Skip to main content
SpringerLink
Log in

The Irregular Linear Quadratic Control Problem for Deterministic Case with Time Delay

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This study deals with the irregular linear quadratic control problem governed by continuous time system with time delay. Linear quadratic (LQ) control for irregular Riccati equation with time delay remains challenging since the controller could not be solved from the equilibrium condition directly. The merit of this paper is that based on a new approach of ‘two-layer optimization’, the controller entries of irregular case with time delay are deduced from two equilibrium conditions in two different layers, which is fundamentally different from the classical regular LQ control. The authors prove that the irregular LQ with time delay is essentially different from the regular case. Specifically, the predictive controller bases on the feedback gain matrix and the state is given in the last part. The presented conclusions are completely new to our best knowledge. Examples is presented to show the effectiveness of the proposed approach.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. John B and Panos J A, Control and communication challenges in networked real-time systems, Proceedings of the IEEE, 2007, 95(1): 9–28.

    Article  Google Scholar 

  2. Miao X and Li L, Adaptive observer-based control for uncertain nonlinear stochastic systems with time-delay, Journal of the Franklin Institute, 2016, 353(14): 3595–3609.

    Article  MathSciNet  Google Scholar 

  3. Zhai C, Liu W M, and Tan F G, Feedback control of a class of coupled-map fuzzy time-delay car-following system, Journal of South China University of Technology, 2017, 45(1): 9–17.

    Google Scholar 

  4. Li N, Wang G, and Wu Z, Linear-quadratic optimal control problem of forward-backward stochastic system with delay, Chinese Control Conference IEEE, 2017, 1822–1827.

  5. Butcher E A, Dabiri A, and Nazari M, Stability and Control of Fractional Periodic Time-Delayed Systems, Time Delay Systems, Springer International Publishing, 2017.

  6. He S, Song J, and Liu F, Robust finite-time bounded controller design of time-delay conic nonlinear systems using sliding mode control strategy, IEEE Transactions on Systems Man & Cybernetics Systems, 2017, 99: 1–11.

    Google Scholar 

  7. Feng Z and Zheng W X, On extended dissipativity of discrete-time neural networks with time delay, IEEE Transactions on Neural Networks & Learning Systems, 2017, 26(12): 3293–3300.

    Article  MathSciNet  Google Scholar 

  8. Wang H, Sun W, and Liu P X, Adaptive intelligent control of nonaffine nonlinear time-delay systems with dynamic uncertainties, IEEE Transactions on Systems Man & Cybernetics Systems, 2017, 47(7): 1474–1485.

    Article  Google Scholar 

  9. Niamsup P and Phat V N, Robust finite-time H∞ control of linear time-varying delay systems with bounded control via Riccati equations, International Journal of Automation & Computing, 2018, 3: 1–9.

    Google Scholar 

  10. Artstein Z, Linear systems with delayed controls: A reduction, IEEE Transactions on Automatic Control, 1982, 27(4): 869–879.

    Article  MathSciNet  Google Scholar 

  11. Alekal Y, Brunovsky P, Chyung D, et al., The quadratic problem for systems with time delays, IEEE Transactions on Automatic Control, 1971, 16(6): 673–687.

    Article  MathSciNet  Google Scholar 

  12. Delfour M C, The linear-quadratic optimal control problem with delays in state and control variables: A state space approach, SIAM Journal on Control and Optimization, 1986, 24(5): 835–883.

    Article  MathSciNet  Google Scholar 

  13. Zhang H, Duan G, and Xie L, Linear quadratic regulation for linear time-varying systems with multiple input delays, Automatica, 2006, 42(9): 1465–1476.

    Article  MathSciNet  Google Scholar 

  14. Zhang H, Lin L, Xu J, et al., Linear quadratic regulation and stabilization of discrete-time systems with delay and multiplicative noise, IEEE Transactions on Automatic Control, 2015, 60(10): 2599–2613.

    Article  MathSciNet  Google Scholar 

  15. Sun J, Li X, and Yong J, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems, SIAM Journal on Control and Optimization, 2016, 54(5): 2274–2308.

    Article  MathSciNet  Google Scholar 

  16. Zhang H and Xu J, Optimal control with irregular performance, SCIENCE CHINA Information Sciences, 2019, DOI: https://doi.org/10.1007/s11432-018-9685-8.

  17. Pazhooh F, Shahraki F, Sadeghi J, et al., Multivariable adaptive neural network predictive control in the presence of measurement time-delay; application in control of Vinyl Acetate monomer process, Journal of Process Control, 2018, 66: 39–50.

    Article  Google Scholar 

  18. Patan K, Two stage neural network modelling for robust model predictive control, ISA Trans, 2018, 72: 56–65.

    Article  Google Scholar 

  19. Penrose R, A generalized inverse of matrices, Mathematical Proceedings of the Cambridge Philosophical Society, 1955, 52: 17–19.

    Article  Google Scholar 

  20. Zhang H and Xu J, Control for Itô stochastic systems with input delay, IEEE Transactions on Automatic Control, 2017, 62(1): 350–365.

    Article  MathSciNet  Google Scholar 

  21. Lim A E B and Zhou X Y, Stochastic optimal LQR control with integral quadratic constraints and indefinite control weights, IEEE Transactions on Automatic Control, 1999, 44: 359–369.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank Professor Huanshui Zhang of Shandong University for his guidance and suggestions. The authors would like to thank the associate editor and the reviewers for their constructive comments and suggestions which improved the quality of the paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Taishan University, Taian, 271000, China

    Tongxing Li, Jing Lei & Peijun Ju

  2. School of Information Science and Engineering, Shandong Agricultural University, Taian, 271000, China

    Lin Li

  3. School of Control Science and Engineering, Shandong University, Jinan, 250061, China

    Nana Jin

Authors
  1. Tongxing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jing Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nana Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Peijun Ju
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tongxing Li.

Additional information

This research was supported by the Natural Science Foundation of Shandong Province under Grant Nos. ZR2018MF019 and ZR2019MF052, the National Natural Science Foundation of China under Grant No. 61873179.

This paper was recommended for publication by Editor SUN Jian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, T., Li, L., Lei, J. et al. The Irregular Linear Quadratic Control Problem for Deterministic Case with Time Delay. J Syst Sci Complex 34, 642–656 (2021). https://doi.org/10.1007/s11424-020-9136-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-020-9136-6

Keywords

Search

Navigation