A discussion on the Hölder and robust finite-time partial stabilizability of Brockett's integrator

Jammazi, Chaker

ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 360-382.

We consider chained systems that model various systems of mechanical or biological origin. It is known according to Brockett that this class of systems, which are controllable, is not stabilizable by continuous stationary feedback (i.e. independent of time). Various approaches have been proposed to remedy this problem, especially instationary or discontinuous feedbacks. Here, we look at another stabilization strategy (by continuous stationary or discontinuous feedbacks) to ensure the asymptotic stability even in finite time for some variables, while other variables do converge, and not necessarily toward equilibrium. Furthermore, we build feedbacks that permit to vanish the two first components of the Brockett integrator in finite time, while ensuring the convergence of the last one. The considering feedbacks are continuous and discontinuous and regular outside zero.

[1] D. Aeyels, Stabilization by smooth feedback of the angular velocity of a rigid body. Syst. Control Lett. 5 (1985) 59-63. | MR | Zbl

[2] D. Aeyels, Stabilization of a class of nonlinear systems by a smooth feedback control. Syst. Control Lett. 5 (1985) 289-294. | MR | Zbl

[3] D. Aeyels and M. Szafranski, Comments on the stabilizablity of angular velocity of rigid body. Syst. Control Lett. 10 (1988) 35-39. | Zbl

[4] V. Andriano, Global feedback stabilization of the angular velocity of symmetric rigid body. Syst. Control Lett. (1993) 361-364. | MR | Zbl

[5] A. Astolfi, Asymptotic stabilization of nonholonomic systems with discontinuous control. Ph.D. thesis, Swiss Federal Institute of Thechnology, Zurich (1996).

[6] A. Astolfi, Discontinuous control of nonholonomic systems. Syst. Control Lett. 27 (1996) 37-45. | MR | Zbl

[7] A. Bacciotti, Local stabilizability of nonlinear control systems. World Scientific (1991). | MR | Zbl

[8] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory. Communications and Control Engineering, Springer-Verlag (2005). | MR | Zbl

[9] L. Beji, A. Abichou and Y. Bestaoui, Position and attitude control of an underactuated autonomous airship. International Journal of Differential Equations and Applications 8 (2004) 231-255. | Zbl

[10] M.K. Bennani and P. Rouchon, Robust stabilization of flat and chained systems, in European Control Conf. (1995).

[11] S.P. Bhat and D.S. Bernstein, Finite-time stability of homogenoues systems, in Procceding of the American Control Conference, Albuquerque, New Mexico (1997) 2513-2514.

[12] S.P. Bhat and D.S. Bernstein, Continuous finite-time stabilization of the translational and rotational double integrators. IEEE Trans. Automat. Contr. 43 (1998) 678-682. | MR | Zbl

[13] S.P. Bhat and D.S. Bernstein, Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38 (2000) 751-766. | MR | Zbl

[14] S.P. Bhat and D.S. Bernstein, Geometric homogeneity with applications to finite-time stability. Math. Control Signals Syst. 17 (2005) 101-127. | MR | Zbl

[15] R.W. Brockett, Asymptotic stability and feedback stabilization, in Differential geometric control theory, Progress in Math. 27 (1983) 181-191. | MR | Zbl

[16] S. Celikovsky and H. Nijmeijer, On the relation between local controllability and stabilizability for a class of nonliner systems. IEEE Trans. Automat. Contr. 42 (1996) 90-94. | MR | Zbl

[17] F.M. Ceragioli, Discontinuous Ordinary Differential Equations and Stabilization. Tesi di dottorato di ricerca in matematica, Consorzio delle universit'a di Cagliari, Firenze, Modena, Perugia e Siena (1999). | Zbl

[18] F.M. Ceragioli, Some remarks on stabilization by means of discontinuous feedbacks. Syst. Control Lett. 45 (2002) 271-281. | MR | Zbl

[19] J.-M. Coron, A necessary condition for feedback stabilization. Syst. Control Lett. 14 (1990) 227-232. | MR | Zbl

[20] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems 5 (1992) 295-312. | MR | Zbl

[21] J.-M. Coron, Relations entre commandabilité et stabilisations non linéaires, in Nonlinear partial differential equations and their applications XI, Collège de France Seminar, Paris (1989-1991), Pitman Res. Notes Math. Ser. 299, Longman Sci. Tech., Harlow (1994) 68-86. | MR | Zbl

[22] J.-M. Coron, Stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws. SIAM J. Control Optim. 33 (1995) 804-833. | MR | Zbl

[23] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs 136. American Mathematical Society (2007). | MR | Zbl

[24] J.-M. Coron and B. D'Andréa Novel, Smooth stabilizing time-varying control laws for a class of nonlinear systems. Applications to mobile robots, in IFAC Nonlinear Control Systems Design, M. Fliess Ed., Bordeaux, France (1992) 413-418.

[25] J.-M. Coron and E.Y. Keraï, Explicit feedbacks stabilizing the attitude of a rigid spacecraft with two torques. Automatica 32 (1996) 669-677. | MR | Zbl

[26] J.-M. Coron and J.-B. Pomet, A remark on the design of time-varying stabilizing feedback laws for controllable systems without drift, in IFAC Nonlinear Control Systems Design, M. Fliess Ed., Bordeaux, France (1992) 397-401.

[27] J.-M. Coron and L. Rosier, A relation between continuous time-varying and discontinuous feedback stabilization. J. Math. Systems Estimation and Control 4 (1994) 67-84. | MR | Zbl

[28] A.L. Fradkov, I.V. Miroshnik and V.O. Nikiforov, Nonlinear and adaptive Control of Complex Systems. Kluwer Academic (2001). | MR | Zbl

[29] W. Haddad, V. Chellaboina and S. Nersesov, A unification between partial stability of state-dependent impulsive systems and stability theory for time-dependent impulsive systems, in Proc. Amer. Contr. Conf. (2003) 4004-4009.

[30] V. Haimo, Finite time controllers. SIAM J. Control Optim. 24 (1986) 760-770. | MR | Zbl

[31] H. Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, Lecture notes in pure and applied Math. 127, S. Elaydi Ed., Proc. Colorado Springs conf. Marcel Dekker Inc., New York (1990) 249-260. | MR | Zbl

[32] Y. Hong, Finite-time stabilization and stabilizability of a class of controllable systems. Syst. Control Lett. 46 (2002) 231-236. | MR | Zbl

[33] Y. Hong and Z.-P. Jiang, Finite-time stabilization of nonlinear systems with parametric and dynamic uncertainties. IEEE Trans. Automat. Contr. 51 (2006) 1950-1956. | MR

[34] Y. Hong, J. Huang and Y. Xu, On an output feedback finite-time stabilization problem. IEEE Trans. Automat. Contr. 46 (2001) 305-309. | MR | Zbl

[35] X. Huang, W. Lin and B. Yang, Global finite-time stabilization of a class of uncertain nonlinear systems. Automatica 41 (2005) 881-888. | MR | Zbl

[36] C. Jammazi, Backstepping and Partial Asymptotic Stabilization. Applications to Partial Attitude Control. International Journal of Control Automation and Systems 6 (2008) 859-872.

[37] C. Jammazi, Finite-time partial stabilizability of chained systems. C. R. Acad. Sci. Paris., Sér. I 346 (2008) 975-980. | MR | Zbl

[38] C. Jammazi, On the partial attitude control of axisymmetric rigid spacecraft, in Intelligent Systems and Automation : 1st Mediterranean Conference on Intelligent Systems and Automation, AIP Conf. Proc. 1019, H. Arioui, R. Marrouki and H.A. Abbassi Eds., Annaba, Algeria (2008) 302-307.

[39] C. Jammazi, Further results on finite-time partial stability and stabilization. Applications to nonlinear control systems, in Intelligent Systems and Automation : 2nd Mediterranean Conference on Intelligent Systems and Automation, AIP Conf. Proc. 1107, L. Beji, S. Otmane and A. Abichou Eds., Zarzis, Tunisia (2009) 111-116.

[40] C. Jammazi, On a sufficient condition for finite-time partial stability and stabilization : Applications. IMA J. Math. Control Inf. 27 (2010) 29-56. | MR | Zbl

[41] C. Jammazi and A. Abichou, Partial stabilizability of an underactuated autonomous underwater vehicle, in Proc. in International Conference “System Identification and Control Problems” SICPRO'07, Moscow Institute of Control (2007) 976-986.

[42] H.K. Khalil, Nonlinear Systems. Prentice Hall (2002). | Zbl

[43] A.L. Kovalev and A.L. Zuyev, On nonasymptotic stabilization of controllable systems, in Proceedings of the 14 International Symposium on Mathematical theory of networks and systems (MTNS), Perpignan, France (2000).

[44] D.A. Lizárraga, P. Morin and C. Samson, Non-robustness of continuous homogeneous stabilizers for affine control systems, in Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, Arizona, USA (1999) 855-860.

[45] M. Maini, P. Morin, J.-B. Pomet and C. Samson, On the robust stabilization of chained systems by continuous feedback, in Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, Arizona, USA (1999) 3472-3477.

[46] R.T. M'Closkey and R.M. Murray, Exponential stabilization of driftless nonlinear control systems using homogeneous feedback. IEEE Trans. Automat. Contr. 42 (1997) 614-628. | MR | Zbl

[47] P. Morin and C. Samson, Time-varying exponential stabilization of a rigid spacecraft with two control torques. IEEE Trans. Automat. Contr. 42 (1997) 528-534. | MR | Zbl

[48] P. Morin, C. Samson, J.-B. Pomet and Z.-P. Jiang, Time-varying feedback stabilization of the attitude of a rigid spacecraft with two controls. Syst. Control Lett. 25 (1995) 375-385. | MR | Zbl

[49] E. Moulay, Une contribution à l'étude de la stabilité en temps fini et de la stabilisation. Ph.D. thesis, L'École Centrale de Lille (2005).

[50] P. Morin, J.-B. Pomet and C. Samson, Development of time-varying feedback stabilization of nonlinear systems, in Nonlinear control design symposium NOLCOS (1998) 587-594.

[51] Y. Orlov, Discontinuous systems - Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions. Communications and Control Engineering, Springer-Verlag (2009). | MR | Zbl

[52] B.E. Paden and S.S. Sastry, A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators. IEEE Trans. Circuits Systems CAS-34 (1987) 73-82. | MR | Zbl

[53] K.Y. Pettersen and O. Egeland, Exponential stabilization of an underactuated surface vessel, in Proc. 35th IEEE Conf. on Decision Control, Kobe, Japan (1996). | Zbl

[54] Z. Qu, Robust control of nonlinear uncertain systems without generalized matching conditions. IEEE Trans. Automat. Contr. 40 (1995) 1453-1460. | MR | Zbl

[55] N. Rouche, P. Habets and P. Laloy, Stability Theory by Lyapunov's Direct Method. Applied Mathematical Sciences, Springer-Verlag (1977). | MR | Zbl

[56] E.P. Ryan, On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J. Control Optim. 32 (1994) 1597-1604. | MR | Zbl

[57] C. Samson, Velocity and torque feedback control of a nonholonomic cart, in Proceedings of International Workshop on Nonlinear and Adaptive Control 162, Springer-Verlag (1991) 125-151. | MR | Zbl

[58] C. Samson, Control of chained systems : Application to path following and time-varying point-stabilization of mobile robots. IEEE Trans. Automat. Contr. 40 (1995) 64-77. | MR | Zbl

[59] E.D. Sontag, Mathematical Control Theory : Determinstic Finite Dimensional Systems, Text in Applied Mathematics 6. Springer-Verlag (1998). | MR | Zbl

[60] E.D. Sontag, Stability and stabilization : Discontinuities and the effect of disturbances, in Nonlinear Analysis, Differential Equations and Control, Proc. NATO Advanced Study Institute, Montreal, F.H. Clarke and R.J. Stern Eds. (1999) 551-598. | MR | Zbl

[61] E.D. Sontag and H.J. Sussmann, Remarks on continuous feedback, in 19th IEEE Conference on Decision and Control, Albuquerque (1980) 916-921.

[62] W. Su and M. Fu, Robust nonlinear control : beyond backstepping and nonlinear forwarding, in IEEE Conference on decision and control (1999) 831-836.

[63] W. Su and M. Fu, Robust stabilization of nonlinear cascaded systems. Automatica 42 (2006) 645-651. | MR | Zbl

[64] H.J. Sussmann, Subanalytic sets and feedback control. J. Differential Equations 31 (1979) 31-52. | MR | Zbl

[65] V.I. Vorotnikov, Partial Stability and Control. Birkhäuser (1998). | MR | Zbl

[66] V.I. Vorotnikov, Partial stability and control : The state-of-the art and development. Autom. Remote Control 66 (2005) 511-561. | MR | Zbl

[67] A.L. Zuyev, On Brockett's condition for smooth stabilization with respect to part of variables, in Proc. European Control Conference ECC'99, Karlsruhe, Germany (1999).

[68] A.L. Zuyev, On partial stabilization of nonlinear autonomous systems : Sufficient conditions and examples, in Proc. of the European Control Conference ECC'01, Porto, Portugal (2001) 1918-1922.