Efficient finite abstraction of mixed monotone systems

Abstract

We present an efficient computational procedure for finite abstraction of discrete-time mixed monotone systems by considering a rectangular partition of the state space. Mixed monotone systems are decomposable into increasing and decreasing components, and significantly generalize the well known class of monotone systems. We tightly overapproximate the one-step reachable set from a box of initial conditions by computing a decomposition function at only two points, regardless of the dimension of the state space. We apply our results to verify the dynamical behavior of a model for insect population dynamics and to synthesize a signaling strategy for a traffic network.

Abstraction refinement and plan revision for control synthesis under high level specifications * *This work was supported by the H2020 ERC Starting Grant BUCOPHSYS, the EU H2020 AEROWORKS project, the EU H2020 Co4Robots project, the Swedish Foundation for Strategic Research, the Swedish Research Council and the KAW Foundation

“…Several methods exist for over-approximating reachable sets for fairly large classes of linear (Kurzhanskiy and Varaiya, 2007;Girard, 2005) and nonlinear systems (Reissig et al, 2016;Coogan and Arcak, 2015).…”

Compositional controller synthesis for vehicular traffic networks

“…The traffic model above is amenable to a scalable method for computing over-approximated transition relations without polytope set operations [8], which we illustrate in Fig. 2.…”

“…We efficiently compute tight rectangular over-approximations (dashed and dotted outlines) of these reachable sets and over-approximate the set of next possible partitions (translucent box) by checking for intersections with the dotted box. The efficiency in the reachability computation arises from the fact that rectangular boxes are characterized by two corner points (see [8] for further details).…”

“…for every intersection v ∈ V, the set L out v is empty or a singleton. This structural assumption ensures that the network exhibits monotone dynamics [8] and contract atomic propositions χ supply n (Σ n ) and χ demand n (∆ n ) hold on lower sets of X n . Monotone systems maintain a partial ordering on states [10].…”

Application of the Hirsch Generic Convergence Statement to the Set‐Point Regulation of Excitable‐Transparent‐Monotone Systems

“…Monotone systems have recently persuaded researchers to focus on such dynamical systems in mathematical and engineering fields (see, e.g., [1][2][3][4][5][6][7][8][9][10][11]). The central property of monotone systems is that their solutions preserve a partial order on the states; hence, they have strong convergence properties.…”