Power-based control: Canonical coordinate transformations, integral and adaptive control

Automatica ◽  
2012 ◽  
Vol 48 (6) ◽  
pp. 1045-1056 ◽  
Author(s):  
Daniel A. Dirksz ◽  
Jacquelien M.A. Scherpen
Keyword(s):  
Adaptive Control ◽  
Coordinate Transformations ◽  
Canonical Coordinate

Poisson Brackets & Angular Momentum

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Generating Functions ◽  
First Integrals ◽  
Lagrangian Mechanics ◽  
Poisson Brackets ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Gauge Transformations ◽  
The Third ◽  
Point Transformations ◽  
Canonical Coordinate

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

Canonical Transformations, Entropy and Quantization

1987 ◽  
Vol 42 (4) ◽  
pp. 333-340 ◽  
Author(s):  
B. Bruhn
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Arbitrary Function ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Integrable Hamiltonian Systems ◽  
Eigenvalue Spectrum ◽  
Complex Phase ◽  
Algebraic Treatment ◽  
Canonical Coordinate

This paper considers various aspects of the canonical coordinate transformations in a complex phase space. The main result is given by two theorems which describe two special families of mappings between integrable Hamiltonian systems. The generating function of these transformations is determined by the entropy and a second arbitrary function which we take to be the energy function. For simple integrable systems an algebraic treatment based on the group properties of the canonical transformations is given to calculate the eigenvalue spectrum of the energy.

Coordinate Transformations and Logical Operations for Minimizing Conservativeness in Coupled Stability Criteria

1994 ◽  
Vol 116 (4) ◽  
pp. 643-649 ◽  
Author(s):  
J. E. Colgate
Keyword(s):  
Frequency Domain ◽  
Unit Circle ◽  
Stability Criterion ◽  
Coordinate Transformation ◽  
Coordinate Transformations ◽  
Stability Criteria ◽  
Small Gain Theorem ◽  
Logical Operations ◽  
Small Gain ◽  
Canonical Coordinate

Often it is desirable to guarantee that a manipulator will remain stable when contacting any member of some set of environments. Coupled stability criteria based on passivity may be used to provide such a guarantee, but may be arbitrarily conservative depending on the environment set. In this paper, two techniques for reducing conservativeness are introduced. The first is based on a canonical coordinate transformation which enables an environment set viewed in the frequency domain to be conformally mapped to the interior of the unit circle. A stability criterion is then derived via the small gain theorem. The second technique uses logical combinations of such criteria to reduce conservativeness further. Both techniques are illustrated with nontrivial examples.

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