Polynomial zonotopes are a non-convex set representation that have many applications, such as reachability analysis of hybrid systems, control in robotics, and verification of nonlinear systems. Such analysis methods often require intersection checking. The usual algorithm for intersection checking is to, first, overapproximate the polynomial zonotope by a zonotope and then, if the overapproximation is too large, split the set and recursively and repeat. Although the overapproximation error in this algorithm converges to the original polynomial zonotope, there are still two problems. First, the overapproximation error is not monotonically decreasing after each split. Second, more critically, the split polynomial zonotopes do not preserve the sparsity structure as the original polynomial zonotope, eliminating any memory and other efficiency benefits made possible by the sparse structure. In this paper, we propose a variation of polynomial zonotopes called denormalized polynomial zonotopes, where each factor variable does not need to be in the range [-1, 1]. We prove this slight modification eliminates the two above issues, while still guaranteeing that overapproximation error will converge to the exact polynomial zonotope. We demonstrate the efficiency improvements through numerical experiments, where in certain cases denormalized polynomial zonotope intersection checking is over 400x faster and uses 550x less memory.
We study control design methods to endow hybrid systems under disturbances with safety guarantees as an inverse-optimality problem. First, we provide sufficient conditions to guarantee inputto-state safety of a hybrid system with disturbance inputs only. Next, given a nominal feedback law, we show that a hybrid system, with inputs and disturbances, can be rendered input-to-state controlled safe under the existence of a control barrier function (CBF) using pointwise min-norm safeguarding feedback laws. Finally, we demonstrate that every CBF is a meaningful value function for a two-player zero-sum hybrid game in the context of safety, and that every pointwise min-norm safeguarding feedback law is optimal for such a game, even though its design is independent of any cost functional. The main results are illustrated in an example.
Hybrid games are a powerful framework for modelling interactions between cyber-physical systems (CPS). While their high level of nondeterminism allows the modelling of complex systems, it also makes many properties undecidable. This paper presents Hybrid Games with Triggers (HGT), an extension that reduces nondeterminism by embedding agents' rational decision-making as triggers conditions that, when met, prompt the agents to act. This approach leads to a time-abstract discrete game with a countable state space, where an agent has a winning strategy in the discrete game if and only if they do in the HGT. Our discretisation preserves both the continuous and discrete dynamics of the original hybrid game, providing a more structured, analysable model without sacrificing dynamic expressivity.
