic input nonlinearities for a general input nonlinearity a correlation analysis based identification algorithm is presented which is shown to be convergent

a survey of recent advances in the numerical treatment of volterra integral and integro differential equations abstract the present survey paper samples recent advances in the numerical analysis of volterra integral equations of the first and second kind and of integro differential equations including equations with weakly singular kernels except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years a fairly extensive bibliography selected to be representative rather than comprehensive complements the paper

a two stage algorithm for identification of nonlinear dynamic systems this paper investigates the two stage stepwise identification for a class of nonlinear dynamic systems that can be described by linear in the parameters models and the model has to be built from a very large pool of basis functions or model terms the main objective is to improve the compactness of the model that is obtained by the forward stepwise methods while retaining the computational efficiency the proposed algorithm first generates an initial model using a forward stepwise procedure the significance of each selected term is then reviewed at the second stage and all insignificant ones are replaced resulting in an optimised compact model with significantly improved performance the main contribution of this paper is that these two stages are performed within a well defined regression context leading to significantly reduced computational complexity the efficiency of the algorithm is confirmed by the computational complexity analysis and its effectiveness is demonstrated by the simulation results

a blind approach to closed loop identification of hammerstein systems this paper is a continuing study of the blind approach for the hammerstein identification in sun et al 1999 and bai and fu 2002 in the framework of a closed loop sampled data system the output is sampled faster than the updating period of the input the parameters of the linear dynamics are consistently estimated from the information of the output only after which the unmeasurable inner signal is uniquely reconstructed the noise effect is explicitly considered in both the parameter and inner signal estimation the estimation of the system orders and time delay are studied on the basis of two groups of basic equations obtained by polyphase decomposition the proposed blind approach is validated and illustrated by a simulated numerical example

numerical solution of non linear volterra integral equations this paper deals with non linear volterra integral equations of the type y x f x 0x h t x y t y x dt convergence criteria are given in the same sense of the maximum and ca norms for the numerical solution of this type of volterra integral equation several numerical methods are compared

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence

insufficiency of the etale brauer manin obstruction towards a simply connected example since poonen s construction of a variety x defined over a number field k for which x k is empty and the etale brauer manin set x bf a k textrm br et is not several other examples of smooth projective varieties have been found for which the etale brauer manin obstruction does not explain the failure of the hasse principle all known examples are constructed using poonen s trick i e they have the distinctive feature of being fibrations over a higher genus curve in particular their albanese variety is non trivial in this paper we construct examples for which the albanese variety is trivial the new geometric ingredient in our construction is the appearance of beauville surfaces assuming the abc conjecture and using geometric work of campana on orbifolds we also prove the existence of an example which is simply connected

bounding the parameters of linear systems with input backlash in this note we present a two stage procedure for deriving parameters bounds of linear systems with input backlash when the output measurement errors are bounded first using steady state input output data parameters of the nonlinear dynamic block are tightly bounded then given a suitable prbs input sequence we evaluate tight bounds on the unmeasurable inner signal which together with noisy output measurements are employed for bounding the parameters of the linear dynamic system

numerical solutions of volterra integral equations with a solution dependent delay on demontre l existence et l unicite des solutions pour l equation ainsi qu une limitation a priori de la solution on definit une solution numerique et on demontre que la solution numerique converge vers la solution de l equation integrale

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

hammerstein system identification through best linear approximation inversion and regularisation abstracthammerstein systems are composed by the cascading of a static nonlinearity and a linear system in this paper a methodology for identifying such systems using a combination of least squares support vector machines ls svm and best linear approximation bla techniques is proposed to do this a novel method for estimating the intermediate variable is presented allowing a clear separation of the identification steps first an approximation to the linear block is obtained through the bla of the system then an approximation to the intermediate variable is obtained using the inversion of the estimated linear block and the known output afterwards a nonlinear model is calculated through ls svm using the estimated intermediate variable and the known input to do this the regularisation capabilities of ls svm play a crucial role finally a parametric re estimation of the linear block is made the method was tested in three examples two of them with hard nonlinearities and was compared with four

brief paper computational complexity analysis of set membership identification of hammerstein and wiener systems this paper analyzes the computational complexity of set membership identification of hammerstein and wiener systems its main results show that even in cases where a portion of the plant is known the problems are generically np hard both in the number of experimental data points and in the number of inputs wiener or outputs hammerstein of the nonlinearity these results provide new insight into the reasons underlying the high computational complexity of several recently proposed algorithms and point out the need for developing computationally tractable relaxations

parameter bounds for armax models from records with bounded errors in variables the errors in variables problem is that of parameter estimation where all observed variables are corrupted by noise so far the computation of parameter bounds for dynamical models has been performed using equation error and output error approaches in this paper parameter bounds for autoregres sive moving average exogenous armax models are derived on the assumption that both input and output are affected by bounded noise parameter bounding by both the bounded equation error and the bounded errors in variables approach is outlined together with the approximation of the feasible parameter region by an orthotope outer bounding set and is tested on simulated data

a minimal k step delay controller for robust tracking of non minimum phase systems abstract this paper considers the problem of look ahead robust tracking for a non minimum phase system under a sensitivity constraint because of non minimum phase zeros the exact tracking of arbitrary reference input is impossible however with an introduction of some delay tracking is possible within any given tolerance 1 0 it is shown in the paper that the minimum delay required to satisfy the tracking performance is decoupled from a sensitivity disturbance rejection constraint moreover it is shown that calculation of the minimum delay reduces to a binary search problem

on the series expansion approach to the identification of hammerstein systems a polynomial identification algorithm for recovering a nonlinearity in the hammerstein system is proposed the estimate employs the legendre orthogonal system with adaptively selected number of terms the global consistency along with rates of convergence are established no assumptions concerning continuity of the nonlinearity or its functional form are made a data driven method using the cross validation technique for selecting the number of terms in the estimate is presented

integral points on subvarieties of semiabelian varieties ii this paper proves a finiteness result for families of integral points on a semiabelian variety minus a divisor generalizing the corresponding result of faltings for abelian varieties combined with the main theorem of the first part of this paper this gives a finiteness statement for integral points on a closed subvariety of a semiabelian variety minus a divisor in addition the last two sections generalize some standard results on closed subvarieties of semiabelian varieties to the context of closed subvarieties minus divisors

affine lines on logarithmic q homology planes with c3 c as motivation in this paper we study somewhat more general affine surfaces let x be a normal algebraic surface defined over c with at worst quotient singularities we call x a logarithmic q homology plane if h x q 0 for all i 0 if x is smooth we simply say that x is a q homology plane similarly if h x z 0 for all i 0 we call x a logarithmic homology plane or simply a homology plane if x is smooth let x be a logarithmic q homology plane then there exists a projective surface w and a reduced effective divisor d such that w d x w is smooth along d and d has simple normal crossings by 10 lemma 2 2 we know

haar wavelet method for some nonlinear volterra integral equations of the first kind we present here a simple efficient haar wavelet method for numerical solution of a class of nonlinear volterra integral equations of the first kind the present method is based on converting nonlinear volterra integral equations of the first kind into linear volterra integral equations of the second kind numerical examples are given to illustrate efficiency and accuracy of the present method comparison with numerical methods in the past has also been done solving some nonlinear first kind volterra integral equations using haar wavelet haar wavelet method is simple and fast solved examples demonstrate the accuracy and efficiency of the present method present method gives better results than the numerical methods described in past necessary accuracy may be obtained by increasing the number of calculation points

identification of wiener type nonlinear systems in a noisy environment wiener systems are widely encountered in engineering and science applications this paper proposes a straightforward approach to the identification of wiener systems in a noisy environment the obtained models are represented in terms of the frequency response for the linear subsystem and the inverse of the static nonlinearity a variety of input excitation signals including random binary and periodic can be used with the proposed method simulation studies using a ph neutralization continuous stirred tank reactor show that excellent results are obtained with relatively short data lengths

resolvents for weakly singular kernels and fractional differential equations the solution r t s of the resolvent equation 1 r t s b t s stb t u r u s du where b t s denotes a given weakly singular matrix is obtained by means of fixed point mappings the result is a series that begins with some singular terms after which the remainder of the terms defines a continuous function in particular the resolvent is calculated for the kernel b t s t s q 1 of the scalar abel integral equation of the second kind 2 x t a t 0t1 t s 1 qx s ds where 0 q 1 it is then used to derive closed form formulas for the resolvents corresponding to q 1 2 and 1 3 furthermore a closed form formula for the resolvent for b t s s t2 s2 q 1 is derived thereby demonstrating that the results of this paper apply not only to kernels of convolution type but also to those of non convolution type as well finally the resolvent for the kernel of 2 is used to find a general solution of a linear fractional differential equation of caputo type

numerical solution of a nonuniquely solvable volterra integral equation using extrapolation methods in this work the numerical solution of a volterra integral equation with a certain weakly singular kernel depending on a real parameter is considered although for certain values of this equation possesses an infinite set of solutions we have been able to prove that euler s method converges to a particular solution it is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize so that general extrapolation algorithms like the e algorithm can be applied to improve the numerical results this is illustrated by means of some examples

parameter identification of discontinuous hammerstein systems the paper deals with the parameter identification of hammerstein systems having discontinuous asymmetric nonlinearities two segment piecewise linear with preload and dead zone application of a simple decomposition technique provides a special form of hammerstein model with a minimum number of parameters the identification is carried out with only one record of measured input output data despite the dead zone to demonstrate the feasibility of the identification method some illustrative examples are included

theta characteristics on algebraic curves the theory of theta characteristics is developed algebraically so that it may be applied to possibly singular and or reducible algebraic curves the configuration of theta characteristics on a curve is described in terms of its singularities with applications to the geometry of plane quartic curves and the problem of appolonius some results on gorenstein local rings are appended 0 introduction the theory of theta characteristics has been studied from different points of view and often inadvertently cf for example the 19th century literature on bitangents to smooth plane quartic curves e g 4 and 10 until recently however virtually all treatments of theta characteristics per se involved the transcendental theory of the theta function associated to a compact riemann surface then in 1971 mumford in 8 gave a completely algebraic treatment of the subject formulating the main results and proving them without reference to the transcendental theory this in turn suggested that theta characteristics could be studied in a more general context for example on possibly singular and or reducible algebraic curves in this paper we undertake such an analysis in 1 following mumford we establish some of the basic facts about theta characteristics algebraically so as to apply to more general curves in fact 1 10 i and 1 13 in their present generality are due to mumford while he assumes throughout 8 that his curve x is smooth his proof seems to apply as given the argument for 1 13 here is somewhat different in order to make the proof more elementary and selfcontained one bonus is the result 1 11 on the subvariety of the moduli space 9thg consisting of curves with large semicanonical linear series this fact seemingly can be proved only via the algebraic set up in 2 we investigate the number and configuration of odd and even theta characteristics on a singular curve the results are expressed in the two main theorems 2 12 and 2 22 giving a more or less complete answer to these questions in terms of the singularities of the curve interestingly a recurrent motif in these arguments is one introduced by mumford in 8 the behavior of the maximal isotropic subspaces for a nondegenerate quadratic form on an even dimensional vector space here the mainspring of theorems 1 13 2 12 and 2 22 is assertion 1 4 below that the sum of the dimensions of the pairwise intersections of any three such subspaces is congruent to n mod 2 received by the editors july 11 1980 and in revised form october 22 1981 1980 mathematics subject classification primary 14h99 14h30 secondary 14h20 14h40 14k25 research partially supported by nsf grant mcs78 04008 aol dl 982 american mathematical society 0002 9947 8 1 0000 0371 10 25

identification of hammerstein nonlinear armax systems two identification algorithms an iterative least squares and a recursive least squares are developed for hammerstein nonlinear systems with memoryless nonlinear blocks and linear dynamical blocks described by armax carma models the basic idea is to replace unmeasurable noise terms in the information vectors by their estimates and to compute the noise estimates based on the obtained parameter estimates convergence properties of the recursive algorithm in the stochastic framework show that the parameter estimation error consistently converges to zero under the generalized persistent excitation condition the simulation results validate the algorithms proposed

identification of hammerstein systems without explicit parameterisation of non linearity this article proposes a new approach to identification of hammerstein systems where a non linearity precedes a linear dynamic system driven by piece wise constant inputs the proposed approach does not require an explicit parameterisation of the non linearity moreover the non linearity does not have to be static but could be the one with finite memories like backlash by exploiting input s piecewise constant property the denominator of the linear system described by an arx model is consistently identified from the information of the output only next a subspace direct equalisation method estimates the unmeasurable inner signal based on the resulted denominator estimate and output measurements contrary to the existing blind approaches the numerator of the linear system is not required which leads to a significant improvement of removing an error propagation on the basis of the estimated inner signal the measured input and output the non linearity and linear system are obtained separately the

controller design oriented model identification method for hammerstein system abstract the paper proposes a new polynomial identification method for the hammerstein system by combining the results of parameter estimation theory and functions approximation theory the method circumvents many disadvantages of the earlier methods it is suitable for producing the controller design oriented model of hammerstein system consistency and convergency analysis are given for the obtained parameter estimates of the system s linear dynamics and approximating polynomial of its nonlinear memoryless element respectively simulation study verifies the effectiveness of the method