1 Introduction

Modern civil aircraft increasingly incorporate electrical onboard systems and equipment. The elimination of further centralized hydraulic circuits allows weight reductions on aircraft level and a better maintainability. Electrically powered flight control actuators already replace their hydraulic equivalents in certain areas. For this purpose, the electro-mechanical actuator (EMA) is a promising enabling technology. It uses an electrical motor, followed by a reduction gearbox to actuate a flight control surface. Predominantly, a permanent magnet synchronous motor (PMSM) is chosen for its superior power density. However, the use of EMAs is still limited to less demanding applications in secondary flight control such as spoilers [15]. One reason is the high hardware complexity of the mechanical drivetrain, the actuator’s electronic and sensor components. This leads to an increase in actuator weight and volume and generates new, complex failure scenarios.

Efficient field-oriented motor control requires an accurate knowledge of the rotor angle. Conventionally, it is measured by a resolver which is mounted on the PMSM rotor shaft as shown in Fig. 1.

Fig. 1
figure 1

Resolver mounted on a PMSM rotor shaft

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Mazzoleni et al. and Di Rito et al. [5, 16] quantify the probability of resolver failures in the range of \(10^{-6}\) or \(10^{-7}\) per flight hour. They both conclude that a simplex resolver design is not reliable enough to meet the safety goals for flight control applications. They propose a dual redundancy or in-lane resolver monitoring, respectively, adding further complexity. The substitution of this sensor including its cabling and affiliated evaluation electronics would thus substantially contribute to a reduction of the EMA hardware complexity and saves costs.

It would allow for a more compact motor unit and facilitates the integration into limited installation spaces. The risk of electrical failure of the safety-critical sensor or its susceptible cabling would be averted. The reduction of rotor inertia furthermore improves the dynamic properties of the PMSM. This paper, therefore, investigates the capabilities of modern sensorless control techniques to provide an accurate estimate of the rotor angle. Specifically, the alternating injection method and the extended electromotive force (EMF) integration method are implemented. They are evaluated on the use case of an aileron actuation system which might reasonably become the first EMA application in primary flight control. The scope of this work is to evaluate the abilities and shortcomings of the selected state-of-the-art sensorless control techniques for this challenging aerospace application. Therefore, extensive laboratory testing is performed using a representative, in-house developed electro-mechanical actuator.

2 Theoretical background

2.1 Physics of the permanent magnet synchronous motor

Physical properties and electrical quantities such as current \(\vec{i}\) and voltage \(\vec{u}\) of a PMSM can be equally represented in three alternative coordinate systems according to Fig. 2. The 3-phase fixed system represents the physical motor windings u, v, w equally spaced by 120 electrical degrees. It can be mathematically simplified when transformed into orthogonal 2-phase fixed coordinates \(\alpha , \beta\). However, in many cases, a representation in 2-phase rotating coordinates is more beneficial. The so-called d-axis is aligned towards the magnetic field of the rotor and rotated by the angle \(\varphi\) in comparison to the \(\alpha\)-axis. The quadrature axis q is orthogonal to the direct axis d in electrical coordinates. A multiplication with the corresponding transformation matrix \({\underline{T}}\) in Appendix A1 allows a conversion between coordinate systems [20].

Fig. 2
figure 2

Coordinate systems

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The physics of a PMSM can be favourably described in rotating coordinates and is given by the voltage equation [9]:

$$\begin{aligned}{} & {} \begin{pmatrix} u_d \\ u_q \end{pmatrix} = & \begin{bmatrix} {R} &{} -\omega _{el} {L_q} \\ \omega _{el}{L_d} &{} {R} \end{bmatrix} \begin{pmatrix} i_d \\ i_q \end{pmatrix} + \begin{bmatrix} {L_{dd}} &{} {L_{dq}} \\ {L_{qd}} &{} {L_{qq}} \end{bmatrix} \dot{\begin{pmatrix} i_d \\ i_q \end{pmatrix}}\nonumber \\{} & \quad {} + \underbrace{ \begin{pmatrix} 0 \\ \omega _{el}{\Psi _{PM}} \end{pmatrix}. }_{\begin{array}{c} \mathrm{{back-EMF}}\ \textbf{e}_{dq} \end{array}} \end{aligned}$$
(1)

Here, the magnetic field angular velocity for the synchronous machine equals the mechanical motor speed multiplied by its number of pole pairs PP:

$$\begin{aligned} \omega _{el}&={PP}\cdot \omega _{me} \nonumber \\ \varphi _{el}&={PP}\cdot \varphi _{me}. \end{aligned}$$
(2)

The phase resistance is given by R. \(L_d\) and \(L_q\) are the absolute inductances in direct and quadrature axes. The differential inductance matrix \({\underline{L}}_{dq}\) characterizes the motor behaviour during dynamic current changes. Equivalent to the absolute inductances, the differential self-inductances \(L_{dd}\) and \(L_{qq}\) show the relationship in d- and q-direction. \(L_{dq} = L_{qd}\) accounts for a cross-coupling between the two axes. The last term in Eq. 1 describes the voltage induced into quadrature axis during a rotor movement and depends on the permanent magnet flux linkage \(\Psi _{PM}\). It is named the back electromotive force or simply back-EMF \(\textbf{e}\). The generated motor torque \(T_\mathrm{{mot}}\) is given in 2-phase rotating coordinates [8]:

$$\begin{aligned} T_\mathrm{{mot}}=\frac{3}{2}\cdot {PP} \cdot ({\Psi _{PM}} \cdot i_q+(L_d-L_q)\cdot i_q \cdot i_d). \end{aligned}$$
(3)

Typically, field-oriented control regulates the direct current component to zero \(i_d^* = 0\) eliminating the reluctance moment.

2.2 Sensorless control techniques

Several decades of intensive research have led to a variety of techniques to estimate the rotor angle of a PMSM [1, 6, 21, 23]. Most methods either evaluate the back-EMF or exploit saliency properties of the motor. The evaluation of the back-EMF requires a minimum rotation speed whereas saliency-based methods function at standstill as well. Consequently, both principles are often combined in hybrid rotor angle observers. This paper makes use of the alternating injection and extended EMF methods explained hereinafter.

2.2.1 Motor saliency

A nonhomogeneous design of the rotor laminations leads to a spatial variation of the magnetic permeability. Especially in PMSMs with interior permanent magnets, the electromagnetic properties vary in d- and q-direction, as illustrated in Fig. 3.

Fig. 3
figure 3

Cross-sectional view of the rotor geometry

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This variation in properties is known as motor saliency and can be modelled in the stationary frame at standstill. Therefore, Eq. 1 is transformed into 2-phase fixed coordinates, and the angular velocity is set to zero:

$$\begin{aligned} \begin{pmatrix} u_\alpha \\ u_\beta \end{pmatrix} = \begin{bmatrix} {R} &{} 0\\ 0 &{} {R} \end{bmatrix} \begin{pmatrix} i_\alpha \\ i_\beta \end{pmatrix} + {\underline{L}}_{\alpha \beta } \dot{\begin{pmatrix} i_\alpha \\ i_\beta \end{pmatrix}}. \end{aligned}$$
(4)

Here, the inductance matrix \({\underline{L}}_{dq}\) is converted into the fixed frame by applying the rotational matrix \({\underline{T}}_{dq\rightarrow \alpha \beta }\) in Eq. 15:

$$\begin{aligned} {\underline{L}}_{\alpha \beta } &={\underline{T}}_{dq \rightarrow \alpha \beta }\cdot {\underline{L}}_{dq}\cdot {\underline{T}}_{dq \rightarrow \alpha \beta }^{-1}. \end{aligned}$$
(5)

Exploiting trigonometric relationships allows to separate the inductance matrix into an isotropic and an anisotropic component \({\underline{L}}_\Sigma\) and \({\underline{L}}_\Delta\) [13]:

$$\begin{aligned} & {\underline{L}}_{\alpha \beta }= \underbrace{ \begin{bmatrix} \frac{{L_{dd}}+{L_{qq}}}{2} &{} 0\\ 0 &{} \frac{{L_{dd}}+{L_{qq}}}{2} \end{bmatrix} }_{\begin{array}{c} {\underline{L}}_{\Sigma } \end{array}} \nonumber \\ & + \quad \underbrace{ \begin{bmatrix} \frac{{L_{qq}}-{L_{dd}}}{2} &{} \frac{{L_{dq}+L_{qd}}}{2}\\ -\frac{{L_{dq}+L_{qd}}}{2} &{} \frac{{L_{qq}}-{L_{dd}}}{2} \end{bmatrix}\cdot \begin{bmatrix} -cos(2\varphi _{el}) &{} -\mathrm{{sin}}(2\varphi _{el})\\ -sin(2\varphi _{el}) &{} \mathrm{{cos}}(2\varphi _{el}) \end{bmatrix} }_{\begin{array}{c} {\underline{L}}_{\Delta }(\varphi _{el}) \end{array}}. \end{aligned}$$
(6)

PMSMs with a significant anisotropic inductance \({\underline{L}}_\Delta\) are classified as salient. The anisotropic component depends on the electrical rotor angle \(\varphi _{el}\), a characteristic which is systematically exploited for rotor angle estimation. A specific voltage pattern is injected into the motor phases and the triggered current response demodulated. Furthermore, the last term in Eq. 6 reveals a twofold periodicity of the anisotropic inductance component. Consequently, saliency-based methods can identify the rotor axis but not its orientation. Therefore, a starting procedure must initially determine its polarity.

2.2.2 Alternating injection

The alternating injection method inserts a high-frequency sinusoidal voltage vector into the assumed rotary motor frame [4]. To avoid unintentional torque generation, it is preferably directed towards the direct rotor axis [14]:

$$\begin{aligned} \begin{pmatrix} u_{{\tilde{d}},\mathrm{{inj}}} \\ u_{{\tilde{q}},\mathrm{{inj}}} \end{pmatrix} = {u_{\mathrm{{inj}}}} \cdot \begin{pmatrix} sin({\omega _{\mathrm{{inj}}}}\cdot t)\\ 0 \end{pmatrix}. \end{aligned}$$
(7)

A bandpass filter can separate the triggered current at the angular injection frequency \(\omega _{\mathrm{{inj}}}\) from the overall current signal content. For salient machines, the alternating current vector is twisted from the injection axis, i.e. the estimated \({\tilde{d}}-\)axis towards the true \(d-\)axis, as shown in Fig. 4.

Fig. 4
figure 4

Exemplary current vectors triggered by alternating voltage injection

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Here, an alternating voltage was injected at various angles while the salient PMSM under test was locked at \(\varphi _{el} = 45^\circ\). For small angles, there is a linear relationship between the estimation error \((\varphi _{el}-{\tilde{\varphi }}_{el})\) and the \({\tilde{q}}-\)component of the current amplitude [13]:

$$\begin{aligned} {\hat{i}}_{{\tilde{q}},\mathrm{{inj}}}=\frac{{L_q}-{L_d}}{{L_d} \cdot {L_q}}\cdot \frac{{{\hat{u}}_\mathrm{{inj}}}}{{\omega _\mathrm{{inj}}}}\cdot (\varphi _{el}-{\tilde{\varphi }}_{el}). \end{aligned}$$
(8)

The estimation error can be reduced to zero in an observer loop, so that the estimated \({\tilde{dq}}\)-axes approach the true rotary dq-frame.

2.2.3 Extended EMF integration

At sufficiently high speeds the rotor angle can be directly deduced from the motor voltage equation in 2-phase fixed coordinates. For salient machines, it is given by [3, 17]:

$$\begin{aligned}{} & {} \begin{pmatrix} u_\alpha \\ u_\beta \end{pmatrix} = \begin{bmatrix} {R} &{} 0\\ 0 &{} {R} \end{bmatrix} \begin{pmatrix} i_\alpha \\ i_\beta \end{pmatrix} + \begin{bmatrix} &{} {L_{dd}} &{} \omega _{el}({L_{dd}}-{L_{qq}})\\ -\omega _{el}({L_{dd}}-{L_{qq}}) &{} {L_{dd}} \end{bmatrix} \dot{\begin{pmatrix} i_\alpha \\ i_\beta \end{pmatrix}} \nonumber \\{} & {} +\underbrace{(({L_d}-{L_q})(\omega _{el}i_d-{\dot{i}}_q)+\omega _{el}{\Psi _{PM}})\cdot \begin{pmatrix} -\mathrm{{sin}}(\varphi _{el})\\ \mathrm{{cos}}(\varphi _{el}) \end{pmatrix}}_{\begin{array}{c} \mathrm{{extended}}\ \mathrm{{EMF}} \ \textbf{ee}_{\alpha \beta } \end{array}}. \end{aligned}$$
(9)

Here, all rotor angle-dependent quantities are merged into one term. It is named the extended EMF \(\textbf{ee}\), emphasizing its wider scope compared to the genuine back-EMF definition in Eq. 1. One way to solve Eq. 9 for the rotor angle is by integrating the phase currents. For this purpose, it is rearranged as follows:

$$\begin{aligned} \begin{pmatrix} \int ee_{\alpha } \,\textrm{d}t\\ \int ee_{\beta } \,\textrm{d}t \end{pmatrix} = \underbrace{\int \begin{pmatrix} u_\alpha \\ u_\beta \end{pmatrix}- \begin{bmatrix} {R} &{} 0\\ 0 &{} {R} \end{bmatrix} \begin{pmatrix} i_\alpha \\ i_\beta \end{pmatrix} \,\textrm{d}t+ \begin{pmatrix} \Psi _{\alpha }(t = 0) \\ \Psi _{\beta }(t = 0) \end{pmatrix}}_{\begin{array}{c} \mathrm{{Flux}}\ \mathrm{{linkage}}\ \mathbf {\Psi }_{\alpha \beta } \end{array}} \nonumber \\ - \begin{bmatrix} {L_{dd}} &{} \omega _{el}({L_{dd}}-{L_{qq}})\\ -\omega _{el}({L_{dd}}-{L_{qq}}) &{} {L_{dd}} \end{bmatrix} \begin{pmatrix} i_\alpha \\ i_\beta \end{pmatrix}. \end{aligned}$$
(10)

The approach is inherently based on stator flux estimation [7]. From Eq. 9, it follows that the rotor angle can be directly calculated using the atan2-operator:

$$\begin{aligned} {\tilde{\varphi }}_{el}=atan2\left( \int ee_{\beta } \,\textrm{d}t, \int ee_{\alpha } \,\textrm{d}t\right) . \end{aligned}$$
(11)

Practically, the method suffers minor errors because of parameter uncertainties and the unknown constant of integration \(\Psi (t \, = \, 0)\). Equation 10 also presumes a constant angular velocity. Rapid speed changes therefore degrade the estimation as well. Furthermore, small sensor offsets inevitably cause an integrator wind-up. This needs to be counteracted with a stabilizing feedback loop exerting a discharging effect on the integrator. Since phase voltages \(u_{\alpha \beta }\) are usually not measured, reference voltages \(u^*_{\alpha \beta }\) can be utilized alternatively, assuming an ideal voltage modulation.

3 Case study

The sensorless control methods are applied on an aileron actuation system for a large civil aircraft in a lab environment [12]. This work, therefore, contributes to recent research activities that aim at deploying electro-mechanical actuators on primary flight control surfaces [2, 11, 19].

3.1 Electro-mechanical test actuator

For validation the linear electro-mechanical actuator in Fig. 5 constructed from commercial off-the-shelf components is used.

Fig. 5
figure 5

Electro-mechanical test actuator

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It is designed to withstand external loads up to \(F_L =26.7\, \textrm{kN}\) and permits a position bandwidth above \(2\,\textrm{Hz}\) for a medium actuator stroke of \(x_\mathrm{{EMA}} = 4\,\textrm{mm}\). The drive motor is a PMSM type NX420EAPR7101 from Parker with a rated power of \(1.5\,\textrm{kW}\). Its voltage characteristics allow for an integration in future high voltage direct current (HVDC) onboard power supply systems with a DC link voltage of \(540\,\textrm{V}\). The interior permanent magnets are arranged in a flux concentrating topology which contributes to a significant motor saliency [10]. The motor shaft is equipped with a resolver measuring the rotor angle. The EMA drivetrain is composed of a two-stage spur gear, followed by a roller screw that converts the rotary movement into a linear actuator stroke. The stroke is measured by a linear variable differential transformer (LVDT) which is integrated into the hollow spindle at the actuator output.

3.2 Aileron test rig

The actuator is integrated into the test rig shown in Fig. 6.

Fig. 6
figure 6

Aileron test rig in single lane operation

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It allows the electro-mechanical actuation of an aileron flight control surface which is here simplified to a stiff, rotating shaft. It can be actuated by two parallel-redundant actuators mimicking the real world application. However, the sensorless control techniques are validated in single lane operation, because no significant interactions between in- and outboard actuator are foreseen. Mechanical parameters are realistically modelled with a spring element on each actuator lane and an inertial disc attached to the aileron shaft. A simple hinge kinematics propagates the linear EMA movement on the aileron. Arbitrary air loads can be applied via two hydraulic cylinders regulating either the actuator load \(F_L\) or the aileron hinge moment \(T_h\). The test rig is highly instrumented with position and force measurements on each lane supplementing the internal actuator sensors. A power supply provides a DC link voltage of \(540\,\textrm{V}\) for the EMA power electronics consisting of a six channel IGBT driver and power module. Phase currents and voltages are obtained from an in-house developed measurement board. The testing facility is operated with a real-time computing system from dSPACE which executes among others the EMA control algorithms.

3.3 Benchmark controller design

The actuator uses a conventional cascade control, which is the most common control architecture for EMAs today. It is parameterized to comply with the performance requirements for the aileron use case, specifically the design points for load and position dynamics as specified in Sect. 3.1. The controller is used as the benchmark against which the new sensorless control algorithms are compared. The benchmark controller scheme is depicted in Fig. 7. It generates phase voltages, so that the actuator position \(x_\mathrm{{EMA}}\) follows its reference signal \(x_\mathrm{{EMA}}^*\).

Fig. 7
figure 7

Simplified architecture of the benchmark controller

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The outer and at the same time least dynamic position control loop uses a proportional feedback of the actuator stroke measured by the LVDT. It feeds the proportional-integral (PI) speed controller which takes the rotor angle derivative as the sensor input. The inner and fastest PI control loop regulates the current in 2-phase rotating coordinates. A current in direct axis \(i_d\) controls the magnetic flux of the PMSM. Since an adjustment of the flux linkage is not intended, \(i_d\) is permanently controlled to zero. In accordance with Eq. 3, the motor torque is directly adjusted via the quadrature current \(i_q\). The maximum reference current and voltages are limited to \(i_q^*=5.4 \,\textrm{A}\) and \(u_{dq}^*=400 \,\textrm{V}\) respectively and thereby limit the maximum power consumption to \(2160 \,\textrm{W}\). This permits the utilization of a portion of the PMSM overload region above its rated power. The resolver signal serves as the input for coordinate transformations between the rotary and fixed motor frame. A space vector pulse width modulation (PWM) generates switching commands for a power inverter to apply the desired reference voltages \(u_{\alpha \beta }^*\) to the three motor phases.

The controller gains are tuned based on a linear system model [11]. Phase and gain margins of \(> 60^\circ\) and \(> 10\,\textrm{dB}\), respectively, for each control loop guarantee a robust stability. In Fig. 8 the Bode plots for the closed-loop frequency response of the three cascaded controllers are visualized. It furthermore depicts the PWM frequency of \(20\,\textrm{kHz}\). The desired position \(-3\,\textrm{dB}\)-bandwidth of \(\ge 2\, \textrm{Hz}\) is achieved in the idealized linear model. A small control overshoot of \(18\,\mathrm {\%}\) is only accepted for the current controller.

Fig. 8
figure 8

Bode plot of the cascade benchmark controller

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4 Sensorless controller design

A hybrid rotor angle estimation is implemented on the laboratory real-time computing system. In a real world application, it would be executed locally on the electronic control unit (ECU) of the actuator and receive commands from the primary flight control computers (PFCC). In the low speed domain including standstill the estimation makes use of the alternating injection method. In the high speed domain, the benefits of the passive extended EMF integration approach are exploited, which functions without any additional signal injection. The transition logic between the implemented techniques is visualized in Fig. 9.

Fig. 9
figure 9

Mode management

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At system start the rotor axis is identified at standstill using alternating injection. Its polarity is determined in a short starting procedure after \(t = 50\, \textrm{ms}\). If necessary, the initial rotor angle is corrected prior to the EMA controller activation at \(t = 100\, \textrm{ms}\). The state machine continuously switches between the low and high speed domain when the estimated rotor speed \({\tilde{\omega }}\) exceeds or falls below the transition threshold. A hysteresis prevents a livelock with repetitive mode changes in the transition region. Starting values for rotor angle \({\tilde{\varphi }}\) and speed \({\tilde{\omega }}\) are transferred between the modes resetting corresponding integrators. This ensures smooth transitions and avoids abrupt value changes.

4.1 Starting procedure

A strong current in the positive direct axis leads to magnetic iron saturation of the stator and rotor laminations [10]. This reduces the differential motor inductance and results in higher current amplitudes triggered by a specific voltage input. Figure 10 confirms this statement by showing the current response following a strong voltage pulse with a reference amplitude of \(400\,\textrm{V}\) and time duration of \(\Delta t = 0.4 \,\textrm{ms}\). The highest current peaks are observable in the positive direct axis for the PMSM of the test actuator. This was experimentally verified at various mechanical rotor angles.

Fig. 10
figure 10

Current amplitudes triggered by strong voltage pulses; plotted in rotating coordinates

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The starting procedure exploits this machine property. Voltage pulses are subsequently injected into the presumed positive and negative direct axes as shown in Fig. 11. At the falling edge of each pulse the current amplitude is measured. If the amplitude following the second pulse exceeds the first one, the initial rotor angle estimate is corrected by \(180^\circ\). The starting procedure is triggered by the overarching mode management as shown in Fig. 9 and is fully automated.

Fig. 11
figure 11

Test trajectory of the starting procedure

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4.2 Alternating injection

The alternating injection method introduced in Sect. 2.2.2 is implemented adopting the approach from Holtz et al. [6]. The block diagram is given in Fig. 12.

Fig. 12
figure 12

Alternating injection method implementation

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A sinusoidal voltage with an amplitude of \(u_\mathrm{{inj}} = 100\,\textrm{V}\) at a frequency of \(f_\mathrm{{inj}} = 2.5\,\textrm{kHz}\) is superimposed onto the direct axis reference voltage. The amplitude was determined empirically taking into account the high voltage motor characteristics. It needs to be high enough to guarantee a satisfactory signal-to-noise ratio of the resulting current response. The current response frequency corresponds to the voltage injection frequency. It is separated from the overall signal content using a finite impulse response (FIR) bandpass filter (BPF). After bandpass filtering a rotor angle estimation error results in a sinusoidal quadrature current signal \(i_{{\tilde{q}},\mathrm{{inj}}}\) in estimated \({\tilde{dq}}\)-rotary frame as previously confirmed in Fig. 4. A multiplication with the positive or negative sign of the direct axis signal content \(i_{{\tilde{d}},\mathrm{{inj}}}\) correlates the sinus half-waves with the injected signal. This way, the correct sign is appointed to the estimation error. The estimated \({\tilde{d}}\)-axis converges towards the true d-axis in a PI observer loop.

The alternating injection method requires a separation between the injection frequency and the frequencies excited by the current controller. Otherwise, the signal demodulation would be significantly disturbed by regular motor control activity. In order to avoid an overlap the bandwidth of the current controller is reduced. Consequentially, the gains of the upper cascaded control loops need to be reduced as well. This leads to a degraded controller dynamics in sensorless operation mode depicted by the solid lines in Fig. 13 for each control loop. They are compared to the original benchmark controllers from Fig. 8 depicted by dotted lines.

Fig. 13
figure 13

Bode plot of the sensorless control mode

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After the reduction of the controller gains, the passband of the BPF is, on one hand, clearly above the bandwidth of the current controller, as shown in the Bode magnitude plot. On the other hand, the BPF passband shows a sufficient margin to the PWM frequency. This guarantees a sufficiently high frequency resolution of the sinusoidal injection signal. Furthermore, the Bode phase plot reveals that the bandpass filter introduces a phase lag in the demodulation of the current response signal. At the injection frequency \(f_\mathrm{{inj}} = 2.5\,\textrm{kHz}\), this phase lag amounts to \(-1665^\circ\) (outside the displayed boundaries of Fig. 13 phase plot). It effectively delays the convergence of the rotor angle estimation and must be taken into account for the tuning of the PI observer gains.

4.3 Extended EMF integration

For the high speed domain, the extended EMF integration method introduced in Sect. 2.2.3 is applied, as shown in Fig. 14. The calculation of the extended EMF is a straightforward implementation of Eq. 10. According to Eq. 11, the atan2-operation already provides the rotor angle. However, it is limited to a \(2\pi\)-interval and affected by sensor noise. A subsequent phase-locked loop (PLL) delivers smooth output signals and additionally provides a continuous rotor speed estimate.

Fig. 14
figure 14

Implementation of the extended EMF integration method

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5 Results

The sensorless control algorithms are extensively tested on the test rig introduced in Sect. 3.2. Their performance is compared with the benchmark controller described in Sect. 3.3.

5.1 Frequency response

The benchmark controller shows a higher \(-3\,\textrm{dB}\)-bandwidth than the sensorless control mode. For medium actuator strokes, it decreases from 3.0 to \(0.6\,\textrm{Hz}\) for the position control loop, as shown in Fig. 15.

Fig. 15
figure 15

Frequency response for medium actuator stroke

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The desired bandwidth for the test actuator of \(\ge 2\, \textrm{Hz}\) is therefore only achieved for the original resolver-based controller mode. The degradation of the dynamic performance is fully explained by the reduction of the controller gains, as illustrated in Fig. 13.

The bandwidth reduction is an inherent disadvantage of sinusoidal signal injection methods and is also documented in the literature. Corley et al. [4] degraded the velocity loop drastically to \(10\,\textrm{Hz}\) in an experimental setup using an injection frequency of \(2\,\textrm{kHz}\). Acarnley et al. [1] reference studies with injection frequencies between \(500\,\textrm{Hz}\) and \(4\,\textrm{kHz}\) compared to \(f_\mathrm{{inj}} = 2.5\,\textrm{kHz}\) in this case study. Despite the degraded performance, Linke et al. [14] conclude that the alternating injection method is suitable for fast position control applying an injection frequency of \(2\,\textrm{kHz}\). However, for primary flight control applications, it can be expected that the decrease in magnitude shown in Fig. 15 would adversely affect the aircraft handling qualities and increase the risk of pilot-induced oscillations. In the future, the wider use of efficient silicon carbide (SiC) MOSFET converters may create the conditions for higher PWM frequencies. This would likewise allow to increase the injection frequency and therefore, improve the dynamic positioning performance.

5.2 Step response

The reduced controller dynamics also manifests itself in an increased rise time following a position step command as presented in Fig. 16. The sensorless control mode moreover shows an increased sensitivity to external actuator loads. In compliance with the requirements neither controller exhibits an overshoot. The remaining steady state error is generally small.

Fig. 16
figure 16

Step response for the benchmark controller (blue) and the sensorless controller (red) in no load operation and under load

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A more detailed analysis of the sensorless controller as presented in Fig. 17 confirms its functionality. Only small errors are observed in both the rotor angle and speed estimation, which holds true for both the alternating injection and the extended EMF integration methods. The transition between these modes is smooth, and there are no significant disruptions of the observer performance.

Fig. 17
figure 17

Detailed analysis of the sensorless controller

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The high agility of the benchmark controller leads to an increased energy consumption during the step response, as shown in the upper diagram of Fig. 18. Electrical power is consumed for both the acceleration and the subsequent braking action of the actuator resulting in two consecutive power peaks. In contrast, the sensorless controller consumes far less energy to perform the actuator movement, as illustrated in the lower diagram of Fig. 18. However, the alternating injection additionally causes a permanent basic energy consumption when activated, which leads to a constant increase in consumed energy near standstill.

Fig. 18
figure 18

Energy consumption for the step response in no load conditions

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5.3 Reference flight profile

The controller performance is validated by applying a representative short-haul flight profile with a duration of approximately 2h. It replicates typical actuator movements and hinge moments for all flight and taxi phases. The aileron deflections and actuator loads are shown in the upper two diagrams of Fig. 19. The cumulated positioning error is assessed with the L1-norm \(\int |x_\mathrm{{EMA}}^*-x_\mathrm{{EMA}}|\ \textrm{d}t\ \mathrm {(m\, s)}\). Especially in flight phases with high aileron activity, the benchmark controller outperforms the sensorless controller because of its capability to follow new position commands faster. This agile operation mode results in short periods with high electrical power demand, reaching the actuator limit of \(2160\,\textrm{W}\). The less dynamic sensorless controller, however, exhibits an increased basic power consumption as a result of additional signal injection. Unlike other industrial applications, the actuator mostly carries out recurrent short stroke movements. As a result, the EMA predominantly operates at low speeds and injection losses occur for most of the flight. This further emphasizes the importance of energy-efficient signal injection methods.

Fig. 19
figure 19

Reference flight profile

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6 Conclusion

This paper investigates the performance of sensorless control for electro-mechanical flight actuators without the need for rotor angle measurement. A hybrid rotor angle estimation is used that is suitable for the entire speed range. At low speeds, including standstill, the alternating injection method is applied, which exploits magnetic saliency of the permanent magnet synchronous motor. Otherwise, the fundamental motor wave is evaluated through the integration of the extended electromotive force. To validate the sensorless control mode, laboratory tests were conducted on a test rig that replicates the aileron actuation system of a large civil aircraft. The results were compared to those obtained with a conventional benchmark controller of the electro-mechanical test actuator. Over long test periods, the estimation algorithm proved to provide accurate rotor angle and speed values, enabling robust EMA position control without the need for a resolver. However, the high frequency current spectrum is allocated to the rotor angle estimator necessitating a reduction of the controller bandwidth. Furthermore, injection losses deteriorate the controller efficiency. For primary flight control such a degradation in performance would affect the aircraft handling qualities and is deemed unacceptable. Future work will hence focus on methodological improvements for the low speed domain in order to dissolve the conflict between robust angle estimation and dynamic current control. This could involve a higher injection frequency, possibly conjoined with an increase of the PWM sampling rate. Injection power losses should be minimized and the usage of passive EMF-based methods expanded to smaller rotation speeds. A thriving research community in this field gives ground for optimism that these challenges may be overcome. An extensive overview of recent methodological developments is given by Wang et al. [23]. Specifically, the so-called arbitrary injection method from Paulus et al. [18] may be a promising alternative to the approach investigated within this paper. A detailed safety assessment in accordance with the methods described in the ARP4761 [22] is still due to show the certifiability of sensorless motor control in flight control EMAs. In the beginning, a sensorless control mode might also serve as a dissimilar redundancy in case of a failure of a conventional resolver.