Nonlinear black-box models in system identification: Mathematical foundations

We discuss several aspects of the mathematical foundations of the nonlinear black-box identification problem. We shall see that the quality of the identification procedure is always a result of a certain trade-off between the expressive power of the model we try to identify (the larger the number of parameters used to describe the model, the more flexible is the approximation), and the stochastic error (which is proportional to the number of parameters). A consequence of this trade-off is the simple fact that a good approximation technique can be the basis of a good identification algorithm. From this point of view, we consider different approximation methods, and pay special attention to spatially adaptive approximants. We introduce wavelet and ‘neuron’ approximations, and show that they are spatially adaptive. Then we apply the acquired approximation experience to estimation problems. Finally, we consider some implications of these theoretical developments for the practically implemented versions of the ‘spatially adaptive’ algorithms.     

Fuzzy logic techniques for mobile robot obstacle avoidance

This paper is concerned with the problem of reactive navigation for a mobile robot in an unknown clustered environment. We will define reactive navigation as a mapping between sensory data and commands. Building a reactive navigation system means providing such a mapping. It can come from a family of predefined functions (like potential fields methods) or it can be built using ‘universal’ approximators (like neural networks). In this paper, we will consider another ‘universal’ approximator: fuzzy logic. We will explain how to choose the rules using a behaviour decomposition approach. It is possible to build a controller working quite well but the classical problems are still there: oscillations and local minima. Finally, we will conclude that learning is necessary for a robust navigation system and fuzzy logic is an easy way to put some initial knowledge in the system to avoid learning from zero.     

Browsing a document collection represented in two- and three-dimensional virtual information space

This paper reports a study of information retrieval performance using an interface in which documents were represented by objects in a virtual environment. Spatial location was determined by semantic content, with inter-object distance representing semantic similarity of documents. The quality of spatial-semantic mapping was manipulated as was the number of dimensions (two versus three) in which document nodes were arranged. Participants were required to browse the information space and identify all documents relevant to a specified topic. Results indicated that participants were able to use three-dimensional spatial mapping of semantic information to facilitate task performance, with performance being better when the quality of the mapping was higher. Strategy differences were identified, with participants adopting a more ‘exhaustive’ approach when searching two-dimensional node arrangements, and a more ‘focused’ approach for three-dimensional arrangements. Cognitive ability was not strongly associated with task performance, but participants of relatively lower cognitive ability tended to out-perform those of higher cognitive ability in three-dimensional conditions. Possible reasons for these findings are discussed.     

Outsourcing strategy in two-stage call centers

This paper considers ‘two-stage’ call centers where some incoming calls are completed by first service while others require an additional second service. Although this type of call center is not uncommon, it has not been dealt with, if any, in the call center literature. In this paper, we introduce the concept of the ‘two-stage’ call center and discuss its features. Furthermore, we develop an effective outsourcing strategy in ‘two-stage’ call centers. To this end, we model ‘two-stage’ service system and propose several call routing structures. The structures are compared through numerical test and conventional queueing theories form the theoretical basis of our study.     

An adaptive neuro-fuzzy model for prediction of student’s academic performance

This paper introduces a systematic approach for the design of a fuzzy inference system based on a class of neural networks to assess the students’ academic performance. Fuzzy systems have reached a recognized success in several applications to solve diverse class of problems. Currently, there is an increasing trend to expand them with learning and adaptation capabilities through combinations with other techniques. Fuzzy systems-neural networks and fuzzy systems-genetic algorithms are the most successful applications of soft computing techniques with hybrid characteristics and learning capabilities. The developed method uses a fuzzy system augmented by neural networks to enhance some of its characteristics like flexibility, speed, and adaptability, which is called the adaptive neuro-fuzzy inference system (ANFIS). New trends in soft computing techniques, their applications, model development of fuzzy systems, integration, hybridization and adaptation are also introduced. The parameters set to facilitate the hybrid learning rules for the constitution of the Sugeno-type ANFIS architecture is then elaborated. The method can produce crisp numerical outcomes to predict the student’s academic performance (SAP). It also provides an alternative solution to deal with imprecise data. The results of the ANFIS model are as robust as those of the statistical methods, yet they encourage a more natural way to interpret the student’s outcomes.     

Soft computing applications in dynamic model identification of polymer extrusion process

This paper proposes the applications of soft computing to deal with the constraints in conventional modelling techniques of the dynamic extrusion process. The proposed technique increases the efficiency in utilising the available information during the model identification. The resultant model can be classified as a ‘grey-box model’ or has been termed as a ‘semi-physical model’ in the context. The extrusion process contains a number of parameters that are sensitive to the operating environment. Fuzzy rule-based system (FRBS) is introduced into the analytical model of extrusion by means of sub-models to approximate those operational-sensitive parameters. In drawing an optimal structure for each sub-model, a hybrid algorithm of genetic algorithm with fuzzy system (GA-fuzzy) has been implemented. The sub-models obtained show advantages such as linguistic interpretability, simpler rule-base and less membership functions (MFs). The developed model is adaptive with its learning ability through the steepest decent error back-propagation algorithm. This ability might help to minimise the deviation of the model prediction when the operational-sensitive parameters adapt to the changing operating environment in the real situation. The model is first evaluated through simulations on the consistency of model prediction with the theoretical analysis. Then, the usefulness of adaptive sub-models during the operation is further explored in existence of prediction error.     

Computational intelligence approaches and linear models in case studies of forecasting exchange rates

Artificial neural networks and fuzzy systems, have gradually established themselves as a popular tool in approximating complicated nonlinear systems and time series forecasting. This paper investigates the hypothesis that the nonlinear mathematical models of multilayer perceptron and radial basis function neural networks and the Takagi–Sugeno (TS) fuzzy system are able to provide a more accurate out-of-sample forecast than the traditional auto regressive moving average (ARMA) and ARMA generalized auto regressive conditional heteroskedasticity (ARMA-GARCH) linear models. Using series of Brazilian exchange rate (R$/US$) returns with 15 min, 60 min and 120 min, daily and weekly basis, the one-step-ahead forecast performance is compared. Results indicate that forecast performance is strongly related to the series’ frequency and the forecasting evaluation shows that nonlinear models perform better than their linear counterparts. In the trade strategy based on forecasts, nonlinear models achieve higher returns when compared to a buy-and-hold strategy and to the linear models.     

White noise theory of robust nonlinear filtering with correlated state and observation noises

In the existing ‘direct’ white noise theory of nonlinear filtering, the state process is still modelled as a Markov process satisfying an Itô stochastic differential equation, while a ‘finitely additive’ white noise is used to model the observation noise. We remove this asymmetry by modelling the state process as the solution of a (stochastic) differential equation with a ‘finitely additive’ white noise as the input. This enables us to introduce correlation between the state and observation noises, and to obtain robust nonlinear filtering equations in the correlated noise case.     

一种多层前馈网参数可分离学习算法

目前大部分神经网络学习算法都是对网络所有的参数同时进行学习.当网络规模较 大时,这种做法常常很耗时.由于许多网络,例如感知器、径向基函数网络、概率广义回归网络 以及模糊神经网络,都是一种多层前馈型网络,它们的输入输出映射都可以表示为一组可变 基的线性组合.网络的参数也表现为二类:可变基中的参数是非线性的,组合系数是线性的. 为此,提出了一个将这二类参数进行分离学习的算法.仿真结果表明,这个学习算法加快了学 习过程,提高了网络的逼近性能.     

A variable structure gradient adaptive algorithm for a class of dynamical systems

A standard assumption in adaptive control is that the parameters being estimated are either constant or vary ‘slowly’ as a function of time. This paper investigates the adaptive control of a class of systems in which the parameters vary as a specified function of state. The dynamic structure of the systems may be either linear or nonlinear. For this class of systems, the state space is separated into distinct subsets. The parameters are then required to remain constant, or be slowly time varying, within the subsets. Given a controller for the system, an analysis of the output error dynamics and the parameter error dynamics leads to a parameter adaptation algorithm with a variable structure. The stability and convergence of both the parameter error and the output tracking error are investigated. An analysis of SISO linear systems with full state information is used to motivate and illustrate the treatment of SISO feedback linearizable systems.     

Fuzzy rule-base driven orthogonal approximation

In this study, orthogonal approximation concept is applied to fuzzy systems. We propose a new useful model adapted from the well-known Sugeno type fuzzy system. The proposed fuzzy model is a generalization of the zero-order Sugeno fuzzy system model. Instead of linear functions in standard Sugeno model, we use nonlinear functions in the consequent part. The nonlinear functions are selected from a trigonometric orthogonal basis. Orthogonal function parameters are trained along with the Sugeno fuzzy system. The proposed model is demonstrated using three simulations—a nonlinear piecewise-continuous scalar function modeling and filtering, nonlinear dynamic system identification, and time series prediction. Finally some performance comparisons are carried out.     

A fuzzy ensemble of parallel polynomial neural networks with information granules formed by fuzzy clustering

In this paper, we introduce a new category of fuzzy models called a fuzzy ensemble of parallel polynomial neural network (FEP²N²), which consist of a series of polynomial neural networks weighted by activation levels of information granules formed with the use of fuzzy clustering. The two underlying design mechanisms of the proposed networks rely on information granules resulting from the use of fuzzy C-means clustering (FCM) and take advantage of polynomial neural networks (PNNs).The resulting model comes in the form of parallel polynomial neural networks. In the design procedure, in order to estimate the optimal values of the coefficients of polynomial neural networks we use a weighted least square estimation algorithm. We incorporate various types of structures as the consequent part of the fuzzy model when using the learning algorithm. Among the diverse structures being available, we consider polynomial neural networks, which exhibit highly nonlinear characteristics when being viewed as local learning models.We use FCM to form information granules and to overcome the high dimensionality problem. We adopt PNNs to find the optimal local models, which can describe the relationship between the input variables and output variable within some local region of the input space.We show that the generalization capabilities as well as the approximation abilities of the proposed model are improved as a result of using polynomial neural networks. The performance of the network is quantified through experimentation in which we use a number of benchmarks already exploited within the realm of fuzzy or neurofuzzy modeling.     

On the persistency of excitation in radial basis function networkidentification of nonlinear systems

Considers radial basis function (RBF) network approximation of a multivariate nonlinear mapping as a linear parametric regression problem. Linear recursive identification algorithms applied to this problem are known to converge, provided the regressor vector sequence has the persistency of excitation (PE) property. The main contribution of this paper is formulation and proof of PE conditions on the input variables. In the RBF network identification, the regressor vector is a nonlinear function of these input variables. According to the formulated condition, the inputs provide PE, if they belong to domains around the network node centers. For a two-input network with Gaussian RBF that have typical width and are centered on a regular mesh, these domains cover about 25% of the input domain volume. The authors further generalize the proposed solution of the standard RBF network identification problem and study affine RBF network identification that is important for affine nonlinear system control. For the affine RBF network, the author formulates and proves a PE condition on both the system state parameters and control inputs.     

Adaptive noise cancellation using enhanced dynamic fuzzy neural networks

In this paper, a novel adaptive noise cancellation algorithm using enhanced dynamic fuzzy neural networks (EDFNNs) is described. In the proposed algorithm, termed EDFNN learning algorithm, the number of radial basis function (RBF) neurons (fuzzy rules) and input-output space clustering is adaptively determined. Furthermore, the structure of the system and the parameters of the corresponding RBF units are trained online automatically and relatively rapid adaptation is attained. By virtue of the self-organizing mapping (SOM) and the recursive least square error (RLSE) estimator techniques, the proposed algorithm is suitable for real-time applications. Results of simulation studies using different noise sources and noise passage dynamics show that superior performance can be achieved.     

Finite element analysis of flexible multibody systems with fuzzy parameters

A computational procedure is presented for predicting the dynamic response and evaluating the sensitivity coefficients of flexible multibody systems whose characteristics include fuzzy parameters. Time-histories of the possibility distributions of the response and the sensitivity coefficients are generated. These coefficients measure the sensitivity of the dynamic response to variations in the material, geometric and external force parameters of the system. The five key components of the procedure are: a) a corotational frame approach used in conjunction with a total Lagrangian formulation; b) beam and shell elements with the Cartesian coordinates of the nodes selected as degrees of freedom, and with continuous inter-element slopes; c) use of an approximate method of extension, based on the α-cut representation, called the ‘vertex method’ for generating the possibility distributions of the desired response quantities and their sensitivity coefficients; d) semi-explicit temporal integration technique for generating the dynamic response; and e) direct differentiation approach for evaluating the sensitivity coefficients. The effectiveness of the procedure and the usefulness of the fuzzy output are demonstrated through numerical examples, including an articulated space structure consisting of beams, shells and revolute joints.     

Some applications of an Implicit Duality Theorem to connections of structures of special types including Dirac and reciprocal structures

In this paper, we present some applications of an Implicit Duality Theorem which was originally a folklore result on Ideal Transformers in Electrical Network Theory. We show, among other things, that results on reciprocal networks (due to Tellegen (Proc. Inst. Radio Engrs. 14 (1953) 265), and Dirac Structures (due to van der Schaft (in: J.W. Polderman, H.L. Trentelman (Eds.), From Intelligent Control to Behavioural Systems. University of Groningen Press) follow as a consequence. These results have the form ‘interconnection of structures of a particular kind yields a structure of the same kind’. Also discussed is the question ‘given smaller structures of a given kind and a desired structure of the same kind, can the former be interconnected to yield the latter?’. We also indicate the range of possible generalizations of this result.     

A fast approach for automatic generation of fuzzy rules bygeneralized dynamic fuzzy neural networks

A fast approach for automatically generating fuzzy rules from sample patterns using generalized dynamic fuzzy neural networks (GD-FNNs) is presented. The GD-FNN is built based on ellipsoidal basis functions and functionally is equivalent to a Takagi-Sugeno-Kang fuzzy system. The salient characteristics of the GD-FNN are: (1) structure identification and parameters estimation are performed automatically and simultaneously without partitioning input space and selecting initial parameters a priori; (2) fuzzy rules can be recruited or deleted dynamically; (3) fuzzy rules can be generated quickly without resorting to the backpropagation (BP) iteration learning, a common approach adopted by many existing methods. The GD-FNN is employed in a wide range of applications ranging from static function approximation and nonlinear system identification to time-varying drug delivery system and multilink robot control. Simulation results demonstrate that a compact and high-performance fuzzy rule-base can be constructed. Comprehensive comparisons with other latest approaches show that the proposed approach is superior in terms of learning efficiency and performance     

Fuzzy membership function based neural networks with applicationsto the visual servoing of robot manipulators

It is shown that there exists a nonlinear mapping which transforms image features and their changes to the desired camera motion without measuring of the relative distance between the camera and the object. This nonlinear mapping can eliminate several difficulties occurring in computing the inverse of the feature Jacobian as in the usual feature-based visual feedback control methods. Instead of analytically deriving the closed form of this mapping, a fuzzy membership function (FMF) based neural network incorporating a fuzzy-neural interpolating network is proposed to approximate the nonlinear mapping, where the structure of the FMF network is similar to that of radial basis function neural network which is known to be very effective in the function approximation. Several FMF networks are trained to be capable of tracking a moving object in the whole workspace along the line of sight. For an effective implementation of the proposed FMF network, an image feature selection process is investigated, and the required fuzzy membership functions are designed. Finally, several numerical examples are presented to show the validity of the proposed visual servoing method     

A new class of intelligent knowledge-based systems with an optimisation-based inference engine

In this paper we describe a new class of intelligent knowledge-based system (IKBS) which can be used principally for managerial decision making applications. This class of applications often requires a framework for knowledge acquisition which allows the system to use the knowledge of several experts. In addition, since in most business decision making the objective is maximise profits, there is a need for an inference engine which allows optimisation to be carried out. The new class of IKBS which is described in this paper has both these properties, i.e., the ability to use the knowledge of multiple experts in a convenient way and an inference engine which by performing optimisations is able to pick out the profit maximising decisions. As an illustration of these concepts, a system for allocation decision making is described. The system ‘Retail-opt’ allows the user to solve problems like allocation of space in retail outlets, allocation of space in mail order catalogues, pricing policy decisions for discounted airline tickets, etc. In the paper, the basic concepts behind ‘Retail-opt’ are described and an application of ‘Retail-opt’ to the problem of retail space allocation in a Scandinavian Department Store is given. A number of other systems which use these concepts for more complicated competitive decision making situations are also described.     

基于量子神经网络的油田水淹层识别方法

针对油藏测井解释中的水淹层识别问题,提出一种量子神经网络模型。该模型用量子旋转门更新量子比特的相位,用受控旋转门实现网络的非线性映射功能。网络可调参数为量子旋转门的旋转角度和受控非门的控制参数。基于梯度下降法设计了学习算法。仿真结果表明,该模型的预测能力优于普通BP网络、模糊神经网络和过程神经网络等其他方法。