Contraction analysis of nonlinear Hamiltonian systems

Abstract

Nonlinear 2 nd -order Hamiltonian dynamics can be decomposed into a hierarchy of two 1 st -order complex component dynamics, allowing their exponential stability to be assessed using basic tools in contraction theory. Exponential convergence rates can be explicitly computed based on the system's damping and the Hessian of its complex action. The results can be used to place state and time-dependent complex contraction rates in a controller or observer design, extending elementary linear time-invariant eigenvalue placement. Thus, they can offer an exact alternative to gain-scheduling, where exponential stability guarantees are local and assume slowly varying eigenvectors of the Jacobian. Finally, Hamiltonian and p.d.e. contraction tools are applied to the exact solution of the classical simultaneous localization and mapping (SLAM) problem in robotics.

A Survey on Aerial Swarm Robotics

“…Many types of model uncertainty can be cast into a bounded perturbation term, including constant unknown time delays [27], [72] and errors arising from heterogeneous dynamics [27], [63]. Recently, incremental stability has been extended to synchronization stability of multiple Itō stochastic nonlinear differential equations [38], [74] with unbounded stochastic disturbances.…”

“…Input-to-State Stability (ISS) is used to study stability of swarm systems with bounded uncertainties [70], [71]. Contraction analysis [72] is used to study global exponential stability of multiple solution trajectories, and hence forms a basis of incremental stability analysis. Contraction-based incremental stability analysis represents an important departure from traditional passivity-based methods using Lyapunov functions, which are concerned primarily with stability of equilibrium points.…”

Dynamic Movement Primitives -A Framework for Motor Control in Humans and Humanoid Robotics

“…In particular, the latter approach also emphasized to remove the explicit time dependency of , such that complicated "clocking" and "reset clock" mechanisms could be avoided, and the combination of policy primitives becomes simplified. Despite the successful application of policy primitives in the mobile robotics domain, so far, it remains a topic of ongoing research [11,12] how to generate and combine primitives in a principled and autonomous way, and how such an approach generalizes to complex movement systems, like human arms and legs.…”

Contraction theory on Riemannian manifolds

“…Slotine et al under the title of contraction theory, involves examining the dynamics of a "virtual displacement" between two infinitesimally separated trajectories. The root piece of literature for this methodology is [1], with a plethora of extensions including graph-theoretic characterizations [2], backstepping design [3], extensions to distributed systems [4], and algorithmic searches for contraction metrics [5]. Rigorous proofs of the main result -a sufficient condition for a vector field to be "contracting" -are varied in style, with some revolving around the use of the matrix measure [6,7] while others utilize the perspective of a contraction metric [8,9,10].…”