Density Functions for Navigation-Function-Based Systems

• Published in: IEEE transactions on automatic control Vol. 53; no. 2; pp. 612 - 617
• Main Authors: Loizou, Savvas G., Jadbabaie, Ali
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Almost gas systems; Applied sciences; Asymptotic properties; Asymptotic stability; Automatic control; Computer science; control theory; systems; Construction; Control system synthesis; Control systems; Control theory. Systems; Density; Density functional theory; density functions; Density measurement; dual Lyapunov techniques; Dynamical systems; Dynamics; Exact sciences and technology; Lyapunov method; Navigation; navigation functions (NFs); Noise measurement; Stability analysis; State-space methods; Tasks; Asymptotic stability; density functions; Zero of function; Navigation; Existence theorem; Density function; Dynamical system; dual Lyapunov techniques; Lyapunov function; Almost gas systems; navigation functions (NFs)
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2008.917745

In this paper, we present a scheme for constructing density functions for systems that are almost globally asymptotically stable (i.e., systems for which all trajectories converge to an equilibrium except for a set of measure zero) using navigation functions (NFs). Although recently-proven converse theorems guarantee the existence of density functions for such systems, such results are only existential and the construction of a density function for almost globally asymptotically stable systems remains a challenging task. We show that for a specific class of dynamical systems that are defined based on an NF, a density function can be easily derived from the system's underlying NF.