Numerical solution of some nonlocal, nonlinear dispersive wave equations

We use a spectral method to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The proposed numerical method is able to capture well the dynamics of the solutions; we use it to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usually connected with integrability, for which there is at present no analytic proof. Thus we study in particular the resolution property of arbitrary initial profiles into sequences of solitary waves for both equations and clean interaction of Benjamin-Ono solitary waves. We also verify numerically that the behaviour of the solution of the Intermediate Long Wave equation as the model parameter tends to the infinite depth limit is the one predicted by the theory.

Error estimates for a fully discrete spectral scheme for a class of nonlinear, nonlocal dispersive wave equations

We analyze a fully discrete spectral method for the numerical solution of the initial- and periodic boundary-value problem for two nonlinear, nonlocal, dispersive wave equations, the Benjamin–Ono and the Intermediate Long Wave equations. The equations are discretized in space by the standard Fourier–Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L2-error bound of spectral accuracy in space and of second-order accuracy in time.

The implications of service-dominant logic and integrated solutions on the sales function

This study explores the implications of an organization moving toward service-dominant logic (S-D logic) on the sales function. Driven by its customers’ needs, a service orientation by its nature requires personal interaction and sales personnel are in an ideal position to develop offerings with the customer. However, the development of S-D logic may require sales staff to develop additional skills. Employing a single case study, the study identified that sales personnel are quick to appreciate the advantages of S-D logic for customer satisfaction and six specific skills were highlighted and explored. Further, three propositions were identified: in an organization adopting S-D logic, the sales process needs to elicit needs at both embedded-value and value-in-use levels. In addition, the sales process needs to coproduce not just goods and service attributes but also attributes of the customer’s usage processes. Further, the sales process needs to coproduce not just goods and service attributes but also attributes of the customer’s usage processes.

A hyperstable neural network for the modelling and control of nonlinear systems

A multivariable hyperstable robust adaptive decoupling control algorithm based on a neural network is presented for the control of nonlinear multivariable coupled systems with unknown parameters and structure. The Popov theorem is used in the design of the controller. The modelling errors, coupling action and other uncertainties of the system are identified on-line by a neural network. The identified results are taken as compensation signals such that the robust adaptive control of nonlinear systems is realised. Simulation results are given.

Neural network enhanced generalised minimum variance self-tuning controller for nonlinear discrete-time systems

A neural network enhanced self-tuning controller is presented, which combines the attributes of neural network mapping with a generalised minimum variance self-tuning control (STC) strategy. In this way the controller can deal with nonlinear plants, which exhibit features such as uncertainties, nonminimum phase behaviour, coupling effects and may have unmodelled dynamics, and whose nonlinearities are assumed to be globally bounded. The unknown nonlinear plants to be controlled are approximated by an equivalent model composed of a simple linear submodel plus a nonlinear submodel. A generalised recursive least squares algorithm is used to identify the linear submodel and a layered neural network is used to detect the unknown nonlinear submodel in which the weights are updated based on the error between the plant output and the output from the linear submodel. The procedure for controller design is based on the equivalent model therefore the nonlinear submodel is naturally accommodated within the control law. Two simulation studies are provided to demonstrate the effectiveness of the control algorithm.

Neural networks for identification of nonlinear systems under random piecewise polynomial disturbances

The problem of identification of a nonlinear dynamic system is considered. A two-layer neural network is used for the solution of the problem. Systems disturbed with unmeasurable noise are considered, although it is known that the disturbance is a random piecewise polynomial process. Absorption polynomials and nonquadratic loss functions are used to reduce the effect of this disturbance on the estimates of the optimal memory of the neural-network model.

Adaptive general predictive controller for nonlinear systems

A nonlinear general predictive controller (NLGPC) is described which is based on the use of a Hammerstein model within a recursive control algorithm. A key contribution of the paper is the use of a novel, one-step simple root solving procedure for the Hammerstein model, this being a fundamental part of the overall tuning algorithm. A comparison is made between NLGPC and nonlinear deadbeat control (NLDBC) using the same one-step nonlinear components, in order to investigate NLGPC advantages and disadvantages.

Nonlinear model structure design and construction using orthogonal least squares and D-optimality design

A very efficient learning algorithm for model subset selection is introduced based on a new composite cost function that simultaneously optimizes the model approximation ability and model robustness and adequacy. The derived model parameters are estimated via forward orthogonal least squares, but the model subset selection cost function includes a D-optimality design criterion that maximizes the determinant of the design matrix of the subset to ensure the model robustness, adequacy, and parsimony of the final model. The proposed approach is based on the forward orthogonal least square (OLS) algorithm, such that new D-optimality-based cost function is constructed based on the orthogonalization process to gain computational advantages and hence to maintain the inherent advantage of computational efficiency associated with the conventional forward OLS approach. Illustrative examples are included to demonstrate the effectiveness of the new approach.

Neurofuzzy design and model construction of nonlinear dynamical processes from data

A common problem in many data based modelling algorithms such as associative memory networks is the problem of the curse of dimensionality. In this paper, a new two-stage neurofuzzy system design and construction algorithm (NeuDeC) for nonlinear dynamical processes is introduced to effectively tackle this problem. A new simple preprocessing method is initially derived and applied to reduce the rule base, followed by a fine model detection process based on the reduced rule set by using forward orthogonal least squares model structure detection. In both stages, new A-optimality experimental design-based criteria we used. In the preprocessing stage, a lower bound of the A-optimality design criterion is derived and applied as a subset selection metric, but in the later stage, the A-optimality design criterion is incorporated into a new composite cost function that minimises model prediction error as well as penalises the model parameter variance. The utilisation of NeuDeC leads to unbiased model parameters with low parameter variance and the additional benefit of a parsimonious model structure. Numerical examples are included to demonstrate the effectiveness of this new modelling approach for high dimensional inputs.

Nonlinear model structure detection using optimum experimental design and orthogonal least squares

A very efficient learning algorithm for model subset selection is introduced based on a new composite cost function that simultaneously optimizes the model approximation ability and model adequacy. The derived model parameters are estimated via forward orthogonal least squares, but the subset selection cost function includes an A-optimality design criterion to minimize the variance of the parameter estimates that ensures the adequacy and parsimony of the final model. An illustrative example is included to demonstrate the effectiveness of the new approach.

Dual-orthogonal radial basis function networks for nonlinear time series prediction

A new structure of Radial Basis Function (RBF) neural network called the Dual-orthogonal RBF Network (DRBF) is introduced for nonlinear time series prediction. The hidden nodes of a conventional RBF network compare the Euclidean distance between the network input vector and the centres, and the node responses are radially symmetrical. But in time series prediction where the system input vectors are lagged system outputs, which are usually highly correlated, the Euclidean distance measure may not be appropriate. The DRBF network modifies the distance metric by introducing a classification function which is based on the estimation data set. Training the DRBF networks consists of two stages. Learning the classification related basis functions and the important input nodes, followed by selecting the regressors and learning the weights of the hidden nodes. In both cases, a forward Orthogonal Least Squares (OLS) selection procedure is applied, initially to select the important input nodes and then to select the important centres. Simulation results of single-step and multi-step ahead predictions over a test data set are included to demonstrate the effectiveness of the new approach.

Optimal piecewise locally linear modeling

Associative memory networks such as Radial Basis Functions, Neurofuzzy and Fuzzy Logic used for modelling nonlinear processes suffer from the curse of dimensionality (COD), in that as the input dimension increases the parameterization, computation cost, training data requirements, etc. increase exponentially. Here a new algorithm is introduced for the construction of a Delaunay input space partitioned optimal piecewise locally linear models to overcome the COD as well as generate locally linear models directly amenable to linear control and estimation algorithms. The training of the model is configured as a new mixture of experts network with a new fast decision rule derived using convex set theory. A very fast simulated reannealing (VFSR) algorithm is utilized to search a global optimal solution of the Delaunay input space partition. A benchmark non-linear time series is used to demonstrate the new approach.

A concept of approximated densities for efficient nonlinear estimation

This paper presents the theoretical development of a nonlinear adaptive filter based on a concept of filtering by approximated densities (FAD). The most common procedures for nonlinear estimation apply the extended Kalman filter. As opposed to conventional techniques, the proposed recursive algorithm does not require any linearisation. The prediction uses a maximum entropy principle subject to constraints. Thus, the densities created are of an exponential type and depend on a finite number of parameters. The filtering yields recursive equations involving these parameters. The update applies the Bayes theorem. Through simulation on a generic exponential model, the proposed nonlinear filter is implemented and the results prove to be superior to that of the extended Kalman filter and a class of nonlinear filters based on partitioning algorithms.

Applying the extended Kalman filter to systems described by nonlinear differential-algebraic equations

This paper describes a method for the state estimation of nonlinear systems described by a class of differential-algebraic equation models using the extended Kalman filter. The method involves the use of a time-varying linearisation of a semi-explicit index one differential-algebraic equation. The estimation technique consists of a simplified extended Kalman filter that is integrated with the differential-algebraic equation model. The paper describes a simulation study using a model of a batch chemical reactor. It also reports a study based on experimental data obtained from a mixing process, where the model of the system is solved using the sequential modular method and the estimation involves a bank of extended Kalman filters.