Continuity and smoothness properties of nonlinear optimization-based feedback controllers

Online optimization-based controllers are becoming increasingly prevalent as a means to control complex high-dimensional nonlinear systems, e.g., bipedal and humanoid robots, due to their ability to balance multiple control objectives subject to input constraints. Motivated by these applications, the goal of this paper is to explore the continuity and smoothness properties of feedback controllers that are formulated as quadratic programs (QPs). We begin by drawing connections between these optimization-based controllers and a family of perturbed nonlinear programming problems commonly studied in operations research. With a view towards robotic systems, some existing results on perturbed nonlinear programming problems are extended and specialized to address conditions that arise when quadratic programs are used to enforce the convergence of control Lyapunov functions (CLFs). The main result of this paper is a novel set of conditions on the continuity of QPs that can be used when a subset of the constraints vanishes. A simulation study of position regulation in the compass gait biped demonstrates how the new conditions of this paper can be applied to more complex robotic systems.