Nonlinear stochastic operators to control the consensus problem in multi-agent systems

Abstract

Consensus problem in multi-agent systems (MAS) has attracted growing interest in recent time. Traditional approaches to controlling consensus problem are often based on linear models, which take their origin from the well known DeGroot model. Various researchers in recent time have proposed nonlinear models simply due to the fact that linear models converge slowly, are characterized with higher number of iterations and at the same time incapable of converging to optimal consensus. Nonlinear models on the other hand, are more efficient, converging faster with lesser number of iterations and to approximate optimal consensus. However, the downside of nonlinear models is that they are often of higher complexity and are setup with restricted conditions. The present concern is to investigate possible nonlinear models with faster convergence to optimal consensus yet with relatively low complexity and more flexible system conditions. The main aim of this research is to control the consensus problem in MAS via doubly stochastic quadratic operator (DSQO). In this research, a nonlinear model of DSQO using majorization theory is investigated to control consensus problem under more flexible conditions as well as lower complexity for specific subclass of the DSQO. The extreme points of doubly stochastic quadratic operator (EDSQO) is then examined for lower complexity. The EDSQO is considered in this case as a special subclass of DSQO sets points in space. Therefore, the vertices of this class are in turn examined to formulate a general solution for the convergence problem. The study focuses on the EDSQO on finite-dimensional simplex (FDS). The general theorems for the DSQO on finite dimensions are derived and presented particularly for finite number of agents. The work carries out a study to define the extreme points of the set of EDSQO on two-dimensional simplex (2DS) and examines the limit behaviour of the trajectories of the EDSQO on FDS. The work then follows up with the dynamic classifications of DSQO on FDS, so as to elaborate a platform class of positive DSQO (PDSQO) which are suitable for reaching a consensus for MAS. The DSQO algorithms are then evaluated based on various kinds of transition matrices and the results are compared to the existing classical consensus algorithms. This presentation also includes the study of higher DSQO degrees (HDSQO) as well as fractional DSQO degrees (FDSQO) for consensus problem in MAS. The research work derives novel low-complexity nonlinear convergence models MPDSQO and MEDSQO which are modified from the DSQO and EDSQO for consensus problem in MAS. In general, the proposed novel protocols of the DSQO have shown and proved to be advantageous over the existing linear and other nonlinear models.