Safety Margins of Inverse Optimal ISSf Controllers

Abstract

We investigate the gain margin of a general nonlinear system under an inverse optimal input-to-state safe (ISSf) controller of the form u=u0(x)+u*(x,u0), where u0 is the nominal control and u* is the inverse optimal safety filter that minimally modifies the nominal controller's unsafe actions over the infinite horizon. By first establishing a converse ISSf-BF theorem, we reveal the equivalence among the achievability of ISSf by feedback, the achievability of inverse optimality, and the solvability of a Hamilton-Jacobi-Isaacs equation associated with the inverse optimal ISSf gain assignment. Then we develop a collection of safety margin results on the overall control u=u0+u*. In the absence of disturbances, we find that standard inverse optimal safe controllers have a certain degree of gain margin. Specifically, when f(x) acts safely but u0 acts unsafely, the gain can be decreased by up to half; and when f(x) acts unsafely, we establish that, if u0 acts safely, the gain can be increased arbitrarily, whereas if u0 acts unsafely, the control recovers the full gain margin [1/2,inf). It is shown, however, that under control gain variation, the safe set of these controllers is locally asymptotically stable, which implies that their safety is sensitive to large but bounded disturbances. To make inverse optimal ISSf controllers robust to gain variation, we propose a gain margin improvement approach at the expense of an increased control effort. This improvement allows the inverse optimal safe control to inherit the standard gain margin of [1/2,inf) without requiring prior knowledge of whether f(x) or u0 acts safely on the safety boundary, while simultaneously ensuring global asymptotic stability of the resulting safe set. In the presence of disturbances, this improvement idea renders inverse optimal ISSf controllers robust to gain variations with the same gain margin of [1/2,inf).