Abstract
In this chapter, the non-identifier based adaptive current control problem is solved for electric synchronous machines.
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Notes
- 1.
The rows are linearly dependent, e.g. row 1 \(+\) row 2 = − row 3.
- 2.
Convention: The \(\alpha \)-axis of the stationary \((\alpha , \beta , 0)\)-reference frame is aligned with the a-axis of the three-phase (a, b, c)-reference frame.
- 3.
A matrix \(\varvec{M} \in \mathbb {R}^{n \times n}\), \(n \in \mathbb {N}\), is normal and orthogonal (or, simply, orthonormal) if and only if the following holds \(\varvec{M}^\top \varvec{M}=\varvec{M}\varvec{M}^\top \) and \(\varvec{M}^\top \varvec{M}=\varvec{I}_n\) (i.e. \(\varvec{M}^\top = \varvec{M}^{-1}\)), respectively (see [39, Def. 3.1.1]).
- 4.
In modern electrical drive systems, depending on the power rating, a relatively high switching frequency \(f_{\mathrm{sw}}\in [2.5, \, 20]\mathrm{kHz}\) is usually utilized to reduce the total harmonic distortion as far as possible (see [302, Sect. 8.4] or [153, Chaps. 5 and 6]).
- 5.
The abbreviations are as follows:Â IGBT = Insulated-gate bipolar transistor and MOSFET = metal-oxide-semiconductor field-effect transistor.
- 6.
Iron losses are neglected (for details see e.g. [303, Sect. 16.7.2]).
- 7.
For vectors \(\varvec{x}^{abc}=(x^a,\, x^b, x^c)^\top \in \mathbb {R}^3\) and \(\varvec{y}^{abc}=(y^a,\, y^b, y^c)^\top \in \mathbb {R}^3\), their cross or vector product is defined by \(\varvec{x}^{abc}\times \varvec{y}^{abc}:= {\tiny \begin{pmatrix} x^by^c- x^cy^b\\ x^cy^a- x^ay^c\\ x^ay^b- x^by^a\end{pmatrix}}\) [39, p. 89].
- 8.
- 9.
Unbounded funnel boundaries are admissible for unsaturated systems [120].
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Hackl, C.M. (2017). Current Control of Electric Synchronous Machines. In: Non-identifier Based Adaptive Control in Mechatronics. Lecture Notes in Control and Information Sciences, vol 466. Springer, Cham. https://doi.org/10.1007/978-3-319-55036-7_14
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DOI: https://doi.org/10.1007/978-3-319-55036-7_14
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