We develop a method to approximate the moments of a discrete-time stochastic polynomial system. Our method is built upon Carleman linearization with truncation. Specifically, we take a stochastic polynomial system with finitely many states and transform it into an infinitedimensional system with linear deterministic dynamics, which describe the exact evolution of the moments of the original polynomial system. We then truncate this deterministic system to obtain a finite-dimensional linear system, and use it for moment approximation by iteratively propagating the moments along the finite-dimensional linear dynamics across time. We provide efficient online computation methods for this propagation scheme with several error bounds for the approximation. Our result also shows that precise values of certain moments can be obtained when the truncated system is sufficiently large. Furthermore, we investigate techniques to reduce the offline computation load using reduced Kronecker power. Based on the obtained approximate moments and their errors, we also provide probability bounds for the state to be outside of given hyperellipsoidal regions. Those bounds allow us to conduct probabilistic safety analysis online through convex optimization. We demonstrate our results on a logistic map with stochastic dynamics and a vehicle dynamics subject to stochastic disturbance.
Sasinee PRUEKPRASERT†a) , Nonmember and Toshimitsu USHIO †b) , Fellow SUMMARY This paper considers an optimal stabilization problem of quantitative discrete event systems (DESs) under the influence of disturbances. We model a DES by a deterministic weighted automaton. The control cost is concerned with the sum of the weights along the generated trajectories reaching the target state. The region of weak attraction is the set of states of the system such that all trajectories starting from them can be controlled to reach a specified set of target states and stay there indefinitely. An optimal stabilizing controller is a controller that drives the states in this region to the set of target states with minimum control cost and keeps them there. We consider two control objectives: to minimize the worst-case control cost (1) subject to all enabled trajectories and (2) subject to the enabled trajectories starting by controllable events. Moreover, we consider the disturbances which are uncontrollable events that rarely occur in the real system but may degrade the control performance when they occur. We propose a linearithmic time algorithm for the synthesis of an optimal stabilizing controller which is robust to disturbances.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.