Abstract. In this paper we propose a new method for reachability analysis of the class of discrete-time polynomial dynamical systems. Our work is based on the approach combining the use of template polyhedra and optimization [1,2]. These problems are non-convex and are therefore generally difficult to solve exactly. Using the Bernstein form of polynomials, we define a set of equivalent problems which can be relaxed to linear programs. Unlike using affine lower-bound functions in [2], in this work we use piecewise affine lower-bound functions, which allows us to obtain more accurate approximations. In addition, we show that these bounds can be improved by increasing artificially the degree of the polynomials. This new method allows us to compute more accurately guaranteed over-approximations of the reachable sets of discrete-time polynomial dynamical systems. We also show different ways to choose suitable polyhedral templates. Finally, we show the merits of our approach on several examples.
Abstract. We describe NLTOOLBOX, a library of data structures and algorithms for reachability computation of nonlinear dynamical systems. It provides the users with an easy way to "program" their own analysis procedures or to solve other problems beyond verification. We illustrate the use of the library for the analysis of a biological model.
Abstract. This paper is concerned with a method for computing reachable sets of linear continuous systems with uncertain input. Such a method is required for verification of hybrid systems and more generally embedded systems with mixed continuous-discrete dynamics. In general, the reachable sets of such systems (except for some linear systems with special eigenstructures) are hard to compute exactly and are thus often over-approximated. The approximation accuracy is important especially when the computed over-approximations do not allow proving a property. In this paper we address the problem of refining the reachable set approximation by adding redundant constraints which allow bounding the reachable sets in some critical directions. We introduce the notion of directional distance which is appropriate for measuring approximation effectiveness with respect to verifying a safety property. We also describe an implementation of the reachability algorithm which favors the constraint-based representation over the vertex-based one and avoids expensive conversions between them. This implementation allowed us to treat systems of much higher dimensions. We finally report some experimental results showing the performance of the refinement algorithm.
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