Reachability computation for polynomial dynamical systems

Abstract

This paper is concerned with the problem of computing the bounded time reachable set of a polynomial discrete-time dynamical system. The problem is well-known for being difficult when nonlinear systems are considered. In this regard, we propose three reachability methods that differ in the set representation. The proposed algorithms adopt boxes, parallelotopes, and parallelotope bundles to construct flowpipes that contain the actual reachable sets. The latter is a new data structure for the symbolic representation of polytopes. Our methods exploit the Bernstein expansion of polynomials to bound the images of sets. The scalability and precision of the presented methods are analyzed on a number of dynamical systems, in comparison with other existing approaches

Occupation measure methods for modelling and analysis of biological hybrid systems

“…We make use of the recent method proposed in Zhao et al (2017), which relies on occupation measures and a sequence semidefinite relaxations to produce a sequence of polynomial controls converging to the optimal solution of a HOCP. There exist other methods which use occupation measures and LMI relaxations to produce both admissible controls and converging outer-approximations of the backward reachable set (BRS) Majumdar et al (2014); Shia et al (2014), or the region of attraction (ROA) Korda et al (2014).Finally, we note that finding a sequence of converging outer-approximations for all valid parameters sets, such as in Streif et al (2013); Dreossi (2016), is another crucial issue in the context of systems biology. When dealing with hybrid systems, Shia et al (2014) can be applied to solve this problem.…”

Kaa: A Python Implementation of Reachable Set Computation Using Bernstein Polynomials

“…The complete set of dynamics and relevant parameters is found in [8] and reachable sets are shown in Figure 3.…”

“…A simpler and easier-to-understand reachable set computation algorithm that utilizes Bernstein polynomials and parallelotopes has been presented in [4]. Several cumulative improvements [5,15,3,7,8] have been proposed to improve its accuracy and efficiency. The tool SAPO [6] which implements these algorithms in C++ is available publicly.…”

“…Several cumulative improvements [5,15,3,7,8] have been proposed to improve its accuracy and efficiency. The tool SAPO [6] which implements these algorithms in C++ is available publicly. In this paper, we present a Python implementation of algorithm presented in [8].…”

“…The tool SAPO [6] which implements these algorithms in C++ is available publicly. In this paper, we present a Python implementation of algorithm presented in [8]. Our motivation for reimplementing the algorithms in Python is two-fold.…”