This paper derives new results in nonlinear system analysis using methods inspired from fluid mechanics and differential geometry. Based on a differential analysis of convergence, these results may be viewed as generalizing the classical Krasovskii theorem, and, more loosely, linear eigenvalue analysis. A central feature is that convergence and limit behavior are in a sense treated separately, leading to significant conceptual simplifications. The approach is illustrated by controller and observer designs for simple physical examples.
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.
Contraction theory is a recently developed control system tool based on an exact differential analysis of convergence. After establishing new combination properties of contracting systems, this paper derives new controller and observer designs for mechanical systems such as aircraft and underwater vehicles. The classical nonlinear strap down algorithm is also shown to be marginally contracting. The relative simplicity of the approach stems from its effective exploitation of the systems' structural specificities.
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