Functional sets with typed symbols : Mixed zonotopes and Polynotopes for hybrid nonlinear reachability and filtering

TL;DR

This paper introduces mixed sets called polynotopes, combining zonotopic and polynomial representations with typed symbols, to improve hybrid nonlinear reachability and filtering in Cyber-Physical Systems.

Contribution

It proposes a novel mixed set framework using typed symbols, extending zonotopes and polynomial zonotopes, with a new polynotopic Kalman filter for hybrid nonlinear systems.

Findings

Effective propagation of mixed interval and logic dependencies

Extension of zonotopic Kalman filters to nonlinear hybrid systems

Numerical examples demonstrating improved accuracy and efficiency

Abstract

Verification and synthesis of Cyber-Physical Systems (CPS) are challenging and still raise numerous issues so far. In this paper, based on a new concept of mixed sets defined as function images of symbol type domains, a compositional approach combining eager and lazy evaluations is proposed. Syntax and semantics are explicitly distinguished. Both continuous (interval) and discrete (signed, boolean) symbol types are used to model dependencies through linear and polynomial functions, so leading to mixed zonotopic and polynotopic sets. Polynotopes extend sparse polynomial zonotopes with typed symbols. Polynotopes can both propagate a mixed encoding of intervals and describe the behavior of logic gates. A functional completeness result is given, as well as an inclusion method for elementary nonlinear and switching functions. A Polynotopic Kalman Filter (PKF) is then proposed as a hybrid nonlinear extension of Zonotopic Kalman Filters (ZKF). Bridges with a stochastic uncertainty paradigm are briefly outlined. Finally, several discrete, continuous and hybrid numerical examples including comparisons illustrate the effectiveness of the theoretical results.