On design of continuous Lyapunov's feedback control

A common approach to Lyapunov's stability control is to design a controller such that a Lyapunov function can be derived for the control system to ensure stability. This procedure often leads to a discontinuous controller. When the controller is implemented, the discontinuous terms are replaced with continuous functions to avoid chattering of the control signal. Two associated problems have been overlooked during this procedure. One is that discontinuous control systems are non-smooth, which violates the fundamental assumptions of solution theories and the applicability of Lyapunov's stability theory is questionable. Another problem is that the replacement of discontinuous terms may weaken stability, which can be critical. In this paper, we discuss proper stability analysis of discontinuous control systems using the extended Lyapunov's second method based on Filippov's solution concept for non-smooth systems. We further propose to utilize the concept of Lyapunov exponents to quantitatively analyze the stability of continuous control systems obtained by replacing the discontinuous terms in the discontinuous controllers. An example involving the stabilization of a two-link non-fixed-base robotic manipulator is presented for demonstration. This research fills the gap in designing continuous Lyapunov's stability controllers regarding limited available Lyapunov functions.