Contraction Theory for Nonlinear Stability Analysis and Learning-based
Control: A Tutorial Overview
- URL: http://arxiv.org/abs/2110.00675v1
- Date: Fri, 1 Oct 2021 23:03:21 GMT
- Title: Contraction Theory for Nonlinear Stability Analysis and Learning-based
Control: A Tutorial Overview
- Authors: Hiroyasu Tsukamoto and Soon-Jo Chung and Jean-Jacques Slotine
- Abstract summary: Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system.
Its nonlinear stability analysis boils down to finding a suitable contraction metric that satisfies a stability condition expressed as a linear matrix inequality.
- Score: 7.918886297003018
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Contraction theory is an analytical tool to study differential dynamics of a
non-autonomous (i.e., time-varying) nonlinear system under a contraction metric
defined with a uniformly positive definite matrix, the existence of which
results in a necessary and sufficient characterization of incremental
exponential stability of multiple solution trajectories with respect to each
other. By using a squared differential length as a Lyapunov-like function, its
nonlinear stability analysis boils down to finding a suitable contraction
metric that satisfies a stability condition expressed as a linear matrix
inequality, indicating that many parallels can be drawn between well-known
linear systems theory and contraction theory for nonlinear systems.
Furthermore, contraction theory takes advantage of a superior robustness
property of exponential stability used in conjunction with the comparison
lemma. This yields much-needed safety and stability guarantees for neural
network-based control and estimation schemes, without resorting to a more
involved method of using uniform asymptotic stability for input-to-state
stability. Such distinctive features permit systematic construction of a
contraction metric via convex optimization, thereby obtaining an explicit
exponential bound on the distance between a time-varying target trajectory and
solution trajectories perturbed externally due to disturbances and learning
errors. The objective of this paper is therefore to present a tutorial overview
of contraction theory and its advantages in nonlinear stability analysis of
deterministic and stochastic systems, with an emphasis on deriving formal
robustness and stability guarantees for various learning-based and data-driven
automatic control methods. In particular, we provide a detailed review of
techniques for finding contraction metrics and associated control and
estimation laws using deep neural networks.
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