A Bayesian approach to modeling and control.

Abstract

This dissertation investigates modeling and control in a Bayesian setting. The methodology assumes that the state of knowledge of the model parameters is characterized by a probability density function, which is updated off-line using experimental input-output data. For given performance specifications and candidate controllers, this naturally leads to such notions as probability of stability and probability of performance, which are then used for controller selection. In the first part of the dissertation, this approach is applied to the area of run-to-run control in semiconductor manufacturing. Analytic formulas for the probability of stability are given for the well-known exponentially weighted moving average run-to-run controller. When considering a more general notion of performance, the Monte Carlo method is used to approximate the probability of performance to a high degree of confidence. This approach to run-to-run control is then illustrated on a virtual plasma etching reactor. Next, the following questions regarding this methodology are investigated. Are scalar measures of identification quality always good predictors of probability of performance? Does improving a scalar measure of deterministic robustness necessarily imply improving the probability of performance? And, is the probability of performance always correct in its ranking of candidate controllers? It turns out that the answer to all three questions is no. The first two answers show that our approach is distinct from classical identification and deterministic robust control. In answering the third question, we show that the probability of misclassification can be made small using both experimental and controller design. In all cases, we illustrate the counterintuitive phenomenon, analyze the underlying mechanism, and a provide general theory to predict their occurrence. A missile autopilot example is used to illustrate these results. Finally, design of experiments for increasing confidence in a predictive claim of control performance is explored. The idea is to select inputs that shape a preposterior distribution of the model parameters such that a certain region in the model parameter space containing a pre-specified percentage of the preposterior density, denoted as an HPD region, is a subset of the region for closed-loop performance. Necessary and sufficient conditions for an HPD region to be a subset of the performance region are given. Using these conditions, a semi-heuristic experimental design algorithm based on the well-known Bayes c-optimal design criterion is then constructed. Roughly speaking, the result of the algorithm is to select inputs that reduce model parameter variance in directions orthogonal to the performance set boundary. To illustrate the results, we revisit the missile autopilot example.