A perturbation approach to the stabilization of nonlinear cascade systems with time-delay

The effect of bounded input perturbations on the stability of nonlinear globally asymptotically stable delay differential equations is analyzed. We investigate under which conditions global stability is preserved and if not, whether semi-global stabilization is possible by controlling the size or shape of the perturbation. These results are used to study the stabilization of partially linear cascade systems with partial state feedback.

Nonlinear thermoacoustics of ducted premixed flames: The influence of perturbation convection speed

When a premixed flame is placed within a duct, acoustic waves induce velocity perturbations at the flame's base. These travel down the flame, distorting its surface and modulating its heat release. This can induce self-sustained thermoacoustic oscillations. Although the phase speed of these perturbations is often assumed to equal the mean flow speed, experiments conducted in other studies and Direct Numerical Simulation (DNS) conducted in this study show that it varies with the acoustic frequency. In this paper, we examine how these variations affect the nonlinear thermoacoustic behaviour. We model the heat release with a nonlinear kinematic G-equation, in which the velocity perturbation is modelled on DNS results. The acoustics are governed by linearised momentum and energy equations. We calculate the flame describing function (FDF) using harmonic forcing at several frequencies and amplitudes. Then we calculate thermoacoustic limit cycles and explain their existence and stability by examining the amplitude-dependence of the gain and phase of the FDF. We find that, when the phase speed equals the mean flow speed, the system has only one stable state. When the phase speed does not equal the mean flow speed, however, the system supports multiple limit cycles because the phase of the FDF changes significantly with oscillation amplitude. This shows that the phase speed of velocity perturbations has a strong influence on the nonlinear thermoacoustic behaviour of ducted premixed flames. © 2013 The Combustion Institute.

Multispacecraft refueling optimization considering the J2 perturbation and window constraints

The optimization of a near-circular low-Earth-orbit multispacecraft refueling problem is studied. The refueling sequence, service time, and orbital transfer time are used as design variables, whereas the mean mission completion time and mean propellant consumed by orbital maneuvers are used as design objectives. The J2 term of the Earth's nonspherical gravity perturbation and the constraints of rendezvous time windows are taken into account. A hybridencoding genetic algorithm, which uses normal fitness assignment to find the minimum mean propellant-cost solution and fitness assignment based on the concept of Pareto-optimality to find multi-objective optimal solutions, is presented. The proposed approach is demonstrated for a typical multispacecraft refueling problem. The results show that the proposed approach is effective, and that the J2 perturbation and the time-window constraints have considerable influences on the optimization results. For the problems in which the J2 perturbation is not accounted for, the optimal refueling order can be simply determined as a sequential order or as the order only based on orbitalplane differences. In contrast, for the problems that do consider the J2 perturbation, the optimal solutions obtained have a variety of refueling orders and use the drift of nodes effectively to reduce the propellant cost for eliminating orbital-plane differences. © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Comparison of acoustic velocity perturbation measurements using PIV vs. two-microphone technique

Understanding combustion instabilities requires accurate measurements of the acoustic velocity perturbation into injectors. This is often accomplished via the use of the two microphone technique, as this only requires two pressure transducers. However, measurements of the actual velocities emerging from the injectors are not often taken, leaving questions regarding the assumptions about the acoustic velocity. A comparison of velocity measured at downstream of the injector with that of two-microphone technique can show the accuracy and limitations of two-microphone technique. In this paper, velocity measurements are taken using both particle image velocimetry (PIV) and the two-microphone technique in a high pressure facility designed for aeroengine injector measurements. The flow is excited using an area modulation device installed on the choked end of the combustion chamber, with PIV measurements enabled by optical access downstream of the injector through a quartz tube and windows. Acoustic velocity perturbations at the injector are determined by considering the Fourier transformed pressure fluctuations for two microphones installed at a known distance upstream of the injector. PIV measurements are realized by seeding the air flow with micrometric water particles under 2.5 bar pressure at ambient temperature. Phase locked velocity fields are realized by synchronizing the acquisition of PIV images with the revolution of the acoustic modulator using the pressure signal measured at the face of injector. The mean velocity fluctuation is calculated as the difference between maximum and minimum velocities, normalized by the mean velocity of the unforced case. Those values are compared with the peak-to-peak velocity fluctuation amplitude calculated by the two-microphone technique. Although the ranges of velocity fluctuations for both techniques are similar, the variation of fluctuation with forcing frequencies diverges significantly with frequency. The differences can be attributed to several limitations associated with of both techniques, such as the quality of the signal, the signal/noise ratio, the accuracy of PIV measurements and the assumption of isentropic flow of the particle velocity from the plenum through the injector. We conclude that two-microphone methods can be used as a reference value for the velocity fluctuation in low order applications such as flame transfer functions, but not for drawing conclusions regarding the absolute velocity fluctuations in the injector. Copyright © 2013 by ASME.

A non-linear perturbation model considering catchment wetness and its application in river flow forecasting

A non-linear perturbation model for river flow forecasting is developed, based on consideration of catchment wetness using an antecedent precipitation index (API). Catchment seasonality, of the form accounted for in the linear perturbation model (the LPM), and non-linear behaviour both in the runoff generation mechanism and in the flow routing processes are represented by a constrained nan-linear model, the NLPM-API. A total of ten catchments, across a range of climatic conditions and catchment area magnitudes, located in China and in other countries, were selected for testing daily rainfall-runoff forecasting with this model. It was found that the NLPM-API model was significantly more efficient than the original linear perturbation model (the LPM). However, restric tion of explicit nan-linearity to the runoff generation process, in the simpler LPM-API form of the model, did not produce a significantly lower value of the efficiency in flood forecasting, in terms of the model efficiency index R-2. (C) 1997 Elsevier Science B.V.

Algorithm Studies on How to Obtain a Conditional Nonlinear Optimal Perturbation (CNOP)

The conditional nonlinear optimal perturbation (CNOP), which is a nonlinear generalization of the linear singular vector (LSV), is applied in important problems of atmospheric and oceanic sciences, including ENSO predictability, targeted observations, and ensemble forecast. In this study, we investigate the computational cost of obtaining the CNOP by several methods. Differences and similarities, in terms of the computational error and cost in obtaining the CNOP, are compared among the sequential quadratic programming (SQP) algorithm, the limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm, and the spectral projected gradients (SPG2) algorithm. A theoretical grassland ecosystem model and the classical Lorenz model are used as examples. Numerical results demonstrate that the computational error is acceptable with all three algorithms. The computational cost to obtain the CNOP is reduced by using the SQP algorithm. The experimental results also reveal that the L-BFGS algorithm is the most effective algorithm among the three optimization algorithms for obtaining the CNOP. The numerical results suggest a new approach and algorithm for obtaining the CNOP for a large-scale optimization problem.

Restructuring of a zooplankton community by perturbation from a wind-forced coastal jet

The impact of transient wind events on an established zooplankton community was observed during a, field survey in a, coastal region off northern Norway in May 2002. A transient wind event induced a coastal jet/filament intrusion of warm, saline water into our survey area where a semi-permanent eddy was present. There was an abrupt change in zooplankton community structure within 4-7 days of the wind event, with a change in the size structure, an increase in lower size classes less than 1 mm in equivalent spherical diameter (ESD) and a decrease in larger size classes greater than 1.5 mm in ESD. The slope of zooplankton biovolume spectra changed from -0.6 to -0.8, consistent with the size shifting towards smaller size classes. This study shows that even well established zooplankton communities are susceptible to restructuring during transient wind events, and in particular when wind forcing induces horizontal currents or filaments.

A Perturbation Scheme for Spherical Arrangements with Application to Molecular Modeling

We describe a software package for computing and manipulating the subdivision of a sphere by a collection of (not necessarily great) circles and for computing the boundary surface of the union of spheres. We present problems that arise in the implementation of the software and the solutions that we have found for them. At the core of the paper is a novel perturbation scheme to overcome degeneracies and precision problems in computing spherical arrangements while using floating point arithmetic. The scheme is relatively simple, it balances between the efficiency of computation and the magnitude of the perturbation, and it performs well in practice. In one O(n) time pass through the data, it perturbs the inputs necessary to insure no potential degeneracies and then passes the perturbed inputs on to the geometric algorithm. We report and discuss experimental results. Our package is a major component in a larger package aimed to support geometric queries on molecular models; it is currently employed by chemists working in "rational drug design." The spherical subdivisions are used to construct a geometric model of a molecule where each sphere represents an atom. We also give an overview of the molecular modeling package and detail additional features and implementation issues.

Asymptotics of Perturbations to the Wave Equation

M. Hieber, I. Wood: Asymptotics of perturbations to the wave equation. In: Evolution Equations, Lecture Notes in Pure and Appl. Math., 234, Marcel Dekker, (2003), 243-252.

Quantum Stratonovich Stochastic Calculus and the Quantum Wong-Zakai Theorem

Gough, John, 'Quantum Stratonovich Stochastic Calculus and the Quantum Wong-Zakai Theorem', Journal of Mathematical Physics. 47, 113509, (2006)

Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions

Gough, John, (2004) 'Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions', Communications in Mathematical Physics 254 pp.498-512 RAE2008

An investigation of post-primary students' images of mathematics

This research study investigates the image of mathematics held by 5th-year post-primary students in Ireland. For this study, “image of mathematics” is conceptualized as a mental representation or view of mathematics, presumably constructed as a result of past experiences, mediated through school, parents, peers or society. It is also understood to include attitudes, beliefs, emotions, self-concept and motivation in relation to mathematics. This study explores the image of mathematics held by a sample of 356 5th-year students studying ordinary level mathematics. Students were aged between 15 and 18 years. In addition, this study examines the factors influencing students‟ images of mathematics and the possible reasons for students choosing not to study higher level mathematics for the Leaving Certificate. The design for this study is chiefly explorative. A questionnaire survey was created containing both quantitative and qualitative methods to investigate the research interest. The quantitative aspect incorporated eight pre-established scales to examine students‟ attitudes, beliefs, emotions, self-concept and motivation regarding mathematics. The qualitative element explored students‟ past experiences of mathematics, their causal attributions for success or failure in mathematics and their influences in mathematics. The quantitative and qualitative data was analysed for all students and also for students grouped by gender, prior achievement, type of post-primary school attending, co-educational status of the post-primary school and the attendance of a Project Maths pilot school. Students‟ images of mathematics were seen to be strongly indicated by their attitudes (enjoyment and value), beliefs, motivation, self-concept and anxiety, with each of these elements strongly correlated with each other, particularly self-concept and anxiety. Students‟ current images of mathematics were found to be influenced by their past experiences of mathematics, by their mathematics teachers, parents and peers, and by their prior mathematical achievement. Gender differences occur for students in their images of mathematics, with males having more positive images of mathematics than females and this is most noticeable with regards to anxiety about mathematics. Mathematics anxiety was identified as a possible reason for the low number of students continuing with higher level mathematics for the Leaving Certificate. Some students also expressed low mathematical self-concept with regards to higher level mathematics specifically. Students with low prior achievement in mathematics tended to believe that mathematics requires a natural ability which they do not possess. Rote-learning was found to be common among many students in the sample. The most positive image of mathematics held by students was the “problem-solving image”, with resulting implications for the new Project Maths syllabus in post-primary education. Findings from this research study provide important insights into the image of mathematics held by the sample of Irish post-primary students and make an innovative contribution to mathematics education research. In particular, findings contribute to the current national interest in Ireland in post-primary mathematics education, highlighting issues regarding the low uptake of higher level mathematics for the Leaving Certificate and also making a preliminary comparison between students who took part in the piloting of Project Maths and students who were more recently introduced to the new syllabus. This research study also holds implications for mathematics teachers, parents and the mathematics education community in Ireland, with some suggestions made on improving students‟ images of mathematics.

Politics, Power, PISA: a genealogy of mathematics education policy at second level in Ireland at the beginning of the 21st century

This thesis traces a genealogy of the discourse of mathematics education reform in Ireland at the beginning of the twenty first century at a time when the hegemonic political discourse is that of neoliberalism. It draws on the work of Michel Foucault to identify the network of power relations involved in the development of a single case of curriculum reform – in this case Project Maths. It identifies the construction of an apparatus within the fields of politics, economics and education, the elements of which include institutions like the OECD and the Government, the bureaucracy, expert groups and special interest groups, the media, the school, the State, state assessment and international assessment. Five major themes in educational reform emerge from the analysis: the arrival of neoliberal governance in Ireland; the triumph of human capital theory as the hegemonic educational philosophy here; the dominant role of OECD/PISA and its values in the mathematics education discourse in Ireland; the fetishisation of western scientific knowledge and knowledge as commodity; and the formation of a new kind of subjectivity, namely the subjectivity of the young person as a form of human-capital-to-be. In particular, it provides a critical analysis of the influence of OECD/PISA on the development of mathematics education policy here – especially on Project Maths curriculum, assessment and pedagogy. It unpacks the arguments in favour of curriculum change and lays bare their ideological foundations. This discourse contextualises educational change as occurring within a rapidly changing economic environment where the concept of the State’s economic aspirations and developments in science, technology and communications are reshaping both the focus of business and the demands being put on education. Within this discourse, education is to be repurposed and its consequences measured against the paradigm of the Knowledge Economy – usually characterised as the inevitable or necessary future of a carefully defined present.