Anytime ellipsoidal over-approximation of forward reach sets of uncertain linear systems

Abstract

Computing tight over-approximation of reach sets of a controlled uncertain dynamical system is a common practice in verification of safety-critical cyber-physical systems (CPS). While several algorithms are available for this purpose, they tend to be computationally demanding in CPS applications since here, the computational resources such as processor availability tend to be scarce, time-varying and difficult to model. A natural idea then is to design "computation-aware" algorithms that can dynamically adapt with respect to the processor availability in a provably safe manner. Even though this idea should be applicable in broader context, here we focus on ellipsoidal over-approximations. We demonstrate that the algorithms for ellipsoidal over-approximation of reach sets of uncertain linear systems, are well-suited for anytime implementation in the sense the quality of the over-approximation can be dynamically traded off depending on the computational time available, all the while guaranteeing safety. We give a numerical example to illustrate the idea, and point out possible future directions.

Certifying the Intersection of Reach Sets of Integrator Agents With Set-Valued Input Uncertainties

“…As discussed in earlier works [9], [11, Sec. VI], having a computational handle on volume is helpful in providing ground truth to quantify the conservatism of numerical algorithms which over-approximate the reach set via simpler geometric shapes such as variants of ellipsoids [28]- [32] or variants of zonotopes [4], [33]- [38].…”

Hausdorff Distance between Norm Balls and their Linear Maps

“…For ω = 3 and t ∈ [0, 2], Fig. 4 shows the time evolution of the numerically estimated Hausdorff distance (25) and the upper bound (29) between the reach sets given by (24) with the same compact convex initial set X 0 ⊂ R 4 , i.e., between X 1 t and X 2 t resulting from the unit p 1 = 2 and p 2 = ∞ norm ball input sets, respectively.…”

“…As such, there exists a vast literature [5,16,25,26,40,41,45,55,57] on reach sets and their numerical approximations. In practice, these sets are of interest because their separation or intersection often imply safety or the lack of it.…”

Convex and Nonconvex Sublinear Regression with Application to Data-driven Learning of Reach Sets

“…To further illustrate the use of support function learning representations for the reach sets, consider the reach sets of two agents A and B with identical dynamics (16), respective inputs…”

“…In other words, different approaches have different interpretations of what does it mean to approximate a set. For example, parametric approximants seek for a simple geometric primitive (e.g., ellipsoid [12]- [16], zonotope [17]- [19] etc.) to serve as a proxy for the set.…”