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<div>The dissertation consists of two research topics.</div><div><br></div><div>The first research direction is to study stability of functional and geometric inequalities. Stability problem is to estimate the deficit of a functional or geometric inequality in terms of the distance from the class of optimizers or a functional that identifies the optimizers. In particular, we investigate the logarithmic Sobolev inequality, the Beckner-Hirschman inequality (the entropic uncertainty principle), and isoperimetric type inequalities for the expected lifetime of Brownian motion. </div><div><br></div><div>The second topic of the thesis is a stochastic representation of fractional integrals and nonlocal operators. We extend the Hardy-Littlewood-Sobolev inequality to symmetric Markov semigroups. To this end, we construct a stochastic representation of the fractional integral using the background radiation process. The inequality follows from a new inequality for the fractional Littlewood-Paley square function. We also prove the Hardy-Stein identity for non-symmetric pure jump Levy processes and the L^p boundedness of a certain class of Fourier multiplier operators arising from non-symmetric pure jump Levy processes. The proof is based on Ito's formula for general jump processes and the symmetrization of Levy processes. <br></div>

The scheduling and Control Group at the University of South Australia has been studying the optimal control of trains for many years, and has developed in-cab devices that help drivers stay on time and minimise energy use. In this thesis, we re-examine the optimal control theory for the train control problem. In particular, we study the optimal control around steep sections of track. To calculate an optimal driving strategy we need a realistic model of train performance. In particular, we need to know a coefficient of rolling resistance and a coefficient of aerodynamic drag. In practice, these coefficients are different for every train and difficult to predict. In the thesis, we study the use of mathematical filters to estimate model parameters from observations of actual train performance.

Quantitative Feedback Theory (QFT) is a control system design methodology founded on the premise that feedback is necessary only because of system uncertainty. Articulating flexible structures, such as flexible manipulators, present a difficult closed-loop control problem. In such servo systems, the coupling of the rigid and flexible modes and the non-minimum phase dynamics severely limit system stability and performance. The difficulties in controlling these structures is exacerbated by the denumerably infinite number of flexible modes and associated difficulties in developing accurate dynamic models for controller design. As such, the control of articulating flexible structures presents a non-trivial testbed for the design of QFT based robust control systems. This dissertation examines the multi-input multi-output (MIMO) QFT based control of an articulating flexible structure and presents an enhancement of the theoretical basis for the MIMO QFT design methodologies. The control problem under consideration is the active vibration control of an articulating single-link flexible manipulator. This is facilitated by an actuation scheme comprised of a combination of spatially discrete actuation, in the form of a DC motor to perform articulation, and spatially distributed actuation, in the form of a piezoelectric transducer for active vibration control. In the process of developing and experimentally validating the QFT based control system, shortcomings in the theoretical basis for the MIMO QFT design methodologies are addressed. Robust stability theorems are developed for the two main MIMO QFT design methodologies, namely the sequential and non-sequential MIMO QFT design methodologies. The theorems complement and extend the existing theoretical basis for the MIMO QFT design methodologies. The dissertation results expose salient features of the MIMO QFT design methodologies and provide connections to other multivariable design methodologies.

Quantitative Feedback Theory (QFT) is a control system design methodology founded on the premise that feedback is necessary only because of system uncertainty. Articulating flexible structures, such as flexible manipulators, present a difficult closed-loop control problem. In such servo systems, the coupling of the rigid and flexible modes and the non-minimum phase dynamics severely limit system stability and performance. The difficulties in controlling these structures is exacerbated by the denumerably infinite number of flexible modes and associated difficulties in developing accurate dynamic models for controller design. As such, the control of articulating flexible structures presents a non-trivial testbed for the design of QFT based robust control systems. This dissertation examines the multi-input multi-output (MIMO) QFT based control of an articulating flexible structure and presents an enhancement of the theoretical basis for the MIMO QFT design methodologies. The control problem under consideration is the active vibration control of an articulating single-link flexible manipulator. This is facilitated by an actuation scheme comprised of a combination of spatially discrete actuation, in the form of a DC motor to perform articulation, and spatially distributed actuation, in the form of a piezoelectric transducer for active vibration control. In the process of developing and experimentally validating the QFT based control system, shortcomings in the theoretical basis for the MIMO QFT design methodologies are addressed. Robust stability theorems are developed for the two main MIMO QFT design methodologies, namely the sequential and non-sequential MIMO QFT design methodologies. The theorems complement and extend the existing theoretical basis for the MIMO QFT design methodologies. The dissertation results expose salient features of the MIMO QFT design methodologies and provide connections to other multivariable design methodologies.

This text presents the development of a four-degree-of-freedom (4DOF) quadcopter prototype that allows the vehicle to rotate around the three axes (yaw, pitch and roll) and linear movement along z-axis (altitude). The goal is to obtain a prototype bench that uses a good amount of components used in commercial quadcopters (sensors and actuators) and use it to apply attitude and altitude controllers, using techniques such as PID, LQR and Sliding-Mode. Starting from the system modeling, its specifications are shown followed by listing the components used, finishing with the development of the controllers and their simulations and applications. / Este texto apresenta o desenvolvimento de um protótipo de quadricóptero com quatro graus de liberdade (4DOF), o qual possibilita a rotação do veículo em torno dos três eixos (yaw, pitch e roll) e o deslocamento ao longo do eixo z (altitude). O objetivo é obter um protótipo de bancada que use a maior quantidade de componentes de um quadricóptero comercial (sensores e atuadores) e usá-lo para a aplicação de controladores de atitude e altitude, utilizando técnicas PID, LQR e Sliding-Mode. Partindo da modelagem do sistema, mostra-se as especificações do mesmo, os componentes utilizados e finaliza-se com o desenvolvimento dos controladores, simulação e aplicação deles.

This text presents the development of a four-degree-of-freedom (4DOF) quadcopter prototype that allows the vehicle to rotate around the three axes (yaw, pitch and roll) and linear movement along z-axis (altitude). The goal is to obtain a prototype bench that uses a good amount of components used in commercial quadcopters (sensors and actuators) and use it to apply attitude and altitude controllers, using techniques such as PID, LQR and Sliding-Mode. Starting from the system modeling, its specifications are shown followed by listing the components used, finishing with the development of the controllers and their simulations and applications. / Este texto apresenta o desenvolvimento de um protótipo de quadricóptero com quatro graus de liberdade (4DOF), o qual possibilita a rotação do veículo em torno dos três eixos (yaw, pitch e roll) e o deslocamento ao longo do eixo z (altitude). O objetivo é obter um protótipo de bancada que use a maior quantidade de componentes de um quadricóptero comercial (sensores e atuadores) e usá-lo para a aplicação de controladores de atitude e altitude, utilizando técnicas PID, LQR e Sliding-Mode. Partindo da modelagem do sistema, mostra-se as especificações do mesmo, os componentes utilizados e finaliza-se com o desenvolvimento dos controladores, simulação e aplicação deles.

Complex Systems-of-Systems, which form the subject matter of the research presented here, may be considered to be systems that are constituted of several components, each of which could themself be regarded as being a complete system in its own right. These component systems are typically largely managed independently of one another. They each have their own intrinsic goals which they are capable of fulfilling autonomously. These goals are additional to any to which they might contribute as part of the overall System-of-Systems that is being investigated. Systems-of-Systems are epitomised by features such as the prevalence of complex webs of mutual interdependency amongst their component systems, the involvement of a broad diversity of stakeholders and the participation of a number of independent decision-makers, each of whom competes with the others for strictly limited resources. Furthermore, the components of Systems-of-Systems may often be found to be widely scattered across an extensive geographical region. Sometimes such systems may even span global distances. Complex Systems-of-Systems of such a nature would appear to be becoming ever more prevalent in the increasingly interconnected and ever shrinking world which is engendered, in particular, by the proliferation of modern information and communication technology. Systems-of-Systems may arise in both commercial and public sector contexts, hence commercial organisations and governments, alike, confront the daunting challenge of planning, establishing and maintaining novel systems of this type. However, it is to be anticipated that significant difficulties might be encountered in applying existing systems engineering methods, which were designed for dealing with far more monolithic types of system, to the treatment of Systems-of- Systems. Instead, the successful achievement of acceptable resolutions to the various Systems-of-Systems issues and conundrums with which the responsible planners can be expected to be confronted would appear to demand the application of sophisticated distributed decision aids, in order to alleviate the the intolerable burden which the decision-makers would otherwise be forced to endure. Such aids need to be able to take a genuinely fair and unbiased account of the interests of a multitude of stakeholders. At the same time, they must be able to satisfactorily accommodate the enforcement of a range of highly involved and intricate constraints upon the various alternative interventions which might potentially be contemplated. In order to underpin the development of decision aids of the kind demanded, it has been sought to establish some basic theoretical foundations for Systems-of-Systems. This body of theory has then been applied to the identification of the methods which might be most suitable for the treatment of Systems-of-Systems questions. Finally, some decision-support tools which are intended to facilitate the implementation of the methods that have been recommended have been described. Any attempt to dictate a single â??bestâ?? intervention to the decision-makers would generally be totally unacceptable. As a result, the objective which has been pursued has been to seek to filter out and identify a relatively sparing number of what would appear to be the most reasonable alternatives from some much more profuse range of possibilities. These may then be afforded more intensive scrutiny by the pertinent decision-makers.