Exponential-Condition-Based Barrier Certificate Generation for Safety Verification of Hybrid Systems

Abstract

A barrier certificate is an inductive invariant function which can be used for the safety verification of a hybrid system. Safety verification based on barrier certificate has the benefit of avoiding explicit computation of the exact reachable set which is usually intractable for nonlinear hybrid systems. In this paper, we propose a new barrier certificate condition, called Exponential Condition, for the safety verification of semi-algebraic hybrid systems. The most important benefit of Exponential Condition is that it has a lower conservativeness than the existing convex condition and meanwhile it possesses the property of convexity. On the one hand, a less conservative barrier certificate forms a tighter over-approximation for the reachable set and hence is able to verify critical safety properties. On the other hand, the property of convexity guarantees its solvability by semidefinite programming method. Some examples are presented to illustrate the effectiveness and practicality of our method.

Robustness of Control Barrier Functions for Safety Critical Control**This work is partially supported by the National Science Foundation Grants 1239055, 1239037 and 1239085.

“…The barrier function and control barrier function considered in this paper are based on [3], [4], and [5]. As in [7], the primary focus is to establish forward invariance of a given set C, which one may interpret as an under approximation of the "initial set" and the "safe set" in previous formulations of barrier functions.…”

“…As in [4], [3] and [7], inequality (5) relaxes the conventional condition by requiring a single superlevel set of h, which is C itself, to be invariant.…”

“…In the original formulation of [2], all sublevel sets of the barrier certificate were required to be invariant because the derivative of the barrier certificate along solutions was required to be non-positive. This condition was relaxed by [3] and [4] so that tighter over-approximations of the reachable set could be obtained, and such that more expressive barrier certificates could be synthesized using semi-definite programming. The key idea there was to only require that a single sublevel set be invariant, namely, the set of points where the barrier certificate was non-positive.…”

“…A second, although less important, difference in the two approaches is that [6] used the more restrictive invariance condition of [1], while [7] used the relaxed condition of [3], appropriately interpreted for the type of barrier function often used in optimization, see [9], where the barrier function is unbounded on the boundary of the allowed set, instead of vanishing on the set boundary.…”

Safety Verification of Hybrid Automata with Transient Safe Modes

“…The above-mentioned works inspire us to treat real-time scheduling as checking finite-time invariance or stability. In this paper, we follow the basic idea of safety verification based on sum-of-squares (SOS) programming [8], [9] and adopt an exponential-conditioned barrier certificate [10], [11] to estimate the maximum time guaranteeing safety of transient safe modes. Beside that, we can use such barrier certificates to verify the soundness of real-time scheduling enlightened from [12].…”