Title: Online Estimation and Adaptive Control for a Class of History Dependent Functional Differential Equations,
Abstract: This paper presents sufficient conditions for the convergence of online
estimation methods and the stability of adaptive control strategies for a class
of history dependent, functional differential equations. The study is motivated
by the increasing interest in estimation and control techniques for robotic
systems whose governing equations include history dependent nonlinearities. The
functional differential equations in this paper are constructed using integral
operators that depend on distributed parameters. As a consequence the resulting
estimation and control equations are examples of distributed parameter systems
whose states and distributed parameters evolve in finite and infinite
dimensional spaces, respectively. suWell-posedness, existence, and uniqueness
are discussed for the class of fully actuated robotic systems with history
dependent forces in their governing equation of motion. By deriving rates of
approximation for the class of history dependent operators in this paper,
sufficient conditions are derived that guarantee that finite dimensional
approximations of the online estimation equations converge to the solution of
the infinite dimensional, distributed parameter system. The convergence and
stability of a sliding mode adaptive control strategy for the history
dependent, functional differential equations is established using Barbalat's
lemma.