On the Boundary of the Robust Admissible Set in State and Input Constrained Nonlinear Systems
Provides a theoretical framework for computing the set of safe initial states in constrained nonlinear systems under disturbances, relevant for safety-critical control.
This paper defines the robust admissible set for nonlinear control systems with bounded disturbances and state/input constraints, and characterizes its boundary via a tangent condition and a saddle-point principle derived from Pontryagin's maximum principle, enabling direct construction of the barrier. The method is demonstrated on an adaptive cruise control example.
In this paper, we consider nonlinear control systems subject to bounded disturbances and to both state and input constraints. We introduce the definition of robust admissible set - the set of all initial states from which the state and input constraints can be satisfied for all times against all admissible disturbances. We focus on its boundary that can be decomposed into the usable part on the state constraint boundary and the barrier, interior to the state constraints. We show that, at the intersection of these two components, the boundary of the robust admissible set must be tangent to the state constraint set and separate the interior of the robust admissible set and its complement, a property that we call the ultimate locally separating hyperplane condition. Moreover, we prove that the barrier must satisfy a saddle-point principle on a Hamiltonian, based on Pontryagin's maximum principle, whose final condition is precisely the ultimate locally separating condition, thus providing a set of differential equations made of the system and its adjoint for a direct construction of the barrier. Lastly, we illustrate our results by calculating the robust admissible set for an adaptive cruise control example.