Lower Bounded Control-Lyapunov Functions

Abstract

Abstract. The well known Brockett condition -a topological obstruction to the existence of smooth stabilizing feedback laws -has engendered a large body of work on discontinuous feedback stabilization. The purpose of this paper is to introduce a class of control-Lyapunov function from which it is possible to specify a (possibly discontinuous) stabilizing feedback law. For control-affine systems with unbounded controls Sontag has described a Lyapunov pair which gives rise to an explicit stabilizing feedback law smooth away from the origin -Sontag's "universal construction" of Artstein's Theorem. In this work we introduce the more general "lower bounded control-Lyapunov function" and a "universal formula" for nonaffine systems. Our "universal formula" is a static state feedback which is measurable and locally bounded but possibly discontinuous. Thus, for the corresponding closed loop system, the classical notion of solution need not apply. To deal with this situation we use the generalized solution due to Filippov.

A Universal Feedback Controller for Discontinuous Dynamical Systems Using Nonsmooth Control Lyapunov Functions

“…Specifically, as will be seen later in the paper, the requirement that there exists z 2 L G VðxÞ such that, for all u 2 R m ; max½L G VðxÞu ¼ zu holds if and only if L G VðxÞ is a singleton. Specifically, for systems of the form (11), note that L f þGu VðxÞ L f VðxÞ þ L G VðxÞu for almost all x and all u, and hence, inf u2U ½max L f VðxÞ þ L G VðxÞu ¼ À1 when x 6 2 R and x 6 ¼ 0, whereas inf u2U ½max L f VðxÞ þ L G VðxÞu < 0 for almost all x 2 R. Hence, Eq. Then, either q À z 6 ¼ 0 or r À z 6 ¼ 0.…”

“…0 (11) where f : R n ! Specifically, we consider discontinuous nonlinear affine dynamical systems of the form _ xðtÞ ¼ f ðxðtÞÞ þ GðxðtÞÞuðtÞ; xð0Þ ¼ x 0 ; a:e: t !…”

“…Finally, we assume that the set L G VðxÞ is single-valued 2 for almost all x 2 R n and that L G VðxÞ 6 ¼ ; at all other points x. (11) if and only if max L f VðxÞ < 0; a:e: x 2 R; (12) where R ¼ D fx 2 R n nf0g : L G VðxÞ ¼ 0g. Specifically, as will be seen later in the paper, the requirement that there exists z 2 L G VðxÞ such that, for all u 2 R m ; max½L G VðxÞu ¼ zu holds if and only if L G VðxÞ is a singleton.…”

“…Furthermore, Rifford [10] addresses the problem of stabilization of globally asymptotically controllable systems wherein the system vector field is locally Lipschitz continuous in the state and uniformly in the control. [11] also provides discontinuous controllers using a Filippov solution framework; however, Hirschorn [11] uses a special closed lower bounded control Lyapunov function which we also do not require here. However, we will not need to consider semiconcavity in what follows.…”

“…[9][10][11][12] to develop a constructive universal feedback control law for discontinuous dynamical systems based on the existence of a nonsmooth control Lyapunov function defined in the sense of generalized Clarke gradients [13] and set-valued Lie derivatives [14]. [9][10][11][12] to develop a constructive universal feedback control law for discontinuous dynamical systems based on the existence of a nonsmooth control Lyapunov function defined in the sense of generalized Clarke gradients [13] and set-valued Lie derivatives [14].…”