Online Inner Approximation of Reachable Sets of Nonlinear Systems with Diminished Control Authority

Abstract

This work presents a method of efficiently computing inner approximations of forward reachable sets for nonlinear control systems with diminished control authority, given an a priori computed reachable set for the nominal system. The method functions by shrinking a precomputed convex reachable set based on a priori knowledge of the system's trajectory deviation growth dynamics. The trajectory deviation growth dynamics determine an upper bound on the minimal deviation between two trajectories emanating from the same point that are generated by control inputs from the nominal and diminished set of control inputs, respectively. These growth dynamics are a function of a given Hausdorff distance bound between the nominal convex space of admissible controls and the possibly unknown impaired space of admissible controls. Because of its relative computational efficiency compared to direct computation of the off-nominal reachable set, this procedure can be applied to onboard fault-tolerant path planning and failure recovery. We consider the implementation of the approximation procedure by way of numerical integration and a root finding scheme, and we present two illustrative examples, namely an application to a control system with quadratic nonlinearities and aircraft wing rock dynamics.

Online Guaranteed Reachable Set Approximation for Systems with Changed Dynamics and Control Authority

“…Unlike in [16], a new hyperrectangular version of the Bihari inequality is required to obtain inner and outer approximations to more generalized reachable sets. As will be specified later, the only requirements imposed on these reachable sets will be that they are nonempty, compact, and connected.…”

“…Compared to [16], this approximation approach yields tighter approximations, since the bounds obtained there are greater or equal to ‖ ‖, as a bound the Euclidean norm of the trajectory deviation is used there.…”

“…(iii) We define = → ( , 0 ) and = → ( , 0 ) for any ∈ [ 0 , 0 + ]. Note that in this proof, unlike in [16], and need not be convex, making the proof more technically challenging. We wish to show that ∶= ⧵ ( ) ⊞ * is a subset of .…”

“…In [16], the case in which the system experiences diminished control authority was considered, i.e., its set of admissible control inputs has shrunk with respect to that of the nominal system. In addition, stringent restrictions on the family of system that could be considered were made in [16], in particular due to the requirement that the reachable set for the nominal and off-nominal set be convex. This ultimately limits the applicability of the theory presented there to a limited set of problems.…”

“…This latter generalization to nonconvex sets requires a significant shift in the way we reason about the minimum deviation between trajectories of the nominal and off-nominal system. In this work, we also consider a change in the system dynamics, and provide methods for obtaining tighter inner and outer approximations for the off-nominal reachable set with respect to what the theory of [16] provides. This latter improvement follows from the fact that the theory in [16] considers the trajectory deviation as expressed by the norm of the states, whereas here we separately consider the deviation in single dimensions of the state.…”