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View CartIndefinite-Quadratic Estimation and Control
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Description
This monograph presents a unified mathematical framework for a wide range of problems in estimation and control. The authors discuss the two most commonly used methodologies: the stochastic H2 approach and the deterministic (worst-case) H∞ approach. Despite the fundamental differences in the philosophies of these two approaches, the authors have discovered that, if indefinite metric spaces are considered, they can be treated in the same way and are essentially the same. The benefits and consequences of this unification are pursued in detail, with discussions of how to generalize well-known results from H2 theory to H∞ setting, as well as new results and insight, the development of new algorithms, and applications to adaptive signal processing.
Keywords: estimation, control, H2 theory, H∞ theory, indefinite metric space, Krein space, Hilbert space, state-space models, linear estimation, array algorithms, game theory
Table of Contents
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Front Matter
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1. Introduction and Motivation
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4. Finite-Horizon H∞ Filtering
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5. Array Algorithms
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6. Several Related Problems
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14. Asymptotic Behavior
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15. Optimal H∞ Solutions
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Back Matter
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Excerpt
One of the most significant accomplishments in control theory during the 1980s and early 1990s has been the development of the theory of H∞ control and estimation. Originally introduced by G. Zames in 1980, H∞ theory is concerned with the design of controllers that are robust with respect to model uncertainty and lack of statistical knowledge on the exogenous signals. In this sense, it can be considered as an outgrowth and extension of the now-classical linear-quadratic-Gaussian (LQG) theory, developed in the 1950s and 1960s, which assumed perfect models and complete statistical knowledge. Whereas such assumptions were reasonable for problems in the guidance and manuevering of space vehicles, to which these theories were first applied, they were much less so in several more mundane industrial problems. Although other approaches to robust control have also been introduced and studied, H∞ theory has attracted the most attention by far and has been developed by many authors using various ingenious methods and employing tools from interpolation theory, operator theory, game theory, circuit theory, and system theory. This abundance of viewpoints poses at least two challenges. One is practical: How to master them all? The other is theoretical: What is the theoretical core of H∞ theory? Rather than attempting to address these questions directly, the theme of this monograph is that such very different solution methods need not be necessary; the basic LQG and Kalman filtering arguments can still be used, provided we set up appropriate control and estimation problems with elements not in a Hilbert space, but in an indefinite-metric (so-called Krein) space.
©1999 SIAM
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