American Institute of Mathematical Sciences

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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

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. Google Scholar

[2]

V. I. Arnold, "Mathematical Models in Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978.  Google Scholar

[3]

A. M. Bloch, "Nonholonomic Mechanics and Control," volume 24 of Interdisciplinary Applied Mathematics, Systems and Control, Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," 2nd edition, Pure and Applied Mathematics, 120, Academic Press, Orlando, FL, 1986.  Google Scholar

[7]

F. Bullo and A. Lewis, "Geometric Control of Mechanical Systems," Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005.  Google Scholar

[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.  Google Scholar

[9]

H. Cendra and S. Grillo, Lagrangian systems with higher order constraints, J. Math. Phys., 48 (2007), 35 pp.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[13]

S. Grillo, "Sistemas Noholónomos Generalizados," Ph.D thesis, Dto. de Matemática, UNSur, 2007. In spanish. Google Scholar

[14]

S. Grillo, Higher order constrained Hamiltonian systems, J. Math. Phys., 50 (2009), 34 pp.  Google Scholar

[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.  Google Scholar

[16]

H. Khalil, "Nonlinear Systems," Upper Saddle River NJ, Prentice Hall, 1996. Google Scholar

[17]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," New York, John Wiley & Son, 1963. Google Scholar

[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).  Google Scholar

[19]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.  Google Scholar

[20]

J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications," New York, Springer-Verlag, 2001. Google Scholar

[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.  Google Scholar

[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). Google Scholar

[23]

D. Pérez, "Sistemas con vínculos de orden superior y su aplicación a la teoría de control automático," Master thesis, Instituto Balseiro, 2007. Google Scholar

[24]

J. Rayleigh, "The Theory of Sound," 2nd edition, Dover Publications, New York, 1945.  Google Scholar

[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.  Google Scholar

[26]

E. Sontag, "Mathematical Control Theory," Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998.  Google Scholar

[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.  Google Scholar

[28]

E. T. Whittaker, "A Treatise on The Analytical Dynamics of Particles and Rigid Bodies," Cambridge University Press, Cambridge, 1937.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundation of Mechanics," New York, Benjaming Cummings, 1985. Google Scholar

[2]

V. I. Arnold, "Mathematical Models in Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978.  Google Scholar

[3]

A. M. Bloch, "Nonholonomic Mechanics and Control," volume 24 of Interdisciplinary Applied Mathematics, Systems and Control, Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," 2nd edition, Pure and Applied Mathematics, 120, Academic Press, Orlando, FL, 1986.  Google Scholar

[7]

F. Bullo and A. Lewis, "Geometric Control of Mechanical Systems," Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005.  Google Scholar

[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.  Google Scholar

[9]

H. Cendra and S. Grillo, Lagrangian systems with higher order constraints, J. Math. Phys., 48 (2007), 35 pp.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[13]

S. Grillo, "Sistemas Noholónomos Generalizados," Ph.D thesis, Dto. de Matemática, UNSur, 2007. In spanish. Google Scholar

[14]

S. Grillo, Higher order constrained Hamiltonian systems, J. Math. Phys., 50 (2009), 34 pp.  Google Scholar

[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.  Google Scholar

[16]

H. Khalil, "Nonlinear Systems," Upper Saddle River NJ, Prentice Hall, 1996. Google Scholar

[17]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," New York, John Wiley & Son, 1963. Google Scholar

[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).  Google Scholar

[19]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.  Google Scholar

[20]

J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis and Applications," New York, Springer-Verlag, 2001. Google Scholar

[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.  Google Scholar

[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). Google Scholar

[23]

D. Pérez, "Sistemas con vínculos de orden superior y su aplicación a la teoría de control automático," Master thesis, Instituto Balseiro, 2007. Google Scholar

[24]

J. Rayleigh, "The Theory of Sound," 2nd edition, Dover Publications, New York, 1945.  Google Scholar

[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.  Google Scholar

[26]

E. Sontag, "Mathematical Control Theory," Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998.  Google Scholar

[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.  Google Scholar

[28]

E. T. Whittaker, "A Treatise on The Analytical Dynamics of Particles and Rigid Bodies," Cambridge University Press, Cambridge, 1937.  Google Scholar

[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.  Google Scholar

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