Computational methods in the fractional calculus of variations and optimal control

The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the lack of analytic methods to solve such fractional problems, numerical techniques are developed. Here, we mainly investigate the approximation of fractional operators by means of series of integer-order derivatives and generalized finite differences. We give upper bounds for the error of proposed approximations and study their efficiency. Direct and indirect methods in solving fractional variational problems are studied in detail. Furthermore, optimality conditions are discussed for different types of unconstrained and constrained variational problems and for fractional optimal control problems. The introduced numerical methods are employed to solve some illustrative examples.

In search of reliable centimeter-level indoor positioning: a smartphone-based approach

This thesis describes the design and implementation of a reliable centimeter-level indoor positioning system fully compatible with a conventional smartphone. The proposed system takes advantage of the smartphone audio I/O and processing capabilities to perform acoustic ranging in the audio band using non-invasive audio signals and it has been developed having in mind applications that require high accuracy, such as augmented reality, virtual reality, gaming and audio guides. The system works in a distributed operation mode, i.e. each smartphone is able to obtain its own position using only acoustic signals. To support the positioning system, a Wireless Sensor Network (WSN) of synchronized acoustic beacons is used. To keep the infrastructure in sync we have developed an Automatic Time Synchronization and Syntonization (ATSS) protocol with a standard deviation of the sync offset error below 1.25 μs. Using an improved Time Difference of Arrival (TDoA) estimation approach (which takes advantage of the beacon signals’ periodicity) and by performing Non-Line-of-Sight (NLoS) mitigation, we were able to obtain very stable and accurate position estimates with an absolute mean error of less than 10 cm in 95% of the cases and a mean standard deviation of 2.2 cm for a position refresh period of 350 ms.