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Lyapunov constraints and global asymptotic stabilization
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 |
References:
[1] |
R. Abraham and J. E. Marsden, "Foundation of Mechanics,", New York, (1985). Google Scholar |
[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). 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.
|
[13] |
S. Grillo, "Sistemas Noholónomos Generalizados,", Ph.D thesis, (2007). Google Scholar |
[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). Google Scholar |
[17] |
S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry,", New York, (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 \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). 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.
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). Google Scholar |
[23] |
D. Pérez, "Sistemas con vínculos de orden superior y su aplicación a la teoría de control automático,", Master thesis, (2007). Google Scholar |
[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). Google Scholar |
[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). 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.
|
[13] |
S. Grillo, "Sistemas Noholónomos Generalizados,", Ph.D thesis, (2007). Google Scholar |
[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). Google Scholar |
[17] |
S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry,", New York, (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 \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). 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.
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). Google Scholar |
[23] |
D. Pérez, "Sistemas con vínculos de orden superior y su aplicación a la teoría de control automático,", Master thesis, (2007). Google Scholar |
[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. |
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