Functional sets with typed symbols : Mixed zonotopes and Polynotopes for hybrid nonlinear reachability and filtering
This work addresses verification and synthesis issues in Cyber-Physical Systems, representing an incremental advancement by extending existing zonotopic methods with typed symbols and hybrid capabilities.
The paper tackles the challenge of verifying and synthesizing Cyber-Physical Systems by introducing mixed sets based on typed symbols, leading to polynotopes that extend sparse polynomial zonotopes. It proposes a Polynotopic Kalman Filter for hybrid nonlinear reachability and filtering, with effectiveness demonstrated through numerical examples.
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.