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Brockett’s First Example: An FAS Approach Treatment

DUAN Guang-Ren   

  1. Center for Control Science and Technology, Southern University of Science and Technology, Shenzhen 518055, China;Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, China
  • Received:2022-02-13 Published:2022-04-13
  • Supported by:
    This paper has been partially supported by the Major Program of National Natural Science Foundation of China under Grant Nos. 61690210, 61690212, the National Natural Science Foundation of China under Grant No. 61333003, and also by the Science Center Program of the National Natural Science Foundation of China under Grant No. 62188101.

DUAN Guang-Ren. Brockett’s First Example: An FAS Approach Treatment[J]. Journal of Systems Science and Complexity, 2022, 35(2): 441-456.

In this note, the well-known Brockett’s first example system is treated with the fully actuated system (FAS) approach. Firstly, it is shown that the system can be exponentially substabilized by a smooth controller in the sense that, except those starting from initial values on the z0-axis of the initial value space, all trajectories of the designed system as well as the control signals decay to zero exponentially. Secondly, global stabilization is realized through a way of enabling the trajectories starting from initial values on the z0-axis also to go to the origin. The idea is to firstly move an initial point on the z0-axis away from the axis using a pre-controller, and then to take over by the designed exponentially sub-stabilizing controller.
[1] Brockett R W, Asymptotic stability and feedback stabilization, Differential Geometric Control Theory, 1983, 27(1): 181–191.
[2] Aeyels D, Stabilization of a class of nonlinear systems by a smooth feedback control, Systems & Control Letters, 1985, 5: 289–294.
[3] Byrnes C I and Isidori A, On the attitude stabilization of rigid spacecraft, Automatica, 1991, 27(1): 87–95.
[4] Krishnan H, Reyhanoglu M, and McClamroch H, Attitude stabilization of a rigid spacecraft using gas jet actuators operating in a failure mode, Proceedings of the 31st IEEE Conference on Decision and Control, Tucson, Arizona, 1992.
[5] Krishnan H, Reyhanoglu M, and McClamroch H, Attitude stabilization of a rigid spacecraft using two control torques: A nonlinear control approach based on the spacecraft attitude dynamics, Automatica, 1994, 30(6): 1023–1027.
[6] M’Closkey R T and Murray R M, Exponential stabilization of driftless nonlinear control systems using homogeneous feedback, IEEE Transactions on Automatic Control, 1997, 42(5): 614–628.
[7] Li H, Yan W, and Shi Y, Continuous-time model predictive control of under-actuated spacecraft with bounded control torques, Automatica, 2017, 75: 144–153.
[8] Irving M and Crouch P E, On sufficient conditions for local asymptotic stability of nonlinear systems whose linearization is uncontrollable, Control Theory Centre Report, University of Warwick, 1983.
[9] Byrnes C I and Isidori A, New results and examples in nonlinear feedback stabilization, Systems & Control Letters, 1989, 12(5): 437–442.
[10] Outbib R, Stabilisation d’une classe de systes affines en controles, Proceedings of European Control Conference, Grenoble, France, 1991.
[11] Wan C J and Bernstein D S, Nonlinear feedback control with global stabilization, Dynamics and Control, 1995, 5(4): 321–346.
[12] Bernstein D S, Nonquadratic cost and nonlinear feedback control, International Journal of Robust and Nonlinear Control, 1993, 3: 211–229.
[13] Reyhanoglu N, Discontinuous feedback stabilization of the angular velocity of a rigid body with two control torques, Proceedings of 35th IEEE Conference on Decision and Control, Kobe, Japan, 1996.
[14] Zhang H H, Wang F, and Trivailo P M, Spin-axis stabilisation of underactuated rigid spacecraft under sinusoidal disturbance, International Journal of Control, 2008, 81(12): 1901–1909.
[15] Cheon Y J, Spin-axis stabilization of gyroless and underactuated rigid spacecraft using modified Rodrigues parameters, Proceedings of SICE Annual Conference, Taipei, Taiwan, 2010.
[16] Jabczyk J, Some comments on stabilizability, Applied Mathematics and optimization, 1989, 19: 1–9.
[17] Duan G R, High-order system approaches — I. Full-actuation and parametric design, Acta Automatica Sinica, 2020, 46(7): 1333–1345(in Chinese).
[18] Duan G R, High-order system approaches — II. Controllability and full-actuation, Acta Automatica Sinica, 2020, 46(8): 1571–1581(in Chinese).
[19] Duan G R, High-order system approaches — III. Super-observability and observer design, Acta Automatica Sinica, 2020, 46(9): 1885–1895(in Chinese).
[20] 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.
[21] 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.
[22] 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.
[23] Duan G R, High-order fully actuated system approaches: Part IV. Adaptive control and highorder backstepping, International Journal of Systems Science, 2020, 52(5): 972–989.
[24] Duan G R, High-order fully actuated system approaches: Part V. Robust adaptive control, International Journal of Systems Science, 2021, 52(10): 2129–2143.
[25] 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.
[26] 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.
[27] 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.
[28] Duan G R, High-order fully actuated system approaches: Part IX. Generalized PID control and model reference tracking, International Journal of Systems Science, 2021, DOI: 10.1080/00207721.2021.1970277.
[29] Duan G R, High-order fully actuated system approaches: Part X. Basics of discrete-time systems, International Journal of Systems Science, 2021, DOI: 10.1080/00207721.2021.1975848.
[30] Duan G R, Discrete-time delay systems: Part 1. Global fully actuated case, Science ChinaInformation Sciences, DOI: 10.1007/s11432-021-3417-3.
[31] Duan G R, Discrete-time delay systems: Part 2. Sub-fully actuated case, Science ChinaInformation Sciences, DOI: 10.1007/s11432-021-3448-1.
[32] Duan G R, Fully actuated system approaches for continuous-time delay systems: Part 1. Systems with state delays only, Science China-Information Sciences, DOI: 10.1007/s11432-021-3459-x.
[33] Duan G R, Fully actuated system approaches for continuous-time delay systems: Part 2. Systems with input delays, Science China-Information Sciences, DOI: 10.1007/s11432-021-3460-y.
[34] Vannelli A and Vidyasagar M, Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems, Automatica, 1985, 21(1): 69–80.
[35] Tibken B, Estimation of the domain of attraction for polynomial systems via LMIs, Proceedings of the 39th IEEE Conference on Decision and Control, 2000, 4: 3860–3864.
[36] Henrion D and Korda M, Convex computation of the region of attraction of polynomial control systems, IEEE Transactions on Automatic Control, 2013, 59(2): 297–312.
[37] Duan G R, Stabilization via fully actuated system approach: A case study, Journal of Systems Science & Complexity, 2022, DOI: 10.1007/s11424-022-2091-7.
[38] Farina L and Rinaldi S, Positive Linear Systems: Theory and Applications, John Wiley & Sons, 2000.
[39] Kaczorek T and Borawski K, Stability of positive nonlinear systems, 201722nd International Conference on Methods and Models in Automation and Robotics (MMAR), Miedzyzdroje, Poland, 2017,
[40] Brockett R W, The early days of geometric nonlinear control, Automatica, 2014, 50(9): 2203–2224.
[41] Duan G R and Zhou B, A frequency-domain approach for converting state-space models into high-order fully actuated models, Journal of Systems Science & Complexity, 2021, DOI: 10.1007/s11424-022-1361-8.
[1] Qing Xu YAN. BOUNDARY STABILIZATION OF TIMOSHENKO BEAM [J]. Journal of Systems Science and Complexity, 2000, 13(4): 376-384.
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