# American Institute of Mathematical Sciences

December  2021, 13(4): 629-646. doi: 10.3934/jgm.2021022

## Explicit solutions of the kinetic and potential matching conditions of the energy shaping method

 1 Centro Atómico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, and CONICET, Argentina 2 CMaLP, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, and CONICET, Argentina 3 CMaLP, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, Argentina

Received  September 2018 Revised  June 2021 Published  December 2021 Early access  August 2021

Fund Project: S. Grillo and L. Salomone thank CONICET for its financial support

In the context of underactuated Hamiltonian systems defined by simple Hamiltonian functions, the matching conditions of the energy shaping method split into two decoupled subsets of equations: the kinetic and potential equations. The unknown of the kinetic equation is a metric on the configuration space of the system, while the unknown of the potential equation are the same metric and a positive-definite function around some critical point of the Hamiltonian function. In this paper, assuming that a solution of the kinetic equation is given, we find conditions (in the $C^{\infty}$ category) for the existence of positive-definite solutions of the potential equation and, moreover, we present a procedure to construct, up to quadratures, some of these solutions. In order to illustrate such a procedure, we consider the subclass of systems with one degree of underactuation, where we find in addition a concrete formula for the general solution of the kinetic equation. As a byproduct, new global and local expressions of the matching conditions are presented in the paper.

Citation: Sergio Grillo, Leandro Salomone, Marcela Zuccalli. Explicit solutions of the kinetic and potential matching conditions of the energy shaping method. Journal of Geometric Mechanics, 2021, 13 (4) : 629-646. doi: 10.3934/jgm.2021022
##### References:
 [1] R. Abraham and J. E. Marsden, Foundation of Mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. [2] V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978. doi: 10.1007/978-1-4757-1693-1. [3] A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376. [4] W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Pure and Applied Mathematics, 63, Academic Press, New York-London, 1975. [5] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7. [6] D. E. Chang, Generalization of the IDA-PBC method for stabilization of mechanical systems, 18th Mediterranean Conference on Control and Automation, MED'10, Marrakech, Morocco, 2010. doi: 10.1109/MED.2010.5547672. [7] D. E. Chang, The method of controlled Lagrangians: Energy plus force shaping, SIAM J. Control Optim., 48 (2010), 4821-4845.  doi: 10.1137/070691310. [8] D. E. Chang, On the method of interconnection and damping assignment passivity-based control for the stabilization of mechanical systems, Regul. Chaotic Dyn., 19 (2014), 556-575.  doi: 10.1134/S1560354714050049. [9] D. E. Chang, Stabilizability of controlled Lagrangian systems of two degrees of freedom and one degree of under-actuation by the energy-shaping method, IEEE Trans. Automat. Control, 55 (2010), 1888-1893.  doi: 10.1109/TAC.2010.2049279. [10] D. E. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. A. Wollsey, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM Control Optim. Calc. Var., 8 (2002), 393-422.  doi: 10.1051/cocv:2002045. [11] B. Gharesifard, Stabilization of systems with one degree of underactuation with energy shaping: A geometric approach, SIAM J. Control Optim., 49 (2011), 1422-1434.  doi: 10.1137/09076698X. [12] H. Goldschmidt, Existence theorems for analytic linear partial differential equations, Ann. of Math. (2), 86 (1967), 246-270.  doi: 10.2307/1970689. [13] S. D. Grillo, L. M. Salomone and M. Zuccalli, Asymptotic stabilizability of underactuated Hamiltonian systems with two degrees of freedom, Russ. J. Nonlinear Dyn., 15 (2019), 309-326.  doi: 10.20537/nd190309. [14] S. D. Grillo, L. M. Salomone and M. Zuccalli, On the relationship between the energy shaping and the Lyapunov constraint based methods, J. Geom. Mech., 9 (2017), 459-486.  doi: 10.3934/jgm.2017018. [15] J. Hamberg, Controlled Lagrangians, symmetries and conditions for strong matching, IFAC Proceedings Volumes, 33 (2000), 57-62.  doi: 10.1016/S1474-6670(17)35547-7. [16] J. Hamberg, General matching conditions in the theory of controlled Lagrangians, Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, 1999. doi: 10.1109/CDC.1999.831306. [17] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. [18] A. D. Lewis, Potential energy shaping after kinetic energy shaping, Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, 2006. doi: 10.1109/CDC.2006.376885. [19] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5. [20] J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis and Applications, Springer-Verlag, New York, 2001. [21] J. G. Romero, A. Donaire and R. Ortega, Robust energy shaping control of mechanical systems, Systems Control Lett., 62 (2013), 770-780.  doi: 10.1016/j.sysconle.2013.05.011. [22] D. Zenkov, Matching and stabilization of linear mechanical systems, Proceedings of 2002 International Symposium of Mathematical Theory of Networks and Systems, South Bend, IN, 2002. Available from: http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=F5FAA6E2AB4998A2C00B8A5CDB0A461C?doi=10.1.1.12.7462&rep=rep1&type=pdf.

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##### References:
 [1] R. Abraham and J. E. Marsden, Foundation of Mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. [2] V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978. doi: 10.1007/978-1-4757-1693-1. [3] A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376. [4] W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Pure and Applied Mathematics, 63, Academic Press, New York-London, 1975. [5] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7. [6] D. E. Chang, Generalization of the IDA-PBC method for stabilization of mechanical systems, 18th Mediterranean Conference on Control and Automation, MED'10, Marrakech, Morocco, 2010. doi: 10.1109/MED.2010.5547672. [7] D. E. Chang, The method of controlled Lagrangians: Energy plus force shaping, SIAM J. Control Optim., 48 (2010), 4821-4845.  doi: 10.1137/070691310. [8] D. E. Chang, On the method of interconnection and damping assignment passivity-based control for the stabilization of mechanical systems, Regul. Chaotic Dyn., 19 (2014), 556-575.  doi: 10.1134/S1560354714050049. [9] D. E. Chang, Stabilizability of controlled Lagrangian systems of two degrees of freedom and one degree of under-actuation by the energy-shaping method, IEEE Trans. Automat. Control, 55 (2010), 1888-1893.  doi: 10.1109/TAC.2010.2049279. [10] D. E. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. A. Wollsey, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM Control Optim. Calc. Var., 8 (2002), 393-422.  doi: 10.1051/cocv:2002045. [11] B. Gharesifard, Stabilization of systems with one degree of underactuation with energy shaping: A geometric approach, SIAM J. Control Optim., 49 (2011), 1422-1434.  doi: 10.1137/09076698X. [12] H. Goldschmidt, Existence theorems for analytic linear partial differential equations, Ann. of Math. (2), 86 (1967), 246-270.  doi: 10.2307/1970689. [13] S. D. Grillo, L. M. Salomone and M. Zuccalli, Asymptotic stabilizability of underactuated Hamiltonian systems with two degrees of freedom, Russ. J. Nonlinear Dyn., 15 (2019), 309-326.  doi: 10.20537/nd190309. [14] S. D. Grillo, L. M. Salomone and M. Zuccalli, On the relationship between the energy shaping and the Lyapunov constraint based methods, J. Geom. Mech., 9 (2017), 459-486.  doi: 10.3934/jgm.2017018. [15] J. Hamberg, Controlled Lagrangians, symmetries and conditions for strong matching, IFAC Proceedings Volumes, 33 (2000), 57-62.  doi: 10.1016/S1474-6670(17)35547-7. [16] J. Hamberg, General matching conditions in the theory of controlled Lagrangians, Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, 1999. doi: 10.1109/CDC.1999.831306. [17] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. [18] A. D. Lewis, Potential energy shaping after kinetic energy shaping, Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, 2006. doi: 10.1109/CDC.2006.376885. [19] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5. [20] J. E. Marsden and T. S. Ratiu, Manifolds, Tensor Analysis and Applications, Springer-Verlag, New York, 2001. [21] J. G. Romero, A. Donaire and R. Ortega, Robust energy shaping control of mechanical systems, Systems Control Lett., 62 (2013), 770-780.  doi: 10.1016/j.sysconle.2013.05.011. [22] D. Zenkov, Matching and stabilization of linear mechanical systems, Proceedings of 2002 International Symposium of Mathematical Theory of Networks and Systems, South Bend, IN, 2002. Available from: http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=F5FAA6E2AB4998A2C00B8A5CDB0A461C?doi=10.1.1.12.7462&rep=rep1&type=pdf.
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