Reachability of Black-Box Nonlinear Systems after Koopman Operator Linearization

Abstract

Reachability analysis of nonlinear dynamical systems is a challenging and computationally expensive task. Computing the reachable states for linear systems, in contrast, can often be done efficiently in high dimensions. In this paper, we explore verification methods that leverage a connection between these two classes of systems based on the concept of the Koopman operator. The Koopman operator links the behaviors of a nonlinear system to a linear system embedded in a higher dimensional space, with an additional set of so-called observable variables. Although, the new dynamical system has linear differential equations, the set of initial states is defined with nonlinear constraints. For this reason, existing approaches for linear systems reachability cannot be used directly. In this paper, we propose the first reachability algorithm that deals with this unexplored type of reachability problem. Our evaluation examines several optimizations, and shows the proposed workflow is a promising avenue for verifying behaviors of nonlinear systems.

Formal Certification Methods for Automated Vehicle Safety Assessment

“…[147] Zonotope approximation fast [167] conservative [174] and convex [90], [174] Ellipsoid approximation quadratic in nature, relatively fast, fits quadratic Lyapunov functions for calculation shape may overly approximate vehicle occupancy [175] Star set approximation Shape more flexible than zonotope or polytope, non-convex, less overapproximation less intuitive [146] Piece-wise linear system equation approximation good accuracy for nonlinear system approximation, proved convergence, preserves the region of limit cycles convergence speed can be improved [168] Semi-definite programming over polynomial representations of continuous functions has efficiency improvement potential slow in computation speed, currently not real-time implementable [94] Differential inclusion of nonlinear dynamic model lower computation complexity accuracy may suffer from dynamic over-approximation, but partition techniques [170] can partially make up this shortcoming [169], [170] Reachable set estimation using human-in-the-loop data sampling fast since data sampling is offline, captures human driving pattern data bias affects the validity of estimated set [139] Controllable set estimation using simulation-generated data high data coverage thanks to simulation relies on heuristic classification criteria for controllability [137], [138] Warm-start of HJI value function Guarantee over-approximation of true BRS, and reduces computation time compared to full initialization the approximation to true BRS is not always exact, sometimes only conservative over-approximation can be achieved [164], [163] Local BRS update significantly reduces computation time compared to full BRS update limited to dealing with static obstacles in the environment [163] Koopman operator linearization of data-driven model Fast reachable set computation, system can be a black box model relatively new approach [171] Taylor model(TM) based flowpipe tight over-approximation Tight over-approximation to nonlinear ODEs, fast calculation n/a [99] Sample-based maxFRS underapproximation…”

“…[138]. Set contour approximation [87], [88], [166], [72], [143], [144], [145], [147], [167], [146], [164], [163], dynamics approximation [168], [94], [169], [170], [99] and sample-based approximation [139], [137], [138], [171], [172] are all viable numerical options. A summary of methods used in primary studies are listed in Table . V. Academic competitions for reachability analysis exist, such as ARCH COMP [173], which is a competition of scientific software in the context of algorithmic verification of continuous and hybrid systems.…”

Enhancing Data-Driven Reachability Analysis using Temporal Logic Side Information

“…Several recent works have proposed performing reachability analysis from data [12]- [20] to overcome the limitation of prior model knowledge. By performing reachability analysis directly from data, we can form a direct link between the actual, historical performance of a robot and our prediction of its reachability, removing the dependency Fig.…”