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Discrete Reachability Analysis With Bounded Error Sets

Set-based reachability analysis is used to guarantee performance and constraint satisfaction of safety-critical systems. Due to uncertainty in disturbances and inputs, reachable sets often grow in complexity with time. While techniques exist to reduce exact reachable sets to lower-order approximations, there has been relatively little attention to bounding approximation error as a set. This article presents a novel method for over-approximating reachable sets of discrete-time linear time-varying dynamics where the same operations employed to compute approximated reachable sets are also used to compute a set bounding the approximation error. The method is generalized so that the trade-off between reachable set complexity and accuracy is highly tunable. For a special case of linear time-invariant dynamics, a closed-form bound on the error set is derived. A numerical example compares the proposed approach to existing methods and demonstrates how it can be tuned to balance between accuracy and complexity of over-approximated reachable sets.