Finite-horizon controllability and reachability for deterministic and stochastic linear control systems with convex constraints

This paper presents a method for rapidly generating controllability and reachability sets for constrained finite horizon Linear Time Varying (LTV) control systems by using convex optimization techniques. Set generation is accomplished by first solving a Semi-Definite Programming (SDP) problem and then solving a series of Second Order Cone Programming (SOCP) problems. Recent advances in convex optimization solvers have made it possible to find the solutions to these problems very quickly. From a geometric stand-point, we first find the largest volume symmetric simplex that fits within the constrained control problem, then grow new simplices out of the faces of the original simplex. This process is repeated until the growing polytope converges to the constraint boundaries of the actual set. Additionally, a method for incorporating stochastic constraints and uncertainties into the deterministic framework is developed by posing the stochastic constraints as chance-constrained constraints. Finally, the controllability set for a two-vehicle Low Earth Orbit (LEO) rendezvous problem with stochastic uncertainties is generated using the new algorithm.