A New Look at Robust Estimation and Identification

A New Look at Robust Estimation and Identification by Keith Evan Schubert Estimation and identification are important areas of almost every problem in science and engineering. A typical way of stating an estimation or identification problem is that there is a system, described by a matrix, A, with inputs, x, and outputs, b. The inputs and outputs could either be matrices or vectors. The equation which describes this is thus Ax = b. The outputs of the system are considered measurable, and from them and the matrix A, it is desired to find the unknown inputs, x. In real systems the equality rarely holds because b is never measured perfectly, modeling and identification do not produce an exact A, and the basic equation Ax = b is a linear approximation. The fundamental problem considered is thus Ax ≈ b, where both A and b are assumed to have errors associated with them. This Dissertation proposes five regression methods to handle the fundamental problem of Ax ≈ b. In particular, let the ”true” system, Atrue, be related to the nominal model, A, by an error matrix EA. Similarly let the true outputs, btrue be related to the measured outputs, b, by Eb. Since the true system is not a mathematical model, the resulting equation is still approximate, (A + EA)x ≈ (b + Eb), but is the best approximation possible. The goal is to find the best x, in the resulting minimization problem, minx ‖(A+EA)x− (b+Eb)‖. Each of the vi five methods in this dissertation consider this problem and makes assumptions on the size and structure of the errors, EA and Eb. All problems are solved using secular equation techniques, so finding the solution corresponds to finding the zero of a possibly multi-dimensional secular equation. The first three methods are extensions of min max regression, which minimizes the cost over x and maximizes it over EA and Eb. The fourth method is the degenerate case (multiple solutions) of min min regression, which minimizes over x, and the errors EA and Eb. The fifth is actually a family of regression problems with rational cost functions based off the backward error criterion of Numerical Analysis.