Skip to main content
SpringerLink
Log in

Brockett’s Second Example: A FAS Approach Treatment

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

Abstract

In this paper, several equivalent forms of the well-known Brockett’s second example system are firstly presented. The stabilization of the system is then treated in the fully actuated system approach. A simple continuous time-invariant sub-stabilizing controller is designed, and the corresponding region of attraction is characterized. As a result, all trajectories of the system starting from the characterized region of attraction are driven exponentially to the origin. Since the region of attraction is very large, the designed sub-stabilizing controller can be directly useful in many practical situations. In cases where the initial values are indeed needed to be chosen out of the region of attraction, extremely simple pre-controllers can be designed, which drive the system trajectories into the designed region of attraction. A simulation of the designed control system is carried out to show the effect 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. Brockett R W, Asymptotic stability and feedback stabilization, Differential Geometric Control Theory, 1983, 27(1): 181–191.

    MathSciNet  MATH  Google Scholar 

  2. Willems J C and Baillieul J, Perspectives in control: A conference honoring the work of Roger W. Brockett, IEEE Control Systems Magazine, 1999, 19(2): 61.

    Google Scholar 

  3. Sontag E D, Feedback stabilization of nonlinear systems, Robust Control of Linear Systems and Nonlinear Control, Birkhäuser Boston, Basel, 1990.

    Google Scholar 

  4. Bacciotti A, Local Stabilizability of Nonlinear Control Systems, World Scientific, Singapore, 1992.

    MATH  Google Scholar 

  5. Zabczyk J, Mathematical Control Theory: An Introduction, 2nd Ed., Springer, Birkhäuser Boston, MA, 2020.

    Book  MATH  Google Scholar 

  6. Hespanha J P and Morse A S, Stabilization of nonholonomic integrators via logic-based switching, Automatica, 1999, 35(3): 385–393.

    Article  MathSciNet  MATH  Google Scholar 

  7. Liberzon D, Switching in Systems & Control, Birkhäuser Boston, Basel, 2003.

    Book  MATH  Google Scholar 

  8. Bloch A and Drakunov S, Stabilization and tracking in the nonholonomic integrator via sliding modes, Systems & Control Letters, 1996, 29(2): 91–99.

    Article  MathSciNet  MATH  Google Scholar 

  9. Floquet T, Drakunov S V, and Perruquetti W, Sliding mode control of extended Heisenberg systems, IFAC Proceedings Volumes, 2004, 37(13): 609–614.

    Article  Google Scholar 

  10. Drakunov S V, Floquet T, and Perruquetti W, Stabilization and tracking control for an extended Heisenberg system with a drift, Systems & Control Letters, 2005, 54(5): 435–445.

    Article  MathSciNet  MATH  Google Scholar 

  11. Defoort M, Floquet T, Perruquetti W, et al., Integral sliding mode control of an extended Heisenberg system, IET Control Theory & Applications, 2009, 3(10): 1409–1424.

    Article  MathSciNet  Google Scholar 

  12. Bloch A M, Drakunov S V, and Kinyon M K, Nonholonomic stabilization and isospectral flows, Proceedings of the 37th IEEE Conference on Decision and Control, 1998, 4: 3581–3586.

    Google Scholar 

  13. Bloch A M, Drakunov S V, and Kinyon M K, Stabilization of nonholonomic systems using isospectral flows, SIAM Journal on Control and Optimization, 2000, 38(3): 855–874.

    Article  MathSciNet  MATH  Google Scholar 

  14. Dolgopolik M V and Fradkov A L, Speed-gradient control of the Brockett integrator, SIAM Journal on Control and Optimization, 2016, 54(4): 2116–2131.

    Article  MathSciNet  MATH  Google Scholar 

  15. Khaneja N and Brockett R, Dynamic feedback stabilization of nonholonomic systems, Proceedings of the 38th IEEE Conference on Decision and Control, 1999, 2: 1640–1645.

    Google Scholar 

  16. Morgansen K A and Brockett R W, Nonholonomic control based on approximate inversion, Proceedings of the 1999 American Control Conference, 1999, 5: 3515–3519.

    Google Scholar 

  17. Zeng S, On the geometric construction of a stabilizing time-invariant state feedback controller for the nonholonomic integrator, Automatica, 2022, 136: 110073.

    Article  MathSciNet  MATH  Google Scholar 

  18. Abbasi W, Stabilization of nonholonomic systems, Doctor’s degree thesis, Capital University of Science and Technology, Islamabad, 2018.

    Google Scholar 

  19. Mansouri A R, Local asymptotic feedback stabilization to a submanifold: Topological conditions, Systems & Control Letters, 2007, 56(7–8): 525–528.

    Article  MathSciNet  MATH  Google Scholar 

  20. Duan G R, High-order system approaches - I. Full-actuation and parametric design, Acta Automatica Sinica, 2020, 46(7): 1333–1345 (in Chinese).

    MATH  Google Scholar 

  21. Duan G R, High-order system approaches - II. Controllability and full-actuation, Acta Automatica Sinica, 2020, 46(8): 1571–1581 (in Chinese).

    MATH  Google Scholar 

  22. Duan G R, High-order system approaches - III. Super-observability and observer design, Acta Automatica Sinica, 2020, 46(9): 1885–1895 (in Chinese).

    MATH  Google Scholar 

  23. Duan G R and Zhou B, Fully actuated system approach for linear systems control: A frequency-domain, Journal of Systems Science & Complexity, 2022, 2022, 35(6): 2046–2061.

    Article  MathSciNet  MATH  Google Scholar 

  24. Duan G R, High-order fully actuated system approaches: Part I. Models and basic procedure, International Journal of Systems Science, 2020, 52(2): 422–435.

    Article  MathSciNet  MATH  Google Scholar 

  25. Duan G R, High-order fully actuated system approaches: Part II. Generalized strict-feedback systems, International Journal of Systems Science, 2020, 52(3): 437–454.

    Article  MathSciNet  MATH  Google Scholar 

  26. Duan G R, High-order fully actuated system approaches: Part III. Robust control and high-order backstepping, International Journal of Systems Science, 2020, 52(5): 952–971.

    Article  MathSciNet  MATH  Google Scholar 

  27. Duan G R, High-order fully actuated system approaches: Part IV. Adaptive control and high-order backstepping, International Journal of Systems Science, 2020, 52(5): 972–989.

    Article  MathSciNet  MATH  Google Scholar 

  28. Duan G R, High-order fully actuated system approaches: Part V. Robust adaptive control, International Journal of Systems Science, 2021, 52(10): 2129–2143.

    Article  MathSciNet  MATH  Google Scholar 

  29. Duan G R, High-order fully actuated system approaches: Part VI. Disturbance attenuation and decoupling, International Journal of Systems Science, 2021, 52(10): 2161–2181.

    Article  MathSciNet  MATH  Google Scholar 

  30. Duan G R, High-order fully actuated system approaches: Part VII. Controllability, stabilizability and parametric design, International Journal of Systems Science, 2021, 52(14): 3091–3114.

    Article  MathSciNet  MATH  Google Scholar 

  31. Duan G R, High-order fully actuated system approaches: Part VIII. Optimal control with application in spacecraft attitude stabilization, International Journal of Systems Science, 2022, 53(1): 54–73.

    Article  MathSciNet  MATH  Google Scholar 

  32. Duan G R, High-order fully actuated system approaches: Part IX. Generalized PID control and model reference tracking, International Journal of Systems Science, 2022, 53(3): 652–674.

    Article  MathSciNet  MATH  Google Scholar 

  33. Duan G R, High-order fully actuated system approaches: Part X. Basics of discrete-time systems, International Journal of Systems Science, 2022, 53(4): 810–832.

    Article  MathSciNet  MATH  Google Scholar 

  34. Duan G R, Discrete-time delay systems: Part 1. Global fully actuated case, Science China-Information Sciences, 2022, 65(8): 1869–1919.

    Article  MathSciNet  Google Scholar 

  35. Duan G R, Discrete-time delay systems: Part 2. Sub-fully actuated case, Science China-Information Sciences, 2022, 65(11): 192201.

    Article  MathSciNet  Google Scholar 

  36. Duan G R, Fully actuated system approaches for continuous-time delay systems: Part 1. Systems with state delays only, Science China-Information Sciences, 2022, 66(1): 112201.

    Article  MathSciNet  Google Scholar 

  37. Duan G R, Fully actuated system approaches for continuous-time delay systems: Part 2. Systems with input delays, Science China-Information Sciences, 2023, 66(2): 112201.

    Article  MathSciNet  Google Scholar 

  38. Duan G R, Stabilization via fully actuated system approach: A case study, Journal of Systems Science & Complexity, 2022, 35(3): 731–747.

    Article  MathSciNet  MATH  Google Scholar 

  39. Duan G R, Brockett’s first example: An FAS approach treatment, Journal of Systems Science & Complexity, 2022, 35(2): 441–456.

    Article  MathSciNet  MATH  Google Scholar 

  40. Sontag E D, Input to state stability: Basic concepts and results, Nonlinear and optimal control theory, Springer, Berlin/Heidelberg, 2008.

    Google Scholar 

  41. M’Closkey R and Murray R, Exponential stabilization of driftless nonlinear control systems using homogeneous feedback, IEEE Transactions on Automatic Control, 1997, 42(5): 614–628.

    Article  MathSciNet  MATH  Google Scholar 

  42. Murray R M and Sastry S S, Nonholonomic motion planning: Steering using sinusoids, IEEE transactions on Automatic Control, 1993, 38(5): 700–716.

    Article  MathSciNet  MATH  Google Scholar 

  43. Wit C C D, Khennouf H, Samson C, et al., Nonlinear control design for mobile robots, Recent Trends in Mobile Robots, World Scientific, Singapore, 1993.

    Google Scholar 

  44. Jiang Z P and Nijmeijer H, Tracking control of mobile robots: A case study in backstepping, Automatica, 1997, 33(7): 1393–1399.

    MathSciNet  MATH  Google Scholar 

  45. Luca A D, Oriolo G, and Vendittelli M, Control of wheeled mobile robots: An experimental overview, RAMSETE: Articulated and Mobile Robotics for Services and Technology, Springer, Berlin/Heidelberg, 2001.

    Google Scholar 

  46. Astolfi A, Discontinuous control of nonholonomic systems, Systems & Control Letters, 1996, 27(1): 37–45.

    Article  MathSciNet  MATH  Google Scholar 

  47. Hauser J, Sastry S, and Meyer G, Nonlinear control design for slightly non-minimum phase systems: Application to V/STOL aircraft, Automatica, 1992, 28(4): 665–679.

    Article  MathSciNet  MATH  Google Scholar 

  48. Martin P, Devasia S, and Paden B, A different look at output tracking: Control of a VTOL aircraft, Automatica, 1996, 32(1): 101–107.

    Article  MATH  Google Scholar 

  49. Khalil H K, High-Gain Observers in Nonlinear Feedback Control, Society for Industrial and Applied Mathematics, Philadelphia, 2017.

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to his Ph.D. students, Weizhen Liu, Guangtai Tian, Qin Zhao, etc., for helping him with reference selection and proofreading, and to his Ph.D. student Tianyi Zhao for working out the simulation results.

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.

Ethics declarations

The author declares no conflict of interest.

Additional information

This paper was partially supported by Shenzhen Key Laboratory of Control Theory and Intelligent Systems under Grant No. ZDSYS20220330161800001, the Science Center Program of the National Natural Science Foundation of China under Grant No. 62188101, and also the Major Program of National Natural Science Foundation of China under Grant Nos. 61690210 and 61690212.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duan, GR. Brockett’s Second Example: A FAS Approach Treatment. J Syst Sci Complex 36, 1789–1808 (2023). https://doi.org/10.1007/s11424-023-2282-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-023-2282-x

Keywords

Search

Navigation