Image Computation for Polynomial Dynamical Systems Using the Bernstein Expansion

Abstract

This paper is concerned with the reachability computation problem for polynomial dynamical systems. Such computations constitute a crucial component in algorithmic verification tools for hybrid systems and embedded software with polynomial dynamics, which have found applications in many engineering domains. We describe two methods for overapproximating reachable sets of such systems; these methods are based a combination of the Bernstein expansion of polynomial functions and a representation of reachable sets by template polyhedra. Using a prototype implementation, the performance of the methods was demonstrated on a number of examples of control systems and biological systems.

Automatic abstraction of non-linear systems using change of bases transformations

“…This has been addressed using a wide variety of techniques in the past, including algebraic and semi-algebraic geometric techniques, interval analysis, constraint propagation and Bernstein polynomials [26,29,21,16,23,19,20,14,22,8]. In particular, the hybridization of non-linear systems is an important approach for converting it into affine systems by subdividing the invariant region into numerous sub-regions and approximating the dynamics as a hybrid system by means of a linear differential inclusion in each region [12,3,7].…”

Reachability Analysis of Polynomial Systems Using Linear Programming Relaxations

“…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.…”

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

“…They form a basis for approximating polynomials over a compact interval, and have nice properties that will be exploited to relax the optimization (3.1) to a linear program. Here, we should mention that a relaxation using Bernstein polynomials was provided in the context of reachability analysis for polynomial dynamical systems [13] and improved in [51]. The novelty in this work is not only the adaptation of these relaxations in the context of polynomial Lyapunov function synthesis but also a new tighter relaxation will be introduced by exploiting the induction relation between Bernstein polynomials.…”