I. Introduction
Consensus problems and collective dynamics have been the subject of significant interest in the control community in recent decades with applications to cooperative and distributed control. Seminal works include [1]–[4]. See [5] for a survey and [4] for some examples of applications. More recently, there has been growing interest in the study of consensus algorithms defined on nonlinear spaces such as Lie groups [6], [7], the n-sphere [8], Grassmannians [9], and Stiefel manifolds [10]. Consensus problems on nonlinear spaces give rise to behaviors and global convergence issues that are not observed in linear models [11]. They are relevant for a number of engineering applications including the design of spatial coordinated motions [12]. A fundamental challenge presented by consensus in nonlinear spaces is due to non-uniqueness of geodesics and topological properties of the underlying space, which result in problems that are fundamentally more complex and interesting than Euclidean analogues. Here, we will use an approach based on positivity theory and monotonicity to study consensus on the circle. In particular, we seek an answer to the following question: Can monotone system design be used to construct consensus algorithms that converge to a given target formation and collective motion on the circle?