Skip to main content
Springer Nature Link
Log in

A FAS approach for stabilization of generalized chained forms: part 2. Continuous control laws

  • Research Paper
  • Published:
Science China Information Sciences Aims and scope Submit manuscript

Abstract

In this paper, continuous time-varying stabilizing controllers for the type of general nonholonomic systems proposed and treated in part 1 are designed using the fully actuated system (FAS) approach. The key step is to differentiate the first scalar equation, and by control of the obtained second-order scalar system, a proportional plus integral feedback form for the first control variable is obtained. With the solution to this designed second-order scalar system, the rest equations in the nonholonomic system form an independent time-varying subsystem which is then handled by the FAS approach. The overall designed controller contains an almost arbitrarily chosen design parameter, and is proven to guarantee the uniformly and globally exponential stability of the closed-loop system. The proposed approach is simple and effective, and is demonstrated with a practical example of ship control.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Duan G R. A FAS approach for stabilization of generalized chained forms: part 1. Discontinuous control laws. Sci China Inf Sci, 2024, 67: 122201

    Article  MathSciNet  Google Scholar 

  2. Brockett R W. Asymptotic stability and feedback stabilization. In: Differential Geometric Control Theory. Boston: Birkhäuser, 1983. 181–191

    Google Scholar 

  3. Duan G R. Brockett’s first example: an FAS approach treatment. J Syst Sci Complex, 2022, 35: 441–456

    Article  MathSciNet  Google Scholar 

  4. Duan G R. Brockett’s second example: a FAS approach treatment. J Syst Sci Complex, 2023, 36: 1789–1808

    Article  MathSciNet  Google Scholar 

  5. Kolmanovsky I, McClamroch N H. Developments in nonholonomic control problems. IEEE Control Syst Mag, 1995, 15: 20–36

    Article  Google Scholar 

  6. Murray R M, Sastry S S. Nonholonomic motion planning: steering using sinusoids. IEEE Trans Automat Contr, 1993, 38: 700–716

    Article  MathSciNet  Google Scholar 

  7. Astolfi A. Discontinuous control of nonholonomic systems. Syst Control Lett, 1996, 27: 37–45

    Article  MathSciNet  Google Scholar 

  8. Rocha E, Castahos F, Moreno J A. Robust finite-time stabilisation of an arbitrary-order nonholonomic system in chained form. Automatica, 2022, 135: 109956

    Article  MathSciNet  Google Scholar 

  9. Mnif F, Metwally K A E. Particle swarm optimisation of a discontinuous control for a wheeled mobile robot with two trailers. Int J Comput Appl Technol, 2011, 41: 169–176

    Article  Google Scholar 

  10. Marchand N, Alamir M. Discontinuous exponential stabilization of chained form systems. Automatica, 2003, 39: 343–348

    Article  MathSciNet  Google Scholar 

  11. Lin W, Pongvuthithum R, Qian C. Control of high-order nonholonomic systems in power chained form using discontinuous feedback. IEEE Trans Automat Contr, 2002, 47: 108–115

    Article  MathSciNet  Google Scholar 

  12. Lin W, Pongvuthithum R. Recursive design of discontinuous controllers for uncertain driftless systems in power chained form. IEEE Trans Automat Contr, 2000, 45: 1886–1892

    Article  MathSciNet  Google Scholar 

  13. Laiou M C, Astolfi A. Discontinuous control of high-order generalized chained systems. Syst Control Lett, 1999, 37: 309–322

    Article  MathSciNet  Google Scholar 

  14. Khennouf H, Wit CCD. On the construction of stabilizing discontinuous controllers for nonholonomic systems. IFAC Proc Volumes, 1995, 28: 667–672

    Article  Google Scholar 

  15. Tian Y P, Li S. Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control. Automatica, 2002, 38: 1139–1146

    Article  MathSciNet  Google Scholar 

  16. Samson C. Control of chained systems application to path following and time-varying point-stabilization of mobile robots. IEEE Trans Automat Contr, 1995, 40: 64–77

    Article  MathSciNet  Google Scholar 

  17. Morin P, Samson C. Control of nonlinear chained systems: from the Routh-Hurwitz stability criterion to time-varying exponential stabilizers. IEEE Trans Automat Contr, 2000, 45: 141–146

    Article  MathSciNet  Google Scholar 

  18. Morin P, Pomet J B, Samson C. Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approximation of Lie brackets in closed loop. SIAM J Control Optim, 1999, 38: 22–49

    Article  MathSciNet  Google Scholar 

  19. Jiang Z P. Iterative design of time-varying stabilizers for multi-input systems in chained form. Syst Control Lett, 1996, 28: 255–262

    Article  MathSciNet  Google Scholar 

  20. Pomet J B, Samson C. Time-Varying Exponential Stabilization of Nonholonomic Systems in Power Form. INRIA Technical Report, 2126, 1993

  21. Xi Z, Feng G, Jiang Z P, et al. A switching algorithm for global exponential stabilization of uncertain chained systems. IEEE Trans Automat Contr, 2003, 48: 1793–1798

    Article  MathSciNet  Google Scholar 

  22. Sordalen O J, Egeland O. Exponential stabilization of nonholonomic chained systems. IEEE Trans Automat Contr, 1995, 40: 35–49

    Article  MathSciNet  Google Scholar 

  23. Jiang Z P, Nijmeijer H. A recursive technique for tracking control of nonholonomic systems in chained form. IEEE Trans Automat Contr, 1999, 44: 265–279

    Article  MathSciNet  Google Scholar 

  24. Duan G R. High-order system approaches: I. Fully-actuated systems and parametric designs (in Chinese). Acta Autom Sin, 2020, 46: 1333–1345

    Google Scholar 

  25. Duan G R. High-order system approaches: II. Controllability and full-actuation (in Chinese). Acta Autom Sin, 2020, 46: 1571–1581

    Google Scholar 

  26. Duan G R. High-order system approaches: III. Observability and observer design (in Chinese). Acta Autom Sin, 2020, 46: 1885–1895

    Google Scholar 

  27. Duan G R. High-order fully actuated system approaches: part I. Models and basic procedure. Int J Syst Sci, 2021, 52: 422–435

    Article  ADS  MathSciNet  Google Scholar 

  28. Duan G R. High-order fully actuated system approaches: part II. Generalized strict-feedback systems. Int J Syst Sci, 2021, 52: 437–454

    Article  ADS  MathSciNet  Google Scholar 

  29. Duan G R. High-order fully actuated system approaches: part III. Robust control and high-order backstepping. Int J Syst Sci, 2021, 52: 952–971

    Article  ADS  MathSciNet  Google Scholar 

  30. Duan G R. High-order fully actuated system approaches: part IV. Adaptive control and high-order backstepping. Int J Syst Sci, 2021, 52: 972–989

    Article  ADS  MathSciNet  Google Scholar 

  31. Duan G R. High-order fully actuated system approaches: part V. Robust adaptive control. Int J Syst Sci, 2021, 52: 2129–2143

    Article  ADS  MathSciNet  Google Scholar 

  32. Duan G R. High-order fully-actuated system approaches: part VI. Disturbance attenuation and decoupling. Int J Syst Sci, 2021, 52: 2161–2181

    Article  ADS  MathSciNet  Google Scholar 

  33. Duan G R. High-order fully actuated system approaches: part VII. Controllability, stabilisability and parametric designs. Int J Syst Sci, 2021, 52: 3091–3114

    Article  ADS  MathSciNet  Google Scholar 

  34. Duan G R. High-order fully actuated system approaches: part VIII. Optimal control with application in spacecraft attitude stabilisation. Int J Syst Sci, 2022, 53: 54–73

    Article  ADS  MathSciNet  Google Scholar 

  35. Duan G R. High-order fully-actuated system approaches: part IX. Generalised PID control and model reference tracking. Int J Syst Sci, 2022, 53: 652–674

    Article  ADS  MathSciNet  Google Scholar 

  36. Duan G R. High-order fully actuated system approaches: part X. Basics of discrete-time systems. Int J Syst Sci, 2022, 53: 810–832

    Article  ADS  MathSciNet  Google Scholar 

  37. Duan G R. Discrete-time delay systems: part 1. Global fully actuated case. Sci China Inf Sci, 2022, 65: 182201

    Article  MathSciNet  Google Scholar 

  38. Duan G R. Discrete-time delay systems: part 2. Sub-fully actuated case. Sci China Inf Sci, 2022, 65: 192201

    Article  MathSciNet  Google Scholar 

  39. Duan G R. Fully actuated system approaches for continuous-time delay systems: part 1. Systems with state delays only. Sci China Inf Sci, 2023, 66: 112201

    Article  MathSciNet  Google Scholar 

  40. Duan G R. Fully actuated system approaches for continuous-time delay systems: part 2. Systems with input delays. Sci China Inf Sci, 2023, 66: 122201

    Article  MathSciNet  Google Scholar 

  41. Duan G R. Robust stabilization of time-varying nonlinear systems with time-varying delays: a fully actuated system approach. IEEE Trans Cybern, 2023, 53: 7455–7468

    Article  PubMed  Google Scholar 

  42. Duan G R. Substability and substabilization: control of subfully actuated systems. IEEE Trans Cybern, 2023, 53: 7309–7322

    Article  PubMed  Google Scholar 

  43. Duan G R. Stabilization via fully actuated system approach: a case study. J Syst Sci Complex, 2022, 35: 731–747

    Article  MathSciNet  Google Scholar 

  44. Kreyszig E. Advanced Engineering Mathematics. 10th ed. New York: Wiley, 1972

    Google Scholar 

  45. Khalil H K. Nonlinear Systems. 3rd ed. Upper Saddle River: Prentice Hall, 2002

    Google Scholar 

  46. Do K D. Global robust adaptive path-tracking control of underactuated ships under stochastic disturbances. Ocean Eng, 2016, 111: 267–278

    Article  Google Scholar 

  47. Fossen T I. Marine Control Systems. Trondheim: Marine Cybernetics, 2002

    Google Scholar 

Download references

Acknowledgements

This work was partially supported by Major Program of National Natural Science Foundation of China (Grant Nos. 61690210, 61690212), National Natural Science Foundation of China (Grant No. 61333003), and Science Center Program of the National Natural Science Foundation of China (Grant No. 62188101). The author is grateful to his Ph.D. students for helping him with reference selection and proofreading. His thanks also go to Drs. Wei SUN, Xiang XU, and Tao LIU for their helpful discussions and comments, and he extends his particular thanks to Dr. Zhongcai ZHANG for helping him work out of the simulation.

Author information

Authors and Affiliations

  1. Shenzhen Key Laboratory of Control Theory and Intelligent Systems, Southern University of Science and Technology, Shenzhen, 518055, China

    Guang-Ren Duan

  2. Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, 150001, China

    Guang-Ren Duan

Authors
  1. Guang-Ren Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guang-Ren Duan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duan, GR. A FAS approach for stabilization of generalized chained forms: part 2. Continuous control laws. Sci. China Inf. Sci. 67, 132201 (2024). https://doi.org/10.1007/s11432-023-3920-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11432-023-3920-8

Keywords

Search

Navigation