Compositional Synthesis of Finite-State Abstractions

Abstract

Controller synthesis techniques for continuous systems with respect to temporal logic specifications typically use a finite-state symbolic abstraction of the system model. Constructing this abstraction for the entire system is computationally expensive, and does not exploit natural decompositions of many systems into interacting components. We describe a methodology for compositional symbolic abstraction to help scale controller synthesis for temporal logic to larger systems. We introduce a new relation, called (approximate) disturbance bisimulation, as the basis for compositional symbolic abstractions. Disturbance bisimulation strengthens the standard approximate alternating bisimulation relation used in control, and extends naturally to systems which are composed of sub-components possibly connected in feedback; disturbance bisimulation handles the feedback signals as disturbances. After proving this composability of disturbance bisimulation for metric systems, we show how one can construct finite-state abstractions compositionally for each component, so that the abstractions are simultaneously disturbance bisimilar to their continuous counterparts. Combining these two results, we can compositionally abstract a network system in a modular way while ensuring that the final composed abstraction is distrubance bisimilar to the original system. We discuss how we get a compositional controller synthesis methodology for networks of such systems against local temporal specifications as a by-product of our construction.

Compositional synthesis of finite abstractions for networks of systems: A small-gain approach

“…Defineσ,ε, andρ ext as follows: Remark 3.7. We emphasize that the proposed small-gain type condition in (3.7) is much more general than the ones proposed in [PPB16,MSSM16]. To be more specific, consider the following system:…”

“…The recent result in [MSSM16] introduces a new system relation, called (approximate) disturbance bisimulation relation, as the basis for the compositional construction of symbolic models. Note that the proposed results in [TI08, PPB16,MSSM16] use the small-gain type conditions and provide complete symbolic models of interconnected systems compositionally. The recent results in [SGZ18] introduce different conditions to handle the compositional construction of complete finite abstractions by leveraging techniques from dissipativity theory [AMP16].…”

“…[PPB16] globally (assumptions 1) and 2) in Theorem 3 in[MSSM16] are continuous-time counterpart of Assumption (A2) in[PPB16]). In particular, those assumptions require existence of K ∞ functions being upper bounded by linear ones and lower bounded by quadratic ones which is impossible.…”

Contract-Based Design of Symbolic Controllers for Safety in Distributed Multiperiodic Sampled-Data Systems

“…Remark 3: Since we are dealing with components equipped with sampled-data controllers that have different sampling periods, existing compositional controller synthesis techniques [28], [30], [31], [27] (which assume that controllers of different components have the same sampling period and reason on discrete-time) would fail. Indeed, continuous-time reasoning is crucial to reason compositionally when sampling occurs asynchronously in the components.…”

“…The authors in [25], [26] presented a compositional abstraction and controller synthesis approach to the class of cascade interconnected systems. In [27], the notion of approximate disturbance simulation was used for compositional abstraction of continuous-time systems, where the states of other components were modeled as disturbance signals.…”