Lyapunov constraints and global asymptotic stabilization

Previous Article
JGM Home
This Issue
Next Article
Embedded geodesic problems and optimal control for matrix Lie groups

June 2011, 3(2): 145-196. doi: 10.3934/jgm.2011.3.145

Lyapunov constraints and global asymptotic stabilization

Sergio Grillo ^1, , Jerrold E. Marsden ^2, and Sujit Nair ^3,

1.	Centro Atmico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, and CONICET, Argentina
2.	Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125, United States
3.	United Technologies Research Center, East Hartford, CT 06118, United States

Received October 2010 Revised June 2011 Published July 2011

Abstract
Related Papers

In this paper, we develop a method for stabilizing underactuated mechanical systems by imposing kinematic constraints (more precisely Lyapunov constraints). If these constraints can be implemented by actuators, i.e., if there exists a related constraint force exerted by the actuators, then the existence of a Lyapunov function for the system under consideration is guaranteed. We establish necessary and sufficient conditions for the existence and uniqueness of constraint forces. These conditions give rise to a system of PDEs whose solution is the required Lyapunov function. To illustrate our results, we solve these PDEs for certain underactuated mechanical systems of interest such as the inertia wheel-pendulum, the inverted pendulum on a cart system and the ball and beam system.

Keywords: Lagrangian and Hamiltonian systems, energy shaping., nonlinear control.

Mathematics Subject Classification: Primary: 93D20; Secondary: 53Z0.

Citation: Sergio Grillo, Jerrold E. Marsden, Sujit Nair. Lyapunov constraints and global asymptotic stabilization. Journal of Geometric Mechanics, 2011, 3 (2) : 145-196. doi: 10.3934/jgm.2011.3.145

References:

[1]	R. Abraham and J. E. Marsden, "Foundation of Mechanics,", New York, (1985).
[2]	V. I. Arnold, "Mathematical Models in Classical Mechanics,", Graduate Texts in Mathematics, 60 (1978).
[3]	A. M. Bloch, "Nonholonomic Mechanics and Control,", volume 24 of Interdisciplinary Applied Mathematics, (2003).
[4]	A. M. Bloch, D. E. Chang, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems II: Potential shaping,, IEEE Trans. Automat. Control, 46 (2001), 1556. doi: 10.1109/9.956051.
[5]	A. M. Bloch, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems I: The first matching theorem,, IEEE Trans. Automat. Control, 45 (2000), 2253. doi: 10.1109/9.895562.
[6]	W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," 2nd edition, Pure and Applied Mathematics, 120,, Academic Press, (1986).
[7]	F. Bullo and A. Lewis, "Geometric Control of Mechanical Systems," Texts in Applied Mathematics, 49,, Springer-Verlag, (2005).
[8]	H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets,, J. Math. Phys., 47 (2006), 2209. doi: 10.1063/1.2165797.
[9]	H. Cendra and S. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys., 48 (2007).
[10]	H. Cendra, A. Ibort, M. de León and D. Martin de Diego, A generalization of Chetaev's principle for a class of higher order non-holonomic constraints,, J. Math. Phys., 45 (2004), 2785. doi: 10.1063/1.1763245.
[11]	D. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. Woolsey, "The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems,", \emph{ESIAM: Control, (2001).
[12]	B. Gharesifard, A. D. Lewis and A.-R. Mansouri, A geometric framework for stabilization by energy shaping: Sufficient conditions for existence of solutions,, Communications for Information and Systems, 8 (2008), 353.
[13]	S. Grillo, "Sistemas Noholónomos Generalizados,", Ph.D thesis, (2007).
[14]	S. Grillo, Higher order constrained Hamiltonian systems,, J. Math. Phys., 50 (2009).
[15]	S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems,, International Journal of Geometric Methods in Modern Physics, (2010). doi: 10.1142/S0219887810004580.
[16]	H. Khalil, "Nonlinear Systems,", Upper Saddle River NJ, (1996).
[17]	S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry,", New York, (1963).
[18]	C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems, Geometrical Structures for Physical Theories, II (Vietri, 1996),, Rend. Sem. Mat. Univ. Pol. Torino \textbf{54} (1996), 54 (1996), 353.
[19]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, 17,, Springer-Verlag, (1994).
[20]	J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications,", New York, (2001).
[21]	R. Ortega, M. W. Spong, F. Gómez-Estern and G. Blankenstein, Stabilization of underactuated mechanical systems via interconnection and damping assignment,, IEEE Trans. Aut. Control, 47 (2002), 1281. doi: 10.1109/TAC.2002.800770.
[22]	D. Pérez, Sistemas noholónomos generalizados y su aplicación a la teoría de control automático mediante vínculos cinemáticos,, Proyecto Integrador, (2006).
[23]	D. Pérez, "Sistemas con vínculos de orden superior y su aplicación a la teoría de control automático,", Master thesis, (2007).
[24]	J. Rayleigh, "The Theory of Sound," 2nd edition,, Dover Publications, (1945).
[25]	A. Shiriaev, J. W. Perram and C. Canudas-de-Wit, Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach,, IEEE Transactions on Automatic Control, 50 (2005), 1164. doi: 10.1109/TAC.2005.852568.
[26]	E. Sontag, "Mathematical Control Theory," Texts in Applied Mathematics, 6,, Springer-Verlag, (1998).
[27]	M. W. Spong, P. Corke and R. Lozano, Nonlinear control of the inertia wheel pendulum,, Automatica, 37 (2001), 1845. doi: 10.1016/S0005-1098(01)00145-5.
[28]	E. T. Whittaker, "A Treatise on The Analytical Dynamics of Particles and Rigid Bodies,", Cambridge University Press, (1937).
[29]	C. Woolsey, C. Reddy, A. Bloch, D. Chang, N. Leonard and J. Marsden, Controlled Lagrangian systems with gyroscopic forcing and dissipation,, European Journal of Control, 10 (2004), 478. doi: 10.3166/ejc.10.478-496.

show all references

References:

[1]	R. Abraham and J. E. Marsden, "Foundation of Mechanics,", New York, (1985).
[2]	V. I. Arnold, "Mathematical Models in Classical Mechanics,", Graduate Texts in Mathematics, 60 (1978).
[3]	A. M. Bloch, "Nonholonomic Mechanics and Control,", volume 24 of Interdisciplinary Applied Mathematics, (2003).
[4]	A. M. Bloch, D. E. Chang, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems II: Potential shaping,, IEEE Trans. Automat. Control, 46 (2001), 1556. doi: 10.1109/9.956051.
[5]	A. M. Bloch, N. E. Leonard and J. E. Marsden, Controlled Lagrangian and the stabilization of mechanical systems I: The first matching theorem,, IEEE Trans. Automat. Control, 45 (2000), 2253. doi: 10.1109/9.895562.
[6]	W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," 2nd edition, Pure and Applied Mathematics, 120,, Academic Press, (1986).
[7]	F. Bullo and A. Lewis, "Geometric Control of Mechanical Systems," Texts in Applied Mathematics, 49,, Springer-Verlag, (2005).
[8]	H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets,, J. Math. Phys., 47 (2006), 2209. doi: 10.1063/1.2165797.
[9]	H. Cendra and S. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys., 48 (2007).
[10]	H. Cendra, A. Ibort, M. de León and D. Martin de Diego, A generalization of Chetaev's principle for a class of higher order non-holonomic constraints,, J. Math. Phys., 45 (2004), 2785. doi: 10.1063/1.1763245.
[11]	D. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. Woolsey, "The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems,", \emph{ESIAM: Control, (2001).
[12]	B. Gharesifard, A. D. Lewis and A.-R. Mansouri, A geometric framework for stabilization by energy shaping: Sufficient conditions for existence of solutions,, Communications for Information and Systems, 8 (2008), 353.
[13]	S. Grillo, "Sistemas Noholónomos Generalizados,", Ph.D thesis, (2007).
[14]	S. Grillo, Higher order constrained Hamiltonian systems,, J. Math. Phys., 50 (2009).
[15]	S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems,, International Journal of Geometric Methods in Modern Physics, (2010). doi: 10.1142/S0219887810004580.
[16]	H. Khalil, "Nonlinear Systems,", Upper Saddle River NJ, (1996).
[17]	S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry,", New York, (1963).
[18]	C.-M. Marle, Kinematic and geometric constraints, servomechanism and control of mechanical systems, Geometrical Structures for Physical Theories, II (Vietri, 1996),, Rend. Sem. Mat. Univ. Pol. Torino \textbf{54} (1996), 54 (1996), 353.
[19]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, 17,, Springer-Verlag, (1994).
[20]	J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications,", New York, (2001).
[21]	R. Ortega, M. W. Spong, F. Gómez-Estern and G. Blankenstein, Stabilization of underactuated mechanical systems via interconnection and damping assignment,, IEEE Trans. Aut. Control, 47 (2002), 1281. doi: 10.1109/TAC.2002.800770.
[22]	D. Pérez, Sistemas noholónomos generalizados y su aplicación a la teoría de control automático mediante vínculos cinemáticos,, Proyecto Integrador, (2006).
[23]	D. Pérez, "Sistemas con vínculos de orden superior y su aplicación a la teoría de control automático,", Master thesis, (2007).
[24]	J. Rayleigh, "The Theory of Sound," 2nd edition,, Dover Publications, (1945).
[25]	A. Shiriaev, J. W. Perram and C. Canudas-de-Wit, Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach,, IEEE Transactions on Automatic Control, 50 (2005), 1164. doi: 10.1109/TAC.2005.852568.
[26]	E. Sontag, "Mathematical Control Theory," Texts in Applied Mathematics, 6,, Springer-Verlag, (1998).
[27]	M. W. Spong, P. Corke and R. Lozano, Nonlinear control of the inertia wheel pendulum,, Automatica, 37 (2001), 1845. doi: 10.1016/S0005-1098(01)00145-5.
[28]	E. T. Whittaker, "A Treatise on The Analytical Dynamics of Particles and Rigid Bodies,", Cambridge University Press, (1937).
[29]	C. Woolsey, C. Reddy, A. Bloch, D. Chang, N. Leonard and J. Marsden, Controlled Lagrangian systems with gyroscopic forcing and dissipation,, European Journal of Control, 10 (2004), 478. doi: 10.3166/ejc.10.478-496.

[1]	Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130
[2]	Sergio Grillo, Leandro Salomone, Marcela Zuccalli. On the relationship between the energy shaping and the Lyapunov constraint based methods. Journal of Geometric Mechanics, 2017, 9 (4) : 459-486. doi: 10.3934/jgm.2017018
[3]	Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1
[4]	B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217
[5]	Morched Boughariou. Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 603-616. doi: 10.3934/dcds.2003.9.603
[6]	Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116
[7]	Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222
[8]	John R. Graef, János Karsai. Oscillation and nonoscillation in nonlinear impulsive systems with increasing energy. Conference Publications, 2001, 2001 (Special) : 166-173. doi: 10.3934/proc.2001.2001.166
[9]	Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861
[10]	Chungen Liu. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 337-355. doi: 10.3934/dcds.2010.27.337
[11]	Janusz Grabowski, Katarzyna Grabowska, Paweł Urbański. Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings. Journal of Geometric Mechanics, 2014, 6 (4) : 503-526. doi: 10.3934/jgm.2014.6.503
[12]	Katarzyna Grabowska. Lagrangian and Hamiltonian formalism in Field Theory: A simple model. Journal of Geometric Mechanics, 2010, 2 (4) : 375-395. doi: 10.3934/jgm.2010.2.375
[13]	M. Motta, C. Sartori. Exit time problems for nonlinear unbounded control systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 137-156. doi: 10.3934/dcds.1999.5.137
[14]	Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063
[15]	Chunjiang Qian, Wei Lin, Wenting Zha. Generalized homogeneous systems with applications to nonlinear control: A survey. Mathematical Control & Related Fields, 2015, 5 (3) : 585-611. doi: 10.3934/mcrf.2015.5.585
[16]	Mikhail Gusev. On reachability analysis for nonlinear control systems with state constraints. Conference Publications, 2015, 2015 (special) : 579-587. doi: 10.3934/proc.2015.0579
[17]	Wei Wang. Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 679-701. doi: 10.3934/dcds.2012.32.679
[18]	Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213
[19]	José F. Cariñena, Irina Gheorghiu, Eduardo Martínez, Patrícia Santos. On the virial theorem for nonholonomic Lagrangian systems. Conference Publications, 2015, 2015 (special) : 204-212. doi: 10.3934/proc.2015.0204
[20]	Anouar Bahrouni, Marek Izydorek, Joanna Janczewska. Subharmonic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1841-1850. doi: 10.3934/dcdss.2019121

2017 Impact Factor: 0.561

American Institute of Mathematical Sciences