Lagrangian and Hamiltonian Methods for Nonlinear Control by Suguru Arimoto (auth.), F. Allgüwer, P. Fleming, P.

By Suguru Arimoto (auth.), F. Allgüwer, P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis, Francesco Bullo, Kenji Fujimoto (eds.)

From the contents

A Differential-Geometric procedure for Bernstein’s Degrees-of-Freedom Problem.- Nonsmooth Riemannian Optimization with purposes to Sphere Packing and Grasping.- Synchronization of Networked Lagrangian Systems.- An set of rules to Discretize One-Dimensional dispensed Port Hamiltonian Systems.- digital Lagrangian development strategy for Infinitedimensional platforms with Homotopy Operators.- Direct Discrete-Time layout for Sampled-Data Hamiltonian regulate Systems.- Kinematic repayment in Port-Hamiltonian Telemanipulation.- Interconnection and Damping project Passivity-Based keep watch over of a Four-Tank System.- in the direction of Power-based keep watch over options for a category of Nonlinear Mechanical Systems.- strength Shaping keep an eye on of Nonlinear structures: A Benchmark Example.- overall strength Shaping regulate of Mechanical structures: Simplifying the Matching Equations through Coordinate Changes.- Simultaneous Interconnection and Damping task Passivity–Based keep an eye on: useful Examples.

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Additional info for Lagrangian and Hamiltonian Methods for Nonlinear Control 2006: Proceedings from the 3rd IFAC Workshop, Nagoya, Japan, July 2006

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G. [10]. Let M denote an n-dimensional Riemannian manifold with Riemannian metric <, > and let df denote the differential of a smooth function f on M . In the sequel we will often write, as usual, Xf := dfX for the Lie derivative of f with respect to a smooth vector field X. A Riemannian or Levi-Civita connection ∇ on M assigns to each pair of vector fields X, Y on M a vector field ∇X Y such that for all smooth functions f on M : (i) (ii) (iii) (iv) (v) (X, Y ) → ∇X Y is bilinear in X, Y . ∇f X Y = f ∇X Y .

Lageman λ = Γ −1 τ, ij Γ := (trPAi PAj )m i,j=1 = γ m i,j=1 and τ = (tr(A1 ξ), · · · , tr(Am ξ)) . The associated Riemannian gradient is easily computable as follows. Theorem 4. Endow C with the Riemannian metric defined by < ·, · >P . The Riemannian gradient grad f (P ) of f on C is m PWP − P − γ ij (tr(Ai PWP) − bi )PAj P. i,j=1 Note that grad f (P ) coincides with the Newton direction with respect to the standard, constant metric < ξ, η >= tr (ξη) [12]. Using the strict convexity of the cost function one then derives the following global convergence result for a variant of the gradient method, which uses the straight lines P + tη as an approximation of the geodesics with respect to < ·, · >P ; [12].

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