Comparative Study Between Backstepping Adaptive and Field Oriented Controls for Doubly Fed Induction Motor

The doubly fed induction motor is powered by two voltage inverters which are connected to two DC bus voltages. Figure 1 illustrates the block diagram of the system studied [12].

1.png

Figure 1. General diagram of the studied system

The DFIM equivalent circuit in the reference (d,q) is shown in Figure 2.

2.png

Figure 2. Electrical diagram of the DFIM

After the Park transformation, the DFIM model is defined by the equations [13, 14]:

• The electrical equations of the DFIM in the reference (d,q) are written as follows:

$\left\{\begin{array}{l}

v_{s d}=R_{s} \cdot i_{s d}+\frac{d}{d t} \psi_{s d}-\frac{d \theta_{s}}{d t} \cdot \psi_{s q} \\

v_{s q}=R_{s} \cdot i_{s q}+\frac{d}{d t} \psi_{s q}+\frac{d \theta_{s}}{d t} \cdot \psi_{s d} \\

v_{r d}=R_{r \cdot} i_{r d}+\frac{d}{d t} \psi_{r d}-\frac{d \theta_{r}}{d t} \cdot \psi_{r q} \\

v_{r q}=R_{r \cdot} i_{r q}+\frac{d}{d t} \psi_{r q}+\frac{d \theta_{r}}{d t} \cdot \psi_{r d}

\end{array}\right.$        (1)

$\left\{\begin{aligned}

i_{s d} &=\frac{1}{\sigma L_{s}} \psi_{s d}-\frac{M}{\sigma L_{s} L_{r}} \cdot \psi_{r d} \\

i_{s q} &=\frac{1}{\sigma L_{s}} \psi_{s q}-\frac{M}{\sigma L_{s} L_{r}} \cdot \psi_{r q} \\

i_{r d} &=-\frac{M}{\sigma L_{s} L_{r}} \psi_{s d}+\frac{1}{\sigma L_{r}} \cdot \psi_{r d} \\

i_{r q} &=-\frac{M}{\sigma L_{s} L_{r}} \psi_{s q}+\frac{1}{\sigma L_{r}} \cdot \psi_{r q}

\end{aligned}\right.$      (2)

with: $\sigma=1-\frac{M}{L_{s} L_{r}} \text { et } M=M_{s r}=M_{r s}$ .

• The magnetic equations of the DFIM in the reference (d,q) are written as follows:

$\left\{\begin{array}{l}

\psi_{s d}=L_{s} \cdot i_{s d}+M \cdot i_{r d} \\

\psi_{s q}=L_{s} \cdot i_{s q}+M \cdot i_{r q} \\

\psi_{r d}=L_{r} \cdot i_{r d}+M \cdot i_{s d} \\

\psi_{r q}=L_{r} \cdot i_{r q}+M \cdot i_{s q}

\end{array}\right.$       (3)

• The mechanical equations of the DFIM are defined by:    

$\left\{\begin{array}{c}

T_{e m}=T_{r}+J \frac{d \Omega}{d t}+f . \Omega \\

T_{e m}=p \cdot\left(\psi_{s q} \cdot \mathrm{i}_{s d}-\psi_{s d} \cdot \mathrm{i}_{s q}\right)

\end{array}\right.$      (4)

The majority of non-linear controls are based on the Lyapunov stability theory. The purpose is to find a control law that makes the derivative of a Lyapunov function, chosen a priori, defined or semi defined negative [17, 18]. The principal difficulty is in the right choice of a suitable Lyapunov function. The Backstepping technique overcomes this difficulty by gradually building a Lyapunov function adapted to the system and allows deducing the control that makes the derivative of this function negative. This ensures that, at all times, the global asymptotic stability of non-linear systems in tracking and regulation [19, 20].

The application of the Backstepping control is done according to the system parameters. If the parameters are known, we used the non-adaptive Backstepping control, else we need an adaptation law, we say that the control used is the adaptive Backstepping control. The adaptive version of Backstepping offers an iterative and systematic method, which allows for non-linear systems of all orders.

4.1 Application of adaptive Backstepping control to the DFIM

The principle objective of non-linear adaptive Backstepping control applied to the DFIM is to allow speed control according to its reference value ($\Omega_{\text {ref }}$) without taking into account external disturbances and internal variations. We propose to remove the classical PI regulators in the scheme of the FOC given in Figure 3 and replace them with adaptive Backstepping control laws. According to electrical, magnetic and mechanical equations of the DFIM, the instant expressions can be written as follows:

• Instantaneous rotor and stator flux:

$\left\{\begin{array}{l}

\frac{d \psi_{r d}}{d t}=v_{r d}+\frac{R_{r \cdot} M}{\sigma \cdot L_{r \cdot L_{s}}} \psi_{s d}-\frac{R_{r}}{\sigma \cdot L_{r}} \psi_{r d}+\psi_{r q} \omega_{r} \\

\frac{d \psi_{r q}}{d t}=v_{r q}+\frac{R_{r \cdot} M}{\sigma \cdot L_{r \cdot} L_{s}} \psi_{s q}-\frac{R_{r}}{\sigma \cdot L_{r}} \psi_{r q}+\psi_{r d} \omega_{r}

\end{array}\right.$    (11)

$\left\{\begin{array}{l}

\frac{d \psi_{s d}}{d t}=v_{s d}+\frac{R_{s} \cdot M}{\sigma \cdot L_{r} \cdot L_{s}} \psi_{r d}-\frac{R_{s}}{\sigma \cdot L_{s}} \psi_{s d}+\psi_{s q} \omega_{s} \\

\frac{d \psi_{s q}}{d t}=v_{s q}+\frac{R_{s} \cdot M}{\sigma \cdot L_{r} \cdot L_{s}} \psi_{r q}-\frac{R_{s}}{\sigma \cdot L_{s}} \psi_{s q}+\psi_{s d} \omega_{s}

\end{array}\right.$     (12)

• Instantaneous mechanical speed:

$\begin{array}{c}

\frac{d \Omega}{d t}=\frac{p \cdot(1-\sigma)}{J \cdot \sigma \cdot M}\left(\psi_{r d} \psi_{s q}-\psi_{r q} \psi_{s d}\right)+ \\

\frac{T_{r}}{J}+\frac{f}{J} \Omega

\end{array}$     (13)

With:

$[X]=\left[\psi_{r d} \psi_{r q} \psi_{s d} \psi_{s q} \Omega\right]^{T}$: Status vector (measurable or estimated).

$[U]=\left[\begin{array}{llll}

v_{r d} & v_{r q} & v_{s d} & v_{s q}

\end{array}\right]^{T}$: Variable of control.

It is evident that the dynamic model of speed is very highly non-linear because of the coupling between velocity and magnetic flux (13). To do this, the study of the stability of the system then consists in searching for a Lyapunov function V(x) of defined sign. Since the DFIM equations are of second order, the implementation is carried out in three steps, one for speed control, the other for magnetic flux control and the last one for estimating system parameters.

• Step 1: Speed controller

The objective of this step is to define the tracking error of the state variable ($e_{\Omega}$) which represents the error between the actual speed Ω (measured or estimated) and the reference speed $\Omega_{r e f}$ , So:

$e_{\Omega}=\Omega_{r e f}-\Omega$    (14)

The combination of the Backstepping method with FOC gives the DFIM control interesting robustness qualities, and further strengthens the Backstepping robustness. Therefore, Eq. (14) becomes as follows:

$\begin{array}{c}

e_{\Omega}=\Omega_{r e f}-\Omega \\

=\Omega_{r e f}-\frac{p \cdot(1-\sigma)}{J \cdot \sigma \cdot M}\left(\psi_{r d} \psi_{s q}\right)+\frac{T_{r}}{J}+\frac{f}{J}

\end{array}$      (15)

To take into account this Eq. (15), the Lyapunov function is defined as follows:

$V_{1}=\frac{1}{2} e_{\Omega}^{2}$    (16)

In order to ensure the system stability, the derivative of the Lyapunov function V₁ must be made negative. To do this, we defined a positive constant (k_Ω) to the derivative of the Eq. (16). After development:

$\begin{array}{l}

\dot{V}_{1} \\

=-K_{\Omega} e_{\Omega}^{2}

\end{array}$$+e_{\Omega \cdot}\left[\begin{array}{c}

K_{\Omega \cdot} e_{\Omega}+\dot{\Omega}_{r e f} \\

-\frac{p \cdot(1-\sigma)}{J \cdot \sigma \cdot M}\left(\psi_{r d} \psi_{s q}\right)+\frac{T_{r}}{J}+\frac{f}{J} \Omega

\end{array}\right]$    (17)

According to this Eq. (17) and the stability condition of Lyapunov function, we consider magnetic flux as the virtual inputs of the DFIM. So:

$\left\{\begin{array}{c}

\psi_{\text {sqref}}=\frac{1}{\frac{p \cdot(1-\sigma)}{J \cdot \sigma \cdot M} \psi_{\text {rdref}}}\left(K_{\Omega \cdot} e_{\Omega}+\frac{T_{r}}{J}+\frac{f}{J} \Omega\right) \\

\psi_{\text {sdref}}=\psi_{s} \\

\psi_{r d_{\text {ref}}}=\psi_{r} \\

\psi_{r q_{\text {ref}}}=0

\end{array}\right.$    (18)

The Eq. (18) is replaced in Eq. (17) and the speed reference $\Omega_{\text {ref }}$ is assumed to be constant. The negativity of the derivative of V₁ is then expressed as follows:

$\dot{V}_{1}=-K_{\Omega} \cdot e_{\Omega}^{2} \leq 0$    (19)

This solution is designed to stabilize the first step. The state of the magnetic flux is chosen as the virtual control of the speed state, then a virtual control of the flux is defined, which will be processed in the second step.

• Step 2: Flux controller

In this step, the errors between the real flux components (measured or estimated) and reference flux are defined, such as:

$\left\{\begin{array}{l}

e_{\psi_{r d}}=\psi_{r d_{-} r e f}-\psi_{r d} \\

e_{\psi_{r d}}=\psi_{r d_{r e f}}-\psi_{r d} \\

e_{\psi_{s d}}=\psi_{s d_{-} r e f}-\psi_{s d} \\

e_{\psi_{s d}}=\psi_{s d_{-} r e f}-\psi_{s d}

\end{array}\right.$     (20)

From the Eq. (20), the real control laws

($v_{s d}, v_{s q}, v_{r d} \text { and } v_{r q}$) of DFIM will be designed. In order to extract these control laws, another term V₂ added to the previous Lyapunov function V₁, such as:

$V_{2}=\frac{1}{2}\left(e_{\Omega}^{2}+e_{\psi_{\mathrm{rd}}}^{2}+e_{\psi_{\mathrm{rq}}}^{2}+e_{\psi_{\mathrm{sd}}}^{2}+e_{\psi_{\mathrm{sq}}}^{2}\right)$   (21)

To guarantee the stability of the system, it is necessary to ensure the negativity of the derivative of the V₂. To do this, the positive constants ($\mathrm{k}_{\psi}>0$) are defined for the derivative of the Eq. (21). After the development:

qqtu_pian_20200710162411.png

(22)

From the Eq. (22) and the stability condition of Lyapunov function, the control laws ($v_{s d}, v_{s q}, v_{r d} \text { and } v_{r q}$) applied to the DFIM in engine mode are obtained, such as:

$\left\{\begin{array}{c}

v_{r d}=K_{\psi_{r d}} e_{\psi_{r d}}+\dot{\psi}_{r}-\frac{R_{r} \cdot M}{\sigma \cdot L_{r \cdot} L_{s}} \psi_{s d}+\frac{R_{r}}{\sigma \cdot L_{r}} \psi_{r d}-\psi_{r q} \omega_{r} \\

v_{r q}=K_{\psi_{r q}} e_{\psi_{r q}}-\frac{R_{r \cdot} M}{\sigma \cdot L_{r} \cdot L_{s}} \psi_{s q}+\frac{R_{r}}{\sigma \cdot L_{r}} \psi_{r q}+\psi_{r d} \omega_{r} \\

v_{s d}=K_{\psi_{s d}} e_{\psi_{s d}}+\dot{\psi}_{s}-\frac{R_{s} \cdot M}{\sigma \cdot L_{r \cdot L_{s}}} \psi_{r d}+\frac{R_{s}}{\sigma \cdot L_{s}} \psi_{s d}-\psi_{s q} \omega_{s} \\

v_{s q}=K_{\psi_{s q}} e_{\psi_{s q}}+\dot{\psi}_{s q_{r e f}}-\frac{R_{s} \cdot M}{\sigma \cdot L_{r \cdot} L_{s}} \psi_{r q}+\frac{R_{s}}{\sigma \cdot L_{s}} \psi_{s q}+\psi_{s d} \omega_{s}

\end{array}\right.$    (23)

The Eq. (23) is replaced in Eq. (22), which implies the negativity of the derivative of V₂, such that:

$\begin{array}{c}

\dot{V}_{2}=\left(\begin{array}{c}

-K_{\Omega} \cdot e_{\Omega}-K_{\psi_{r d}} e_{\psi_{r d}}-K_{\psi_{r q}} e_{\psi_{r q}} \\

-K_{\psi_{s d}} e_{\psi_{s d}}-K_{\psi_{s q}} e_{\psi_{s q}}

\end{array}\right) \\

\leq 0

\end{array}$    (24)

The Eq. (24) implies asymptotic stability towards the origin and the output of the DFIM then follows its reference value.

In the above equations, the control laws are developed under the hypothesis that the DFIM parameters are invariant. This hypothesis is not always true. In fact, one of the problems encountered in controlling the DFIM concerns the variation of its internal parameters and external disturbances due to changes in temperature, the level of magnetic saturation in the machine and the variation in the load torque. The use of estimators improves the robustness of the machine against parametric variations and measurement noise.

• Step 3: Parameters estimation

To design the non-linear adaptive Backstepping control, the real parameters vector of the DFIM is replaced by its estimate. In this case, the control laws given by Eq. (23) will be reinforced by terms that will compensate for the transitions of the estimated parameters.

We have:

$\left\{\begin{array}{c}

\dot{e}_{\Omega}=\hat{a}_{1}\left(\psi_{r} \psi_{s}\right)+\frac{T_{r}}{J}+\frac{f}{J} \Omega \\

\dot{e}_{\psi_{r d}}=\dot{\psi}_{r}-v_{r d}-\hat{a}_{2} \psi_{s d}+\hat{a}_{3} \psi_{r d}-\psi_{r q} \omega_{r} \\

\dot{e}_{\psi_{r q}}=-v_{r q}-\hat{a}_{2} \psi_{s q}+\hat{a}_{3} \psi_{r q}+\psi_{r d} \omega_{r} \\

\dot{e}_{\psi_{s d}}=-v_{s d}+\dot{\psi}_{s}-\hat{a}_{4} \psi_{r d}+\hat{a}_{5} \psi_{s d}-\psi_{s q} \omega_{s} \\

\dot{e}_{\psi_{s q}}=-v_{s q}+\dot{\psi}_{s q_{r e f}}-\hat{a}_{4} \psi_{r q}+\hat{a}_{5} \psi_{s q}+\psi_{s d} \omega_{s}

\end{array}\right.$    (25)

With:

$\begin{aligned}

\hat{a}_{1}=\frac{p \cdot(1-\hat{\sigma})}{J \cdot \hat{\sigma} \cdot M} & ; \hat{a}_{2}=\frac{\hat{R}_{r} \cdot M}{\hat{\sigma} \cdot \hat{L}_{r} \cdot \hat{L}_{s}} ; \hat{a}_{3}=\frac{\hat{R}_{r}}{\hat{\sigma} \cdot \hat{L}_{r}} ; \\

\hat{a}_{4} &=\frac{\hat{R}_{s} \cdot M}{\hat{\sigma} \cdot \hat{L}_{r} \cdot \hat{L}_{s}} ; \hat{a}_{5}=\frac{\widehat{R}_{s}}{\hat{\sigma} \cdot \hat{L}_{s}}

\end{aligned}$

From the Eq. (26), a new function of Lyapunov V₂ is defined, such as:

$\begin{array}{l}V_{2}=\frac{1}{2}\left(e_{\Omega}^{2}+e_{\psi_{\mathrm{rd}}}^{2}+e_{\psi_{\mathrm{rq}}}^{2}+e_{\psi_{\mathrm{sd}}}^{2}+e_{\psi_{\mathrm{sq}}}^{2}+\frac{\tilde{T}_{r}^{2}}{\gamma_{r}}+\frac{\tilde{a}_{1}^{2}}{\gamma_{1}}+\frac{\tilde{a}_{2}^{2}}{\gamma_{2}}+\right. \\\left.\frac{\tilde{a}_{3}^{2}}{\gamma_{3}}+\frac{\tilde{a}_{4}^{2}}{\gamma_{4}}+\frac{\tilde{a}_{5}^{2}}{\gamma_{5}}\right)

\end{array}$      (26)

To ensure the system stability, the negativity of the derivative of the Eq. (27) must be guaranteed. To do this, the positive constants ($\mathrm{k}_{\psi}$)>0 is added to the derivative of this equation. The following expression is obtained after the calculation:

27.png

(27)

From Eq. (7), the control laws ($v_{s d}, v_{s q}, v_{r d} \text { and } v_{r q}$) applied to DFIM are obtained, such as:

$\left\{\begin{array}{c}

v_{r d}=K_{\psi_{r d}} e_{\psi_{r d}}+\dot{\psi}_{r}-\tilde{a}_{2}+\tilde{a}_{3} \psi_{r d}-\psi_{r q} \omega_{r} \\

v_{r q}=K_{\psi_{r q}} e_{\psi_{r q}}-\tilde{a}_{2} \psi_{s q}+\tilde{a}_{3} \psi_{r q}+\psi_{r d} \omega_{r} \\

v_{s d}=K_{\psi_{s d}} e_{\psi_{s d}}+\dot{\psi}_{s}-\tilde{a}_{4} \psi_{r d}+\tilde{a}_{5} \psi_{s d}-\psi_{s q} \omega_{s} \\

v_{s q}=K_{\psi_{s q}} e_{\psi_{s q}}+\dot{\psi}_{s q_{r e f}}-\tilde{a}_{4} \psi_{r q}+\tilde{a}_{5} \psi_{s q}+\psi_{s d} \omega_{s}

\end{array}\right.$    (28)

The Eq. (29) is replaced in Eq. (28), which simplifies the dynamics of the V₂ derivative as follows:

29.png

(29)

From the Eq. (29), the adaptation expressions of the internal and external parameters of DFIM are obtained as follows:

• Expression of adaptation of the load torque:

$\tilde{T}_{r}=\int \dot{\tilde{T}}_{r}=\int-\frac{e_{\Omega}}{J} \cdot \gamma_{r}$    (30)

• Expression of adaptation of stator and rotor resistors:

$\left\{\begin{array}{c}

\tilde{R}_{r}=\int \dot{\tilde{R}}_{r}=\int \dot{\tilde{\sigma}} \cdot \dot{\tilde{L}}_{r} \cdot \dot{\tilde{a}}_{3} \\

=\int-\dot{\tilde{\sigma}} \cdot \dot{\tilde{L}}_{r \cdot} \cdot \gamma_{3}\left(\psi_{r q} e_{\psi_{r q}}+\psi_{r d} e_{\psi_{r d}}\right) \\

\tilde{R}_{s}=\int \dot{\tilde{R}}_{s}=\int \tilde{\sigma} \cdot \dot{\tilde{L}}_{s} \cdot \dot{\tilde{a}}_{5} \\

=\int-\dot{\tilde{\sigma}} \cdot \dot{\tilde{L}}_{s} \cdot \gamma_{5}\left(\psi_{z q} e_{\psi_{z q}}+\psi_{s d} e_{\psi_{s d}}\right)

\end{array}\right.$    (31)

• Adaptation expression of stator and rotor inductances:

$\left\{\begin{array}{l}

\tilde{L}_{r}=\int \dot{\tilde{L}}_{r}=\int M \cdot \frac{\dot{\tilde{a}}_{5}}{\dot{\tilde{a}}_{4}}=\int-\frac{\gamma_{5}\left(\psi_{s d} e_{\psi_{s d}}+\psi_{s q} e_{\psi_{s q}}\right)}{\gamma_{4}\left(\psi_{r q} e_{\psi_{s q}}+\psi_{r d} e_{\psi_{s d}}\right)} \\

\tilde{L}_{s}=\int \dot{\tilde{L}}_{s}=\int M \cdot \frac{\dot{\tilde{a}}_{3}}{\dot{\tilde{a}}_{2}}=\int-\frac{\gamma_{3}\left(\psi_{r q} e_{\psi_{r q}}+\psi_{r d} e_{\psi_{r d}}\right)}{\gamma_{2}\left(\psi_{s q} e_{\psi_{r q}}+\psi_{s d} e_{\psi_{r d}}\right)}

\end{array}\right.$    (32)

With:

$\widetilde{\sigma}=\int \dot{\tilde{\sigma}}=\int \frac{p}{p+J . M. \dot{\widetilde{a}}_{1}}=\int \frac{p}{p+\left(J . M \cdot \gamma_{1} \cdot e_{\Omega}\left(\psi_{r} \cdot \psi_{s}\right)\right)}$    (33)

7.png

Figure 7. Synoptic schema of non-linear adaptive backstepping control applied to DFIM

Figure 7 gives the principal scheme adopted for the non-linear adaptive Backstepping control applied to the DFIM. The blocks of the adaptive Backstepping control calculate the magnetic flux representing the reference flux obtained from the speed error. The calculation of control laws is based on the error between the actual and reference flux.

4.2 Simulation result

In order to test the performances and robustness of the system, the same series of numerical simulations made in the previous section are used.

4.2.1 Performance tests

In this section, two performance tests are performed:

• Speed step of $\Omega_{r e f}=157 \mathrm{rad} / \mathrm{s}$ (equivalent to 1500 tr/min) with the application of load torque at t=2s (Figure 8).
• Test of a trapezoidal speed with introduction of load torque at t=1s (Figure 9).

The simulation results obtained in Figure 8 illustrate the dynamic response of the speed and torque by applying the non-linear adaptive Backstepping control. For a Speed step, the speed follows its reference value with a static error equal to 0.12%. At the start, the speed responds with a response time $t_{r}(\Omega)$=138ms, a starting torque T_em= 6N.m and null overshoot. Figure (8.b) shows that at time t =2s, when applying the load torque, the electromagnetic torque follows its reference value with a band of ∆T_em=±1N.m, because the regulator reacts instantly to the reference electromagnetic torque. The speed reaches its set-point with a rejection time almost equal to 70ms. The speed is still sensitive to disturbances in the load torque with a relative drop of 0.255% for T_r=10N.m.

Figure 9 shows the simulation results of Backstepping Control applied to the DFIM tested by a trapezoidal speed set-point with the application of load torque T_r =10N.m at t=1s. these results indicate that:

• The speed indicated in Figures (9.a and 9.b) follows its reference value with a dynamic error of 0.25rad/s. Figure also shows that the speed is not very sensitive to disturbances, as they make small peaks with a relative drop of 0.255% for the torque T_r=10N.m. As well as the application or removal of load torque.
• Figure (9.c) shows that the electromagnetic torque follows its reference value, it stabilizes on the resistive torque, because the controller reacts instantly on the reference torque, depending on the case of an acceleration or deceleration of the rotational speed.
• According to Figures (9.d and 9.e) the direct components of the stator and rotor flux ψ_rd have an aperiodic response without overshoot and a response time of t_r(ψrd)=40ms and t_r(ψsd)= 70ms, for quadrature components ψ_rq et ψ_sq stays almost equal to zero.

The amplitude of stator currents increases in proportion to the machine load. During the evolution of the set-points and in particular during the reversal of the rotation direction, with the variation of the load torque, oscillations are observed at the level of the quantities as shown in Figure (9.f and 9.g).

8a.png

(a) Rotation speed

8b.png

(b) Electromagnetic torque

Figure 8. Simulation results with speed step

9a.png

(a) Rotation speed

9b.png

(b) Speed error

9c.png

(c) Electromagnetic torque

9d.png

(d) Rotor flux

9e.png

(e) Stator flux

9f.png

(f) Rotor currents

9g.png

(g) Stator currents

Figure 9. Simulation results with a trapezoidal

4.2.2 Robustness tests

To verify the performance and asymptotic stability of the adaptive Backstepping control on the Rotation Speed, changes are applied to the model parameters of the DFIM used. The results of the simulations are obtained for different robustness tests of the control against parametric variations of the motor, depending on the variation of the resistances (heating cases) and inductances (saturation cases). Figure 10 represent the dynamic behavior of the system in this test.

With regard to robustness to parametric variations, the analysis of the simulation results in Figure 10 shows the superiority of the Backstepping control. Despite these variations, the tracking and regulation behavior remains remarkable. However, a change in the resistance or inductance of its nominal value has no influence on the response time, which remains constant and equal to t=138ms. On the contrary, the rise time changes. The static error of the system is lower. The study of index response indicates that our system remains stable with a response time around the nominal response time without becoming unstable or showing a significant overshoot during parametric variations.

10a.png

(a) Rotor resistance

10b.png

(b) Stator resistance

10c.png

(c) Rotor inductance

10d.png

(d) Stator inductance

Figure 10. Robustness test for a variation

For this paper, the FOC and Backstepping control strategies for the doubly fed induction motor are presented. The purpose is to make a comparative study between these two control approaches. After testing both techniques using the Matlab/Simulink environment, it concludes that the adaptive Backstepping control offers better performance compared to FOC in terms of static and dynamic error, response time, overshoot and robustness. In addition, another major advantage of the Backstepping control is the insensitivity against parametric changes of the machine. Future work will focus on the implementation of the Backstepping control using an FPGA-based test bench in our laboratory.