Sensorless Control of Permanent Magnet Synchronous Machine with Magnetic Saliency Tracking Based on Voltage Signal Injection ^†

Abstract

This paper presents a sensorless control method of a permanent magnet synchronous machine (PMSM) with magnetic saliency estimation. This is based on a high-frequency injection (HFI) technique applied on the modified PMSM model in the γδ reference frame. Except for sensorless control, an emphasis is placed on the magnetic saliency estimation to indicate a practical approach in tracking PMSM inductance variations. The magnetic saliency is determined using calculations embedded in the speed and position algorithm through current measurements. A notable characteristic of the modified PMSM model is that the corresponding rotor flux integrates both permanent magnet and saliency term fluxes. In applying a HFI technique for sensorless control, the structure of the PMSM flux model is formatted accordingly. A novel inductance matrix is derived that is completely compatible with the HFI methodology, since its elements include terms of angle error differential and average inductances. In addition, a sliding mode observer (SMO) is designed to estimate the speed and angle of rotor flux based on equivalent control applying a smooth function of the angle error instead of a sign one to reduce the chattering phenomenon. The control strategy is principally based on the adequacy of the proposed modified model and on the appropriateness of the SMO structure to successfully track the rotor flux position with the required stability and accuracy. Simulation results demonstrate the performance of the PMSM sensorless control verifying the effectiveness of the proposed algorithm to detect PMSM saliency, speed and position in steady state and transient modes successfully.

Notation

• u_d, u_q = dq axis stator voltages
• i_d, i_q = dq axis stator currents
• λ_d, λ_q = dq axis stator magnetic fluxes
• λ_m = rotor magnetic flux
• L_d, L_q = dq axis inductances
• ΣL =(L_d+L_q)/2 = average inductance
• ΔL = (L_d-L_q)/2 = differential inductance
• r_s = stator resistance
• u_γ, u_δ= γδ axis stator voltages
• i_γ, i_δ= γδ axis stator currents
• λ_γ, λ_δ= γδ axis stator magnetic fluxes
• p = number of pole pairs
• θ = θ_e = electrical angular position
• ω = ω_e = electrical angular speed

1. Introduction

Permanent magnet synchronous machines (PMSM) are used to an increasing rate in a wide rage of industrial applications due to their very appealing properties. Among them are the high efficiency and dynamics, the high torque inertia ratio, the small size and the very low torque ripple. For internal PMSM (IPMSM) in particular, low or even zero-speed sensorless operation is feasible, while field weakening allows extension of operation at speeds above the nominal. Efficient operation of synchronous machines implies very low energy losses compensating their higher initial cost. In a high-performance PMSM operation, advanced control methods are applied, such as field oriented control (FOC), aiming to achieve smooth rotation over the entire speed range, fast response and full torque control [1,2]. However, these PMSM control methods need an accurate rotor position to fulfil these control requirements. Typically, rotor position sensors, such as optical encoders or magnetic resolvers are used to directly perform rotor angle measurement. Nevertheless, the mounted sensors increase the total cost and mainly introduce noise reducing the reliability of the implemented control particularly in electrically hostile environment. As a consequence, an increasing interest was created in sensorless control techniques that have the aim of employing indirect techniques for the rotor position estimation instead of using mechanical sensors.

In the literature, a plethora of PMSM sensorless approaches have been proposed that could be mainly classified into two strategies: the fundamental excitation and saliency and signal injection [3,4,5,6,7]. The first strategy has been established based on the state observer methodology using only measurements of fundamental excitation variables such as stator voltages and currents. Appropriately designed back electromotive force (back-EMF or BEMF) or magnetic flux observers estimate the rotor position and speed information [8,9,10,11,12]. In particular, the back-EMF estimation methods are normally capable of providing accurate positions in a limited speed range from the middle to high-speeds [1]. At low speed range, the induced back-EMF is relatively very small degrading seriously the estimation accuracy, since the amplitude of back-EMF is comparable to the added measurement noise [1,2,13]. By contrast with a fundamental strategy, the saliency and signal injection-based methods are applied to detect rotor flux position by means of PMSM spatial inductance variations. At low speed range even at zero speed operation, a rotor position can be accurately estimated through injecting voltage signals of high frequency, as it is shown in Figure 1 [4,5,6,14,15]. This PMSM sensorless strategy is based on the rotor magnetic anisotropy or on the presence of magnetic saliency, i.e., (L_d − L_q) ≠ 0. Although high-frequency injection (HFI) methods are very efficient in terms of angle estimation accuracy, the control performance is strongly affected by the PMSM magnetic, mechanical parameters and the mathematical model used. Furthermore, the injected high-frequency signal may introduce a considerable audible noise that could not be tolerable [16]. Moreover, the proposed PMSM model plays a key role in the proposed sensorless approach. Since the rotor position cannot be detected, the d–q axis mathematical model cannot be applied directly. Most approaches are based on the estimation of the PMSM variables such as the back electromotive force, in the stationary reference frame αβ. In a typical sensorless control scheme, the rotor speed and position are obtained from estimated PMSM parameters or variables, such as back-EMF or rotor flux, implementing state estimators such as a Luenberger, sliding mode or phase-locked-loop (PLL)-type observer [4,17]. Several developed observers were based on PMSM current model, which may cause instability issues in some speed ranges due to introduced model assumptions [4,17]. In avoiding such model simplifications, the magnetic saliency and the back-EMF terms are included into the so-called extended EMF [1,18,19]. However, the observers in the αβ stationary reference frame suffer from phase delay between the real and the estimated EMF, since the real extended EMF is a sinusoidal signal [18,20]. This undesired phase delay is due to the observer operation, which is similar to signal filtering. Despite the αβ stationary frame, PMSM variables, such as flux and extended EMF, are constant quantities in the dq (direct-quadrature) synchronous rotating reference frame that, however, cannot be applied directly. An advantageous alternative proposal is to express PMSM mathematical model in the γδ reference frame, which is rotating at estimated angular velocity $\widehat{\omega }$ and lagging behind the dq synchronous reference frame by electrical angle $\overline{\theta }$, i.e., the angle difference between dq and γδ (see Figure 2) [1,21,22]. Among the benefits gained from transforming the PMSM model to γδ reference frame is its suitability for sensorless control allowing the estimation of variables in a rotating frame rather than in stationary frame. Particularly, the γδ-modified model is more convenient regarding the magnetic saliency and it could be applied on both salient-pole (Ld ≠ Lq) and nonsalient-pole PMSM (Ld = Lq = Ls) [1,21]. Observers developed in the γδ reference frame are able to provide the angle error $\overline{\theta }$ between dq and γδ reference frames instead of the rotor angle θ. In addressing the phase delay issue, the PMSM model in the γδ reference frame is preferable, since the γδ frame is associated with the rotor flux. As a result, the phase delay between dq and γδ variables is very small considered as negligible [1].

Typically, PMSM are classified into two common types of permanent magnet (PM) machines, namely the surface permanent magnet synchronous machine (SPMSM or SPM) and internal permanent magnet synchronous machine (IPMSM or IPM). The magnetic saliency of a machine is defined as the difference between d-axis and q-axis inductances, i.e., (Ld − Lq). In SPMSM, the magnets are mounted on the surface of the rotor, whereas the magnets of IPMSM are buried inside the rotor. Since the permeability of permanent magnets is very low, it can be considered as equal to the air permeability along the flux paths. Therefore, the effective air gap remains the same in the magnetic flux paths of the d-axis and q-axis for SPMSM [23]. As a result, the inductance measured at the machine terminal is constant regardless of the rotor position, i.e., L_d = L_q, implying that the magnetic saliency of SPMSM is zero or very low, i.e., (Ld − Lq) ≅ 0. In contrast, the effective air gap in the magnetic flux path of IPMSM differs between the d-axis and q-axis depending on the rotor position, since the permanent magnets have lower permeability than iron. Hence, the machine inductance fluctuates implying a magnetic saliency different than zero, i.e., (Ld − Lq) ≠ 0. Measuring the inductance changes allows the estimation of rotor position. As a result, it is possible to detect the rotor position using inductance saliency (sensorless control or open speed control loop) [24,25,26]. On the other hand, the quantification of PMSM magnetic saliency is a very complex and difficult problem requiring data analysis of experimental measurements or finite element methods (FEM). Mainly, the inductance variance is associated with the magnetic saturation or rotor geometric saliency, which are the most common sources of magnetic saliency. Besides these, the rotor eccentricity, the eddy currents and the slotting of the rotor and stator may also cause magnetic saliency [15,23,24]. In dq axes, the synchronous inductances can be expressed as the sum of the leakage and magnetizing inductances, i.e., Ld = Lld + Lmd, Lq = Llq + Lmq, where Lld, Llq are leakage inductances and Lmd, Lmq represent the magnetizing inductances. Leakage inductances are associated with the slots, teeth and faces’ magnetic leakage, while magnetizing inductances are associated with the main magnetic flux passing through the air gap. Considering magnetic saturation and rotor geometric saliency, both of them influence the leakage and the magnetizing inductances [15,24,25,26]. The signal injection methods are able to detect both types of saliencies with injecting frequencies usually in the range from 0.5 kHz to 2 kHz [27]. However, the high-frequency injection may decrease the precision of estimation due to the current controller bandwidth limitations in a closed loop. Therefore, it is preferable to generate the harmonic voltage injection in the inverter stage [23].

In this work, a new mathematical model of PMSM is proposed expressed in a modified form for rotor flux equations and referred to the γδ estimated rotating frame. The PMSM model in γδ is much more convenient for sensorless control methods including the saliency and signal injection strategies [21]. The development of the PMSM modified model in γδ and the change of the stator flux to obtain a novel stator inductance matrix L_γδ constitute the main theoretical contributions of the present study. Saliency depended terms, such as (L_d − L_q)i_d, are embedded into the modified rotor flux in γδ allowing minimization the PMSM model approximations. Despite of the extended EMF or saliency back-EMF model, the modified γδ model is advantageous, since speed has no effect on the modified rotor flux instead [1,12,21,28,29]. As is proven, the γδ inductance matrix L_γδ is transformed from the original L_dq based on the angle difference between dq and γδ. In particular, it is also proven that the elements of the γδ inductance matrix L_γδ are functions of the average inductance, the differential inductance and angle error. This property is of great importance, since the presented analysis of the derived matrix L_γδ verifies the appropriateness of the γδ reference frame in the design procedure. Due to the injection of the high frequency voltage, the resulting high-frequency (HF) stator current contains two components with positive and negative frequencies, i.e., current vectors rotate in opposite directions. After appropriate processing and filtering, the PMSM rotor position information is extracted through a sliding mode observer (SMO) employing the equivalent control methodology [30]. Since chattering avoidance is important in sliding mode applications, the sgn(.) function is substituted by the smooth function tanh(.). Such a continuous approximation is very efficient allowing considerable reduction of chattering effect. Also, a notable advantageous property of the proposed solution is that the PMSM stator inductances in dq and magnetic saliency (L_d − L_q) = 2ΔL can be reliably calculated based on the magnitudes of the derived HF current components. Figure 1 presents in details the total control scheme based on the γδ modified model of PMSM aiming to estimate the rotor speed and position and calculate magnetic saliency. The present sensorless control includes the measurement, the estimation and the control phases. In the measurement phase, only the stator currents i_a and i_b are needed. Estimation of rotor speed and angle is curried out at observer after processing the current signals (modulation, low-pass filter (LPF) and band-pass filter (BPF)). The desired control is implemented using tree proportional-integral controllers: two for current control (inner loops) and one for speed control (outer loop). Injection succeeds adding the HF voltage signals to the γδ voltage components derived from current controllers. After obtaining the reference voltages u_α* and u_β*, a voltage source inverter (VSI) produces the desired voltage to feed the PMSM by means of appropriate modulation, such as Space Vector Pulse Width Modulation (SVPWM). The evolved algorithm permits the accurate speed/position estimation and on-line calculation of stator impedances and saliency for monitoring or fault detection [31,32,33,34,35]. Finally, Simulink/Matlab is used to examine and evaluate the proposed algorithm indicating very satisfactory results.

The rest of the paper is organized as follows. In Section 2, an analysis of the modified PMSM model is presented in dq and γδ reference frame emphasizing in inductance matrix and magnetic saliency. The high-frequency injection and rotor speed and position observer is described in Section 3. In deriving the stator inductances and saliency, the relations between filtered current signals are analyzed in Section 4. Simulation set-up and results are presented and discussed in Section 5, while Section 6 concludes the presented work.

2. Analysis of Modified Permanent Magnet Synchronous Machine (PMSM) Model

2.1. Modified PMSM Voltage and Flux/Current Model in Synchronous Reference Frame dq

The PMSM mathematical model depends mainly on the type, i.e., IPMSM or SPMSM, the geometric properties, and the reference frame used. In next paragraphs, a salient-pole permanent magnet (PM) synchronous machine is considered, where the air gap of the flux path is varying due to the presence of magnetic saliency, L_d ≠ L_q, and the induced back EMF (BEMF) is sinusoidal. The following equations express the PMSM model in the synchronous reference frame dq:

$$u_{dq}=r_s{i}_{dq}+\omega J_s{\lambda }_{dq}+{\dot{\lambda }}_{dq},$$

(1)

$${\lambda }_{dq}=L_{dq}{i}_{dq}+{\lambda }_{mdq},$$

(2)

where

$$i_{dq}=\left[\begin{array}{c}{u}_d\\ u_q\end{array}\right],{\lambda }_{dq}=\left[\begin{array}{c}{\lambda }_d\\ {\lambda }_q\end{array}\right],J_s=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],L_{dq}=\left[\begin{array}{cc}{L}_d& 0\\ 0& L_q\end{array}\right]\mathrm{and}{\lambda }_{mdq}=\left[\begin{array}{c}{\lambda }_m\\ 0\end{array}\right],$$

(3)

Considering the magnetic saliency term (L_d − L_q), the flux terms of the PMSM model in dq could be expressed in a symmetric form, i.e., the stator flux matrix λ_dq in Equation (2) can be equivalently rewritten as:

$${\lambda }_{dq}=\left[\begin{array}{cc}{L}_q& 0\\ 0& L_q\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}{\lambda }_m+\left(L_d-L_q\right)i_d\\ 0\end{array}\right]=\left[\begin{array}{cc}{L}_q& 0\\ 0& L_q\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}{\lambda }_{mm}\\ 0\end{array}\right]=L_{qq}{i}_{dq}+{\lambda }_{mmdq},$$

(4)

Here, the λ_mm is defined as the PMSM modified rotor flux in dq, which depends on the permanent magnet flux λ_m, the magnetic saliency (Ld − Lq), and the d-axis stator current i_d (see Figure 2). In practice, for small i_d currents, the modified rotor flux λ_mm is mainly dominated by λ_m.

2.2. Modified PMSM Voltage and Flux/Current Model in γδ

The modified rotor flux vector is schematically shown in Figure 2 for αβ, dq and γδ reference frames. By definition, the γδ reference frame is an arbitrary reference frame, which is rotating at an estimated angular velocity $\widehat{\omega }$ and lagging behind the dq reference frame by the electrical angle determined as $\Delta \theta =\overline{\theta }=\theta -\widehat{\theta }$ [1,19]. The γδ PMSM model is obtained through transforming the corresponding dq model by means of the transformation matrix K_Δθ defined in Equation (5), which depends on the angle difference between the dq and γδ rotating reference frames [1,21].

2.3. Modified PMSM Voltage and Flux/Current Model in γδ

The modified rotor flux vector is schematically shown in Figure 2 for αβ, dq and γδ reference frames. By definition, the γδ reference frame is an arbitrary reference frame, which is rotating at an estimated angular velocity $\widehat{\omega }$ and lagging behind the dq reference frame by the electrical angle determined as $\Delta \theta =\overline{\theta }=\theta -\widehat{\theta }$ [1,21]. The γδ PMSM model is obtained through transforming the corresponding dq model by means of the transformation matrix K_Δθ defined in Equation (5), which depends on the angle difference between the dq and γδ rotating reference frames [1,21],

$$K_{\Delta \theta }=\left[\begin{array}{cc}\cos \overline{\theta }& -\sin \overline{\theta }\\ \sin \overline{\theta }& \cos \overline{\theta }\end{array}\right],$$

(5)

Multiplying from the left both parts of Equations (1), (4) by K_Δθ, the γδ model of PMSM for stator voltage and flux is written as follows:

$$\begin{gathered} K_{\Delta \theta }{u}_{dq}=K_{\Delta \theta }{r}_s{i}_{dq}+K_{\Delta \theta }\omega J_s{\lambda }_{dq}+K_{\Delta \theta }{\dot{\lambda }}_{dq}\iff u_{\gamma \delta }=r_s{i}_{\gamma \delta }+\omega J_s{\lambda }_{\gamma \delta }+\left(-\dot{\overline{\theta }}{J}_s{\lambda }_{\gamma \delta }+{\dot{\lambda }}_{\gamma \delta }\right)\iff \\ {u}_{\gamma \delta }=r_s{i}_{\gamma \delta }+\omega J_s{\lambda }_{\gamma \delta }+\left(-\dot{\overline{\theta }}{J}_s{\lambda }_{\gamma \delta }+{\dot{\lambda }}_{\gamma \delta }\right)\iff u_{\gamma \delta }=r_s{i}_{\gamma \delta }+\widehat{\omega }{J}_s{\lambda }_{\gamma \delta }+{\dot{\lambda }}_{\gamma \delta }, \end{gathered}$$

(6)

and

$$K_{\Delta \theta }{\lambda }_{dq}=K_{\Delta \theta }{L}_{qq}{K}_{\Delta \theta }^{-1}{K}_{\Delta \theta }{i}_{dq}+K_{\Delta \theta }{\lambda }_{mmdq}\iff {\lambda }_{\gamma \delta }=L_{qq}{i}_{\gamma \delta }+{\lambda }_{mm\gamma \delta },$$

(7)

Here, λ_mmγδ represents the modified rotor magnetic flux in γδ defined by:

$$\begin{gathered} {\lambda }_{mm\gamma \delta }=K_{\Delta \theta }{\lambda }_{mmdq}=\left[\begin{array}{cc}\cos \overline{\theta }& -\sin \overline{\theta }\\ \sin \overline{\theta }& \cos \overline{\theta }\end{array}\right]{\lambda }_{mmdq}=\left[\begin{array}{cc}\cos \overline{\theta }& -\sin \overline{\theta }\\ \sin \overline{\theta }& \cos \overline{\theta }\end{array}\right]\left[{\lambda }_m+\left(L_d-L_q\right)i_d\right]\left[\begin{array}{c}1\\ 0\end{array}\right] \\ =\left[{\lambda }_m+\left(L_d-L_q\right)i_d\right]\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]=\left[\begin{array}{c}{\lambda }_m\cos \overline{\theta }\\ {\lambda }_m\sin \overline{\theta }\end{array}\right]+\left(L_d-L_q\right)\left[\begin{array}{c}{i}_d\cos \overline{\theta }\\ i_d\sin \overline{\theta }\end{array}\right], \end{gathered}$$

(8)

It is noted that the right hand-side of Equation (8) contains a term depended on magnetic saliency (L_d − L_q), i_d current and the angle difference Δθ. For HFI methods, it is convenient to replace i_d with its equivalent expressed as function of the currents in γδ and the angle difference Δθ.

2.4. PMSM Inductance Matrix L_γδ and Magnetic Saliency

Now taking into account that $i_d=i_{\gamma }\cos \overline{\theta }+i_{\delta }\sin \overline{\theta }$ and ΔL=(L_d − L_q)/2, the term that includes the magnetic saliency (L_d − L_q) in Equation (8) is rewritten as:

$$\begin{gathered} \left(L_d-L_q\right)\left[\begin{array}{c}{i}_d\cos \overline{\theta }\\ i_d\sin \overline{\theta }\end{array}\right]=\left(L_d-L_q\right)\left[\begin{array}{c}{i}_{\gamma }{\left(\cos \overline{\theta }\right)}^2+i_{\delta }\left(\sin \overline{\theta }\cos \overline{\theta }\right)\\ i_{\gamma }\left(\sin \overline{\theta }\cos \overline{\theta }\right)+i_{\delta }{\left(\sin \overline{\theta }\right)}^2\end{array}\right] \\ =\frac{1}{2}\left(L_d-L_q\right)\left[\begin{array}{cc}\left(1+\cos 2\overline{\theta }\right)& \sin 2\overline{\theta }\\ \sin 2\overline{\theta }& \left(1-\cos 2\overline{\theta }\right)\end{array}\right]\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]=\Delta L_{\gamma \delta }{i}_{\gamma \delta }, \end{gathered}$$

(9)

where ΔL_γδ is the PMSM differential inductance matrix in γδ depended which depends on the magnetic saliency (L_d − L_q) and the angle difference Δθ. This is defined as:

$$\Delta L_{\gamma \delta }=\left[\begin{array}{cc}\Delta L\left(1+\cos 2\overline{\theta }\right)& \Delta L\sin 2\overline{\theta }\\ \Delta L\sin 2\overline{\theta }& \Delta L\left(1-\cos 2\overline{\theta }\right)\end{array}\right],$$

(10)

After substituting Equation (9) into Equation (8), the modified rotor magnetic flux λ_mmγδ is written as the sum of partial flux terms:

$${\lambda }_{mm\gamma \delta }={\lambda }_{m\gamma \delta }+\Delta L_{\gamma \delta }{i}_{\gamma \delta },$$

(11)

where

$${\lambda }_{m\gamma \delta }={\lambda }_m\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right],$$

(12)

In the same manner, the stator magnetic flux λ_γδ in Equation (7) could be rewritten as follows:

$${\lambda }_{\gamma \delta }=L_{qq}{i}_{\gamma \delta }+{\lambda }_{mm\gamma \delta }=L_{qq}{i}_{\gamma \delta }+{\lambda }_{m\gamma \delta }+\Delta L_{\gamma \delta }{i}_{\gamma \delta }=\left(L_{qq}+\Delta L_{\gamma \delta }\right)i_{\gamma \delta }+{\lambda }_{m\gamma \delta }=L_{\gamma \delta }{i}_{\gamma \delta }+{\lambda }_{m\gamma \delta },$$

(13)

Here, the inductance matrix L_γδ is defined as L_γδ = L_qq +ΔL_γδ. Using Equation (10) and taking into account that ΣL = L_q +Δ L, the inductance matrix L_γδ in Equation (13) could be also written as:

$$\begin{gathered} L_{\gamma \delta }=L_{qq}+\Delta L_{\gamma \delta }=\left[\begin{array}{cc}{L}_q+\Delta L\left(1+\cos 2\overline{\theta }\right)& \Delta L\sin 2\overline{\theta }\\ \Delta L\sin 2\overline{\theta }& L_q+\Delta L\left(1-\cos 2\overline{\theta }\right)\end{array}\right] \\ =\left[\begin{array}{cc}\mathsf{\Sigma }L+\Delta L\cos 2\overline{\theta }& \Delta L\sin 2\overline{\theta }\\ \Delta L\sin 2\overline{\theta }& \mathsf{\Sigma }L-\Delta L\cos 2\overline{\theta }\end{array}\right]=\left[\begin{array}{cc}\mathsf{\Sigma }L& 0\\ 0& \mathsf{\Sigma }L\end{array}\right]+\left[\begin{array}{cc}\Delta L\cos 2\overline{\theta }& \Delta L\sin 2\overline{\theta }\\ \Delta L\sin 2\overline{\theta }& -\Delta L\cos 2\overline{\theta }\end{array}\right] \\ =\mathsf{\Sigma }L\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+\Delta L\left[\begin{array}{cc}\cos 2\overline{\theta }& \sin 2\overline{\theta }\\ \sin 2\overline{\theta }& -\cos 2\overline{\theta }\end{array}\right], \end{gathered}$$

(14)

Equation (14) implies that the L_γδ depends on the average inductance ΣL, differential inductance ΔL and the double of angle error 2Δθ [21]. It should be noted that the same result is obtained for the inductance matrix L_γδ by means of direct transformation of the inductance matrix L_dq to γδ reference frame, i.e., $L_{\gamma \delta }=K_{\Delta \theta }{L}_{dq}{K}_{\Delta \theta }^{-1}$.

Integrating both parts of Equation (6) and solving for λ_γδ, it is:

$${\lambda }_{\gamma \delta }={\displaystyle \underset{0}{\overset{t}{\int }}\left(u_{\gamma \delta }-r_s{i}_{\gamma \delta }-\widehat{\omega }{J}_s{\lambda }_{\gamma \delta }\right)dt}$$

(15)

Substituting Equations (15) in (13) and solving for i_γδ, it results in:

$$\begin{gathered} L_{\gamma \delta }{i}_{\gamma \delta }+{\lambda }_{m\gamma \delta }={\displaystyle \underset{0}{\overset{t}{\int }}\left(u_{\gamma \delta }-r_s{i}_{\gamma \delta }-\widehat{\omega }{J}_s{\lambda }_{\gamma \delta }\right)dt}\iff \\ {i}_{\gamma \delta }={\left(L_{\gamma \delta }\right)}^{-1}\left[{\displaystyle \underset{0}{\overset{t}{\int }}\left(u_{\gamma \delta }-r_s{i}_{\gamma \delta }-\widehat{\omega }{J}_s{\lambda }_{\gamma \delta }\right)dt}-{\lambda }_{m\gamma \delta }\right], \end{gathered}$$

(16)

Additionally, the inverse matrix of L_γδ is defined by:

$${\left(L_{\gamma \delta }\right)}^{-1}=\frac{1}{\det \left(L_{\gamma \delta }\right)}adj\left(L_{\gamma \delta }\right)=\frac{1}{L_d{L}_q}\left[\begin{array}{cc}\mathsf{\Sigma }L-\Delta L\cos 2\overline{\theta }& -\Delta L\sin 2\overline{\theta }\\ -\Delta L\sin 2\overline{\theta }& \mathsf{\Sigma }L+\Delta L\cos 2\overline{\theta }\end{array}\right],$$

(17)

Here the determinant det (L_γδ) and adjugate adj (L_γδ) are calculated as follows:

$$\begin{gathered} \det \left(L_{\gamma \delta }\right)=L_q^2+\left[L_q\Delta L\left(1+\cos 2\overline{\theta }+1-\cos 2\overline{\theta }\right)\right]+{\left(\Delta L\right)}^2\left[1-{\left(\cos 2\overline{\theta }\right)}^2-{\left(\sin 2\overline{\theta }\right)}^2\right] \\ =L_q^2+2L_q\Delta L+\Delta L\left(1-1\right)=L_q\left(L_q+2\Delta L\right)=L_q\left[L_q+\left(L_d-L_q\right)\right]=L_q{L}_d\ne 0, \end{gathered}$$

(18)

and

$$\begin{gathered} adj\left(L_{\gamma \delta }\right)=\left[\begin{array}{cc}{L}_q+\Delta L\left(1-\cos 2\overline{\theta }\right)& -\Delta L\sin 2\overline{\theta }\\ -\Delta L\sin 2\overline{\theta }& L_q+\Delta L\left(1+\cos 2\overline{\theta }\right)\end{array}\right] \\ =\left[\begin{array}{cc}\mathsf{\Sigma }L-\Delta L\cos 2\overline{\theta }& -\Delta L\sin 2\overline{\theta }\\ -\Delta L\sin 2\overline{\theta }& \mathsf{\Sigma }L+\Delta L\cos 2\overline{\theta }\end{array}\right], \end{gathered}$$

(19)

Observing the terms on right hand-side of Equation (17), it is noted that the γδ currents in Equation (16) are functions of L_dL_q, ΣL, ΔL and Δθ. This means that the existence of angle error information in the stator currents allows the rotor angle detection through appropriate processing of the γδ current signal.

3. High-Frequency Injection (HFI) of Stator Voltage for Rotor Position Estimation

3.1. Analysis of High Frequency Stator Current in γδ Reference Frame

The variation of stator inductance due to rotor angle change implies the presence of magnetic saliency. For the injection methods, the magnetic saliency property is very important, enabling the estimation of the PMSM rotor speed and position. Measuring and processing the PMSM response of the additional HF current signal permits accurate rotor angle estimation and calculation of magnetic saliency. There are three main voltage-injection methods, namely the injection of a sinusoidal voltage signal expressed in the αβ stationary frame, the injection of a sinusoidal voltage signal in the dq frame, and the injection of discrete voltage signal in the form of pulses in the dq frame. The important characteristic of the mentioned methods is their advantage to estimate the rotor flux position at extremely low and even zero speeds. In this work, the injected voltage u_iγδ is a continuous sinusoidal signal superposed on the fundamental supply frequency and expressed in γδ-estimated frame. The injected voltage vector u_iγδ rotates with amplitude u_im and angular speed ω_i = 2πf_i in relation to γδ. Therefore the HF voltage u_iγδ is expressed by:

$$u_{i\gamma \delta }=u_{im}\left[\begin{array}{c}-\sin {\theta }_i\\ \cos {\theta }_i\end{array}\right]=u_{im}\left[\begin{array}{c}-\sin {\omega }_{i}t\\ \cos {\omega }_{i}t\end{array}\right],$$

(20)

where

$${\theta }_i={\displaystyle \underset{0}{\overset{t}{\int }}{\omega }_{i}dt},$$

(21)

Taking in account Equations (15) and (16), the resulting stator flux and current component due to to injected HF voltage could be calculated accordingly as follows:

$${\lambda }_{i\gamma \delta }={\displaystyle \underset{0}{\overset{t}{\int }}\left(u_{i\gamma \delta }-r_s{i}_{i\gamma \delta }-\widehat{\omega }{J}_s{\lambda }_{i\gamma \delta }\right)dt},$$

(22)

and

$$L_{\gamma \delta }{i}_{i\gamma \delta }={\lambda }_{i\gamma \delta }\iff i_{i\gamma \delta }={\left(L_{\gamma \delta }\right)}^{-1}{\lambda }_{i\gamma \delta },$$

(23)

Substituting Equation (22) into (23), it is:

$$i_{i\gamma \delta }={\left(L_{\gamma \delta }\right)}^{-1}\left[{\displaystyle \underset{0}{\overset{t}{\int }}\left(u_{i\gamma \delta }-r_s{i}_{i\gamma \delta }-\widehat{\omega }{J}_s{\lambda }_{i\gamma \delta }\right)dt}\right],$$

(24)

Supposing that the frequency f_i is large enough, such that $\widehat{\omega }<<{\omega }_i$, r_s << ω_iL_d and r_s << ω_iL_q, terms as the HF voltage drop on the stator resistance could be eliminated. Also, since the integral of injected voltage u_iγδ is the dominant term into Equation (24), the HF current calculation is simplified as:

$$\begin{gathered} i_{i\gamma \delta }\cong {\left(L_{\gamma \delta }\right)}^{-1}\left[{\displaystyle \underset{0}{\overset{t}{\int }}{u}_{i\gamma \delta }dt}\right]=\frac{1}{L_d{L}_q}\left[\begin{array}{cc}\mathsf{\Sigma }L-\Delta L\cos 2\overline{\theta }& -\Delta L\sin 2\overline{\theta }\\ -\Delta L\sin 2\overline{\theta }& \mathsf{\Sigma }L+\Delta L\cos 2\overline{\theta }\end{array}\right]\frac{u_{im}}{{\omega }_i}\left[\begin{array}{c}\cos {\omega }_{i}t\\ \sin {\omega }_{i}t\end{array}\right] \\ =\frac{u_{im}}{{\omega }_i{L}_d{L}_q}\left[\begin{array}{c}\mathsf{\Sigma }L\cos {\omega }_{i}t-\Delta L\cos 2\overline{\theta }\cos {\omega }_{i}t-\Delta L\sin 2\overline{\theta }\sin {\omega }_{i}t\\ -\Delta L\sin 2\overline{\theta }\cos {\omega }_{i}t+\mathsf{\Sigma }L\sin {\omega }_{i}t+\Delta L\cos 2\overline{\theta }\sin {\omega }_{i}t\end{array}\right] \\ =\frac{u_{im}}{{\omega }_i{L}_d{L}_q}\left[\begin{array}{c}\mathsf{\Sigma }L\cos {\omega }_{i}t-\Delta L\cos \left(2\overline{\theta }-{\omega }_{i}t\right)\\ \mathsf{\Sigma }L\sin {\omega }_{i}t-\Delta L\sin \left(2\overline{\theta }-{\omega }_{i}t\right)\end{array}\right], \end{gathered}$$

(25)

Applying Euler’s formula, it will be:

$$\begin{gathered} i_{i\gamma \delta }=\frac{u_{im}}{{\omega }_i{L}_d{L}_q}\left\{\mathsf{\Sigma }L\left(\cos {\omega }_{i}t+j\sin {\omega }_{i}t\right)-\Delta L\left[\cos \left(2\overline{\theta }-{\omega }_{i}t\right)+j\sin \left(2\overline{\theta }-{\omega }_{i}t\right)\right]\right\} \\ =\frac{u_{im}}{{\omega }_i{L}_d{L}_q}\left[\mathsf{\Sigma }L{e}^{j{\omega }_{i}t}-\Delta L{e}^{j\left(2\overline{\theta }-{\omega }_{i}t\right)}\right]=i_{i\gamma \delta p}+i_{i\gamma \delta n}, \end{gathered}$$

(26)

where i_iγδp and i_iγδn are the high-frequency current components defined as:

$$i_{i\gamma \delta p}=\frac{u_{im}}{{\omega }_i{L}_d{L}_q}\mathsf{\Sigma }L{e}^{j{\omega }_{i}t},$$

(27)

and

$$i_{i\gamma \delta n}=-\frac{u_{im}}{{\omega }_i{L}_d{L}_q}\Delta L{e}^{j\left(2\overline{\theta }-{\omega }_{i}t\right)},$$

(28)

Here, i_iγδp and i_iγδn represent the HF stator current components with positive and negative frequencies, respectively. As expected, the superimposed stator current vector consists of two separate vector components, where i_iγδp is the positively rotating vector with angular speed ω_i and i_iγδn is the negatively rotating vector with angular speed [–ω_i + d(2Δθ)/dt]. It is worth noting that the average inductance is included into Equation (27), while the differential inductance ΔL and the double error of rotor position 2Δθ are included in Equation (28). Applying signal processing and filtering, the information regarding rotor position, stator inductance and magnetic saliency could be retrieved.

3.2. Angle Error between dq and γδ Reference Frames

Based on Equation (26) the overall estimation procedure of the rotor angle error is divided into two steps (see Figure 3). The sensorless algorithm is firstly focused on isolating the angle error information included into Equation (26), whereas in the second step it aimed to estimate rotor position through sliding mode observer (SMO). For separation of the involved error angle Δθ, both parts of Equation (26) are multiplyied by $e^{j{\omega }_{i}t}$, i.e.,

$$i_{i\gamma \delta }{e}^{j{\omega }_{i}t}=\frac{u_{im}}{{\omega }_i{L}_d{L}_q}\left[\mathsf{\Sigma }L{e}^{j{\omega }_{i}t}-\Delta L{e}^{j\left(2\overline{\theta }-{\omega }_{i}t\right)}\right]e^{j{\omega }_{i}t}=\frac{u_{im}}{{\omega }_i{L}_d{L}_q}\left[\mathsf{\Sigma }L{e}^{j2{\omega }_{i}t}-\Delta L{e}^{j2\overline{\theta }}\right],$$

(29)

The result in Equation (29) shows that the modulation through the multiplication by the unity vector $e^{j{\omega }_{i}t}$ leads to a vector consisting of two signals which rotate at 2ω_i and $2d\overline{\theta }/dt$. Now, passing the resulting HF current signal in Equation (29) through a low-pass filter (LPF), the obtained current signal consists only of components proportional to $\sin 2\overline{\theta }$ and $\cos 2\overline{\theta }$ in a form as follows:

$${\left[i_{i\gamma \delta }{e}^{j{\omega }_{i}t}\right]}_{LPF}=i_{LPF}{}_{i\gamma \delta }=-\frac{u_{im}\Delta L}{{\omega }_i{L}_d{L}_q}{e}^{j2\overline{\theta }}=k_i\left[\begin{array}{c}\cos 2\overline{\theta }\\ \sin 2\overline{\theta }\end{array}\right]\left[\begin{array}{c}{k}_i\cos 2\widehat{\theta }\\ k_i\sin 2\widehat{\theta }\end{array}\right]\simeq k_{i}2\overline{\theta },$$

(30)

where

$$k_i=-\frac{u_{im}\Delta L}{{\omega }_i{L}_d{L}_q},$$

(31)

This implies that angle information could be recovered from the angle difference obtained through filtering off directly the modulated HF stator current in Equation (29). Obviously, the low-pass filtered signal in Equation (30) is in a convenient form to be utilized for speed and position estimation. In the next step, the LPF output signals are used as inputs of a SMO to derive the speed and position of the PMSM rotor flux (see Figure 3). In a similar manner, the first part containing the HF current signal in Equation (29) is also isolated after passing the signal $i_{i\gamma \delta }{e}^{j{\omega }_{i}t}$ through a band-pass filter (BPF). Thus, BPF output consists only of the HF current component given by:

$${\left[i_{i\gamma \delta }{e}^{j{\omega }_{i}t}\right]}_{BPF}=i_{BPF}{}_{i\gamma \delta }=\frac{u_{im}\mathsf{\Sigma }L}{{\omega }_i{L}_d{L}_q}{e}^{j2{\omega }_{i}t}=k_j\left[\begin{array}{c}\cos 2{\omega }_{i}t\\ \sin 2{\omega }_{i}t\end{array}\right],$$

(32)

where

$$k_j=\frac{u_{im}\mathsf{\Sigma }L}{{\omega }_i{L}_d{L}_q},$$

(33)

Considering Equations (31) and (33), it follows that k_i depends on the inductances L_d, L_q and differential inductance, whereas k_j represents the amplitude of the BPF output depending on the inductances L_d, L_q and their average. The block diagram of current signal processing is shown in Figure 3. As illustrated, only the δ component of LPF output is needed for speed/position estimation, while the parameters k_i and k_j from both LPF and BPF are used to compute the inductances L_d, L_q and saliency (L_d − L_q).

3.3. Angle Error of PMSM Rotor Flux in γδ Reference Frame

Although, there are available two orthogonal signals from Equation (29), $k_i\cos 2\overline{\theta }$ and $k_i\sin 2\overline{\theta }$, the speed and position estimation algorithm requires only the $k_i\sin 2\overline{\theta }$ component. Figure 4 shows the observer structure for speed and position tracking in details. Considering the i_LPFiδ component from Equation (30), the estimated angle of rotor flux is related to the real one through the following relation (3rd Ptolemy’s identity/the difference formula for sine, see Figure 4):

$$i_{LPFi\delta }=k_i\sin 2\overline{\theta }=k_i\sin \left(2\theta -2\widehat{\theta }\right)=k_i\sin 2\theta \cos 2\widehat{\theta }-k_i\cos 2\theta \sin 2\widehat{\theta },$$

(34)

Equation (34) is interpreted schematically in Figure 4 implying how the angle error is equivalently connected to real and estimated speed by means of i_LPFiδ. If the difference between the estimated and actual rotor flux positions is small, the current i_LPFiδ can be approximated by $k_{i}2\overline{\theta }$ because $\sin 2\overline{\theta }\cong 2\overline{\theta }=2\Delta \theta =2\theta -2\widehat{\theta }$. This implies that i_LPFiδ is almost directly proportional to Δθ for the small angle difference.

3.4. Design of Sliding Mode Observer (SMO) for Estimation of Rotor Flux Angle and Speed

Among the proposed observer schemes, the sliding mode observers (SMO) are widely studied in the literature and applied on a plethora of industrial applications. The main advantage of sliding mode methodology is that provides robustness and fast convergence in finite time under system disturbances and modeling uncertainties. In designing the SMO, its structure typically includes two steps, namely the sliding manifold and control law. Here, the sliding manifold design is determined choosing the rotor angle and speed errors, $\overline{\theta }$ and $\overline{\omega }$, as sliding surfaces. Also the control law consists of the sign function $\mathrm{sgn}2\overline{\theta }$, while the observer gains are defined as γ_iθ and γ_iω for angle and speed tracking, respectively. In addition, the principles of equivalent control approach are also used in sliding mode observer stability after reaching phase. Based on Equation (34), the proposed angle/speed observer is determined as follows:

$$\dot{\widehat{\theta }}=\widehat{\omega }+{\gamma }_{i\theta }\mathrm{sgn}2\overline{\theta },$$

(35)

and

$$\dot{\widehat{\omega }}={\gamma }_{i\omega }\mathrm{sgn}2\overline{\theta },$$

(36)

where $\mathrm{sgn}2\overline{\theta }$ is the signum function $2\overline{\theta }$. In Figure 4, the input and output signals of SMO are schematically illustrated using an equivalent block diagram.

3.4.1. SMO Dynamics and Stability

Using the above sliding manifold and Equations (35) and (36), the SMO dynamics could be expressed as:

$$\dot{\overline{\theta }}=\dot{\theta }-\dot{\widehat{\theta }}=\omega -\widehat{\omega }-{\gamma }_{i\theta }\mathrm{sgn}2\overline{\theta }=\overline{\omega }-{\gamma }_{i\theta }\mathrm{sgn}2\overline{\theta },$$

(37)

and

$$\dot{\overline{\omega }}=\dot{\omega }-\dot{\widehat{\omega }}=-{\gamma }_{i\omega }\mathrm{sgn}2\overline{\theta },$$

(38)

Let us suppose that γ_iθ is large enough, such that:

$${\gamma }_{i\theta }>>\left|\overline{\omega }\right|\ge \overline{\omega },$$

(39)

i.e., γ_iθ is an upper bound of $\overline{\omega }$, then:

$$\left(37\right)\stackrel{\dot{\overline{\theta }}=0}{\to }\overline{\omega }={\gamma }_i{}_{\theta }{\left(\mathrm{sgn}2\overline{\theta }\right)}_{eq}\iff {\left(\mathrm{sgn}2\overline{\theta }\right)}_{eq}=\frac{\overline{\omega }}{{\gamma }_i{}_{\theta }},$$

(40)

Here, the term (.)_eq represents the equivalent control. Eventually the equivalent control could be used as input for the speed estimation procedure. Substituting Equation (36) into (38) it results that:

$$\dot{\overline{\omega }}=-{\gamma }_i{}_{\omega }{\left(\mathrm{sgn}2\overline{\theta }\right)}_{eq}=-\frac{{\gamma }_i{}_{\omega }}{{\gamma }_i{}_{\theta }}\overline{\omega },$$

(41)

This stability analysis implies that the SMO defined in Equations (35) and (36) is asymptotically stable with errors tending to zero. Considering Equation (41), it is obvious that the developed approach of sliding mode observer is implemented as a reduced order asymptotic observer by means of the equivalent control.

Considering the reaching phase, the sliding surface $\overline{\theta }=0$ is reached at limited time t_r given by:

$$t_r\le \frac{2V^{1/2}\left(0\right)}{\xi }=\frac{\sqrt{2}\left|\overline{\theta }\left(0\right)\right|}{\left({\gamma }_i{}_{\theta }-P_d\right)},$$

(42)

where

$$\xi =\sqrt{2}\left({\gamma }_i{}_{\theta }-P_d\right),$$

(43)

The P_d represents the upper limit of the bounded disturbance regarding $\overline{\omega }$. Also taking in account Equation (41), a general solution converges to zero asymptotically, described by:

$$\dot{\overline{\omega }}=-\frac{{\gamma }_i{}_{\omega }}{{\gamma }_i{}_{\theta }}\overline{\omega }\iff \overline{\omega }\left(t\right)=\overline{\omega }\left(0\right)e^{-\left({\gamma }_{i\omega }/{\gamma }_{i\theta }\right)t},$$

(44)

Inspecting Equation (42), it obvious that the maximum time t_r is directly proportional to the initial angle error $\left|\overline{\theta }\left(0\right)\right|$, while it is directly inverserly proportional to the gain γ_iθ. In addition, the speed error converges asymptotically faster to zero as the rate (γ_/ω/γ_iθ) increases [30].

3.4.2. Approximation of sgn(.) Function-Chattering Reduction

Although a sliding mode observer offers advadages and enhanced stability properties, the applied control signal causes high-frequency oscillations after the system states, $\overline{\theta }$ and $\overline{\omega }$, reach the sliding surfaces. Such undesirable oscillations are called a chattering phenomenon and they affect negatively sliding mode applications. As a consequence, the chattering excites the system-unmodeled dynamics leading probably to large estimation errors or observer malfunction. Conventionally, the control law is substituted by a smooth continuous function to approximate the discontinuous sign function. This could succeed in dealing with the serious disadvantage of chattering phenomenon. Among the suggested solutions, the hyperbolic tangent tanh(.) is a relatively simple smooth approximation of the sign function in reducing chattering, since $\mathrm{sgn}2\overline{\theta }\approx \tanh \left(k2\overline{\theta }\right)$ for k >> 1.

4. Estimation of PMSM Inductances and Magnetic Saliency

4.1. Calculating the Parameters k_i and k_j

Since the γδ rotating reference frame is orthogonal, the magnitude of the parameter k_i is calculated as the hypotenuse of a right triangle whose legs are the components of the LPF output (Pythagorean Theorem), that is the square root of the squares of i_LPFγ and i_LPFδ currents in Equation (30), i.e.,

$$k_i=-\left[\sqrt{{\left(i_{LPF\gamma }\right)}^2+{\left(i_{LPF\delta }\right)}^2}\right]\mathrm{sgn}(\Delta L),$$

(45)

Alternatively, the k_i can also estimated from i_LPFiγ in Equation (30), as the angle error tends to zero (i.e., $k_i\cos 2\overline{\theta }=k_i$). The sign of k_i is opposite of this of ΔL, i.e., for ΔL > 0, it is k_i < 0 or for ΔL < 0, it is k_i > 0. Normally, if the PMSM is operating as a motor, it will be ΔL < 0 and thus k_i > 0. In the same manner, the parameter k_j is positive and it is obtained from the components of the BPF output in γδ frame, i.e.,

$$k_j=\left[\sqrt{{\left(i_{BPF\gamma }\right)}^2+{\left(i_{BPF\delta }\right)}^2}\right],$$

(46)

Regarding Equations (45) and (46), it results that magnitudes of both parameters k_i and k_j essentially represent the amplitudes of the low and high frequency current signals in Equation (29). Also comparing the magnitudes and absolute values of k_i and k_j, it is k_j > k_i and |k_j| > |k_i|, since ΣL > −ΔL and ΣL > |ΔL|.

4.2. Expressing dq Impedances as Functions of the k_i and k_j Parameters

Inspecting Equations (31) and (33), it can be noted that k_i and k_j are directly analogous to ΔL and ΣL respectively with ratio coefficient ±u_m/(ω_iL_dL_q). Additionally, the sum and difference of ΣL and ΔL are equal to L_d and L_q i.e., (ΣL + ΔL) = [(L_d + L_q)/2] + [(L_d − L_q)/2] = L_d and (ΣL − ΔL) = [(L_d + L_q)/2] − [(L_d − L_q)/2] = L_q. This implies that the d-axis and q-axis inductances of PMSM can be easily calculated in terms of the k_i and k_j parameters. Adding by parts Equations (31) and (33), it follows that:

$$k_j+k_i=\frac{u_{im}\left(\mathsf{\Sigma }L-\Delta L\right)}{{\omega }_i{L}_d{L}_q}=\frac{u_{im}{L}_q}{{\omega }_i{L}_d{L}_q}=\frac{u_{im}}{{\omega }_i{L}_d},$$

(47)

Now solving Equation (47) for L_d, the d-axis inductance is given as

$$\stackrel{(47)}{\to }{L}_d=\frac{u_{im}}{{\omega }_i\left(k_j+k_i\right)},$$

(48)

Also, subtracting by parts Equation (31) from Equation (33), the following results:

$$k_j-k_i=\frac{u_{im}\left(\mathsf{\Sigma }L+\Delta L\right)}{{\omega }_i{L}_d{L}_q}=\frac{u_{im}{L}_d}{{\omega }_i{L}_d{L}_q}=\frac{u_{im}}{{\omega }_i{L}_q},$$

(49)

Subsequently, solving Equation (49) for L_q, this is:

$$\stackrel{(49)}{\to }{L}_q=\frac{u_{im}}{{\omega }_i\left(k_j-k_i\right)},$$

(50)

4.3. Expressing the PMSM Magnetic Saliency (L_d − L_q) as Function of Parameters k_i and k_j

Substituting L_d and L_q from Equations (48) and (50), the magnetic saliency can be written as follows:

$$\left(L_d-L_q\right)=2\Delta L=\frac{u_{im}}{{\omega }_i}\left[\frac{1}{\left(k_j+k_i\right)}+\frac{1}{\left(k_j-k_i\right)}\right]=\frac{u_{im}}{{\omega }_i}\left[\frac{2k_j}{\left(k_j^2-k_i^2\right)}\right],$$

(51)

Equation (51) allows the calculation of PMSM magnetic saliency using the parameters k_i and k_j. After accomplishing the calculations from Equation (45) to Equation (51), it is feasible to estimate both inductances L_d, L_q and magnetic saliency. The analysis above implies that the stator inductance changes or even magnetic fault diagnosis could be monitored by means of the proposed HFI method.

5. Simulation Results and Discussion

5.1. Description of Simulated PMSM Model and Control System

For test and evaluation purposes, a vector control is employed, whose block diagram is depicted in Figure 1. Modeling and design of the proposed sensorless control scheme is implemented using Simulink/Matlab application. Mainly, the total model structure including dynamics is described in Figure 1. The simulated PMSM model is based on the primitive Equations (5)–(19) associated with the γδ reference frame, while observer model uses Equations (30) and (34)–(36) to estimate both rotor speed and position. In addition, stator inductances and magnetic saliency are calculated using Equations (45)–(51) after suitable processing, i.e., modulating and filtering the stator currents (see Figure 3 and Figure 5). The simulated model was tested and verified using the PMSM parameters listed in

Table 1. Parameters of Permanent Magnet Synchronous Machine (PMSM).

Symbol	Quantity	Expressed in SI
S	Apparent power	5.5 kVA
cosφ	Electric power coefficient	0.8
Vl-l	Line to line voltage	380 V
rs	Stator resistance	2.5 Ω
Lmd	d-axis magnetizing inductance	0.360 H
Ld	d-axis inductance	0.400 H
Lq	q-axis inductance	0.210 H
λm	Permanent Magnet Flux	0.5 Vs (or Wb)
J	Moment of inertia	0.089 kgm2
p	Magnetic pole pairs	1
ωm	Mechanical angular speed	3000 rpm

. Considering Equation (31) and L_d > L_q, i.e., ΔL > 0, the sign of k_i is negative in this particular machine. A three-leg VSI drives the PMSM fed with 400 V dc. In simulation tests, the output voltage of the VSI is modulated by means of SVPWM algorithm, while the switching frequency is set at 5 kHz. As it is demonstrated in Figure 1, the HF voltage signals are added with the u_γ^* and u_δ^* voltage references. In aiming to precisely estimate the rotor position, the frequency and amplitude of the injected signal is set at 1 kHz and 50 V, respectively. Considering the control law of the SMO, the parameter k of the hyperbolic tangent function $\tanh (k2\overline{\theta })$ is set equal to 10, while the observer performance has been attained for gains γ_iθ = 40 and γ_iω = 5. A diagram of the developed PMSM model in γδ is demonstrated in Figure 6.

In future work, hardware implementation of the studied approach should be based on a development board equipped with powerful digital signal processor (DSP) or multi processor unit (MPU). For example, a single-board solution is the DS1104 research and development (R&D) Controller Board of dSPACE. The Real-Time Interface (RTI) software provided allows direct implementation of the developed Simulink models on the real-time hardware. Data collection, communication and control are succeeded in through the available interfaces (A/D or D/A converter channels) including the PWM outputs.

5.2. Response at Very Low Speed and Standstill

Simulation results are presented in Figure 7 without external torque disturbance. The reference speed is changed stepwise from 0 rad/s to π/2 rad/s (0.5 Hz) and at t₂ = 2s it changed from π/2 rad/s to 0 rad/s (standstill). Figure 7a,b shows the estimated rotor speed and stator currents, respectively, while HF stator currents i_iγδ, the modulated and i_LPFiγ are displayed in Figure 7c. In Figure 7a, it is demonstrated that the SMO converges very fast with accurate speed estimation. Also, the response of stator current i_γδ is shown in Figure 7b including its HF components. The k_i in Figure 7c is obtained from the i_LPFiγ from Equation (30), i.e., after LP filtering the γ-component of modulated HF current. At standstill operation without external torque disturbance in particular, the observer error remains very small preserving its excellent performance. The inspection of the iγδ current waveforms shows that the observer-controller system behaves well with a very small chattering introduced in iγ due to the fast switching of the control input in the SMO. However the usage of hyperbolic tangent function, tanh(.), has greatly improved the currents’ response while keeping the advantageous characteristics of the SMO.

5.3. Very Low Speed Response with Torque Load

Simulation results presented in Figure 7d-f show the PMSM speed response during speed changes in the presence of external torque 1 Nm. Here the speed is changed stepwise from 0 rad/s to π/2 rad/s (0.25 Hz) and from π/2 rad/s to -π rad/s (0.5 Hz) at t₂ = 2s. Also the external torque of 1 Nm is applied at t₁ = 1 s and it is removed at t₃ = 3 s. The rotor speed and angle are shown in Figure 7d,e, respectively, while the HF stator currents including modulated and k_i are shown in Figure 7f. The very small speed and angle errors show the robustness of the proposed estimation scheme. Also, as expected the frequency of derived current signals is double that of the injected one after modulation. It is worth noting here that during the transition from π/2 rad/s to -π rad/s the observer behavior is robust and stable even in presence of torque disturbance. The observer keeps converging fast with very small angle errors between the synchronous dq (real) and γδ estimated reference frames.

5.4. Flux and Torque Response with Saliency Estimation at Very Low Speed

The stator flux and torque responses are presented in Figure 8b,c respectively. Here, the speed changes stepwise from 0 rad/s to π rad/s and from π rad/s to −π rad/s at t₂ = 2 s in presence of 1 Nm as external torque disturbance applied at t₁ = 1 s and then removed at t₃ = 3 s. An estimation of the saliency 2ΔL is demonstrated in Figure 8a, while HF stator currents with estimation of k_i parameter are shown in Figure 8d. It can be observed that the estimated saliency is very close to the real one, implying the accuracy of the proposed estimation scheme. However this accuracy is mostly affected on the information extracted for both k_i and k_j parameters.

6. Conclusions

A novel sensorless algorithm was developed and tested for the speed and position of a PMSM based on HFI methodology. Using the γδ modified PMSM model, the proposed scheme was evaluated as an effective sensorless approach embedding appropriately the magnetic saliency terms of the modified rotor flux into a new inductance matrix in γδ. Applying directly a HF voltage signal to a VSI, the resulting stator current was utilized to extract rotor angle information through LPF and sliding mode observer. Based on the advantages of the SMO structure, the speed/position observer converges very fast in finite time even for zero speed command at presence of torque disturbance (low and very low speed range, 0.5–0Hz). In addition, the LP and BP filtered signals were used in a simple manner to track stator inductances and magnetic saliency. Simulation results demonstrate the estimation scheme efficiency verifying the observer robustness at very low and even at standstill operation. The proposed algorithm performed well with exceptional convergence characteristics providing accurate estimates of dq inductances and saliency.