Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview

Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under contraction metric defined with uniformly positive definite matrix, the existence which results in necessary and sufficient characterization incremental exponential stability multiple solution trajectories respect each other. By using squared length as Lyapunov-like function, its analysis boils down finding suitable that satisfies condition expressed linear matrix inequality, indicating many parallels can be drawn between well-known systems for systems. Furthermore, takes advantage superior robustness property used conjunction comparison lemma. This yields much-needed safety guarantees neural network-based control estimation schemes, without resorting more involved method uniform asymptotic input-to-state stability. Such distinctive features permit systematic construction via convex optimization, thereby obtaining explicit bound on distance time-varying target trajectory perturbed externally due disturbances learning errors. The objective this paper therefore present tutorial overview advantages deterministic stochastic systems, emphasis deriving formal various learning-based data-driven automatic methods. In particular, we provide detailed review techniques metrics associated laws deep networks.