The Extended Λ-Method for Controlled Lagrangian Systems

Chang, Dong Eui

doi:10.3182/20050703-6-cz-1902.00754

Cited by 3 publications

(2 citation statements)

References 12 publications

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“…Several studies have been dedicated to derive the solution of these matching equations as in Gomez-Estern et al (2001), Acosta et al (2005), Viola et al (2007), and references therein. Moreover, some methods are also proposed in Hamberg (1999), Bloch et al (2000Bloch et al ( , 2001, Ortega et al (2002), Chang (2005), and Auckly and Kapitansky (2006). As identified from these works, there are plenty of difficulties while solving related PDEs; hence, the stabilization issue is still considered as a difficult problem for the underactuated case.…”

Section: Introductionmentioning

confidence: 99%

Stabilization of a class of underactuated Euler Lagrange system using an approximate model

Yıldız

Goren-Sumer

2021

Transactions of the Institute of Measurement and Control

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The energy shaping method, Controlled Lagrangian, is a well-known approach to stabilize the underactuated Euler Lagrange (EL) systems. In this approach, to construct a control rule, some nonlinear and nonhomogeneous partial differential equations (PDEs), which are called matching conditions, must be solved. In this paper, a method is proposed to obtain an approximate solution of these matching conditions for a class of underactuated EL systems. To develop this method, the potential energy matching condition is transformed to a set of linear PDEs using an approximation of inertia matrices. Hence, the assignable potential energy function and the controlled inertia matrix both are constructed as a common solution of these PDEs. Subsequently, the gyroscopic and dissipative forces are determined as the solution for kinetic energy matching condition. Conclusively, the control rule is constructed by adding energy shaping rule and additional dissipation injection to provide asymptotic stability. The stability analysis of the closed-loop system which used the control rule derived with the proposed method is also provided. To demonstrate the success of the proposed method, the stability problem of the inverted pendulum on a cart is considered.

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Section: Introductionmentioning

confidence: 99%

Stabilization of a class of underactuated Euler Lagrange system using an approximate model

Yıldız

Goren-Sumer

2021

Transactions of the Institute of Measurement and Control

View full text Add to dashboard Cite

show abstract

“…The energy shaping method for stabilization of mechanical systems has been widely used for the last twenty years both on the Lagrangian side and Hamiltonian side; Acosta et al [2005], Auckly et al [2000], Bloch et al [2001Bloch et al [ , 1997Bloch et al [ , 2000, Bloch and Marsden [1990], Chang [2005Chang [ , 2007a, Chang et al [2002], Fantoni et al [2000], Hamberg [2000], Ortega et al [2002], van der Schaft [1986], Woolsey et al [2004], Zenkov [2000]. The basic idea in this method can be summarized from the Lagrangian viewpoint as follows: given a mechanical system, we find a feedback control such that the closed-loop system can be represented by a new mechanical system connected with a dissipative force and a gyroscopic force, and the energy of the second mechanical system attains a non-degenerate minimum at an equilibrium of interest.…”

Section: Introductionmentioning

confidence: 99%