Skip to main content
Top
Published in: Journal of Applied Mathematics and Computing 1-2/2017

18-02-2016 | Original Research

PID-type iterative learning control for impulsive ordinary differential equations

Authors: Zhuoyan Gao, Shengda Liu, JinRong Wang

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2017

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, we consider PID-type iterative learning control law for nonlinear impulsive ordinary differential equations. We obtain convergence results for open-loop and closed-loop iterative learning schemes with initial error in the sense of \(\lambda \)-norm. Finally, a numerical example is given to demonstrate the validity of the design methods.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Uchiyama, M.: Formulation of high-speed motion pattern of a mechanical arm by trial. Trans. Soc. Instrum. Control Eng. 14, 706–712 (1978)CrossRef Uchiyama, M.: Formulation of high-speed motion pattern of a mechanical arm by trial. Trans. Soc. Instrum. Control Eng. 14, 706–712 (1978)CrossRef
2.
go back to reference Arimoto, S., Kawamura, S.: Bettering operation of robots by learning. J. Robot. Syst. 1, 123–140 (1984)CrossRef Arimoto, S., Kawamura, S.: Bettering operation of robots by learning. J. Robot. Syst. 1, 123–140 (1984)CrossRef
3.
go back to reference Arimoto, S.: Mathematical theory of learning with applications to robot control. In: Narendra, K.S. (ed.) Adaptive and Learning Systems: Theory and Applications, pp. 379–388. Yale University, New Haven (1985) Arimoto, S.: Mathematical theory of learning with applications to robot control. In: Narendra, K.S. (ed.) Adaptive and Learning Systems: Theory and Applications, pp. 379–388. Yale University, New Haven (1985)
4.
go back to reference Bien, Z., Xu, J.X.: Iterative Learning Control Analysis: Design Integration and Applications. Springer, New York (1998)CrossRef Bien, Z., Xu, J.X.: Iterative Learning Control Analysis: Design Integration and Applications. Springer, New York (1998)CrossRef
5.
go back to reference Chen, Y.Q., Wen, C.: Iterative Learning Control: Convergence, Robustness and Applications. Springer, New York (1999)CrossRefMATH Chen, Y.Q., Wen, C.: Iterative Learning Control: Convergence, Robustness and Applications. Springer, New York (1999)CrossRefMATH
6.
go back to reference Norrlof M.: Iterative learning control: Analysis, design, and Experiments, Linkoping Studies in Science and Technology, Dissertations, No. 653, Sweden, (2000) Norrlof M.: Iterative learning control: Analysis, design, and Experiments, Linkoping Studies in Science and Technology, Dissertations, No. 653, Sweden, (2000)
7.
8.
go back to reference Xu, J.X., Panda, S.K., Lee, T.H.: Real-time iterative learning control: Design and applications, advances in industrial control. Springer, Berlin (2009) Xu, J.X., Panda, S.K., Lee, T.H.: Real-time iterative learning control: Design and applications, advances in industrial control. Springer, Berlin (2009)
9.
10.
go back to reference Hou, Z., Xu, J., Yan, J.: An iterative learning approach for density control of freeway traffic flow via ramp metering. Transport. Res. Part C Emerg. Technol. 16, 71–97 (2008)CrossRef Hou, Z., Xu, J., Yan, J.: An iterative learning approach for density control of freeway traffic flow via ramp metering. Transport. Res. Part C Emerg. Technol. 16, 71–97 (2008)CrossRef
11.
go back to reference Wang, Y., Gao, F., Doyle III, F.J.: Survey on iterative learning control, repetitive control, and run-to-run control. J. Process Control 19, 1589–1600 (2009)CrossRef Wang, Y., Gao, F., Doyle III, F.J.: Survey on iterative learning control, repetitive control, and run-to-run control. J. Process Control 19, 1589–1600 (2009)CrossRef
12.
go back to reference de Wijdeven, J.V., Donkers, T., Bosgra, O.: Iterative Learning Control for uncertain systems: Robust monotonic convergence analysis. Automatica 45, 2383–2391 (2009)MathSciNetCrossRefMATH de Wijdeven, J.V., Donkers, T., Bosgra, O.: Iterative Learning Control for uncertain systems: Robust monotonic convergence analysis. Automatica 45, 2383–2391 (2009)MathSciNetCrossRefMATH
14.
go back to reference Ruan, X., Bien, Z.Z., Wang, Q.: Convergence characteristics of proportional-type iterative learning control in the sense of Lebesgue-\(p\) norm. IET Control Theory Appl. 6, 707–714 (2012)MathSciNetCrossRef Ruan, X., Bien, Z.Z., Wang, Q.: Convergence characteristics of proportional-type iterative learning control in the sense of Lebesgue-\(p\) norm. IET Control Theory Appl. 6, 707–714 (2012)MathSciNetCrossRef
15.
go back to reference Ruan, X., Zhao, J.: Convergence monotonicity and speed comparison of iterative learning control algorithms for nonlinear systems. IMA J. Math. Control Inf. 30, 473–486 (2013)MathSciNetCrossRefMATH Ruan, X., Zhao, J.: Convergence monotonicity and speed comparison of iterative learning control algorithms for nonlinear systems. IMA J. Math. Control Inf. 30, 473–486 (2013)MathSciNetCrossRefMATH
16.
go back to reference Li, Y., Chen, Y.Q., Ahn, H.S.: Fractional-order iterative learning control for fractional-order linear systems. Asian J. Control 13, 1–10 (2011)MathSciNetCrossRefMATH Li, Y., Chen, Y.Q., Ahn, H.S.: Fractional-order iterative learning control for fractional-order linear systems. Asian J. Control 13, 1–10 (2011)MathSciNetCrossRefMATH
17.
go back to reference Li, Y., Chen, Y.Q., Ahn, H.S., Tian, G.: A survey on fractional-order iterative learning control. J. Optim. Theory Appl. 156, 127–140 (2013)MathSciNetCrossRefMATH Li, Y., Chen, Y.Q., Ahn, H.S., Tian, G.: A survey on fractional-order iterative learning control. J. Optim. Theory Appl. 156, 127–140 (2013)MathSciNetCrossRefMATH
18.
go back to reference Lan, Y.H., Zhou, Y.: Iterative learning control with initial state learning for fractional order nonlinear systems. Comput. Math. Appl. 64, 3210–3216 (2012)MathSciNetCrossRefMATH Lan, Y.H., Zhou, Y.: Iterative learning control with initial state learning for fractional order nonlinear systems. Comput. Math. Appl. 64, 3210–3216 (2012)MathSciNetCrossRefMATH
19.
go back to reference Lan, Y.H., Zhou, Y.: \(D\)-type iterative learning control for fractional order linear time-delay systems. Asian J. Control 15, 669–677 (2013)MathSciNetCrossRefMATH Lan, Y.H., Zhou, Y.: \(D\)-type iterative learning control for fractional order linear time-delay systems. Asian J. Control 15, 669–677 (2013)MathSciNetCrossRefMATH
20.
go back to reference Owens, D.H.: Norm Optimal Iterative Learning Control: Iterative Learning Control. Springer, London (2016)CrossRefMATH Owens, D.H.: Norm Optimal Iterative Learning Control: Iterative Learning Control. Springer, London (2016)CrossRefMATH
21.
go back to reference Jin, X.: Adaptive iterative learning control for high-order nonlinear multi-agent systems consensus tracking. Syst. Control Lett. 89, 16–23 (2016)MathSciNetCrossRefMATH Jin, X.: Adaptive iterative learning control for high-order nonlinear multi-agent systems consensus tracking. Syst. Control Lett. 89, 16–23 (2016)MathSciNetCrossRefMATH
22.
go back to reference Li, Y., Jiang, W.: Fractional order nonlinear systems with delay in iterative learning control. Appl. Math. Comput. 257, 546–552 (2015)MathSciNetMATH Li, Y., Jiang, W.: Fractional order nonlinear systems with delay in iterative learning control. Appl. Math. Comput. 257, 546–552 (2015)MathSciNetMATH
23.
go back to reference Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)CrossRefMATH Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)CrossRefMATH
24.
go back to reference Bainov, D.D., Lakshmikantham, V., Simeonov, P.S.: Theory of Impulsive Differential Equations. vol. 6 of Series in Modern Applied Mathematics. World Scientific, Singapore (1989)MATH Bainov, D.D., Lakshmikantham, V., Simeonov, P.S.: Theory of Impulsive Differential Equations. vol. 6 of Series in Modern Applied Mathematics. World Scientific, Singapore (1989)MATH
25.
go back to reference Liu, S., Wang, J., Wei, W.: A study on iterative learning control for impulsive differential systems. Commun. Nonlinear Sci. Numer. Simul. 24, 4–10 (2015)MathSciNetCrossRef Liu, S., Wang, J., Wei, W.: A study on iterative learning control for impulsive differential systems. Commun. Nonlinear Sci. Numer. Simul. 24, 4–10 (2015)MathSciNetCrossRef
26.
go back to reference Ma, F., Li, C.: Open-closed-loop PID-type iterative learning control for linear systems with initial state error. J. Vib. Control 17, 1791–1797 (2010)MathSciNetMATH Ma, F., Li, C.: Open-closed-loop PID-type iterative learning control for linear systems with initial state error. J. Vib. Control 17, 1791–1797 (2010)MathSciNetMATH
27.
go back to reference Samoilenko, A.M., Perestyuk, N.A.: Stability of solutions of differential equations with impulse effect. Differ. Equ. 13, 1981–1992 (1977) Samoilenko, A.M., Perestyuk, N.A.: Stability of solutions of differential equations with impulse effect. Differ. Equ. 13, 1981–1992 (1977)
Metadata
Title
PID-type iterative learning control for impulsive ordinary differential equations
Authors
Zhuoyan Gao
Shengda Liu
JinRong Wang
Publication date
18-02-2016
Publisher
Springer Berlin Heidelberg
Published in
Journal of Applied Mathematics and Computing / Issue 1-2/2017
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-016-0995-x

Other articles of this Issue 1-2/2017

Journal of Applied Mathematics and Computing 1-2/2017 Go to the issue

Premium Partner