Feedback Refinement Relations for the Synthesis of Symbolic Controllers

Abstract

We present an abstraction and refinement methodology for the automated controller synthesis to enforce general predefined specifications. The designed controllers require quantized (or symbolic) state information only and can be interfaced with the system via a static quantizer. Both features are particularly important with regard to any practical implementation of the designed controllers and, as we prove, are characterized by the existence of a feedback refinement relation between plant and abstraction. Feedback refinement relations are a novel concept introduced in this paper. Our work builds on a general notion of system with set-valued dynamics and possibly non-deterministic quantizers to permit the synthesis of controllers that robustly, and provably, enforce the specification in the presence of various types of uncertainties and disturbances. We identify a class of abstractions that is canonical in a well-defined sense, and provide a method to efficiently compute canonical abstractions. We demonstrate the practicality of our approach on two examples.Comment: This work has been accepted for publication in the IEEE Trans. Automatic Control. v3: Definition VIII.2 corrected; plus minor modifications; accepted versio

The Challenges in Specifying and Explaining Synthesized Implementations of Reactive Systems

“…In an inverse fashion one can incrementally construct the complex implementation starting with a coarse abstraction and gradually refine it with respect to a given partial order, that forces a correct construction. Inspired by counterexample guided abstraction refinement, a series of incremental synthesis procedures have been investigated [40,41,44]. In each stage, refinement suggestions give information on what behavior is added or excluded from the implementation.…”

Abstraction refinement and plan revision for control synthesis under high level specifications

“…Several methods exist for over-approximating reachable sets for fairly large classes of linear (Kurzhanskiy and Varaiya, 2007;Girard, 2005) and nonlinear systems (Reissig et al, 2016;Coogan and Arcak, 2015).…”

“…To control the sampled system S τ with the controller C a obtained in Algorithm 5, the systems S τ = (X τ , U τ , −→ τ ) and S a = (X a , U a , −→ a ), defined in Sections 2.1 and 3 respectively, must satisfy a feedback refinement relation as defined below, adapted from Reissig et al (2016). Definition 10.…”

Approximately symbolic models for a class of continuous-time nonlinear systems

“…piany@kth.se, dimos@kth.se by over-approximating the behavior of the concrete system. However, as recently pointed out in [25], this approach may result in a refinement complexity issue.…”