UC San Diego

The ubiquity of dynamical systems in society has brought the subject matter of their safety to the fore. From small personal drones to autonomous vehicles and jumbo jets, the safety of dynamical systems has become critical and of utmost importance. In this dissertation, we develop control strategies for enforcing safety — characterized by state constraints — in nonlinear dynamical systems of two kinds.

First, systems with input delays are considered. This class of systems is particularly relevant because delays abound in engineering applications and their effect can be catastrophic if not systematically addressed. We develop control strategies for enforcing safety in (i) systems with a single, time-varying input delay across all input channels, and (ii) systems with distinct, constant input delays across input channels. The control strategies developed utilize a nontrivial combination of state-predictors with a safe feedback law designed for the corresponding delay- free system. In the case of systems with distinct input delays, we introduce algorithms that enforce safety using a subset of shorter-delayed input channels whenever possible, allowing safety guarantees to be made before longer input delays have been compensated. This is a feature that is especially beneficial when input delays are of significantly different lengths.

Next, a prescribed-time safety (PTSf) algorithm is developed for systems where safety is required for a finite, a priori prescribed time. In contrast to existing algorithms that only permit an asymptotic approach to (but not arrival at) the boundary of a safe set, PTSf allows a system to reach the boundary of a safe set — at an infinitely soft rate — at the end of the prescribed time, irrespective of initial condition. This feature is of interest in systems where the desired nominal behavior is prescribed-time stabilization to a set point on the boundary of the safe set. In addition to enforcing safety only for a prescribed time, PTSf is developed for safety constraints of arbitrarily high relative degree. With the aid of time-varying backstepping, and an explicit initial control gain selection criteria, PTSf retains the entirety of a safe set with no restriction on a system’s initial condition.