Skip to main content
SpringerLink
Log in

SPM for Time-Delayed Nonlinear Optimal Control Problems

  • Chapter
  • First Online:
Symplectic Pseudospectral Methods for Optimal Control

Part of the book series: Intelligent Systems, Control and Automation: Science and Engineering ((ISCA,volume 97))

Abstract

The existence of time delays makes it hard to obtain analytical solutions for the optimal control problem of time-delay systems, especially for systems involving high nonlinearity and/or complicated constraints. Hence, various numerical methods to solve time-delayed optimal control problems (TDOCPs) are developed. In this chapter, we focus on TDOCPs with a single state delay and the corresponding SPM is developed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Dadebo S, Luus R (1992) Optimal control of time-delay systems by dynamic programming. Opt Control Appl Methods 13(1):29–41

    Article  MathSciNet  Google Scholar 

  2. Livk I, Rehbock V (2007) Optimal control of a batch crystallization process. J Indus Manag Optimiz 3(3):585–596

    MathSciNet  MATH  Google Scholar 

  3. Cai G, Huang J (2002) Optimal control method with time delay in control. J Sound Vib 251(3):383–394

    Article  MathSciNet  Google Scholar 

  4. Cai G, Huang J, Yang S (2003) An optimal control method for linear systems with time delay. Comput Struct 81(15):1539–1546

    Article  MathSciNet  Google Scholar 

  5. Khan H, Liao S, Mohapatra RN et al (2009) An analytical solution for a nonlinear time-delay model in biology. Commun Nonlinear Sci Numer Simul 14(7):3141–3148

    Article  Google Scholar 

  6. Shamsara E, Shamsara J, Afsharnezhad Z (2016) Optimal control therapy and vaccination for special HIV-1 model with delay. Theory Biosci 135(4):217–230

    Article  Google Scholar 

  7. Kharatishvili GL (1961) The maximum principle in the theory of optimal process with time-lags. Dokl Akad Nauk SSSR 136:39–42

    Google Scholar 

  8. Kharatishvili GL (1967) A maximum principle in external problems with delays, mathematical theory on control. Academic Press, New York, NY

    MATH  Google Scholar 

  9. Halanay A (1968) Optimal controls for systems with time lag. SIAM J Control 6(2):215–234

    Article  MathSciNet  Google Scholar 

  10. Göllmann L, Kern D, Maurer H (2010) Optimal control problems with delays in state and control variables subject to mixed control–state constraints. Opt Control Appl Methods 30(4):341–365

    Article  MathSciNet  Google Scholar 

  11. Hoseini SM (2020) Optimal control of linear pantograph-type delay systems via composite Legendre method. J Franklin Inst 357(9):5402–5427

    Article  MathSciNet  Google Scholar 

  12. Elnagar GN, Kazemi MA (2001) Numerical solution of time-delayed functional differential equation control systems. J Comput Appl Math 130(1–2):75–90

    Article  MathSciNet  Google Scholar 

  13. Maleki M, Hadi-Vencheh A (2010) Combination of non-classical pseudospectral and direct methods for the solution of brachistochrone problem. Int J Comput Math 87(8):1847–1856

    Article  MathSciNet  Google Scholar 

  14. Maleki M, Hashim I (2014) Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems. J Franklin Inst 351(2):811–839

    Article  MathSciNet  Google Scholar 

  15. Perng MH (1986) Direct approach for the optimal control of linear time-delay systems via shifted Legend re polynomials. Int J Control 43(6):1897–1904

    Article  Google Scholar 

  16. Maleki MA, Hashim IA, Abbasbandy SB (2012) Solution of time-varying delay systems using an adaptive collocation method. Appl Math Comput 219(4):1434–1448

    MathSciNet  MATH  Google Scholar 

  17. Wang X, Peng H, Zhang S et al (2017) A symplectic local pseudospectral method for solving nonlinear state-delayed optimal control problems with inequality constraints. Int J Robust Nonlinear Control 28(6):2097–2120

    Article  MathSciNet  Google Scholar 

  18. Wang X, Liu J, Dong X et al (2020) A symplectic pseudospectral method for constrained time-delayed optimal control problems and its application to biological control problems. Optimization. https://doi.org/10.1080/02331934.2020.1786568

    Article  Google Scholar 

  19. Haddadi N, Ordokhani Y, Razzaghi M (2012) Optimal control of delay systems by using a hybrid functions approximation. J Optim Theory Appl 153(2):338–356

    Article  MathSciNet  Google Scholar 

  20. Khellat F (2009) Optimal control of linear time-delayed systems by linear Legendre multiwavelets. J Optim Theory Appl 143(1):107–121

    Article  MathSciNet  Google Scholar 

  21. Banks HT, Burns JA (1978) Hereditary control problems: numerical methods based on averaging approximations. SIAM J Control Optim 16(2):169–208

    Article  MathSciNet  Google Scholar 

  22. Bonalli R, Hérissé B, Trélat E (2017) Solving optimal control problems for delayed control-affine systems with quadratic cost by numerical continuation. In: 2017 American control conference, 24–26 May 2017. Seattle, WA, USA

    Google Scholar 

  23. Chen C, Sun D, Chang C (2000) Numerical solution of time-delayed optimal control problems by iterative dynamic programming. Opt Control Appl Methods 21(3):91–105

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, Dalian University of Technology, Dalian, Liaoning, China

    Xinwei Wang & Haijun Peng

  2. War Research Institute, Academy of Military Sciences, Beijing, China

    Jie Liu

Authors
  1. Xinwei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jie Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haijun Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinwei Wang .

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Wang, X., Liu, J., Peng, H. (2021). SPM for Time-Delayed Nonlinear Optimal Control Problems. In: Symplectic Pseudospectral Methods for Optimal Control. Intelligent Systems, Control and Automation: Science and Engineering, vol 97. Springer, Singapore. https://doi.org/10.1007/978-981-15-3438-6_6

Download citation

Publish with us

Policies and ethics

Search

Navigation