Online Adaptation of Two-Parameter Inverter Model in Sensorless Motor Drives

This article designs parameter adaptation algorithms for online simultaneous identification of a two-parameter sigmoid inverter model for compensating inverter nonlinearity to reduce the voltage error in flux estimation for a position sensorless motor drive. The inverter model has two parameters, <inline-formula><tex-math notation="LaTeX">$a_2$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$a_3$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$a_2$</tex-math></inline-formula> is “plateau voltage” and <inline-formula><tex-math notation="LaTeX">$a_3$</tex-math></inline-formula> is a shape parameter that mainly accounts for the stray capacitor effect. Parameter <inline-formula><tex-math notation="LaTeX">$a_3$</tex-math></inline-formula> is identified by the <inline-formula><tex-math notation="LaTeX">$(6k\pm 1)$</tex-math></inline-formula>th order harmonics in measured current. Parameter <inline-formula><tex-math notation="LaTeX">$a_2$</tex-math></inline-formula> is identified by the amplitude mismatch of the estimated active flux. It is found that the classic linear flux estimator, i.e., the hybrid of voltage model and current model, cannot be used for <inline-formula><tex-math notation="LaTeX">$a_2$</tex-math></inline-formula> identification. This article proposes to use a saturation function based nonlinear flux estimator to build an effective indicator for <inline-formula><tex-math notation="LaTeX">$a_2$</tex-math></inline-formula> error. The coupled identifiability of the two parameters is revealed and analyzed, which was not seen in literature. The concept of the low current region where the two-way coupling between <inline-formula><tex-math notation="LaTeX">$a_2$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$a_3$</tex-math></inline-formula> occurs is established. In theory, it is suggested to stop the inverter identification in the low current region. However, the experimental results in which dc bus voltage variation and load change are imposed have shown the effectiveness of the proposed online inverter identification and compensation method, even in low current region.


I. INTRODUCTION
T HE voltage source inverter along with the pulsewidth modulation (PWM) plays a significant role in commercialized adjustable speed motor drives. Fig. 1 shows a typical three-phase three-wire inverter topology with three half-bridges. Given  fact that inverter terminal voltage sensors are absent in most motor drives, when the motor drives are operating in position sensorless mode, the commanded controller voltage u * xg will be used instead of the actual terminal voltage u xg for state (e.g., active flux [1]) observation, where g denotes the center of the dc bus capacitor and x is the phase name placeholder, x = a, b, c.
In practice, nonideal properties of the power switches, i.e., dead-time, turn-ON/OFF time, conduction voltage drop, and stray capacitor, will result in an undesired inverter voltage drop D x (i x ) as a nonlinear function of phase current i x ; thus, the inverter's actual terminal voltage u xg is which motivates the need of a feedforward compensation term D x to cancel the inverter voltage drop D x , and, hence, the commanded inverter voltage u * * xg should be which results in the compensation error as follows: The voltage errorD x influences the performance of model-based position sensorless drive in at least two aspects. First, the fundamental component inD x will cause errors in both the amplitude and angle of the estimated active flux. Second, the (6k ± 1)th order harmonics inD x will cause fluctuation in both position and speed estimation. In addition, deteriorated performance due toD x is also observed in magnetic saliency based sensorless drives [2]- [4].
The compensation voltageD x can be obtained using various methods, which can be classified into datasheet model [2], offline measurement, and online compensation. The offline measurement is often executed at motor standstill, and the inverter voltage-current characteristics, i.e., the U-I curve, are recorded for lookup table (LUT) or curve fitting purposes [3], [5], [6]. Particularly in [3], both current amplitude and angle dependencies ofD x are measured and stored as a 2-D LUT. Besides, it is also possible to obtain the U-I curve when motor is running, e.g., at 16.67-Hz stator frequency, using repetitive control, which results in a U-I curve that additionally includes the hysteresis effect [7,Fig. 14].
As a matter of fact, inverter characteristics depend not only on load current but also on temperature [8]. Moreover, the dc bus voltage is not always stiff, especially for applications involving batteries. Even though the dc bus voltage can be measured, there is a nonlinear mapping between U-I curve and dc bus voltage, especially at low currents. The above facts motivate the need of online compensation.
Online compensation methods can be classified into invasive methods and noninvasive methods. Invasive methods involve additional excitation to the motor. In a rotating carrier signal injection-based sensorless drive, voltage error's negative effect on the position estimation can be detected from the positivesequence carrier current [4]. In [9], the dead-time is identified online by actively switching between continuous PWM and discontinuous PWM.
The noninvasive methods are further divided into invertermodel-free methods [10]- [13] and online parameter adaptation. For example, the inverter voltage drop D x can be reproduced by a time-delay control based disturbance observer [10] without using any inverter model. The disturbance observer method is reported to have deteriorated performance at higher speeds [11], and it relies on measured speed for calculating electromotive force (EMF). Sensorless inverter-model-free compensation can be achieved through iterative learning [12], given the fact that the harmonics in estimated d-axis EMF are periodic. In [13], the compensation based on single noise canceler is designed to minimize the measured sixth-order current harmonics caused by imperfect compensation. However, the inverter-model-free property means the compensation could be exact only at steady state, and there will be undesired transients in compensation voltage whenever motor operating condition changes.
In order to achieve inverter-model-based online compensation in sensorless drive, online parameter adaptation is proposed, for which an inverter model is indispensable. 1) In [14], [17], and [18], the square waveform compensation voltageD x (sign(i x ); a 2 ) depends on current polarity and is modeled with a single plateau voltage parameter a 2 . The plateau voltage a 2 can be extracted from the commanded d-axis voltage that regulates the d-axis current to zero against the disturbance voltage caused by the inverter, if the d-axis position is available [17], [18]. This idea is later modified and tested in a sensorless drive [14]. 2) In [15], a saturation function modelD x (i x ; a 2 , k S ) is used, where the slope k S is assumed known, and a 2 is identified online by applying recursive least square method to the q-axis voltage equation. 3) A trapezoidal voltage modelD x (θ x ; a 2 , θ t ) is proposed by Park and Sul [16], which is mapped to the current phasor angle θ x for compensating online the inverter voltage drop. This model depends on two parameters, i.e., the ramp region angle θ t and the plateau voltage a 2 . By assuming a 2 is known, the θ t is updated to minimize the sixth-order current harmonics [16] or the sum of 6th-and 12th-order harmonics [19] in a synchronous dq-frame. The key take-away here is that the inverter can be modeled with only two parameters, i.e., plateau voltage a 2 and shape parameter, e.g., θ t or k S . The plateau voltage parameter a 2 is equal to 3 V with the distortion voltage V defined in [20], and the shape parameter mainly accounts for effect of stray capacitor [21, Section 7.1.3], [22]. Table I summarizes the differences among different online parameter adaptation methods. The method in [14] relies solely on plateau voltage a 2 , but the current polarity-based compensation voltage is prone to suffering large voltage error when wrong polarity is detected, especially for motor with small inductance (e.g., high speed motor). In literature, only the researches in [16] and [19] have identified the shape parameter, but it has two main issues that need improvement. First, there exists undesired transient process in θ t whenever there is a change in current vector amplitude. This phenomenon implies that the optimal value of θ t is a function of current amplitude and is closely related to the fact that the shape of trapezoidal voltage is solely determined by θ t regardless of current amplitude values. Second, the fundamental voltage error is a function of θ t . That is, as θ t keeps increasing, there will be more and more loss in the fundamental component of the trapezoidal voltage, as shown in [16,Fig. 8].
Motivated by the two issues regarding the trapezoidal voltage method [16], this article proposes 1) to use current value based inverter modelD x (i x ) instead of the current phasor angle based inverter modelD x (θ x ) and 2) to identify online a 2 for compensating fundamental voltage error. Conventionally, a 2 is calculated from voltage equation and the current derivative must be properly dealt with. For example, [15] uses the q-axis voltage equation as an identification model so that the q-axis current derivative can be neglected, while the time-delay approach [10] uses the αβ-frame voltage equations to calculate the disturbance voltage so that the current derivative needs to be approximated with numerical differentiation plus low-pass filter. This article, however, integrates the voltage equation and proposes to use a dedicated adaptive nonlinear flux estimator for online identification of a 2 . Compared to [15], our proposed method does not rely on the knowledge of speed and does not assume the shape parameter is known. Compared to [10], our proposed method does not need to implement pure differentiation.
Aside from the classification-based literature review above, it should be pointed out that the inverter nonlinearity compensation is really hardware-related. For example, some inverter shows very steep change in D x (i.e., very large dD x di x ) near i x = 0 (see, e.g., [23]); and in this case, extra care must be taken to obtain the correct current polarity, for example, by instantaneous back  [23] or by current prediction at switching instant [21,Section 7.1.4]. This article, on the other hand, studies the general problem of online adaptation of inverter model parameters.
The contributions of this article are 1) to analyze the parameter identifiability of an inverter model with respect to motor operating conditions, 2) to analyze the coupled identifiability for the two inverter parameters, and 3) to propose a complete parameter adaptation scheme using coherent demodulation (CD) and a modified saturation-based flux estimator.

II. PROPOSED ONLINE COMPENSATION METHOD
Letˆand * denote the estimated and commanded value, respectively. Assuming the current control error i x − i * x can be neglected, the proposed current value dependent compensation voltagê is equipped with two online parameter adaptation algorithms (PAAs) for updating the plateau voltage estimateâ 2 and the sigmoid shape parameter estimateâ 3 . The a 2 -PAA is driven by estimated flux amplitude error where s = d dt is differential operator; ψ 2 = [ψ α2 , ψ β2 ] T ∈ R 2 is the output of an active flux estimator; B is defined as the dc bias in the estimated active flux modulusK Active ψ 2 with respect to the motor's active flux parameter K Active ; the motor operating mode variable m equals 1 for motoring and equals −1 for regenerating; γ a2 is adaptation gain; τ ψ2 is the time constant of the low-pass filter.
The a 3 -PAA is driven by current harmonics (cf., [16] and [19]) where I 6 , I 12 , and I 18 denote the amplitudes of the 6th-, 12th-, and 18th-order harmonic current that are obtained from the CD in (6b) with h = 6, 12, and 18; w 6 , w 12 , and w 18 are constant weights; (6c) shows that the sum of harmonic currents I Σh is obtained by transforming αβ-frame current i = [i α , i β ] T into the direct axis defined by the phase a current's commanded phasor angle θ * a [19]; (6d) shows that θ * a depends on the dq-frame current commands i * d , i * q , and ψ 2 's angle:θ d arctan2(ψ β2 , ψ α2 ); γ a3 is the adaptation gain; and τ cd is the time constant of the low-pass filter.
A. Two-Parameter Identifiability of a 2 and a 3 The desired system behaviors due to parameter mismatch are that the fundamental voltage error is only caused byâ 2 error and that the (6k ± 1)th-order harmonic voltage error is solely due toâ 3 error. Unfortunately, the identifiability of the two parameters a 2 and a 3 is coupled. As a result, the adaptation gains γ a2 and γ a3 should be carefully designed, and both PAAs for a 2 and a 3 should be suspended when commanded phase current amplitude I * x is too low or more specifically when I * x < 6/â 3 . 1) Motivation for a 3 -Identifiability: The identifiability of shape parameter a 3 depends on the harmonics in the measured I Σh . On the one hand, the identifiability of the parameter a 3 relies on the sinusoidal back EMF assumption, in a sense that the (6k±1)th-order harmonics detected in measured current are solely due to inverter voltage error. On the other hand, the a 3 -identifiability will become weak during low current region (LCR) because the compensation voltage has no (6k±1)thorder harmonics anymore when I * x is too low. The time-domain shape of the phase compensation voltageD x (i * x (t);â 2 ,â 3 ) depends on both its shape parameterâ 3 and the commanded load current i * x (t). As shown in Fig. 2(a), when I * x is large,D x in time domain is rich in harmonic contents so that error inâ 3 causes remarkable current amplitude ripple and a 3 can be identified. On the other hand, when I * x is very low in Fig. 2(b),D x in time domain looks like a pure sinusoidal so that the error inâ 3 would cause very limited 6kth-order harmonics in the measured , plotted against the normalized current peak valueâ 3 I * x . (a) Fundamental and (b) (6k ± 1)-th harmonic components. Note that 1.21 V is the maximal fundamental voltage that D x (∞; 1,â 3 ) can provide, and the fundamental component ofD x being equal to 95% of its maximum always corresponds toâ 3 I * x = 6.
I Σh , and, as a result, the PAA for a 3 should cease because the identifiability of a 3 via current harmonics is lost.
The problem now becomes how we can decide at what current level the PAA for a 3 should stop, and how we can generally decide this current level for any inverter. The answer to this problem is trivial (i.e., I * x > 0) if only a 3 is being identified. However, we are going to propose that the current level for stoppingâ 3 adaptation should take into account the coupled identifiability between a 2 and a 3 because whenâ 2 is erroneous, a 3 cannot be identified.
2) a 2 -Identifiability: The identifiability of the plateau voltage a 2 depends on the fact that using mismatchedâ 2 will lead to fundamental voltage error inD x [see (3)] in state observation, and this fundamental voltage error can be revealed by calculating flux amplitude error B in (5b).
3) Coupled Identifiability: Recall thatâ 2 serves as a scaling factor in the two-parameter inverter model (4), which implies thatâ 2 error would cause both fundamental and (6k ± 1)thorder harmonic voltage error. The effect ofâ 2 can be studied by plotting the harmonic contents ofD x (i * x (t);â 2 ,â 3 ) at different a 2 values, i.e.,â 2 = 1, 2, 3 V, as shown in Fig. 3, where the horizontal axis of I * x is normalized by a factor of a 3 . This normalization allows our analysis to be generally valid for any inverter and implies that the effect ofâ 3 and I * x onD x is equivalent.
From Fig. 3(a), it is learned that the fundamental component voltage is simply scaled byâ 2 . From Fig. 3(b), it is seen that the ratio of harmonic to fundamental does not depend onâ 2 at all, as the three traces at differentâ 2 values of 1, 2, 3 V overlap for each harmonic. In summary,â 2 acts as a scaling factor, andâ 2 error would result in voltage errorD x at all harmonics; while a change inâ 3 will mainly cause a change in harmonics contents inD x ifâ 3 I * x is large enough. As a result, bothâ 2 error andâ 3 error could lead to harmonics in the measured I Σh , which means that the two-parameter identifiability is coupled in terms of current harmonics. When the proposed PAAs are implemented in practice, erroneousâ 2 would cause divergedâ 3 results.
To enable decoupled identification, the convergence ofâ 2 is of higher priority, and γ a2 should be selected to be relatively large. Afterâ 2 fast converges, the convergence ofâ 3 is then guaranteed by its single-parameter identifiability.

4) Loss of a 2 -Identifiability in Lowâ 3 I *
x Region: Unfortunately, the decoupled identification is not effective for all working conditions. According to Fig. 3(a), for a fixedâ 2 value, the fundamental component inD x (t) drastically decreases aŝ a 3 I * x reduces below 6 and is equal to 95% × 1.21â 2 when a 3 I * x = 6. It is desired that the fundamental component inD x (t) always equals 1.21â 2 for anyâ 3 I * x . However, whenâ 3 I * x < 6 (i.e., when current is low), erroneousâ 3 will lead to biasedâ 2 results. This is because during lowâ 3 I * x region, erroneousâ 3 is also causing fundamental voltage error inD x .
As a result, during lowâ 3 I * x region, bothâ 2 error andâ 3 error could lead to fundamental error inD x for state observation, which means the two-parameter identifiability becomes coupled in terms of the flux amplitude error (or fundamental voltage error). Since we cannot differentiate the fundamental voltage error contribution betweenâ 2 andâ 3 , the PAAs should stop whenâ 3 I * x < 6.

B. Main Proposition
Taking the two-parameter identifiability into account, this article proposes to stop both PAAs when the normalized commanded current peak valueâ 3 I * x is less than 6, which corresponds to the situation where the fundamental component of D x (t) has reduced down to 95% due to reduction inâ 3 I * x . Main Proposition: The PAAs (5a) and (6a) should be implemented with their adaptation gains satisfying the following requirements: In summary, the two-parameter identifiability is revealed in terms of the single-parameter identifiability and the decoupled identifiability for each parameter as follows. The a 2 parameter identifiability is understood by the following.
1) The fact thatâ 2 error will cause fundamental error inD x that leads to flux amplitude error B.
2) The assumption that currentâ 3 I * x is large enough such that there is less than 5% reduction in fundamental component ofD x (t) at any fixedâ 2 value, as is indicated in Fig. 3(a). This imposes (7b). The a 3 parameter identifiability is understood by the following.
1) The fact that when currentâ 3 I * x is large enough,â 3 error will cause harmonics error inD x that will lead to remarkable harmonics in the measured I Σh (6c); but when currentâ 3 I * x becomes lower and lower, the harmonics contents gradually vanish, as shown in Fig. 3(b).
2) The assumption thatâ 2 has almost converged such that the harmonics detected in the measured I Σh are mainly due toâ 3 error rather thanâ 2 error. This imposes (7a). Note that from Fig. 3(b), (7c) can be relaxed to a value lower than 6 if only a 3 is being identified. However, since

III. NONLINEAR ACTIVE FLUX ESTIMATOR
The PAA (5a) for a 2 relies on a flux estimator that can convert voltage errorD x into nonzero dc bias B in ψ 2 . According to our study, the classical linear flux estimator (e.g., the voltage model and current model fusion method [24]) fails to produce nonzero dc bias B (which implies that a 2 cannot be identified from B) when the drive is regenerating (m = −1) andâ 2 < a 2 in Fig. 4(a) or when the drive is motoring (m = 1) andâ 2 > a 2 in Fig. 4(b). This undesired B = 0 phenomenon is due to the proportional-integral (PI) correction terms used in the flux estimator. The PI correction is always active and is forcing the amplitude mismatch between voltage model and current model to be zero, which is to blame for the B = 0 phenomenon.
As an alternative, we propose to use a nonlinear flux estimator whose saturation correction action is not always triggered. In particular, the saturation function based flux estimator that originates in [25] is modified for producing a nonzero B. The key property is that the nonlinear flux estimator behaves as a pure integrator as long as the flux estimate component ψ α2 (or ψ β2 ) is within the range [− , ].

A. Saturation Function Based Flux Estimator
A saturation function Sat(·) is added to the voltage model active flux estimator that is corrected by an offset voltage estimatê where u * = [u * α , u * β ] T is αβ-frame voltage command, i = [i α , i β ] T is measured αβ-frame current, R is stator resistance, L q is q-axis inductance, and Sat([x α , x β ] T ) limits its input vector's components x α , x β within the range [− , ], i.e.,

Sat
x Ideally, the limit is set to K Active . To understand the working principles of (8), consider an example scenario, where the α-axis offset voltage errorũ α,offset = u α,offset −û α,offset exists and is negative. After integration, the negativeũ α,offset will lower the entire waveform of ψ α2 , such that the lower bound (− ) is reached, as shown in Fig. 5. From Fig. 5, t 0 and t 2 are used to denote the time instants of the flux estimate zero-crossings, and the α-axis flux estimate extrema are defined by which will be used to build the offset voltage estimateû offset to close a feedback loop that eliminates the unknown dc bias u offset in the calculated EMF.

B. Proposed Exact Offset Voltage Calculation Method
This article proposes to exploit the "saturation time" to directly compute the value of offset voltage error. Graphical definitions of the saturation time concept are presented in Fig. 5, where t α,sat,min denotes the time duration when ψ α2 reaches the lower bound (− ) within [t 0 , t 2 ]. Since in Fig. 5, only the lower bound (− ) is reached; so the lower bound saturation time t α,sat,min = 0 and the upper bound saturation time t α,sat,max = 0. With α-axis saturation times available, the α-axis offset voltage error can be directly calculated as which means that "it takes (t 2 − t 0 − t α,sat,min − t α,sat,max ) [s] for the uncompensated offset voltageũ α,offset [V] to contribute to a flux dc bias of 1 2 (ψ α2,min + ψ α2,max ) [Wb] in the α-axis flux estimate ψ α2 ." Finally, our proposed offset voltage estimate for α-axis iŝ with k i being a positive scalar. The analysis for β-axis is the same as α-axis, andû β,offset is defined accordingly. This means the nonlinear flux estimator only introduces componentwise correction, while the flux amplitude based correction used in the conventional linear flux estimator depends on both α-axis and β-axis components of the flux estimate.

C. Simulated Waveforms of DC Bias B
The simulation results of ψ 2 from (8) usingû offset from (12) are shown in Fig. 6. It is observed that "theâ 2 error indicator," mB, is negative whenã 2 = a 2 −â 2 > 0 in Fig. 6(a), while mB is positive whenã 2 < 0 in Fig. 6(b). This reveals the working principle of the PAA (5a), i.e., to updateâ 2 by driving the error indicator mB to zero. Fig. 7 are tested in this section and will be referred to as "INV1" and "INV2" in the sequel.

Two self-built inverters shown in
1) INV1 uses the 600 V, 30 A, insulated-gate bipolar transistor based intelligent power module, FNB43060T2 from ON semiconductor. 2) INV2 uses the 1200 V, 120 A, silicon carbide metal-oxide-semiconductor field-effect transistor, CAS120M12BM2 from Cree.

A. Offline Measurement and Curve Fitting Results
The U-I curves of INV1 and INV2 are measured at V dc = 150 V, 300 V, as shown in Fig. 8. The measured data are fitted to function f (i x ; a 1 , a 2 , a 3 ) = a 1 i x +D x (i x ; a 2 , a 3 ). If the fitting error is small enough, we can useD x from (4) with the fitted a 2 and a 3 as the inverter model at a certain dc bus voltage in replacement of an LUT, for both INV1 and INV2. However, the fitted a 1 values annotated in Fig. 8 are found to be much larger  than motor nominal resistance (R = 1.1 Ω), and compensation using the fitted a 2 and a 3 leads to deteriorated sensorless control performance.

B. Low Current Region and Current Rating Matching
From Fig. 8, it is found that a 2 increases and a 3 decreases as the dc bus voltage V dc gets higher. Besides, INV2, whose current rating is four times as large as that of INV1, has much smaller a 3 value and has much wider LCR, up to approximately 1.5 A at V dc = 300 V. Here, LCR means the i x range (denoted by [0, According to the parametric Fourier analysis in Fig. 3, the upper bound of LCR can be estimated by the equation: a 3 I LCR = 6. When V dc = 300 V, the I LCR values of INV1 and INV2, based on the curve fitting results, are 0.3 and 1.5 A, respectively, which can be visually justified with Fig. 8. The upper bounds of the LCRs of INV1 and INV2 are approximately 1.0% and 1.25% of the rated current of the power switches, respectively. The above fact implies that the proposed PAAs should be generally applicable if the inverter current rating and the motor current  rating are well matched, in which case the motor's phase peak current I * x will be larger than I LCR only to fight against friction torque. For this ideal situation, the mechanism to avoid parameter adaptation during loss of two-parameter identifiability in our main proposition will not be needed.

V. EXPERIMENTAL VALIDATION OF THE PROPOSED PAAS
This article investigates the challenging situation where the motor's peak current I x is near I LCR , such that normalized peak current a 3 I x is near 6. This is achieved by using a test motor whose rated current is much lower than the inverter current rating. By doing this, we are going to show that (7b) and (7c) in our main proposition are conservative, and the PAAs are found to be effective down to much lower current thanâ 3 I * x = 6.

A. Test Bench Setup
INV1 and INV2 drive two 750 W, 3 Arms, 4 pole pairs, 3phase surface-mounted permanent magnet servo motors whose shafts are mechanically coupled. A dc power supply is used as the shared dc bus for both inverters. The period of the space vector pulsewidth modulation (SVPWM) is T PWM = 0.1 ms. Dead-time is 5 μs in Figs. 10 and 11. INV2 drives the test motor, and INV1 drives the load motor. The parameters of the two motors are: R = 1.1 Ω, d-axis inductance L d = 5 mH = L q , and permanent magnet flux linkage [1]. The load motor is vector controlled with a fixed speed command of −300 r/min and its speed regulator output limit i * q,max is set to 3 A in Fig. 10 and is switched between 3 and 1.5 A every 8 s in Fig. 11.

B. Block Diagram and System Synthesis
The block diagram of the PAAs-based inverter voltage drop compensation scheme is shown in Fig. 9. The SVPWM module outputs gate signals to control the inverter based on the input voltage command u * * as well as the measured V dc . Amplitude-invariant Clarke transformation converts phase quantities u * * xn , x = a, b, c, into u * * , where u * * xn consists of the torque controller's output voltage u * xn and the compensation voltageD x . Note thatD x is a function of phase current command i * x and is parameterized by the two inverter parameters, a 2 andâ 3 , whereâ 2 is updated by PAA in (5a) andâ 3 by PAA in (6a). The PAA forâ 2 relies onK Active , the amplitude of the active flux estimate ψ 2 in (8), and the preidentified parameter K Active . The PAA forâ 3 depends on the detection of the 6th-, 12th-, and 18th-order current harmonics in I Σh , the direct-axis component of measured current i in the commanded current vector-oriented frame from (6b). The total computational cost for executing the two PAAs in a digital signal processor with system clock frequency of 200 MHz is 4.3 μs and becomes 3.1 μs if w 12 = w 18 = 0.
Finally, a critically damped cascaded speed observer (see, e.g., [26]) tuned by its observer bandwidth parameter ω ob is implemented to reconstruct electrical speedω r fromθ d .

C. Adaptation of Two Parameters at Various V dc Values
The experimental results of sensorless operation with PAAs using INV1 at various dc bus voltage values are shown in Fig. 10, where experimentally recorded waveforms at different speed observer bandwidths (ω ob ) are compared. 1) Low Speed Observer Bandwidth: Fig. 10(a) corresponds to the case of the low speed observer bandwidth ω ob = 30 rad/s. At the beginning (t = 0) when V dc = 50 V, position errorθ d = θ d −θ d is oscillating and is peaking over 0.5 rad because initial values ofâ 2 andâ 3 are erroneous. The oscillation inθ d is reduced when the motor is loaded at t = 3 s and is minimized at t = 7 s when the PAAs are applied to updateâ 2 andâ 3 . At t = 14, 19, and 25 s, V dc is increased to 100, 150, and 300 V, respectively. It can be seen that large V dc change (from 150 to 300 V) causes a large oscillation in bothθ d and the encoder measured speed signal ω r . After t = 30 s, V dc decreases to 150, 100, and 50 V in order. The waveform ofâ 2 shows analogy to that of V dc .
When ω ob = 30 rad/s is relatively low, the speed controller's load disturbance rejection ability is poor, but the sensorless system shows robustness against sudden voltage errorD x caused by step change in V dc , such that the oscillation in speed waveform quickly diminishes in Fig. 10(a).
2) High Speed Observer Bandwidth: Fig. 10(b) further shows the results when ω ob = 150 rad/s. When ω ob is large, the load rejection ability is improved, but the robustness against voltage error disturbance is degraded. On the one hand, the spike in actual speed waveform is almost eliminated when load is applied at t = 3 s. On the other hand, the system falls into severe oscillation for approximately 4 s, when V dc steps from 150 to 300 V. The oscillation is caused by the slow convergence ofâ 2 as well as the oscillation inâ 3 waveform because the identifiability of a 2 and a 3 is coupled in terms of current harmonics ifâ 2 does not quickly converge, as is analyzed in Section II-A3.

D. Adaptation of Two Parameters Against Load Changes
The experiment in Fig. 10 is continued with the dc bus voltage fixed at 150 V, but the load motor's torque current now switches between 1.5 and 3 A every 8 s to test the PAAs' behavior under load change; 1) when load motor torque current is 1.5 A, the test motor current is about I x ≈ 0.5 A; 2) when load motor torque current is 3 A, the test motor current is about I x ≈ 1.5 A as shown in the first waveform in Fig. 11. Recall that the LCR upper bound of the test inverter, INV1, is I LCR = 1.5 A. In other words, the experiment in Fig. 11 further investigates the effectiveness of the proposed PAAs under the nonideal situation where peak current I x ≈ 0.5 A, which is less than When t ∈ [0, 33] s, the nominal value of resistance R = 1.1 Ω is used in (8). It is observed thatâ 2 converges to different values corresponding to different loads. This is an expected behavior becauseâ 2 andâ 3 are coupled when motor peak current I x is lower than I LCR . By "coupled," we mean that bothâ 2 andâ 3 can affect the fundamental voltage component inD x .

E. Robustness Against Stator Resistance Error
As a bonus advantage, the proposed PAAs-based online inverter voltage drop compensation in Fig. 1 can also reject the disturbance in flux estimation caused by resistance error and keep the sensorless system stable when an erroneousR is used.
When t ∈ [33, 100] s in Fig. 11, the resistance value used in (8) is detuned to be 50% and 200% of the nominal value, at t = 33 s and t = 60 s, respectively. At the instant when the resistance detuning happens, theθ d waveform shows some spikes but converges to zero quickly. With the aid of the online adaptation ofâ 2 andâ 3 and the associated voltage compensationD x , the sensorless system does not lose stability. Particularly,â 2 converges to a higher value when used resistanceR is too small and converges to a lower value when used resistanceR is too large.

VI. DISCUSSION
This section provides discussion on experiment in terms of disturbance study, comparative study, and sensitivity study.

A. Analysis of the Disturbances
Disturbances are observed in the profiles of position errorθ d , speeds ω r ,ω r , and q-axis current i q in Figs. 10 and 11. We are now going to investigate the transient disturbances via zoomedin plots and study the steady-state periodic disturbances with Fourier analysis. In Fig. 12, when dc bus voltage step changes to 150 V, a large voltage errorD x results, which disturbs the position error θ d . Asâ 2 andâ 3 are updated,θ d converges toward zero, and the compensation voltagesD α andD β decrease remarkably in magnitude. The actual speed ω r is drastically disturbed when ω ob is low in Fig. 12(a) but stays close to speed command when ω ob is high in Fig. 12(b). The comparison shows that the robustness of sensorless speed control is dependent on ω ob .   In Fig. 13, when load changes, there is an undesired transient disturbance inθ d , ω r , andω r , which is caused by the voltage error corresponding to the transients ofâ 2 andâ 3 . Current amplitude I x is within LCR of the inverter; so a 2 and a 3 cannot be accurately identified because the parameter identifiability of a 2 and a 3 is coupled in LCR.
2) Steady-State Periodic Disturbances: The Fourier analysis of steady-state waveforms in Fig. 11 is shown in Fig. 14. From Fig. 14, the following observations can be made.
1) The speed disturbance at synchronous frequency (f 1 = 13.3 Hz) is due to dc offset in measured current [25].
3) The speed disturbance at 4.5f 1 = 60 Hz is due to rotor manufacturing tolerances, which implies that the four-pole-pair servo motor has 18 slots, resulting in the (k × 18 4 )th-order harmonics in speed [27]. 4) The speed disturbance at 2.25f 1 = 30 Hz is due to stator and rotor manufacturing tolerances, according to our finite element simulation studies of an 18-slot, eight-pole motor with both manufacturing tolerances at both stator teeth and rotor magnets. 5) The speed disturbance at 11.6, 15, and 23.2 Hz is due to sensorless control because those harmonics are absent in ω r when motor is controlled using encoder.

B. Comparative Study
This subsection conducts comparative study to further highlight the advantages of the proposed online adaptation method. To this end, we have implemented the trapezoidal compensation voltage method proposed in [19] for comparison. The experimental results are shown in Figs. 15 and 16. When implementing the method in [19], a fixed plateau voltageâ 2 is used, which is only accurate at V dc = 150 V; its shape parameter, i.e., the ramp region angle θ t ∈ [0 • , 25 • ] is defined in [19].
From Fig. 15, when i q ≈ −2 A and θ t = 19 • , it is observed that the peak-to-peak value of position errorθ d of the proposed method is approximately 0.2 rad, which is almost half of that of the existing method. This is because large θ t is causing much loss in fundamental component ofD x . A step load change is applied at t ≈ 6.3 s, and both methods show similar performance as i q has become large enough for INV2, which can be understood by the fact that when current is high enough, the shape of the compensation voltage for both methods will approach a square waveform. Fig. 15 shows that a change in the shape parameter (â 3 or θ t ) often leads to a change in fundamental component ofD x . For our proposed method, the fundamental voltage error is actively compensated by online adaptation ofâ 2 , but for the method from [19], the fundamental voltage error is not controlled and becomes larger when θ t is larger, causing larger peak-to-peak value ofθ d in Fig. 15(b).
In Fig. 16(a), for the proposed method, the peak-to-peak value ofθ d is less than 0.4 rad, and there is no apparent difference before and after V dc changes because the change in distortion  voltage D x is compensated by online adaptation ofâ 2 . As a comparison, in Fig. 16(b), θ t is updated to mitigate voltage error due to erroneous plateau voltage when V dc drops to 100 V, but the sensorless system falls into oscillation giving a peak-to-peak value ofθ d that is over 1.2 rad.

C. Sensitivity Study
This subsection discusses experimental behavior due to L quncertainty, dead-time settings, and speed commands. 1) Parameter Uncertainty in L q : In Section VI-B, we have chosen to use the peak-to-peak value of position errorθ d as performance metric, rather than the mean value ofθ d . This is because the mean value ofθ d is dependent on uncertainty in L q . As is shown in Fig. 17(a), when the q-axis inductance valueL q we used in the controller is different from the actual one L q , position errorθ d will result.
The experimental measurement of L q -uncertainty is shown in Fig. 17(b), where the mean position error is plotted against the L q -uncertainty under two different current levels. The influence of L q -uncertainty on position estimation accuracy is found to be dependent on motor current.
Another issue due to L q -uncertainty we find from Fig. 17(a) is that the projection of i to d -axis is not zero, denoted by i d , meaning that there is a change in the actual active flux amplitude K Active = K E + (L d − L q )i d if L d = L q . This will affect our saturation function based flux estimator whose limit is calculated as = K E + (L d − L q ) × 0 A. From Fig. 17(b), it is found that theL q value that minimizes the mean position error is 7.5 mH which is larger than L d . This implies that is not equal to K Active in our experiment. But the test motor has small inductance; so the change in K Active is ignored in our study.
2) Different Dead-Time Settings: More experimental results are performed to record estimatedâ 2 ,â 3 values under different dead-time settings, as shown in Fig. 18. It is found that bothâ 2 ,â 3 increase as the dead-time increases. According to [20], one realizes the slope of the curves in Fig. 18(a) is approximately proportional to V dc times dead-time.

3) Lowest Operating Speed:
According to our experiment, the sensorless drive is able to stably operate down to 100 r/min when i q is 3 A, or 150 r/min when i q is 4 A.

VII. CONCLUSION
This article followed the design process for a generic parameter adaptive system. 1) Model the phenomenon based on the physics or at least provide an approximated way of representation. 2) Decide which parameters need to be identified online.
3) Study the single parameter-identifiability. The most intuitive way is to look at which variables show "abnormal behaviors" when the single parameter is erroneous. The PAAs are dynamics to minimize the error indicators. 4) If there is more than one unknown parameter, the coupled identifiability must be analyzed. Luckily, if there is only one-way coupling, prioritized adaptation rates should suffice; whereas, unfortunately, if there is two-way coupling, it is suggested to not simultaneously identify these two parameters. 5) Close the loop by using the online updated parameters. This procedure was executed in this article as follows. 1) The inverter voltage drop is modeled using sigmoid function for engineering purposes. 2) The plateau voltage a 2 due to dead-time and the shape parameter a 3 due to stray capacitor are to be identified.
3) The error indicators, i.e., mB and I Σh , are built upon the abnormally behaved variables, i.e., ψ 2 and i. In particular, it is revealed that a nonlinear flux estimator must be used to build an effective error indicator mB. PAAs are designed to reduce mB and I Σh . 4) When motor's peak current I x is within inverter's LCR, two-way coupling becomes severe; but if I x is large enough, it is valid to assume that only one-way coupling exists between the identifiability of a 2 and a 3 . 5) The identifiedâ 2 andâ 3 are used to compensate inverter voltage drop. The proposed PAAs-based inverter compensation scheme improves the robustness of sensorless algorithm against not only voltage error but also resistance error. It was theoretically suggested to suspend the PAAs when operating in inverter's LCR. However, as have been shown with experimental results, given the fact that the inverter current rating is 120 A and the motor current rating is only 3 Arms, the PAAs still work, even though the inverter parameters might never converge to their actual values anymore. This result raises a question to engineers designing the adaptive system-which is more desired, the accurate parameter identification or the robustness against uncertainty (of dc bus voltage and stator resistance)? This article pointed out that both accuracy and robustness can be preserved by selecting a motor whose current rating matches inverter's current rating.