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Research Papers

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

[+] Author and Article Information

School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332

Professor
School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332

Alternatively, we can consider Krasovskii solutions of Eq. (2) wherein the possible misbehavior of the derivative of the state on null measure sets is not ignored; that is, $K[Fφ](x)$ is replaced with $K[Fφ](x)=∩δ>0co¯{Fφ(Bδ(x))}$ and where $Fφ$ is assumed to be locally bounded.

The assumption that $LGV(x)$ is single-valued is necessary. Specifically, as will be seen later in the paper, the requirement that there exists $z¯∈LGV(x)$ such that, for all $u∈Rm,max[LGV(x)u]=z¯u$ holds if and only if $LGV(x)$ is a singleton. To see this, let $q,r∈LGV(x)$, with q≠r, and assume, ad absurdum, that the required $z¯$ exists. Then, either $q-z¯≠0$ or $r-z¯≠0$. Assume $q-z¯≠0$ and let $uT=q-z¯$. Then, $qu-z¯u=(q-z¯)u=(q-z¯)(q-z¯)T=∥q-z¯∥22>0$. Hence, $qu>z¯u$, which leads to a contradiction.

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 5, 2013; final manuscript received September 9, 2014; published online November 7, 2014. Assoc. Editor: Yang Shi.

J. Dyn. Sys., Meas., Control 137(4), 041005 (Apr 01, 2015) (6 pages) Paper No: DS-13-1432; doi: 10.1115/1.4028593 History: Received November 05, 2013; Revised September 09, 2014

## Abstract

The consideration of nonsmooth Lyapunov functions for proving stability of feedback discontinuous systems is an important extension to classical stability theory since there exist nonsmooth dynamical systems whose equilibria cannot be proved to be stable using standard continuously differentiable Lyapunov function theory. For dynamical systems with continuously differentiable flows, the concept of smooth control Lyapunov functions was developed by Artstein to show the existence of a feedback stabilizing controller. A constructive feedback control law based on a universal construction of smooth control Lyapunov functions was given by Sontag. Even though a stabilizing continuous feedback controller guarantees the existence of a smooth control Lyapunov function, many systems that possess smooth control Lyapunov functions do not necessarily admit a continuous stabilizing feedback controller. However, the existence of a control Lyapunov function allows for the design of a stabilizing feedback controller that admits Filippov and Krasovskii closed-loop system solutions. In this paper, we develop a constructive 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 and set-valued Lie derivatives.

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## Figures

Fig. 1

Phase portrait of the open-loop nonsmooth harmonic oscillator

Fig. 2

Phase portrait of the closed-loop nonsmooth harmonic oscillator

Fig. 3

State trajectories of the closed-loop system versus time

Fig. 4

Control trajectories of the closed-loop system versus time

Fig. 5

Phase portrait of the open-loop system

Fig. 6

Phase portrait of the closed-loop system

Fig. 7

State trajectories of the closed-loop system versus time

Fig. 8

Control trajectories of the closed-loop system versus time

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