1 research outputs found
Moment Analysis of Stochastic Hybrid Systems Using Semidefinite Programming
This paper proposes a semidefinite programming based method for estimating
moments of a stochastic hybrid system (SHS). For polynomial SHSs -- which
consist of polynomial continuous vector fields, reset maps, and transition
intensities -- the dynamics of moments evolve according to a system of linear
ordinary differential equations. However, it is generally not possible to solve
the system exactly since time evolution of a specific moment may depend upon
moments of order higher than it. One way to overcome this problem is to employ
so-called moment closure methods that give point approximations to moments, but
these are limited in that accuracy of the estimations is unknown. We find lower
and upper bounds on a moment of interest via a semidefinite program that
includes linear constraints obtained from moment dynamics, along with
semidefinite constraints that arise from the non-negativity of moment matrices.
These bounds are further shown to improve as the size of semidefinite program
is increased. The key insight in the method is a reduction from stochastic
hybrid systems with multiple discrete modes to a single-mode hybrid system with
algebraic constraints. We further extend the scope of the proposed method to a
class of non-polynomial SHSs which can be recast to polynomial SHSs via
augmentation of additional states. Finally, we illustrate the applicability of
results via examples of SHSs drawn from different disciplines