475 research outputs found
Game Refinement Relations and Metrics
We consider two-player games played over finite state spaces for an infinite
number of rounds. At each state, the players simultaneously choose moves; the
moves determine a successor state. It is often advantageous for players to
choose probability distributions over moves, rather than single moves. Given a
goal, for example, reach a target state, the question of winning is thus a
probabilistic one: what is the maximal probability of winning from a given
state?
On these game structures, two fundamental notions are those of equivalences
and metrics. Given a set of winning conditions, two states are equivalent if
the players can win the same games with the same probability from both states.
Metrics provide a bound on the difference in the probabilities of winning
across states, capturing a quantitative notion of state similarity.
We introduce equivalences and metrics for two-player game structures, and we
show that they characterize the difference in probability of winning games
whose goals are expressed in the quantitative mu-calculus. The quantitative
mu-calculus can express a large set of goals, including reachability, safety,
and omega-regular properties. Thus, we claim that our relations and metrics
provide the canonical extensions to games, of the classical notion of
bisimulation for transition systems. We develop our results both for
equivalences and metrics, which generalize bisimulation, and for asymmetrical
versions, which generalize simulation
Approximation Metrics for Discrete and Continuous Systems
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
A New Simulation Metric to Determine Safe Environments and Controllers for Systems with Unknown Dynamics
We consider the problem of extracting safe environments and controllers for
reach-avoid objectives for systems with known state and control spaces, but
unknown dynamics. In a given environment, a common approach is to synthesize a
controller from an abstraction or a model of the system (potentially learned
from data). However, in many situations, the relationship between the dynamics
of the model and the \textit{actual system} is not known; and hence it is
difficult to provide safety guarantees for the system. In such cases, the
Standard Simulation Metric (SSM), defined as the worst-case norm distance
between the model and the system output trajectories, can be used to modify a
reach-avoid specification for the system into a more stringent specification
for the abstraction. Nevertheless, the obtained distance, and hence the
modified specification, can be quite conservative. This limits the set of
environments for which a safe controller can be obtained. We propose SPEC, a
specification-centric simulation metric, which overcomes these limitations by
computing the distance using only the trajectories that violate the
specification for the system. We show that modifying a reach-avoid
specification with SPEC allows us to synthesize a safe controller for a larger
set of environments compared to SSM. We also propose a probabilistic method to
compute SPEC for a general class of systems. Case studies using simulators for
quadrotors and autonomous cars illustrate the advantages of the proposed metric
for determining safe environment sets and controllers.Comment: 22nd ACM International Conference on Hybrid Systems: Computation and
Control (2019
Control refinement for discrete-time descriptor systems: a behavioural approach via simulation relations
The analysis of industrial processes, modelled as descriptor systems, is
often computationally hard due to the presence of both algebraic couplings and
difference equations of high order. In this paper, we introduce a control
refinement notion for these descriptor systems that enables analysis and
control design over related reduced-order systems. Utilising the behavioural
framework, we extend upon the standard hierarchical control refinement for
ordinary systems and allow for algebraic couplings inherent to descriptor
systems.Comment: 8 pages, 3 figure
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