On the Composition of Discrete and Continuous-time Assume-Guarantee Contracts for Invariance

Abstract

Many techniques for verifying invariance properties are limited to systems of moderate size. In this paper, we propose an approach based on assume-guarantee contracts and compositional reasoning for verifying invariance properties of a broad class of discrete-time and continuous-time systems consisting of interconnected components. The notion of assumeguarantee contracts makes it possible to divide responsibilities among the system components: a contract specifies an invariance property that a component must fulfill under some assumptions on the behavior of its environment (i.e. of the other components). We define weak and strong semantics of assumeguarantee contracts for both discrete-time and continuous-time systems. We then establish a certain number of results for compositional reasoning, which allow us to show that a global invariance property of the whole system is satisfied when all components satisfy their own contract. Interestingly, we show that the weak satisfaction of the contract is sufficient to deal with cascade compositions, while strong satisfaction is needed to reason about feedback composition. Specific results for systems described by differential inclusions are then developed. Throughout the paper, the main results are illustrated using simple examples.

Contracts as specifications for dynamical systems in driving variable form

“…where x g ∈ X g , d g ∈ D g , and w g ∈ W. Note that both the assumptions (19) and guarantees (20) are systems of the same form as Σ in (17) and that they share the same space of external variables. Finally, X a , D a , X g , and D g above are all finite-dimensional vector spaces.…”

“…The relevance of contracts is given by their use as formal specifications for systems Σ as in (17). This is made explicit using the following definition.…”

“…Remark 3: Whereas the concept of contracts was originally proposed in the scope of software engineering in [14], Definition 3 is inspired by assume/guarantee contracts in formal methods developed in [6], see also the recent book [7] for a detailed discussion. We note that these works consider models of computation that are inherently discrete in nature, such that the theory developed in [6], [7] is not applicable to continuous dynamical systems as in (17).…”

Contract based Design of Symbolic Controllers for Vehicle Platooning

“…A continuous-time assume-guarantee contract for Σ is a tuple C = (A W , G X , G Y ) where A W ⊆ W is a set of assumptions on the external input and G X ⊆ X , G Y ⊆ Y are sets of guarantees on the state and output, respectively. Intuitively, an assume-guarantee contract is strongly satisfied when if the system's input belongs to A W up to a time instant t, then the system's state belongs to G X at least until t, and the system's output belongs to G Y until t + δ with δ > 0, see [3].…”

“…ACM ISBN 978-1-4503-5642-8/18/04. https://doi.org/10.1145/3178126.3187001 the general framework introduced in [3], for compositional reasoning and assign an assume-guarantee contract to each vehicle. In Section 4, we explain how symbolic control techniques can be used to synthesize controllers enforcing the specified contracts.…”