Uncertainty Propagation in Abstracted Systems

Abstract

In this chapter, it is shown that given a system and its abstraction, the evolution of uncertain initial conditions in the original system is, in some sense, matched by the evolution of the uncertainty in the abstracted system. In other words, it is shown that the concept of Φ-related vector fields extends to the case of stochastic initial conditions where the probability density function (pdf) for the initial conditions is known. In the deterministic case, the Φ mapping commutes with the system dynamics. In this chapter, it is shown that in the case of stochastic initial conditions, the induced mapping, Φ_pdf, commutes with the evolution of the pdf according to the Liouville equation. It is also shown that a control system abstraction can capture the time evolution of the uncertainty in the original system by an appropriate choice of control input. Application of the convservation law results in a partial differential equation known as the Liouville equation, for which a closed form solution is known. The solution provides the time evolution of the initial pdf which can be followed by the abstracted system.