An angular speed and position FLL-based estimator using linear Hall-effect sensors

This paper proposes using a frequency-locked loop-based detector to estimate rotational speed and angle position for an electric machine rotor shaft. The measurement system consists of arrays of permanent magnets fixed to the rotor shaft together with linear Hall-effect sensors attached to a fixed frame. Parametric uncertainties on the sensor assembly lead to significantly noisy signals, exhibiting unbalance and harmonic distortion. To accurately estimate rotational speed and angle, it is proposed to use a frequency-locked loop scheme based on a fourth-order harmonic oscillator (FOHO) to allow the processing of the symmetric components, thus dealing with the unbalance. The scheme also includes an adaptive law to reconstruct the fundamental frequency. Moreover, a harmonic compensation mechanism comprising parallel FOHOs is included; each FOHO is tuned at the spectral component under concern for its cancellation. The proposed algorithm delivers a clean estimate of the positive sequence fundamental component despite the disturbances at the signals provided by the Hall-effect sensor, which is used to reconstruct the rotation angle. The described approach could enhance low-cost sensing solutions in applications where position feedback is mandatory and sensorless control is impossible, not requiring special installation considerations.

Button-shaped sensor permanent magnets (PM) are placed on the rear end of the rotor shaft. In particular, it is convenient that the number of sensor PMs matches the number of motor magnets (N m ), where the number of pole pairs is defined as: and the pole pitch angle is defined as (2) To measure the axial flux of the sensor PMs array, the motor cover (fixed frame) holds a sensor board with N h linear HESs. The location of these sensors obeys the sensor pitch angle. For a two-sensor configuration, transducers are placed in quadrature; hence, Similarly, three-sensor setups use a balanced three-phase distribution, and thus where index k can exploit periodicity and fit the sensors in 154 the board without geometrical interference.

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Industrial applications feature multiple HES configura-     The depicted acquired signals show significant variability concerning the air gap g. As shown in Figure 4, if perfect sinusoidal waveforms are assumed, then their amplitude can be adjusted to a decaying exponential function of g, where and negative sequences will be present, making the proposed 213 two-phase modeling suitable for the problem at hand.

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It is noteworthy that systems comprising more than two 215 phases can also be mapped to the two-phases fixed frame.

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To this end, we appealed to the three-phase voltage classical (αβ) using the Clarke's transformation T : Notice that vector notation is being used, where vectors and where the superscript sign stands for the rotation direction, 246 and thus it is used to identify the sequence. 247 First, consider that the signal v αβ involves only one operating frequency coming from the operation angular speed, then a model can be derived for v αβ considering that (also referred to as phasors), and θ is the angle. The model is 250 derived by taking the time derivative of v αβ as follows: where ω =θ, J is a skew symmetric matrix, i.e., J = −J 252 and J J = I, where I is the 2 × 2 identity matrix.

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An auxiliary variable representing the difference whose time derivative is given, in its turn, by which is an equation that completes the model description.

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Summarizing, the dynamic model of the unbalanced signal at one specific angular speed ω can now be written in its complete form as follows: which represents a fourth-order harmonic oscillator (FOHO).

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Notice that both state variables v αβ and ϕ αβ hold a direct 261 relationship with the positive and negative sequences given Moreover, since the matrix in (13) is invertible, then the pos-264 itive and negative sequences can be reconstructed as follows:

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Based on the FOHO structure of the above-described model in (11) and (12), we propose the following FOHO-based adaptive estimator (FOHO-AE) to approximate the fundamental component of the incoming voltage signal: proposed observer estimates allv α ,v β ,φ α , andφ β at once.

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The estimation law forω follows from the Lyapunov's approach. For this, first, we obtain the so called error model to account for the deviation of states' trajectories (the estimates) toward the signals' actual values, in other words, to account for the errors that must be approaching zero. The error model is obtained after expressing system (15)-(16) in terms of the Next, according to Lyapunov's approach, we proposed the 274 following quadratic storage function: whose time derivative is given bẏ where we used the propertyṽ αβ Jφ αβ = −φ αβ Jṽ αβ . Furthermore, it can be seen that which is referred to as the fundamental frequency estima- If the measured signal contains additional components in 296 the frequency spectrum other than the fundamental, then the 297 model can be extended as follows: where H is a set comprising the indexes of all the har-299 monic components of the measured signal, e.g., H = 300 {1, 3, 2/5, 3/5, · · · }, that is, the spectral components of in-

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A model for the k-th harmonic component can be obtained as an extension of (11)-(12) by including the index k as follows:v which is referred to as the FOHO k . Finally, by extending (14), the positive and negative sequences of the k-th harmonic component can be derived as follows: Analogously, we propose to extend (15)-(16) to estimate the kω voltage component by including the k-th index as follows: v αβ,k = kωJφ αβ,k + γ kṽαβ,k , k ∈ H, which is referred to as FOHO-AE k .

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Notice that the estimatedω, which is common to every 307 FOHO-AE k , is obtained by the FFE. However, as described is to estimate the harmonic components under concern, add 316 them, and then subtract their sum from the measured signal.

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The effect is equivalent to clean the measured v αβ so as where an amount of π is added to getθ e ∈ [0, 2π].

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Remark 1. Notice that,ω e =θ e must match the estimated 337 frequencyω obtained in the FFE.

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The stability analysis above presented yielded the conditions on the control parameters γ 1 , λ > 0 to preserve stability; however, it is still necessary to tune the control parameters to guarantee a preferred dynamic performance of the proposed estimator. Therefore, for a first tuning approach, we propose to consider a linearization of the system model, assuming an ideal operation of the HESs, which is equivalent to consider a balanced and uncontaminated measured voltage. In other words, ϕ αβ = v αβ = v + αβ,1 , where both amplitude and frequency are constant. To simplify the notation, the harmonic order and the sequence superscripts are omitted in what follows. Based on these considerations, the observer can be rewritten asv The error model, i.e., the model in terms of the increments, is given bẏ since ω is assumed constant.

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The error model can now be transformed to yield a more familiar structure by considering the following coordinates transformation: Out of which, the above error model is transformed tȯ where || · || 2 stands for the squared amplitude.

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Out of these, we propose the following tuning rules which These explicit tuning rules, which respond to such a perfor-361 mance criterion, represent, however, a first rough approxima-362 tion only. Therefore, they require further fine-tuning, which 363 is usually performed by trial-and-error. The above tuning guidelines focus on the frequency estimation dynamics only. Therefore, to complement the tuning process, the behavior of the FOHO-AE k has to be further analyzed as described next. Once again, if we consider only the positive sequence component, then the FOHO-AE k can be rewritten using complex scalar terms as follows:

C. TEMPERATURE VARIATION ANALYSIS
Since (ṽ α,k + jṽ β,k ) = (v α,k + jv β,k ) − (v α,k + jv β,k ), then 387 the system can be rewritten in the Laplace domain as which is a band-pass filter (BPF) centered atω, whose only 389 tunable parameter γ k determines its bandwidth (BW) and the 390 cutoff frequency (where the stopband starts). It is essential to 391 notice thatω must be constant to enable the transfer function 392 representation, which entails an additional assumption to the 393 previous analysis.

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Now, after substitution of s → jΩ, the magnitude of (50) 395 is given by therefore, γ k can be set to satisfy a desired BW. Solving for 397 Ω in the previous expression yields which leads to a frequency interval [Ω 1 , Ω 2 ] where the mag-399 nitude in the passband overpasses M . Notice that high values 400 of γ k enlarge the bandwidth of (50), while it becomes more 401 "frequency selective" for smaller γ k .

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The experimental setup in Section V was used to verify the AE 1 to estimate the frequency, whose γ 1 is more significant.

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As γ k imposes a response speed/selectivity trade-off, it is 504 necessary to tune it according to the intended application.

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Notice that the estimation of ω em resulted from the back-506 EMF signals υ abc at the motor's stator, which were barely 507 disturbed as shown in Figure 11. Besides, v αβ was measured 508 from the 2-HESs configuration, exhibiting large unbalancing 509 and subharmonic distortion. Figure 12 shows that, despite  processing method, while the calibration process mentioned 524 in Section II was avoided. The resulting signals were largely 525 disturbed, as shown in Figure 13, where it is possible to 526 directly compare υ abc and v abc as three HESs were used.

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Signals v abc were first mapped to v αβ using (6) as shown 528 in Figure 13. This process not only highlights the ability of 529 the FLL-HES to manage heavily distorted input signals, but 530 also outlines it as an enabler of sensors re-purposing. 531 Figure 15 shows that, despite exhibiting larger distur-