Uncertainty, complication and optimal model construction

Abstract

I consider the problem of determining optimal knowledge partitions in problems described by complicated models with associated embedded optimization requirements. The particular models studied are linear control models in which the embedded optimization is that of observer/ controller design. The existence of an optimal knowledge partition rests on the tradeoff between the gain in accuracy achieved by more knowledge about a system offset by the increased complication that knowledge typically brings ("information oyerload" ). Thus, even though increased knowledge I will always result in a true optimum that is superior, the increased complication (and nonzero computation costs) may make finding that optimum practically impossible. With the reduction of complication achieved by eliminating knowledge, one automatically experiences an increase of uncertainty. Thus, the possibility of actually correctly reducing complication requires that one have a theory capable of handling the uncertainty. In control theory, while control under uncertainty is an extremely important field with many deep results, until now there was no theory providing a nonperturbative optimal control law for uncertain Stochastic Linear Quadratic Regulator systems. My first result creates such a theory. In order to obtain an explicit example of how one can determine an optimal knowledge partition, I study a simple control system - the iso-spectral system. For this system, it is explicitly demonstrated that the controller design problem is equivalent to a spin-glass of Sherrington-Kirkpatric type. Using this result, I prove that a nontrivial optimal knowledge partition can exist and give a prescription for determining locally optimal optimal knowledge partitions. That a model as simple as the iso-spectral system leads to an optimization problem equivalent to finding the ground state of a spin--glass is an indication that the possible existence of an optimal knowledge partition should always be considered.