Avoiding geometric intersection operations in reachability analysis of hybrid systems

Abstract

Although a growing number of dynamical systems studied in various fields are hybrid in nature, the verification of properties, such as stability, safety, etc., is still a challenging problem. Reachability analysis is one of the promising methods for hybrid system verification, which together with all other verification techniques faces the challenge of making the analysis scale with respect to the number of continuous state variables. The bottleneck of many reachability analysis techniques for hybrid systems is the geometrically computed intersection with guard sets. In this work, we replace the intersection operation by a nonlinear mapping onto the guard, which is not only numerically stable, but also scalable, making it possible to verify systems which were previously out of reach. The approach can be applied to the fairly common class of hybrid systems with piecewise continuous solutions, guard sets modeled as halfspaces, and urgent semantics, i.e. discrete transitions are immediately taken when enabled by guard sets. We demonstrate the usefulness of the new approach by a mechanical system with backlash which has 101 continuous state variables.

Time-Triggered Conversion of Guards for Reachability Analysis of Hybrid Automata

“…The complete system dynamics, controller, and initial set description are available in another work [2], and here we only provide a brief description.…”

“…Again, by removing the parts of the reachable set that are already hit by other guards, one can extract cases with a single guard intersection, as studied in this work (also see Sec. 5.5 of [2]). The only di erence is that now a tree of possible next discrete states is spanned instead of consecutive next discrete states as in case 1).…”

“…Research on intersection in flowpipe construction involves techniques which avoid the intersection operation by employing a nonlinear mapping onto the guard [2]. Continuization methods [3,4,12] eliminate intersections using abstractions that get rid of fast-switching dynamics or eliminate guard intersections between similar continuous dynamics, as performed for m either in this work.…”

“…In particular, we leverage time-triggered transitions [2,8]. In a system with time-triggered transitions, the discrete mode change occurs after a certain amount of time has elapsed.…”

Reachability analysis of nonlinear systems using conservative polynomialization and non-convex sets

“…The extension to hybrid systems is performed as in [1], which computes the intersection with the guard set x9 = 100 geometrically. Integrating the nonlinear reachability into a more efficient approach [2], which avoids computationally expensive guard intersections, is part of future work. The computation time is 1296 s, where again almost all the time (1155 s) is spent on interval arithmetic.…”