Lyapunov stability theory of nonsmooth systems

Abstract

Abstract-This paper develops nonsmooth Lyapunov stability theory and LaSalle's invariance principle for a class of nonsmooth Lipschitz continuous Lyapunov functions and absolutely continuous state trajec· tories. Computable tests based on Filipov's differential inclusion and Clarke's generalized gradient are derived. The primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right-hand side such as in nonsmooth dynamic systems or variable structure control.

Nonsmooth Lyapunov functions and discontinuous Carathéodory systems

“…Next a LaSalle-like invariance principle is proved (see Shevitz and Paden, 1994, Ryan, 1998, Bacciotti and Ceragioli, 1999 for other versions of the invariance principle). In order to get it, some regularity for the vector field is needed.…”

“…This notion is analogous to the notion of set-valued derivative of a map with respect to a differential inclusion introduced in Shevitz and Paden (1994) and improved in Bacciotti and Ceragioli (1999) (note that in Bacciotti and Ceragioli (1999), Clarke regular functions instead of nonpathological functions were used; the analogous set-valued derivative for nonpathological functions has been studied in Ceragioli (2000)). …”

“…We consider nonsmooth Lyapunov functions: nonsmoothness of Lyapunov functions is in fact unavoidable when studying stability properties of discontinous systems. In particular, Lipschitz continuous Lyapunov functions which are also nonpathological (Valadier, 1989) are considered and a notion of derivative introduced in Shevitz and Paden (1994), improved in Bacciotti and Ceragioli (1999) and then used in Ceragioli (1999, 2003), is used throughout the paper. The paper consists of two main section.…”

Input-to-state stability and interconnections of discontinuous dynamical systems

“…the example of the extended flower system at the end of the paper). The seminal book [8] and the paper [21] provide extensions of stability analysis for discontinuous dynamical systems using non-smooth Lipschitz continuous Lyapunov functions. We will extend this work towards ISS and ISS interconnection 3 results using a generalized Filippov's solution concept.…”

Friction compensation in a controlled one-link robot using a reduced-order observer

“…Firstly, in order to check condition (a), we use a candidate Lyapunov function V = 1 2 Jx 2 3 (see (Filippov, 1988) and (Shevitz and Paden, 1994) for details on Lyapunov analysis for differential inclusions). Its time-derivativeV = Jẋ 3 x 3 obeyṡ…”