State constrained patchy feedback stabilization

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March 2015, 5(1): 141-163. doi: 10.3934/mcrf.2015.5.141

State constrained patchy feedback stabilization

Fabio S. Priuli ^1,

1.	Istituto per le Applicazioni del Calcolo “M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini 19, I-00185 Roma, Italy

Received November 2013 Revised April 2014 Published January 2015

Abstract
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We construct a patchy feedback for a general control system on $\mathbb{R}^d$ which realizes practical stabilization to a target set $\Sigma$, when the dynamics is constrained to a given set of states $S$. The main result is that $S$--constrained asymptotically controllability to $\Sigma$ implies the existence of a discontinuous practically stabilizing feedback. Such a feedback can be constructed in ``patchy'' form, a particular class of piecewise constant controls which ensure the existence of local Carathéodory solutions to any Cauchy problem of the control system and which enjoy good robustness properties with respect to both measurement errors and external disturbances.

Keywords: Asymptotic controllability, patchy feedback, state constraint, robustness., stabilization.

Mathematics Subject Classification: 34A34, 34A36, 34H05, 34H15, 93D15, 93D0.

Citation: Fabio S. Priuli. State constrained patchy feedback stabilization. Mathematical Control and Related Fields, 2015, 5 (1) : 141-163. doi: 10.3934/mcrf.2015.5.141

References:

[1]	F. Ancona and A. Bressan, Patchy vector fields and asymptotic stabilization, ESAIM Control Optim. Calc. Var., 4 (1999), 445-471. doi: 10.1051/cocv:1999117.
[2]	F. Ancona and A. Bressan, Flow stability of patchy vector fields and robust feedback stabilization, SIAM J. Control Optim., 41 (2002), 1455-1476. doi: 10.1137/S0363012901391676.
[3]	F. Ancona and A. Bressan, Nearly time optimal stabilizing patchy feedbacks, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 24 (2007), 279-310. doi: 10.1016/j.anihpc.2006.03.010.
[4]	F. Ancona and A. Bressan, Patchy feedbacks for stabilization and optimal control: General theory and robustness properties, in Geometric Control and Nonsmooth Analysis, Ser. Adv. Math. Appl. Sci., 76, World Sci. Publ., Hackensack, NJ, 2008, 28-64. doi: 10.1142/9789812776075_0002.
[5]	A. M. Bloch and S. Drakunov, Stabilization and tracking in the nonholonomic integrator via sliding modes, Systems Control Lett., 29 (1996), 91-99. doi: 10.1016/S0167-6911(96)00049-7.
[6]	A. Bressan, Singularities of stabilizing feedbacks, Rend. Sem. Mat. Univ. Politec. Torino, 56 (1998), 87-104.
[7]	A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.
[8]	A. Bressan and F. S. Priuli, Nearly optimal patchy feedbacks, Discr. Cont. Dyn. Systems Series A, 21 (2008), 687-701. doi: 10.3934/dcds.2008.21.687.
[9]	R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory (eds. R. W. Brockett, R. S. Millman and H. J. Sussmann), Progr. Math., 27, Birkhaüser, Boston, MA, 1983, 181-191.
[10]	F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998.
[11]	F. H. Clarke, L. Rifford and R. J. Stern, Feedback in state constrained optimal control, ESAIM Control Optim. Calc. Var., 7 (2002), 97-133. doi: 10.1051/cocv:2002005.
[12]	F. H. Clarke and R. J. Stern, State constrained feedback stabilization, SIAM J. Control Optim., 42 (2003), 422-441. doi: 10.1137/S036301290240453X.
[13]	F. H. Clarke and R. J. Stern, Lyapunov feedback characterizations state constrained controllability stabilization, Systems and Control Letters, 54 (2005), 747-752. doi: 10.1016/j.sysconle.2004.11.013.
[14]	J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, 5 (1992), 295-312. doi: 10.1007/BF01211563.
[15]	J.-M. Coron, On the stabilization in finite time of locally controllable systems by means of continuous time-varying feedback law, SIAM J. Control Optim., 33 (1995), 804-833. doi: 10.1137/S0363012992240497.
[16]	S. Drakunov and V. I. Utkin, Sliding-mode observers: Tutorial, in Proceedings of the 34th IEEE Conference of Decision and Control, IEEE Publications, 4, Piscataway, NJ, 1995, 3376-3378. doi: 10.1109/CDC.1995.479009.
[17]	R. Goebel, C. Prieur and A. R. Teel, Hybrid feedback control and robust stabilization of nonlinear systems, IEEE Trans. Autom. Control, 52 (2007), 2103-2117. doi: 10.1109/TAC.2007.908320.
[18]	R. Goebel and A. R. Teel, Direct design of robustly asymptotically stabilizing hybrid feedback, ESAIM Control Optim. Calc. Var., 15 (2009), 205-213. doi: 10.1051/cocv:2008023.
[19]	R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbb{R}^{N}$, Nonlinear Anal., 3 (1979), 145-154. doi: 10.1016/0362-546X(79)90044-0.
[20]	E. D. Sontag, Stability and stabilization: Discontinuities and the effect of disturbances, in Nonlinear Analysis, Differential Equations, and Control (eds. F. H. Clarke and R. J. Stern), NATO Sci. Ser. C Math. Phys. Sci., 528, Kluwer Academic, Dordrecht, 1999, 551-598.
[21]	E. D. Sontag and H. J. Sussmann, Remarks on continuous feedback, in Proceedings of the 19th IEEE Conference on Decision and Control, IEEE Publications, Piscataway, NJ, 1980, 916-921. doi: 10.1109/CDC.1980.271934.
[22]	H. J. Sussmann, Subanalytic sets and feedback control, J. Differential Equations, 31 (1979), 31-52. doi: 10.1016/0022-0396(79)90151-7.

show all references

References:

[1]	F. Ancona and A. Bressan, Patchy vector fields and asymptotic stabilization, ESAIM Control Optim. Calc. Var., 4 (1999), 445-471. doi: 10.1051/cocv:1999117.
[2]	F. Ancona and A. Bressan, Flow stability of patchy vector fields and robust feedback stabilization, SIAM J. Control Optim., 41 (2002), 1455-1476. doi: 10.1137/S0363012901391676.
[3]	F. Ancona and A. Bressan, Nearly time optimal stabilizing patchy feedbacks, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 24 (2007), 279-310. doi: 10.1016/j.anihpc.2006.03.010.
[4]	F. Ancona and A. Bressan, Patchy feedbacks for stabilization and optimal control: General theory and robustness properties, in Geometric Control and Nonsmooth Analysis, Ser. Adv. Math. Appl. Sci., 76, World Sci. Publ., Hackensack, NJ, 2008, 28-64. doi: 10.1142/9789812776075_0002.
[5]	A. M. Bloch and S. Drakunov, Stabilization and tracking in the nonholonomic integrator via sliding modes, Systems Control Lett., 29 (1996), 91-99. doi: 10.1016/S0167-6911(96)00049-7.
[6]	A. Bressan, Singularities of stabilizing feedbacks, Rend. Sem. Mat. Univ. Politec. Torino, 56 (1998), 87-104.
[7]	A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.
[8]	A. Bressan and F. S. Priuli, Nearly optimal patchy feedbacks, Discr. Cont. Dyn. Systems Series A, 21 (2008), 687-701. doi: 10.3934/dcds.2008.21.687.
[9]	R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory (eds. R. W. Brockett, R. S. Millman and H. J. Sussmann), Progr. Math., 27, Birkhaüser, Boston, MA, 1983, 181-191.
[10]	F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998.
[11]	F. H. Clarke, L. Rifford and R. J. Stern, Feedback in state constrained optimal control, ESAIM Control Optim. Calc. Var., 7 (2002), 97-133. doi: 10.1051/cocv:2002005.
[12]	F. H. Clarke and R. J. Stern, State constrained feedback stabilization, SIAM J. Control Optim., 42 (2003), 422-441. doi: 10.1137/S036301290240453X.
[13]	F. H. Clarke and R. J. Stern, Lyapunov feedback characterizations state constrained controllability stabilization, Systems and Control Letters, 54 (2005), 747-752. doi: 10.1016/j.sysconle.2004.11.013.
[14]	J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, 5 (1992), 295-312. doi: 10.1007/BF01211563.
[15]	J.-M. Coron, On the stabilization in finite time of locally controllable systems by means of continuous time-varying feedback law, SIAM J. Control Optim., 33 (1995), 804-833. doi: 10.1137/S0363012992240497.
[16]	S. Drakunov and V. I. Utkin, Sliding-mode observers: Tutorial, in Proceedings of the 34th IEEE Conference of Decision and Control, IEEE Publications, 4, Piscataway, NJ, 1995, 3376-3378. doi: 10.1109/CDC.1995.479009.
[17]	R. Goebel, C. Prieur and A. R. Teel, Hybrid feedback control and robust stabilization of nonlinear systems, IEEE Trans. Autom. Control, 52 (2007), 2103-2117. doi: 10.1109/TAC.2007.908320.
[18]	R. Goebel and A. R. Teel, Direct design of robustly asymptotically stabilizing hybrid feedback, ESAIM Control Optim. Calc. Var., 15 (2009), 205-213. doi: 10.1051/cocv:2008023.
[19]	R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbb{R}^{N}$, Nonlinear Anal., 3 (1979), 145-154. doi: 10.1016/0362-546X(79)90044-0.
[20]	E. D. Sontag, Stability and stabilization: Discontinuities and the effect of disturbances, in Nonlinear Analysis, Differential Equations, and Control (eds. F. H. Clarke and R. J. Stern), NATO Sci. Ser. C Math. Phys. Sci., 528, Kluwer Academic, Dordrecht, 1999, 551-598.
[21]	E. D. Sontag and H. J. Sussmann, Remarks on continuous feedback, in Proceedings of the 19th IEEE Conference on Decision and Control, IEEE Publications, Piscataway, NJ, 1980, 916-921. doi: 10.1109/CDC.1980.271934.
[22]	H. J. Sussmann, Subanalytic sets and feedback control, J. Differential Equations, 31 (1979), 31-52. doi: 10.1016/0022-0396(79)90151-7.

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