Remarks on Energy Based Control of a Nonlinear, Underactuated, MIMO and Unstable Benchmark
In the last decade, energy based control theory has undergone a significant breakthrough in dealing with underactated mechanical systems with two successful and similar tools, controlled Lagrangians and controlled Hamiltanians (IDA-PBC). However, because of the complexity of these tools, successful case studies are lacking, in particular, MIMO cases. The seminal theoretical paper of controlled Lagrangians proposed by Bloch and his colleagues presented a benchmark example–a 4 d.o.f underactuated pendulum on a cart but a detailed and completed design is neglected. To compensate this ignorance, the note revisit their design idea by addressing explicit control functions for a similar device motivated by a vector thrust body hovering in the air. To the best of our knowledge, this system is the first MIMO, underactuated example that is stabilized by using energy based tools at the courtesy of the original design idea. Some observations are given based on computer simulation.
[1] A. Bloch, D. Chang, N. Leonard, and J. Marsden, "Controlled lagragians
and the stabilisation of mechanical systems ii: potential shaping," IEEE
transaction on automatic control, vol. 46, pp. 1556-1571, 2001.
[2] A. Bloch, P.Krishnaprasad, J. Marsden, and G. S. de Alvarez, "Stabilization
of rigid body dynamics by internal and external torques," Automatica,
vol. 28, pp. 745-756, 1992.
[3] A. Bloch, N. Leonard, and J. Marsden, "Stabilization of mechanical
systems using controlled lagrangians," in Proc. 36th IEEE Conference
of Decision and Control,, San Diego, CA, USA, 1997, pp. 2356-2361.
[4] "Matching and stabilization by the method of controlled lagrangians,"
in Proc. of the 37th IEEE Conference on Decision and
Control, Tampa, FL, USA, 1998, pp. 1446-1451.
[5] "Controlled lagragians and the stabilisation of mechanical systems
i:the first matching theorem," IEEE transaction on automatic control,
vol. 45, pp. 2253-2269, 2000.
[6] A. Woolsey and N. Leonard, "Stabilizing underwater vehicle motion using
internal rotors," Automatica, vol. 38, pp. 2053-2062, 2002.
[7] D. Auckly and L. Kapitanski, "On the λ-equations for matching control
laws," SIAM Journal on control and optimization, vol. 37, pp. 1372-1388,
2002.
[8] R. Ortega, A. Loria, P. J. Nicklasson, and H. Sira-Ramirez, Passivity-
based control of Euler-Lagrange systems. Springer-Verlag, 1998.
[9] A. V. D. Schaft, L2 gain and passivity techniques in nonlinear cntrol,
Communication & Control Engineering Series. Springer-Verlag, 2000.
[10] R. Ortega, W. Spong, F. Gomez-Estern, and G. Blankenstein, "Stabilization
of a class of underactuated mechanical systems via interconnection
and damping assignment," IEEE transaction on automatic control, vol. 47,
pp. 1218-1233, 2002.
[11] G. Blankenstein, R. Ortega, and A. V. D. Schaft, "The matching
conditions of controlled lagrangian and ida-passivity based control,"
International Journal of Control, vol. 75, pp. 645-665, 2002.
[12] D. Chang and J. Marsden, "The equivalence of of controlled lagrangian
and controlled hamiltonian systems," ESAIM: Control, Optimisation and
Calculus of Variations, vol. 8, pp. 393-422, 2002.
[13] C. Reddy, W. Whitacre, and C. Woolsey. "Controlled lagrangians with
gyroscopic forcing: an experimental application" In Proceedings of the
2004 American Control Conference, 511-516, Boston, Massachuseetts,
2004.
[14] F. G'omez-Estern and A. van der Schaft, "Physical damping in idapbc
controlled underactuated mechanical systems" In European Journal
on Control: Special Issue on Hamiltonian and Lagrangian Methods for
Nonlinear Control, 10:451-468, 2004.
[15] G. Liu, I. Mareels, and D. Neˇsi'c, "A note on the control of a spherical
inverted pendulum," in Proc. of 7th IFAC Symposium on Nonlinear
Control Systems, Pretoria, South Africa, pages 838-843, 2007.
[1] A. Bloch, D. Chang, N. Leonard, and J. Marsden, "Controlled lagragians
and the stabilisation of mechanical systems ii: potential shaping," IEEE
transaction on automatic control, vol. 46, pp. 1556-1571, 2001.
[2] A. Bloch, P.Krishnaprasad, J. Marsden, and G. S. de Alvarez, "Stabilization
of rigid body dynamics by internal and external torques," Automatica,
vol. 28, pp. 745-756, 1992.
[3] A. Bloch, N. Leonard, and J. Marsden, "Stabilization of mechanical
systems using controlled lagrangians," in Proc. 36th IEEE Conference
of Decision and Control,, San Diego, CA, USA, 1997, pp. 2356-2361.
[4] "Matching and stabilization by the method of controlled lagrangians,"
in Proc. of the 37th IEEE Conference on Decision and
Control, Tampa, FL, USA, 1998, pp. 1446-1451.
[5] "Controlled lagragians and the stabilisation of mechanical systems
i:the first matching theorem," IEEE transaction on automatic control,
vol. 45, pp. 2253-2269, 2000.
[6] A. Woolsey and N. Leonard, "Stabilizing underwater vehicle motion using
internal rotors," Automatica, vol. 38, pp. 2053-2062, 2002.
[7] D. Auckly and L. Kapitanski, "On the λ-equations for matching control
laws," SIAM Journal on control and optimization, vol. 37, pp. 1372-1388,
2002.
[8] R. Ortega, A. Loria, P. J. Nicklasson, and H. Sira-Ramirez, Passivity-
based control of Euler-Lagrange systems. Springer-Verlag, 1998.
[9] A. V. D. Schaft, L2 gain and passivity techniques in nonlinear cntrol,
Communication & Control Engineering Series. Springer-Verlag, 2000.
[10] R. Ortega, W. Spong, F. Gomez-Estern, and G. Blankenstein, "Stabilization
of a class of underactuated mechanical systems via interconnection
and damping assignment," IEEE transaction on automatic control, vol. 47,
pp. 1218-1233, 2002.
[11] G. Blankenstein, R. Ortega, and A. V. D. Schaft, "The matching
conditions of controlled lagrangian and ida-passivity based control,"
International Journal of Control, vol. 75, pp. 645-665, 2002.
[12] D. Chang and J. Marsden, "The equivalence of of controlled lagrangian
and controlled hamiltonian systems," ESAIM: Control, Optimisation and
Calculus of Variations, vol. 8, pp. 393-422, 2002.
[13] C. Reddy, W. Whitacre, and C. Woolsey. "Controlled lagrangians with
gyroscopic forcing: an experimental application" In Proceedings of the
2004 American Control Conference, 511-516, Boston, Massachuseetts,
2004.
[14] F. G'omez-Estern and A. van der Schaft, "Physical damping in idapbc
controlled underactuated mechanical systems" In European Journal
on Control: Special Issue on Hamiltonian and Lagrangian Methods for
Nonlinear Control, 10:451-468, 2004.
[15] G. Liu, I. Mareels, and D. Neˇsi'c, "A note on the control of a spherical
inverted pendulum," in Proc. of 7th IFAC Symposium on Nonlinear
Control Systems, Pretoria, South Africa, pages 838-843, 2007.
@article{"International Journal of Electrical, Electronic and Communication Sciences:61702", author = "Guangyu Liu", title = "Remarks on Energy Based Control of a Nonlinear, Underactuated, MIMO and Unstable Benchmark", abstract = "In the last decade, energy based control theory has undergone a significant breakthrough in dealing with underactated mechanical systems with two successful and similar tools, controlled Lagrangians and controlled Hamiltanians (IDA-PBC). However, because of the complexity of these tools, successful case studies are lacking, in particular, MIMO cases. The seminal theoretical paper of controlled Lagrangians proposed by Bloch and his colleagues presented a benchmark example–a 4 d.o.f underactuated pendulum on a cart but a detailed and completed design is neglected. To compensate this ignorance, the note revisit their design idea by addressing explicit control functions for a similar device motivated by a vector thrust body hovering in the air. To the best of our knowledge, this system is the first MIMO, underactuated example that is stabilized by using energy based tools at the courtesy of the original design idea. Some observations are given based on computer simulation.
", keywords = "Controlled Lagrangian, Energy Shaping, Spherical Inverted Pendulum, Controlled Hamiltonian.", volume = "2", number = "12", pages = "2859-8", }
