Feedback Integrators

Abstract

A new method is proposed to numerically integrate a dynamical system on a manifold such that the trajectory stably remains on the manifold and preserves first integrals of the system. The idea is that given an initial point in the manifold we extend the dynamics from the manifold to its ambient Euclidean space and then modify the dynamics outside the intersection of the manifold and the level sets of the first integrals containing the initial point such that the intersection becomes a unique local attractor of the resultant dynamics. While the modified dynamics theoretically produces the same trajectory as the original dynamics, it yields a numerical trajectory that stably remains on the manifold and preserves the first integrals. The big merit of our method is that the modified dynamics can be integrated with any ordinary numerical integrator such as Euler or Runge-Kutta. We illustrate this method by applying it to three famous problems: the free rigid body, the Kepler problem and a perturbed Kepler problem with rotational symmetry. We also carry out simulation studies to demonstrate the excellence of our method and make comparisons with the standard projection method, a splitting method and Störmer-Verlet schemes.

On controller design for systems on manifolds in Euclidean space

“…Remark The technique of embedding into ambient Euclidean space and transversal stabilization was successfully tested in creating feedback integrators for structure‐preserving numerical integration of the dynamics of uncontrolled dynamical systems. This technique is extended to control systems in this paper.…”

“…The embedding technique has been also applied in control theory. For example, it was used to produce a simple proof of the Pontryagin maximum principle on manifolds and was combined with the transversal stabilization technique to yield feedback‐based structure‐preserving numerical integrators for simulation of dynamical systems . A series of relevant works have been made by Maggiore and his collaborators on local transverse feedback linearizability of control‐invariant submanifolds and virtual holonomic constraints.…”

“…For example, it was used to produce a simple proof of the Pontryagin maximum principle on manifolds 11 and was combined with the transversal stabilization technique to yield feedback-based structure-preserving numerical integrators for simulation of dynamical systems. 12 A series of relevant works have been made by Maggiore and his collaborators on local transverse feedback linearizability of control-invariant submanifolds and virtual holonomic constraints. [13][14][15] The focus of Maggiore is placed on the creation of a submanifold for a given system and its transversal stabilization via feedback for path-following controller synthesis, whereas our work in this paper is focused on embedding and extending a state-space manifold of a given system into Euclidean space and its transversal stabilization for tracking controller synthesis.…”

Feedback Integrators for Nonholonomic Mechanical Systems

“…Hence, by Theorem 2.1 in [3], every trajectory of (54b) starting in V −1 ([0, c]) remains in V −1 ([0, c]) for all future time and asymptotically converges to V −1 (0) as t → ∞. Also, V −1 (0) is an invariant set of (54b).…”

Controller design for systems on manifolds in Euclidean space

“…For example, it was used to produce a simple proof of the Pontryagin maximum principle on manifolds, 5 and was combined with the transversal stabilization technique to yield feedback-based structure-preserving numerical integrators for simulation of dynamical systems. 6 A series of relevant works have been made by Maggiore and his collaborators on local transverse feedback linearizability of control-invariant submanifolds and virtual holonomic constraints. 17,20,21 The focus of Maggiore is placed on creation of a submanifold for a given system and its transversal stabilization via feedback for path-following controller synthesis, whereas our work in this paper is focused on embedding and extending a state space manifold of a given system into Euclidean space and its transversal stabilization for tracking controller synthesis.…”