Robust recursive estimation in the presence of heavy-tailed observation noise

Includes bibliographical references (p. 33-41).Supported by the U.S. Army Research Office fellowship. ARO-DAAL03-86-G-0017 Supported by the U.S. Air Force Office of Scientific Research. AFOSR-85-0227 AFOSR-89-0276Irvin C. Schick and Sanjoy K. Mitter

On Control and Estimation of Large and Uncertain Systems

This thesis contains an introduction and six papers about the control and estimation of large and uncertain systems. The first paper poses and solves a deterministic version of the multiple-model estimation problem for finite sets of linear systems. The estimate is an interpolation of Kalman filter estimates. It achieves a provided energy gain bound from disturbances to the point-wise estimation error, given that the gain bound is feasible. The second paper shows how to compute upper and lower bounds for the smallest feasible gain bound. The bounds are computed via Riccati recursions. The third paper proves that it is sufficient to consider observer-based feedback in output-feedback control of linear systems with uncertain parameters, where the uncertain parameters belong to a finite set. The paper also contains an example of a discrete-time integrator with unknown gain. The fourth paper argues that the current methods for analyzing the robustness of large systems with structured uncertainty do not distinguish between sparse and dense perturbations and proposes a new robustness measure that captures sparsity. The paper also thoroughly analyzes this new measure. In particular, it proposes an upper bound that is amenable to distributed computation and valuable for control design. The fifth paper solves the problem of localized state-feedback L2 control with communication delay for large discrete-time systems. The synthesis procedure can be performed for each node in parallel. The paper combines the localized state-feedback controller with a localized Kalman filter to synthesize a localized output feedback controller that stabilizes the closed-loop subject to communication constraints. The sixth paper concerns optimal linear-quadratic team-decision problems where the team does not have access to the model. Instead, the players must learn optimal policies by interacting with the environment. The paper contains algorithms and regret bounds for the first- and zeroth-order information feedback

Acknowledgement Misspecification in Macroeconomic Theory

We explore methods for confronting model misspecification in macroeconomics. We construct dynamic equilibria in which private agents and policy makers recognize that models are approximations. We explore two generalizations of rational expectations equilibria. In one of these equilibria, decision makers use dynamic evolution equations that are imperfect statistical approximations, and in the other misspecification is impossible to detect even from infinite samples of time-series data. In the first of these equilibria, decision rules are tailored to be robust to the allowable statistical discrepancies. Using frequency domain methods, we show that robust decision makers treat model misspecification like time-series econometricians.

Model Predictive Control meets robust Kalman filtering

Model Predictive Control (MPC) is the principal control technique used in industrial applications. Although it offers distinguishable qualities that make it ideal for industrial applications, it can be questioned its robustness regarding model uncertainties and external noises. In this paper we propose a robust MPC controller that merges the simplicity in the design of MPC with added robustness. In particular, our control system stems from the idea of adding robustness in the prediction phase of the algorithm through a specific robust Kalman filter recently introduced. Notably, the overall result is an algorithm very similar to classic MPC but that also provides the user with the possibility to tune the robustness of the control. To test the ability of the controller to deal with errors in modeling, we consider a servomechanism system characterized by nonlinear dynamics

ESTIMATION-BASED SOLUTIONS TO INCOMPLETE INFORMATION PURSUIT-EVASION GAMES

Differential games are a useful tool both for modeling conflict between autonomous systems and for synthesizing robust control solutions. The traditional study of games has assumed decision agents possess complete information about one another’s strategies and numerical weights. This dissertation relaxes this assumption. Instead, uncertainty in the opponent’s strategy is treated as a symptom of the inevitable gap between modeling assumptions and applications. By combining nonlinear estimation approaches with problem domain knowledge, procedures are developed for acting under uncertainty using established methods that are suitable for applications on embedded systems. The dissertation begins by using nonlinear estimation to account for parametric uncertainty in an opponent’s strategy. A solution is proposed for engagements in which both players use this approach simultaneously. This method is demonstrated on a numerical example of an orbital pursuit-evasion game, and the findings motivate additional developments. First, the solutions of the governing Riccati differential equations are approximated, using automatic differentiation to obtain high-degree Taylor series approximations. Second, constrained estimation is introduced to prevent estimator failures in near-singular engagements. Numerical conditions for nonsingularity are approximated using Chebyshev polynomial basis functions, and applied as constraints to a state estimate. Third and finally, multiple model estimation is suggested as a practical solution for time-critical engagements in which the form of the opponent’s strategy is uncertain. Deceptive opponent strategies are identified as a candidate approach to use against an adaptive player, and a procedure for designing such strategies is proposed. The new developments are demonstrated in a missile interception pursuit-evasion game in which the evader selects from a set of candidate strategies with unknown weights