Delay dynamics of a levitating motor with two-limit control strategy

https://doi.org/10.1016/j.ijnonlinmec.2020.103645Get rights and content

Highlights

  • Analysis of DDE is performed in magnetic levitation context.

  • Simplified second order DDE is constructed from original sixth order system.

  • DDE model can predict the stability criteria for the original system.

Abstract

In a recent work (Shayak, 2019), I have proposed a new comparator-based control algorithm for a magnetically levitated motor. The rotor dynamics are governed by a sixth order nonlinear differential equation, whose stability analysis is treated as given. Here we consider this device from a dynamical systems viewpoint. We first present a simplified model which is a second order nonlinear delay differential equation. We then see that in this equation, a fixed point which is unstable in the absence of control gets converted to a small-amplitude stable limit cycle in its presence. Extrapolating from the simplified model, we find that insufficient damping can create an instability but it can be countered by increasing the inverter voltage, decreasing the inverter switching period, and relaxing the displacement tolerance value. Extensive simulation results show that all predictions made on the basis of the simplified model are applicable to the original system.

Introduction

Magnetic levitation is an age-old problem (the oldest reference [1] is a patent dating from 1912) but one with surprisingly few convincing solutions. Earnshaw’s theorem states that it is impossible to achieve stable spatial confinement of a time-independent charge or current configuration using time-independent electric and magnetic fields and materials of positive permittivity and permeability. This theorem is the fundamental obstacle in the path to development of a maglev system.

Theoretically the easiest solution to the problem is to use the ultimate negative-permeability material, the superconductor [2], [3], [4], which completely expels magnetic field from inside its bulk. Practically, materials superconduct only at temperatures of order −200 °C, and that makes this approach implausible except in spacecraft applications. A variant of superconducting levitation is the floating frog experiment (frog is diamagnetic) by Geim [5] which is impossible to implement on a larger scale.

A different solution to the Earnshaw problem uses eddy currents or induced currents. A regular conductor exposed to a high-frequency oscillating magnetic field acts like a superconductor and expels the field from its bulk, giving rise to levitation forces. Such configurations have been analysed in References [6], [7], [8]. There are several problems with the practical implementation of such a setup. One is that despite a good flux exclusion, eddy currents often give rise to a significant drag torque on a rotating levitator — the drag decays as the inverse of the rotation speed, which implies constant drag power. The second is that pumping the eddy currents through the conductor dissipates quite a lot of energy which must be supplied externally. Finally, it is often found that the restoring and damping forces experienced in an eddy current levitator are insufficient compared to the system size.

A new principle of stabilization was shown by the Levitron [9], [10], [11] – a spinning top which can float above a static magnetic block. The physics of Levitron has been explained as a “static equilibrium in a potential energy field arising dynamically from the adiabatic coupling of the [top’s] spin with the magnetic field of the base” [11]. Despite the many physical subtleties of the Levitron, it has so far not proven possible to extend the concept to practically useful sizes and scales.

A principle different from all of these is to use control. This is sometimes considered physically inelegant, and a brute force solution; nevertheless, it might well be the most practical solution to the Earnshaw problem. Moreover, control does not always imply a ham-handed approach to the system dynamics — an unsubtle philosophy often converts to bulky, complicated, and/or unreliable apparatus. On the other hand, an approach which is based on a solid dynamical understanding of the system can result in a minimum-interference apparatus which does what is required but nothing beyond that. There is no better demonstrator of this than the control algorithms used for induction motors.

The first algorithm invented to control an induction motor was field-oriented control (FOC), which was invented by Blaschke [12] and Hasse [13] in the early 1970s. This is (in retrospect) a crude approach that compensates for its lack of subtlety with its complexity. FOC measures the existing motor currents, calculates what will be required to achieve the required levels of magnetic field and torque, and uses an inverter to synthesize those new currents. Although FOC resulted in excellent performance, it was troublesome to implement and the apparatus often broke down. Research on improving FOC continued over the next few years but the adoption of induction motors in industry remained sporadic at best. Despite their drawbacks of presence of mechanical commutator and low power-to-weight ratio, dc motors remained popular on account of their tractable uncontrolled handling characteristics and simple control strategies.

The situation underwent a radical change in 1986 when Takahashi and Noguchi invented direct torque control (DTC) [14]. In one fell swoop, DTC was able to make the induction motor the mainstay of industry and railways. The revolution occurred because DTC is vastly simpler to implement than FOC. The inverter in DTC is a three-phase two-level voltage source — something which can only turn each phase ON and OFF. The control apparatus consists of a set of comparators which determine whether the magnetic field and the torque are less than, greater than or equal to their set values and a switching table which determines the required inverter state in terms of the comparator outputs.

We shall now look at some of the recent developments in the field of controlled magnetic levitators, which are also called active magnetic bearings if the levitating object rotates in one place instead of translating in space. Further, since we will be dealing with levitating (or bearingless) motors, let us narrow down our focus to those. Refs. [15], [16] have proposed a novel design of a permanent magnet motor with controlled axial force. A similar design may be found in Ref. [17]. Refs. [18], [19] on the other hand propose an architecture with radial control. The latter features a six-pole permanent magnet on the rotor and two sets of windings on the stator – a single phase six-pole winding and a separate three-phase four-pole one. A similar approach but with different polarities has been demonstrated in Ref. [20]. In Refs. [21], [22] the two separate windings have been integrated into one structure.

In all these works however, the control principle is basically the same and is very similar to FOC for the induction motor. In a recent work [23], I have proposed an equivalent of DTC for the levitating motor — like the original DTC it is supplied by a discrete inverter and uses comparators and a switching table to select the voltage vector at each instant. The details of the implementation of the strategy can be found from Ref. [23] – what that has no space for is the nonlinear dynamic model and analysis at the heart of the proposal. Accordingly, I shall elaborate this aspect in the present Article.

Section snippets

The proposed control algorithm

In this Section 1 will summarize the control algorithm proposed in Ref. [23]. The stator of the proposed device has two concentric windings – a dipolar winding to generate torque and a quadrupolar winding to achieve confinement. The dipolar rotor is not fixed to be coaxial with the stator (as in a conventional motor) but can move about inside the stator. We consider a two degree-of-freedom model and focus on confinement of the rotor in the plane perpendicular to the symmetry axis, and use the

A simplified model

To make progress into the analysis of ((1.1)-3) we construct a simplified model. Since a chopper-controlled dc motor [24] is a simplified version of a DTC-controlled induction motor [25], we consider a system where the currents follow dc-like dynamics and there is only one rotor displacement variable X. This reduces the vector dynamics of the actual system to a scalar model. Let us say there are two currents i1 and i2 which satisfy the differential equations i1+γ1i1=V1,i2+γ2i2=V2, where γ1

Simulation results

All simulations are based on the original system Eq. ((1.1)-3) and not on the simplified model. For the simulations I have chosen a switching period T=0.0003. At the start of each switching period, we have the initial conditions X (0), vx(0), Y (0), vy(0), Ks2(0), and the three indicator variables s4(0), sx(0) and sy(0). Basis these indicators we select the quadrupolar voltage vector V2 (Table 1). We then use ((1.1)-3) to obtain X (1), vx(1), Y (1), vy(1) and Ks2(1) at the end of the interval.

Conclusion

In Ref. [23] I have proposed a new control strategy for an active magnetic bearing, based on Takahashi and Noguchi’s direct torque control. It features measuring the displacement and comparing with tolerance values, determining a reference force and comparing with the actual force, and then triggering a switching table. There are no arbitrary current synthesizers, no PID controllers and no online coordinate transformations from synchronous to stator frame. Here, we saw how a second order delay

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

I would thank to thank the anonymous Reviewers of International Journal of Nonlinear Mechanics whose comments have resulted in substantial improvements to the quality of this manuscript.

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