Approximation Metrics for Discrete and Continuous Systems

Abstract

Established system relationships for discrete systems, such as language inclusion, simulation, and bisimulation, require system observations to be identical. When interacting with the physical world, modeled by continuous or hybrid systems, exact relationships are restrictive and not robust. In this paper, we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion, approximate simulation, and approximate bisimulation relations. We define a hierarchy of approximation pseudo-metrics between two systems that quantify the quality of the approximation, and capture the established exact relationships as zero sections. Our approximation framework is compositional for a synchronous composition operator. Algorithms are developed for computing the proposed pseudo-metrics, both exactly and approximately. The exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations, or dually, the solution of a static game whose cost is the so-called branching distance between the systems. Approximations for the pseudo-metrics can be obtained by considering Lyapunov-like functions called simulation and bisimulation functions. We illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This journal article is available at ScholarlyCommons: http://repository.upenn.edu/cis_papers/343 782 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 5, MAY 2007 Approximation Metrics for Discrete and Continuous SystemsAntoine Girard and George J. Pappas, Senior Member, IEEE Abstract-Established system relationships for discrete systems, such as language inclusion, simulation, and bisimulation, require system observations to be identical. When interacting with the physical world, modeled by continuous or hybrid systems, exact relationships are restrictive and not robust. In this paper, we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion, approximate simulation, and approximate bisimulation relations. We define a hierarchy of approximation pseudo-metrics between two systems that quantify the quality of the approximation, and capture the established exact relationships as zero sections. Our a...

Verification Using Simulation

“…Moreover, we have shown [11] that it is the smallest (or minimal) bisimulation metric which is the analog for metrics of the largest (or maximal) bisimulation relation for relations. Though it is possible to compute the minimal bisimulation metric for systems with a finite number of states [10], it becomes more problematic for systems with an infinite number of states.…”

“…The branching distance defined in [10] and [11] as the smallest function (but not necessarily metric) d satisfying the functional equation:…”

“…Whereas choosing metrics may not be natural for purely combinatorial discrete problems, they are very natural for continuous and hybrid systems. Having a notion of distance between states and observations, enables us to build on the recently developed framework of approximate bisimulation metrics [10][11][12][13]. Bisimulation metrics measure how far two states are from being bisimilar, thus enabling the quantification of error between trajectories originating from approximately bisimilar states.…”

Robustness of Temporal Logic Specifications

“…First, in the simplest case where we just simulate the response of a system, we can derive bounds for the magnitude of the disturbances that the system can tolerate while still satisfying the same MTL specification. Second, we can use approximation metrics [21] in order to verify a system using simulations [22].…”

Approximate Simulation Relations for Hybrid Systems

“…The notion of language approximation is much more adequate in this context. In (Girard and Pappas, 2005c), we proposed a framework for system approximation based on approximate versions of simulation relations. Instead of requiring that the observations of a system and its approximation are and remain equal, we require that they are and remain arbitrarily close.…”