Convexity + curvature: Tools for the global stabilization of nonlinear systems with control inputs subject to magnitude and rate bounds

Abstract

The aim of this paper is to address the global asymptotic stabilization (gas) of affine systems with control inputs subject to magnitude and rate bounds, in the framework of ArtsteinSontag’s control Lyapunov function (clf) approach. These bounds are defined by compact (convex) control value sets (cvs) U with 0 ∈ intU . Convex Analysis together with Differential Geometry allow us to reveal the intrinsic geometry involved in the clf stabilization problem, and to solve it, if an optimal control ω(x) exists. The existence and uniqueness of ω(x) depends on convexity properties of cvs U ; whereas its regularity and boundedness of its differential is achieved in terms of the curvature of U . However, in view that control ω(x) is singular, we redesign it to derive an explicit formula for regular damping feedback controls fulfilling magnitude and rate bounds that render gas a class of affine systems.