On SK1of curves and kahler differentials

This paper deals with flatness-based control of a salient permanent-magnet synchronous motor in the bond graph (BG) domain. It develops two main points. The first proposes and develops a new flat outputs identification procedure valid for multiple-input non-linear BG models without elements in derivative causality assignment. This procedure exploits a variational (tangent) BG model obtained by using Kähler differentials. The second deals with control design based on physical system decomposition into electrical, mechanical, and coupling submodels. Each loop of the decomposition tracks a reference for the local flat output of the corresponding subsystem. This decomposition enables the designing of a control block for each submodel by means of system inversion using the concept of bicausality. Then, the resulting blocks are concatenated in order to build the global controller. Finally, the global stability of the feedback system for both cases (known and unknown load torques) is tested and the control scheme is assessed through simulations in order to illustrate the performances of the method.

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Some Commutative Algebra

Asymptotic Differential Algebra and Model Theory of Transseries ◽

10.23943/princeton/9780691175423.003.0002 ◽

2017 ◽

Author(s):

Matthias Aschenbrenner ◽

Lou van den Dries ◽

Joris van der Hoeven

Keyword(s):

Commutative Algebra ◽

Principal Ideal ◽

Noetherian Rings ◽

Local Rings ◽

Zariski Topology ◽

Algebraic Differential Equations ◽

Field Extensions ◽

Notational Convention ◽

Kähler Differentials ◽

Integrally Closed

This chapter provides a background on commutative algebra and gives a self-contained proof of Johnson's Theorem 5.9.1 on regular solutions of systems of algebraic differential equations. It presents the facts on regular local rings and Kähler differentials needed for Theorem 5.9.1. It also recalls a common notational convention concerning a commutative ring R and an R-module M, with U and V as additive subgroups of R and M. Other topics include the Zariski topology, noetherian rings and spaces, rings and modules of finite length, integral extensions and integrally closed domains, Krull's Principal Ideal Theorem, differentials, and derivations on field extensions.

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