Cooperative Control of Dynamical Systems With Application to Autonomous Vehicles

TLDR

Dynamical systems are heterogeneous and can be transformed into canonical forms with finite relative degrees. The paper proposes a matrix‑theory framework to analyze and design cooperative controls for groups of dynamical systems with intermittent, dynamic, local sensing and communication. The framework models sensing/communication with a time‑varying binary matrix, employs augmentation of irreducible matrices and lower triangulation of reducible matrices to study how local‑output‑feedback cooperative control shapes group behavior, and illustrates the approach with formation control of nonholonomic chained systems. The authors derive a necessary and sufficient condition for convergence of a multiplicative sequence of reducible row‑stochastic matrices, and show that simple gain choices yield cooperative behaviors such as single‑group, multi‑group, adaptive, and formation control.

Abstract

In this paper, a new framework based on matrix theory is proposed to analyze and design cooperative controls for a group of individual dynamical systems whose outputs are sensed by or communicated to others in an intermittent, dynamically changing, and local manner. In the framework, sensing/communication is described mathematically by a time-varying matrix whose dimension is equal to the number of dynamical systems in the group and whose elements assume piecewise-constant and binary values. Dynamical systems are generally heterogeneous and can be transformed into a canonical form of different, arbitrary, but finite relative degrees. Utilizing a set of new results on augmentation of irreducible matrices and on lower triangulation of reducible matrices, the framework allows a designer to study how a general local-and-output-feedback cooperative control can determine group behaviors of the dynamical systems and to see how changes of sensing/communication would impact the group behaviors over time. A necessary and sufficient condition on convergence of a multiplicative sequence of reducible row-stochastic (diagonally positive) matrices is explicitly derived, and through simple choices of a gain matrix in the cooperative control law, the overall closed-loop system is shown to exhibit cooperative behaviors (such as single group behavior, multiple group behaviors, adaptive cooperative behavior for the group, and cooperative formation including individual behaviors). Examples, including formation control of nonholonomic systems in the chained form, are used to illustrate the proposed framework.