Accounting for uncertainty in control-relevant statistics

To make appropriate decisions based on common indices used in control, both the point estimates and their uncertainties must be known. Many control-relevant statistics, such as model predictions, gain margins and other frequency domain quantities, are functions of parameters of process models. Confidence regions for these quantities are most often calculated under the assumption that these quantities have an asymptotic limiting normal distribution. These confidence regions may be erroneous, and very misleading, as the asymptotic results ignore the influence of parameter nonlinearities. In addition, proximity of the model parameters to stability/invertibility boundaries also distorts the confidence regions from those predicted from asymptotic theory. Generalized profiling is a flexible numerical method for constructing confidence intervals and confidence regions for model parameters, and functions of model parameters. Applications in nonlinear regression [D. Bates, D. Watts, Nonlinear Regression Analysis and Its Applications, John Wiley & Sons, New York, 1988] indicate that it provides a much more accurate representation of uncertainty in those instances when the asymptotic uncertainty results are inaccurate or misleading. Generalized profiling is based on the likelihood approach to quantifying uncertainty. The numerical construction of these likelihood uncertainty regions requires solution to a series of constrained optimization problems. Computationally efficient diagnostic tests, motivated by profiling, are developed. These can be effectively employed as screening tools to indicate when the asymptotic results are most likely to be inadequate.