Pole assignment control design for time–varying time–delay systems using radial basis functions

Systems with time-varying time delays present a particularly challenging control problem. They have been observed across a wide array of domains, from hydraulic actuators to insulin delivery control systems. Control systems that address system time-delays, nonlinearities and uncertainty are the subject of much research but, whilst the specific concept of varying time delays is sometimes acknowledged (for example in the control of hydraulic manipulators), this appears to be less widely investigated than some other types of nonlinearity. In part motivated by recent research into internal multi-model control, as similarly applied to systems with unknown time-varying delays, the present work utilises a Gaussian radial basis function to switch between two or more partial controllers. Each partial controller is based on a linear model with a (time-invariant) time delay. The new algorithm is developed and evaluated via simulation using a non-minimal state space (NMSS) framework, with pole assignment as the design criterion. Simulation results suggest that it yields improved performance in comparison to a simpler switching approach and the equivalent linear control system. However, laboratory examples and further research into robustness and stability is required in the next step

Adaptive filtering of evoked potentials with radial-basis-function neural network prefilter

Evoked potentials (EPs) are time-varying signals typically buried in relatively large background noise. To extract the EP more effectively from noise, we had previously developed an approach using an adaptive signal enhancer (ASE) (Chen et al., 1995). ASE requires a proper reference input signal for its optimal performance. Ensemble- and moving window-averages were formerly used with good results. In this paper, we present a new method to provide even more effective reference inputs for the ASE. Specifically, a Gaussian radial basis function neural network (RBFNN) was used to preprocess raw EP signals before serving as the reference input. Since the RBFNN has built-in nonlinear activation functions that enable it to closely fit any function mapping, the output of RBFNN can effectively track the signal variations of EP. Results confirmed the superior performance of ASE with RBFNN over the previous method.published_or_final_versio

A numerical study of viscous vortex rings using a spectral method

Viscous, axisymmetric vortex rings are investigated numerically by solving the incompressible Navier-Stokes equations using a spectral method designed for this type of flow. The results presented are axisymmetric, but the method is developed to be naturally extended to three dimensions. The spectral method relies on divergence-free basis functions. The basis functions are formed in spherical coordinates using Vector Spherical Harmonics in the angular directions, and Jacobi polynomials together with a mapping in the radial direction. Simulations are performed of a single ring over a wide range of Reynolds numbers (Re approximately equal gamma/nu), 0.001 less than or equal to 1000, and of two interacting rings. At large times, regardless of the early history of the vortex ring, it is observed that the flow approaches a Stokes solution that depends only on the total hydrodynamic impulse, which is conserved for all time. At small times, from an infinitely thin ring, the propagation speeds of vortex rings of varying Re are computed and comparisons are made with the asymptotic theory by Saffman. The results are in agreement with the theory; furthermore, the error is found to be smaller than Saffman's own estimate by a factor square root ((nu x t)/R squared) (at least for Re=0). The error also decreases with increasing Re at fixed core-to-ring radius ratio, and appears to be independent of Re as Re approaches infinity). Following a single ring, with Re=500, the vorticity contours indicate shedding of vorticity into the wake and a settling of an initially circular core to a more elliptical shape, similar to Norbury's steady inviscid vortices. Finally, we consider the case of leapfrogging vortex rings with Re=1000. The results show severe straining of the inner vortex core in the first pass and merging of the two cores during the second pass

Pricing Financial Derivatives using Radial Basis Function generated Finite Differences with Polyharmonic Splines on Smoothly Varying Node Layouts

In this paper, we study the benefits of using polyharmonic splines and node layouts with smoothly varying density for developing robust and efficient radial basis function generated finite difference (RBF-FD) methods for pricing of financial derivatives. We present a significantly improved RBF-FD scheme and successfully apply it to two types of multidimensional partial differential equations in finance: a two-asset European call basket option under the Black--Scholes--Merton model, and a European call option under the Heston model. We also show that the performance of the improved method is equally high when it comes to pricing American options. By studying convergence, computational performance, and conditioning of the discrete systems, we show the superiority of the introduced approaches over previously used versions of the RBF-FD method in financial applications

A meshless, integration-free, and boundary-only RBF technique

Based on the radial basis function (RBF), non-singular general solution and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation techniques for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are very mathematically simple. In this study, the RBFs are employed to approximate the inhomogeneous terms via the DRM, while non-singular general solution leads to a boundary-only RBF formulation for homogenous solution. The present scheme is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of nonsingular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does no require the artificial boundary and results in the symmetric system equations under certain conditions. The efficiency and utility of this new technique are validated through a number of typical numerical examples. Completeness concern of the BKM due to the only use of non-singular part of complete fundamental solution is also discussed

A robust extension to the triple plane pressure mode matching method by filtering convective perturbations

Time-periodic CFD simulations are widely used to investigate turbomachinery components. The triple-plane pressure mode matching method (TPP) developed by Ovenden and Rienstra extracts the acoustic part in such simulations. Experience shows that this method is subject to significant errors when the amplitude of pseudo-sound is high compared to sound. Pseudo-sound are unsteady pressure fluctuations with a convective character. The presented extension to the TPP improves the splitting between acoustics and the rest of the unsteady flow field. The method is simple: i) the acoustic eigenmodes are analytically determined for a uniform mean flow as in the original TPP; ii) the suggested model for convective pressure perturbations uses the convective wavenumber as axial wavenumber and the same orthogonal radial shape functions as for the acoustic modes. The reliability is demonstrated on the simulation data of a low-pressure fan. As acoustic and convective perturbations are separated, the accuracy of the results increases close to sources, allowing a reduction of the computational costs by shortening the simulation domain. The extended method is as robust as the original one--giving the same results for the acoustic modes in absence of convective perturbations.Comment: Accepted 15-05-11 by International Journal of Aeroacoustics to be published in the special issue focusing on turbomachinery aeroacoustic