Online estimation and adaptive control for a class of history dependent functional differential equations
- 21 Downloads
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.
KeywordsOnline estimation Adaptive control Functional differential equations
We would like to thank the anonymous referees for their constructive input and valuable suggestions that helped us improve the manuscript.
- 5.Corduneaunu, C.: Integral Equations and Applications. Cambridge University Press, Cambridge (2008)Google Scholar
- 8.Dadashi, S., Bobade, P., Kurdila, A.J.: Error estimates for multiwavelet approximations of a class of history dependent operators. In: 2016 IEEE 55th Conference on Decision and Control (CDC) (2016)Google Scholar
- 9.Demetriou, M.A.: Adaptive parameter estimation of abstract parabolic and hyperbolic distributed parameter systems. Ph.D. thesis, Departments of Electrical-Systems and Mathematics, University of Southern California, Los Angeles, CA (1993)Google Scholar
- 19.Khalil, H.: Nonlinear Systems, 3rd edn. Prentice-Hall, Upper Saddle River (2003)Google Scholar
- 20.Krasovskii, N.N.: On the application of the second method of A.M. Lyapunov to equations with time delays. Prikl. Mat. Mekh. 20, 315–327 (1956)Google Scholar
- 24.Lewis, F.L., Dawson, D., Abdallah, C.: Robot Manipulator Control: Theory and Practice. Marcel Dekker Inc, New York (2004)Google Scholar
- 27.Rudakov, V.P.: Qualitative theory in a Banach space, Lyapunov–Krasovskii functionals, and generalization of certain problems. Ukr. Mat. Zh. 30(1), 130–133 (1978)Google Scholar
- 28.Rudakov, V.P.: On necessary and sufficient conditions for the extendability of solutions of functional-differential equations of the retarded type. Ukr. Mat. Zh. 26(6), 822–827 (1974)Google Scholar
- 31.Siciliano, B., Sciavicco, L., Villani, L., Oriolo, G.: Robotics: Modeling, Planning, and Control. Springer, London (2010)Google Scholar
- 32.Spong, M., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control. Wiley, New York (2005)Google Scholar