Approximating Viability Kernels of High-Dimensional Linear Systems

Using concepts from Viability Theory, this paper presents an algorithm to determine a safety property of linear and time invariant systems. A number of Robotics scenarios can be framed within Viability Theory, namely applications concerning autonomous vehicles. Often the navigation problem consists in keeping the trajectory of the system inside some compact set representing motion constraints. This means computing the viability kernel (i.e., the largest set of initial states for which it is guaranteed that there are available controls that maintain the trajectories inside the constraint set) out of the admissible/safe areas and, implicitly, find the controls that keep the robot inside such sets. This represents a form of controllability of the system. An algorithm from the literature, based on sampling the boundary of the viability kernel, was modified to yield a faster algorithm. The distinctive feature of the extended algorithm is the use of zonotopes to represent sets. Simulations for systems with dimensions between $n=5$ and $n=100$ are presented and illustrate the relative performance improvement. The algorithm is illustrated with a chain of integrators. A comparison in performance between the original and the extended algorithms suggests that the latter has a better performance for a sufficiently wide range of system dimensions.