The matching conditions of controlled Lagrangians and IDA-passivity based control

被引:128
作者
Blankenstein, G [1 ]
Ortega, R
VanDer Schaft, AJ
机构
[1] Ecole Polytech Fed Lausanne, Dept Math, CH-1015 Lausanne, Switzerland
[2] SUPELEC, Signaux & Syst Lab, CNRS, F-91192 Gif Sur Yvette, France
[3] Univ Twente, Fac Math Sci, NL-7500 AE Enschede, Netherlands
关键词
D O I
10.1080/00207170210135939
中图分类号
TP [自动化技术、计算机技术];
学科分类号
0812 ;
摘要
This paper discusses the matching conditions resulting from the controlled Lagrangians method and the interconnection and damping assignment passivity based control (IDA-PBC) method. Both methods have been presented recently in the literature as means to stabilize a desired equilibrium point of an Euler-Lagrange, respectively Hamiltonian, system. In the context of mechanical systems with symmetry, the original controlled Lagrangians method is reviewed, and an interpretation of the matching assumptions in terms of the matching of kinetic and potential energy is given. Secondly, both methods are applied to the general class of underactuated mechanical systems and it is shown that the controlled Lagrangians method is contained in the IDA-PBC method. The Lambda-method as described in recent papers for the controlled Lagrangians method, transforming the matching conditions (a set of non-linear PDEs) into a set of linear PDEs, is discussed. The method is used to transform the matching conditions obtained in the IDA-PBC method into a set of quadratic and linear PDEs. Finally, the extra freedom obtained in the IDA-PBC method (with respect to the controlled Lagrangians method) is used to discuss the integrability of the closed-loop system. Explicit conditions are derived under which the closed-loop Hamiltonian system is integrable, leading to the introduction of gyroscopic terms.
引用
收藏
页码:645 / 665
页数:21
相关论文
共 24 条
[1]  
ANDREEV F, 2000, MATCHING LINEAR SYST
[2]  
[Anonymous], THEORIE MATRICES
[3]  
[Anonymous], P 38 IEEE C DEC CONT
[4]  
[Anonymous], CRM P LECT NOTES
[5]  
Arnold V. I., 1978, Mathematical methods of classical mechanics
[6]  
Auckly D, 2000, COMMUN PUR APPL MATH, V53, P354
[7]  
BLOCH A, 2001, UNPUB ESAIM CONTROL
[8]  
Bloch AM, 1998, IEEE DECIS CONTR P, P1446, DOI 10.1109/CDC.1998.758490
[9]  
Bloch AM, 1997, IEEE DECIS CONTR P, P2356, DOI 10.1109/CDC.1997.657135
[10]   Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping [J].
Bloch, AM ;
Chang, DE ;
Leonard, NE ;
Marsden, JE .
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, 2001, 46 (10) :1556-1571