General matching conditions in the theory of controlled Lagrangians

Hamberg, Johan

doi:10.1109/cdc.1999.831306

Cited by 62 publications

(65 citation statements)

References 2 publications

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“…The work of [2], [22], and [23] studies the CL method from the point of view of matching Lagrangians defined in terms of general metric tensors. This has the advantage of generality and gives geometric insight into the problem, but it has the disadvantage that one is left with a rather general PDE to be solved in order to make the method effective in applications.…”

Section: B History and Related Literaturementioning

confidence: 99%

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Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

Bloch

Chang

Leonard

et al. 2001

IEEE Trans. Automat. Contr.

302

232

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Abstract-We extend the method of controlled Lagrangians (CL) to include potential shaping, which achieves complete state-space asymptotic stabilization of mechanical systems. The CL method deals with mechanical systems with symmetry and provides symmetry-preserving kinetic shaping and feedback-controlled dissipation for state-space stabilization in all but the symmetry variables. Potential shaping complements the kinetic shaping by breaking symmetry and stabilizing the remaining state variables. The approach also extends the method of controlled Lagrangians to include a class of mechanical systems without symmetry such as the inverted pendulum on a cart that travels along an incline.

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Section: B History and Related Literaturementioning

confidence: 99%

“…Then, there is an explicit feedback control such that becomes an asymptotically stable equilibrium. The control is given in (22) and (20) with parameters chosen to satisfy the following three conditions: 1) should be chosen to have a maximum at ; 2) ; 3)…”

Section: Asymptotic Stabilizationmentioning

confidence: 99%

Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

Bloch

Chang

Leonard

et al. 2001

IEEE Trans. Automat. Contr.

302

232

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“…This leads to interesting questions regarding the modified symplectic structures involved and we shall look at this in a future publication. The matching conditions derived in this paper are explicit; for more general, but less explicit conditions, see [2] and [17].…”

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confidence: 98%

Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem

Bloch

Leonard

Marsden

2000

IEEE Trans. Automat. Contr.

446

380

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Abstract-We develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled Lagrangians. The procedure involves making structured modifications to the Lagrangian for the uncontrolled system, thereby constructing the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system, where new terms in these equations are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods can be used to find control gains that yield closed-loop stability.In this paper we use kinetic shaping to preserve symmetry and only stabilize systems modulo the symmetry group. In the sequel to this paper (Part II), we extend the technique to include potential shaping and we achieve stabilization in the full phase space.The procedure is demonstrated for several underactuated balance problems, including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane.

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“…There are already some ad hoc approaches to this problem in the presence of additional structure. Indeed, the "kinetic shaping" techniques of [2] (see also [6]) may be seen as providing local equivalence to a desired form of the closed-loop system, remaining in the setting of the Levi-Civita connection. In coming to grips with the local equivalence problem, one will have to understand how the local invariants of an affine connection interact with the control vector fields Y.…”

Section: Future Directionsmentioning

confidence: 99%

The category of affine connection control systems

Lewis

Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)

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The category of affine connection control systems is one whose objects are control affine systems whose drift vector field is the geodesic spray of an affine connection, and whose control vector fields are vertical lifts to the tangent bundle of vector fields on configuration space. This class of system includes a large and important collection of mechanical systems. The morphisms (feedback transformations) in this category are investigated.

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General matching conditions in the theory of controlled Lagrangians

Cited by 62 publications

References 2 publications

Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem

The category of affine connection control systems

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