Nonlinear Dynamics

, Volume 92, Issue 3, pp 1431–1451 | Cite as

Online estimation and adaptive control for a class of history dependent functional differential equations

Original Paper
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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. Well-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.

Keywords

Online estimation Adaptive control Functional differential equations 

Notes

Acknowledgements

We would like to thank the anonymous referees for their constructive input and valuable suggestions that helped us improve the manuscript.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringVirginia TechBlacksburgUSA
  2. 2.Department of Engineering Science and MechanicsVirginia TechBlacksburgUSA

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