Numerical solutions to noisy systems

Abstract

A numerical method for rigorous overapproximation of a solution set of an input-affine system whose inputs represent some bounded noise is presented. The method gives high order error for a single time step and a uniform bound on the error over the finite time interval. The approach is based on the approximations of inputs by linear functions at each time step. We derive the single-step error in the one-dimensional case, and give the formula for the error in higher dimensions. As an illustration of the theory presented, a rigorous numerical result is given.978-1-4244-7746-3/10/$26.00 ©2010 IEEE

Ongoing Work on Automated Verification of Noisy Nonlinear Systems with Ariadne

“…where f : R n × R m → R n is a smooth function, V is a compact set and v(t) is a measurable function known as the disturbance input. In particular, [19] discusses how to compute the reachable set for nonlinear control systems which are affine with respect to noisy inputs. Also, a reasonable assumption in practice is that noisy inputs are elements of a box whose vector components are intervals.…”

“…A recent publication [13] proposes a method for computing outer approximations of reachable sets for nonlinear control systems by constructing convex polyhedral enclosures of reachable sets; it produces upper and lower bounds via polyhedra and demonstrates the efficiency of the proposed algorithm through several examples. Since all the examples are input-affine systems, we plan to compare this approach to the implementation of [19] within Ariadne.…”

“…It also has been successfully used for dominance checking of controllers [3] and even for correct-by-construction code generation [7]. This paper discusses the ongoing work aimed at extending the dynamical model used in Ariadne to differential inclusions, based on the work of [19], in order to perform reachability analysis in the presence of noisy inputs. While the most straightforward application of differential inclusions is for modeling system uncertainty, it is worth remarking that they can be used also to support contractbased design [16]: given a complex system, we can replace the actual input of an automaton with an input having partially defined behavior.…”

Accurate reachability analysis of uncertain nonlinear systems

“…Logarithmic norms and component-wise, one-sided Lipschitz conditions are used in [30,45], while the algorithms in [4] and [1] use a conservative linear, respectively, polynomial approximation of the right-hand-side of (1). Polynomial approximations in combination with interval remainders, so-called Taylor models [12], to over-approximate reachable sets of nonlinear systems with time-varying inputs are developed and analyzed in [55]. Also, in principle, the algorithms to compute validated solutions of initial value problems based on interval arithmetic and a Taylor series expansion of the flow function as discussed in [38] can be used to over-approximate reachable sets of (1), by modeling the system as an autonomous system (without inputs) with interval parameters.…”

“…Compared to the schemes in [1,4] it is guaranteed to converge under suitable assumptions. Compared to other non-hybridization based algorithms that compute rigorous enclosures and are guaranteed to converge in the context of nonlinear systems with time-varying inputs or nonlinear differential inclusions, our approach is not based on a computational demanding uniform discretization of the state space [41], nor is our approach limited to additive disturbances [45] or input affine systems [55].…”