Variable speed drive for the sensorless PWM control of an AC motor by exploiting PWM-induced artefacts

[0001] The present application claims priority to U.S. provisional utility patent application Ser. No. 62/905,663 filed on Sep. 25, 2019 and to European application serial no. 20305748.4 filed on Jul. 2, 2020, which are both incorporated herewith in their entirety.

[0002] This disclosure generally pertains to the field of motor control. More precisely, it relates to Variable Speed Drives (VSDs), which are used to control the operation of Alternating Current (AC) electric motors. The focus is on VSDs that rely on Pulse-Width Modulation (PWM) to have continuous control over the speed of the controlled AC motor.

[0003] VSDs are typically used as industrial drives in factories, installations, HVAC systems and the like to control e.g. the position, speed and/or torque of an electric motor that is dedicated to a particular task, such as e.g. the operation of a fan or the hoisting of a load.

[0004] In order to perform a closed-loop control of an AC motor, a VSD needs to have real-time information on the operating status of the AC motor. This information might for example be the instantaneous angular position and/or angular velocity of the motor's rotor.

[0005] The VSD may obtain this information from dedicated sensors, which are arranged on the motor and monitor the motor's operating status. However, fitting an electric motor with such sensors adds to the complexity and size of the whole drive assembly. The required sensors and sensor cabling also increase the price and reduce the reliability.

[0006] This is why so-called “sensorless” VSDs have become more and more common. In these sensorless VSDs, the motor's operating status is estimated on the basis of measurements of the motor's drive current. No additional external sensors are used. In order to improve the estimations, in particular when the motor operates at low velocity, it is a standard procedure to inject an external high-frequency probing signal into the motor's drive voltage.

[0007] The article “Adding virtual measurements by signal injection” by Pascal Combes et al., published in 2016 in the proceedings of the 2016 American Control Conference on pages 999 ff., conceptualizes and generalizes the signal injection technique for the sensorless control of electric motors at low speed.

[0008] Signal injection is an effective method, but it comes at a price: the ripple it creates may in practice yield unpleasant acoustic noise and excite unmodeled dynamics. In particular, in the very common situation where the electric motor is fed by a PWM inverter, the frequency of the injected probing signal may not be as high as desired so as not to interfere with the PWM (typically, it cannot exceed 500 Hz in an industrial drive with a 4 kHz-PWM frequency).

[0009] The article “A novel approach for sensorless control of PM machines down to zero speed without signal injection or special PWM technique” by C. Wang et al., IEEE Transactions on Power Electronics, Vol. 19, No. 6, November 2004, pages 1601 ff., proposes to measure the phase current ripples induced by conventional PWM to derive the rotor position and speed of a PWM-controlled electric motor. However, this approach is based on the measurement of current derivatives, which requires specialized current sensors and is sensitive to noise.

[0010] In view of the above, it is an object of the present disclosure to provide a PWM-based variable speed drive with an improved sensorless AC motor control without signal injection.

[0011] According to the present disclosure, this object is achieved with a variable speed drive for the closed loop control of the operation of an AC electric motor based on a given control law, the variable speed drive comprising:

• • [0012] an output terminal for delivering a controlled alternating drive voltage to the controlled AC electric motor;
  • [0013] a solid-state power inverter for generating the drive voltage;
  • [0014] a drive controller for controlling the generation of the drive voltage by the power inverter; and
  • [0015] a drive current sensing device for measuring the instantaneous intensity of the drive current taken up by the controlled AC electric motor, and for providing the resulting measurements as a drive current intensity signal to the drive controller,
    wherein the drive controller includes:
  • [0016] a pulse-width modulation generator;
  • [0017] a control law module storing the given control law; and
  • [0018] a state variable estimation module for estimating the instantaneous value of at least one state variable of the controlled AC electric motor,
    wherein the control law module is adapted to, based on the stored control law and state variable estimates provided by the estimation module, compute a target voltage signal and output the computed target voltage signal to the pulse-width modulation generator,
    wherein the pulse-width modulation generator is adapted to:
  • [0019] approximate the received target voltage signal with a pulse-width modulated inverter control signal;
  • [0020] control the operation of the power inverter using the inverter control signal, thereby obtaining the drive voltage;
  • [0021] compute, based on the deviation between the inverter control signal and the target voltage signal, a state variable estimation support signal; and
  • [0022] output the computed state variable estimation support signal to the state variable estimation module, and
    wherein the state variable estimation module is adapted to:
  • [0023] estimate the instantaneous value of a state variable of the AC electric motor based on the received state variable estimation support signal and the drive current intensity signal provided by the drive current sensing device; and
  • [0024] output the resulting state variable estimate to the control law module.

[0025] By modifying the PWM generator so that it computes an estimation support signal based on the PWM, and by providing this estimation support signal to the state variable estimator, the state variable estimator has supplementary information, which it can use to improve its state variable estimations.

[0026] Optionally, the variable speed drive according to the present disclosure can have the following features, separately or in combination one with the others:

• • [0027] the pulse-width modulation generator is adapted to compute the state variable estimation support signal based on a pulse-width modulation inherent disturbance signal, which is obtained by subtracting the target voltage signal from the inverter control signal;
  • [0028] the pulse-width modulation generator is adapted to compute the state variable estimation support signal by integrating the disturbance signal to obtain the primitive of the disturbance signal;
  • [0029] the variable speed drive is adapted to rely on a single feedback to perform closed loop control of the AC electric motor, namely the drive current intensity signal provided by the drive current sensing device;
  • [0030] the variable speed drive is adapted to control the operation of the AC electric motor without the injection of a dedicated probing signal into the drive voltage;
  • [0031] the drive controller further includes an analog-to-digital converter for converting the drive current intensity signal into a digital signal prior to its input into the state variable estimation module;
  • [0032] the state variable estimation module is adapted to estimate the instantaneous value of the rotor position of the electric motor based on the received state variable estimation support signal and the drive current intensity signal provided by the drive current sensing device;
  • [0033] the pulse-width modulation generator is adapted to apply three-phase pulse-width modulation with single carrier to generate the inverter control signal;
  • [0034] the pulse-width modulation generator is adapted to apply three-phase pulse-width modulation with interleaved carriers to generate the inverter control signal.

[0035] According to a further aspect, the present disclosure also relates to an electric drive assembly comprising a synchronous reluctance motor or a permanent-magnet synchronous motor and a variable speed drive as defined above for controlling the motor.

[0036] According to yet a further aspect, the present disclosure also relates to a method of controlling, in a closed loop, the operation of an AC electric motor based on a given control law, the method comprising the following steps:

• [0037] a) measuring the instantaneous intensity of the drive current taken up by the controlled AC electric motor;
• [0038] b) estimating the instantaneous value of a state variable of the AC electric motor using the measured drive current intensity;
• [0039] c) computing, based on the given control law and the estimated state variable, a target voltage signal;
• [0040] d) approximating the computed target voltage signal with a pulse-width modulated inverting control signal;
• [0041] e) computing, based on the deviation between the inverting control signal and the target voltage signal, a state variable estimation support signal;
• [0042] f) generating, by voltage inversion, a controlled alternating drive voltage using the inverting control signal; and
• [0043] g) delivering the generated drive voltage to the controlled AC electric motor;
  wherein the state variable estimation according to step b) relies on the state variable estimation support signal computed in step e) as an additional input together with the drive current intensity measured in step a).

[0047] FIG. 1 is a schematic diagram of an electric drive assembly 100 according to the present disclosure. The electric drive assembly 100 comprises a variable speed drive, or VSD, 200 and an AC electric motor 300.

[0048] The electric drive assembly 100 may be used in diverse industrial settings. For example, it may drive a fan of a heating, ventilation and air conditioning (HVAC) system. As another example, it may also be used to drive a water pump of a sewage installation. Many other industrial applications can be envisaged by the skilled person.

[0049] Preferably, the AC electric motor 300 is a synchronous motor, such as a permanent magnet synchronous motor, or PMSM, or a synchronous reluctance motor, or SynRM.

[0050] The purpose of the variable speed drive 200 is to control the proper operation of the electric motor 300. Thanks to the variable speed drive 200, the motor 300 can be operated at the right speed at the right time, depending on the application. The variable speed drive 200 may also allow to control the torque output of the electric motor 300 to its load.

[0051] The variable speed drive 200 controls the electric motor 300 in a closed loop. This means that the variable speed drive 200 constantly receives feedback on the instantaneous status of the motor 300 during the control of the motor. The variable speed drive 200 adjusts its control of the electric motor 300 based on a given control law. The specifics of the control law depend on the type of application of the electric motor 300.

[0052] With reference to FIG. 1, the variable speed drive 200 comprises an output terminal 210, a solid-state power inverter 220, a drive controller 230, and a drive current sensing device or current sensor 240.

[0053] The variable speed drive 200 is electrically connected to the electric motor 300 via its output terminal 210. The power output 210 delivers a controlled alternating drive voltage u_pwm to the AC electric motor 300. The drive voltage u_pwm is a modulated signal whose amplitude is determined by the DC voltage Vbus applied to the power inverter 220. The modulation frequency of the drive voltage u_pwm depends on the switching frequency of the power inverter 220. The modulated drive voltage u_pwm emulates an ideal sinusoidal drive voltage whose amplitude and frequency determine the operation of the electric motor 300.

[0054] The power inverter 220 generates the drive voltage u_pwm by chopping up a DC voltage with the help of solid-state switches T1, T2.

[0055] The skilled person will note that the diagram of FIG. 1 shows a single phase control. This is only for simplification. Typically, the electric motor 300 will be a three-phase motor. In this case, the power inverter 220 generates a drive voltage for each of the three phases of the motor.

[0056] The current sensor 240 of the VSD 200 measures the instantaneous intensity of the drive current taken up by the electric motor 300. The current sensor 240 provides its measurements as a drive current intensity signal i_s to the drive controller 230.

[0057] According to the present disclosure, the motor control by the VSD 200 is a so-called “sensorless” control. This means that the control feedback entirely relies on the current measurements provided by the current sensor 240. There are no external sensors mounted on the motor 300, such as shaft encoders and the like, to provide feedback to the VSD 200 on the motor status.

[0058] The drive controller 230 controls the generation of the drive voltage u_pwm by the power inverter 220. This is done on the basis of an inverter control signal M provided by the drive controller 230 to the power inverter 220.

[0059] The drive controller 230 may be implemented as a microcontroller or a field programmable gate array (FPGA).

[0060] According to the present disclosure, the drive controller 230 includes a pulse-width modulation, or PWM, generator 232, a control law module 234 storing the given control law, a state variable estimation module 236, and an analog to digital converter, or ADC, 238.

[0061] The control law module 234 is adapted to, based on the stored control law and state variable estimates z₀ to z_n provided by the estimation module 236, compute a target voltage signal u_s and output the computed target voltage signal u_s to the PWM generator 232.

[0062] The target voltage signal u_s represents the analog voltages that must be applied to the stator windings of the electric motor 300 to obtain the desired speed or torque from the motor 300.

[0063] Since the variable speed drive 200 relies on pulse-width modulation, the target voltage signal u_s is not directly applied to the electric motor 300. Rather, it is fed to the PWM generator 232 to be approximated by the pulse-width modulated inverter control signal M, which in turn is used to control the power inverter 220.

[0064] The pulse-width modulation generator 232 may apply three-phase pulse-width modulation with single carrier to generate the inverter control signal M (i.e. the approximation of the target voltage signal u_s).

[0065] Alternatively, the PWM generator may also apply three-phase pulse-width modulation with interleaved carriers to generate the inverter control signal M.

[0066] The PWM generator 232 may of course also use other PWM schemes to generate the inverter control signal M.

[0067] According to the present disclosure, the PWM generator 232 has the particularity that it computes, based on the deviation between the inverter control signal M and the target voltage signal u_s, a state variable estimation support signal s₁, and outputs the computed state variable estimation support signal s₁ to the state variable estimation module 236.

[0068] The state variable estimation module or estimator 236 estimates the instantaneous value of one or more state variables of the AC electric motor 300 based on the drive current intensity signal i_s provided by the drive current sensor 240.

[0069] As shown in FIGS. 1 and 2, the estimator 236 may estimate several state variables z₀ to z_n. These state variables may e.g. correspond to the rotor position of the electric motor, the angular velocity of the motor's rotor, etc.

[0070] According to the present disclosure, the estimator 236 also uses the estimation support signal s₁ to estimate the value of at least one of the state variables z₀ to z_n.

[0071] The estimator 236 provides the state variable estimates z₀ to z_n to the control law module 234. The control law module 234 uses these estimates in the stored control law in order to determine the target voltage signal u_s.

[0072] As shown in the figures, the drive controller 230 may also include an analog to digital converter 238. The purpose of the ADC 238 is to convert the analog current signal i_s provided by the current sensor 240 into a digital signal that can be processed by the estimator 236.

[0073] FIG. 2 shows the signal flow between the different components of the electric drive assembly 100 of FIG. 1. The PWM generator 232 receives the target voltage signal u_s from the control law module 234. Using pulse-width modulation, it approximates the target voltage signal u_s by an inverter control signal M (u, t/ε). The inverter control signal M is fed to the power inverter 220. Based on this control signal M, the inverter 220 delivers a drive voltage u_pwm to the electric motor 300. With the current sensor 240, the currents in the motor's stator windings are measured and digitized in the ADC 238. The digitized current signal is then fed to the estimator 236. The estimator 236 also receives the estimation support signal s₁ (u, t/ε) from the PWM generator 232. The estimator 236 provides estimates of different motor state variables z₀ to z_n based on the received inputs.

[0074] An important aspect of the present disclosure is the enhanced PWM generator 232, which not only generates the inverter control signal M, but also the state variable estimation support signal s₁. With the help of the estimation support signal s₁, the estimation module 236 can extract supplementary information from the current signal i_s to improve the state variable estimations.

[0075] The PWM generator 232 computes the state variable estimation support signal s₁ based on a pulse-width modulation inherent disturbance signal s₀. In fact, the present disclosure relies on the insight that the inverter control signal M generated by the PWM generator 232 can be modelled as a superposition of the target voltage signal u_s and a disturbance signal s₀. Indeed, the inverter control signal M is a series of rectangular voltage pulses of varying widths, which on average, corresponds to the desired target voltage signal u_s. In other words, the inverter control signal M can be regarded as the desired target voltage signal u_s with an added voltage “ripple”. This ripple or PWM disturbance in the voltage creates a disturbance in the stator flux of the motor 300, which in turn creates a disturbance in the measured current i_s.

[0076] The present disclosure takes advantage of this unintended side-effect of the pulse width modulation. The ripple/artefact induced by the pulse-width modulation in the measured currents is used in the state variable estimation. This improves the estimates and thus the control of the electric motor 300.

[0077] A measure of the pulse-width modulation disturbance signal s₀ (i.e. the ripple) can be obtained by subtracting the target voltage signal u_s from the inverter control signal M. The difference between the target voltage signal us and the inverter control signal M can then be integrated to obtain the primitive s₁ of the disturbance signal s₀.

[0078] These calculations, namely the subtraction and the integration, are done by the PWM generator 232. As shown in the figures, the PWM generator 232 provides the result s₁ to the estimator 236.

[0079] It can be mathematically shown (see the remainder of the present disclosure) that the primitive s₁ is a useful input for the estimator 236 to determine the instantaneous value of a state variable of the electric motor 300, such as the rotor position θ.

[0080] The variable speed drive of the present disclosure is particularly useful for the control of synchronous electric motors at low speeds. By suitably using the excitation provided by the pulse-width modulation itself, the variable speed drive of the present disclosure has the same benefits as a conventional variable speed drive relying on an external excitation signal, without the drawbacks of increased acoustic noise and potential interference with the pulse width modulation.

[0081] In the variable speed drive of the present disclosure, standard pulse-width modulation does not need to be modified.

[0082] Also, the estimates by the estimation module 236 only require current measurements from standard current sensors. There is no need for current measurements at extremely precise instants, which is prone to measurement errors and very impractical in an industrial drive.

[0083] Furthermore, the variable speed drive of the present disclosure also does not require specialized sensors capable of measuring current derivatives, as in other known solutions.

[0084] The teachings of the present disclosure may also be applied to the control of other types of actuators. For example, one may think of controlling the operation of an electromagnet in a magnetic bearing, or of a solenoid valve of a hydraulic or pneumatic cylinder.

• [0085] Consider the control system modeled by the evolution equation
  ż=f(z)+g(z)u
  y_α=h(z)  (1)
  where z is the internal state vector of the system, u is the control input vector, and y_α is the measurement vector. In several practical applications (in particular electromechanical devices fed by PWM inverter), the control input is not applied directly, but through a fast-periodic modulation which yields u only in the mean. The actual system is therefore
  {dot over (x)}=f(x)+g(x)(u,t/ϵ)
  y=h(x)
  where x is the state vector of the system disturbed by the modulator and y the corresponding measurement. ϵ>0 is a known small number, and is 1-periodic and has mean u with respect to its second argument, i.e., (ν, σ)=(ν, σ+1) and ∫₀¹(ν, σ)dσ=ν. The system can clearly be written as
  {dot over (x)}=f(x)+g(x)(u+s₀(u,t/ϵ))
  y=h(x)
  where s₀ defined by s₀(ν, σ):=(ν,σ)−ν is 1-periodic and has zero mean with respect to its second argument, i.e., s₀(ν, σ)=s₀(ν, σ+1) and ∫₀¹s₀(ν, σ)dσ=0. In other words, we consider a generalized kind of signal injection, where the probing signal s₀ is generated by the modulation process itself; notice s₀ depends on u, which causes several difficulties.
• [0086] Thanks to a mathematical analysis based on the cherry of second-order averaging, it can be shown that the measured signal y may be written as
  y(t)=y_α(t)+ϵy_ν(t)s₁(u(t),t/ϵ)+O(ϵ²)
  where defined by s₁, defined by s₁(ν, σ):=∫₀^σs₀(ν, s)ds−∫₀¹∫₀^σs₀(ν, s)ds dσ, is the zero-mean primitive of s₀ with u considered as a parameter; the “actual measurement” y_α and the “virtual measurement” y_νs₁s₁^T can be interpreted as outputs of the “ideal” system without modulation (1), namely
  y_α=h(z)  (2)
  y_νs₁s₁^T(u)=h′(z)g(z)s₁s₁^T(u)  (3)
  Notice the square matrix s₁s₁^T(u)=∫₀¹(s₁s₁^T)(u, σ)dσ is in general invertible (see “Multiphase PWM” hereafter), so that we simply have y_ν=h′(z)g(z).
• [0087] The overall purpose of the invention is to extract from the measured signal y estimates of y_α and y_ν which can be used to control the system. All is then as if one would want to control the “ideal” system without modulation (1) with a feedback law relying not only on the “actual measurement” (2), but also on the “virtual measurement” (3). The problem is then simpler thanks to the supplementary information (3); in particular in the case of sensorless control electric motors at low velocity, this supplementary information gives access to the rotor position, hence is instrumental in the design of the control law. The invention consists mainly of two parts:
  • [0088] 1) an enhanced modulator, generating not only the modulating signal , but also the zero-mean primitive s₁ of s₀
  • [0089] 2) estimators and extracting not only y_α, but also y_ν from the measured signal y, using the knowledge of s1.
• [0090] A possible implementation for the estimators is

[0091] $\begin{array}{c}\left(t\right)=\frac{3}{2ϵ}{\int }_{t-ϵ}^{t}y\left(\tau \right)d\tau -\frac{1}{2ϵ}{\int }_{t-2ϵ}^{t-ϵ}y\left(\tau \right)\mathrm{dt}\\ \left(t\right)\stackrel{\mathrm{—––}}{s_1{s}_1^T}\left(u\left(t\right)\right)=\frac{1}{{ϵ}^2}{\int }_{t-ϵ}^t\left(y\left(\tau \right)-\left(\tau \right)\right)s_1\left(u\left(\tau \right),\tau /ϵ\right)d\tau \end{array}$
related to the desired signals by
(t)=y_α(t)+(ϵ²)
(t)s₁s₁^T(u(t))=y_ν(t)s₁s₁^T(u(t))+(ϵ).
Notice these estimators are periodic low-pass filters with FIR (Finite Impulse response) many variants relying on different periodic low-pass FIR filters are possible.

[0092] Generating analog physical power signals is extremely impractical, since a lot of power must be dissipated in the amplifier. Pulse Width Modulation (PWM) addresses the issue by using the transistors in saturation mode. Indeed, transistors are more efficient when they are used in saturation than when they are used in their linear range. The desired value is realized in average by adjusting the width of the pulses (hence the name of the technique).

[0093] The simplest way to realize a PWM modulator is to compare the analog signal u with a triangular carrier c oscillating between −u_max and u_max follows:

[0094] $\begin{array}{c}{u}_{\mathrm{PWM}}\left(t\right)=\{\begin{array}{c}-u_{\max }\mathrm{if}c\left(t\right)>u\left(t\right)\\ u_{\max }\mathrm{if}c\left(t\right)<u\left(t\right)\end{array}=u_{\max }\mathrm{sign}\left(u\left(t\right)-c\left(t\right)\right)\\ \mathrm{where}\\ c\left(t\right)=\{\begin{array}{c}4u_{\max }\left(\frac{t}{T}-k+\frac{1}{4}\right)\mathrm{if}\frac{t}{T}\in \left[k-\frac{1}{2},k\right]\\ -4u_{\max }\left(\frac{t}{T}-k-\frac{1}{4}\right)\mathrm{if}\frac{t}{T}\in \left[k,k+\frac{1}{2}\right]\end{array}\end{array}$

[0095] This is the natural sampling PWM demonstrated in FIG. 10 where the carrier is shown as continuous line c, the analog signal is shown as dashed line u and the PWM signal is shown as dotted line Upwm.

[0096] One of the disadvantages of the natural sampling PWM is that the pulses are not symmetrical. To address this issue, we can first sample the analog signal to obtain u[k]=u(kT) and then apply the PWM modulation on the sampled signal. With this PWM scheme we can check that

[0097] $\begin{array}{c}\frac{1}{T}{\int }_{\left(k-\frac{1}{2}\right)T}^{\left(k+\frac{1}{2}\right)T}{u}_{\mathrm{PWM}}\left(t\right)\mathrm{dt}=\frac{1}{T}({\int }_{\left(k-\frac{1}{2}\right)T}^{\left(\frac{u\left[k\right]}{4u_{\max }}+k-\frac{1}{4}\right)T}{u}_{\max }\mathrm{dt}-\\ {\int }_{\left(\frac{u\left[k\right]}{4u_{\max }}+k-\frac{1}{4}\right)T}^{\left(-\frac{(u\left[k\right]}{4u_{\max }}+k+\frac{1}{4}\right)T}{u}_{\mathrm{ma}x}\mathrm{dt}+\\ {\int }_{\left(-\frac{u\left[k\right]}{4u_{\max }}+k+\frac{1}{4}\right)T}^{\left(k+\frac{1}{2}\right)T}{u}_{\max }\mathrm{dt})\\ =\frac{u_{\max }}{T}(\frac{u\left[k\right]}{4u_{\max }}+\frac{1}{4}+2\frac{u\left[k\right]}{4u_{\max }}-\frac{1}{2}+\\ \frac{u\left[k\right]}{4u_{\max }}+\frac{1}{4})\\ =u\left[k\right]\end{array}$

[0098] The expression of u_PWM can be rewritten under the form u_PWM=u+ũ with

[0099] $\tilde{u}\left(\frac{t}{T},u\right)=u_{\max }\mathrm{sign}\left(u-c\left(\frac{t}{T}\right)\right)-u,$
which is a zero-mean fast-varying periodic signal with a dependence on the control.

[0100] When controlling poly-phase electrical devices, n>1 analog references must be modulated.

[0101] In this case the s₁ will be a vector with n lines and s₁s₁^T is a n×n matrix.

[0102] Traditionally, to facilitate implementation, the n modulators use the same carrier. In this case, when two of the references are equal, this control scheme leads to two equal components in s₁, which means that s₁s₁^T, will not be invertible, which means that we do not obtain all the information in h′(z)g(z). Besides, when two components are close from one another s₁s₁^T will be poorly conditioned, which complexities the signal processing.

[0103] To get more information and improve the conditioning of s₁s₁^T, we can use a different carrier for each reference. In this case, the components of s₁ are always independent. Consequently, the matrix s₁s₁^T, is always invertible and well-conditioned.

[0104] Using Clarke transformation Matrices

[0105] $𝒞:=\frac{2}{3}\left(\begin{array}{ccc}1& -1/2& -1/2\\ 0& \sqrt{3}/2& -\sqrt{3}/2\end{array}\right),{𝒞}^{-1}=\left(\begin{array}{cc}1& 0\\ -1/2& \sqrt{3}/2\\ -1/2& -\sqrt{3}/2\end{array}\right)$
and the rotation

[0106] $ℛ\left(\theta \right):=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$
the model of the Synchronous Reluctant Motor (SynRM) is given by

[0107] $\begin{array}{c}\frac{d{\varphi }_{\mathrm{SDQ}}}{\mathrm{dt}}=ℛ\left(-\theta \right)𝒞u_{\mathrm{Sabc}}-R_s{L}^{-1}{\varphi }_{\mathrm{SDQ}}-J{\mathrm{\omega \varphi }}_{\mathrm{SDQ}}\\ \frac{d\theta }{\mathrm{dt}}=\omega \\ t_{\mathrm{Sabc}}={𝒞}^{-1}ℛ\left(\theta \right)L^{-1}{\varphi }_{\mathrm{SDQ}}\end{array}$

[0108] The state of this system is ϕ_SDQ, the vector of the stator flux in field-oriented DQ frame, and θ, the angular position of the rotor. The vector of stator voltages in physical abc frame, u_Sabc is the control input, while ω, the rotor speed, is a disturbance input, which must be obtained to achieve a proper control of the SynRM. When “sensorless” control is used the sole available measurement is the vector of stator currents in physical abc frame, t_Sabc. The parameters of the model are the stator resistance R_s and the matrix of inductances

[0109] $L=\left(\begin{array}{cc}{L}_D& 0\\ 0& L_Q\end{array}\right).$

[0110] Since PWM is used, the voltage can be written u_Sabc(t)=ū_Sabc(t)+ũ_Sabc(t/ϵ, ū_Sabc(t)).

[0111] Rewriting the system in the time scale σ:=t/ϵ, we obtain

[0112] $\begin{array}{c}\frac{d{\varphi }_{\mathrm{SDQ}}}{d\sigma }=ϵ\left(ℛ\left(-\theta \right)𝒞{\stackrel{\_}{u}}_{\mathrm{Sabc}}+ℛ\left(-\theta \right)𝒞{\stackrel{\_}{u}}_{\mathrm{Sabc}}\left(\sigma ,{\stackrel{\_}{u}}_{\mathrm{Sabc}}\right)-R_s{L}^{-1}{\varphi }_{\mathrm{SDQ}}-J{\mathrm{\omega \varphi }}_{\mathrm{SDQ}}\right)\\ \frac{d\theta }{d\sigma }=\mathrm{ϵ\omega }\\ t_{\mathrm{Sabc}}={𝒞}^{-1}ℛ\left(\theta \right)L^{-1}{\varphi }_{\mathrm{SDQ}}\end{array}$

[0113] Which is like the standard form of averaging. Applying the averaging procedure, we can show that the PWM disturbance in the voltages creates a disturbance in the stator flux, which becomes ϕ_SDQ=ϕ_SDQ+ϵ(−θ)Ũ_Sabc(t/ϵ,ū_Sabc), where Ũ_Sabc is the primitive of ũ_Sabc, with zero-mean over a PWM method. The flux disturbance in turn creates a disturbance in the measured currents which becomes

[0114] $t_{\mathrm{Sabc}}={\stackrel{\_}{t}}_{\mathrm{Sabc}}+{\mathrm{ϵ𝒞}}^{-1}\underset{s\left(\theta \right)}{\underbrace{ℛ\left(\theta \right)L^{-1}ℛ\left(-\theta \right)}}𝒞{\tilde{U}}_{\mathrm{Sabc}}\left(t/ϵ,{\stackrel{\_}{u}}_{\mathrm{Sabc}}\right)+O\left({ϵ}^2\right)$

[0115] The undisturbed variables follow the original model

[0116] $\begin{array}{c}\frac{d{\stackrel{\_}{\varphi }}_{\mathrm{SDQ}}}{\mathrm{dt}}=ℛ\left(-\theta \right)𝒞{\stackrel{\_}{u}}_{\mathrm{Sa\beta }}-R_s{L}^{-1}{\stackrel{\_}{\varphi }}_{\mathrm{SDQ}}-J\omega {\stackrel{\_}{\varphi }}_{\mathrm{SDQ}}\\ \begin{array}{c}\frac{d\theta }{dt}=\omega \\ {\stackrel{\_}{t}}_{S\mathrm{\alpha \beta }}={𝒞}^{-1}ℛ\left(\theta \right)L^{-1}{\stackrel{\_}{\varphi }}_{\mathrm{SDQ}}\end{array}\end{array}$

[0117] Thanks to the proposed estimators, provided Ũ_SabcŨ_Sabc^T is the invertible (see “Poly-phase PWM” above), we can recover t_Sabc and (θ) from t_Sabc, which allow to run the or control law, which computes ū_Sαβ from t_Sαβ, and retrieve θ using (θ)=(θ)L⁻¹(−θ), as can be done with HF injection, but without the need of an additional disturbance.

[0118] Abstract: We show that for PWM-operated devices, it is possible to benefit from signal injection without an external probing signal, by suitably using the excitation provided by the PWM itself. As in the usual signal injection framework conceptualized in [1], an extra “virtual measurement” can be made available for use in a control law, but without the practical drawbacks caused by an external signal.

[0119] Signal injection is a control technique which consists in adding a fast-varying probing signal to the control input. This excitation creates a small ripple in the measurements, which contains useful information if properly decoded. The idea was introduced in [2], [3] for controlling electric motors at low velocity using only measurements of currents. It was later conceptualized in [1] as a way of producing “virtual measurements” that can be used to control the system, in particular to overcome observability degeneracies. Signal injection is a effective method, see e.g. applications to electromechanical devices along these lines in [4], [5], but it comes at a price; the ripple it creates may in practice yield unpleasant acoustic noise and excite unmodeled dynamics, in particular in the very common situation when the device is fed by a Pulse Width Modulation (PWM) inverter; indeed, the frequency of the probing signal may not be as high as desired so as not to interfere with the PWM (typically, it can not exceed 500 Hz in an industrial drive with a 4 kHz-PWM frequency).

[0120] The goal of this paper is to demonstrate that for PWM-operated devices, it is possible to benefit from signal injection without an external probing signal, by using the excitation provided by the PWM itself, as e.g. in [6]. More precisely, consider the Single-Input Single-Output system
{dot over (x)}=f(x)+g(x)u,  (1a)
y=h(x),  (1b)
where u is the control input and y the measured output. We first show in section II that when the control is impressed through PWM, the dynamics may be written as

[0121] $\begin{array}{cc}\stackrel{.}{x}=f\left(x\right)+g\left(x\right)\left(u+s_0\left(u,\frac{t}{\varepsilon }\right)\right),& \left(2\right)\end{array}$
with s₀ 1-periodic and zero-mean in the second argument i.e. s₀(u, σ+1)=s₀(u, σ) and ∫₀¹s₀(u, σ)dσ=0 for all u; ε is the PWM period, hence assumed small. The difference with usual signal injection is that the probing signal s₀ generated by the modulation process now depends not only on time, but also on the control input u. This makes the situation more complicated, in particular because s₀ can be discontinuous in both its arguments. Nevertheless, we show in section III that the second-order averaging analysis of [1] can be extended to this case. In the same way, we show in section IV that the demodulation procedure of [1] can be adapted to make available the so-called virtual measurement
y₀:=H₁(x):=εh′(x)g(x),
in addition to the actual measurement y₀:=H₀(x):=h(x). This extra signal is likely to simplify the design of a control law, as illustrated on a numerical example in section V.

[0122] Finally, we list some definitions used throughout the paper, S denotes a function of two variables, which is T-periodic in the second argument, i.e. S(ν, σ+T)=S(ν, σ) for all ν:

• • [0123] the mean of S in the second argument is the function (of one variable)

[0124] $\stackrel{\_}{S}\left(v\right):=\frac{1}{T}{\int }_0^{T}S\left(v,\sigma \right)d\sigma ;$
S has zero mean in the second argument if S is identically zero

• • [0125] if S has zero mean in the second argument, its zero-mean primitive in the second argument is defined by

[0126] $S_1\left(v,\tau \right):={\int }_0^{\tau }S\left(v,\sigma \right)d\sigma -\frac{1}{T}{\int }_0^T{\int }_0^{\tau }S\left(v,\sigma \right)d\sigma d\tau ;$
notice S₁ is T-periodic in the second argument because S has zero mean in the second argument

• • [0127] the moving average M(k) of k is defined by

[0128] $M\left(k\right)\left(t\right):=\frac{1}{\varepsilon }{\int }_{t-\varepsilon }^{t}k\left(\tau \right)d\tau$

• • [0129] _∞ denotes the uniform “big O” symbol of analysis, namely ƒ(z, ε)=_∞(ε^p) if |ƒ(z, ε)|≤Kε^p for ε small enough, with K>0 independent of z and ε.

[0130] When the control input u in (1a) is impressed through a PWM process with period ε, the resulting dynamics reads

[0131] $\begin{array}{cc}\stackrel{.}{x}=f\left(x\right)+g\left(x\right)ℳ\left(u,\frac{t}{\varepsilon }\right),& \left(3\right)\end{array}$
with 1-periodic and mean u in the second argument; the detailed expression for is given below. Setting s₀(u, σ):=(u, σ)−u, (3) obviously takes the form (2), with s₀ 1-periodic and zero-mean in the second argument.

[0132] Classical PWM with the period ε and range [−u_m, u_m] is obtained by comparing the input signal u to the ε-periodic sawtooth carrier defined by

[0133] $c\left(t\right):=\{\begin{array}{cc}{u}_m+4w\left(\frac{t}{\varepsilon }\right)& \mathrm{if}-\frac{u_m}{2}\le w\left(\frac{t}{\varepsilon }\right)\le 0\\ u_m-4w\left(\frac{t}{\varepsilon }\right)& \mathrm{if}0\le w\left(\frac{t}{\varepsilon }\right)\le \frac{u_m}{2}\end{array};$
the 1-periodic function

[0134] $w\left(\sigma \right):=u_m\mathrm{mod}\left(\sigma +\frac{1}{2},1\right)-\frac{u_m}{2}$
wraps the normalized time

[0135] $\sigma =\frac{t}{\varepsilon }\mathrm{to}\left[-\frac{u_m}{2},\frac{u_m}{2}\right].$
If u varies slowly enough, it crosses the carrier c exactly once on each rising and filling ramp, at times t₁^u>t₂^u such that

[0136] $u\left(t_1^u\right)=u_m+4w\left(\frac{t_1^u}{\varepsilon }\right)$ $u\left(t_2^u\right)=u_m-4w\left(\frac{t_2^u}{\varepsilon }\right).$

[0137] The PWM-encoded signal is therefore given by

[0138] $u_{\mathrm{pwm}}\left(t\right)=\{\begin{array}{cc}{u}_m& \mathrm{if}-\frac{u_m}{2}<w\left(\frac{t}{\varepsilon }\right)\le w\left(\frac{t_1^u}{\varepsilon }\right)\\ -u_m& \mathrm{if}w\left(\frac{t_1^u}{\varepsilon }\right)<w\left(\frac{t}{\varepsilon }\right)\le w\left(\frac{t_2^u}{\varepsilon }\right)\\ u_m& \mathrm{if}w\left(\frac{t_2^u}{\varepsilon }\right)<w\left(\frac{t}{\varepsilon }\right)\le \frac{u_m}{2}\end{array}.$
FIG. 3 illustrates the signals u, c and u_pwm. The function

[0139] $\begin{array}{c}ℳ\left(u,\sigma \right):=\{\begin{array}{cc}{u}_m& \mathrm{if}-2u_m<4w\left(\sigma \right)\le u-u_m\\ -u_m& \mathrm{if}u-u_m<4w\left(\sigma \right)\le u_m-u\\ u_m& \mathrm{if}{u}_m-u<4w\left(\sigma \right)\le 2u_m\end{array}\\ =u_m+\mathrm{sign}\left(u-u_m-4w\left(\sigma \right)\right)+\mathrm{sign}\left(u-u_m+4w\left(\sigma \right)\right),\end{array}$
which is obviously 1-periodic and with mean u with respect to its second argument, therefore completely describes the PWM process since

[0140] $u_{\mathrm{pwm}}\left(t\right)=ℳ\left(u\left(t\right),\frac{t}{\varepsilon }\right).$

[0141] Finally, the induced zero-mean probing signal is

[0142] $\begin{array}{c}{s}_0\left(u,\sigma \right):=ℳ\left(u,\sigma \right)-u\\ =u_m-u+\mathrm{sign}\left(\frac{u-u_m}{4}-w\left(\sigma \right)\right)+\mathrm{sign}\left(\frac{u-u_m}{4}-w\left(\sigma \right)\right),\end{array}$
and its zero-mean primitive in the second argument is

[0143] $s_1\left(u,\sigma \right):=\left(1-\frac{u}{u_m}\right)w\left(\sigma \right)-\frac{u-u_m}{4}-w\left(\sigma \right)+\frac{u-u_m}{4}-w\left(\sigma \right).$

[0144] Remark 1: As s₀ is only piecewise continuous, one might expect problems to define the “solutions” of (2). But as noted above, if the input u(t) of the PWM encoder varies slowly enough, its output

[0145] $u_{\mathrm{pwm}}\left(t\right)=ℳ\left(u\left(t\right),\frac{t}{\varepsilon }\right)$
will have exactly two discontinuities per PWM period. Chattering is therefore excluded, which is enough to ensure the existence and uniqueness of the solutions of (2), see [7], without the need for the more general Filipov theory [8]. Of course, we assume (without loss of generality in practice) that ƒ, g and h in (1) are smooth enough.

[0146] Notice also s₁ is continuous and piecewise C¹ in both its arguments. The regularity in the second argument was to be expected as s₁(u, ·) is a primitive of s₀(u, ·); on the other hand, the regularity in the first argument stems from the specific form of s₀.

[0147] Section III-A outlines the overall approach and states the main Theorem 1, which is proved in the somewhat technical section III-B. As a matter of fact, the proof can be skipped without losing the main thread; suffice to say that if s₀ were Lipschitz in the first argument, the proof would essentially be an extension of the analysis by “standard” second-order averaging of [1], with more involved calculations

A. Main Result

[0148] Assume we have designed a suitable control law
ū=α(η,Y,t)
{dot over (η)}=α(η,Y,t)₁
where ηε^q, for the system

[0149] $\stackrel{.}{\stackrel{\_}{x}}=f\left(\stackrel{\_}{x}\right)+g\left(\stackrel{\_}{x}\right)\stackrel{\_}{u},\stackrel{\_}{Y}=H\left(\stackrel{\_}{x}\right):=\left(\begin{array}{c}h\left(\stackrel{\_}{x}\right)\\ \varepsilon h^{\prime }\left(\stackrel{\_}{x}\right)g\left(\stackrel{\_}{x}\right)\end{array}\right).$
By “suitable”, we mean the resulting closed-loop system
{dot over (x)}=ƒ(x)+g(x)α(η,H(x),t)  (4a)
{dot over (η)}=α(η,H(x),t)  (4b)
has the desired exponentially stable behavior. We have changed the notations of the variables with · to easily distinguish between the solutions of (4) and of (7) below. Of course, this describes an unrealistic situation:

• • [0150] PWM is not taken into account
  • [0151] the control law is not implementable, as it uses not only the actual output y_α=h(x), but also the a priori not available virtual output y_v=εh′(x)g(x).

[0152] Define now (up to _∞(ε²)) the function
H(x,η,σ,t):=H(x−εg(x)s₁(α(η,H(x),t),σ))+_∞(ε²),  (5)
where s₁ is the zero mean primitive of s₀ in the second argument, and consider the control law

[0153] $\begin{array}{cc}u=\alpha \left(\eta ,\stackrel{\_}{H}\left(x,\eta ,\frac{t}{\varepsilon },t\right),t\right)& \left(6a\right)\\ \stackrel{.}{\eta }=a\left(\eta ,\stackrel{\_}{H}\left(x,\eta ,\frac{t}{\varepsilon },t\right),t\right).& \left(6b\right)\end{array}$
The resulting closed-loop system, including PWM, reads

[0154] $\begin{array}{cc}\stackrel{.}{x}=f\left(x\right)+g\left(x\right)ℳ\left(\alpha \left(\eta ,\stackrel{\_}{H}\left(x,\eta ,\frac{t}{\varepsilon },t\right),t\right),\frac{t}{\varepsilon }\right)& \left(7a\right)\\ \stackrel{.}{\eta }=a\left(\eta ,\stackrel{\_}{H}\left(x,\eta ,\frac{t}{\varepsilon },t\right),t\right).& \left(7b\right)\end{array}$
Though PWM is now taken into account the control law (6) still seems to contain unknown terms.
Nevertheless, it will turn out from the following result that it can be implemented.

[0155] Theorem 1: Let (x(t), η(t)) be the solution of (7) starting from (x₀, η₀), and define u(t):=α(η(t), H(x(t)), t) and y(t):=H(x(t)); let (x(t), η(t)) be the solution of (4) starting from (x₀−εg(x₀). s₁(u(0), 0), η₀), and define ū(t):=α(η(t), H(x(t)),t). Then, for t≥0,

[0156] $\begin{array}{cc}x\left(t\right)=\stackrel{\_}{x}\left(t\right)+\varepsilon g\left(\stackrel{\_}{x}\left(t\right)\right)s_1\left(\stackrel{\_}{u}\left(t\right),\frac{t}{\varepsilon }\right)+{𝒪}_{\infty }\left({\varepsilon }^2\right)& \left(8a\right)\\ \eta \left(t\right)=\stackrel{\_}{\eta }\left(t\right)+{𝒪}_{\infty }\left({\varepsilon }^2\right)& \left(8b\right)\\ y\left(t\right)=H_0\left(\stackrel{\_}{x}\left(t\right)\right)+H_1\left(\stackrel{\_}{x}\left(t\right)\right)s_1\left(\stackrel{\_}{u}\left(t\right),\frac{t}{\varepsilon }\right)+{𝒪}_{\infty }\left({\varepsilon }^2\right).& \left(8c\right)\end{array}$

[0157] The practical meaning of the theorem is the following. As the solution (x(t), η(t))) is piecewise C¹, we have by Taylor expansion using (8a)-(8b) that u(t)=ū(t)+_∞(ε²). In the same way, as s₁ is also piecewise C¹, we have

[0158] $s_1\left(u\left(t\right),\frac{t}{\varepsilon }\right)=s_1\left(\stackrel{\_}{u}\left(t\right),\frac{t}{\varepsilon }\right)+{𝒪}_{\infty }\left({\varepsilon }^2\right).$
As a consequence, we can invert (8a)-(8b), which yields

[0159] $\begin{array}{cc}\stackrel{\_}{x}\left(t\right)=x\left(t\right)-\varepsilon g\left(x\left(t\right)\right)s_1\left(u\left(t\right),\frac{t}{\varepsilon }\right)+{𝒪}_{\infty }\left({\varepsilon }^2\right)& \left(9a\right)\\ \stackrel{\_}{\eta }\left(t\right)=\eta \left(t\right)+{𝒪}_{\infty }\left({\varepsilon }^2\right).& \left(9b\right)\end{array}$
Using this into (5), we then get

[0160] $\begin{array}{cc}\begin{array}{c}\stackrel{\_}{H}\left(x\left(t\right),\eta \left(t\right),\frac{t}{\varepsilon },t\right)=H\left(x\left(t\right)-\varepsilon g\left(x\left(t\right)\right)s_1\left(u\left(t\right),\frac{t}{\varepsilon }\right)\right)+{𝒪}_{\infty }\left({\varepsilon }^2\right),\\ =H\left(\stackrel{\_}{x}\left(t\right)\right)+{𝒪}_{\infty }\left({\varepsilon }^2\right).\end{array}& \left(10\right)\end{array}$
On the other hand, we will see in section IV that, thanks to (8c), we can produce an estimate Ŷ=H(x)+_∞(ε²). This means the PWM-fed dynamics (3) acted upon by the implementable feedback
u=α(η,Ŷ,t)
{dot over (η)}=α(η,Ŷ,t).
behaves exactly as the closed-loop system (4) except for the presence of a small ripple (described by (8a)-(8b)).

[0161] Remark 2: Notice that, according to Remark 1, H₀(x(t)) and H₁(x(t)) in (8c) may be as smooth as desired (the regularity is inherited from only f, g, h, α, a); on the other hand

[0162] $s_1\left(u\left(t\right),\frac{t}{\varepsilon }\right)$
is only continuous and piecewise C¹. Nevertheless, this is enough to justify all the Taylor expansions performed in the paper.
Proof of Theorem 1

[0163] Because of the lack of regularity of s₀, we must go hack to the fundamentals of the second-order averaging theory presented in [9, chapter 2] (with slow time dependence [9, section 3.3]). We first introduce two ad hoc definitions.

[0164] Definition 1: A function φ(X, σ) is slowly-varying in average if there exists λ>0 such that for ε small enough,
∫_α^α+T∥φ(p(εσ)+ε^kq(σ),σ)−φ(p(εσ),σ)∥dσ≤λTε^k,
where p, q are continuous with q bounded; α and T>0 are arbitrary constants. Notice that if φ is Lipschitz in the first variable then it is slowly-varying in average. The interest of this definition is that it is satisfied by s₀.

[0165] Definition 2: A function φ is _∞(ε³) in average if there exists K>0 such that ∥∫₀^αϕ(q(s), s)ds∥≤k ε³σ for all σ≥0. Clearly, if ϕ is _∞(ε³) then it is _∞(ε³) in average.

[0166] The proof of Theorem 1 follows the same steps as [9, chapter 2], but with weaker assumptions. We first rewrite (7) in the fast timescale σ:=t/ε as

[0167] $\begin{array}{cc}\frac{\mathrm{dX}}{d\sigma }=\varepsilon F\left(X,\sigma ,\mathrm{\varepsilon \sigma }\right).\mathrm{where}X:=\left(x,\eta \right)\mathrm{and}F\left(X,\sigma ,\tau \right):=\left(\begin{array}{c}f\left(x\right)+g\left(x\right)ℳ\left(\alpha \left(\eta ,\stackrel{\_}{H}\left(x,\eta ,\sigma ,\tau \right),\tau \right)\right)\\ a\left(\eta ,\stackrel{\_}{H}\left(x,\eta ,\sigma ,\tau \right),\tau \right)\end{array}\right).& \left(11\right)\end{array}$
Notice F is 1-periodic in the second argument. Consider also the so-called averaged system

[0168] $\begin{array}{cc}\frac{d\stackrel{\_}{X}}{d\sigma }=\varepsilon \stackrel{\_}{F}\left(\stackrel{\_}{X},\mathrm{\varepsilon \sigma }\right).& \left(12\right)\end{array}$
where F is the mean F in the second argument.

[0169] Define the near-identity transformation

[0170] $\begin{array}{cc}X=\tilde{X}+\varepsilon W\left(\tilde{X},\sigma ,\mathrm{\varepsilon \sigma }\right),\mathrm{where}\tilde{X}:=\left(\tilde{x},\tilde{\eta }\right)\mathrm{and}W\left(\tilde{X},\sigma ,\tau \right):=\left(\begin{array}{c}g\left(\tilde{x}\right)\\ 0\end{array}\right)s_1\left(\alpha \left(\tilde{\eta },H\left(\tilde{x},\tilde{\eta },\sigma ,\tau \right),\tau \right),\sigma \right).& \left(13\right)\end{array}$
Inverting (13) yields
{tilde over (X)}=X−εW(X,σ,εσ)+_∞(ε²).  (14)
By lemma 1, this transformation puts (11) into

[0171] $\begin{array}{cc}\frac{d\tilde{X}}{d\sigma }=\varepsilon \stackrel{\_}{F}\left(\tilde{X},\mathrm{\varepsilon \sigma }\right)+{\varepsilon }^2\Phi \left(\tilde{X},\sigma ,\mathrm{\varepsilon \sigma }\right)+\varphi \left(\tilde{X},\sigma ,\mathrm{\varepsilon \sigma }\right);& \left(15\right)\end{array}$
Φ is periodic and zero-mean in the second argument, and slowly-varying in average, and ϕ is _∞(ε³) in average.

[0172] By lemma 2, the solutions X(σ) and {tilde over (X)}(σ) of (12) and (15), starting from the same initial conditions, satisfy
{tilde over (X)}(σ)=X(σ)+_∞(ε³).

[0173] As a consequence, the solution X(σ) of (11) stating from X₀ and the solution X(σ) of (12) starting from X₀−εW(X₀, 0, 0) are related by X(σ)=X(σ)+εW(X(σ), σ, εσ)+_∞(ε²), which is exactly (8a)-(8b). Inserting (8a) in y=h(x) and Taylor expanding yields (8c).

[0174] Remark 3: If s₀ were differentiable in the first variable, Φ would be Lipschitz and ϕ would be on a in (15), hence the averaging theory of [9] would directly apply.

[0175] Remark 4. In the sequel, we prove for simplicity only the estimation {tilde over (X)}(σ)=X(σ)+(ε²) on a timescale 1/ε. The continuation to infinity follows from the exponential stability of (4), exactly as in [1, Appendix].

[0176] In the same way, lemma 2 is proved without slow-time dependence, the generalization being obvious as in [9, section 3.3].

[0177] Lemma 1: The transformation (13) puts (11) into (15), where Φ is periodic and zero-mean in the second argument, and slowly-varying in average, ϕ is _∞(ε³) in average.

[0178] Proof: To determine the expression for d{tilde over (X)}/dσ, the objective is to compute dX/dσ as a function of {tilde over (X)} with two different methods. On the one hand we replace X with its transformation (13) in the closed-loop system (11), and on the other hand we differentiate (13) with respect to σ.

[0179] We first compute s₀(α(η, H(x, η, σ, εσ), εσ), σ) as a function of {tilde over (X)}=({tilde over (x)}, {tilde over (η)}). Exactly as in (10), with ({tilde over (x)}, {tilde over (η)}) replacing (x, η), and (14) replacing (9), we have
H(x,η,σ,εσ)=H({tilde over (x)})+_∞(ε²).
Therefore, by Taylor expansion
α(η,H(x,η,σ,εσ),εσ)=α({tilde over (η)},H({tilde over (x)}),εσ)+ε²K_α({tilde over (X)},σ),
with K_α bounded. The lack of regularity of s₀ prevents further Taylor expansion; nonetheless, we still can write
s₀(α(η,H(x,η,σ,εσ),εσ),σ)=s₀(α({tilde over (η)},H({tilde over (x)}),εσ)+ε²K_α({tilde over (X)},σ),σ).
Finally inserting (13) into (11) and Taylor expanding, yields after tedious but straightforward computations,

[0180] $\begin{array}{cc}\frac{\mathrm{dX}}{d\sigma }=\varepsilon \stackrel{\_}{F}\left(\tilde{X},\mathrm{\varepsilon \sigma }\right)+\varepsilon G\left(\tilde{X}\right)s_0^{a,+}\left(\widetilde{·}\right)+{\varepsilon }^2\stackrel{\_}{F}\left(\tilde{X},\mathrm{\varepsilon \sigma }\right)G\left(X\right)s_1^a\left(\widetilde{·}\right)+{\varepsilon }^2{G}^{\prime }\left(\tilde{X}\right)G\left(\tilde{X}\right)s_1^a\left(\widetilde{·}\right)s_0^{a,+}\left(\widetilde{·}\right)+{𝒪}_{\infty }\left({\varepsilon }^3\right);& \left(16\right)\end{array}$
we have introduced the following notations

[0181] $\left(\widetilde{·}\right):=\left(\tilde{X},\sigma ,\mathrm{\varepsilon \sigma }\right)$ $s_i^a\left(\widetilde{·}\right):=s_i\left(\alpha \left(\tilde{\eta },H\left(\tilde{x}\right),\mathrm{\varepsilon \sigma }\right),\sigma \right),s_0^{a,+}\left(\widetilde{·}\right):=s_0\left(\alpha \left(\tilde{\eta },H\left(\tilde{x}\right),\mathrm{\varepsilon \sigma }\right)+{\varepsilon }^2{K}_a\left(\tilde{X},\sigma \right),\sigma \right)$ ${\mathrm{\Delta s}}_0^a\left(\widetilde{·}\right):=s_0^{a,+}\left(\widetilde{·}\right)-s_0\left(\widetilde{·}\right)$ $G\left(X\right):=\left(\begin{array}{c}g\left(x\right)\\ 0\end{array}\right)$ $\stackrel{\_}{F}\left(X,\mathrm{\varepsilon \sigma }\right):=\left(\begin{array}{c}f\left(x\right)+g\left(x\right)\alpha \left(\eta ,H\left(x\right),\mathrm{\varepsilon \sigma }\right)\\ \alpha \left(\eta ,H\left(x\right),\mathrm{\varepsilon \sigma }\right)\end{array}\right).$

[0182] We now time-differentiate (13), which reads with the previous notations
X={tilde over (X)}+εG({tilde over (X)})s₁^α({tilde over (·)}).
This yields

[0183] $\begin{array}{cc}\frac{\mathrm{dX}}{d\sigma }=\frac{d\tilde{X}}{d\sigma }+{\varepsilon G}^{\prime }\left(\tilde{X}\right)\frac{d\tilde{X}}{d\sigma }{s}_1^a\left(\widetilde{·}\right)+\varepsilon G\left(\tilde{X}\right){\partial }_1{s}_1^{\alpha }\left(\widetilde{·}\right)\frac{d\tilde{X}}{d\sigma }+\varepsilon G\left(\tilde{X}\right)s_0^a\left(\widetilde{·}\right)+{\varepsilon }^{2}G\left(\tilde{X}\right){\partial }_3{s}_1^a\left(\widetilde{·}\right),& \left(17\right)\end{array}$
since ∂₂s₁^α=s₀^α. Now assume {tilde over (X)} satisfies

[0184] $\begin{array}{cc}\frac{d\tilde{X}}{d\sigma }=\varepsilon \stackrel{\_}{F}\left(\tilde{X},\varepsilon \sigma \right)+\mathrm{\varepsilon G}\left(\tilde{X}\right){\Delta s}_0^a\left(\widetilde{·}\right)+{\varepsilon }^2\Psi \left(\widetilde{·}\right),& \left(18\right)\end{array}$
where Ψ({tilde over (·)}) is yet to be computed. Inserting (18) into (17),

[0185] $\begin{array}{cc}\frac{\mathrm{dX}}{d\sigma }=\varepsilon \stackrel{\_}{F}\left(\tilde{X},\varepsilon a\right)+\varepsilon G\left(\tilde{X}\right){\mathrm{\Delta s}}_0^a\left(\widetilde{·}\right)+{\varepsilon }^2\Psi \left(\widetilde{·}\right)+{\varepsilon }^2{G}^{\prime }\left(\tilde{X}\right)\stackrel{\_}{F}\left(\tilde{X},\mathrm{\varepsilon \sigma }\right)s_1^a\left(\widetilde{·}\right)+{ϵ}^2{G}^{\prime }\left(\tilde{X}\right)G\left(\tilde{X}\right){\mathrm{\Delta s}}_0^a\left(\widetilde{·}\right)s_1^a\left(\widetilde{·}\right)+{\varepsilon }^{2}G\left(\tilde{X}\right){\partial }_1{s}_1^a\left(\widetilde{·}\right)\stackrel{\_}{F}\left(\tilde{X},\mathrm{\varepsilon \sigma }\right)+{\varepsilon }^{2}G\left(\tilde{X}\right){\partial }_1{s}_1^a\left(\widetilde{·}\right)G\left(\tilde{X}\right){\mathrm{\Delta s}}_0^a\left(\widetilde{·}\right)+\varepsilon G\left(\tilde{X}\right)s_0^a\left(\widetilde{·}\right)+{\varepsilon }^{2}G\left(\tilde{X}\right){\partial }_3{s}_1^a\left(\widetilde{·}\right)+O_{\infty }\left({\varepsilon }^3\right).& \left(19\right)\end{array}$
Next, equation (19) and (16), Ψ satisfies

[0186] $\begin{array}{cc}\Psi \left(\widetilde{·}\right)=\left[\stackrel{\_}{F},G\right]\left(\tilde{X},\mathrm{\varepsilon \sigma }\right)s_1^a\left(\widetilde{·}\right)+G^{\prime }\left(\tilde{X}\right)G\left(\tilde{X}\right)s_0^a\left(\widetilde{·}\right)s_1^a\left(\widetilde{·}\right)-G\left(\tilde{X}\right){\partial }_1{s}_1^a\left(\widetilde{·}\right)\stackrel{\_}{F}\left(\tilde{X},\mathrm{\varepsilon \sigma }\right)-G\left(\tilde{X}\right){\partial }_3{s}_1^{\alpha }\left(\widetilde{·}\right)-G\left(\tilde{X}\right){\partial }_1{s}_1^a\left(\widetilde{·}\right)G\left(\tilde{X}\right){\mathrm{\Delta s}}_0^a\left(\widetilde{·}\right).& \left(20\right)\end{array}$
This gives the expressions of Φ and ϕ in (15),

[0187] $\Phi \left(\widetilde{·}\right):=\left[\stackrel{\_}{F},G\right]\left(\tilde{X},\mathrm{\varepsilon \sigma }\right)s_1^a\left(\widetilde{·}\right)+G^{\prime }\left(\tilde{X}\right)G\left(\tilde{X}\right)s_0^a\left(\widetilde{·}\right)s_1^a\left(\widetilde{·}\right)-G\left(\tilde{X}\right){\partial }_1{s}_1^a\left(\widetilde{·}\right)\stackrel{\_}{F}\left(\tilde{X},\mathrm{\varepsilon \sigma }\right)-G\left(\tilde{X}\right){\partial }_3{s}_1^a\left(\widetilde{·}\right),\varphi \left(\widetilde{·}\right):={\varepsilon }^2{\Psi }_1\left(\widetilde{·}\right)+\varepsilon G\left(\tilde{X}\right){\mathrm{\Delta s}}_0^a,\mathrm{with}$ ${\Psi }_1\left(\widetilde{·}\right):=-G\left(\tilde{X}\right){\partial }_1{s}_1^a\left(\widetilde{·}\right)G\left(\tilde{X}\right){\mathrm{\Delta s}}_0^a\left(\widetilde{·}\right).$

[0188] The last step is to check that Φ and ϕ satisfy the assumptions of the lemma. Since s₀^α, s₁^α, ∂₁s₁^αand ∂₂s₁^α are periodic and zero-mean in the second argument, and slowly-varying in average, so is Φ.There remains to prove that ϕ=_∞(ε³) in average. Since Δs₀^α is slowly-varying in average,
∫₀^σ∥Δs₀^α({tilde over (·)})(s))∥ds≤λ₀σε².
with λ₀>0, G being bounded by a constant c_g, this implies
∥∫₀^σεG({tilde over (X)}(s))Δs₀^α({tilde over (·)})(s))dsμ≤c_gλ₀σε³.
Similarly, ∂₁s₁ being bounded by c₁₁, Ψ₁ satisfies
∥∫₀^σε²Ψ₁({tilde over (·)}(s))ds∥≤c_g²c₁₁λ₀σε₀ε³.
Summing the two previous inequalities yields
∥∫₀^σϕ({tilde over (·)}(s))ds∥≤λ₀c_g(1+c₁₁c_gε₀)σε³,
which concludes the proof.

[0189] Lemma 2: Let X(σ) and {tilde over (X)}(σ) be respectively the solutions of (12) and (15) starting at 0 from the same initial conditions. Then, for all σ≥0
{tilde over (X)}(σ)=X(σ)+_∞(ε²).

[0190] Proof: Let E(σ):={tilde over (X)}(σ)−X(σ). Then,

[0191] $\begin{array}{c}E\left(\sigma \right)={\int }_0^{\sigma }\left[\frac{d\tilde{X}}{\mathrm{d\sigma }}\left(s\right)-\frac{d\stackrel{\_}{X}}{\mathrm{d\sigma }}\left(s\right)\right]\mathrm{ds}\\ =\varepsilon {\int }_0^{\sigma }\left[F\left(\tilde{X}\left(s\right)\right)-F\left(\stackrel{\_}{X}\left(s\right)\right)\right]\mathrm{ds}+{\varepsilon }^2{\int }_0^{\sigma }\Phi \left(\widetilde{·}\left(s\right)\right)\mathrm{ds}+\\ {\int }_0^{\sigma }\varphi \left(\widetilde{·}\left(s\right)\right)\mathrm{ds}\end{array}$
As F Lipschitz with constant λ_F,
ε∫₀^σ∥F({tilde over (X)}(s))−F(X(s))∥ds≤ελ_F∫₀^σ∥E(s)∥ds.
On the other hand, there exists by lemma 3 c₁ such that
ε²∥∫₀^σΦ({tilde over (·)})(s))ds∥≤c₁ε²
Finally, as ϕ is _∞(ε³) in average, there exists c₂ such that
μ∫₀^σϕ({tilde over (·)}(s))ds∥≤c₂ε³σ.
The summation of these estimations yields
∥E(σ)∥≤ελ_F∫₀^σ∥E(s)∥ds+c₁ε²+c₂ε³σ.
Then by Gronwall's lemma [9, Lemma 1.3.3]

[0192] $E\left(\sigma \right)\le \left(\frac{c_2}{{\lambda }_p}+c_1\right)e^{{\lambda }_F\sigma }{\varepsilon }^2,$
which means {tilde over (X)}=X+_∞(ε²).

[0193] The following lemma is an extension of Besjes' lemma [9, Lemma 2.8.2] when φ is no longer Lipschitz, but only slowly-varying in average.

[0194] Lemma 3: Assume φ(X, σ) is T-periodic and zero-mean in the second argument, bounded, and slowly-varying in average. Assume the solution X(σ) of {dot over (X)}=_∞(ε) is defined for 0≤σ≤L/ε. There exists c₁>0 such that
∥∫₀^σφ(X(s),s)ds∥≤c₁.

[0195] Proof: Along the lines of [9], we divide the interval [0, t] in m subintervals [0, T], . . . , [(m−1)T, mT] and a remainder [mT, t]. By splitting the integral on those intervals, we write

[0196] ${\int }_0^{\sigma }\phi \left(\varkappa \left(s\right),s\right)\mathrm{ds}=\sum _{i=0}^m{\int }_{\left(i-1\right)T}^{\mathrm{iT}}\phi \left(\varkappa \left(\left(i-1\right)T\right),s\right)\mathrm{ds}+\sum _{i=0}^m{\int }_{\left(i-1\right)T}^{\mathrm{iT}}\left[\phi \left(\varkappa \left(s\right),s\right)-\phi \left(\varkappa \left(\left(i-1\right)T\right),s\right)\right]\mathrm{ds}+{\int }_{\mathrm{mT}}^{\sigma }\phi \left(\varkappa \left(s\right),s\right)\mathrm{ds},$
where each of the integral in the first sum are zero as φ is periodic with zero mean. Since φ is bounded, the remainder is also bounded by a constant c₂>0. Besides

[0197] $\begin{array}{c}x\left(s\right)=x\left(\left(i-1\right)T\right)+{\int }_{\left(i-1\right)T}^{}\stackrel{.}{\varkappa }\left(\tau \right)d\tau \\ =\varkappa \left(\left(i-1\right)T\right)+\varepsilon q\left(s\right),\end{array}$
with q continuous and bounded. By hypothesis, there exists λ>0 such that 0≤i≤m,
∫_(i−1)T^T∥φ(x(s),s)−φ(x((i−1)T),s)∥ds≤λTε
Therefore by summing the previous estimations,
∥∫₀^σφ(x(s),s)ds∥≤mλTε+o₂,
with mT≤t≤L/ε, consequently mλTε+o₂≤λL+o₂; which concludes the proof.

[0198] From (8c), we can write the measured signal y as

[0199] $y\left(t\right)=y_a\left(t\right)+y_v\left(t\right)s_1\left(u\left(t\right),\frac{t}{\varepsilon }\right)+O_{\infty }\left({\varepsilon }^2\right),$
where the signal u feeding the PWM encoder is known. The following result shows y_α and y_ν can he estimated from y, for use in a control law as described in section III-A.

[0200] Theorem 2: Consider the estimators ŷ_α and ŷ_u defined by

[0201] ${\hat{y}}_a\left(t\right):=\frac{3}{2}M\left(y\right)\left(t\right)-\frac{1}{2}M\left(y\right)\left(t-\varepsilon \right)$ $k_{\Delta }\left(\tau \right):=\left(y\left(\tau \right)-{\hat{y}}_n\left(\tau \right)\right)s_1\left(u\left(\tau \right),\frac{\tau }{\varepsilon }\right)$ ${\hat{y}}_v\left(t\right):=\frac{M\left(k_{\Delta }\right)\left(t\right)}{\stackrel{\_}{s_1^2}\left(u\left(t\right)\right)},$
where M:yε⁻¹∫₀^εy(τ)dτ is the moving average operator, and s₁² the mean of s₁² in the second argument (cf end of section I). Then,
ŷ_α(t)=y_α(t)+_∞(ε²)  (21a)
ŷ_ν(t)=y_ν(t)′_∞(ε²).  (21b)
Recall that by construction y_ν(t)=_∞(ε), hence (21b) is essentially a first-order estimation; notice also that s₁²(u(t)) is always non-zero when u(t) does not exceed the range of the PWM encoder.

[0202] Proof: Taylor expanding y_α, y_ν, u and s₁ yields

[0203] $y_a\left(t-\tau \right)=y_a\left(t\right)-r{\stackrel{.}{y}}_a\left(t\right)+{O_{\infty }\left(\tau \right)}^2$ $y_{\nu }\left(t-\tau \right)=y_v\left(t\right)+O_{\infty }\left(\varepsilon \right)O_{\infty }\left(\tau \right)$ $\begin{array}{c}{s}_1\left(u\left(t-\tau \right),\sigma \right)=s_1\left(u\left(t\right)+O_{\infty }\left(\tau \right),\sigma \right)\\ =s_1\left(u\left(t\right),\sigma \right)+O_{\infty }\left(\tau \right);\end{array}$
in the second equation, we have used y_ν(t)=_∞(ε). The moving average y_α then reads