Lyapunov constraints and global asymptotic stabilization

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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
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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, Benjaming Cummings, 1985.
[2]	V. I. Arnold, "Mathematical Models in Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978.
[3]	A. M. Bloch, "Nonholonomic Mechanics and Control," volume 24 of Interdisciplinary Applied Mathematics, Systems and Control, Springer-Verlag, New York, 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), 1556-71. 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), 2253-70. 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, Orlando, FL, 1986.
[7]	F. Bullo and A. Lewis, "Geometric Control of Mechanical Systems," Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005.
[8]	H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets, J. Math. Phys., 47 (2006), 2209-38. doi: 10.1063/1.2165797.
[9]	H. Cendra and S. Grillo, Lagrangian systems with higher order constraints, J. Math. Phys., 48 (2007), 35 pp.
[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), 2785-2801. 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," ESIAM: Control, Optimisation and Calculus of Variations, 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), 353-398.
[13]	S. Grillo, "Sistemas Noholónomos Generalizados," Ph.D thesis, Dto. de Matemática, UNSur, 2007. In spanish.
[14]	S. Grillo, Higher order constrained Hamiltonian systems, J. Math. Phys., 50 (2009), 34 pp.
[15]	S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems, International Journal of Geometric Methods in Modern Physics, 2010. In press. doi: 10.1142/S0219887810004580.
[16]	H. Khalil, "Nonlinear Systems," Upper Saddle River NJ, Prentice Hall, 1996.
[17]	S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," New York, John Wiley & Son, 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 54 (1996), 353-364; Various approaches to conservative and nonconservative non-holonomic systems, Rep. Math. Phys., 42 (1998), 211-229 (MR1656282).
[19]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.
[20]	J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications," New York, Springer-Verlag, 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), 1281-1233. 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, Carrera de Ingeniería Mecánica del Instituto Balseiro, (2006).
[23]	D. Pérez, "Sistemas con vínculos de orden superior y su aplicación a la teoría de control automático," Master thesis, Instituto Balseiro, 2007.
[24]	J. Rayleigh, "The Theory of Sound," 2nd edition, Dover Publications, New York, 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), 1164-1176. doi: 10.1109/TAC.2005.852568.
[26]	E. Sontag, "Mathematical Control Theory," Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998.
[27]	M. W. Spong, P. Corke and R. Lozano, Nonlinear control of the inertia wheel pendulum, Automatica, 37 (2001), 1845-1851. 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, Cambridge, 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), 478-496. doi: 10.3166/ejc.10.478-496.

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References:

[1]	R. Abraham and J. E. Marsden, "Foundation of Mechanics," New York, Benjaming Cummings, 1985.
[2]	V. I. Arnold, "Mathematical Models in Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978.
[3]	A. M. Bloch, "Nonholonomic Mechanics and Control," volume 24 of Interdisciplinary Applied Mathematics, Systems and Control, Springer-Verlag, New York, 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), 1556-71. 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), 2253-70. 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, Orlando, FL, 1986.
[7]	F. Bullo and A. Lewis, "Geometric Control of Mechanical Systems," Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005.
[8]	H. Cendra and S. Grillo, Generalized nonholonomic mechanics, servomechanisms and related brackets, J. Math. Phys., 47 (2006), 2209-38. doi: 10.1063/1.2165797.
[9]	H. Cendra and S. Grillo, Lagrangian systems with higher order constraints, J. Math. Phys., 48 (2007), 35 pp.
[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), 2785-2801. 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," ESIAM: Control, Optimisation and Calculus of Variations, 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), 353-398.
[13]	S. Grillo, "Sistemas Noholónomos Generalizados," Ph.D thesis, Dto. de Matemática, UNSur, 2007. In spanish.
[14]	S. Grillo, Higher order constrained Hamiltonian systems, J. Math. Phys., 50 (2009), 34 pp.
[15]	S. Grillo, F. Maciel and D. Pérez, Closed-loop and constrained mechanical systems, International Journal of Geometric Methods in Modern Physics, 2010. In press. doi: 10.1142/S0219887810004580.
[16]	H. Khalil, "Nonlinear Systems," Upper Saddle River NJ, Prentice Hall, 1996.
[17]	S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," New York, John Wiley & Son, 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 54 (1996), 353-364; Various approaches to conservative and nonconservative non-holonomic systems, Rep. Math. Phys., 42 (1998), 211-229 (MR1656282).
[19]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.
[20]	J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications," New York, Springer-Verlag, 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), 1281-1233. 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, Carrera de Ingeniería Mecánica del Instituto Balseiro, (2006).
[23]	D. Pérez, "Sistemas con vínculos de orden superior y su aplicación a la teoría de control automático," Master thesis, Instituto Balseiro, 2007.
[24]	J. Rayleigh, "The Theory of Sound," 2nd edition, Dover Publications, New York, 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), 1164-1176. doi: 10.1109/TAC.2005.852568.
[26]	E. Sontag, "Mathematical Control Theory," Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998.
[27]	M. W. Spong, P. Corke and R. Lozano, Nonlinear control of the inertia wheel pendulum, Automatica, 37 (2001), 1845-1851. 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, Cambridge, 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), 478-496. doi: 10.3166/ejc.10.478-496.

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