control of chaotic dynamical systems using radial basis function network approximators abstract this paper presents a general control method based on radial basis function networks rbfns for chaotic dynamical systems for many chaotic systems that can be decomposed into a sum of a linear and a nonlinear part under some mild conditions the rbfn can be used to well approximate the nonlinear part of the system dynamics the resulting system is then dominated by the linear part with some small or weak residual nonlinearities due to the rbfn approximation errors thus a simple linear state feedback controller can be devised to drive the system response to a desirable set point in addition to some theoretical analysis computer simulations on two representative continuous time chaotic systems the duffing and the lorenz systems are presented to demonstrate the effectiveness of the proposed method

albanese maps and off diagonal long time asymptotics for the heat kernel we discuss long time asymptotic behaviors of the heat kernel on a non compact riemannian manifold which admits a discontinuous free action of an abelian isometry group with a compact quotient a local central limit theorem and the asymptotic power series expansion for the heat kernel as the time parameter goes to infinity are established by employing perturbation arguments on eigenvalues and eigenfunctions of twisted laplacians our ideas and techniques are motivated partly by analogy with floque bloch theory on periodic schrodinger operators for the asymptotic expansion we make careful use of the classical laplace method in the course of a discussion we observe that the notion of albanese maps associated with the abelian group action is closely related to the asymptotics a similar idea is available for asymptotics of the transition probability of a random walk on a lattice graph the results obtained in the present paper refine our previous ones 4 in the asymptotics the euclidean distance associated with the standard realization of the lattice graph which we call the albanese distance plays a crucial role

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

large deviation and the tangent cone at infinity of a crystal lattice we discuss a large deviation property of a periodic random walk on a crystal lattice in view of geometry and relate it to a rational convex polyhedron in the first homology group of a finite graph which as we shall observe has remarkable combinatorial features and shows up also in the gromov hausdorff limit of a crystal lattice

adaptive neural control for a class of switched nonlinear systems in this paper adaptive neural control is presented for a class of switched nonlinear systems with switching jumps and uncertainties in both system models and switching signals condition on limiting variation of the lyapunov function is given for input to state asymptotic stability of switched systems with switching jumps the control objective is achieved uniformly with respect to a class of switching signals the coupled difficulties from the discrepancy between control gains and switching jumps are overcome by discontinuous adaptive neural control combined with the classical adaptive control smooth approximations of the discontinuous controls are then presented for a systematic design procedure

process modelling and fault diagnosis using fuzzy neural networks abstract process modelling and fault diagnosis using fuzzy neural networks are studied in this paper in fuzzy network based modelling the process operation is partitioned into several fuzzy operating regions within each region a local model of simple form for example a low order linear model is used to model the process the global model output is obtained through the centre of gravity defuzzification which is essentially the interpolation of local model outputs a fuzzy neural network can be used to implement such a fuzzy model process knowledge is used to initially divide the process operation into several fuzzy operating regions and to set up the initial fuzzification layer weights process input output data are used to train the network through training membership functions of fuzzy operating regions are refined and local models are learnt this technique has been successfully applied to the modelling of ph dynamics in a continuous stirred tank reactor cstr the fuzzy network for fault diagnosis is obtained by adding a fuzzification layer to a conventional feed forward network the fuzzification layer converts the increments in on line measurements and controller outputs into three fuzzy sets increase steady and decrease the following layers then classify the symptoms represented by these fuzzy sets into various fault categories the fuzzification layer can compress training data and thereby ease training effort robustness of the diagnosis system is enhanced by adopting a fuzzy approach in representing abnormalities in the process the technique has been successfully applied to a cstr system

comparison of some neural network and scattered data approximations the inverse manipulator kinematics example this paper compares the application of five different methods for the approximation of the inverse kinematics of a manipulator arm from a number of joint angle cartesian coordinate training pairs the first method is a standard feedforward neural network with error backpropagation learning the next two methods are derived from an extended kohonen map algorithm that we combine with shepard interpolation for the forward computation we compare the method of ritter et al for the learning of the extended kohonen map to our own scheme based on gradient descent optimization we also study three scattered data approximation algorithms they include two variants of the radial basis function rbf method hardy s multiquadrics and gaussian rbf we further develop our own local polynomial fit method that could be considered as a modification of mclain s method we propose extensions to the considered scattered data approximation algorithms to make them suitable for vector valued multivariable functions such as the mapping of cartesian coordinates into joint angle coordinates

discrete time model reference adaptive control of nonlinear dynamical systems using neural networks a multilayer discrete time neural net nn controller is presented for the direct model reference adaptive control of a class of multi input multi output mim0 nonlinear dynamical systems the nonlinear dynamical system is assumed to be controllable and its state vector is available for measurement the nn controller exhibits learning while functioning features instead of learning then control so that control is immediate with no explicit learning phase needed the tracking error between the output of a nonlinear plant and an ideal linear model converges within a very short time persistence of excitation pe is not needed linearity in the parameters is not required and certainty equivalence is not used this overcomes several limitations of standard adaptive control the novel weight tuning paradigm for the nn controller is based on the well known delta rule but includes a modification to the learning rate parameter plus a correction term it guarantees tracking as well as bounded nn weights in non i

robust h control of multiple time delay uncertain nonlinear system using fuzzy model and adaptive neural network in this paper a robust control method which combines fuzzy model based control with adaptive neural network control is presented for a class of uncertain nonlinear system with multiple time delay the fuzzy t s model with multiple time delay is adopted for the approximate modeling of the nonlinear system with some unknown uncertainties and fuzzy model based h control law is designed by means of lmi method a full adaptive rbf neural network is added to the fuzzy h control in order to guarantee the robust stability of the controlled system the effect of the unknown uncertainties and the error caused by fuzzy modeling can be overcome by adaptive tuning of the weights centers and widths of the rbf neural network on line and no constraint or matching conditions are required the stability of the designed closed loop system is thus proved the proposed method is applied to a multiple time delay nonlinear chaotic system and the simulation results show that the proposed method cannot only stabilize the chaos systems but has strong robustness against uncertainties and external disturbance

a general fuzzy scheme for nonlinear processes with stability analysis abstract in this paper we develop a general fuzzy control scheme for nonlinear processes assuming little knowledge about the dynamics of the controlled process the proposed scheme starts by probing the process at different points in its operating region to generate a fuzzy quantisation a simple local controller is then designed at each fuzzy locality a fuzzy inference mechanism then links up tje local controllers to form a global controller which can be further refined by the learning algorithm by employing a newly developed structure adaptive fuzzy modelling scheme the appropriate fuzzy rule base for the inference mechanism can be extracted stably and efficiently the conditions for the stability of the global controller are rigourously established simulation results are presented to illustrate the effectiveness of the scheme

completion and continuation of nonlinear traffic time series a probabilistic approach when dealing with nonlinear time series of car traffic on highways one of the outstanding problems to be solved is completion and continuation of data in space and time to this end the underlying process is decomposed into stochastic and deterministic components the former is approximated by gaussian white noise while the latter refers apart from always existing trends to the space and time dependent jam propagation process jams are modelled in terms of dynamical bayesian networks with radial basis functions involved the models developed are used to tackle travel time estimation and prediction results are obtained for one of the most crowded traffic areas of europe namely the ring like highway around cologne

direct adaptive control of partially known nonlinear systems a direct adaptive control strategy for a class of single input single output nonlinear systems is presented the major advantage of the proposed method is that a detailed dynamic nonlinear model is not required for controller design the only information required about the plant is measurements of the state variables the relative degree and the sign of a lie derivative which appears in the associated input output linearizing control law unknown controller functions are approximated using locally supported radial basis functions that are introduced only in regions of the state space where the closed loop system actually evolves lyapunov stability analysis is used to derive parameter update laws which ensure under certain assumptions the state vector remains bounded and the plant output asymptotically tracks the output of a linear reference model the technique is successfully applied to a nonlinear biochemical reactor model

improved data visualisation through nonlinear dissimilarity modelling inherent to state of the art dimension reduction algorithms is the assumption that global distances between observations are euclidean despite the potential for altogether non euclidean data manifolds we demonstrate that a non euclidean manifold chart can be approximated by implementing a universal approximator over a dictionary of dissimilarity measures building on recent developments in the field this approach is transferable across domains such that observations can be vectors distributions graphs and time series for instance our novel dissimilarity learning method is illustrated with four standard visualisation datasets showing the benefits over the linear dissimilarity learning approach

weighted gradient enhanced kriging for high dimensional surrogate modeling and design optimization a novel formulation of gradient enhanced surrogate model called weighted gradient enhanced kriging is proposed and used in combination with the cheap gradients obtained by the adjoint method to a

fast and efficient second order method for training radial basis function networks this paper proposes an improved second order iso algorithm for training radial basis function rbf networks besides the traditional parameters including centers widths and output weights the input weights on the connections between input layer and hidden layer are also adjusted during the training process more accurate results can be obtained by increasing variable dimensions initial centers are chosen from training patterns and other parameters are generated randomly in limited range taking the advantages of fast convergence and powerful search ability of second order algorithms the proposed iso algorithm can normally reach smaller training testing error with much less number of rbf units during the computation process quasi hessian matrix and gradient vector are accumulated as the sum of related sub matrices and vectors respectively only one jacobian row is stored and used for multiplication instead of the entire jacobian matrix storage and multiplication memory reduction benefits the computation speed and allows the training of problems with basically unlimited number of patterns several practical discrete and continuous classification problems are applied to test the properties of the proposed iso training algorithm

a hybrid constructive algorithm for single layer feedforward networks learning single layer feedforward networks slfns have been proven to be a universal approximator when all the parameters are allowed to be adjustable it is widely used in classification and regression problems the slfn learning involves two tasks determining network size and training the parameters most current algorithms could not be satisfactory to both sides some algorithms focused on construction and only tuned part of the parameters which may not be able to achieve a compact network other gradient based optimization algorithms focused on parameters tuning while the network size has to be preset by the user therefore trial and error approach has to be used to search the optimal network size because results of each trial cannot be reused in another trial it costs much computation in this paper a hybrid constructive hc algorithm is proposed for slfn learning which can train all the parameters and determine the network size simultaneously at first by combining levenberg marquardt algorithm and least square method a hybrid algorithm is presented for training slfn with fixed network size then with the hybrid algorithm an incremental constructive scheme is proposed a new randomly initialized neuron is added each time when the training entrapped into local minima because the training continued on previous results after adding new neurons the proposed hc algorithm works efficiently several practical problems were given for comparison with other popular algorithms the experimental results demonstrated that the hc algorithm worked more efficiently than those optimization methods with trial and error and could achieve much more compact slfn than those construction algorithms

improved time series prediction with a new method for selection of model parameters a new method for model selection in prediction of time series is proposed apart from the conventional criterion of minimizing rms error the method also minimizes the error on the distribution of singularities evaluated through the local holder estimates and its probability density spectrum predictions of two simulated and one real time series have been done using kernel principal component regression kpcr and model parameters of kpcr have been selected employing the proposed as well as the conventional method results obtained demonstrate that the proposed method takes into account the sharp changes in a time series and improves the generalization capability of the kpcr model for better prediction of the unseen test data

layered neural networks with gaussian hidden units as universal approximations a neural network with a single layer of hidden units of gaussian type is proved to be a universal approximator for real valued maps defined on convex compact sets of rn

admissibility of linear estimators with respect to inequality constraints under matrix loss function in this paper we investigate the admissibility of linear estimators in the multivariate linear model with respect to inequality constraints under matrix loss function the necessary and sufficient conditions for a linear estimator to be admissible in the class of homogeneous linear estimators and the class of inhomogeneous linear estimators are obtained respectively

standard realizations of crystal lattices via harmonic maps an eells sampson type theorem for harmonic maps from a finite weighted graph is employed to characterize the equilibrium configurations of crystals it is thus observed that the mimimum principle frames symmetry of crystals

random walks on periodic graphs this paper concerns random walks on periodic graphs embedded in the d dimensional euclidian space r d and obtains asymptotic expansions of the green functions of them up to the second order term which expressed fairly explicitly are easily computable for many examples the result is used to derive an asymptotic form of the hitting distribution of a hyperplane of codimension one which involves not only the first but also second order terms of the expansion of the green function we also give similar expansions of the transition probabilities of the walks

admissibility of linear estimators in multivariate linear models with respect to inequality constraints this paper studies the admissibility of both homogeneous and inhomogeneous linear estimators in multivariate linear models with respect to inequality constraints the relationship between homogeneous and inhomogeneous admissible linear estimators with respect to two different inequality constraints is obtained

admissible linear estimators in linear models with respect to inequality constraints abstract admissibility of linear estimators is characterized in linear models e y x d y v with an unknown multidimensional parameter v varying in the cartesian product c v where c is a halfspace and v is a given set of nonnegative definite symmetric matrices the relation between admissibility of inhomogeneous and homogeneous linear estimators is discussed and some sufficient and necessary conditions for admissibility of an inhomogeneous linear estimator are given some results were extended to the case where c is a given polyhedral convex cone

svm multiregression for nonlinear channel estimation in multiple input multiple output systems this paper addresses the problem of multiple input multiple output mimo frequency nonselective channel estimation we develop a new method for multiple variable regression estimation based on support vector machines svms a state of the art technique within the machine learning community for regression estimation we show how this new method which we call m svr can be efficiently applied the proposed regression method is evaluated in a mimo system under a channel estimation scenario showing its benefits in comparison to previous proposals when nonlinearities are present in either the transmitter or the receiver sides of the mimo system

long time asymptotics of non symmetric random walks on crystal lattices abstract in the present paper we study long time asymptotics of non symmetric random walks on crystal lattices from a view point of discrete geometric analysis due to kotani and sunada 11 25 we observe that the euclidean metric associated with the standard realization of the crystal lattice called the albanese metric naturally appears in the asymptotics in the former half of the present paper we establish two kinds of functional central limit theorems for random walks we first show that the brownian motion on the euclidean space with the albanese metric appears as the scaling limit of the usual central limit theorem for the random walk next we introduce a family of random walks which interpolates between the original non symmetric random walk and the symmetrized one we then capture the brownian motion with a constant drift of the asymptotic direction on the euclidean space with the albanese metric associated with the symmetrized random walk through another kind of central limit theorem for the family of random walks in the latter half of the present paper we give a spectral geometric proof of the asymptotic expansion of the n step transition probability for the non symmetric random walk this asymptotic expansion is a refinement of the local central limit theorem obtained by sunada 22 23 and is a generalization of the result in 11 for symmetric random walks on crystal lattices to non symmetric cases

design of asymptotic estimators an approach based on neural networks and nonlinear programming a methodology to design state estimators for a class of nonlinear continuous time dynamic systems that is based on neural networks and nonlinear programming is proposed the estimator has the structure of a luenberger observer with a linear gain and a parameterized in general nonlinear function whose argument is an innovation term representing the difference between the current measurement and its prediction the problem of the estimator design consists in finding the values of the gain and of the parameters that guarantee the asymptotic stability of the estimation error toward this end if a neural network is used to take on this function the parameters i e the neural weights are chosen together with the gain by constraining the derivative of a quadratic lyapunov function for the estimation error to be negative definite on a given compact set it is proved that it is sufficient to impose the negative definiteness of such a derivative only on a suitably dense grid of sampling points the gain is determined by solving a lyapunov equation the neural weights are searched for via nonlinear programming by minimizing a cost penalizing grid point constraints that are not satisfied techniques based on low discrepancy sequences are applied to deal with a small number of sampling points and hence to reduce the computational burden required to optimize the parameters numerical results are reported and comparisons with those obtained by the extended kalman filter are made

higher order statistics based radial basis function networks for signal enhancement in this paper a higher order statistics hos based radial basis function rbf network for signal enhancement is introduced in the proposed scheme higher order cumulants of the reference signal were used as the input of hos based rbf an hos based supervised learning algorithm with mean square error obtained from higher order cumulants of the desired input and the system output as the learning criterion was used to adapt weights the motivation is that the hos can effectively suppress gaussian and symmetrically distributed non gaussian noise the influence of a gaussian noise on the input of hos based rbf and the hos based learning algorithm can be mitigated simulated results indicate that hos based rbf can provide better performance for signal enhancement under different noise levels and its performance is insensitive to the selection of learning rates moreover the efficiency of hos based rbf under the nonstationary gaussian noise is stable

asymptotic behavior of the transition probability of a random walk on an infinite graph ideas cultivated in spectral geometry are applied to obtain an asymptotic property of a reversible random walk on an infinite graph satisfying a certain periodic condition in the course of our argument we employ perturbation theory for the maximal eigenvalues of twisted transition operator as a result an asymptotic of the probabilityp n x y that a particle starting atxreachesyat timenasngoes to infinity is established

existence and uniqueness results for neural network approximations some approximation theoretic questions concerning a certain class of neural networks are considered the networks considered are single input single output single hidden layer feedforward neural networks with continuous sigmoidal activation functions no input weights but with hidden layer thresholds and output layer weights specifically questions of existence and uniqueness of best approximations on a closed interval of the real line under mean square and uniform approximation error measures are studied a by product of this study is a reparametrization of the class of networks considered in terms of rational functions of a single variable this rational reparametrization is used to apply the theory of pade approximation to the class of networks considered in addition a question related to the number of local minima arising in gradient algorithms for learning is examined