Safety Verification of Nonlinear Polynomial System via Occupation Measures

Abstract

In this paper, we introduce a flexible notion of safety verification for nonlinear autonomous systems by measuring how much time the system spends in given unsafe regions. We consider this problem in the particular case of nonlinear systems with a polynomial dynamics and unsafe regions described by a collection of polynomial inequalities. In this context, we can quantify the amount of time spent in the unsafe regions as the solution to an infinite-dimensional linear program (LP). This LP measures the volume of the unsafe region with respect to the occupation measure of the system trajectories. Using Lasserre hierarchy, we approximate the solution to the infinite-dimensional LP using a sequence of finitedimensional semidefinite programs (SDPs). The solutions to the SDPs in this hierarchy provide monotonically converging upper bounds on the optimal solution to the infinite-dimensional LP. Finally, we validate the performance of our framework using numerical simulations.

Peak Estimation and Recovery with Occupation Measures

“…This section proves equivalence between Measure and Function programs, based on methods in [11,3]. Let K be a cone with dual cone K * , where K * = {v | v, x ≥ 0, ∀x ∈ K} for linear functionals v [2].…”

“…Previous convex work in this area includes [5], which offers sum-of-squares (SOS) criteria for producing upper bounds. Results for safety analysis are presented in [3]. Specialized results for analyzing impulse responses of linear systems are in [4].…”

“…Figure6demonstrates safety margins on the following system f (x) = [x 2 , −x 1 − x 2 + x3 1 /3] from[18] with an infinite-horizon. Trajectories originate from X 0 : (x 1 − 1.5) 2 + x 2 2 ≤ 0.4 2 .…”

“…As mentioned in Sec. 2.4,[18] cannot distinguish between unsafety and insufficient degree for θ = 3π/4.Safety verification by occupation measures has been presented in[3] by solving a measure program to find the expected amount of time trajectories spend in X u . Trajectories never enterX u if E u [t] = E[t | x(t) ∈ X u ]= 0 for any LMI relaxation.…”