On the Control of Affine Systems with Safety Constraints: Relaxed In-Block Controllability

TL;DR

This paper introduces the concept of relaxed in-block controllability (RIBC) for affine systems within polytopes, addressing cases where full control is not possible due to safety constraints, and provides geometric conditions for RIBC.

Contribution

It defines RIBC for affine systems with safety constraints and establishes necessary and sufficient geometric conditions for its feasibility.

Findings

Necessary conditions for RIBC are identified.

Sufficient conditions for RIBC are established.

Illustrative examples demonstrate the main results.

Abstract

We consider affine systems defined on polytopes and study the cases where the systems are not in-block controllable with respect to the given polytopes. That are the cases in which we cannot fully control the affine systems within the interior of a given polytope, representing the intersection of given safety constraints. Instead, we introduce in this paper the notion of relaxed in-block controllability (RIBC), which can be useful for the cases where one can distinguish between soft and hard safety constraints. In particular, we study whether all the states in the interior of a given polytope, formed by the intersection of soft safety constraints, are mutually accessible through the interior of a given bigger polytope, formed by the intersection of hard safety constraints, by applying uniformly bounded control inputs. By exploring the geometry of the problem, we provide necessary conditions for RIBC. We then show when these conditions are also sufficient. Several illustrative examples are also given to clarify the main results.