Time-Delay Systems: Analysis and Control using the Lambert W Function.

Time-delay systems can arise due to inherent time-delays in the system or a deliberate introduction of time-delays into the system for control purposes. Such systems frequently occur in engineering and science. Time-delays can cause significant problems (e.g., instability and inaccuracy) and, thus, limit and degrade achievable performance. Time-delay terms lead to an infinite number of roots of the characteristic equation, and make analysis difficult using classical methods, especially, in determining stability and designing stabilizing controllers. Thus, such problems have been addressed mainly by using approximate, numerical, and graphical methods. However, such approaches constitute limitations, for example, on accuracy and robustness. The objective of this research is to develop an effective approach to analyze and control time-delay systems. Using the LambertWfunction, free and forced analytical solutions to delay differential equations are derived. The main advantage of this solution approach lies in the fact that the solution has an analytical form expressed in terms of system parameters and, thus, one can explicitly determine how each parameter affects each eigenvalue and the solution. Also, each eigenvalue in the infinite eigenspectrum is associated individually with a branch of the LambertWfunction. Solutions are obtained, for the first time, for systems of delay differential equations using the matrix Lambert W function. The obtained solutions are used to analyze essential system properties, such as stability, controllability and observability, and to design controllers for stabilizing systems, improving robustness and/or meeting time-domain specifications. Then, these methods are applied to biological systems to analyze the immune system via eigenvalue sensitivity analysis, to automotive powertrain systems to design feedback control with observers for improvements in fuel economy and emissions, and to manufacturing processes to improve productivity via stability analysis. The newly developed approach based on the matrix Lambert W function provides a tool for analysis and control, which is accurate (i.e., no approximation of timedelay terms), robust (i.e., no prediction of responses from models), and easy to implement (i.e., no need for complex nonlinear controllers).Ph.D.Mechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64756/1/syjo_1.pd

Delay-dependent Stability of Genetic Regulatory Networks

Genetic regulatory networks are biochemical reaction systems, consisting of a network of interacting genes and associated proteins. The dynamics of genetic regulatory networks contain many complex facets that require careful consideration during the modeling process. The classical modeling approach involves studying systems of ordinary differential equations (ODEs) that model biochemical reactions in a deterministic, continuous, and instantaneous fashion. In reality, the dynamics of these systems are stochastic, discrete, and widely delayed. The first two complications are often successfully addressed by modeling regulatory networks using the Gillespie stochastic simulation algorithm (SSA), while the delayed behavior of biochemical events such as transcription and translation are often ignored due to their mathematically difficult nature. We develop techniques based on delay-differential equations (DDEs) and the delayed Gillespie SSA to study the effects of delays, in both continuous deterministic and discrete stochastic settings. Our analysis applies techniques from Floquet theory and advanced numerical analysis within the context of delay-differential equations, and we are able to derive stability sensitivities for biochemical switches and oscillators across the constituent pathways, showing which pathways in the regulatory networks improve or worsen the stability of the system attractors. These delay sensitivities can be far from trivial, and we offer a computational framework validated across multiple levels of modeling fidelity. This work suggests that delays may play an important and previously overlooked role in providing robust dynamical behavior for certain genetic regulatory networks, and perhaps more importantly, may offer an accessible tuning parameter for robust bioengineering

Differential geometric MCMC methods and applications

This thesis presents novel Markov chain Monte Carlo methodology that exploits the natural representation of a statistical model as a Riemannian manifold. The methods developed provide generalisations of the Metropolis-adjusted Langevin algorithm and the Hybrid Monte Carlo algorithm for Bayesian statistical inference, and resolve many shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlation structure. The performance of these Riemannian manifold Markov chain Monte Carlo algorithms is rigorously assessed by performing Bayesian inference on logistic regression models, log-Gaussian Cox point process models, stochastic volatility models, and both parameter and model level inference of dynamical systems described by nonlinear differential equations