Abstract
In this paper, we study efficient approaches to reachability analysis for discrete-time nonlinear dynamical systems when the dependencies among the variables of the system have low treewidth. Reachability analysis over nonlinear dynamical systems asks if a given set of target states can be reached, starting from an initial set of states. This is solved by computing conservative over approximations of the reachable set using abstract domains to represent these approximations. However, most approaches must tradeoff the level of conservatism against the cost of performing analysis, especially when the number of system variables increases. This makes reachability analysis challenging for nonlinear systems with a large number of state variables. Our approach works by constructing a dependency graph among the variables of the system. The tree decomposition of this graph builds a tree wherein each node of the tree is labeled with subsets of the state variables of the system. Furthermore, the tree decomposition satisfies important structural properties. Using the tree decomposition, our approach abstracts a set of states of the high dimensional system into a tree of sets of lower dimensional projections of this state. We derive various properties of this abstract domain, including conditions under which the original high dimensional set can be fully recovered from its low dimensional projections. Next, we use ideas from message passing developed originally for belief propagation over Bayesian networks to perform reachability analysis over the full state space in an efficient manner. We illustrate our approach on some interesting nonlinear systems with low treewidth to demonstrate the advantages of our approach.
Highlights
Reachability analysis asks whether a target set of states is reachable over a finite or infinite time horizon, starting from an initial set for a dynamical system
We study reachability analysis using the idea of tree decompositions over the dependency graph of a dynamical system
We discuss some interesting ways in which precision can be regained by carefully analyzing this situation. We combine these ideas together into an approach for reachability analysis of nonlinear systems using a grid domain that represents complex non convex sets as a union of fixed size cells using a gridding of the state-space. Such a domain would be prohibitively expensive, we show that the tree decomposition abstract domain can drastically cut down on the complexity of computing reachable set overapproximations in this domain, yielding precise reachable set estimation for some nonlinear systems with low treewidth
Summary
Reachability analysis asks whether a target set of states is reachable over a finite or infinite time horizon, starting from an initial set for a dynamical system. We discuss some interesting ways in which precision can be regained by carefully analyzing this situation We combine these ideas together into an approach for reachability analysis of nonlinear systems using a grid domain that represents complex non convex sets as a union of fixed size cells using a gridding of the state-space. Such a domain would be prohibitively expensive, we show that the tree decomposition abstract domain can drastically cut down on the complexity of computing reachable set overapproximations in this domain, yielding precise reachable set estimation for some nonlinear systems with low treewidth. Sankaranarayanan whose dependency graphs do not have sufficiently low treewidth can still be tackled in a conservative manner using some ideas from this paper
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