Controller Synthesis for Linear System With Reach-Avoid Specifications

We address the problem of synthesizing provably correct controllers for linear systems with reach-avoid specifications. We show that, once a tracking controller is fixed, the reachable states from an initial neighborhood, subject to any disturbance, can be over-approximated by a sequence of ellipsoids, with that are independent of the open-loop controller. Hence, the open-loop controller can be synthesized independently to meet the reach-avoid specification for an initial neighborhood. Exploiting several computational geometry techniques, we reduce the openloop controller synthesis problem to satisfiability over quantifierfree linear real arithmetic. The number of linear constraints in the satisfiability problem is linear to number of hyperplanes as the surfaces of the polytopic obstacles and goad sets. The overall synthesis algorithm, computes a tracking controller, and then iteratively covers the entire initial set to find open-loop controllers for initial neighborhoods. The algorithm is sound and, for a class of robust systems, is also complete. We implement this synthesis algorithm in a tool REALSYN and show that it scales to several high-dimensional systems with complex reach-avoid specifications.