Composite control of non-linear singularly perturbed systems: meeting a design objective

Using the integral manifold approach, a composite control—the sum of a fast control and a slow control—is derived for a particular class of non-linear singularly perturbed systems. The fast control is designed completely at the outset, thus ensuring the stability of the fast transients of the system and, furthermore, the existence of the integral manifold. A new method is then presented which simplifies the derivation of a slow control such that the singularly perturbed system meets a preselected design objective to within some specified order of accuracy. Though this approach is, by its very nature, ad hoc, the underlying procedure is easily extended to more general classes of singularly perturbed systems by way of three examples.

Optimal control of a class of discrete–continuous non-linear systems — decomposition and hierarchical structure

A technique is derived for solving a non-linear optimal control problem by iterating on a sequence of simplified problems in linear quadratic form. The technique is designed to achieve the correct solution of the original non-linear optimal control problem in spite of these simplifications. A mixed approach with a discrete performance index and continuous state variable system description is used as the basis of the design, and it is shown how the global problem can be decomposed into local sub-system problems and a co-ordinator within a hierarchical framework. An analysis of the optimality and convergence properties of the algorithm is presented and the effectiveness of the technique is demonstrated using a simulation example with a non-separable performance index.

Linear-wavelet models for non-linear identification applied to a pressure plant

A nonlinear regression structure comprising a wavelet network and a linear term is proposed for system identification. The theoretical foundation of the approach is laid by proving that radial wavelets are orthogonal to linear functions. A constructive procedure for building such models is described and the approach is tested with experimental data.

Efficient non-linear data assimilation in geophysical fluid dynamics

New ways of combining observations with numerical models are discussed in which the size of the state space can be very large, and the model can be highly nonlinear. Also the observations of the system can be related to the model variables in highly nonlinear ways, making this data-assimilation (or inverse) problem highly nonlinear. First we discuss the connection between data assimilation and inverse problems, including regularization. We explore the choice of proposal density in a Particle Filter and show how the ’curse of dimensionality’ might be beaten. In the standard Particle Filter ensembles of model runs are propagated forward in time until observations are encountered, rendering it a pure Monte-Carlo method. In large-dimensional systems this is very inefficient and very large numbers of model runs are needed to solve the data-assimilation problem realistically. In our approach we steer all model runs towards the observations resulting in a much more efficient method. By further ’ensuring almost equal weight’ we avoid performing model runs that are useless in the end. Results are shown for the 40 and 1000 dimensional Lorenz 1995 model.

Enhanced second order non-linear optical properties of silica sol-gel thin films through photo-induced poling

Sol-gel derived inorganic materials are of interest as hosts for non-linear optically active guest molecules and they offer particular advantages in the field of non-linear optics. Orientationally ordered glasses have been prepared using a sol-gel system based on tetramethoxysilane, methyltrimethoxysilane and a non-linear optical chromophore Disperse Red 1. The novel technique of photo-induced poling was used to generate enhanced levels of polar order. The level of enhancement is strongly dependent on the extent of gelation and an optimum preparation time of ∼100 h led to an enhancement factor of ∼5. Films prepared in this manner exhibited a high stability of the polar order.

Are property prices non-linear? An investigation of the behaviour of US REITs and UK property company shares

Linear models of market performance may be misspecified if the market is subdivided into distinct regimes exhibiting different behaviour. Price movements in the US Real Estate Investment Trusts and UK Property Companies Markets are explored using a Threshold Autoregressive (TAR) model with regimes defined by the real rate of interest. In both US and UK markets, distinctive behaviour emerges, with the TAR model offering better predictive power than a more conventional linear autoregressive model. The research points to the possibility of developing trading rules to exploit the systematically different behaviour across regimes.

Composite control of non-linear singularly perturbed systems: a geometric approach

Using a geometric approach, a composite control—the sum of a slow control and a fast control—is derived for a general class of non-linear singularly perturbed systems. A new and simpler method of composite control design is proposed whereby the fast control is completely designed at the outset. The slow control is then free to be chosen such that the slow integral manifold of the original system approximates a desired design manifold to within any specified order of ε accuracy.

Composite control of non-linear singularly perturbed systems: a geometric approach

Using a geometric approach, a composite control—the sum of a slow control and a fast control—is derived for a general class of non-linear singularly perturbed systems. A new and simpler method of composite control design is proposed whereby the fast control is completely designed at the outset. The slow control is then free to be chosen such that the slow integral manifold of the original system approximates a desired design manifold to within any specified order of ε accuracy.