Information theoretic aspects of the control and the mode estimation of stochastic systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2004.Includes bibliographical references (leaves 131-137).(cont.) parallel with a communication paradigm and deriving an analysis of performance. In our approach, the switching system is viewed as an encoder of the mode, which is interpreted as the message, while a probing signal establishes a random code. Using a distortion function, we define an uncertainty ball where the estimates are guaranteed to lie with probability arbitrarily close to 1. The radius of the uncertainty ball is directly related to the entropy rate of the switching process.In this thesis, we investigate three problems: the first broaches the control under information constraints in the presence of uncertainty; in the second we derive a new fundamental limitation of performance in the presence of finite capacity feedback; while the third studies the estimation of Hidden Markov Models. Problem 1: We study the stabilizability of uncertain stochastic systems in the presence of finite capacity feedback. We consider a stochastic digital link that sends words whose size is governed by a random process. Such link is used to transmit state measurements between the plant and the controller. We derive necessary and sufficient conditions for internal and external stabilizability of the feedback loop. In addition, stability in the presence of uncertainty in the plant is analyzed using a small-gain argument. Problem 2: We address a fundamental limitation of performance for feedback systems, in the presence of a communication channel. The feedback loop comprises a discrete-time, linear and time-invariant plant, a channel, an encoder and a decoder which may also embody a controller. We derive an inequality of the form L ̲>[or equal to] [epsilon]max ... - C[channel] where L ̲is a measure of disturbance rejection, A is the open loop dynamic matrix and Cchannel is the Shannon capacity of the channel. Our measure L ̲is non-negative and smaller L ̲indicates better rejection (attenuation), while L ̲= 0 signifies no rejection. Additionally, we prove that, under a stationarity assumption, L ̲admits a log-sensitivity integral representation. Problem 3: We tackle the problem of mode estimation in switching systems. From the theoretical point of view, our contribution is twofold: creating a framework that has a clearby Numo Miguel Lara Cintra Martins.Ph.D

New Jersey Institute of Technology 1977-1979 Graduate Division Programs

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Hierarchical Stability and Chaotic Motion of Gravitational Few-Body Systems

In this thesis the hierarchical stability and chaotic motion of the classical few body system are studied, and then extended into the framework of the relativistic theory of gravitation. Because of the importance of integrability to both hierarchical stability and Hamiltonian chaos, a general discussion is also given on integrals and symmetries using the modern language of differential geometry. The study of this thesis is closely related to the stability problem of our Solar System and the mass transfer process of compact binary star systems. The approach carried out is both computational and theoretical. The computational part is a systematical investigation of the hierarchical stability (no drastic change in orbital elements or of the hierarchy) of the general 3-body problem, in comparison with the Hill-type stability. The importance of eccentricity in relation to stability is manifest, and the complexity of the phase space structure and fractal nature of the boundary between regular and chaotic regions are reflected in this study. The theoretical work is a continuation of the investigations of the effects of integrals on possible motions. Using a canonical transformation method, a stronger inequality is found for the spatial 3-body problem, giving better estimation of the Hill-type stability regions. It is proved that a Hill-type stability guarantees one of the three hierarchical stability conditions. This classical study is then developed into an inequality method establishing restrictions of symmetries (integrals) on possible motions. The method is first applied to gravitational systems in general relativity and their post-Newtonian approximations. The thesis is split into part I, a general introduction and discussion of the relevant methods, and part II, the original research and main body of the thesis. In chapter 1 a general introduction to the problem of the Solar System's stability is given, with an emphasis on Roy's hierarchical stability and the divergence problem of classical perturbation theory due to chaos. Chapter 2 is a review of the theory of Hamiltonian chaos, presented at a level of comprehending chaos mathematically. The importance of number theory, infinite series and integrability to chaos is emphasised. The geometrical method of studying nonlinear dynamical systems is introduced; classical perturbation theory is used to comprehend the KAM theorem. Particular attention is paid to coordinate-free interpretation of the integrability and separability conditions. In this chapter, a collection of integrable and chaotic systems is given because of their conceptual value to later chapters. Based on the Toda and Henon-Heiles Hamiltonian systems, a discussion is given on the general relationship of a system to its truncated system. This suggests a similar situation for the geodesic motion in Kerr geometry. Chapter 3 is the last chapter of part I on chaos. In this chapter we study the history of chaotic dynamics and its impact on science in general. Although it is standard to study quantization of regular and chaotic motions, the present author pays particular attention to a philosophical compatibility between the theory of chaotic attractors and quanium mechanics. Noting that the two revolutionary theories were born at almost the same time, and that Poincare was a contributor to both theories, the present author carries out a historical search for a possible mutual influence in the development of the theories. However, it is found that such a connection is surprisingly tenuous. The original work is included in part II. The classical 3-body problem is studied in chapters 4 and 5; and the relativistic few-body problem is studied in chapters 6 and 7. In chapter 4, we first review the previous approaches on the Hill-type stability of the general 3-body problem. It is found that all results of previous studies are equivalent and do not go beyond a direct use of Sundman's inequality. Zare's (1976) canonical transformation study on the coplanar 3-body problem is modified and applied to the spatial problem, thus obtaining inequalities stronger than Sundman's. These inequalities determine the best possible Hill-type stability regions for the general 3-body problem, although the critical configurations and the value of (C2H)c cannot be improved. In this approach, it is found that the moment of inertia ellipse of the system may be used to simplify the calculation. Because of this, it is hoped that the same stronger inequalities may also apply to systems with more than three bodies. (Abstract shortened by ProQuest.)

Applications of robust optimal control to decision making in the presence of uncertainty

This thesis is concerned with robustness of decision making in financial economics. Feedback control models developed in engineering are applied to three separate though linked problems in order to examine the role and impact of robustness in the creation and application of decision rules. Three problems are examined using robust optimal control techniques to evaluate the impact of robustness and stability in financial economic models. The first problem examines the use of linear models of robust optimal control in the pricing of castastrophe based derivatives and finds its relative performance to be superior to the popular jump diffusion and stochastic volatility models in the pricing of these emerging instruments. The novelty of the approach arises from the examination of the impact of robustness and stability of the pricing solution. The second problem involves robustness and stability of hedging. An alternative method of creating hedging rules is developed. The method is based on robust control Lyapunov functions that are simple, robust and stable in operation, yet in practice are not so conservative that they eliminate all trading gains. The third problem involves the development of robust control policies for managing risk, using non-linear robust optimal control techniques to provide clear evidence of superior performance of robust models when compared with existing VAR and EVT approaches to risk management. The novelty in the approach arises from the development of a simple and powerful risk management metric

Gravitational radiation theory

A survey is presented of current research in the theory of gravitational radiation. The mathematical structure of gravitational radiation is stressed. Furthermore, the radiation problem is treated independently from other problems in gravitation. The development proceeds candidly through three points of view - scalar, rector, and tensor radiation theory - and the corresponding results are stated

Proceedings of the Workshop on Applications of Distributed System Theory to the Control of Large Space Structures

Two general themes in the control of large space structures are addressed: control theory for distributed parameter systems and distributed control for systems requiring spatially-distributed multipoint sensing and actuation. Topics include modeling and control, stabilization, and estimation and identification