Flow*: An Analyzer for Non-linear Hybrid Systems

Abstract

Abstract. The tool FLOW* performs Taylor model-based flowpipe construction for non-linear (polynomial) hybrid systems. FLOW* combines well-known Taylor model arithmetic techniques for guaranteed approximations of the continuous dynamics in each mode with a combination of approaches for handling mode invariants and discrete transitions. FLOW* supports a wide variety of optimizations including adaptive step sizes, adaptive selection of approximation orders and the heuristic selection of template directions for aggregating flowpipes. This paper describes FLOW* and demonstrates its performance on a series of nonlinear continuous and hybrid system benchmarks. Our comparisons show that FLOW* is competitive with other tools. Overview of FLOW*In this paper, we present the FLOW* tool to generate flowpipes for non-linear hybrid systems using Taylor Models (TMs). TMs were originally proposed by Berz and Makino [1] to represent functions by means of higher-order Taylor polynomial expansions, bloated by an interval to represent the approximation error. TMs support functional operations such as addition, multiplication, division, derivation and antiderivation. Guaranteed integration techniques can utilize TMs to provide tight flowpipe over-approximations to non-linear ODEs, with each flowpipe segment represented by a TM [2]. However, these techniques do not naturally extend to non-linear hybrid systems consisting of multiple modes and discrete transitions (jumps). Figure 1 presents a schematic diagram of the major components of FLOW*. FLOW* accepts (i) A hybrid system model file which describes the modes, the polynomial dynamics associated with each mode and the transitions between modes; (ii) A specification file includes TM flowpipes with the state space and unsafe set specifications. For a model file, FLOW* performs a flowpipe construction for a specified time horizon [0, T ] and a maximum jump depth J such that the flowpipe set is an over-approximation of the states which can be reached in [0, T ] with at most J jumps. FLOW* also checks whether the flowpipe intersects the unsafe set and outputs a visualization of the set of reachable states using polyhedral over-approximations of the computed TM flowpipes. FLOW* is extensible in quite simple ways. Our TM output can be parsed in by other tools, including FLOW* itself to check multiple properties incrementally.

Mean Field Approximation of Uncertain Stochastic Models

“…The first class of methods manipulates directly sets of states, finitely represented, for instance, as polytopes [25], ellipsoids [26], or zonotopes [27], or by relying on interval arithmetic [28] combined with constraint solving [29], [30], exploiting satisfaction modulo solvers over reals [31], [32]. The dynamics of the system is lifted at the set level, so that one computes the evolution of the reachable set under the action of the dynamics.…”

“…Another family of methods, very common for differential inclusions, is based on interval-based methods combined with constrain solving to control the over-approximation introduced by interval computations. Among them, we recall [29] and [30]. Other methods for integrating differential inclusions rely on error bounding [36].…”

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

“…These methods work by propagating regions of states, which can be represented using constraint polyhedra [26], support functions [32], orthogonal polyhedra [23], zonotopes [5], Taylor models [21,44] or ellipsoids [17,37]. These representations have been implemented in powerful analysis tools for hybrid automata including HyTech [34], Ariadne [13], Flow* [20], PHAVer [26], SpaceEx [27], CORA [1], and Hylaa [11].…”

Using Rigorous Simulation to Support ISO 26262 Hazard Analysis and Risk Assessment

“…Notably, SpaceEx [14] is capable of simulating large systems (comprising hundreds of variables) provided that dynamics, resets and guards are affine. Tools such as Flow* [15], based on Taylor Models or those based on Taylor Integrators [16], are capable of simulating models with non-linear dynamics, guards and resets with good precision. Common to these tools is that they process models expressed in the hybrid automaton [17] formalism.…”