Risk-Bounded Control Using Stochastic Barrier Functions

Abstract

In this paper, we study Stochastic Control Barrier Functions (SCBFs) to enable the design of probabilistic safe real-time controllers in presence of uncertainties and based on noisy measurements. Our goal is to design controllers that bound the probability of a system failure in finite-time to a given desired value. To that end, we first estimate the system states from the noisy measurements using an Extended Kalman filter, and compute confidence intervals on the filtering errors. Then, we account for filtering errors and derive sufficient conditions on the control input based on the estimated states to bound the probability that the real states of the system enter an unsafe region within a finite time interval. We show that these sufficient conditions are linear constraints on the control input, and, hence, they can be used in tractable optimization problems to achieve safety, in addition to other properties like reachability, and stability. Our approach is evaluated using a simulation of a lane-changing scenario on a highway with dense traffic.

Safe Control in the Presence of Stochastic Uncertainties

“…PID [22]), optimization-based controllers (e.g. some robust controllers [12] and nonlinear controllers [23]), safe control methods [2], [5], [6], [7], [8], [24], and some hybrid of those. One of the major design considerations for K is system safety.…”

“…Properties associated with entrance to a set have been studied from the perspective of stochastic stability in different contexts. 6 However, stochastic stability is not applicable in our setting for two reasons: the state rarely stays in a set indefinitely after reaching it; and the initial state is often predetermined and cannot be chosen. 7 Instead, we consider a weaker notion than stability, formally defined below, which requires re-entry for any initial condition but does not require invariance upon re-entry.…”

“…Equivalently, even if the state does not belong to a forward convergent set M with respect to (1) at some time, it will enter M in finite 5 A forward invariant set is also known as positive invariant set in the existing literature [23]. 6 Stochastic stability is defined by, given M, the existence of a set M 0 such that x 0 ∈ M 0 ⇒ xt ∈ M, ∀t ≥ 0 with probability one [25,Chapter 2]. Stochastic stability requires invariance with probability one, and the invariance only need to hold for some set of (not all) initial conditions.…”

“…Given the challenges in obtaining exact probability, existing literature has provided insightful upper and lower bounds for safe probabilities [2], [5], [6], [7], [8]. Their approaches can be roughly classified into ones based on martingale inequalities and ones that convert a stochastic problem into a deterministic one.…”

“…To bound the probability of safety, the former approach constructs super/sub-martingales that bound the evolution of the barrier function values over time [2], [5], [6], [7]. In this approach, the constructed martingales do not use the complete distribution of the system dynamics, thus resulting in a loose bound.…”