Discrete Abstractions of Nonlinear Systems Based on Error Propagation Analysis

Abstract

This paper proposes a computational method for the feasibility check and design of discrete abstract models of nonlinear dynamical systems. First, it is shown that a given discretetime dynamical system can be transformed into a finite automaton by embedding a quantizer into its state equation. Under this setting, a sufficient condition for approximate bisimulation in infinite steps of time between the concrete model and its discrete abstract model is derived. The condition takes the form of a set of linear inequalities and thus can be checked efficiently by a linear programming solver. Finally, the iterative refinement algorithm, which generates a discrete abstract model under a given error specification, is proposed. The algorithm is guaranteed to terminate in finite iterations.Index Terms-Discrete abstraction, nonlinear systems, quantized systems.

Parametrization of completeness in symbolic abstraction of bounded input linear systems

“…In fact, satisfying the zero deviation condition is very difficult if not impossible because of state-quantization induced symbolic approximations. So in approximately similar symbolic abstraction [9,11,12,13], the zero deviation condition is relaxed as:…”

“…The extant methodology of approximately similar symbolic abstraction (discussed in [9,11,12,13]) establishes soundness between symbolic model and system model while also specifying the proximity parameter, which is the precision bound of the approximate simulation relation [9,11,12,13]. Parametrization of the amount of deviation (proximity) between a system model and its symbolic model in addition to demonstrating soundness of the model is the advantage of approximately similar symbolic abstraction.…”

“…Also, a good methodology of symbolic abstraction allows for adjusting the proximity to a very small amount. In this regard, the methodologies discussed in [9,11,12,13] permit sound symbolic abstraction with arbitrarily small proximity. However, a stronger method of abstraction, discussed in [9], can construct an approximately bisimilar (not just similar) finite symbolic model to a time-quantized system model of a globally asymptotically stable system, in which case the abstraction is complete in addition to being sound and proximate.…”

“…Although proximity has been parametrized through the notion of approximate simulation [9,11,12,13], no attempt has been made until now to parametrize completeness. Parametrization of completeness would be useful while abstracting bounded input unstable systems that, in many cases, can not have fully complete (plus sound) state-time quantized symbolic models.…”

“…As such, soundness, completeness and proximity conditions are met by the symbolic model of a GAS system constructed by the procedure discussed in [9]. Regarding unstable systems and also those systems that meet a stabilizing condition [11,12] but are possibly unstable, there has been work on abstracting the systems into similar symbolic models i.e. based on approximate simulation, but not on approximate bisimulation [11,12,13].…”