Computing abstractions of nonlinear systems

Sufficiently accurate finite state models, also called symbolic models or discrete abstractions, allow one to apply fully automated methods, originally developed for purely discrete systems, to formally reason about continuous and hybrid systems, and to design finite state controllers that provably enforce predefined specifications. We present a novel algorithm to compute such finite state models for nonlinear discrete-time and sampled systems which depends on quantizing the state space using polyhedral cells, embedding these cells into suitable supersets whose attainable sets are convex, and over-approximating attainable sets by intersections of supporting half-spaces. We prove a novel recursive description of these half-spaces and propose an iterative procedure to compute them efficiently. We also provide new sufficient conditions for the convexity of attainable sets which imply the existence of the aforementioned embeddings of quantizer cells. Our method yields highly accurate abstractions and applies to nonlinear systems under mild assumptions, which reduce to sufficient smoothness in the case of sampled systems. Its practicability in the design of discrete controllers for nonlinear continuous plants under state and control constraints is demonstrated by an example.

💡 Research Summary

The paper addresses the long‑standing challenge of constructing finite‑state (symbolic) models for nonlinear discrete‑time and sampled‑data systems in a way that is both accurate and computationally tractable. Such models enable the application of automated verification and synthesis techniques—originally devised for purely discrete systems—to continuous‑time and hybrid dynamics, thereby allowing the design of controllers that provably satisfy complex specifications (e.g., safety, reachability, LTL).

The authors propose a novel algorithm that proceeds in three conceptual stages. First, the continuous state space is partitioned into polyhedral cells. Rather than using the cells directly, each cell is embedded into a carefully designed polyhedral superset whose reachable (attainable) set under the system dynamics remains convex. The paper supplies new sufficient conditions for this convexity: for sampled‑data systems it suffices that the underlying vector field be twice continuously differentiable (C²) and that the sampling interval be sufficiently small; under these assumptions the flow map over one sampling step is 1‑Lipschitz, guaranteeing that the image of a convex set stays convex.

Second, the reachable set of each superset is over‑approximated by an intersection of supporting half‑spaces. The key technical contribution is a recursive description of these half‑spaces. Starting from the half‑spaces that define the superset itself, the algorithm propagates them forward one time step at a time using a linearisation of the dynamics together with a bounded nonlinear remainder term. At each step new half‑spaces are generated, and redundant ones are eliminated by simple inclusion tests. This recursion yields a sequence of polyhedral over‑approximations that converge to a user‑specified precision ε (measured, for example, by the Hausdorff distance).

Third, the algorithm iteratively refines the half‑space collection until the desired precision is reached. Because each iteration only manipulates a modest number of half‑spaces, the computational burden scales linearly with the number of cells N and with the average number k of half‑spaces per cell, i.e., O(N·k·n) where n is the system dimension. This is a dramatic improvement over naïve global polyhedral over‑approximations, which suffer from exponential growth in the number of facets.

The paper includes a thorough complexity analysis and an implementation based on MATLAB/Simulink together with the SCOTS toolbox. Two benchmark examples illustrate the method’s effectiveness. The first is a second‑order nonlinear system x⁺ = x + τ(−x³ + u) with bounded input u∈