Abstract: An automatic tuner for control systems that produces as output values for parameters of an arbitrary controller. The controller is in a control loop so that its output effects changes in actuators and regulates a physical process. The controller consists of either linear and nonlinear controller components or a combination of both. The tuner has a nonlinear approximator that has been optimized off-line. The off-line optimization is done without supervised learning so that desired outputs of neural network do not need to be available, and separate optimization to generate the desired outputs is not necessary. The off-line optimization can also rely on optimization criteria that are arbitrary: differentiability, convexity, even continuity of criteria are not required. The off-line optimization ensures robustness of generated controller parameters so that the input process characteristics do not need to be highly accurate. The off-line optimization is performed in such a way that the input parameters that relate to desired closed-loop system behavior include robustness parameters that can be used to effect tradeoffs between robust and nominal performance. The inputs to the nonlinear approximator consist of two sets of input parameters, either of which may be empty. A first set of input parameters can include parameters that relate to process characteristics. A second set of input parameters can include parameters that relate to desired closed-loop system behavior. The output values may be proportional and/or integral and/or derivative gains for PID-like controllers. The output values otherwise may be parameters for delay-compensation controllers, parameters for controllers that consist of lead-lag terms in combination with PID controllers, parameters for higher-order linear controllers, discrete variables that select between different control structures, or parameters for nonlinear controllers of predetermined structure. The nonlinear approximator may be implemented as a compositional sigmoidal mapping, a multilayer perception structure, a fuzzy logic model, a radial basis function network, a polynomial expansion, or other parametrized nonlinear structure.

Abstract: This paper investigates the application of a radial basis function (RBF) neural network to the prediction of field strength based on topographical and morphographical data. The RBF neural network is a two-layer localized receptive field network whose output nodes from a combination of radial activation functions computed by the hidden layer nodes. Appropriate centers and connection weights in the RBF network lead to a network that is capable of forming the best approximation to any continuous nonlinear mapping up to an arbitrary resolution. Such an approximation introduces best nonlinear approximation capability into the prediction model in order to accurately predict propagation loss over an arbitrary environment based on adaptive learning from measurement data. The adaptive learning employs hybrid competitive and recursive least squares algorithms. The unsupervised competitive algorithm adjusts the centers while the recursive least squares (RLS) algorithm estimates the connection weights. Because these two learning rules are both linear, rapid convergence is guaranteed. This hybrid algorithm significantly enhances the real-time or adaptive capability of the RBF-based prediction model. The applications to Okumura's (1968) data are included to demonstrate the effectiveness of the RBF neural network approach.

Abstract: A recursive orthogonal least squares (ROLS) algorithm for multi-input, multi-output systems is developed in this paper and is applied to updating the weighting matrix of a radial basis function network. An illustrative example is given, to demonstrate the effectiveness of the algorithm for eliminating the effects of ill-conditioning in the training data, in an application of neural modelling of a multi-variable chemical process. Comparisons with results from using standard least squares algorithms, in batch and recursive form, show that the ROLS algorithm can significantly improve the neural modelling accuracy. The ROLS algorithm can also be applied to a large data set with much lower requirements on computer memory than the batch OLS algorithm.

Abstract: This paper considers a parametric nonlinear least square (NLS) optimization problem. Unlike a classical NLS problem statement, we assume that a nonlinear optimized system depends on two arguments: an input vector and a parameter vector. The input vector can be modified to optimize the system, while the parameter vector changes from one optimization iteration to another and is not controlled. The optimization process goal is to find a dependence of the optimal input vector on the parameter vector, where the optimal input vector minimizes a quadratic performance index. The paper proposes an extension of the Levenberg-Marquardt algorithm for a numerical solution of the formulated problem. The proposed algorithm approximates the nonlinear system in a vicinity of the optimum by expanding it into a series of parameter vector functions, affine in the input vector. In particular, a radial basis function network expansion is considered. The convergence proof for the algorithm is presented. The proposed approach is applied to task-level learning control of a two-link flexible arm. Each evaluation of the system in the optimization process means completing a controlled motion of the arm.

Abstract: Incremental Net Pro (IncNet Pro) with local learning feature and statistically controlled growing and pruning of the network is introduced. The architecture of the net is based on RBF networks. Extended Kalman Filter algorithm and its new fast version is proposed and used as learning algorithm. IncNet Pro is similar to the Resource Allocation Network described by Platt in the main idea of the expanding the network. The statistical novel criterion is used to determine the growing point. The Bi-radial functions are used instead of radial basis functions to obtain more flexible network.

Abstract: This paper presents an axiomatic approach for building RBF neural networks and also proposes a supervised learning algorithm based on gradient descent for their training. This approach results in a broad variety of admissible RBF models, including those employing Gaussian radial basis functions. The form of the radial basis functions is determined by a generator function. A sensitivity analysis explains the failure of gradient descent learning on RBF networks with Gaussian radial basis functions, which are generated by an exponential generator function. The same analysis verifies that RBF networks generated by a linear generator function are much more suitable for gradient descent learning. Experiments involving such RBF networks indicate that the proposed gradient descent algorithm guarantees fast learning and very satisfactory generalization ability.

Abstract: This paper proposes a united training method of the link weights of the Gaussian radial basis function networks (GRBFN) and the shape parameter α of the RBF. The training method corresponding to the former is a kind of recursive least squares backpropagation (RLS-BP) learning algorithm which is an accurately recursive method, the training method corresponding to the latter is an adaptive gradient descending (AGD) searching algorithm which is an approximately approaching method. We use the one-dimensional images of radar targets to study the effect of the shape parameter α on the rate of recognition, and survey the changes of the shape parameter αs of radial basis functions corresponding to different hidden nodes, and present the judgement confidence curves of different radar targets. In addition, the forgotten factor λ which makes the effects on the speed of convergence is also discussed. The experimental results are presented.

Abstract: The radial basis function (RBF) network is an efficient function approximator. Theoretical researches focus on the capabilities of the network to reach an optimal solution. Unfortunately, few results concerning the design, and training of the network are available. When dealing with a specific application, the performances of the network dramatically depend on the number of neurons and on the distribution of the hidden neurons in the input space. Generally, the network resulting from learning applied to a predetermined architecture, is either insufficient or over-complicated. In this study, we focus on genetic learning for the RBF network applied to prediction of chaotic time series. The centers and widths of the hidden layer neurons basis function-defined as the barycenter and distance between two input patterns-are coded into a chromosome. It is shown that the basis functions which are also coded as a parameter of the neurons provide an additional degree of freedom resulting in a smaller optimal network. A direct matrix inversion provides the weights between the hidden layer and the output layer and avoids the risk of getting stuck into a local minimum. The performances of a network with Gaussian basis functions is compared with those of a network with genetic determination of the basis functions on the Mackey-Glass delay differential equation.

Abstract: On-line learning is examined for the radial basis function network, an important and practical type of neural network. The evolution of generalization error is calculated within a framework which allows the phenomena of the learning process, such as the specialization of the hidden units, to be analyzed. The distinct stages of training are elucidated, and the role of the learning rate described. The three most important stages of training, the symmetric phase, the symmetry-breaking phase, and the convergence phase, are analyzed in detail; the convergence phase analysis allows derivation of maximal and optimal learning rates. As well as finding the evolution of the mean system parameters, the variances of these parameters are derived and shown to be typically small. Finally, the analytic results are strongly confirmed by simulations.

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Abstract: Radial basis functions can be combined into a network structure that has several advantages over conventional neural network solutions. However, to operate effectively the number and positions of the basis function centres must be carefully selected. Although no rigorous algorithm exists for this purpose, several heuristic methods have been suggested. In this paper a new method is proposed in which radial basis function centres are selected by the mean-tracking clustering algorithm. The mean-tracking algorithm is compared with k means clustering and it is shown that it achieves significantly better results in terms of radial basis function performance. As well as being computationally simpler, the mean-tracking algorithm in general selects better centre positions, thus providing the radial basis functions with better modelling accuracy.

Abstract: The dual reciprocity boundary element method traditionally uses the linear radial basis function r for interpolation Recently, however, the use of the r function has been questioned both in relation to accuracy and in relation to the number and position of internal nodes required to obtain satisfactory solutions Much research has been done in an attempt to fix criteria for choosing which approximation function should be used One of the alternatives recently suggested is the augmented thin plate spline function, which consists of a thin plate spline function, r2 log(r), augmented with the first three terms of a Pascal triangle expansion In this paper families of similar functions are obtained by augmenting radial basis functions with appropriate global expansions: these functions will be called hybrid approximate functions It will be shown that using an appropriate hybrid function accurate results can be obtained for many body forces or pseudo body forces in elasticity without the need for internal nodes© 1997 John Wiley & Sons, Ltd

Abstract: The effects of a neural network’s topology on its performance are well known, yet the question of finding optimal configurations automatically remains largely open. This paper proposes a solution to this problem for RBF networks. A self-optimising approach, driven by an evolutionary strategy, is taken. The algorithm uses output information and a computationally efficient approximation of RBF networks to optimise the K-means clustering process by co-evolving the two determinant parameters of the network’s layout: the number of centroids and the centroids’ positions. Empirical results demonstrate promise.

Abstract: A new approach to nonlinear Projection to Latent Structures (PLS) modelling using Radial Basis Function (RBF) neural networks to provide a nonlinear inner relationship is described, along with a hybrid optimisation technique for training the networks. Results are given showing an improvement in modelling performance over linear PLS for a variety of problems. An application of the technique to fault detection on a validated model of an industrial distillation plant is also demonstrated.

Abstract: Modeling data sets by mixtures is a common technique in many pattern recognition applications. The expectation maximization (EM) algorithm for mixture decomposition suffers from the disadvantage that the number of components in the mixture needs to be specified. In this paper, we propose a new objective function, the minimum of which gives the number of components automatically. The proposed method, known as the agglomerative Gaussian mixture decomposition algorithm, is then used to determine the number of hidden nodes in a radial basis function network. We present results on real data sets which indicate that the proposed method is not sensitive to initialization and gives better classification rates.

Abstract: A radial basis function network (RBFN) based power system stabilizer (PSS) is presented in this paper to improve the dynamic stability of multimachine power systems. The proposed RBFN is trained over a wide range of operating conditions in order to re-tune the parameters of the PSS in real-time. Time domain simulations of a multimachine power system with different operating conditions subject to a three phase fault are studied and investigated. The performance of the proposed RBFN PSS is compared to that of conventional power system stabilizer (CPSS). The results show the good damping characteristics of the proposed RBFN PSS over a wide range of operating conditions.

Abstract: A chaotic field generator is represented by a non-linear equation. Its generating function is modeled empirically by a statistical non-parametric estimator. The estimator corresponds to a radial basis function neural network which learns from a record of a field given in some initial domain to predict the field distribution elsewhere. The performance of the generator is demonstrated by prediction of a chaotic series and a regular as well as a chaotic surface.

Abstract: This paper proposes a framework for constructing and training growing radial basis function (GRBF) neural networks. The GRBF network grows in the process of training by splitting one of the prototypes that determine the locations of the radial basis functions. Two splitting criteria are proposed to determine which prototype to split at each growing cycle. The proposed hybrid learning scheme provides the framework for incorporating existing algorithms in the training of GRBF networks. A supervised learning scheme based on the minimization of the localized class-conditional variance is also proposed. GRBF neural networks are evaluated and tested on pattern classification applications with very satisfactory results.

Abstract: A new algorithm for general robust function approximation by an artificial neural network is presented. The basis for this work is Fritzke's supervised growing cell structures approach (1993) which combines supervised and unsupervised learning. It is extended by the capability of resampling the function under examination automatically, and by the definition of a new error measure which enables an accurate approximation of arbitrary goal functions.

Abstract: The paper presents a systematic approach for designing a radial basis function (RBF) network based adaptive power system stabilizer using orthogonal least squares learning algorithm. The training patterns are generated over a wide range in machine real/reactive power output and terminal voltage using linearized model of the system. Investigations reveal that the required number of RBF centers heavily depend on the spread factor /spl beta/ and tolerance expressed as sum of squared errors. Studies reveal that the dynamic performance of the system with radial basis function network adaptive power system stabilizer (RBFAPSS) is superior to that with a conventional PSS. Moreover, RBFAPSS provides optimum performance for a wide range in loading conditions and large perturbations.

Abstract: This paper considers a problem of bioreactor control, which is formulated in Anderson and Miller (1990) and Ungar (1990) as a benchmark problem for application of neural network-based adaptive control algorithms. A completely adaptive control of this strongly nonlinear system is achieved with no a priori knowledge of its dynamics. This becomes possible thanks to a novel architecture of the controller, which is based on an affine Radial Basis Function network approximation of the sampled-data system mapping. Approximation with such network could be considered as a generalization of a standard practice to linearize a nonlinear system about the working regime. As the network is affine in the control components, it can be inverted with respect to the control vector by using fast matrix computations. The considered approach includes several features, recently introduced in some advanced process control algorithms. These features-multirate sampling, on-line adaptation, and Radial Basis Function approximation of the system nonlinearity-are crucial for the achieved high performance of the controller.