Analysis and Control of Perturbed Density Systems

TL;DR

This paper introduces the concept of density systems, analyzing how density functions influence system behavior and stability, and proposes adaptive control laws for systems with known and unknown parameters under disturbances.

Contribution

It develops a new framework for density systems and designs adaptive control laws to ensure system outputs remain within specified sets.

Findings

Theoretical analysis of density systems and their stability properties.

Design of adaptive control laws for systems with disturbances.

Numerical simulations demonstrating the effectiveness of the proposed methods.

Abstract

The paper investigates dynamical systems for which the derivative of some positive-definite function along the solutions of this system depends on so-called density function. In turn, such dynamical systems are called density systems. The density function sets the density of the space, where the system is evolved, and affects the behaviour of the original system. For example, this function can define (un)stable regions and forbidden regions where there are no system solutions. The density function can be used in the design of new adaptive control laws with the formulation of appropriate new control goals, e.g., stabilization in given bounded or semi-bounded sets. To design a novel adaptive control law that ensures the system outputs in a given set, systems with known and unknown parameters under disturbances are considered. All theoretical results and conclusions are illustrated by numerical simulations.