Skip to main content
SpringerLink
Log in

Discounted-Cost Linear Quadratic Output Regulation of Switched Linear Systems

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper studies the discounted-cost linear quadratic output regulator design for a class of switched linear systems by designing the controllers for each subsystem and the switching signal. The distinguishing feature of the proposed method is that the designed discounted-cost linear quadratic output regulator will achieve not only the desired optimization index, but also the exponentially convergent of the state of the closed-loop switched systems. The embedding transformation technique is used to transform the studied problem of switched system into a traditional optimal control problem. Then, it is shown the bang-bang-type solution of the embedded optimal control problem is the optimal solution to the original problems for both the finite horizon and infinite horizon. Both the switching signal and the controller for each subsystem are designed simultaneously. An example of boost converter is shown to illustrate the effectiveness of the proposed result.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. B.D.O. Anderson, J.B. Moore, Optimal Control: Linear Quadratic Methods (Prentice Hall, Upper Saddle River, 1990)

    MATH  Google Scholar 

  2. P.K. Anh, P.T. Linh, D.D. Thuan, S. Trenn, Stability analysis for switched discrete-time linear singular systems. Automatica 119, 1–9 (2020)

    Article  MathSciNet  Google Scholar 

  3. D. Antunes, W.P.M.H. Heemels, Linear quadratic regulation of switched systems using informed policies. IEEE Trans. Autom. Control 62(6), 2675–2688 (2017)

    Article  MathSciNet  Google Scholar 

  4. H. Bijl, T.B. Schon, Optimal controller/observer gains of discounted-cost LQG systems. Automatica 101, 471–474 (2017)

    Article  MathSciNet  Google Scholar 

  5. G.S. Deaecto, J.C. Geromel, F.S. Garcia, J.A. Pomilio, Switched affine systems control design with application to dc–dc converters. IET Control Theory Appl. 4(7), 1201–1210 (2010)

    Article  Google Scholar 

  6. J. Fisher, R. Bhattacharya, Linear quadratic regulation of systems with stochastic parameter uncertainties. Automatica 45(12), 2831–2841 (2009)

    Article  MathSciNet  Google Scholar 

  7. J. Fu, R. Ma, T. Chai, Z. Hu, Dwell-time-based standard \({H}^{\infty }\) control of switched systems without requiring internal stability of subsystems. IEEE Trans. Autom. Control 64(7), 3019–3025 (2019)

    Article  MathSciNet  Google Scholar 

  8. Y. Jin, Y. Zhang, Y. Jing, J. Fu, An average dwell-time method for fault-tolerant control of switched time-delay systems and its application. IEEE Trans. Ind. Electron. 66(4), 3139–3147 (2019)

    Article  Google Scholar 

  9. Z. Li, H. Gao, H.R. Karimi, Stability analysis and \({H^{\infty }}\) controller synthesis of discrete-time switched systems with time delay. Syst. Control Lett. 66, 85–93 (2014)

    Article  MathSciNet  Google Scholar 

  10. S. Liu, L. Xie, H. Zhang, Linear quadratic regulation for discrete-time systems with multiple delays in single input channel. Int. J. Robust Nonlinear Control 41(2), 13336–13341 (2008)

    Google Scholar 

  11. A.Y. Lu, G.H. Yang, Stabilization of switched systems with all modes unstable via periodical switching laws. Automatica 122, 1–8 (2020)

    Article  MathSciNet  Google Scholar 

  12. R. Ma, J. Fu, T. Chai, Dwell-time-based observers design for unknown inputs switched linear systems without requiring strong detectability of subsystems. IEEE Trans. Autom. Control 62(8), 4215–4221 (2017)

  13. Y. Miladi, N. Derbel, M. Feki, Optimal control based on multiple models approach of chaotic switched systems, application to a stepper motor. Int. J. Autom. Control 15(2), 240–258 (2021)

    Article  Google Scholar 

  14. P. Riedinger, A switched LQ regulator design in continuous time. IEEE Trans. Autom. Control 59(5), 1322–1328 (2014)

    Article  MathSciNet  Google Scholar 

  15. S.A.A. Rizvi, Z. Lin, Output feedback Q-learning control for the discrete-time linear quadratic regulator problem. IEEE Trans. Neural Netw. Learn. Syst. 30(5), 1523–1536 (2018)

    Article  MathSciNet  Google Scholar 

  16. I. Rusnak, Least mean squares error based filter of linear system with prescribed convergence rate. In: 2016 IEEE International Conference on the Science of Electrical Engineering (ICSEE), pp. 1–5 (2016)

  17. C. Seatzu, D. Corona, A. Giua, A. Bemporad, Optimal control of continuous-time switched affine systems. IEEE Trans. Autom. Control 51(5), 726–741 (2006)

    Article  MathSciNet  Google Scholar 

  18. X. Su, P. Shi, L. Wu, Y.D. Song, Fault detection filtering for nonlinear switched stochastic systems. IEEE Trans. Autom. Control 61(5), 1310–1315 (2016)

    Article  MathSciNet  Google Scholar 

  19. A.R. Syed Ali, L. Zongli, Reinforcement learning-based linear quadratic regulation of continuous-time systems using dynamic output feedback. IEEE Trans. Cybern. 50(11), 4670–4679 (2019)

    Google Scholar 

  20. J. Tan, W. Wang, J. Yao, Finite-time stability and boundedness of switched systems with finite-time unstable subsystems. Circuits Syst. Signal Process. 38(7), 2931–2950 (2019)

    Article  Google Scholar 

  21. L. Vachhani, CORDIC as a switched nonlinear system. Circuits Syst. Signal Process. 39(4), 3234–3249 (2019)

    MATH  Google Scholar 

  22. Q. Wang, Z. Wu, Y. Chen, Controller design for switched systems with non-symmetrical input saturation. Circuits Syst. Signal Process. 40(10), 136–153 (2021)

    Article  Google Scholar 

  23. Q. Wang, Z.G. Wu, P. Shi, H. Yan, Y. Chen, Stability analysis and control for switched system with bounded actuators. IEEE Trans. Syst. Man Cybern. Syst. 50(11), 4506–4512 (2018)

    Article  Google Scholar 

  24. Q. Wang, H. Yu, Z.G. Wu, G. Chen, Stability analysis for input saturated discrete-time switched systems with average dwell-time. IEEE Trans. Syst. Man Cybern. Syst. 51(1), 412–419 (2018)

    Article  Google Scholar 

  25. R. Wang, B. Xue, L. Hou, S. Fei, J. Zhao, Quasi-time-dependent \(l_{2}-l_{\infty }\) filtering of discrete-time switched systems with admissible edge-dependent average dwell time. Circuits Syst. Signal Process. 39(4), 4320–4338 (2020)

    Article  Google Scholar 

  26. Y. Wang, Y. Chang, A.F. Alkhateeb, N.D. Alotaibi, Adaptive fuzzy output-feedback tracking control for switched nonstrict-feedback nonlinear systems with prescribed performance. Circuits Syst. Signal Process. 40(1), 88–113 (2021)

    Article  Google Scholar 

  27. A.G. Wu, Y.Y. Qian, W. Liu, V. Sreeram, Linear quadratic regulation for discrete-time antilinear systems: an anti-Riccati matrix equation approach. J. Franklin Inst. 353(5), 1041–1060 (2016)

    Article  MathSciNet  Google Scholar 

  28. G. Wu, J. Sun, J. Chen, Optimal linear quadratic regulator of switched systems. IEEE Trans. Autom. Control 14(8), 2898–2904 (2018)

    MathSciNet  MATH  Google Scholar 

  29. L. Wu, J. Lam, Weighted \({H}^{\infty }\) filtering of switched systems with time-varying delay: average dwell time approach. Circuits Syst. Signal Process. 28(6), 1017–1036 (2009)

    Article  MathSciNet  Google Scholar 

  30. L. Wu, R. Yang, P. Shi, X. Su, Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings. Automatica 59, 206–215 (2015)

    Article  MathSciNet  Google Scholar 

  31. L. Wu, W.X. Zheng, H. Gao, Dissipativity-based sliding mode control of switched stochastic systems. IEEE Trans. Autom. Control 58(3), 785–791 (2012)

    Article  MathSciNet  Google Scholar 

  32. W. Xu, Z.G. Feng, J.W. Peng, K.F.C. Yiu, Optimal switching for linear quadratic problem of switched systems in discrete time. Automatica 78, 185–193 (2017)

    Article  MathSciNet  Google Scholar 

  33. S. Yin, H. Gao, J. Qiu, O. Kaynak, Descriptor reduced-order sliding mode observers design for switched systems with sensor and actuator faults. Automatica 76, 282–292 (2017)

    Article  MathSciNet  Google Scholar 

  34. H. Zhang, W. Wang, L. Xie, Linear quadratic regulation for continuous-time systems with time-varying delay. IFAC Proc. 41(2), 8779–8784 (2008)

    Google Scholar 

  35. W. Zhang, J. Hu, A. Abate, Infinite-horizon switched LQR problems in discrete time: a suboptimal algorithm with performance analysis. IEEE Trans. Autom. Control 57(7), 1815–1821 (2012)

    Article  MathSciNet  Google Scholar 

  36. X. Zhao, X. Liu, S. Yin, H. Li, Improved results on stability of continuous-time switched positive linear systems. Automatica 50(2), 614–621 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work of Ruicheng Ma was partially supported by National Natural Science Foundation of China (62073157), Scientific Research Fund of Educational Department of Liaoning Province (LZD201901) and Liaoning Revitalization Talents Program (XLYC1807012).

Author information

Authors and Affiliations

  1. Mathematics, Beijing University of Technology, Beijing, 100022, China

    Aoxue Xiang

  2. Mathematics, Liaoning University, Shenyang, 110036, China

    Ruicheng Ma & Yetian Fan

Authors
  1. Aoxue Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruicheng Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yetian Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ruicheng Ma.

Ethics declarations

Availability of data and materials

The datasets supporting the conclusions of this article are included within the article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiang, A., Ma, R. & Fan, Y. Discounted-Cost Linear Quadratic Output Regulation of Switched Linear Systems. Circuits Syst Signal Process 41, 5414–5427 (2022). https://doi.org/10.1007/s00034-022-02039-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-022-02039-x

Keywords

Search

Navigation