Pre-orders for reasoning about stability

Pavithra Prabhakar, Geir E Dullerud, Mahesh Viswanathan

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Pre-orders between processes, like simulation, have played a central role in the verification and analysis of discretestate systems. Logical characterization of such pre-orders have allowed one to verify the correctness of a system by analyzing an abstraction of the system. In this paper, we investigate whether this approach can be feasibly applied to reason about stability properties of a system. Stability is an important property of systems that have a continuous component in their state space; it stipulates that when a system is started somewhere close to its ideal starting state, its behavior is close to its ideal, desired behavior. In [6], it was shown that stability with respect to equilibrium states is not preserved by bisimulation and hence additional continuity constraints were imposed on the bisimulation relation to ensure preservation of Lyapunov stability. We first show that stability of trajectories is not invariant even under the notion of bisimulation with continuity conditions introduced in [6]. We then present the notion of uniformly continuous simulations - namely, simulation with some additional uniform continuity conditions on the relation-that can be used to reason about stability of trajectories. Finally, we show that uniformly continuous simulations are widely prevalent, by recasting many classical results on proving stability of dynamical and hybrid systems as establishing the existence of a simple, obviously stable system that simulates the desired system through uniformly continuous simulations.

Original languageEnglish (US)
Title of host publicationHSCC'12 - Proceedings of the 15th ACM International Conference on Hybrid Systems
Subtitle of host publicationComputation and Control
Pages197-206
Number of pages10
DOIs
StatePublished - 2012
Event15th ACM International Conference on Hybrid Systems: Computation and Control, HSCC'12 - Beijing, China
Duration: Apr 17 2012Apr 19 2012

Other

Other15th ACM International Conference on Hybrid Systems: Computation and Control, HSCC'12
CountryChina
CityBeijing
Period4/17/124/19/12

Keywords

  • Bisimulations
  • Pre-orders
  • Stability
  • Uniform continuity
  • Verification

ASJC Scopus subject areas

  • Computer Networks and Communications
  • Computer Science Applications

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