Continuous automata: bridging the gap between discrete and continuous time system models

The principled use of models in design and maintenance of a system is fundamental to the engineering methodology. As the complexity and sophistication of systems increase so do the demands on the system models required to design them. In particular the design of agent systems situated in the real world, such as robots, will require design models capable of expressing discrete and continuous changes of system parameters. Such systems are referred to as mode-switching or hybrid systems.This thesis investigates ways in which time is represented in automata system models with discretely and continuously changing parameters. Existing automaton approaches to hybrid modelling rely on describing continuous change at a sequence of points in time. In such approaches the time that elapses between each point is chosen non- deterministically in order to ensure that the model does not over-step a discrete change. In contrast, the new approach this thesis proposes describes continuous change by a continuum of points which can naturally and deterministically capture such change. As well as defining the semantics of individual models the nature of the temporal representation is particularly important in defining the composition of modular com­ponents. This new approach leads to a clear compositional semantics based on the synchronization of input and output values.The main contribution of this work is the derivation of a limiting process which provides a theoretical foundation for this new approach. It not only provides a link between dis­crete and continuous time representations, but also provides a basis for deciding which continuous time representations are theoretically sound. The resulting formalism, the Continuous I/O machine, is demonstrated to be comparable to Hybrid Automata in expressibility, but its representation of time gives it a much stronger compositional semantics based on the discrete synchronous machines from which it is derived.TThe conclusion of this work is that it is possible to define an automaton model that describes a continuum of events and that this can be effectively used to model complete mode-switching physical systems in a modular fashion

Stable Realization of a Delay System Modeling a Convergent Acoustic Cone

This paper deals with the physical modeling and the digital time simulation of acoustic pipes. We will study the simplified case of a single convergent cone. It is modeled by a linear system made of delays and a transfer function which represents the wave reflection at the entry of the cone. According to [1], the input/output relation of this system is causal and stable whereas the reflection function is unstable. In the continuous time-domain, a first state space representation of this delay system is done. Then, we use a change of state to separate the unobservable subspace and its orthogonal complement, which is observable. Whereas the unobservable part is unstable, it is proved that the observable part is stable, using the D-Subdivision method. Thus, isolating this latter observable subspace, to build the minimal realization, defines a stable system. Finally, a discrete-time version of this system is derived and is proved to be stable using the Jury criterion. The main contribution of this work is neither the minimal realization of the system nor the proofs of stability, but it is rather the solving of an old problem of acoustics which has heen achieved using standard tools of automatic control

Influence of sampling rate on the calculated fidelity of an aircraft simulation

One of the factors that influences the fidelity of an aircraft digital simulation is the sampling rate. As the sampling rate is increased, the calculated response of the discrete representation tends to coincide with the response of the corresponding continuous system. Because of computer limitations, however, the sampling rate cannot be increased indefinitely. Moreover, real-time simulation requirements demand that a finite sampling rate be adopted. In view of these restrictions, a study was undertaken to determine the influence of sampling rate on the response characteristics of a simulated aircraft describing short-period oscillations. Changes in the calculated response characteristics of the simulated aircraft degrade the fidelity of the simulation. In the present context, fidelity degradation is defined as the percentage change in those characteristics that have the greatest influence on pilot opinion: short period frequency omega, short period damping ratio zeta, and the product omega zeta. To determine the influence of the sampling period on these characteristics, the equations describing the response of a DC-8 aircraft to elevator control inputs were used. The results indicate that if the sampling period is too large, the fidelity of the simulation can be degraded

Processes and continuous change in a SAT-based planner

AbstractThe TM-LPSAT planner can construct plans in domains containing atomic actions and durative actions; events and processes; discrete, real-valued, and interval-valued fluents; reusable resources, both numeric and interval-valued; and continuous linear change to quantities. It works in three stages. In the first stage, a representation of the domain and problem in an extended version of PDDL+ is compiled into a system of Boolean combinations of propositional atoms and linear constraints over numeric variables. In the second stage, a SAT-based arithmetic constraint solver, such as LPSAT or MathSAT, is used to find a solution to the system of constraints. In the third stage, a correct plan is extracted from this solution. We discuss the structure of the planner and show how planning with time and metric quantities is compiled into a system of constraints. The proofs of soundness and completeness over a substantial subset of our extended version of PDDL+ are presented

Riemann-Stieltjes integrals with respect to fractional Brownian motion and applications

In this dissertation we study Riemann-Stieltjes integrals with respect to (geometric) fractional Brownian motion, its financial counterpart and its application in estimation of quadratic variation process. From the point of view of financial mathematics, we study the fractional Black-Scholes model in continuous time. We show that the classical change of variable formula with convex functions holds for the trajectories of fractional Brownian motion. Putting it simply, all European options with convex payoff can be hedged perfectly in such pricing model. This allows us to give new arbitrage examples in the geometric fractional Brownian motion case. Adding proportional transaction costs to the discretized version of the hedging strategy, we study an approximate hedging problem analogous to the corresponding discrete hedging problem in the classical Black-Scholes model. Using the change of variables formula result, one can see that fractional Brownian motion model shares some common properties with continuous functions of bounded variation. We also show a representation for running maximum of continuous functions of bounded variations such that fractional Brownian motion does not enjoy this property

Foundations and modelling of dynamic networks using Dynamic Graph Neural Networks: A survey

Dynamic networks are used in a wide range of fields, including social network analysis, recommender systems, and epidemiology. Representing complex networks as structures changing over time allow network models to leverage not only structural but also temporal patterns. However, as dynamic network literature stems from diverse fields and makes use of inconsistent terminology, it is challenging to navigate. Meanwhile, graph neural networks (GNNs) have gained a lot of attention in recent years for their ability to perform well on a range of network science tasks, such as link prediction and node classification. Despite the popularity of graph neural networks and the proven benefits of dynamic network models, there has been little focus on graph neural networks for dynamic networks. To address the challenges resulting from the fact that this research crosses diverse fields as well as to survey dynamic graph neural networks, this work is split into two main parts. First, to address the ambiguity of the dynamic network terminology we establish a foundation of dynamic networks with consistent, detailed terminology and notation. Second, we present a comprehensive survey of dynamic graph neural network models using the proposed terminologyComment: 28 pages, 9 figures, 8 table

A Discrete Time Presentation of Quantum Dynamics

Inspired by the discrete evolution implied by the recent work on loop quantum cosmology, we obtain a discrete time description of usual quantum mechanics viewing it as a constrained system. This description, obtained without any approximation or explicit discretization, mimics features of the discrete time evolution of loop quantum cosmology. We discuss the continuum limit, physical inner product and matrix elements of physical observables to bring out various issues regarding viability of a discrete evolution. We also point out how a continuous time could emerge without appealing to any continuum limit.Comment: 20 pages, RevTex, no figures. Additional Clarifications added. Version accepted for publication in Class. Quant. Gra

About Lorentz invariance in a discrete quantum setting

A common misconception is that Lorentz invariance is inconsistent with a discrete spacetime structure and a minimal length: under Lorentz contraction, a Planck length ruler would be seen as smaller by a boosted observer. We argue that in the context of quantum gravity, the distance between two points becomes an operator and show through a toy model, inspired by Loop Quantum Gravity, that the notion of a quantum of geometry and of discrete spectra of geometric operators, is not inconsistent with Lorentz invariance. The main feature of the model is that a state of definite length for a given observer turns into a superposition of eigenstates of the length operator when seen by a boosted observer. More generally, we discuss the issue of actually measuring distances taking into account the limitations imposed by quantum gravity considerations and we analyze the notion of distance and the phenomenon of Lorentz contraction in the framework of ``deformed (or doubly) special relativity'' (DSR), which tentatively provides an effective description of quantum gravity around a flat background. In order to do this we study the Hilbert space structure of DSR, and study various quantum geometric operators acting on it and analyze their spectral properties. We also discuss the notion of spacetime point in DSR in terms of coherent states. We show how the way Lorentz invariance is preserved in this context is analogous to that in the toy model.Comment: 25 pages, RevTe