Contraction analysis of nonlinear Hamiltonian systems

摘要

Nonlinear 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nd</sup> -order Hamiltonian dynamics can be decomposed into a hierarchy of two 1 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">st</sup> -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.