Abstract
The aim of the present tutorial paper is to recall notions from manifold calculus and to illustrate how these tools prove useful in describing system-theoretic properties. Special emphasis is put on embedded manifold calculus (which is coordinate-free and relies on the embedding of a manifold into a larger ambient space). In addition, we also consider the control of non-linear systems whose states belong to curved manifolds. As a case study, synchronization of non-linear systems by feedback control on smooth manifolds (including Lie groups) is surveyed. Special emphasis is also put on numerical methods to simulate non-linear control systems on curved manifolds. The present tutorial is meant to cover a portion of the mentioned topics, such as first-order systems, but it does not cover topics such as covariant derivation and second-order dynamical systems, which will be covered in a subsequent tutorial paper.
Highlights
The theory of dynamical systems whose state spaces possess the structures of curved manifolds has been applied primarily in physics
The observation at the core of such applications is that those dynamical systems whose descriptive variables are bound to one another by non-linear holonomic constraints may be studied by means of the rich variety of mathematical tools provided by manifold calculus and may be framed in the class of dynamical systems on manifold
The present tutorial paper does not cover a number of subjects, such as the covariant derivation of vector fields, continuous-time second-order dynamical systems arising from a Lagrangian framework nor higher-order discrete-time dynamical systems, nor the key topics related to manifold curvature
Summary
The theory of dynamical systems whose state spaces possess the structures of curved manifolds has been applied primarily in physics (especially to mathematically describe the theory of general relativity at the beginning of the 20th century). As a specific applied field, the time synchronization of first-order dynamical systems on curved state manifolds by non-linear control will be surveyed. Section introduces the notion of Riemannian Hessian, which stems from a second-order approximation of a manifold-to-scalar function, and recalls optimization algorithms that extend the Newton method to look for a zero of a vector field. The present tutorial paper does not cover a number of subjects, such as the covariant derivation of vector fields, continuous-time second-order dynamical systems arising from a Lagrangian framework nor higher-order discrete-time dynamical systems, nor the key topics related to manifold curvature. These topics will be the subject of a forthcoming tutorial paper
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