Robust Experiment Design

This Thesis addresses the problem of robust experiment design, i.e., how to design an input signal to maximise the amount of information obtained from an experiment given limited prior knowledge of the true system. The majority of existing literature on experiment design specifically considers optimal experiment design, which, typically depends on the true system parameters, that is, the very thing that the experiment is intended to find. This obviously gives rise to a paradox. The results presented in this Thesis, on robust experiment design, are aimed at resolving this paradox. In the robust experiment design problem, we assume that the parameter vector is a-priori known to belong to a given compact set, and study the design of an input spectrum which maximises the worst case scenario over this set. We also analyse the problem from a different perspective where, given the same assumption on the parameter vector, we examine cost functions that give rise to an optimal input spectrum independent of the true system features. As a first approach to this problem we utilise an asymptotic (in model order) expression for the variance of the system transfer function estimator. To enable the extension of these results to finite order models, we digress from the main topic and develop several fundamental integral limitations on the variance of estimated parametric models. Based on these results, we then return to robust experiment design, where the input design problems are reformulated using the fundamental limitations as constraints. In this manner we establish that our previous results, obtained from asymptotic variance formulas, are valid also for finite order models. Robustness issues in experiment design also arise in the area of `identification for (robust) control'. In particular, a new paradigm has recently been developed to deal with experiment design for control, namely `least costly experiment design'. In the Thesis we analyse least costly experiment design and establish its equivalence with the standard formulation of experiment design problems. Next we examine a problem involving the cost of complexity in system identification. This problem consists of determining the minimum amount of input power required to estimate a given system with a prescribed degree of accuracy, measured as the maximum variance of its frequency response estimator over a given bandwidth. In particular, we study the dependence of this cost on the model order, the required accuracy, the noise variance and the size of the bandwidth of interest. Finally, we consider the practical problem of how to optimally generate an input signal given its spectrum. Our solution is centered around a Model Predictive Control (MPC) based algorithm, which is straightforward to implement and exhibits fast convergence that is empirically verified.