Contraction Metrics in Adaptive Nonlinear Control

Lyapunov stability theory is the bedrock of direct adaptive control. Fundamentally, Lyapunov stability requires constructing a distance-like function which must decrease with time to ensure stability. Feedback linearization, backstepping, and sum-of-squares optimization are common approaches for constructing such a distance function, but require the system to possess certain inherent/structural properties or involves solving a non-convex optimization problem. These restrictions/complexities arise because Lyapunov stability theory relies on constructing an explicit distance function. This work uses contraction metrics to derive an adaptive controller for stabilizable nonlinear systems by constructing a distance-like function differentially rather than explicitly. Because stabilizability is in fact equivalent to the existence of a contraction metric, the proposed approach is significantly more general than available results in the literature. In particular, the method can be applied to underactuated systems. More broadly, it can also be used in transfer learning where a feedback controller has been carefully learned for a nominal system, but needs to remain effective in the presence of significant but structured variations in parameters. Simulation results illustrate the approach.