Modelling and control of nonlinear systems using Gaussian processes with partial model information

Gaussian processes are gaining increasing popularity among the control community, in particular for the modelling of discrete time state space systems. However, it has not been clear how to incorporate model information, in the form of known state relationships, when using a Gaussian process as a predictive model. An obvious example of known prior information is position and velocity related states. Incorporation of such information would be beneficial both computationally and for faster dynamics learning. This paper introduces a method of achieving this, yielding faster dynamics learning and a reduction in computational effort from O(Dn2) to O((D - F)n2) in the prediction stage for a system with D states, F known state relationships and n observations. The effectiveness of the method is demonstrated through its inclusion in the PILCO learning algorithm with application to the swing-up and balance of a torque-limited pendulum and the balancing of a robotic unicycle in simulation. © 2012 IEEE.

An efficient approach of modelling the flexural cracking behaviour of un-notched plain concrete prisms subject to monotonic and cyclic loading

In the field of vibration-based damage detection of concrete structures efficient damage models are needed to better understand changes in the vibration properties of cracked structures. These models should quantitatively replicate the damage mechanisms in concrete and easily be used as damage detection tools. In this paper, the flexural cracking behaviour of plain concrete prisms subject to monotonic and cyclic loading regimes under displacement control is tested experimentally and modelled numerically. Four-point bending tests on simply supported un-notched prisms are conducted, where the cracking process is monitored using a digital image correlation system. A numerical model, with a single crack at midspan, is presented where the cracked zone is modelled using the fictitious crack approach and parts outside that zone are treated in a linear-elastic manner. The model considers crack initiation, growth and closure by adopting cyclic constitutive laws. A multi-variate Newton-Raphson iterative solver is used to solve the non-linear equations to ensure equilibrium and compatibility at the interface of the cracked zone. The numerical results agree well with the experiments for both loading scenarios. The model shows good predictions of the degradation of stiffness with increasing load. It also approximates the crack-mouth-opening-displacement when compared with the experimental data of the digital image correlation system. The model is found to be computationally efficient as it runs full analysis for cyclic loading in less than 2. min, and it can therefore be used within the damage detection process. © 2013 Elsevier Ltd.

A value mapping tool for sustainable business modelling

Purpose: Although business models that deliver sustainability are increasingly popular in the literature, few tools that assist in sustainable business modelling have been identified. This paper investigates how businesses might create balanced social, environmental and economic value through integrating sustainability more fully into the core of their business. A value mapping tool is developed to help firms create value propositions better suited for sustainability. Design/methodology/approach: In addition to a literature review, six sustainable companies were interviewed to understand their approaches to business modelling, using a case study approach. Building on the literature and practice, a tool was developed which was pilot tested through use in a workshop. The resulting improved tool and process was subsequently refined through use in 13 workshops. Findings: A novel value mapping tool was developed to support sustainable business modelling, which introduces three forms of value (value captured, missed/destroyed or wasted, and opportunity) and four major stakeholder groups (environment, society, customer, and network actors). Practical implications: This tool intends to support business modelling for sustainability by assisting firms in better understanding their overall value proposition, both positive and negative, for all relevant stakeholders in the value network. Originality/value: The tool adopts a multiple stakeholder view of value, a network rather than firm centric perspective, and introduces a novel way of conceptualising value that specifically introduces value destroyed or wasted/ missed, in addition to the current value proposition and new opportunities for value creation. © Emerald Group Publishing Limited.

Automated computational modelling for solid mechanics

We present some recent developments in automated computational modelling with an emphasis on solid mechanics applications. The automation process permits an abstract mathematical model of a physical problem to be translated into computer code rapidly and trivially, and can lead to computer code which is faster than hand-written and optimised code. Crucial to the approach is ensuring that mathematical abstractions inherent in the mathematical model are inherited by the software library. © Springer Science+Business Media B.V. 2008.

Modelling and validation of wet steam flow in a low pressure steam turbine

Results of numerical investigations of the wet steam flow in a three stage low pressure steam turbine test rig are presented. The test rig is a scale model of a modern steam turbine design and provides flow measurements over a range of operating conditions which are used for detailed comparisons with the numerical results. For the numerical analysis a modern CFD code with user defined models for specific wet steam modelling is used. The effect of different theoretical models for nucleation and droplet growth are examined. It is shown that heterogeneous condensation is highly dependent on steam quality and, in this model turbine with high quality steam, a homogeneous theory appears to be the best choice. The homogeneous theory gives good agreement between the test rig traverse measurements and the numerical results. The differences in the droplet size distribution of the three stage turbine are shown for different loads and modelling assumptions. The different droplet growth models can influence the droplet size by a factor of two. An estimate of the influence of unsteady effects is made by means of an unsteady two-dimensional simulation. The unsteady modelling leads to a shift of nucleation into the next blade row. For the investigated three stage turbine the influence due to wake chopping on the condensation process is weak but to confirm this conclusion further investigations are needed in complete three dimensions and on turbines with more stages. Copyright © 2011 by ASME.

First principles interatomic potential for tungsten based on Gaussian process regression

An accurate description of atomic interactions, such as that provided by first principles quantum mechanics, is fundamental to realistic prediction of the properties that govern plasticity, fracture or crack propagation in metals. However, the computational complexity associated with modern schemes explicitly based on quantum mechanics limits their applications to systems of a few hundreds of atoms at most. This thesis investigates the application of the Gaussian Approximation Potential (GAP) scheme to atomistic modelling of tungsten - a bcc transition metal which exhibits a brittle-to-ductile transition and whose plasticity behaviour is controlled by the properties of $\frac{1}{2} \langle 111 \rangle$ screw dislocations. We apply Gaussian process regression to interpolate the quantum-mechanical (QM) potential energy surface from a set of points in atomic configuration space. Our training data is based on QM information that is computed directly using density functional theory (DFT). To perform the fitting, we represent atomic environments using a set of rotationally, permutationally and reflection invariant parameters which act as the independent variables in our equations of non-parametric, non-linear regression. We develop a protocol for generating GAP models capable of describing lattice defects in metals by building a series of interatomic potentials for tungsten. We then demonstrate that a GAP potential based on a Smooth Overlap of Atomic Positions (SOAP) covariance function provides a description of the $\frac{1}{2} \langle 111 \rangle$ screw dislocation that is in agreement with the DFT model. We use this potential to simulate the mobility of $\frac{1}{2} \langle 111 \rangle$ screw dislocations by computing the Peierls barrier and model dislocation-vacancy interactions to QM accuracy in a system containing more than 100,000 atoms.

Modelling and comparison of trapped fields in (RE)BCO bulk superconductors for activation using pulsed field magnetization

The ability to generate a permanent, stable magnetic field unsupported by an electromotive force is fundamental to a variety of engineering applications. Bulk high temperature superconducting (HTS) materials can trap magnetic fields of magnitude over ten times higher than the maximum field produced by conventional magnets, which is limited practically to rather less than 2 T. In this paper, two large c-axis oriented, single-grain YBCO and GdBCO bulk superconductors are magnetized by the pulsed field magnetization (PFM) technique at temperatures of 40 and 65 K and the characteristics of the resulting trapped field profile are investigated with a view of magnetizing such samples as trapped field magnets (TFMs) in situ inside a trapped flux-type superconducting electric machine. A comparison is made between the temperatures at which the pulsed magnetic field is applied and the results have strong implications for the optimum operating temperature for TFMs in trapped flux-type superconducting electric machines. The effects of inhomogeneities, which occur during the growth process of single-grain bulk superconductors, on the trapped field and maximum temperature rise in the sample are modelled numerically using a 3D finite-element model based on the H-formulation and implemented in Comsol Multiphysics 4.3a. The results agree qualitatively with the observed experimental results, in that inhomogeneities act to distort the trapped field profile and reduce the magnitude of the trapped field due to localized heating within the sample and preferential movement and pinning of flux lines around the growth section regions (GSRs) and growth sector boundaries (GSBs), respectively. The modelling framework will allow further investigation of various inhomogeneities that arise during the processing of (RE)BCO bulk superconductors, including inhomogeneous Jc distributions and the presence of current-limiting grain boundaries and cracks, and it can be used to assist optimization of processing and PFM techniques for practical bulk superconductor applications. © 2014 IOP Publishing Ltd.

Computational modelling of building damage due to tunnelling

Underground constructions in soft ground may lead to settlement damage to existing buildings. In The Netherlands the situation is particularly complex, because of the combination of soft soil, fragile pile foundations and brittle, unreinforced masonry façades. The tunnelling design process in urban areas requires a reliable risk damage assessment. In the engineering practice the current preliminary damage assessment is based on the limiting tensile strain method (LTSM). Essentially this is an uncoupled analysis, in which the building is modelled as an elastic beam subject to imposed Greenfield settlements and the induced tensile strains are compared with a limit value for the material. The soil-structure interaction is included only as a ratio between the soil and the building stiffness. In this paper, a coupled approach is evaluated. The soil-structure interaction in terms of normal and shear behaviour is represented by interface elements and a cracking model for masonry is included. This project aims to improve the existing damage classification system for masonry buildings subjected to tunnel-induced settlement, in order to evaluate the necessity of strengthening techniques or mitigation measures.

Modelling raindrop impact and splash erosion processes within a spatial cell: a stochastic approach

A new approach is proposed to simulate splash erosion on local soil surfaces. Without the effect of wind and other raindrops, the impact of free-falling raindrops was considered as an independent event from the stochastic viewpoint. The erosivity of a single raindrop depending on its kinetic energy was computed by an empirical relationship in which the kinetic energy was expressed as a power function of the equivalent diameter of the raindrop. An empirical linear function combining the kinetic energy and soil shear strength was used to estimate the impacted amount of soil particles by a single raindrop. Considering an ideal local soil surface with size of I m x I m, the expected number of received free-failing raindrops with different diameters per unit time was described by the combination of the raindrop size distribution function and the terminal velocity of raindrops. The total splash amount was seen as the sum of the impact amount by all raindrops in the rainfall event. The total splash amount per unit time was subdivided into three different components, including net splash amount, single impact amount and re-detachment amount. The re-detachment amount was obtained by a spatial geometric probability derived using the Poisson function in which overlapped impacted areas were considered. The net splash amount was defined as the mass of soil particles collected outside the splash dish. It was estimated by another spatial geometric probability in which the average splashed distance related to the median grain size of soil and effects of other impacted soil particles and other free-falling raindrops were considered. Splash experiments in artificial rainfall were carried out to validate the availability and accuracy of the model. Our simulated results suggested that the net splash amount and re-detachment amount were small parts of the total splash amount. Their proportions were 0.15% and 2.6%, respectively. The comparison of simulated data with measured data showed that this model could be applied to simulate the soil-splash process successfully and needed information of the rainfall intensity and original soil properties including initial bulk intensity, water content, median grain size and some empirical constants related to the soil surface shear strength, the raindrop size distribution function and the average splashed distance. Copyright (c) 2007 John Wiley & Sons, Ltd.

Process control in agile supply chain network.

The work comprises a new theoretical development applied to aid decision making in an increasingly important commercial sector. Agile supply, where small volumes of high margin, short life cycle innovative products are offered, is increasingly carried out through a complex global supply chain network. We outline an equilibrium solution in such a supply chain network, which works through limited cooperation and coordination along edges (links) in the network. The links constitute the stochastic modelling entities rather than the nodes of the network. We utilise newly developed phase plane analysis to identify, model and predict characteristic behaviour in supply chain networks. The phase plane charts profile the flow of inventory and identify out of control conditions. They maintain quality within the network, as well as intelligently track the way the network evolves in conditions of changing variability. The methodology is essentially distribution free, relying as it does on the study of forecasting errors, and can be used to examine contractual details as well as strategic and game theoretical concepts between decision-making components (agents) of a network. We illustrate with typical data drawn from supply chain agile fashion products.

On Models, Modelling and the Distinctive Nature of Model-Based Reasoning

M.H. Lee, On Models, Modelling and the Distinctive Nature of Model-Based Reasoning, AI Communications, 12 (3), pp127-137.1999.

Comparison of deterministic, stochastic and fuzzy logic uncertainty modelling for capacity extension projects of DI/WFI pharmaceutical plant utilities with variable/dynamic demand

The last 30 years have seen Fuzzy Logic (FL) emerging as a method either complementing or challenging stochastic methods as the traditional method of modelling uncertainty. But the circumstances under which FL or stochastic methods should be used are shrouded in disagreement, because the areas of application of statistical and FL methods are overlapping with differences in opinion as to when which method should be used. Lacking are practically relevant case studies comparing these two methods. This work compares stochastic and FL methods for the assessment of spare capacity on the example of pharmaceutical high purity water (HPW) utility systems. The goal of this study was to find the most appropriate method modelling uncertainty in industrial scale HPW systems. The results provide evidence which suggests that stochastic methods are superior to the methods of FL in simulating uncertainty in chemical plant utilities including HPW systems in typical cases whereby extreme events, for example peaks in demand, or day-to-day variation rather than average values are of interest. The average production output or other statistical measures may, for instance, be of interest in the assessment of workshops. Furthermore the results indicate that the stochastic model should be used only if found necessary by a deterministic simulation. Consequently, this thesis concludes that either deterministic or stochastic methods should be used to simulate uncertainty in chemical plant utility systems and by extension some process system because extreme events or the modelling of day-to-day variation are important in capacity extension projects. Other reasons supporting the suggestion that stochastic HPW models are preferred to FL HPW models include: 1. The computer code for stochastic models is typically less complex than a FL models, thus reducing code maintenance and validation issues. 2. In many respects FL models are similar to deterministic models. Thus the need for a FL model over a deterministic model is questionable in the case of industrial scale HPW systems as presented here (as well as other similar systems) since the latter requires simpler models. 3. A FL model may be difficult to "sell" to an end-user as its results represent "approximate reasoning" a definition of which is, however, lacking. 4. Stochastic models may be applied with some relatively minor modifications on other systems, whereas FL models may not. For instance, the stochastic HPW system could be used to model municipal drinking water systems, whereas the FL HPW model should or could not be used on such systems. This is because the FL and stochastic model philosophies of a HPW system are fundamentally different. The stochastic model sees schedule and volume uncertainties as random phenomena described by statistical distributions based on either estimated or historical data. The FL model, on the other hand, simulates schedule uncertainties based on estimated operator behaviour e.g. tiredness of the operators and their working schedule. But in a municipal drinking water distribution system the notion of "operator" breaks down. 5. Stochastic methods can account for uncertainties that are difficult to model with FL. The FL HPW system model does not account for dispensed volume uncertainty, as there appears to be no reasonable method to account for it with FL whereas the stochastic model includes volume uncertainty.

Experimental quantification and modelling of attrition of infant formulae during pneumatic conveying

Infant formula is often produced as an agglomerated powder using a spray drying process. Pneumatic conveying is commonly used for transporting this product within a manufacturing plant. The transient mechanical loads imposed by this process cause some of the agglomerates to disintegrate, which has implications for key quality characteristics of the formula including bulk density and wettability. This thesis used both experimental and modelling approaches to investigate this breakage during conveying. One set of conveying trials had the objective of establishing relationships between the geometry and operating conditions of the conveying system and the resulting changes in bulk properties of the infant formula upon conveying. A modular stainless steel pneumatic conveying rig was constructed for these trials. The mode of conveying and air velocity had a statistically-significant effect on bulk density at a 95% level, while mode of conveying was the only factor which significantly influenced D[4,3] or wettability. A separate set of conveying experiments investigated the effect of infant formula composition, rather than the pneumatic conveying parameters, and also assessed the relationships between the mechanical responses of individual agglomerates of four infant formulae and their compositions. The bulk densities before conveying, and the forces and strains at failure of individual agglomerates, were related to the protein content. The force at failure and stiffness of individual agglomerates were strongly correlated, and generally increased with increasing protein to fat ratio while the strain at failure decreased. Two models of breakage were developed at different scales; the first was a detailed discrete element model of a single agglomerate. This was calibrated using a novel approach based on Taguchi methods which was shown to have considerable advantages over basic parameter studies which are widely used. The data obtained using this model compared well to experimental results for quasi-static uniaxial compression of individual agglomerates. The model also gave adequate results for dynamic loading simulations. A probabilistic model of pneumatic conveying was also developed; this was suitable for predicting breakage in large populations of agglomerates and was highly versatile: parts of the model could easily be substituted by the researcher according to their specific requirements.

Mathematical modelling and optimisation of the formulation and manufacture of aggregate food products

In this PhD study, mathematical modelling and optimisation of granola production has been carried out. Granola is an aggregated food product used in breakfast cereals and cereal bars. It is a baked crispy food product typically incorporating oats, other cereals and nuts bound together with a binder, such as honey, water and oil, to form a structured unit aggregate. In this work, the design and operation of two parallel processes to produce aggregate granola products were incorporated: i) a high shear mixing granulation stage (in a designated granulator) followed by drying/toasting in an oven. ii) a continuous fluidised bed followed by drying/toasting in an oven. In addition, the particle breakage of granola during pneumatic conveying produced by both a high shear granulator (HSG) and fluidised bed granulator (FBG) process were examined. Products were pneumatically conveyed in a purpose built conveying rig designed to mimic product conveying and packaging. Three different conveying rig configurations were employed; a straight pipe, a rig consisting two 45° bends and one with 90° bend. It was observed that the least amount of breakage occurred in the straight pipe while the most breakage occurred at 90° bend pipe. Moreover, lower levels of breakage were observed in two 45° bend pipe than the 90° bend vi pipe configuration. In general, increasing the impact angle increases the degree of breakage. Additionally for the granules produced in the HSG, those produced at 300 rpm have the lowest breakage rates while the granules produced at 150 rpm have the highest breakage rates. This effect clearly the importance of shear history (during granule production) on breakage rates during subsequent processing. In terms of the FBG there was no single operating parameter that was deemed to have a significant effect on breakage during subsequent conveying. A population balance model was developed to analyse the particle breakage occurring during pneumatic conveying. The population balance equations that govern this breakage process are solved using discretization. The Markov chain method was used for the solution of PBEs for this process. This study found that increasing the air velocity (by increasing the air pressure to the rig), results in increased breakage among granola aggregates. Furthermore, the analysis carried out in this work provides that a greater degree of breakage of granola aggregates occur in line with an increase in bend angle.