Simplified conditions for matching and for generalized matching in the theory of controlled Lagrangians

Hamberg, Johan

doi:10.1109/acc.2000.876957

Cited by 9 publications

(8 citation statements)

References 11 publications

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“…The CL method has been applied to various examples, such as, the cart multipendulum system (n-unactuated inverted pendulums on an actuated cart) [2,3], some inverted pendulums [6], the Furuta's pendulum [7], nonholonomic mechanical systems [15,16], and the rigid spacecraft with one actuated rotor [17]. On the IDA-PBC method side, there are the ballbeam system and inertia wheel pendulum [10], cart pendulum system [9,11], and vertical takeoff and landing (VTOL) aircraft [11].…”

Section: Introductionmentioning

confidence: 99%

Controller design for 2-DOF underactuated mechanical systems based on controlled Lagrangians and application to Acrobot control

2009

Front. Electr. Electron. Eng. China

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On the basis of controlled Lagrangians, a controller design is proposed for underactuated mechanical systems with two degrees of freedom. A new kinetic energy equation (K-equation) independent of the gyroscopic forces is found due to the use of their property. As a result, the necessary and sufficient matching condition comprises the new K-equation and the potential energy equation (P-equation) cascaded, the regular condition, and the explicit gyroscopic forces. Further, for two classes of input decoupled systems that cover the main benchmark examples, the new K-equation, respectively, degenerates from a quasilinear partial differential equation (PDE) into an ordinary differential equation (ODE) under some choice and into a homogeneous linear PDE with two kinds of explicit general solutions. Benefiting from one of the general solutions, the obtained smooth state feedback controller for the Acrobots is of a more general form. Specifically, a constant fixed in a related paper by the system parameters is converted into a controller parameter ranging over an open interval along with some new nonlinear terms involved. Unlike what is mentioned in the related paper, some categories of the Acrobots cannot be stabilized with the existing interconnection and damping assignment passivity based control (IDA-PBC) method. As a contribution, the system can be locally asymptotically stabilized by the selection of the new controller parameter except for only one special case.

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Section: Introductionmentioning

confidence: 99%

Controller design for 2-DOF underactuated mechanical systems based on controlled Lagrangians and application to Acrobot control

2009

Front. Electr. Electron. Eng. China

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“…Finally, we would like to remark that recently some results have been obtained in Hamberg (2000 b) and Zenkov et al (2000) extending the method of controlled Lagrangians to also include the matching and stabilization of Euler±Lagrange systems with (non-holonomic) contraints.…”

Section: Introductionmentioning

confidence: 99%

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

Blankenstein

Ortega²,

Schaft³

2002

International Journal of Control

163

151

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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 ¶-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.

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“…Proposition 3: The systems (1) and (2) match if and only if the matching conditions (12) hold. In that case, the state feedback control law is explicitly given by…”

Section: Propositionmentioning

confidence: 99%

“…Finally, we would like to remark that recently some results have been obtained in 12,13 extending the method of controlled Lagrangians to also include the matching and stabilization of Euler-Lagrange systems with (nonholonomic) constraints.…”

Section: Controlled Lagrangiansmentioning

confidence: 99%

Matching in the Method of Controlled Lagrangians and IDA-Passivity Based Control

Blankenstein

Schaft

2005

Nonlinear and Adaptive Control

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This paper reviews the method of controlled Lagrangians 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 system, respectively Hamiltonian system, by searching for a stabilizing structure preserving feedback law. The conditions under which two Euler-Lagrange or Hamiltonian systems are equivalent under feedback are called the matching conditions (consisting of a set of nonlinear PDEs).Both methods are applied to the general class of underactuated mechanical systems and it is shown that the IDA-PBC method contains the controlled Lagrangians method as a special case by choosing an appropriate closed-loop interconnection structure. Moreover, explicit conditions are derived under which the closed-loop Hamiltonian system is integrable, leading to the introduction of gyroscopic terms.The λ-method as introduced in recent papers for the controlled Lagrangians method transforms the matching conditions into a set of linear PDEs. In this paper the method is extended, transforming the matching conditions obtained in the IDA-PBC method into a set of quasi-linear and linear PDEs.

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Simplified conditions for matching and for generalized matching in the theory of controlled Lagrangians

Cited by 9 publications

References 11 publications

Controller design for 2-DOF underactuated mechanical systems based on controlled Lagrangians and application to Acrobot control

Controller design for 2-DOF underactuated mechanical systems based on controlled Lagrangians and application to Acrobot control

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

Matching in the Method of Controlled Lagrangians and IDA-Passivity Based Control

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