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Title:Design of minimax controllers for nonlinear systems using cost-to-come methods
Author(s):Didinsky, Garry
Doctoral Committee Chair(s):Basar, Tamer
Department / Program:Engineering, Electronics and Electrical
Discipline:Engineering, Electronics and Electrical
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Engineering, Electronics and Electrical
Abstract:This thesis develops and utilizes the cost-to-come methodology for the construction of minimax controllers for nonlinear systems with partial-state information (PSI), which are subjected to deterministic uncertainty. It introduces the notion of a cost-to-come function and shows how it leads to necessary and sufficient conditions for the existence of a minimax controller. These conditions are the existence of the cost-to-come function and the existence of a solution to an auxiliary full-information (FI) minimax problem. It is also proven that a solution of the FI problem leads directly to a solution of the original PSI problem. The significance of this result is that the auxiliary FI problem can be solved using better-known dynamic programming (DP) techniques.
Generally, the auxiliary problem has an infinite-dimensional state, which may make the cost-to-come methodology impractical. Toward resolving this impediment, the thesis first identifies two classes of problems for which cost-to-come methods offer significant insights: one, problems that satisfy a certainty-equivalence principle (which is attributed to the case when a minimax controller can be chosen to be a full-state information (FSI) minimax controller with the state replaced by a "worst-case" value of the state), and two, problems for which the cost-to-come function can be characterized by a finite number of parameters (which is attributed to the case where the auxiliary FI problem has a finite-dimensional state). Outside these two classes, bounding techniques can be used to reduce the problem complexity, at the expense of some possible performance degradation. By bounding the FSI cost-to-go function from above, the certainty-equivalence principle can be generalized to gain some insight into the structure of cost-bounding controllers, which may lead to alternative parameter design methods. Bounding the cost-to-come function by finite-dimensional structured cost-to-come functions allows the construction of an auxiliary FI problem, which leads to a cost-bounding controller policy.
As an application of the cost-to-come methodology, a class of affine-quadratic (AQ) disturbance attenuation problems is considered in some detail. This class contains H$\sp\infty$-optimal control and filtering problems, problems of parameter identification and nonlinear adaptive control, and memoryless control problems. For the former subclass, cost-to-come methods provide an alternative derivation of the well-known necessary and sufficient conditions and of a minimax policy. For the latter three subclasses, new algorithms are developed that lead either directly or recursively to optimal solutions. Some numerical examples and simulation studies are provided to illustrate the theoretical results.
Issue Date:1995
Type:Text
Language:English
URI:http://hdl.handle.net/2142/23641
Rights Information:Copyright 1995 Didinsky, Garry
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9522104
OCLC Identifier:(UMI)AAI9522104


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