Control of chaotic dynamical systems using radial basis function network approximators

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.     

Scale-based clustering using the radial basis function network

This paper shows how scale-based clustering can be done using the radial basis function network (RBFN), with the RBF width as the scale parameter and a dummy target as the desired output. The technique suggests the "right" scale at which the given data set should be clustered, thereby providing a solution to the problem of determining the number of RBF units and the widths required to get a good network solution. The network compares favorably with other standard techniques on benchmark clustering examples. Properties that are required of non-Gaussian basis functions, if they are to serve in alternative clustering networks, are identified. This work, on the whole, points out an important role played by the width parameter in RBFN, when observed over several scales, and provides a fundamental link to the scale space theory developed in computational vision.     

Multiplication-free radial basis function network

For the purpose of adaptive function approximation, a new radial basis function network is proposed which is nonlinear in its parameters. The goal is to reduce significantly the computational effort for a serial processor, by avoiding multiplication in both the evaluation of the function model and the computation of the parameter adaptation. The approximation scheme makes use of a grid-based Gaussian basis function network. Due to the local support of digitally implemented Gaussian functions the function representation is parametric local and therefore well suited for an online implementation on a microcomputer. A gradient descent based nonlinear learning algorithm is presented and the convergence of the algorithm is proved.     

Sequential approximate multi-objective optimization using radial basis function network

In industrial design optimization, objectives and constraints are generally given as implicit form of the design variables, and are evaluated through computationally intensive numerical simulation. Under this situation, response surface methodology is one of helpful approaches to design optimization. One of these approaches, known as sequential approximate optimization (SAO), has gained its popularity in recent years. In SAO, the sampling strategy for obtaining a highly accurate global minimum remains a critical issue. In this paper, we propose a new sampling strategy using sequential approximate multi-objective optimization (SAMOO) in radial basis function (RBF) network. To identify a part of the pareto-optimal solutions with a small number of function evaluations, our proposed sampling strategy consists of three phases: (1) a pareto-optimal solution of the response surfaces is taken as a new sampling point; (2) new points are added in and around the unexplored region; and (3) other parts of the pareto-optimal solutions are identified using a new function called the pareto-fitness function. The optimal solution of this pareto-fitness function is then taken as a new sampling point. The upshot of this approach is that phases (2) and (3) add sampling points without solving the multi-objective optimization problem. The detailed procedure to construct the pareto-fitness function with the RBF network is described. Through numerical examples, the validity of the proposed sampling strategy is discussed.     

Adaptive radial basis function neural network control with variable variance parameters

The paper presents a direct adaptive control architecture for a class of nonlinear dynamic systems, which are either ill defined or rather complex. The direct adaptive architecture employs radial basis function (RBF) neural network (NN) systems to reconstruct the ideal feedback linearization control. With the modified adaptation algorithm proposed herein, the on-line function approximation capability of the RBF NN a system is enhanced to remove the auxiliary control term and switching element in a conventional RBF-NN-based controller; simultaneously, the tracking performance is upgraded. Global asymptotic stability of the on-line algorithm is established in the Lyapunov sense to guarantee that the tracking error can converge to a small neighbourhood of the origin. Simulation validations for an inverted pendulum system are finally performed to verify the effectiveness of the proposed controller and the theoretical discussion.     

Blind equalization using a predictive radial basis function neural network

In this paper, we propose a novel blind equalization approach based on radial basis function (RBF) neural networks. By exploiting the short-term predictability of the system input, a RBF neural net is used to predict the inverse filter output. It is shown here that when the prediction error of the RBF neural net is minimized, the coefficients of the inverse system are identical to those of the unknown system. To enhance the identification performance in noisy environments, the improved least square (ILS) method based on the concept of orthogonal distance to reduce the estimation bias caused by additive measurement noise is proposed here to perform the training. The convergence rate of the ILS learning is analyzed, and the asymptotic mean square error (MSE) of the proposed predictive RBF identification method is derived theoretically. Monte Carlo simulations show that the proposed method is effective for blind system identification. The new blind technique is then applied to two practical applications: equalization of real-life radar sea clutter collected at the east coast of Canada and deconvolution of real speech signals. In both cases, the proposed blind equalization technique is found to perform satisfactory even when the channel effects and measurement noise are strong.     

Median radial basis function neural network

Radial basis functions (RBFs) consist of a two-layer neural network, where each hidden unit implements a kernel function. Each kernel is associated with an activation region from the input space and its output is fed to an output unit. In order to find the parameters of a neural network which embeds this structure we take into consideration two different statistical approaches. The first approach uses classical estimation in the learning stage and it is based on the learning vector quantization algorithm and its second-order statistics extension. After the presentation of this approach, we introduce the median radial basis function (MRBF) algorithm based on robust estimation of the hidden unit parameters. The proposed algorithm employs the marginal median for kernel location estimation and the median of the absolute deviations for the scale parameter estimation. A histogram-based fast implementation is provided for the MRBF algorithm. The theoretical performance of the two training algorithms is comparatively evaluated when estimating the network weights. The network is applied in pattern classification problems and in optical flow segmentation.     

Online identification using radial basis function neural network coupled with KPCA

In this paper, we are going to propose an online radial basis function (RBF) neural network algorithm without any preprocessing step. Then a kernel principal component analysis (KPCA) is coupled with the proposed online RBF neural network algorithm. Indeed, the KPCA method is used as a preprocessing step to reduce the feature dimension which fed to the RBF neural network. Reducing memory requirements of the models makes RBF neural network training efficient and fast. These two proposed algorithms are applied, with success, for identification of a mobile robot position. The simulation results present that the used sigmoid function as a kernel, compared to other kernel functions, which gives an excellent model and a minimum mean square error.     

Self-organizing radial basis function network for real-time approximation of continuous-time dynamical systems

Real-time approximators for continuous-time dynamical systems with many inputs are presented. These approximators employ a novel self-organizing radial basis function (RBF) network, which varies its structure dynamically to keep the prescribed approximation accuracy. The RBFs can be added or removed online in order to achieve the appropriate network complexity for the real-time approximation of the dynamical systems and to maintain the overall computational efficiency. The performance of this variable structure RBF network approximator with both Gaussian RBF (GRBF) and raised-cosine RBF (RCRBF) is analyzed. The compact support of RCRBF enables faster training and easier output evaluation of the network than that of the network with GRBF. The proposed real-time self-organizing RBF network approximator is then employed to approximate both linear and nonlinear dynamical systems to illustrate the effectiveness of our proposed approximation scheme, especially for higher order dynamical systems. The uniform ultimate boundedness of the approximation error is proved using the second method of Lyapunov.     

Identification of nonlinear discrete-time systems using raised-cosine radial basis function networks

An effective technique for identifying nonlinear discrete-time systems using raised-cosine radial basis function (RBF) networks is presented. Raised-cosine RBF networks are bounded-input bounded-output stable systems, and the network output is a continuously differentiable function of the past input and the past output. The evaluation speed of an n-dimensional raised-cosine RBF network is high because, at each discrete time, at most 2ⁿ RBF terms are nonzero and contribute to the output. As a consequence, raised-cosine RBF networks can be used to identify relatively high-order nonlinear discrete-time systems. Unlike the most commonly used RBFs, the raised-cosine RBF satisfies a constant interpolation property. This makes raised-cosine RBF highly suitable for identifying nonlinear systems that undergo saturation effects. In addition, for the important special case of a linear discrete-time system, a first-order raised-cosine RBF network is exact on the domain over which it is defined, and it is minimal in terms of the number of distinct parameters that must be stored. Several examples, including both physical systems and benchmark systems, are used to illustrate that raised-cosine RBF networks are highly effective in identifying nonlinear discrete-time systems.     

Adaptive control of underwater vehicle-manipulator systems using radial basis function networks

This paper deals with a control scheme for underwater vehicle-manipulator systems with the dynamics of thrusters in the presence of uncertainties in system parameters. We have developed two controllers that overcome thruster nonlinearities, which cause an uncontrollable system: one is a regressor-based adaptive controller and the other is a robust controller. However, the structure of the adaptive controller is very complex due to the feedforward terms including the regressors of dynamic system models, and the error feedback gains of the robust controller with a good control performance are excessively high due to the lack of feedforward terms. In this paper we develop an adaptive controller that uses radial basis function networks instead of the feedforward terms. The replacement leads to a moderately high gain controller whose structure is simpler than that of the regressor-based adaptive controller.     

On modeling microscopic vehicle fuel consumption using radial basis function neural network

Energy conservation is one of the central challenges for the transportation system today. A variety of microscopic vehicle fuel consumption models have been developed to support eco-friendly transport strategies. However, most existing models are regression based and are sensitive to the vehicle-specific parameters and the operating conditions, therefore, an expensive and time-consuming calibration procedure is always indispensable in these models’ application. In this paper, we propose an artificial neural network-based model to avoid the calibration problem. The main works include: (1) collect extensive field datasets such as large-scale controller area network bus to reflect the local transportation environment’s fuel consumption characteristics; (2) conduct correlational analysis to identify the key fuel consumption influence factors; (3) develop a radial basis function neural network-based learning model to capture the nonlinear relationship between the key factors and the corresponding fuel consumption values based on the collected training datasets. The proposed model can give a reasonable prediction of instantaneous fuel consumption without calibration. The effectiveness of the proposed model is validated from a combination of both in-lab and field experiments.     

基于径向基函数神经网络的网络流量识别模型

提出了一种基于径向基函数神经网络的网络流量识别方法。根据实际网络中的流量数据,建立了一个基于RBF神经网络的流量识别模型。先介绍了RBF神经网络的结构设计及学习算法,针对RBF神经网络在隐节点过多的情况下算法过于复杂的缺点,采用了优化的算法计算隐含层节点。仿真实验证明,该模型具有较好的准确率、低复杂度、高识别效果和良好的自适应性。     

Construction of a radial basis function network using an evolutionary algorithm for grade estimation in a placer gold deposit

This paper presents a study highlighting the predictive performance of a radial basis function (RBF) network in estimating the grade of an offshore placer gold deposit. In applying the radial basis function network to grade estimation of the deposit, several pertinent issues regarding RBF model construction are addressed in this study. One of the issues is the selection of the RBF network along with its center and width parameters. Selection was done by an evolutionary algorithm that utilizes the concept of cooperative coevolutions of the RBFs and the associated network. Furthermore, the problem of data division, which arose during the creation of the training, calibration and validation of data sets for the RBF model development, was resolved with the help of an integrated approach of data segmentation and genetic algorithms (GA). A simulation study conducted showed that nearly 27% of the time, a bad data division would result if random data divisions were adopted in this study. In addition, the efficacy of the RBF network was tested against a feed-forward network and geostatistical techniques. The outcome of this comparative study indicated that the RBF model performed decisively better than the feed-forward network and the ordinary kriging (OK).     

Flank wear prediction in drilling using back propagation neural network and radial basis function network

In the present work, two different types of artificial neural network (ANN) architectures viz. back propagation neural network (BPNN) and radial basis function network (RBFN) have been used in an attempt to predict flank wear in drills. Flank wear in drill depends upon speed, feed rate, drill diameter and hence these parameters along with other derived parameters such as thrust force, torque and vibration have been used to predict flank wear using ANN. Effect of using increasing number of sensors in the efficacy of predicting drill wear by using ANN has been studied. It has been observed that inclusion of vibration signal along with thrust force and torque leads to better prediction of drill wear. The results obtained from the two different ANN architectures have been compared and some useful conclusions have been made.     

Non-linear systems identification using radial basis functions

This paper investigates the identification of discrete-time non-linear systems using radial basis functions. A forward regression algorithm based on an orthogonal decomposition of the regression matrix is employed to select a suitable set of radial basis function centers from a large number of possible candidates and this provides, for the first time, fully automatic selection procedure for identifying parsimonious radial basis function models of structure-unknown non-linear systems. The relationship between neural networks and radial basis functions is discussed and the application of the algorithms to real data is included to demonstrate the effectiveness of this approach.     

Design of compliant mechanism using hyper radial basis function network and reanalysis formulation

The design of compliant mechanisms often involves multiple minima. Since non-gradient type global optimization techniques cannot handle as many design variables as the commonly used, gradient based SIMP method, a hyper radial basis function network is used in this paper for Particle Swarm Optimization. An exact reanalysis formulation is presented which computes the strain energy from the solution of the mutual energy problem. The design of motion inverter is presented to demonstrate the proposed approach. The use of hyper radial basis function network results in checkerboard-free, smooth topology using very crude finite element analysis mesh, whereas the reanalysis formulation cuts down the analysis time into half. Thus the design process is made computationally efficient and less dependent on human interpretation of the optimal topology.     

Position control of a mobile inverted pendulum system using radial basis function network

This article presents the implementation of position control of a mobile inverted pendulum (MIP) system by using the radial basis function (RBF) network. The MIP has two wheels to move on the plane and to balance the pendulum. The MIP is a nonlinear system whose dynamics is nonholonomic. The goal of this study was to control the MIP to maintain the balance of the pendulum while tracking a desired position of the cart. The reference compensation technique scheme is used as a neural network control method for the MIP. The back-propagation learning algorithm of the RBF network is derived for online learning and control. The control algorithm has been embedded on a DSP 2812 board to achieve real-time control. Experimental results are conducted and show successful control performances of both balancing and tracking the desired position of the MIP.     

Prediction of reservoir sensitivity using RBF neural network with trainable radial basis function

Reservoir sensitivity prediction is an important basis for designing reservoir protection program scientifically and exploiting oil and gas resources efficiently. Researchers have long endeavored to establish a method to predict reservoir sensitivity, but all of the methods have some limitations. Radial basis function (RBF) neural network, which provided a powerful technique to model non-linear mapping and the learning algorithm for RBF neural networks, corresponds to the solution of a linear problem, therefore it is unnecessary to establish an accurate model or organize rules in large number, and it enjoys the advantages such as simple network structure, fast convergence rate, and strong approximation ability, etc. However, different radial basis function has different non-linear mapping ability, and different data require different radial basis functions. Nowadays, the choice of radial basis function in the network is based on experience or test result only, which exerts a great adverse impact on the network performance. In this study, a new RBF neural network with trainable radial basis function was proposed by the linear combination of common radial basis functions. The input parameters of the network were the influence factors of reservoir sensitivity such as porosity and permeability, etc. The output parameter was the corresponding sensitivity index. The network was trained and tested with the data collected from our own experiments. The results showed that the new RBF neural network is effective and improved, of which the accuracy is obviously higher than the network with single radial basis function for the prediction of reservoir sensitivity.     

用蛙跳算法优化RBF神经网络参数的研究

针对径向基函数（Radial Basis Functions,RBF）神经网络结构参数确定问题,提出了一种基于蛙跳算法优化RBF神经网络参数的新方法。将RBF神经网络参数组成一个多维向量,作为蛙跳算法中的参数进行优化。以适应度函数为标准,在可行解空间中搜索最优解,并对蛙跳算法进行了改进。非线性函数逼近实验结果表明,该优化算法相对标准遗传优化算法、粒子群优化算法有较小的均方误差,具有更好的逼近能力。