Computing Safe Sets of Linear Sampled-Data Systems

Abstract

Leveraging autonomous systems in safety-critical applications requires formal robustness guarantees against uncertainties. We address this issue by computing safe terminal sets with corresponding safety-preserving terminal controllers, which ensure robust constraint satisfaction for an infinite time horizon. To maximize the region of operation, we also construct as large as possible safe initial sets that can be safely steered into the safe terminal set in finite time. We use scalable reachability analysis and convex optimization to efficiently compute safe sets of sampled-data systems. These systems are composed of a physical plant evolving in continuous time and a digital controller being implemented in discrete time. We further verify the effectiveness of our robust control approach using a simple double-integrator system and a vehicle-platooning benchmark.

Online Guaranteed Reachable Set Approximation for Systems with Changed Dynamics and Control Authority

“…For some , , , ∈ ℝ, we will denote the ordering < < < by the shorthand notation , , , . We have: Having considered all cases, in all admissible scenarios it follows that the statement in (11) is false. This in turn contradicts the claim that there exists ∈ ⧵ such that ∉ ( ) ⊞ * .…”

“…Inner approximations of reachable sets have received comparatively less attention than outer approximations [7], but have recently seen use in path-planning problems with collision avoidance [8], as well as viability kernel computation [9], which can in turn be used for guaranteed trajectory planning [10]. Another application lies in safe set determination, in which one aims to obtain an inner approximation of the maximal robust control invariant set [11]. Methods for determining inner approximations of reachable sets have been based on various principles, including relying on polynomial inner approximation of the nonlinear system dynamics using interval calculus [12], ellipsoid calculus [13], and viscosity solutions to HJB equations [14].…”

Compositions of Multiple Control Barrier Functions Under Input Constraints

“…Second, we present an iterative algorithm to find a viability domain parameterized by these decoupled CBFs. That is, instead of using zonotopes, [17], [18], polytopes [19], [20], or other parameterized functions [22], [24], [25], this paper expresses viability domains in terms of an intersection of CBF sets. As each CBF forbids state trajectories from crossing the boundary of its own CBF set, our algorithm focuses on verifying Nagumo's condition only at the states where the boundaries of multiple CBF sets intersect.…”

A Linear Programming Approach to Computing Safe Sets for Software Rejuvenation

“…Also, they introduce the hardware root of trust, which is a secure onboard module that hosts the necessities for implementing software rejuvenation; the secure execution interval, representing the period where external communications are disabled; and the safety controller, which is executed after the software refresh. Likewise, another relevant topic in the literature is that of the computation of safe sets, e.g., a safe robust invariant terminal set is defined along with a maximum safe initial set in [14]. Due to the computational complexity of polytope operators [15], [16], these sets typically adopt simple representations, e.g., zonotopes [14] and ellipsoids [10].…”