Abstract
The aim of this work is to present a method for tuning the parameters of PD controller under the influences of the uncertainties, in order to stabilize the position of a rotor supported by active magnetic bearings (AMBs). The uncertainties are relative to mass, transverse and polar moment of inertia of the rotor. The introduction of the uncertainties is due to an incomplete modeled dynamic of the system or in the case the system being subjected to a parametric variation. The presence of the uncertainties produces a set of differences among the values of the output. Poles displacement method is used to reach the asymptotically stability condition characterized by a periodic oscillation during the transient response as a consequence of the impulse input. In this way we carried out some particular condition under graphical representation which helps making a prevision when the phenomena of instability occurs. In the present approach the poles displacement is obtained by imposing respectively the condition on the real part, which must be negative, and in the discriminant of a second degree equation, which must be less than zero, both depend on the uncertainties and the angular speed of the rotor. All calculations are performed through a 4-axis AMB rigid rotor to validate the PD controller method rule introduced in this work.
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Abbreviations
- Ω:
-
Angular velocity
- f mag_to_g (t):
-
Vector of force components acting on center of mass of shaft
- i c (t):
-
Vector of current components
- q g (t):
-
Vector of center of mass displacements
- q b (t):
-
Vector of bearing section displacements
- q sensor (t):
-
Vector of displacement captured by sensors
- x(t) :
-
State vector
- y(t) :
-
Output vector
- A:
-
Dynamic matrix
- B:
-
Input-state matrix
- C:
-
State-output matrix
- B Θmag :
-
Transformation matrix of magnetic bearing force
- B mag :
-
Transformation coordinates matrix of magnetic bearing force on center of mass of rotor
- B Θdisp :
-
Transformation matrix of magnetic bearing Section’s displacement
- B sensor :
-
Transformation coordinates of displacements of sensor and center of mass of shaft
- G:
-
Gyroscopic matrix
- K I :
-
Current gains matrix
- K s :
-
Displacement gains matrix
- M:
-
Mass matrix
- P m :
-
Maximum percentage of mass uncertainty
- \( P_{{I_{T} }} \) :
-
Maximum percentage of transverse inertia uncertainty
- \( P_{{I_{P} }} \) :
-
Maximum percentage of polar inertia uncertainty
- Δm :
-
Normalized mass uncertainty
- ΔG :
-
Normalized polar inertia uncertainty
- α:
-
Slope of magnets
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Barbaraci , G., Virzi’ Mariotti, G. (2013). Influence of Uncertainties on PD Tuning. In: Dobre, G. (eds) Power Transmissions. Mechanisms and Machine Science, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6558-0_15
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DOI: https://doi.org/10.1007/978-94-007-6558-0_15
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