Overview of Fundamental Frequency Sensorless Algorithms for AC Motors: A Unified Perspective

This article uses the active flux concept to review fundamental frequency sensorless algorithms for both induction and permanent magnet motors in one framework. Fundamentally, sensorless torque estimation can be directly solved using voltage-current model (VM) estimator or indirectly solved using current-speed model (CM) estimator. The latter turns the torque estimation problem into a speed estimation problem. The stator flux in VM and the $d$ -axis angle in CM are deemed as the two sets of original states for the sensorless drive. Through the change of states, the direct torque estimation can be realized via observer designs, whereas the speed dependency in the dynamics of the unknown state [e.g., active flux and electromotive force (emf)] gives rise to a class of speed estimation methods, known as model reference adaptive system (MRAS). The idea of a general speed observer is proposed to summarize various separate speed estimation methods needed for direct torque estimation. It is suggested to adopt inherently sensorless designs such that two-way coupling between torque estimation and speed estimation is avoided. For induction motors, it turns out that the unmodeled voltage in the active flux dynamics reveals current flowing in rotor bars and can be further modeled, for which the solutions to regeneration instability problem are discussed, and change of states is recommended to attain global stability. Finally discussed are the results of the slow reversal test, where local weak observability of ac motors can be potentially preserved.


I. INTRODUCTION
Sensorless or self-sensing control is typically referring to the speed regulation of inverter-fed ac motors without using a mechanical sensor for speed (i.e., tachogenerator) or position (e.g., an encoder, a resolver, and a Hall sensor), which is especially desired for high-speed motors (see [1]), motors with large-diameter or hollow shafts, and motors used in adverse environments [2]. Inside sensorless control, a sensorless algorithm provides torque feedback for field oriented control (FOC) [3], [4] or direct torque control (DTC) [5], [6].
Recently, there have been several excellent reviews [7]- [9] on sensorless control of PM motors with differentiated focuses on the applications [7], the saliency-based invasive methods [8], and the nonideal conditions [9]. Among these works, the fundamental frequency sensorless algorithms are covered with limited details. In our opinion, the nuanced differences and numerous variants of those algorithms can be understood fairly easily if one pays attention to the choice of state variables and the considerations behind the choice.
This article reviews all fundamental frequency models that can be adopted for designing sensorless algorithms, and provides a solid foundation to understand all design variants for both PM motors and induction motors, with the aid of the active flux concept [10]. As a matter of fact, in the induction motor context, the active flux is no new concept and is the natural choice of state (i.e., rotor flux) with the inverseequivalent circuit [11], and it has long been found useful in analysis, e.g., of DTC [12, eq. (7)]. In the salient PM motor context, active flux has other names in the literature, such as fictitious PM flux [13], linear flux [14], and extended flux [15].
The key feature that makes active flux so important is the fact that it allows to model ac motor with only one inductance, L q , which dominates how fast current can change.

A. Problem Formulation
Motor's electrical angular rotor speed ω r is governed by Newton's second law of motion J s n −1 pp sω r = T em − T L (1)  (2) with i q = −i α sin γ d +i β cos γ d being the q-axis current, where the FOC formulation is derived by substituting the following relation between ψ s and ψ A [10]: Here, the polar coordinate definition of active flux ψ A assumes a fundamental frequency model, and the angle of active flux vector ψ A is used as the d-axis angle for field orientation in FOC. Sensorless control of system (1) is challenging because neither the input T em nor the output ω r is known. The objective of a sensorless algorithm is to estimate T em and ω r . The motion equation (1) cannot be used for torque estimation because there exists an unknown term T L , and therefore, we shall seek other models.

B. Choice of Original States
Now, we are faced with the first choice of state variables. If we accept FOC formulation, making ψ A and γ d the states, active flux in (3) can be described by the current-speed model (CM) with i d = i α cos γ d + i β sin γ d being the d-axis current, R req the inverse-circuit rotor resistance, ω sγ d the synchronous speed, and ω sl the slip speed. For induction motors, there is a low-pass filter (LPF) with a time constant of ((L d − L q )/R req ) applied to the coil excitation in (4a), and only in this case, the slip relation ω sl = (R req i q /ψ A ) is valid. For inverter-fed PM motors, it is valid to simply put R req = ∞, ω sl = 0, and ω = ω r .
If the DTC formulation is adopted, making ψ s the state, as per Faraday's law, the stator voltage equation in αβ-frame serves as the voltage-current model (VM) of active flux where e s sψ s is the stator electromotive force (emf) and u is the αβ-frame stator voltage. Therefore, the torque estimation problem is solved by either: 1) estimating d-axis position γ d and ψ A using CM (4) or 2) estimating stator flux ψ s (and ψ A ) using VM (5).
Determining the d-axis angle in a feedforward fashion as (4b) is called indirect field orientation (IFO). The IFO solves the torque estimation indirectly via speed estimation. Counterintuitively, the VM in the dq-frame can be used to estimate field speed ω. To see this, applying the Park transformation P(γ d ) to VM (5) yields where e d,ss and e q,ss are steady-state auxiliary emfs in the dqframe and can be used for estimating ω. Fig. 1 shows the equivalent circuit of the dq model (6). In Fig. 1(a), (4a) has been represented as a parallel RL circuit and a mutual linkage, and the PM flux linkage ψ PM has been modeled as a product of a virtual current source i f and a mutual inductance L f . Fig. 1(b) implies that the active power passing through motor air gap is (3/2)ωψ A i q [cf. (2)]. 1 On the other hand, implementing γ d = arctan2(ψ βA , ψ αA ) is called direct field orientation (DFO). The IFO and DFO use the same information, i.e., measured voltage and current, but in different manners. The key difference is that the DFO can be made explicitly independent of motor speed and speed error because there is no speed variable in (5), which facilitates speed-independent direct torque estimation.

C. Change of States
Alternatively, we may select current i and active flux ψ A as the new states for torque estimation. Taking time derivative of (3) derives the active flux dynamics in (7a), and substituting (3) into the VM (5) gives the output dynamics in (7b) where E u denotes the unmodeled dynamics. From (7), ac motor appears to be modeled with one inductance, L q . As we will see later, ψ A is only one example of many choices of state changes, and alternative choices include the "active emf" e A sψ A or the extended emf e E in the Appendix. Nevertheless, the key idea here is that, by explicitly selecting the system output i as state, the torque estimation problem can now be solved by designing: 1) a full-or reduced-order state observer using (7) or 2) a disturbance observer using only the output equation (7b).

D. Coupling With Speed Estimation
From (7), the synchronous speed ω sγ d appears to be an inherent parameter of ac motors. As a result, speed estimation can be achieved by: 1) estimating ω r (and T L ) as state using (1) and sγ d = ω or 2) estimating ω as a constant parameter (note ω r = ω − ω sl ).
Direct torque estimation using (5) offers a possibility for being independent on its ensuing separate speed estimation, which is preferred as stabilities of torque estimation and speed estimation can be analyzed separately. On the other hand, the speed dependency, e.g., in model (6) and (7a), require joint estimation of torque and speed, hence resulting in complicated stability analysis. In fact, there is a convention in most sensorless PM motor literature, that is, the stability analysis of the two-way coupling between the joint estimation of torque and speed is not performed. The first part of this article will follow this convention to help the readers better understand the literature, while rigorous analysis of the interconnected estimators is reviewed in the second part (see Section V), dedicated to sensorless induction motors. Fortunately, the induction motor-oriented analysis can be potentially transferred to a PM motor, as exemplified in [16].

II. DESIGN IN ORIGINAL STATES
A stator flux estimate can be obtained by integrating e s , but the estimate (1/s)e s is unbounded in practice. There are two approaches to stabilize the integration, i.e., the time-domain approach and the frequency-domain approach, where several steady-state assumptions are often resorted to, including the following.

A. VM Estimator in Time Domain
Letˆdenote the estimated value. As shown in Fig. 2, the integrator is stabilized at both its input and output where the flux compensationD ψ is used to offset the center of the estimated flux trajectory, and the voltage compensation D is added to remove the unmodeled dc drift D in e s , e.g., due to thermal drift in analog devices. When there is a drift D ψ = t 0 Ddt in the flux estimate, the resulting errors in position and speed estimation can be derived as in [17, eqs. (19) and (21)]. Typical generation processes ofD ψ andD are summarized in Fig. 3.

1) Correction by Flux Trajectory Center:
A corollary of the constant flux amplitude assumption is that an accurate α-axis flux estimate results in a sinusoidal waveform whose maximum and minimum add up to null. In other words, the center of the flux estimate over one electrical cycle, i.e., should be 0. One can directly useD ψ in (9)  2) Correction by Amplitude Mismatch: Assuming that ψ A andψ A are, respectively, available from (4a) and Fig. 2, the active flux amplitude mismatch ε =ψ A − ψ A can be used to generate a voltage compensation, as shown in Fig. 3(c) [10], The flux amplitude mismatch ε =ψ A − ψ A from (11) has an equivalent current error form, as shown in (10b), if a current estimateî is introduced as (10a)î which is used, e.g., in [29] to build a sliding mode (SM) variant of (11).
Note that, for induction motors, the calculated ψ A is often rendered as the flux command [35], [36]. The VM estimator (8) plus correction (11) describes the socalled hybrid flux estimator that outputs the sum of high-pass filtered VM estimate and low-pass filtered "CM" estimate and leads to the classical interpretation that VM is used for high speeds and "CM" is used for low speeds [22]- [25], [27]. It is worth pointing out that there exists an abusive use of the term "CM." In the PM motor literature, the "CM" estimate is often written in the dq-frame and does not provide any angle information (see [27, eq. (10)]), which is distinct from the CM (4) expressed in polar-coordinates.
Discussions for selection of k 1 , k 2 are documented in [21]- [23], and it is suggested in [21], [22], and [35] to set k 2 to zero. Coincidentally, a contribution from the control community also suggests to not implement the k 2 integral term but in a slightly different form from (11), as shown in Fig. 3(d), to facilitate the stability analysis [37], [38] However, note that the k 2 = 0 design results in a nonzero bias in flux estimate in practice. The integrator (k 2 /s) or another dedicated "offset extractor" [36] can be used to eliminate the flux bias. The VM estimator (8) corrected by (11) or (12) is speedindependent, but it is possible to introduce speed-dependency by transforming (8) into the dq-frame. In [39] and [40], the current error in (10) is used to build a speed-adaptive VM estimator in the dq-frame.
3) State Observer for Flux Offset D ψ : Assume that there is an offset D ψ existing in the biased VM flux estimatê ψ s , which satisfies the following steady-state speed-dependent dynamics [41]: where it is assumed that D ψ is slowly varying, and the orthogonality between the αand β-axes of the flux holds. Fig. 3(e) shows a state observer using (13), which is also known as the frequency-adaptive observer in the literature [27]. The observer tuning (or pole placement) in [41] is dependent onω; thus, sensitivity with respect to speed error has been analyzed. One can understand the VM along with its frequency-adaptive observer as a "single tune integrator" with a mandatory dependency onω [42], [43]. Readers are referred to [9, Table 4] for a review of frequency-adaptive observers for eliminating emf harmonics.

B. VM Estimator in Frequency-Domain
In order to stabilize the pure integration, a high-pass filter (HPF) can be added to the output of the VM estimator [46], which is equivalent to replacing the integrator with an LPF which is speed-independent if a fixed cutoff frequency ω c is used. In fact, it is reported that placing the LPF pole −ω c to be close to zero is sufficient for the zero-speed operation of the induction motor [47], but it is also recommended to adopt the speed-dependent tuning ω c = k|ω|, k > 0 to adjust observer damping with respect to operating speed [48], [49]. The LPF in (14) is stable but introduces undesired lag and gain. The two equivalent forms of an integrator, as shown in Fig. 4, serve as a starting point to design two types of frequency-domain compensations for (14).
1) Multiplicative Compensation: Let j = √ −1. Allow an abuse of notation between a complex number and the R 2 vector, i.e., e s = [e αs , e βs ] T = e αs + je βs . Then, the multiplicative compensation can be introduced at a steady state as follows [cf. Fig. 4

(a)] [48]-[53]
which is speed-dependent and can be regarded as a single-tune integrator in the sense that its frequency response atω is the same as (1/s). Estimator (15) is also called statically compensated VM [49], [54] as the compensation is only exact at steady state. It is suggested in [51] to apply the static compensation to the stator emf e s first before going through the LPF. Advice for selecting k in (15) is given in [49], [51], and [52]. In [55], three speed-dependent cascaded LPFs are used to recover the desired frequency response of a pure integrator. The presence of the skew-symmetric matrix J in (15) means that, e.g., β-axis emf is used for compensating α-axis dynamics and vice versa, which implies that the orthogonality has been assumed. However, if some signal phase shift network is introduced, the compensation can be accomplished within the same axis, and therefore, even elliptical flux trajectory can be tracked [56].
2) Additive Compensation: In Fig. 4(b), the output signal ψ s is unbounded, and an idea to stabilize the system is to use a bounded signal in replace ofψ s that is being fed-back to (ω c /(s + ω c )). Typical ideas to generate a stable flux signal have been presented in Fig. 5. The componentwise limiter in Fig. 5(a) distorts the flux waveform, while the amplitude limiter in Fig. 5(b) does not correct flux angle [57]. Fig. 5(c) exploits the orthogonality between flux and emf to estimate the flux amplitude [57], for which deteriorated performance in transients has been reported [21]. Fig. 5(d) shows a VM estimator utilizing the CM estimator output as the stable signal [35]. Fig. 5(d) is an example of using both the amplitude and angle information of the CM (4) (in polar coordinate) for correcting VM estimator, while the compensation (11) only  utilizes the amplitude information of the CM, as an effort to avoid speed dependency.

C. CM Estimator
Theoretically, the CM (4) can calculate flux amplitude and angle without any voltage information, but it is practically impossible to be implemented without access to the VM equation. Typically, CM is implemented by estimating speed in the IFO dq-frame, i.e., the d-axis angle of the reference frame is determined as the integral of a speed estimate: 1) CM With Constant dq-Frame Current: Assume that si d = si q = sψ A = 0, and the field speed can then be directly calculated at the current step according to (6) ω = e q,ss /ψ A , if e d,ss = 0.
2) CM With Dynamic dq-Frame Current: It is possible to remove the assumption si d = si q = 0 for (17). To this end, the d-axis correction is driven by d- , and the q-axis emf e q,ss is replaced with an active emf amplitude estimateê A,ss that is proportional to the integral of q-axis current errorĩ where note k 2 is originally set to zero in [58]. Note that the one-step current predictions are embedded in (18) as L q sî q = e q,ss −ê A,ss and L q sî d = e d,ss , as clarified in Fig. 6 has been suggested in [63]. In addition, a closed-loop current observer can be implemented instead of the open-loop current predictions [64], [65], and moreover, in [64], the motion equation (1) is exploited to modify (18a) for better dynamic performance if inertia J s is provided. It is one of the most interesting observations in the field of sensorless control that (when ψ A is constant) the open-loop q-axis current prediction in (18b) contains field speed error information, and the open-loop d-axis current prediction in (18a) contains position error information (see [35, Fig. 6], [60], and [62, eq. (9.26)]). This can be further generalized for timevarying ψ A scenario as "basic relations for sensorless flux estimation" [49, eq. (13)].
3) Other CM Estimators: The CM estimatorγ d = (1/s)ω turns torque estimation problem into a speed estimation problem given an d-axis angle estimateγ d . Therefore, the dq frame speed-adaptive VM estimator in [39] and [40] can be used to establish a CM estimator, as done in [66]. Technically speaking, the CM-inspired speed estimation prefers to be implemented in the dq-frame. Using no correction in the CM estimator (i.e.,γ d = (1/s)ω + 0) is generally suggested [64], but outlier does exist, e.g., [67, eq. (17)].

III. DESIGN VIA CHANGE OF STATES
The current dynamics (7b) are disturbed by an unknown emf disturbance e A sψ A . There are two ensuing ideas to deal with it, i.e., a state observer and a disturbance observer. The latter does not need a model for disturbance and, thus, is potentially speed-independent, while the former becomes speed-dependent as it models the disturbance as either a flux or emf state.

A. State Observer for Flux
Using (7), a flux state observer can be implemented in full-order form [13], [14], [68], and [69], or reduced-order form [70]. Fig. 7(a) shows that a full-order observer is corrected by current errorĩ =i −î with correction functions Fig. 7(b) implies that a reduced-order observer is in effect corrected by the current time-derivative error (si −sî ) with the gain matrix K ∈ R 2×2 .
1) Full-Order Flux Observer Variants: In [13], the observer is analyzed by the Lyapunov stability theory by finding the positive-definite matrix for the Kalman-Yakubovich lemma. In [69] and [71], the SM flux state observer that has nonlinear f 1 , f 2 is proposed, and the pole placement for robustness improvement is detailed in [71]. In [68], the flux state observer is implemented in γ δ-frame, and in particular, the speed difference between dq-frame and γ δ-frame that arises during speed transients is estimated by a separate speed state observer.
2) Reduced-Order Flux Observer Design: In [70], the reduced-order flux observer is analyzed in the misaligned γ δframe and is described by a rotatory differential operator: P(γ )s P −1 (γ ) = s I + (sγ) J , where the stability analysis (that suggests K = k 1 I −k 2 sign(ω) J) holds only if speed is already known:ω = ω.

B. State Observer for EMF
Differentiating the active flux dynamics (7a) yields the active emf dynamics (19b) and its corresponding output equation (19a) where the unmodeled dynamics appear during speed transients but are often neglected in state observer design. The d-axis angle is recovered by integrating emf: e A dt, or directly calculated as Note that the constant active flux amplitude (i.e., sψ A = 0) has been implicitly assumed in (20). Note that the extended emf e E in the Appendix can also be substituted in (20). From the perspective of state observer design, the differences between an active emf observer and an extended emf observer are minor, so there is no need to differentiate between the two, and in fact, their reduced-order state observers share the same form where the subscript X is a placeholder of the letter A or E. The implementation of active emf observers is shown in Fig. 8. Full-order emf state observers are widely studied [72]- [78], while the reduced-order form stays as a classic [79], [80]. The key difference between full order and reduced order lies in how they treat speed error. The full-order observer reconstructs a current estimateî and calculates the output errorĩ that is used to tune a speed estimateω. This requires thatĩ is sensitive to speed error. On the other hand, since current i is measured, there is no need to reconstruct an estimateî , and the unknown state is directly estimated by (21), assuming that a speed estimateω is available. This needs the state estimateê X to exhibit robustness against speed error via careful observer pole placement [80].
1) Full-Order EMF Observer Variants: Linear corrections f 1 and f 2 are beneficial for analysis and tuning [68], [72]- [74], In [72], the linear system analysis is conducted to tune the observer. The γ δ-frame implementation of the emf observer can be found in [68], where included is an interesting study that analyzes the consequences of using constant speed assumption in an emf observer. In [73], the influence of dc offset on emf estimation is analyzed. In [74], observer steady-state errors are analyzed considering current/votlage error, parameter uncertainty, and filtering. Nonetheless, nonlinear SM corrections are also proposed [75]- [78], among which time-varying gain is used in [77] to reduce chattering. It is worth noting that excessive correction in current estimate dynamics reducesĩ 's sensitivity to speed error. In an extreme case,ĩ is forced to be zero (e.g., trapped in SM surface) so that there is no way to extract speed information fromĩ (but is still possible to extract from the correction terms).
2) Reduced-Order EMF Observer Designs: By assuming that se A = 0, Tomita et al. [79] proposed a reduced-order observer, for which the speed-dependent pole placement was mandatory and was derived by analyzing the H ∞ norm of the transfer function from the unmodeled dynamics to the estimated emf error. Similar H ∞ norm-based pole placement is used in [80] to design a reduced-order observer for the extended emf model (37) to preserve its robustness against speed error.

C. Disturbance Observer
The idea of classical disturbance observer (DO-also abbreviated as DOB in literature) is to simply solve for the unknown term using the output equation (7b) or (19a), and to avoid pure differentiation, an LPF is inserted The reduced-order state observer can be deemed as a special case of (22) if LPF(s) is generalized to an R 2×2 transfer function matrix In replace of the LPF(s) in (22) e s − L q si (23) where the speed dependency has been introduced.
In addition to the classical form (22), DO has an alternative form being a current observer with output correction f (ĩ) where the correction term f (·) or its filtered version serves as the emf estimate, as shown in Fig. 9. In this DO, the active emf e A is treated as a disturbance with assumed dynamics, and the correction f shall be modified accordingly when the assumed dynamics change. As shown in Fig. 10, the PI correction assumes that e A is constant, the proportional-resonant (PR) correction assumes that e A is sinusoidal with an angular speed ofω, and the SM correction in Fig. 10(c) and (d) assumes that e A is bounded (and the bound is known).
To our interest, we shall introduce a general form of the dynamic correction function f α (s) ∈ R for the α-axis as follows: f α (s) = k 1 |ĩ α | κ 1 sign ĩ α + k 2 |ĩ α | κ 2 sign ĩ α dt. (25) The correction f α is said to be nondynamic if k 2 = 0 and said to be dynamic if k 2 > 0. "Dynamic" means that f α has an internal state. In fact, since the integral term in (25) can be considered as an "extended state," the DO with dynamic correction is also called an extended state observer (ESO). If κ 1 < 1, the correction is a nonlinear function ofĩ α , causing undesired dither in emf estimate, known as chattering.
1) Linear DO: From (25), assuming constant emf disturbance se A = 0 and putting κ 1 = κ 2 = 1 result in linear DO. In [81], nondynamic correction is implemented. In [82], dynamic PI correction is used. The linear DO is deemed to be poorly damped if constant correction gains k 1 and k 2 are used [9]. Frequency-domain formulation of the DO with PI correction is proposed in [83], which is further generalized for salient PM motors in [73].
The constant emf assumption se A = 0 is not true [see (19b)], but it becomes reasonable if one transforms (24) into the dq-frame such that dq-frame emf is modeled as dc disturbance, and the correction f becomes P f . In [84]- [86], P f is implemented as a PI regulator. In [87], classical DO form (22) with LPF is used to estimate extended emf e E in the dq-frame. 3 Note that speed dependency is introduced in the dq-frame linear DO.
2) SMDO: The working principle of SMDO is to use the correction term f to forceî to track i. To make this happen, the norm of f should be larger than that of e A no matter how small the output errorĩ is. For example, the classical SMDO uses a nondynamic SM control law: f = k 1 sign(S) with simple SM surface S =ĩ and constant SM gain k 1 such that the norm f equals to k 1 as long asĩ = 0. Since ||e A || is proportional to ω, and to assure a wide operating speed range, large SM gain k 1 is mandatory, implying that DO performance varies as operating speed changes.
By using large but constant SM gain k 1 , SMDO is speedindependent, but there are excessive noises in the "raw" estimated emf disturbance f , known as chattering issues. To mitigate the chattering, it is proposed to use a "milder" SM control law than the discontinuous correction sign(·), such as the saturation function [88] and the sigmoid function [89]- [91]. Another idea is to design f as a linear combination of SM correction and proportional correction [88], [91]. However, those "milder" SM control laws break the Lyapunov stability of SMDO.
Alternatively, without modifying the SM control law, an additional LPF can be applied to f , beforeγ d is extracted. The phase delay caused by the LPF should be compensated for different speeds [92]- [94]. Given a constant SM gain, it is suggested that the pole of the LPF should rely onω [88], as indicated in Fig. 9.
To avoid speed-dependency, it is recommended to implement f as a dynamic correction that puts signum inside an integral like the k 2 term in (25). This integral introduces an additional state, resulting in a second-order SMDO, and the newly introduced state can be interpreted as an estimate of the emf appearing in the current dynamics. For example, the supertwisting algorithm can be used to estimate the emf [95]- [97] [98, Sec. V], and the resulting continuous α-axis correction is f α = k 1 |ĩ α | 1/2 sign(ĩ α ) + k 2 sign(ĩ α )dt, where the integral term serves as a continuous estimate of the emf, as shown in Fig. 10(d). As a comparison, the second-order SMDO proposed in [99] uses discontinuous α-axis correction as f α = k 1 sign(ĩ α ) + k 2 sign(ĩ α )dt, as shown in Fig. 10(c). Note that the continuity of f α is decided by the k 1 term, and also note applying variable gain k 1 |ĩ α | 1/2 to sign(ĩ α ) does not eliminate chattering because the derivative of |ĩ α | 1/2 is infinite whenĩ α = 0. The chattering is reduced because of using a smaller k 1 value, and a nonzero k 2 term (that takes advantage of the history of output errorĩ ).
So far, all SMDOs reviewed above use current errorĩ as the SM surface S. Improvement is expected by further designing the SM surface. For example, SM surface involving the integral of current error ĩ dt can eliminate the reaching phase [100]- [102] [103, p. 287]. SM surfaces involving the derivative of current error sĩ promise "convergence in finite time," such as the fast terminal SM manifold [104] and the nonsingular terminal SM manifold [93]. Note that the derivative can be reconstructed by the supertwisting algorithm [93].
Besides, the SM gain can be designed to be a function of S [102], which is also known as the reaching law design. A less common idea to reduce SM gain is to execute the numerical integral of SMDO at a frequency that is three times as high as the pulsewidth modulation (PWM) frequency [92], [105].
While most SMDO designs are speed-independent, it is possible to introduce speed dependency. First, similar to the dq-frame linear DO, one can implement SMDO in the dq-frame, and the dq-frame correction P f in [106] is implemented as a combination of proportional correction and SM correction. Second, in [107], in addition to the dynamic correction P f , a speed-dependent SM surface that consists of current error and its integral is further selected. Third, the SMDO in [108] is designed based on the speed-dependent output equation of the extended emf model (37a). Finally, there are also examples of using speed-dependent tuning for second-order SMDO [97, eq. (10)].

IV. SEPARATE SPEED ESTIMATION
This section focuses on DTC or DFO controlled drive such that torque estimation is already done, and an estimate of some unknown state (ê X ,ψ A ,γ d ) is available. This estimate serves as the reference input (denoted by ϑ or ϑ) to the separate speed estimation. "Separate" means that the speed estimation design assumes the reference input signal is independent of the speed estimate, i.e., it is inherently sensorless [49], neglecting speed error's influence on reference input.

A. Direct Calculation
Steady-state dynamics can be used to directly calculate speed, which provides a static but instantaneous speed estimate.

1) Direct Calculation From EMF:
Speed is simply the emf amplitude divided by the flux amplitude [93], [95]. Note that this direct calculation motivates the CM estimator (16).
2) Direct Calculation From Position: Speed estimate can be calculated by the forward difference of the flux angleγ d [14], [17]. An additional LPF is embedded to reduce the amplified noise [17].

B. General Observer for Speed
Assuming that a scalar reference signal denoted as ϑ γ d is available from prior flux/emf estimation, then the following (n +1)th-order general state observer for ϑ summarizes a wide class of speed estimation methods: where the output error isθ = ϑ−θ. Similar generalization like (26) has been studied in the literature, known as generalized PI observer (GPIO) [116] or generalized ESO (GESO) [117]. The position observer (26) does not define a speed estimate, and there are potentially n+1 variants of speed estimates [118] Table I gives a list of example implementations of (26). Fig. 11 shows block diagrams of (26) and (27) when n = 2 and j 0 = 0, 1.
Alternatively, one can design a state observer for the q-axis current by redefining ϑ and f in (26) accordingly for either PM motors [22] or induction motors [110]. Such a scalar output observer can be considered as a reduced-order implementation of the original natural observer [125], or it can be understood as an ELO implementation of the q-axis current observer [65]. The difference is that the natural observer for ϑ = i q utilizes motor's active power error |u q |ĩ q as output error, which, however, makes observer tuning to be dependent on the q-axis voltage u q , thus time-varying [22].
2) Phase-Locked Loop (PLL): PLL is widely used for extracting frequency information from the d-axis angle error. During the formulation of PLL, various types of position error signals can be exploited, such as the q-axis voltage [52],  the angle of γ δ-frame extended emf [85], the d-axis current error [65], and the forward difference of high-frequency components of the αβ-frame current [120]. PLL for induction motors would need to add an additional slip relation [49] such that f (ϑ) = ω sl . Speed error during speed transients is inevitable because typical PLL is the second-order system that assumes constant speed [126], which is known as a type-2 system [41]. The transfer function from actual position to estimated position can be found in [85, eq. (39)] and [87, eq. (18)]. The PLL speed error during speed transients is analyzed in [82]. In [30], an additional PI term driven by torque error is further added to the PLL-based speed estimate. PLL can be generalized for higher order to track time-varying speed [70, eq. (24)], and see also [34] and [87, eq. (22)] for an application of type-3 PLL for tracking ramp speed signal.

3) Extended Kalman Filter (EKF):
In [75] and [76], EKF is applied to a third-order system with position, speed, and acceleration as states, without needing inertia parameter J s . This EKF is similar to type-3 PLL but has time-varying gains.

C. Model Reference Adaptive System (MRAS)
Two different concepts of MRASs are shown in Fig. 12. Fig. 12(a) shows an MRAS whose reference and adjustable model are, respectively, the actual motor and the speedadaptive full-order state observer. Fig. 12(b) shows an MRAS whose reference and adjustable model are, respectively, the reduced-order state observer and a speed-adaptive redundant observer.
2) Speed Adaptation Law: Adaptation law for field speed iŝ ω = PI(s) × Regressor × Output error (29) where the output error can be the current errorĩ from full-order observer [see Fig. 12(a)], the flux/emf mismatchθ from redundant observer [see Fig. 12(b)], or any other mismatch, e.g., the active flux amplitude mismatch ε in (10b) [66]. Speed adaptation assumes that speed is a constant parameter, but an inertia-dependent speed-adaptation law that includes an additional torque term is also proposed [65], [67], [109]. The selection of the regressor is critical, and it often depends on how speed appears in the adopted model: example choices include active flux [13] and extended emf [72]. It is shown in [66] that decoupling of analysis between flux estimation and speed estimation is achieved via linearized model-based observer tuning and wise selection of speed regressor. However, the fact is that the generation of speed regressor offers a degree of freedom to possibly find Lyapunov function, hence eliminating the need for linearization, which will be reviewed in Section V.

V. LOSE THE CONSTANT ψ A ASSUMPTION
As per (4a), ψ A is time-varying as long as (L d − L q )i d varies, and a change in ψ A leads to unmodeled dynamics E u in (7a) and (19b), which will affect the performance of state observer and any other method relying on constant ψ A assumption. In practice, time-varying ψ A could be beneficial for improving operating efficiency or dynamic performance. For example, at high speeds, a faster speed dynamic process can be achieved if the flux amplitude is first weakened such that more dc bus voltage becomes available for producing torque current [127].
For PM motors, the strategy is mainly to compensate for the unmodeled dynamics. In [15], an angle compensation that takes into account the unmodeled dynamics in (19b) when ψ A varies is proposed. In [13], si d in the unmodeled dynamics E u in (7a) is compensated for by its estimate.
For induction motors, the voltage E u is closely related to the voltage drop on the rotor resistance, and hence, it can, in fact, be modeled. This section is dedicated to induction motors.

A. Induction Motor Model With Time-Varying ψ A
For induction motors, the unmodeled dynamics E u satisfy in which (4a) and (3) have been substituted, and from which the slip relation ω sl = (R req i q /ψ A ) is derived. The model with stator current and active flux as states is derived from (5), (7a), and (30) which is exactly the inverse-circuit induction motor model, with active flux ψ A being equivalent to the rotor flux, L d denoting stator inductance, L q designating transient leakage inductance, and (R req /(L d − L q ))I − ω r J . There are two facts that make induction motors unique.
1) Active flux amplitude ψ A is not constant and is maintained by stator excitation for non-PM motors. 2) There is a slip speed ω sl difference between field speed ω and rotor speed ω r , i.e., ω r = ω − ω sl . In other words, the induction motor is neither a PM motor nor a synchronous motor. These two facts correspond to the two unique features of sensorless induction motors.
1) There is a chance for estimated flux amplitudeψ A to collapse [128]. There is also a chance for a change in flux amplitude being misinterpreted as a change in flux angle, which is revealed in the "basic relations for sensorless flux estimation" [49, eq. (13)]. As a result, an unstable sensorless operation that is not observed in PM motors might happen (e.g., at the low-speed regenerating operation).
2) The zero-speed operation under load does not mean the zero-frequency operation for induction motors.

B. Common Act of Choosing Active Flux as State
Many studies have been devoted to achieving speed estimation using the model (31). Note that the estimated flux is seldom used for Park transformation, and in most cases, the IFO d-axis angle (1/s)ω is used to define the dq-frame of an induction motor.
For better dynamic performance, ω r is further treated as an observer state in the dq-frame SM state observer [138] with d-axis current error chosen as output error. A different approach, however, is to put all the correction terms into the dynamics of the load torque estimateT L and use an exact copy of motor dynamics (31) and (1) to establish a so-called natural observer [125].
2) Stability Challenges at Low-Speed Regeneration: It is quite a challenge to stabilize the full-order state observer for (31) in low-speed regeneration even when speed is observable. Classical designs are flawed. For example, the stable speed adaptation law derived from the Lyapunov function [131] depends on the unknown flux error; and the hyperstability analysis in [129] depends on the assumption that the ratio between flux error norm and current error norm has a finite upper bound (so that the unknown flux error can be replaced by current error). Careful observer gain designs based on the linearized model (see [132], [135], [139], and [140]) and the positive real property [141] are proposed for extending stable operating region. Alternatively, it is also effective to redesign the speed adaptation law [133], [142], [143], [144, eq. (26)]. Readers are referred to [145] for a survey of the stabilization methods.
3) MRAS: In the field of sensorless induction motor drive, MRAS is jargon for a system with the VM (5) being the reference model and the new CM (31c) being the adjustable model. The pioneering work on MRAS in [46] (see also the follow-up works [146], [147]) chooses VM estimator in (14) as reference model and implement CM with high-pass filtered i as adjustable model, as shown in Fig. 14(a). As a result, the obtained flux estimate has significant delays with respect to the actual flux; thus, it cannot be used for DFO. This is a good example showing the spirit of the IFO nature of sensorless induction motors that an accurate estimate of flux/emf is not needed because the IFO d-axis angle estimate only depends  [148].
Alternatively, one can stabilize the VM estimator using the mismatchθ between outputs of VM and CM estimators as the correction. Fig. 14(b) shows an example of proportional correction kθ [149]. The MRAS implementation in [23] has made it clear that the amplitude mismatch is used to stabilize the VM estimator, while the angle mismatch is used to tuneω used in the CM estimator. In [150], the VM is transformed to an estimated dq-frame to derive a generalized slip relation in terms of the VM correction gains. The dualreference observer proposed in [151] is also an MRAS and is an example of implementing the amplitude correction (11) in its current error form (10b) in a time-varying ψ A model. In [147], the speed adaptation law is implemented as an SM control law. In [152], the correction in VM is replaced with a supertwisting-based dynamic correction. Readers are referred to [153] for a dedicated review of MRAS variants having different choices of output errors.

C. State Observer Design via Change of States
To attack the regeneration instability challenge, an ideal solution is to find a globally stable speed-adaptive observer design. In the classics of adaptive observer design [154]- [156], stable observer design is often proposed for a class of state-affine systems that do not describe the induction motor dynamics (31). In fact, it has been shown in [157] that, in order to find a Lyapunov function for the full-order state observer with (i, ψ A ) as states, the observer coefficients must be dependent on the actual speed, implying that globally stable design does not exist. Thus, to obtain global stability, one needs to either find a proper state transformation that can describe the induction motor as a system in the adaptive observer form [155] s i where ∈ R 4 is the regressor of speed that consists of only known signals (i, u) and x is the new unknown state; or develop advanced observer design applicable to a wider class of systems that allow unknown speed regressor (x) [158], [159]. 1) Change of States: A list of state transformations for induction motors is given in Table II. Model 1 is (31). Integrator back-stepping-based observer design is proposed for model 2 (x = e A ) in [98,Sec. III], where speed is assumed as a known signal in the stability proof. In [144], an attempt to apply adaptive observer design proposed in [154] to model 3 (x = −e A ) has ended with requiring infinite gain to attain asymptotical stability, where the signal si is obtained using a state variable filter-which can be avoided if model 4 (x = Ωψ A ) is used instead [144]. The research using model 5 (x = ψ s ) is summarized in the monograph [160], and the key property of choosing stator flux as the unknown internal state is that there is no unknown variable ω r appearing in the dynamics of unknown state x. Finally, model 6 (x = Ωψ s ) is in adaptive observer form (32), and therefore, the existence of globally stable speed-adaptive observer for induction motor is an established fact [161]- [163], where filtering the original speed regressor is found essential [163].
2) Advances in High-Gain Observer Design: The advances in the high-gain observer (HGO) design [158], [159] accept a wider class of systems than (32), and application of this HGO to model 6 is studied in [164]. Moreover, the speed ω r and load torque T L can be treated as states so that model 4 has been extended with a third R 2 state as z = −s(ω r J ψ A ) [165]. The requirement on the partitioned matrix needed in [158] and [165] is later removed in [159], which allows one to design HGO in terms of the two scalar states ω r and T L instead of the R 2 vector z [159]. Note that the speed regressor redesign from [156] has facilitated the above Lyapunov stability-based HGO design [158], [159].

D. Disturbance Observer Design via Change of States
Substituting (31c) into (31a) will introduce a disturbance variable denoted as x = Ωψ A . Fig. 15 shows the general DO for estimating this disturbance, where SM correction is widely used [111]- [115], and linear dynamic correction is also studied [166]. Alternatively, one can design a full-order SMDO for model 1 or model 5 with the speed-dependent term ω r Jψ A or ω r Jψ s being treated as disturbance [151].

VI. CONSIDERATIONS FOR LOW-SPEED OPERATIONS
The sufficient conditions for preserving ac motor observability provide guidance for stable operation near sγ d = 0. The state vector [i α , i β , ω, γ d ] T of a PM motor is locally weakly observable if the following inequality holds [167]: The state vector [i α , i β , ψ αA , ψ βA , ω, T L ] T of an induction motor is locally weakly observable if the inequality holds [165], [167] where the slip speed has been substituted. To preserve the observability at ω = 0, both (33) and (34) suggest that an auxiliary observability vector in the dq-frame 4 should rotate or change its direction with respect to the d-axis.
In particular, it can be shown that the local weak observability of a surface-mounted PM motor holds with nonzero acceleration at ω = 0, i.e., zero-frequency crossing [168]. Slow speed reversal test has been recommended to test stability margin of a sensorless drive near the zero-frequency operation [145]. The CM estimator can survive the test with the help of the load-dependent i d excitation [54], where the d-axis angle error still shows trends of divergence during zero-frequency crossing. It is suggested to use high-frequency signal injection to assist zero-frequency crossing for the dqframe VM estimator in [39]. However, the αβ-frame VM estimator shows a smooth d-axis angle waveform even during zero-frequency crossing [22], [36], [41], [110]. Note that only voltage compensation is needed in [22], [36], and [110], while the speed-dependent flux compensation is used in [41]. The SMDO proposed in [88] is also found to be able to perform the test but shows increased noises in speed estimate compared to its VM counterpart [22]. The globally stable speed-adaptive observer also shows speed divergence near a zero-frequency crossing because the persistency of excitation is lost at dc excitation [162]. The typical acceleration rate of the slow reversal is between 50 and 100 r/min/s, whereas a successful 5r/min/s test is reported in [110] and [162], and it can go down to 2 r/min/s if deliberate flux weakening is allowed [162]. As a final note, compared with operating at standstill (see [22] and [41]), operating extremely near zero frequency (e.g., 1 r/min [10]) is considered a more challenging working condition.

VII. CONCLUSION
The act to use the same torque expression for all ac motors provides us a unified perspective to review fundamental frequency sensorless algorithms based on the model of torqueproducing flux, i.e., active flux. The resulting equivalent circuit models the effect of the PM, the saliency, and the rotor conductors.
The torque estimation problem can be solved directly or indirectly, which corresponds to a DFO or IFO sensorless drive. The direct torque estimation includes the VM estimator and the observers, such that the d-axis angle is computed from αβ components of flux or emf vector. The indirect torque estimation problem is turned into speed estimation problem via the CM estimator, i.e.,γ d = (1/s)ω. The CM estimator enables speed estimation in IFO dq-frame, but any other speed estimation method, such as MRAS and speed-adaptive state observer, can be used-which is often seen in IFO-controlled induction motors.
No method is perfect. VM is critically stable, CM estimator has algebraic loop issues, state observer is disturbed by estimated speed error, and disturbance observer suffers from undesired dither in estimation. Nevertheless, advances to each method have been made in the literature.
1) The VM estimator is analyzed in the dq frame to derive operating speed-dependent stability results [38], implying that stability is guaranteed with speed-dependent tuning.
2) The CM estimator is linearized for analysis, from which "complete stability" is attained with proper tuning [54], [169]. 3) Challenges have been reported in designing stable state observers for the model using active flux as states, while the studies that involve change of states are fruitful, e.g., globally stable speed-adaptive observer (except zero stator frequency condition) [16], [159], [162]. 4) Dynamic SM correction of higher order has long been found able to eliminate chattering in SMDO (see [170, ch. 6]).
Unlike an IFO drive, a DFO drive needs to estimate speed separately. A general observer is proposed to summarize various speed estimation algorithms, including PLL, ELO, EKF, GPIO, GESO, and SMDO, which differs in terms of linear or nonlinear corrections, time-invariant or time-varying gains, and their assumed motion dynamics. It is worth mentioning that, for the same general observer of order n, there are potentially n + 1 variants of speed estimates having different characteristics in terms of dynamic performance and noise attenuation.
The applicability of sensorless algorithms depends on working conditions and motor types. Speed estimation methods assuming constant speed are more suited for motors with large inertia, while high dynamic performance can be obtained with higher order speed estimation. For ultrahigh-speed applications with high switching frequencies, the CM estimator is recommended for its minimum computational cost. For high-saliency motors and non-PM motors, as the active flux amplitude is regulated in real-time, designs that take into account the dynamic active flux amplitude should be considered. For lowspeed applications, the local weak observability has suggested adopting a high saliency PM motor or induction motor from a theoretical point of view.

APPENDIX HISTORY OF EXTENDED EMF MODEL
By manipulating inductance, the dq-model (6) is rewritten Amplitude of the extended emf e E + L d si q = u q − Ri q − ωL q i d .
Note that current dynamics (37a) become explicitly dependent on ω.
Historically, it is not easy to design sensorless algorithms for salient-pole PM motors, owing to the position-dependent inductance that leads to a highly nonlinear model. The salientpole PM motor can be treated as a nonsalient one if one assumes that (L d − L q )i ψ PM [80]. The idea of extended emf was timely proposed to relax the need for this low saliency assumption.
For PM motors with high saliency, inductance variation caused by magnetic saturation should be further considered. Generally speaking, the dq inductances should be modeled as a function of the dq currents: L d = L d (i d ) and L q = L q (i q ). However, the active flux model (6) only needs L q (i q ) and is modified to take the differential inductance (see [171]) into account as follows: sψ A + L q si d = u d − Ri d + ωL q i q ωψ A + L q + dL q di q i q si q = u q − Ri q − ωL q i d (38) where we have assumed that (dL q /di d ) = 0. In contrast, the extended emf model (36) would need to use both L d (i d ) and L q (i q ).