Institute for Systems Research
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Item Robust H∞ Output Feedback Control of Bilinear Systems(1996) Teolis, C.A.; Yuliar, S.; James, Matthew R.; Baras, John S.; ISRThe study of robust nonlinear control has attracted increasing interest over the last few years. Progress has been aided by the recent entension [FM91, Jam92] of the linear quadratic results [Jac73, Whi81] linking the theories of L2 gain control (nonlinear H∞ control), different games, and the stochastic risk sensitive control. Most of the previous research conducted in the area of robust nonlinear control has focused on the case where full state information is available. Thus, previously little attention has been given to the problem of robust nonlinear control via output feedback. In this paper we address the problem of robust H∞ output feedback control for the special case of bilinear systems.Item Robust and Risk-Sensitive Output Feedback Control for Finite State Machines and Hidden Markov Models(1994) Baras, John S.; James, Matthew R.; ISRThe purpose of this paper is to develop a framework for designing controllers for finite state systems which are robust with respect to uncertainties. A deterministic model for uncertainties is introduced, leading to a dynamic game formulation of the robust control problem. This problem is solved using an appropriate information state. A risk-sensitive stochastic control problem is formulated and solved for Hidden Markov Models, corresponding to situations where the model for the uncertainties is stochastic. The two problems are related using small noise limits.Item Partially Observed Differential Games, Infinite Dimensional HJI Equations, and Nonlinear HControl(1994) James, Matthew R.; Baras, John S.; ISRThis paper presents new results for partially observed nonlinear differential games, which are applied to the nonlinear output feedback Hrobust control problem. Using the concept of information state, we solve these problems in terms of an infinite dimensional partial differential equation, viz., the Hamilton-Jacobi-Isaacs equation for partial observed differential games. We give definitions of smooth and viscosity solutions, and prove that the value function is a viscosity solution of the HJI equation. We prove a verification theorem, which implies that the optimal controls are separated in that they depend on the observations through the information state. This constitutes a separation principle for partially observed differential games. We also present some new results concerning the certainty equivalence principle.Item Robust Output Feedback Control for Discrete - Time Nonlinear Systems(1993) James, Matthew R.; Baras, John S.; ISRIn this paper we present a new approach to the solution of the output feedback robust control problem. We employ the recently developed concept of information state for output feedback dynamic games, and obtain necessary and sufficient conditions for the solution to the robust control problem expressed in terms of the information state. The resulting controller is an information state feedback controller, and is intrinsically infinite dimensional. Stability results are obtained using the theory of dissipative systems, and indeed, our results are expressed in terms of dissipation inequalities.Item Output Feedback Risk - Sensitive Control and Differential Games for Continuous - Time Nonlinear Systems(1993) James, Matthew R.; Baras, John S.; Elliott, Robert J.; ISRIn this paper we carry out a formal analysis of an output feedback risk-sensitive stochastic control problem. Using large deviation limits, this problem is related to a deterministic output feedback differential game. Both problems are solved using appropriate information states. The use of an information state for the game problem is new, and is the principal contribution of our work. Our results have implications for the nonlinear robust stabilization problem.Item Risk-Sensitive Control and Dynamic Games for Partially Observed Discrete - Time Nonlinear Systems(1992) James, Matthew R.; Baras, John S.; Elliott, Robert J.; ISRIn this paper we solve a finite-horizon partially observed risk- sensitive stochastic optimal control problem for discrete-time nonlinear systems, and obtain small noise and small risk limits. The small noise limit is interpreted as a deterministic partially observed dynamic game, and new insights into the optimal solution of such game problems are obtained. Both the risk-sensitive stochastic control problem and the deterministic dynamic game problem are solved using information states, dynamic programming, and associated separated policies. A certainty equivalence principle is also discussed. Our results have implications for the nonlinear robust stabilization problem. The small risk limits is a standard partially observed risk neutral stochastic optimal control problem.Item Asymptotic Nonlinear Filtering and Large Deviations with Application to Observer Design(1988) James, Matthew R.; Baras, J.; ISRAn important problem in control theory is the design of observers for nonlinear control systems. By observer we mean a deterministic dynamical system which uses observed information to compute an estimate of the state of the control system in such as way that the error decays to zero. Baras and Krishnaprasad have proposed that an observer design might result from a study of an asymptotic nonlinear filtering problem obtained by adding small noise terms to the equations defining the control system. The purpose of this thesis is to study this asymptotic filtering problem and to develop observer designs based on their idea. Asymptotic nonlinear filtering problems have been studied by several authors, and are closely related to large deviations (Wentzell-Freidlin theory). We prove using vanishing viscosity and control theoretic methods a logarithmic limit result for solutions of the Zakai equation. This limit is characterized by a Hamilton-Jacobi equation which, as noted by Hijab, arises in Mortensen's deterministic minimum energy estimation. We make a careful study of this equation in the light of the relatively recent theory of viscosity solutions due to Crandall and Lions. We study the weak limit of the conditional measures and filter. Inspired by Hijab's large deviation result for pathwise conditional measures, we obtain a large deviation principle "in probability" for the conditional measures, and also a large deviation principle for the distributions of these measures. This asymptotic analysis suggests that the limiting filter is a candidate observer. We present an exact infinite dimensional observer for uncontrolled observable systems. In the case of uncontrolled nonlinear dynamics and linear observations, Bensoussan obtained a finite dimensional observer which is an approximation to the limiting filter. A detectability condition was used to prove exponential decay of the error, provided the initial condition lies in a bounded region. We extend his approach to the general case of controlled nonlinear dynamics and nonlinear observation. In particular, we obtain an observer for a class of fully nonlinear systems with no constraints on the initial conditions. The Benes case is considered.Item Asymptotic Nonlinear Filtering and Large Deviations with Application to Observer Design.(1988) James, Matthew R.; ISRAn important problem in control theory is the design of observers for nonlinear control systems. By observer we mean a deterministic dynamical-system which uses observed information to compute an estimate of the state of the control system in such a way that the error decays to zero. Baras and Krishnaprasad have proposed that an observer design might result from a study of an asymptotic nonlinear filtering problem obtained by adding small noise terms to the equations defining the control system. The purpose of this thesis is to study this asymptotic filtering problem and to develop observer designs based on their idea.Item Controllability and Observability of Nonlinear Systems.(1987) James, Matthew R.; ISRThese tutorial notes discuss the basic ideas in the theory of controllability and observability for nonlinear control systems. The theory treated is primarily due to Hermann and Krener. The first section gives a short overview of the issues, followed by section 2 which reviews distributions, codistributions, and the Frobenius Theorem. Section 3 deals with controllability. Chow's Theorem is presented, before beginning the Hermann-Krener theory. Finally, section 4 discusses the Hermann-Krener formulation of observability. A number of examples and illustrations are provided.Item Introduction to Manifolds, Lie Groups, and Estimation on the Circle.(1987) James, Matthew R.; ISRThese lecture notce introduce some basic concepts concerning smooth manifolds and Lie groups, since they are often the natural mathematical setting for studying nonlinear estimation and control problems. After reviewing some multivariable calculus, we discuss manifolds, derivative maps, vector fields and flows. Then Lie groups and Lie algebras are defined and discussed. Finally, we discuss the paper by Lo and Willsky which treats nonlinear estimation problems on the circle. A number of examples and illustrations are provided.