A Small Gain Theorem for Parametric Assume-Guarantee Contracts

Abstract

The problem of verifying properties of large, networked cyberphysical systems (CPS) is beyond the reach of most computational tools today. Two common "divide-and-conquer" techniques for CPS verification are assume-guarantee contracts from the formal methods literature and input-output properties from the control theory literature. Combining these two approaches, we first introduce the notion of a parametric assume-guarantee contract, which lets one reason about system behavior abstractly in a parameter domain. We next show how a finite gain property can be encoded in this form and provide a generalized small-gain theorem for parametric assume-guarantee contracts. This theorem recovers the classical small gain theorem as a special case and its derivation highlights the connection between assume-guarantee reasoning and small-gain results. This new small-gain theorem applies to behaviors beyond bounded deviation from a nominal point to include a fragment of linear temporal logic with parametrized predicates that can encode safety, recurrence, and liveness properties. Our results are validated with an example which certifies that the interconnection of two freeway segments experiences intermittent congestion.

Compositional Set Invariance in Network Systems with Assume-Guarantee Contracts

“…Next, we present a framework that gives performance guarantee to the network system based on assume-guarantee reasoning. First, the definition of assume-guarantee contract is formally defined, which is adopted from [19].…”

Contracts as specifications for dynamical systems in driving variable form

“…Here, recall that A ⊗ and G ⊗ represent Σ 1 ⊗ Σ 2 as defined in (11). Now, take any (x ⊗ 1 , x ⊗ 2 ) ∈ π X1×X2 (S) and let x ∈ X be such that (x, x ⊗ 1 , x ⊗ 2 ) ∈ S. By the reasoning above (46), we have Cx = C 1 x ⊗ 1 and Cx = C 2 x ⊗ 2 , leading to C 1 x ⊗ 1 = C 2 x ⊗ 2 .…”

“…As such, the contracts in [17] do not allow for including dynamics in the specification. Richer contracts, called parametric assume/guarantee contracts, are defined in [11], allowing for expressing input-output gain properties. However, only discrete-time systems are considered in this work.…”

Metamathematics for Systems Design

“…In the study of CPS and hybrid systems, pursuit of compositionality via a certain form of ''contract'' is nowadays widespread [29,57,90,94]. One use of contracts typical of CPS is for separating safety from efficiency: the system architecture is hierarchical with the higher, symbolic level addressing safety and the lower, numeric level addressing efficiency; the higher level determines high-level policies such as discrete actions (as in maneuver automata [29]) or so-called safety envelopes (as in [90]); and the lower level synthesizes a controller that complies with the high-level policies and at the same time aims at efficiency.…”