Completeness of Lyapunov Abstraction

In this work, we continue our study on discrete abstractions of dynamical systems. To this end, we use a family of partitioning functions to generate an abstraction. The intersection of sub-level sets of the partitioning functions defines cells, which are regarded as discrete objects. The union of cells makes up the state space of the dynamical systems. Our construction gives rise to a combinatorial object - a timed automaton. We examine sound and complete abstractions. An abstraction is said to be sound when the flow of the time automata covers the flow lines of the dynamical systems. If the dynamics of the dynamical system and the time automaton are equivalent, the abstraction is complete. The commonly accepted paradigm for partitioning functions is that they ought to be transversal to the studied vector field. We show that there is no complete partitioning with transversal functions, even for particular dynamical systems whose critical sets are isolated critical points. Therefore, we allow the directional derivative along the vector field to be non-positive in this work. This considerably complicates the abstraction technique. For understanding dynamical systems, it is vital to study stable and unstable manifolds and their intersections. These objects appear naturally in this work. Indeed, we show that for an abstraction to be complete, the set of critical points of an abstraction function shall contain either the stable or unstable manifold of the dynamical system.Comment: In Proceedings HAS 2013, arXiv:1308.490

Model Reduction by Moment Matching for Linear Switched Systems

Two moment-matching methods for model reduction of linear switched systems (LSSs) are presented. The methods are similar to the Krylov subspace methods used for moment matching for linear systems. The more general one of the two methods, is based on the so called "nice selection" of some vectors in the reachability or observability space of the LSS. The underlying theory is closely related to the (partial) realization theory of LSSs. In this paper, the connection of the methods to the realization theory of LSSs is provided, and algorithms are developed for the purpose of model reduction. Conditions for applicability of the methods for model reduction are stated and finally the results are illustrated on numerical examples.Comment: Sent for publication in IEEE TAC, on October 201

Model Reduction of Linear Switched Systems by Restricting Discrete Dynamics

We present a procedure for reducing the number of continuous states of discrete-time linear switched systems, such that the reduced system has the same behavior as the original system for a subset of switching sequences. The proposed method is expected to be useful for abstraction based control synthesis methods for hybrid systems

Stochastic Analysis of Synchronization in a Supermarket Refrigeration System

Display cases in supermarket systems often exhibit synchronization, in which the expansion valves in the display cases turn on and off at exactly the same time. The study of the influence of switching noise on synchronization in supermarket refrigeration systems is the subject matter of this work. For this purpose, we model it as a hybrid system, for which synchronization corresponds to a periodic trajectory. Subsequently, we investigate the influence of switching noise. We develop a statistical method for computing an intensity function, which measures how often the refrigeration system stays synchronized. By analyzing the intensity, we conclude that the increase in measurement uncertainty yields the decrease at the prevalence of synchronization.Comment: In Proceedings HAS 2014, arXiv:1501.0540