Analysis of Jump Linear Systems Driven by Lumped Processes

Safety critical control systems such as flight control systems use fault-tolerant technology to minimize the effect of faults and increase the dependability of the system. In fault-tolerant systems, the system availability process indicates the current operational mode of an interconnection of digital logic devices. It is a process that results from the transformation of the stochastic processes characterizing the availability of the devices forming the system. To assess safety critical control systems, the following measures of performance will be considered: the steady-state mean output power, the mean output energy, the mean time to failure and the mean time to repair. For this assessment it is important to determine the characteristics of the system availability process since both stability and the aforementioned measure of performance are directly dependent on it. When the system availability process results from a transformation of a homogeneous Markov chain, it is well-known that the resulting process is not necessarily a homogeneous Markov chain. In particular, when the Markov chain characterizing the faults is a zeroth order Markov chain, it is shown that the availability process results in another zeroth order Markov chain. Thus, all the results which are known to analyze closed-loop systems driven by a homogeneous Markov chain can be applied to the zeroth order Markov chain. However, simpler formulas that do not trivially follow from these Markov chain results can be derived in this case. Part of this dissertation is dedicated to the derivation of these new formulas. On the other hand, when the system availability results in either a non-homogeneous Markov chain or a non-Markov chain, the existing Markov results can not be directly applied. This problem is addressed here. The necessity for better integration of the fault tolerant and the control designs for safety critical systems is also addressed. The dependability of current designs is primarily assessed with measures of the interconnection of fault tolerant devices: the reliability metrics that include the mean time to failure and the mean time to repair. These metrics do not directly take into account the interaction of the fault tolerant components with the dynamics of the system. In this dissertation, a first step to better integrate fault tolerant and the control designs for safety critical systems is made. These are the problems that motivated this work. Therefore, the goals of this dissertation are: to develop a suitable methodology to analyze a jump linear system when the driving process is characterized by a zeroth order Markov chain, a non-homogeneous Markov chain and a non-Markov chain; and to integrate the analysis of jump linear systems with the reliability theory for network architectures

Performance Analysis of Recoverable Flight Control Systems Subject to Neutron-Induced Upsets Using Hybrid Dynamical Models

It has been observed that atmospheric neutrons can produce single-event upsets in digital flight control hardware. Potentially, they can reduce system performance and introduce a safety hazard. One experimental system-level approach investigated to help mitigate the effects of these upsets is NASA Langley\u27s Recoverable Computer System. It employs rollback error recovery using dual-lock-step processors together with new fault tolerant architectures and communication subsystems. In this dissertation, a class of stochastic hybrid dynamical models, which consists of a jump-linear system and a stochastic finite-state automaton, is used to describe the performance of a Boeing 737 aircraft system in closed-loop with a Recoverable Computer System. The jump-linear system models the switched dynamics of the closed-loop system due to the presence of controller recoveries. Each dynamical model in the jump-linear system was obtained separately using system identification techniques and high fidelity flight simulation software. The stochastic finite-state automaton approximates the recovery logic of the Recoverable Computer System. The upsets process is modeled by either an independent, identically distributed process or a first-order Markov chain. Mean-square stability and output tracking performance of the recoverable flight control system are analyzed theoretically via a model-equivalent Markov jump-linear system of the stochastic hybrid model. The model was validated using data from a controlled experiment at NASA Langley, where simulated neutron-induced upsets were injected into the system at a desired rate. The effects on the output tracking performance of a simulated aircraft were then directly observed and quantified. The model was then used to analyze a neutron-based experiment on the Recoverable Computer System at the Los Alamos National Laboratory. This model predicts that the experimental flight control system, when functioning as designed, will provide robust control performance in the presence of neutron-induced single-event upsets at normal atmospheric levels

Pre-Averaging Based Estimation of Quadratic Variation in the Presence of Noise and Jumps: Theory, Implementation, and Empirical Evidence

This paper provides theory as well as empirical results for pre-averaging estimators of the daily quadratic variation of asset prices. We derive jump robust inference for pre-averaging estimators, corresponding feasible central limit theorems and an explicit test on serial dependence in microstructure noise. Using transaction data of different stocks traded at the NYSE, we analyze the estimators’ sensitivity to the choice of the pre-averaging bandwidth and suggest an optimal interval length. Moreover, we investigate the dependence of pre-averaging based inference on the sampling scheme, the sampling frequency, microstructure noise properties as well as the occurrence of jumps. As a result of a detailed empirical study we provide guidance for optimal implementation of pre-averaging estimators and discuss potential pitfalls in practice.Quadratic Variation, Market Microstructure Noise, Pre-averaging, Sampling Schemes, Jumps

Nonparametric Markovian Learning of Triggering Kernels for Mutually Exciting and Mutually Inhibiting Multivariate Hawkes Processes

In this paper, we address the problem of fitting multivariate Hawkes processes to potentially large-scale data in a setting where series of events are not only mutually-exciting but can also exhibit inhibitive patterns. We focus on nonparametric learning and propose a novel algorithm called MEMIP (Markovian Estimation of Mutually Interacting Processes) that makes use of polynomial approximation theory and self-concordant analysis in order to learn both triggering kernels and base intensities of events. Moreover, considering that N historical observations are available, the algorithm performs log-likelihood maximization in $O(N)$ operations, while the complexity of non-Markovian methods is in $O(N^{2})$. Numerical experiments on simulated data, as well as real-world data, show that our method enjoys improved prediction performance when compared to state-of-the art methods like MMEL and exponential kernels