Higher-order algorithms for nonlinearly parameterized adaptive control

A set of new adaptive control algorithms is presented. The algorithms are applicable to linearly parameterized systems and to nonlinearly parameterized systems satisfying a certain monotonicity requirement. A variational perspective based on the Bregman Lagrangian (Wibisono, Wilson, & Jordan, 2016) is adopted to define a general framework that systematically generates higher-order in-time speed gradient algorithms. These general algorithms reduce to a recently developed method as a special case, and naturally extend to composite methods that combine perspectives from both adaptive control and system identification. A provocative connection between adaptive control algorithms for nonlinearly parameterized systems and algorithms for isotonic regression and provable neural network learning is utilized to extend these higher-order algorithms to the realm of nonlinearly parameterized systems. Modifications to all presented algorithms inspired by recent progress in distributed stochastic gradient descent algorithms are developed. Time-varying learning rate matrices based on exponential forgetting/bounded gain forgetting least squares can be stably employed in the higher-order context, and conditions for their applicability in the nonlinearly parameterized setting are provided. A consistent theme of our contribution is the exploitation of strong connections between classical nonlinear adaptive control techniques and recent progress in optimization and machine learning.