Computing descent direction of MTL robustness for non-linear systems

Abstract

Abstract-The automatic analysis of transient properties of nonlinear dynamical systems is a challenging problem. The problem is even more challenging when complex state-space and timing requirements must be satisfied by the system. Such complex requirements can be captured by Metric Temporal Logic (MTL) specifications. The problem of finding system behaviors that do not satisfy an MTL specification is referred to as MTL falsification. This paper presents an approach for improving stochastic MTL falsification methods by performing local search in the set of initial conditions. In particular, MTL robustness quantifies how correct or wrong is a system trajectory with respect to an MTL specification. Positive values indicate satisfaction of the property while negative values indicate falsification. A stochastic falsification method attempts to minimize the system's robustness with respect to the MTL property. Given some arbitrary initial state, this paper presents a method to compute a descent direction in the set of initial conditions, such that the new system trajectory gets closer to the unsafe set of behaviors. This technique can be iterated in order to converge to a local minimum of the robustness landscape. The paper demonstrates the applicability of the method on some challenging nonlinear systems from the literature. I. INTRODUCTIONA number of applications can only be accurately modeled using nonlinear dynamical models. Typical such applications include analog circuits [1]-[3] and biological and medical systems [4]- [7]. A common theme of all the aforementioned applications is the need to verify transient or periodic properties of the system. Such properties might involve sequencing of events, conditional reachability and invariants and realtime constraints and can be formally captured using temporal logics [4], [8].Unfortunately, for complex nonlinear systems, these types of properties are hard -if not impossible -to verify algorithmically. Therefore, recent research efforts have been invested in property falsification methods [9]-[12]. In falsification, the space of operating conditions and/or inputs is searched in order to find an initial condition and/or parameter that will force the system to exhibit an unsafe behavior with respect to the formal requirement. In turn, the unsafe system trajectory can be used in order to manually or automatically modify the system to achieve the desired system behavior and performance [13], [14].In [10], [15], the temporal logic falsification problem is converted into an optimization (minimization) problem based on the notion of robustness of temporal logics [16]. Essentially, a system trajectory with negative robustness is one that proves the existence of unsafe system behaviors.

Fly-by-Logic: Control of Multi-Drone Fleets with Temporal Logic Objectives

“…While the examples in this paper show the good performance of our method, the method itself is not without limitations: a) The high-level optimization (12) at the heart of this approach is a non-convex problem and the solvers used guarantee convergence only to a local optima, which may have a negative robustness value. Parallel instances of the solver with different initial starting points can alleviate this problem in practice.…”

“…Stochastic heuristics like [11] have also been used for STL missions, but offer very few or no guarantees. Non-smooth optimization has also been explored, but only for safety properties [12].…”

“…2.2), it is itself non-differentiable as a function of the trajectory and the control inputs. A priori, this necessitates the use of Mixed-Integer Programming solvers [8], non-smooth optimizers [16], or stochastic heuristics [11] to solve P ρ . However, it was recently shown in [10] that it is more efficient and more reliable to instead approximate the non-differentiable objective ρ ϕ by a smooth (infinitely differentiable) functionρ ϕ and solve the resulting optimization problem P using Sequential Quadratic Programming.…”

“…We use a shrinking horizon approach. At time 0, the waypoints are computed by solving (12) and given to the quadrotors to track. Then every T f seconds, estimates for p, v, a are obtained, and the optimization is solved again, while fixing all variables for previous time instants to their observed/computed values, to generate new waypoints for the remaining duration of the trajectory.…”

A trajectory splicing approach to concretizing counterexamples for hybrid systems

“…As a result, our approach can exploit the continuous sensitivity to the initial conditions in each mode, while at the same time dealing with discontinuities due to transitions. A recent extension of the robustness-guided falsification approach uses subgradient methods to find descent directions for the robustness metric [1]. However, this approach is currently restricted to purely continuous systems.…”