1 Introduction

Bearingless motor is a new type of motor which combines traditional motor with magnetic bearing. The rotor can be suspended by electromagnetic force, eliminating friction with the rotating shaft and the need for lubricating oil. This results in advantages such as reduced mechanical friction, less lubrication requirements, and longer lifespan. As a result, it has promising applications in aerospace and bioengineering [1, 2]. As a new type of bearingless motor, PMa-BSynRM inherits the advantages of permanent magnet-assisted synchronous reluctance motor and bearingless motor. It cannot only suspend the rotor, but also make full use of permanent magnet torque and reluctance torque, so that less permanent magnet is used to maintain a high torque density. PMa-BSynRM has attracted attention due to its excellent performance [3].

With the development of bearingless motor, sensorless control is getting more and more attention because it can successfully reduce the dependence on mechanical sensors and improve the reliability of the system. Due to the complexity of PMa-BSynRM rotor motion, sensorless control is divided into speed sensorless control [4] and displacement sensorless control [5, 6]. Speed sensorless control, which eliminates the speed sensor, helps minimize the axial length, volume, and cost of the PMa-BSynRM while still ensuring stable motor rotation.

The core of speed sensorless control is to indirectly estimate the speed by controlling easily measured quantities such as voltage and current in the system. Existing sensorless control technologies can be divided into two categories: (1) methods based on high-frequency signal injection [7, 8], which are usually only applied to low-speed operation control. (2) A model-based approach that relies on an estimate of the flux linkage or back electromotive force (EMF). At present, the method based on back EMF estimation is dominant in the middle and high-speed area. These methods typically include a back EMF estimator and a cascade position/velocity extractor to extract the rotor speed and position. Sensorless control algorithms based on back EMF extraction can be implemented by a variety of methods, such as sliding mode observer [9], model reference adaptive system (MRAS) [10], and extended Kalman filter (EKF) [11]. The SMO exhibits robust adaptability to unmodeled dynamics and interference. However, it suffers from the chattering phenomenon caused by the discontinuity of the switching function. The design of the MRAS aims to ensure system stability, which involves a more intricate process and exhibits greater sensitivity to parameter variations. The EKF method has strong disturbance resistance, but each switching cycle requires a large amount of data and calculations, demanding high processor performance.

Extended state observer (ESO) shows great application prospect in sensorless control because of its high estimation accuracy, anti-interference ability, and robustness to parameter change. ESO can be divided into linear ESO (LESO) and nonlinear ESO (NESO). In [12], the NESO is proposed for the rotor speed and position estimation for sensorless interior permanent magnet synchronous motor drives, and an optimization parameter configuration method is deployed to expand the bandwidth of the NESO. However, nonlinear functions are mostly used in NESO with complex structure and numerous parameters, which is not conducive to engineering applications [13]. LESO parameters are associated with bandwidth frequency, which makes the physical meaning of LESO parameters more intuitive [14]. Although LESO solves the problem of parameter setting and stability analysis, the transfer function of traditional LESO shows second-order low-pass characteristics, resulting in the estimated back EMF phase lagging behind the actual back EMF phase [15]. Increased bandwidth reduces phase delay, but also reduces LESO's ability to suppress high-frequency noise. In [16], the rotor position signal estimated by the extended back EMF-based sensorless control method is utilized as the subject variable of ESO to obtain accurate rotor position information, eliminating the need for the low-pass filter and phase-locked loop design. In [17], an adaptive bandwidth tuning scheme is proposed to tune parameters of the ESO online to obtain ideal back EMF estimation performance in wide-speed operation range. In [18], a frequency adaptive LESO is proposed to accurately estimate the back EMF. The gain of the observer is designed according to the transfer function of a pre-designed second-order complex coefficient filter, and its stability is guaranteed by the generalized Routh criterion. In [19], a fast sinusoidal interference estimator is combined with linear ESO to obtain the back EMF. This method effectively improves the estimation ability of the back potential, but the anti-DC interference performance is not obvious. Based on the idea of this method, this paper improves the LESO, which has the advantages of clear physical concept and high degree of freedom.

The speed sensorless control method of PMa-BSynRM based on ELESO is proposed in this paper. In this method, the proportional resonance controller is responsible for adaptive tracking of back EMF, while the LESO reinforces resistance to DC disturbance, thereby reducing the influence of the PMa-BSynRM suspension force system on the torque system and minimizing speed fluctuations to enhance speed estimation accuracy.

The structure of this paper is as follows. In Section 2 the main operating mechanism of PMa-BSynRM and the mathematical model of torque system considering the influence of suspension force system are given. In Sect. 3, a speed sensorless control method of PMa-BSynRM based on ELESO is proposed. The experimental platform is established in Sect. 4, where the effects of proposed ELESO on PMa-BSynRM speed sensorless control are investigated. In Sect. 5, the conclusion is drawn.

2 Basic principle and mathematical model of PMA-BSynRM

2.1 Basic principle of levitation force generation

The research object of this paper is a PMa-BSynRM, the basic structure of which is shown in Fig. 1. The stator of PMa-BSynRM has 24 slots. There are two sets of windings inside: The outer layer is the torque winding, the pole pairs are 2, the inner layer is the suspension force winding, the pole pairs are 1.

Fig. 1
figure 1

Structure diagram of PMa-BSynRM

Full size image

According to the conditions of radial suspension force, the difference between pole pairs PM of torque winding and pole pairs PB of suspension force winding is 1, that is, PM = PB ± 1 [3]. The rotation direction and angular velocity of the torque winding are the same as that of the suspension winding. The torque winding provides the magnetic flux ΦM, which is used to generate the tangential force required for the motor to rotate, while the radial magnetic field is in a stable state. The suspension force winding is used to produce a magnetic flux that breaks the original radial equilibrium magnetic flux ΦB. The stable suspension of PMa-BSynRM rotor is realized by changing the current direction of the suspension force winding and generating a controllable unbalanced magnetic field.

2.2 Mathematical model of the PMa-BSynRM

The mathematical model of PMa-BSynRM is presented in this section. Suspension force winding current iBd, iBq, torque winding current iMd, iMq are all d-q-axis components in synchronous rotating coordinate system.

The mathematical model of PMa-BSynRM radial suspension force can be established by Maxwell tensor method. When the radial x-axis and y-axis deviations of the rotor are x and y, respectively, the radial x-axis suspension force Fx and radial y-axis suspension force Fy can be expressed as

$$ \left[ {\begin{array}{*{20}c} {F_{{\text{x}}} } \\ {F_{{\text{y}}} } \\ \end{array} } \right] = (k_{{\text{M}}} + k_{{\text{L}}} )\left[ {\begin{array}{*{20}c} {\psi_{{{\text{Md}}}} } & {\psi_{{{\text{Mq}}}} } \\ {\psi_{{{\text{Mq}}}} } & { - \psi_{{{\text{Md}}}} } \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} {i_{{{\text{Bd}}}} } \\ {i_{{{\text{Bq}}}} } \\ \end{array} } \right] + k_{{\text{C}}} \left[ {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right] $$
(1)

where iBd and iBq are the current components on the d and q axes of the suspension force winding, respectively, ψMd and ψMq are the equivalent flux links of the permanent magnet on the d and q axes of the torque winding, respectively, kM is the Maxwell force constant, kL is the Lorentz force constant, and kC is the constant coefficient.

Considering the coupling relationship between torque winding and suspension force winding, and the influence of rotor eccentricity, the d-q-axis component of PMa-BSynRM torque and suspension force winding flux is expressed [20]

$$ \left[ \begin{gathered} \psi_{{{\text{M}}{d}}} \hfill \\ \psi_{{{\text{M}}{q}}} \hfill \\ \psi_{{{\text{B}}{d}}} \hfill \\ \psi_{{{\text{B}}{q}}} \hfill \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}c} {L_{{{\text{Md}}}} } & {0} & {L_{{\text{C}}} x} & { - L_{{\text{C}}} y} \\ {0} & {L_{{{\text{Mq}}}} } & {L_{{\text{C}}} y} & {L_{{\text{C}}} x} \\ {L_{{\text{C}}} x} & {L_{{\text{C}}} y} & {L_{{\text{B}}} } & {0} \\ { - L_{{\text{C}}} y} & {L_{{\text{C}}} x} & {0} & {L_{{\text{B}}} } \\ \end{array} } \right] \cdot \left[ \begin{gathered} i_{{{\text{M}}{d}}} + i_{{f}} \\ i_{{{\text{M}}{q}}} \\ i_{{{\text{B}}{d}}} \\ i_{{{\text{B}}{q}}} \\ \end{gathered} \right] $$
(2)

where ψMd, ψMq are the d- and q-axis equivalent air gap flux linkages of the torque windings, respectively, ψBd, ψBq are the d- and q-axis equivalent air gap flux linkages of the suspension force windings, respectively, LMd, LMq are the d- and q-axis self-inductances of the torque windings, respectively, LB is the self-inductance of the suspension force windings, Lc is the suspension force constant.

The magnetic co-energy equation of the system can be obtained from the stator winding flux equation.

$$ \begin{gathered} W_{{\text{m}}} { = }\frac{1}{2}\left[ {\begin{array}{*{20}c} {i_{{{\text{Md}}}} + i_{{\text{f}}} } & {i_{{{\text{Mq}}}} } & {i_{{{\text{Bd}}}} } & {i_{{{\text{Bq}}}} } \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} {\psi_{{{\text{Md}}}} } & {\psi_{{{\text{Mq}}}} } & {\psi_{{{\text{Bd}}}} } & {\psi_{{{\text{Bq}}}} } \\ \end{array} } \right]^{{\text{T}}} \\ = \frac{1}{2}[L_{{{\text{Md}}}} (i_{{{\text{Md}}}} + i_{{\text{f}}} )^{2} + L_{{{\text{Mq}}}} i_{{{\text{Mq}}}}^{2} + L_{{\text{B}}} (i_{{{\text{Bd}}}}^{2} + i_{{{\text{Bq}}}}^{2} )] \\ \, + [i_{{{\text{Md}}}} i_{{{\text{Bd}}}} + i_{{{\text{Mq}}}} i_{{{\text{Bq}}}} + i_{{\text{f}}} i_{{{\text{Bd}}}} ]L_{{\text{C}}} x \\ \, + [ - i_{{{\text{Md}}}} i_{{{\text{Bq}}}} + i_{{{\text{Mq}}}} i_{{{\text{Bd}}}} - i_{{\text{f}}} i_{{{\text{Bq}}}} ]L_{{\text{C}}} y \\ \end{gathered} $$
(3)

Since the kinetic energy of the PMa-BSynRM during operation can be equivalent to the integral of the torque to the rotation angle, the virtual displacement method is used to differentiate θ in the (3), and the electromagnetic torque expression obtained is as follows

$$ \begin{gathered} T_{{\text{e}}} = \frac{3}{2}P_{{\text{M}}} [\psi_{{\text{f}}} i_{{{\text{Mq}}}} + (L_{{{\text{Md}}}} - L_{{{\text{Mq}}}} )i_{{{\text{Md}}}} i_{{{\text{Mq}}}} ] + P_{{\text{B}}} (xL_{{\text{C}}} i_{{\text{f}}} i_{{{\text{Bd}}}} \\ + yL_{{\text{C}}} i_{{\text{f}}} i_{{{\text{Bq}}}} ) + \left[ {\begin{array}{*{20}c} {i_{{{\text{Md}}}} } & {i_{{{\text{Mq}}}} } \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} {L_{{\text{C}}} x} & { - L_{{\text{C}}} y} \\ {L_{{\text{C}}} y} & {L_{{\text{C}}} x} \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} { - i_{{{\text{Bd}}}} } \\ {i_{{{\text{Bq}}}} } \\ \end{array} } \right] \\ \end{gathered} $$
(4)

According to the rotor dynamics theory, the rotor’s motion equation can be obtained as

$$ \left\{ \begin{gathered} m\ddot{x} = F_{{\text{x}}} - f_{{\text{x}}} \hfill \\ m\ddot{y} = F_{{\text{y}}} - mg - f_{{\text{y}}} \hfill \\ \frac{\pi }{30}\frac{{J\dot{n}}}{{P_{{\text{M}}} }} = T_{{\text{e}}} - T_{{\text{L}}} \hfill \\ \end{gathered} \right. $$
(5)

where m is the mass of the rotor, fx and fy are the interference force applied along the x- and y-direction, respectively, g is the gravity constant, J is the inertia of the rotor motion, n is the rotating velocity of the rotor, TL is the load torque.

2.3 The mathematical model of speed system considering the coupling of suspension force system

It can be found that in (4), the values of the latter two items are related to rotor eccentricity, so in the control process, only by stably controlling the rotor suspension and making it in a balanced position can a more stable torque output be obtained. However, in the actual system, due to the influence of processing technology and other problems, the rotor has unbalanced vibration, and its actual displacement can be approximated as [21]

$$ \left\{ \begin{gathered} x(t) = A\cos (\omega_{r} t + \psi ) \hfill \\ y(t) = A\sin (\omega_{r} t + \psi ) \hfill \\ \end{gathered} \right. $$
(6)
$$ A = \frac{{m\varepsilon \omega _{r}^{3} \cos \varphi }}{{\omega _{r} [K_{C} (k_{p} + k_{d} ) - K_{M} - 2m\omega _{r}^{2} ]\cos \psi + K_{C} k_{i} \sin \psi }} $$
(7)
$$ \psi = \phi + \arctan \left[ {\frac{{K_{C} k_{{\text{i}}} }}{{K_{C} (k_{p} + k_{d} )\omega_{r} - K_{M} \omega_{r} - 2m\omega_{r}^{3} }}} \right] $$
(8)

Combined with the above formula, the torque expression of PMa-BSynRM can be written as

$$ T_{{\text{e}}} = T_{{{\text{em}}}} + T_{{{\text{ed}}}} $$
(9)

where Tem is the electromagnetic torque of the motor torque system, while Ted is the influence of the suspension force system coupling on the electromagnetic torque, which can be regarded as a disturbance. Combined with the effects of unbalanced vibration, the perturbation torque expression can be written as

$$ \begin{aligned} T_{{{\text{ed}}}} = & \left( {P_{{\text{B}}} L_{{\text{C}}} i_{{\text{f}}} i_{{{\text{Bd}}}} - L_{{\text{C}}} i_{{{\text{Md}}}} i_{{{\text{Bd}}}} + L_{{\text{C}}} i_{{{\text{Mq}}}} i_{{{\text{Bq}}}} } \right)x \\ &+ (P_{{\text{B}}} L_{{\text{C}}} i_{{\text{f}}} i_{{{\text{Bd}}}} - L_{{\text{C}}} i_{{{\text{Mq}}}} i_{{{\text{Bd}}}} + L_{{\text{C}}} i_{{{\text{Md}}}} i_{{{\text{Bd}}}} )y \\ \end{aligned} $$
(10)

Further sorting can be obtained

$$ T_{{\text{d}}} = D\sin (\omega_{{\text{r}}} t + \xi ) $$
(11)

where \(B = P_{{\text{B}}} L_{{\text{C}}} i_{{\text{f}}} i_{{{\text{Bd}}}} - L_{{\text{C}}} i_{{{\text{Md}}}} i_{{{\text{Bd}}}} + L_{{\text{C}}} i_{{{\text{Mq}}}} i_{{{\text{Bq}}}}\), \(C = P_{{\text{B}}} L_{{\text{C}}} i_{{\text{f}}} i_{{{\text{Bd}}}} - L_{{\text{C}}} i_{{{\text{Mq}}}} i_{{{\text{Bd}}}} + L_{{\text{C}}} i_{{{\text{Md}}}} i_{{{\text{Bd}}}}\), \(D = \sqrt {B^{2} A^{2} + C^{2} A^{2} }\).

From the above formula, it can be seen that the speed system will introduce a disturbance with the same frequency as the speed. When the motor is running at low speed, because the current of the torque winding is very small, the suspension winding has a great influence on the torque system, which cannot be ignored in order to make the motor run smoothly.

3 Proposed sensorless control based on ELESO

In this section, the complex vector model of PMa-BSynRM torque system is presented, and the PMa-BSynRM speed sensorless control method based on LESO and the proposed ELSO is presented, and the stability of the proposed method is analyzed.

3.1 PMa-BSynRM complex vector model of torque system

Before the design of the PMa-BSynRM speed sensorless controller, it is assumed that the rotor can be stably suspended in the center of rotation, that is, the influence of the suspension force system on the torque system is ignored. Since PMa-BSynRM rotor has obvious salient pole characteristics, its torque system model is modeled using effective flux method, and the torque system voltage expression is as follows

$$ {\mathbf{u}}_{{{\text{dq}}}}^{{\text{M}}} = (R_{{{\text{Ms}}}} + sL_{{{\text{Mq}}}} + j\omega_{{\text{e}}} L_{{{\text{Mq}}}} ){\mathbf{i}}_{{{\text{dq}}}}^{{\text{M}}} + (s + j\omega_{{\text{e}}} )\psi_{{\text{f}}}^{{\text{a}}} $$
(12)

where \({\mathbf{u}}_{{{\text{dq}}}}^{{\text{M}}}\) and \({\mathbf{i}}_{{{\text{dq}}}}^{{\text{M}}}\) are the complex vector of voltage and current, \({\mathbf{u}}_{{{\text{dq}}}}^{{\text{M}}} = u_{{{\text{Md}}}} + ju_{{{\text{Mq}}}}\), \({\mathbf{i}}_{{{\text{dq}}}}^{{\text{M}}} = i_{{{\text{Md}}}} + ji_{{{\text{Mq}}}}\), s is the Laplace operator, \(\psi_{{\text{f}}}^{{\text{a}}}\) is the effective flux, represents the flux that actually generates the total torque, including the reluctance torque component, which expression is

$$ \psi_{{\text{f}}}^{{\text{a}}} = \psi_{{\text{f}}} + (L_{{{\text{Md}}}} - L_{{{\text{Mq}}}} )i_{{{\text{Md}}}} $$
(13)

To avoid the accumulation of angle errors during coordinate transformation, the sensorless control algorithm is designed in the stationary coordinate system, and the correspondence between the complex vectors in the synchronous coordinate system and the stationary coordinate system is

$$ {\mathbf{u}}_{\alpha \beta }^{{\text{M}}} = (R_{{{\text{Ms}}}} + sL_{{{\text{Mq}}}} ){\mathbf{i}}_{\alpha \beta }^{{\text{M}}} + e^{j\theta } (s + j\omega_{{\text{e}}} )\psi_{{\text{f}}}^{{\text{a}}} $$
(14)

If the last term is defined as the complex vector of the back potential, then the complex vector voltage equation under the two-phase static shafting is

$$ {\mathbf{u}}_{\alpha \beta }^{{\text{M}}} = (R_{{{\text{Ms}}}} + sL_{{{\text{Mq}}}} ){\mathbf{i}}{}_{\alpha \beta }^{{\text{M}}} + {\mathbf{e}}_{\alpha \beta } $$
(15)

At steady state, \(s\psi_{{\text{f}}}^{{\text{a}}}\) is equal to zero, and the component of the complex vector of the back potential is

$$ {\mathbf{e}}_{\alpha \beta } = e_{{\upalpha }} + je_{{\upbeta }} = - \omega_{{\text{e}}} \psi_{{\text{f}}}^{{\text{a}}} \sin \theta_{{\text{e}}} + j\omega_{{\text{e}}} \psi_{{\text{f}}}^{{\text{a}}} \cos \theta_{{\text{e}}} $$
(16)

It can be seen that (16) contains the position information of the rotor, so the sensorless control of the torque system can be designed according to (16).

3.2 Rotor position estimation method based on LESO

With current \({\mathbf{i}}_{{{\text{dq}}}}^{{\text{M}}}\) as the state variable, according to (15), the state space model of the system can be expressed as

$$ \frac{{{\text{d}}{\mathbf{i}}_{\alpha \beta }^{{\text{M}}} }}{{{\text{d}}t}} = b_{{\text{i}}} {\mathbf{u}}_{\alpha \beta }^{{\text{M}}} + A_{0} {\mathbf{i}}{}_{\alpha \beta }^{{\text{M}}} + {\mathbf{E}}_{\alpha \beta } $$
(17)

where \({\mathbf{E}}_{\alpha \beta } = - b_{{\text{i}}} {\mathbf{e}}_{\alpha \beta }\), bi = 1/LMq, A = − Rs/Lq.

Back EMF calculation vector

$$ {\mathbf{E}}_{\alpha \beta } = {\mathbf{E}}_{\alpha \beta }^{{{\text{ideal}}}} + {\mathbf{f}}_{{\text{e}}} $$
(18)

Even though the sources (internal/external) and manifestations (periodic/aperiodic) of perturbations are different, they are eventually transformed into similar forms under the normalization idea of ESO's total perturbations, which simplifies system design and analysis.

Taking the total disturbance fe, ideal back EMF \({\mathbf{E}}_{\alpha \beta }^{{{\text{ideal}}}}\) as new state variables, (17) can be written as

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\text{d}}}{{{\text{d}}t}}{\mathbf{i}}_{\alpha \beta }^{{\text{M}}} = A_{0} {\mathbf{i}}_{\alpha \beta }^{{\text{M}}} + {\mathbf{E}}_{\alpha \beta }^{{{\text{ideal}}}} + {\mathbf{f}}_{{\text{e}}} + b_{{{\text{i}}0}} {\mathbf{u}}_{\alpha \beta } } \hfill \\ {\frac{{\text{d}}}{{{\text{d}}t}}{\mathbf{f}}_{{\text{e}}} = {\mathbf{x}}_{{\text{e}}} } \hfill \\ {\frac{{\text{d}}}{{{\text{d}}t}}{\mathbf{E}}_{\alpha \beta }^{{{\text{ideal}}}} = {\mathbf{y}}_{e} } \hfill \\ \end{array} } \right. $$
(19)

where A0 is the estimated value of A0, bi0 is the estimate of bi, xe is the differential vector of fe, ye is the differential vector of the ideal back EMF \({\mathbf{E}}_{\alpha \beta }^{{{\text{ideal}}}}\).

$$ {\mathbf{f}}_{{\text{e}}} = (A - A_{0} ){\mathbf{i}}_{\alpha \beta }^{{\text{M}}} + (b_{{\text{i}}} - b_{{{\text{i0}}}} ){\mathbf{u}}_{\alpha \beta }^{{\text{M}}} - b_{{\text{i}}} ({\mathbf{f}}_{{{\text{dc}}}} + {\mathbf{f}}_{{\text{h}}} ) $$
(20)

According to (19), a LESO-based back EMF observer is constructed

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\text{d}}}{{{\text{d}}t}}{\hat{\mathbf{i}}}_{\alpha \beta }^{{\text{M}}} = {\mathbf{A}}_{0} {\hat{\mathbf{i}}}_{\alpha \beta }^{{\text{M}}} + ({\hat{\mathbf{E}}}_{\alpha \beta }^{{{\text{ideal}}}} + {\hat{\mathbf{f}}}_{{\text{e}}} ) + b_{{{\text{i}}0}} {\mathbf{u}}_{\alpha \beta }^{{\text{M}}} + \beta_{{{\text{e1}}}} {{\varvec{\upvarepsilon}}}_{{\text{e}}} } \hfill \\ {\frac{{\text{d}}}{{{\text{d}}t}}({\hat{\mathbf{E}}}_{\alpha \beta }^{{{\text{ideal}}}} + {\hat{\mathbf{f}}}_{{\text{e}}} ) = \beta_{{{\text{e2}}}} {{\varvec{\upvarepsilon}}}_{{\text{e}}} } \hfill \\ \end{array} } \right. $$
(21)

where \({{\varvec{\upvarepsilon}}}_{{\text{e}}} = {\mathbf{i}}_{\alpha \beta }^{{\text{M}}} - {\hat{\mathbf{i}}}_{\alpha \beta }^{{\text{M}}}\) is the current error vector, \({\mathbf{A}}_{0} {\hat{\mathbf{i}}}_{\alpha \beta }^{{\text{M}}}\) is a known kinetic model, \({\hat{\mathbf{i}}}_{\alpha \beta }^{{\text{M}}} ,{\hat{\mathbf{E}}}_{\alpha \beta }^{{{\text{ideal}}}}\) are the observations of \({\mathbf{i}}_{\alpha \beta }^{{\text{M}}} ,{\mathbf{E}}_{\alpha \beta }^{{{\text{ideal}}}}\), respectively, βe1, βe2 are the gains of linear ESO.

Since the back EMF and the total disturbance can only be observed by the same method (that is, integral), combined with (21), the existence of coupling makes the observer only estimate \({\hat{\mathbf{E}}}_{\alpha \beta }\), the back EMF with disturbance,

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\text{d}}}{{{\text{d}}t}}{\hat{\mathbf{i}}}_{\alpha \beta }^{{\text{M}}} = {\mathbf{A}}_{0} {\hat{\mathbf{i}}}_{\alpha \beta }^{{\text{M}}} + {\hat{\mathbf{E}}}_{\alpha \beta } + b_{{{\text{i}}0}} {\mathbf{u}}_{\alpha \beta }^{{\text{M}}} + \beta_{{{\text{e1}}}} {{\varvec{\upvarepsilon}}}_{{\text{e}}} } \hfill \\ {\frac{{\text{d}}}{{{\text{d}}t}}{\hat{\mathbf{E}}}_{\alpha \beta } = \beta_{{{\text{e2}}}} {{\varvec{\upvarepsilon}}}_{{\text{e}}} } \hfill \\ \end{array} } \right. $$
(22)

The transfer function reflecting the observed performance is equal to the transfer function between the estimated back EMF and the actual back EMF, which can be derived as

$$ \frac{{\hat{e}_{{\alpha {(}\beta {)}}} }}{{e_{{\alpha {(}\beta {)}}} }} = \frac{{\beta_{e2} }}{{s^{2} + ( - A + \beta_{{{\text{e1}}}} )s + \beta_{{{\text{e}}2}} }} $$
(23)

In order to ensure the stability and rapid convergence of the observer, the closed-loop pole is assigned to the same point − ω0, and the gain coefficient of the extended observer can be set as follows

$$ L = [\begin{array}{*{20}c} {\beta_{{{\text{e1}}}} } & {\beta_{{{\text{e2}}}} } \\ \end{array} ]^{{\text{T}}} = [\begin{array}{*{20}c} {2\omega_{0} + A} & {\omega_{0}^{2} } \\ \end{array} ]^{{\text{T}}} $$
(24)

where ω0 is the bandwidth of the extended observer (Fig. 2).

Fig. 2
figure 2

Bode diagram based on LESO observer

Full size image

As can be seen from (23), the transfer function is similar to a second-order low-pass filter, and the graph is the corresponding Bode graph. The linear observer uses only pure integrator, and regards any disturbance in the cutoff frequency and the ideal back potential as the estimated back potential. The signal outside the cutoff frequency is attenuated rapidly, and it does have certain inhibition ability for the higher harmonic disturbance in the back potential.

3.3 Rotor position estimation method based on ELESO

In this section, a sensorless control algorithm for medium and high speed based on ELESO is presented. The proposed ELESO can be designed as

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\text{d}}}{{{\text{d}}t}}{\hat{\mathbf{i}}}_{\alpha \beta }^{{\text{M}}} = {\mathbf{A}}_{0} {\hat{\mathbf{i}}}_{\alpha \beta }^{{\text{M}}} + {\hat{\mathbf{E}}}_{\alpha \beta }^{{{\text{ideal}}}} + {\hat{\mathbf{f}}}_{{\text{e}}} + b_{{{\text{i}}0}} {\mathbf{u}}_{\alpha \beta }^{{\text{M}}} + \beta_{{{\text{e1}}}} {{\varvec{\upvarepsilon}}}_{{\text{e}}} } \hfill \\ {\frac{{\text{d}}}{{{\text{d}}t}}{\hat{\mathbf{f}}}_{{\text{e}}} = \beta_{{{\text{e2}}}} {{\varvec{\upvarepsilon}}}_{{\text{e}}} } \hfill \\ {{\hat{\mathbf{E}}}_{\alpha \beta }^{{{\text{ideal}}}} (s) = G_{{{\text{QPR}}}} (s) \cdot \varepsilon_{\alpha \beta } (s)} \hfill \\ \end{array} } \right. $$
(25)

where GQPR(s) is the transfer function of the quasi-proportional resonant (QPR) controller, which is embedded into LESO to enhance the bandwidth of the original observer and improve the estimation accuracy of the back EMF. When the resonant frequency ω0 is input to estimate the rotational speed, the rotational speed adaptive can be realized. The structure diagram of sensorless control algorithm based on ELESO is shown in Fig. 3.

Fig. 3
figure 3

Structure diagram of position sensorless control algorithm based on ELESO

Full size image

The transfer function between the estimated back EMF and the actual back EMF obtained by the proportional resonance controller is

$$ G_{{{\text{OA}}}} (s) = \frac{{{\hat{\mathbf{E}}}_{\alpha \beta }^{{{\text{ideal}}}} }}{{{\hat{\mathbf{E}}}_{\alpha \beta } }} = \frac{{sG_{{{\text{QPR}}}} (s)}}{{s^{2} + [\beta_{{{\text{e}}1}} - A + G_{{{\text{QPR}}}} (s)]s + \beta_{{{\text{e}}2}} }} $$
(26)

Combined with (16), it can be obtained

$$ {\hat{\mathbf{E}}}_{\alpha \beta }^{{{\text{ideal}}}} = G_{{{\text{OA}}}} (s){\hat{\mathbf{E}}}_{\alpha \beta }^{{{\text{ideal}}}} + G_{{{\text{OA}}}} (s){\mathbf{f}}_{{\text{e}}} $$
(27)

As can be seen from the above equation, the estimated back EMF comes from two parts: disturbance and ideal back EMF, and the transfer function representing the observation performance and anti-disturbance ability has the same form GOA(s), whose Bode diagram is shown in Fig. 4 (ω0 = 100π rad/s).

Fig. 4
figure 4

Bode diagram based on ELESO observer

Full size image

It can be seen that the phase shift is zero at the given resonant frequency. The amplitude decays rapidly on the right side of the resonant point. Therefore, the back EMF observer based on ELESO has certain characteristics of resisting high-frequency interference. With the increase of the bandwidth of the linear part, the gain on the left side of the resonant point decreases gradually. It can be seen that the existence of the linear part improves the ability of the back EMF observer to resist DC disturbance. In any operating state of the motor, the resonant frequency of the QPR is always locked to the frequency of the back EMF (obtained by the PLL), so the back EMF observer is like a sliding bandpass filter that can track the back EMF at any speed with high gain.

3.4 Stability analysis

In practical applications, the stability analysis in the discrete domain is needed because the microprocessor is used for discrete control, and the parameters of the observer depend on the time step.

According to the form of GOA(s), the open-loop transfer function with unit negative feedback can be written directly

$$ G_{{{\text{OAO}}}} (s) = \frac{{{\hat{\mathbf{E}}}_{\alpha \beta }^{{{\text{ideal}}}} }}{{{\hat{\mathbf{E}}}_{\alpha \beta } }} = \frac{{sG_{{{\text{QPR}}}} (s)}}{{s^{2} + [\beta_{{{\text{e}}1}} - A]s + \beta_{{{\text{e}}2}} }} $$
(28)

When the estimated motor speed is input to the resonant frequency ω0 of QPR to achieve self-adaptation, the discrete open-loop transfer function of the proposed observer can be obtained by bilinear variation method.

$$ G_{{{\text{OAO}}}} (z) = Z\left[ {k_{{\text{p}}} \frac{{s(s^{2} + 2\omega_{{\text{c}}} (1 + k_{{\text{r}}} /k_{{\text{p}}} )s + \hat{\omega }_{{\text{e}}}^{2} )}}{{(s^{2} + (\beta_{{{\text{e}}1}} - A)s + \beta_{{{\text{e}}2}} )(s^{2} + 2\omega_{{\text{c}}} s + \hat{\omega }_{{\text{e}}}^{2} )}}} \right] $$
(29)

According to (29), when the open-loop gain kp increases from 0 to ∞, the discrete closed-loop root locus of the proposed ELESO with different parameters is shown in Fig. 5. In Fig. 5, the proposed observer trajectories are shown when the rotor speed ωe increases from 0 to 1200π rad/s in steps of 300π rad/s and when the observer bandwidth ω0 increases from 0 to 1000 rad/s in steps of 250 rad/s with other parameters constant. It can be seen that the characteristic root of the closed-loop observer is always kept within the Z-plane unit circle and the system has a large stable margin.

Fig. 5
figure 5

The proposed discrete observer root locus. (a) Increase of rotor speed. (b) Increase of observer bandwidth

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4 Simulation results

In order to verify the effectiveness of the proposed method, the numerical simulations for the PMa-BSynRM system have been performed. The proposed speed sensorless control scheme based on ELESO is shown in Fig. 6. In these tests, the sensorless control method based on ELESO is compared with the sensorless control method based on LESO. The same saturation limit is imposed in two methods simultaneously. By adjusting the control parameters of each method, better results are obtained. In addition, two standard proportional integral controllers are used in two current tracking loops. For traditional LESO and proposed ELESO, the gain ω is set to 6500 rad/s, ELESO controller parameters ωc = π rad/s, kr = 90, kp = 0.5. PI controller parameters of traditional PLL kp = 200, ki = 11,000.

Fig. 6
figure 6

Vector control block diagram of the PMa-BSynRM

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To compare the control performance of LESO and ELESO controls, the position estimates of PMa-BSynRM running at 1000 r/min and 3000 r/min are given in Figs. 7 and 8. Figures 7a and 8a show that the rotor position estimation error of the traditional LESO-based control method increases with the increase of rotational speed, and the average position estimation error at 1000 r/min and 3000 r/min is 0.1048rad and 0.3914rad, respectively. In Fig. 7b and 8b, the average error of rotor position estimation for the ELESO-based control method is 0.0397 rad and 0.1989 rad, respectively. According to the experimental results, it can be concluded that when LESO method is used for speed sensorless control, the phase of the back EMF lags due to the low-pass filtering characteristics of the observer. After the phase-locked loop action, the lag of the back EMF is reflected in the angle error. In addition, consistent with theoretical analysis, with the increase of rotor speed, the phase lag is more serious. By using the proposed ELESO method, the estimated back EMF is close to the actual at different speeds, the rotor position error is smaller, and the angle error only increases slightly with the increase of speed, which has a good speed adaptive ability. Therefore, the proposed speed sensorless control algorithm can effectively improve the accuracy of rotor position estimation.

Fig. 7
figure 7

Simulation results of steady state at 1000 r/min. (a) LESO. (b) ELESO

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Fig. 8
figure 8

Simulation results of steady state at 3000 r/min. (a) LESO. (b) ELESO

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In PMa-BSynRM speed sensorless control simulation tests, rotor radial displacement waveforms based on LESO method and ELESO method are shown in Fig. 9. Unbalanced vibration exists in the rotor of bearingless motor during rotation, and its vibration frequency corresponds to the mechanical frequency of the rotor [21]. When the rotor speed is 1000 r/min, the rotor vibration frequency is 16.7 Hz, and the rotor vibration amplitude in the x- and y-direction is 23 µm, respectively. When the rotor speed is 3000 r/min, the rotor vibration frequency is 50 Hz, and the rotor vibration amplitude in the x- and y-directions is 35 µm, respectively. The radial displacement waveforms of PMa-BSynRM rotor based on ELSO speed sensorless control are shown in Fig. 10b. When the rotor speed is 1000 r/min, the rotor vibration amplitude in the x- and y-direction is 16 µm, respectively. When the rotor speed is 3000 r/min, the rotor vibration amplitude in the x- and y-direction is 26 µm, respectively. Under the two control methods, the radial displacement of the rotor is much less than the 0.25 mm clearance between the rotor and the auxiliary bearing, which realizes the stable suspension operation of the rotor. The suspension force performance of the proposed LESO control method is superior to the traditional LESO control method, mainly due to the coupling between the PMa-BSynRM torque system and the suspension force system, as analyzed in Sect. 2. When the rotational speed is more stable, it helps to stabilize the suspension. In addition, the rotor angle also needs to be estimated in the control of the suspension force system, as shown in Fig. 6. Therefore, the accuracy of rotor angle estimation by speed sensorless control algorithm will also affect the suspension performance of the PMa-BSynRM.

Fig. 9
figure 9

Simulation results of the rotor radial displacement waveforms. (a) LESO. (b) ELESO

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Fig. 10
figure 10

Experimental platform

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5 Experimental validations

In this section, the proposed speed sensorless control strategy is verified on the experimental platform of a PMa-BSynRM, as shown in Fig. 10. One end of the PMa-BSynRM is supported by a three-degree-of-freedom aligning ball bearing, and the other end is equipped with an auxiliary bearing, which can realize two-degree-of-freedom suspension. The real-time rotor position information is collected via eddy current sensors that have been conditioned through an interface circuit board before also being sent to ADC ports on DSP for data acquisition. The basic parameters of the PMa-BSynRM used in the experiment are shown in Table 1. The controller utilizes TMS320F28335 with an interrupt frequency of 10 kHz to execute both the vector control algorithm for torque and the suspension control algorithm within the same interrupt process. The I-f starting mode is adopted to realize the starting purpose, and the speed threshold of I-f control transition to sensorless control is set to 300 r/min.

Table 1 Parameters of the prototype
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5.1 Steady-state experiment

In this section, the steady-state performance of the traditional LESO and the proposed ELESO are compared. The actual rotor position and estimated rotor position error of the PMa-BSynRM under LESO and ELESO speed sensorless control methods are depicted in Figs. 11 and 12, respectively, at speeds of 1000 r/min and 3000 r/min.

Fig. 11
figure 11

Steady-state test results at 1000 r/min. (a) LESO. (b) ELESO

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Fig. 12
figure 12

Steady-state test results at 3000 r/min. (a) LESO. (b) ELESO

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Figures 11a and 12a show that the rotor position estimation error of traditional LESO-based control method increases with the increase of rotational speed, and the mean position estimation error at 1000 r/min and 3000 r/min is 0.1351 rad and 0.5024 rad, respectively. In Figs. 11b and 12b, the mean error of rotor position estimation for the ELESO-based control method is 0.0493 rad and 0.2198 rad, respectively. Compared with LESO method, the rotor position estimation error based on ELESO method decreases by 63.5% at 1000 r/min and by 56.25% at 3000 r/min.

In the PMa-BSynRM speed sensorless control experiment, the rotor radial displacement waveform based on LESO method and ELESO method is shown in Fig. 13. The radial displacement waveforms of PMa-BSynRM rotor based on LESO speed sensorless control method are shown in Fig. 13a. When the rotor speed is 1000 r/min and 3000 r/min, the rotor vibration frequency is 16.7 Hz and 50 Hz, and the rotor vibration amplitude in the x- and y-direction is 24 µm and 25 µm, 38 µm and 44 µm, respectively. The radial displacement waveform of PMa-BSynRM rotor based on ELSO speed sensorless control is shown in Fig. 13b. When the rotor speed is 1000 r/min and 3000 r/min, the rotor vibration amplitude in the x- and y-direction is 14 µm and 12 µm, 41 µm and 33 µm, respectively. It can be seen that compared with the simulation, the harmonic content of the experimental waveform is larger, because the rotor vibration is caused by many factors, and the simulation usually only considers the unbalanced vibration as the main factor.

Fig. 13
figure 13

Radial displacement waveforms of the rotor. (a) LESO. (b) ELESO

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The rotor position waveforms of PMa-BSynRM low-speed sensorless control experiment based on ELESO are presented in Fig. 14. At 250 r/min, the error of rotor forward estimation is 0.1971 rad, and the error of rotor reverse estimation is 0.2223 rad. Since ELESO's speed sensorless control method estimates rotor position based on back EMF, it is not suitable for low-speed control. However, due to ELESO's good noise interference ability, stable speed sensorless control can still be achieved at 250 r/min forward rotation and reverse rotation.

Fig. 14
figure 14

Steady-state test results at 250 r/min. (a) Forward. (b) Reverse

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The radial displacement waveforms of PMa-BSynRM at 250 r/min with speed sensorless control are given in Fig. 15. Because accurate rotor position is needed to generate stable suspension force in PMa-BSynRM suspension force control, the accuracy of rotor position estimation is also the key to stable suspension of the rotor. It can be seen that PMa-BSynRM is turning forward at 250 r/min, and the rotor vibration amplitude in the x- and y-direction is 8 m, respectively. When the rotor is inverted at 250 r/min, the vibration amplitude in x- and y-direction is 10 m and 11 m. The rotor vibrates at 50 Hz. At low speed, the unbalance vibration force caused by the mass unbalance of the rotor is small. Therefore, different from 1000 r/min and 3000 r/min, rotor vibration at this time is caused by dead zone of the inverter, and the frequency is 6 times of the current frequency [22].

Fig. 15
figure 15

Radial displacement waveforms of the rotor. (a) Forward. (b) Reverse

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5.2 Transient experiment

The transient performance of the conventional LESO method and the proposed ELESO method for speed sensorless control is compared in this section, considering variations in acceleration and radial displacement conditions.

In the acceleration experiment, the speed command changes in the order of 600–1000–3000 r/min. The actual speed, estimated speed, and position estimation error are shown in Fig. 16. The speed estimation performance is excellent, with a maximum speed estimation error of 13 r/min. The position estimation error increases with the increase of rotational speed, and the maximum error is only about 0.1144 rad, which verifies the good dynamic performance of the proposed method.

Fig. 16
figure 16

Acceleration test results and the speed command changes in the sequence of 600–1000-3000 r/min

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According to the analysis in Sect. 2 due to the coupling between the torque system and the suspension force system of bearingless motor, the change of the working condition of the suspension force system will affect the torque system. Under the change of rotor radial displacement, the speed waveform of PMa-BSynRM without speed sensing control of traditional LESO and proposed ELESO is given in Fig. 17. During the experiment, the rotor displacement in the x direction is referred to as a 10 Hz square wave, and the radial displacement waveform and reference waveform in the x direction of the rotor are shown in Fig. 17. As shown in Fig. 17, the change of rotor radial displacement will affect the speed and cause the speed fluctuation, which is caused by the coupling between the speed system and the suspension force system of the bearingless motor. The speed fluctuation of the proposed ELESO method is significantly smaller than that of the traditional LESO method, which is consistent with the analysis in Sect. 3. In the disturbance experiment caused by the change of radial displacement, the performance of the proposed speed sensorless control method is superior to that of the traditional LESO method, which once again verifies the excellent performance of the proposed method.

Fig. 17
figure 17

Disturbance experiment

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6 Conclusion

In this paper, a speed sensorless control method of PMa-BSynRM is proposed, which can effectively reduce the interference of PMa-BSynRM torque system and suspension force system coupling to speed control, thereby enhancing the performance of speed control. Through the proposed ELESO, which embeds a QPR controller in LESO to estimate the back EMF of PMa-BSynRM and subsequently achieve speed estimation. The speed sensorless control of PMa-BSynRM can be enhanced by configuring LESO and QPR controllers in ELESO to effectively mitigate DC and high-frequency disturbances. The proposed control method is verified by PMa-BSynRM experimental platform. The rotor position estimation error based on ELESO method decreases by 63.5% at 1000 r/min and by 56.25% at 3000 r/min. The experimental results demonstrate that the proposed PMa-BSynRM speed sensorless control method, based on ELESO, exhibits reduced observation error and minimal impact on suspension force coupling compared to the traditional LESO approach. The proposed method is not only suitable for PMa-BSynRM, but also can be used for reference for other types of bearingless motors.