**2.1.** *IM* **model**

In [4] the following *IM* model is proposed, in the fixed (*α*,*β*) frame:

$$\begin{cases} \dot{i}\_{s\mathbf{s}} &= -\frac{R\_{s}l\_{r}^{2} + R\_{r}M\_{sr}^{2}}{\sigma L\_{s}L\_{r}^{2}} \dot{i}\_{s\mathbf{s}} + \frac{M\_{sr}}{\sigma L\_{s}L\_{r}} \left(\frac{R\_{r}}{L\_{r}}\phi\_{r\mathbf{s}} + p\,\Omega\phi\_{r\boldsymbol{\beta}}\right) + \frac{1}{\sigma L\_{s}}v\_{s\mathbf{s}}\\ \dot{i}\_{s\boldsymbol{\beta}} &= -\frac{R\_{s}L\_{r}^{2} + R\_{r}M\_{sr}^{2}}{\sigma L\_{s}L\_{r}^{2}} \dot{i}\_{s\boldsymbol{\beta}} + \frac{M\_{sr}}{\sigma L\_{s}L\_{r}} \left(\frac{R\_{r}}{L\_{r}}\phi\_{r\boldsymbol{\beta}} - p\,\Omega\phi\_{r\boldsymbol{\alpha}}\right) + \frac{1}{\sigma L\_{s}}v\_{s\boldsymbol{\beta}}\\ \dot{\phi}\_{r\boldsymbol{\alpha}} &= \frac{M\_{s}rR\_{r}}{L\_{r}}\dot{i}\_{s\mathbf{s}} - \frac{R\_{r}}{L\_{r}}\phi\_{r\mathbf{s}} - p\,\Omega\phi\_{r\boldsymbol{\beta}}\\ \dot{\phi}\_{r\boldsymbol{\beta}} &= \frac{M\_{s}rR\_{r}}{L\_{r}}\dot{i}\_{s\boldsymbol{\beta}} - \frac{R\_{r}}{L\_{r}}\phi\_{r\boldsymbol{\beta}} + p\,\Omega\phi\_{r\boldsymbol{\alpha}}\\ \dot{\Omega} &= \frac{pM\_{s}r}{L\_{r}}\left(\phi\_{r\boldsymbol{\alpha}}\boldsymbol{i}\_{s\boldsymbol{\beta}} - \phi\_{r\boldsymbol{\beta}}\boldsymbol{i}\_{s\boldsymbol{\alpha}}\right) - \frac{f}{f}\,\Omega - \frac{1}{f}T\_{l} \end{cases} \tag{1}$$

<sup>1</sup> The microprocessors may be dedicated to many process tasks as supervision process, communication process in addition to the considered task

As the mechanical position and magnetic variables are unknown, *d* − *q* frame is well appropriate for sensorless observer based control design.

### *IM* **parameters:**

2 Will-be-set-by-IN-TECH

The third strategy that is a powerful observer that can estimate simultaneously variables and parameters of a large class of nonlinear systems doesn't require a very high performance processor for real time implementation but they are often tested at high speed in sensorless

However for our best of knowledge, examination of the literature on the third strategy shows that the real time computation constraints with a cheapest microprocessors or microprocessors not specially allowed to this task<sup>1</sup> are not taken into account to deal with industrial

Meanwhile, compared with other observers, sliding mode technic [20] have attractive advantages of robustness against matching disturbances and, insensitivity to some specific variation of parameters in sliding mode behavior. However, the chattering effect (that is inherent to standard first sliding mode technic) is often an obstacle for practical applications. Higher-Order Sliding Modes (see for example [2], [18] and [5]) are one of the solutions which does not compromise robustness and avoid filtering of estimated variables as considered by

In this chapter, a second order sliding mode observer for the *IM* without mechanical sensor is presented for the open problem of sensorless *IM* drives at very low frequency. This observer converges in finite time and is robust to the variation of parameters. To illustrate the proposed observer, firstly a very simple case is presented in order to exemplified the tuning parameters. Then, to highlight the technological interest of the proposed method and also show the difficulties due to real time computation constraints when a basic microprocessors are used,

This paper is organized as follows: the section 2 recalls both *IM* model and unobservability phenomena of *IM*. In section 3 the super twisting algorithm (second sliding mode observer) is first presented in a simple case and then applied for sensorless *IM*. After that the section 4 proposes a discrete version of the super twisting observer. In section 5 the experimental results of the proposed observer carried out in an industrial framework are presented. Some

*IM* whereas the main difficulties are mainly at very low frequencies [10], [8].

applications of sensorless *IM* including very low frequencies drives.

other methods.

an industrial application is proposed.

**2. Technical background**

⎧

˙

˙

*φ*˙

*is<sup>α</sup>* <sup>=</sup> <sup>−</sup> *RsL*<sup>2</sup>

*is<sup>β</sup>* <sup>=</sup> <sup>−</sup> *RsL*<sup>2</sup>

*<sup>φ</sup>*˙*r<sup>α</sup>* <sup>=</sup> *MsrRr Lr*

*<sup>r</sup><sup>β</sup>* <sup>=</sup> *MsrRr Lr*

<sup>Ω</sup>˙ <sup>=</sup> *pMsr JLr*

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

addition to the considered task

**2.1.** *IM* **model**

conclusions and remarks are drawn in section VII.

In [4] the following *IM* model is proposed, in the fixed (*α*,*β*) frame:

*is<sup>α</sup>* +

*is<sup>β</sup>* +

(*φrαis<sup>β</sup>* <sup>−</sup> *<sup>φ</sup>rβisα*) <sup>−</sup> *<sup>f</sup>*

*Msr σLsLr* ( *Rr Lr*

*Msr σLsLr* ( *Rr Lr*

<sup>1</sup> The microprocessors may be dedicated to many process tasks as supervision process, communication process in

*<sup>J</sup>* <sup>Ω</sup> <sup>−</sup> <sup>1</sup> *J Tl*

*φr<sup>α</sup>* − *p* Ω*φr<sup>β</sup>*

*φr<sup>β</sup>* + *p* Ω*φr<sup>α</sup>*

*<sup>φ</sup>r<sup>α</sup>* <sup>+</sup> *<sup>p</sup>* <sup>Ω</sup>*φrβ*) + <sup>1</sup>

*<sup>φ</sup>r<sup>β</sup>* <sup>−</sup> *<sup>p</sup>* <sup>Ω</sup>*φrα*) + <sup>1</sup>

*σLs vs<sup>α</sup>*

*σLs vs<sup>β</sup>*

(1)

*<sup>r</sup>* + *RrM*<sup>2</sup> *sr*

*<sup>r</sup>* + *RrM*<sup>2</sup> *sr*

> *is<sup>α</sup>* <sup>−</sup> *Rr Lr*

> *is<sup>β</sup>* <sup>−</sup> *Rr Lr*

*σLsL*<sup>2</sup> *r*

*σLsL*<sup>2</sup> *r*


### *IM* **variables :**


In order to construct the proposed observer for an industrial application, we work with a per unit model, under the following equations :

$$\begin{cases}
\dot{\mathbf{x}}\_1 = -\gamma \mathbf{x}\_1 + \theta \left( b \mathbf{x}\_3 + c \mathbf{x}\_5 \mathbf{x}\_4 \right) + \tilde{\xi} \, v\_1 \\
\dot{\mathbf{x}}\_2 = -\gamma \mathbf{x}\_2 + \theta \left( b \mathbf{x}\_4 - c \mathbf{x}\_5 \mathbf{x}\_3 \right) + \tilde{\xi} \, v\_2 \\
\dot{\mathbf{x}}\_3 = a \, \mathbf{x}\_1 - b \, \mathbf{x}\_3 - c \, \mathbf{x}\_5 \mathbf{x}\_4 \\
\dot{\mathbf{x}}\_4 = a \, \mathbf{x}\_2 - b \, \mathbf{x}\_4 + c \, \mathbf{x}\_5 \mathbf{x}\_3 \\
\dot{\mathbf{x}}\_5 = h \, (\mathbf{x}\_3 \mathbf{x}\_2 - \mathbf{x}\_4 \mathbf{x}\_1) - d \, \mathbf{x}\_5 - e \, T\_l
\end{cases} \tag{2}$$

With the following parameters:

*<sup>x</sup>*<sup>1</sup> <sup>=</sup> *is<sup>α</sup> Iref <sup>x</sup>*<sup>2</sup> <sup>=</sup> *is<sup>β</sup> Iref <sup>x</sup>*<sup>3</sup> <sup>=</sup> *<sup>ω</sup>ref <sup>φ</sup>r<sup>α</sup> Vref <sup>x</sup>*<sup>4</sup> <sup>=</sup> *<sup>ω</sup>ref <sup>φ</sup>r<sup>β</sup> Vref <sup>x</sup>*<sup>5</sup> <sup>=</sup> *<sup>p</sup>* <sup>Ω</sup> *ωref <sup>a</sup>* <sup>=</sup> *MsrIref <sup>ω</sup>ref TrVref <sup>b</sup>* <sup>=</sup> <sup>1</sup> *τr <sup>c</sup>* <sup>=</sup> *<sup>ω</sup>ref <sup>d</sup>* <sup>=</sup> *fv J <sup>e</sup>* <sup>=</sup> *<sup>p</sup> J ωref <sup>h</sup>* <sup>=</sup> *<sup>p</sup>*<sup>2</sup> *MsrIref Vref Jω*<sup>2</sup> *ref Lr <sup>θ</sup>* <sup>=</sup> *KVref Iref ωref <sup>ξ</sup>* <sup>=</sup> *Vref σ LsIref <sup>σ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>M</sup>*<sup>2</sup> *sr LsLr <sup>γ</sup>* <sup>=</sup> *RsL*<sup>2</sup> *<sup>r</sup>* + *RrM*<sup>2</sup> *sr σLsL*<sup>2</sup> *r <sup>τ</sup><sup>r</sup>* <sup>=</sup> *Lr Rr <sup>K</sup>* <sup>=</sup> *Msr σLsLr*

Thus for the sake of homogeneity, hereafter experimental results will be given in per-unit (p.u.).

## **2.2. Observability**

The *IM* observability has been studied by several authors (see for example [3], [13], [9]). In [9], it is proved that the IM observability cannot be established in the particular case when fluxes Φ*rα*, Φ*r<sup>β</sup>* and speed Ω are constant, even if we use the higher derivatives of currents. This is a sufficient and necessary condition for lost of observability.

This operating case match to the following physically interpretation:

*Constant fluxes* (*φ*˙*r<sup>α</sup>* = *φ*˙ *<sup>r</sup><sup>β</sup>* = 0)

With *ω<sup>s</sup>* the stator voltage pulsation and *Tem* the electromagnetic torque.

$$
\omega\_s = p\Omega + \frac{R\_r T\_{em}}{p\Phi\_{rd}^2} = 0 \tag{3}
$$

Considering the following system:

With *f*(*x*, *t*) a bounded function.

*e*˙1 and *e*˙2. Thus

With

And

Thus

Where

*x*˜2 �−→ *x*2.

hereafter according to figure 2.

⎧ ⎪⎪⎨

*x*˙1 = *x*<sup>2</sup> *x*˙2 = *f*(*x*, *t*) *y* = *h*(*x*) = *x*<sup>1</sup>

For system (6), a second order sliding mode observer is designed in the following way:

1 <sup>2</sup> *sign*(*e*1)

With *λ*, *α* > 0 and *e*<sup>1</sup> = *x*<sup>1</sup> − *x*ˆ1.

The efficiency of the this strategy depends on coefficients *α* and *λ*. For second order system (6)we show convergence of estimated variables (*x*ˆ1,*x*ˆ2) to (*x*1,*x*2) by studying dynamics errors

*x*ˆ1 = *e*<sup>2</sup> − *λ*|*x*<sup>1</sup> − *x*ˆ1|

*x*ˆ2 = *f*(*x*, *t*) − *αsign*(*x*<sup>1</sup> − *x*ˆ1)

*<sup>f</sup>*(*x*, *<sup>t</sup>*) <sup>∈</sup> [−*<sup>f</sup>* <sup>+</sup>, *<sup>f</sup>* +], *<sup>e</sup>*<sup>2</sup> <sup>=</sup> *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*ˆ2

*<sup>e</sup>*¨1 <sup>=</sup> *<sup>f</sup>*(*x*, *<sup>t</sup>*) <sup>−</sup> *<sup>α</sup>sign*(*e*1) <sup>−</sup> <sup>1</sup>

*<sup>e</sup>*¨1 <sup>∈</sup> [−*<sup>f</sup>* <sup>+</sup>, *<sup>f</sup>* +] <sup>−</sup> *<sup>α</sup>sign*(*e*1) <sup>−</sup> <sup>1</sup>

*<sup>f</sup>* <sup>+</sup> <sup>=</sup> *max*{ *<sup>f</sup>*(*x*, *<sup>t</sup>*)}

Conditions on *λ* and *α* that permit a convergence in finite time of (*e*˙1,*e*1) to (0,0) are derived

*Proposition*: For any initial conditions *x*(0), *x*ˆ(0), there exists a choice of *λ* and *α* such that the error dynamics *e*˙1 and *e*˙2 converge to zero in finite time and by consequence *x*ˆ1 �−→ *x*<sup>1</sup> and

1

2 *λ*|*e*1| − 1 <sup>2</sup> *e*˙1

> 2 *λ*|*e*1| − 1 <sup>2</sup> *e*˙1

<sup>2</sup> *sign*(*x*<sup>1</sup> − *x*ˆ1)

Industrial Application of a Second Order Sliding Mode Observer for Speed and Flux Estimation in Sensorless Induction Motor

*x*ˆ1 = *x*ˆ2 + *λ*|*e*1|

*x*ˆ2 = *αsign*(*e*1)

(6)

363

(7)

(8)

⎪⎪⎩

� ˙

� *<sup>e</sup>*˙1 <sup>=</sup> *<sup>x</sup>*˙1 <sup>−</sup> ˙

*<sup>e</sup>*˙2 <sup>=</sup> *<sup>x</sup>*˙2 <sup>−</sup> ˙

˙

where Φ<sup>2</sup> *rd* <sup>=</sup> *<sup>φ</sup>*<sup>2</sup> *<sup>r</sup><sup>α</sup>* + *<sup>φ</sup>*<sup>2</sup> *<sup>r</sup><sup>β</sup>* is the square of the direct flux in (d, q) frame.

*Constant speed* (Ω˙ = 0)

$$T\_{em} = f\Omega + T\_l \tag{4}$$

Thanks to previous equations, we obtain:

$$T\_l = -\left(f + \frac{p^2 \Phi\_{rd}^2}{R\_r}\right) \Omega \tag{5}$$

**Figure 1.** Inobservability curve

The unobservability curve in the map (*Tl*, Ω) is shown in figure (1).

Obviously, the observability is lost gradually when we approach this curve [9].

## **3. Second order sliding mode observer**

### **3.1. Super twisting algorithm: An academic example**

Sliding modes were used at first, as a control technique, but in the recent years it presented as a very good tool for observer design [17], [17], [5].

Considering the following system:

$$\begin{cases} \dot{\mathfrak{x}}\_1 = \mathfrak{x}\_2 \\ \dot{\mathfrak{x}}\_2 = f(\mathfrak{x}, t) \\ y\_1 = h(\mathfrak{x}) = \mathfrak{x}\_1 \end{cases} \tag{6}$$

With *f*(*x*, *t*) a bounded function.

For system (6), a second order sliding mode observer is designed in the following way:

$$\begin{cases}
\dot{\mathfrak{k}}\_1 = \mathfrak{k}\_2 + \lambda |e\_1|^{\frac{1}{2}} \text{sign}(e\_1) \\
\dot{\mathfrak{k}}\_2 = \operatorname{asigm}(e\_1)
\end{cases} \tag{7}$$
 
$$\text{With } \lambda, \mathfrak{a} > 0 \quad \text{and} \quad e\_1 = \mathfrak{x}\_1 - \mathfrak{x}\_1.$$

The efficiency of the this strategy depends on coefficients *α* and *λ*. For second order system (6)we show convergence of estimated variables (*x*ˆ1,*x*ˆ2) to (*x*1,*x*2) by studying dynamics errors *e*˙1 and *e*˙2.

Thus

4 Will-be-set-by-IN-TECH

The *IM* observability has been studied by several authors (see for example [3], [13], [9]). In [9], it is proved that the IM observability cannot be established in the particular case when fluxes Φ*rα*, Φ*r<sup>β</sup>* and speed Ω are constant, even if we use the higher derivatives of currents. This is a

> *RrTem p*Φ<sup>2</sup> *rd*

= 0 (3)

Ω (5)

*Tem* = *f* Ω + *Tl* (4)

sufficient and necessary condition for lost of observability.

*<sup>r</sup><sup>β</sup>* = 0)

This operating case match to the following physically interpretation:

With *ω<sup>s</sup>* the stator voltage pulsation and *Tem* the electromagnetic torque.

*Tl* = −

The unobservability curve in the map (*Tl*, Ω) is shown in figure (1).

**3. Second order sliding mode observer**

a very good tool for observer design [17], [17], [5].

**3.1. Super twisting algorithm: An academic example**

Obviously, the observability is lost gradually when we approach this curve [9].

Sliding modes were used at first, as a control technique, but in the recent years it presented as

*Tl*

*<sup>f</sup>* <sup>+</sup> *<sup>p</sup>*2Φ<sup>2</sup> *rd Rr*

> −

Ω

*<sup>f</sup>* <sup>+</sup> *<sup>p</sup>*2Φ<sup>2</sup> *rd Rr*

*ω<sup>s</sup>* = *p*Ω +

*<sup>r</sup><sup>β</sup>* is the square of the direct flux in (d, q) frame.

**2.2. Observability**

*Constant fluxes* (*φ*˙*r<sup>α</sup>* = *φ*˙

*rd* <sup>=</sup> *<sup>φ</sup>*<sup>2</sup>

*Constant speed* (Ω˙ = 0)

**Figure 1.** Inobservability curve

*<sup>r</sup><sup>α</sup>* + *<sup>φ</sup>*<sup>2</sup>

Thanks to previous equations, we obtain:

where Φ<sup>2</sup>

$$\begin{cases} \dot{\boldsymbol{e}}\_1 = \dot{\boldsymbol{x}}\_1 - \dot{\boldsymbol{x}}\_1 = \boldsymbol{e}\_2 - \lambda |\boldsymbol{x}\_1 - \hat{\boldsymbol{x}}\_1|^\frac{1}{2} sign \, n(\boldsymbol{x}\_1 - \hat{\boldsymbol{x}}\_1) \\\\ \dot{\boldsymbol{e}}\_2 = \dot{\boldsymbol{x}}\_2 - \dot{\boldsymbol{x}}\_2 = f(\mathbf{x}, t) - asign(\mathbf{x}\_1 - \hat{\mathbf{x}}\_1) \end{cases} \tag{8}$$

With

*<sup>f</sup>*(*x*, *<sup>t</sup>*) <sup>∈</sup> [−*<sup>f</sup>* <sup>+</sup>, *<sup>f</sup>* +], *<sup>e</sup>*<sup>2</sup> <sup>=</sup> *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*ˆ2

And

$$\ddot{e}\_1 = f(\mathbf{x}, t) - \alpha \dot{s} g n(e\_1) - \frac{1}{2} \lambda |e\_1|^{-\frac{1}{2}} \dot{e}\_1$$

Thus

$$\vec{e}\_1 \in [-f^+, f^+] - \kappa \text{sign}(e\_1) - \frac{1}{2}\lambda |e\_1|^{-\frac{1}{2}}\vec{e}\_1$$

Where

$$f^+ = \max\{f(\mathbf{x}, t)\}$$

Conditions on *λ* and *α* that permit a convergence in finite time of (*e*˙1,*e*1) to (0,0) are derived hereafter according to figure 2.

*Proposition*: For any initial conditions *x*(0), *x*ˆ(0), there exists a choice of *λ* and *α* such that the error dynamics *e*˙1 and *e*˙2 converge to zero in finite time and by consequence *x*ˆ1 �−→ *x*<sup>1</sup> and *x*˜2 �−→ *x*2.

**Figure 2.** Upper bound of finite time convergence curve.

*Proof*: Consider system (6). To show the convergence of (*x*ˆ1, *x*ˆ2) to (*x*1, *x*2) (ie., (*e*1,*e*2) → (0, 0)), we need to show that

$$\frac{|\dot{e}\_1(T\_2)|}{|\dot{e}\_1(0)|} < 1\tag{9}$$

**First quadrant**: *e*<sup>1</sup> > 0 and *e*˙1 > 0

*e*˙1 = 0 (point B in figure 2).

Then we can compute *e*(*T*1) as follows

**Second quadrant**: *e*<sup>1</sup> > 0 and *e*˙1 < 0

Since *e*˙1 is negative, then

In this case, *<sup>e</sup>*¨1 <sup>=</sup> <sup>−</sup>*<sup>f</sup>* <sup>+</sup> <sup>−</sup> *<sup>α</sup>sign*(*e*1) <sup>−</sup> *<sup>λ</sup>*

**Computing of** *e*1(*T*1) From (10), we have

Which implies that

And

*e*1(*T*1).

By choosing

*<sup>e</sup>*˙1 <sup>≥</sup> 0. The rising trajectory is given by *<sup>e</sup>*¨1 <sup>=</sup> <sup>−</sup>(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +).

Starting from point A of figure 2 the trajectory of *e*˙1 = *f*(*e*1) is in the first quadrant *e*<sup>1</sup> ≥ 0 and

we ensure that *e*¨1 < 0 and hence *e*˙1 decreases and tends towards the y-axis, corresponding to

*<sup>e</sup>*¨1 <sup>=</sup> <sup>−</sup>(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +)

*t* 2

2 <sup>1</sup>(0) <sup>2</sup>(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +) <sup>+</sup>

From (13), since *e*˙1(*T*1) = 0, we obtain the necessary time for going from *A* to *B* with *B* =

*<sup>T</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup>*˙1(0)

<sup>2</sup> <sup>+</sup> *<sup>e</sup>*˙1(0)*<sup>t</sup>*

*<sup>e</sup>*1(*t*) = <sup>−</sup>(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +)

*<sup>e</sup>*1(*T*1) = <sup>−</sup>(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +) *<sup>e</sup>*˙

<sup>2</sup> <sup>|</sup>*e*1<sup>|</sup> − 1 <sup>2</sup> *e*˙1

<sup>1</sup> ) <sup>&</sup>gt; <sup>−</sup>*λ*<sup>1</sup>

<sup>|</sup>*e*˙1(*t*)| ≤ <sup>2</sup>(*<sup>α</sup>* <sup>+</sup> *<sup>f</sup>* <sup>+</sup>

<sup>2</sup> <sup>|</sup>*e*1<sup>|</sup> − 1

1 ) *<sup>λ</sup>* <sup>|</sup>*e*1(*t*)<sup>|</sup>

1

<sup>=</sup> *<sup>e</sup>*˙ 2 <sup>1</sup>(0)

becomes negative (*e*¨1 < 0) on making a good choice of *α* which leads to

(*α* + *f* <sup>+</sup>

*α* > *f* <sup>+</sup> (12)

Industrial Application of a Second Order Sliding Mode Observer for Speed and Flux Estimation in Sensorless Induction Motor

365

*<sup>e</sup>*˙1(*t*) = <sup>−</sup>(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +)*<sup>t</sup>* <sup>+</sup> *<sup>e</sup>*˙1(0) (13)

(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +) (14)

<sup>2</sup> *e*˙1 (16)

<sup>2</sup> (17)

*e*¨ 2 <sup>1</sup>(0) (*<sup>α</sup>* − *<sup>f</sup>* +)

<sup>2</sup>(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +) (15)

of figure 2, where *e*˙1(*T*2) = *C* and *e*˙1(0) = *A*.

Figure 2 illustrates the finite time convergence behavior of the proposed observer for system 6. In what follows we will give the error trajectory for each quadrant in the worst cases.

Let consider the system's dynamic *e*¨1

$$\ddot{e}\_1 = f(\mathbf{x}, t) - a \dot{s} g n(e\_1) - \frac{\lambda}{2} |e\_1|^{-\frac{1}{2}} \dot{e}\_1 \tag{10}$$

with *<sup>d</sup>* <sup>|</sup>*x*<sup>|</sup> *dt* <sup>=</sup> *xsign* ˙ (*x*).

Equation (10) leads to

$$\vec{e}\_1 \in \left[ -f^+, f^+ \right] - a \text{sign}(e\_1) - \frac{\lambda}{2} |e\_1|^{-\frac{1}{2}} \vec{e}\_1 \tag{11}$$

where

$$f^+ = \max(f(t, \mathfrak{x})),$$

## **First quadrant**: *e*<sup>1</sup> > 0 and *e*˙1 > 0

Starting from point A of figure 2 the trajectory of *e*˙1 = *f*(*e*1) is in the first quadrant *e*<sup>1</sup> ≥ 0 and *<sup>e</sup>*˙1 <sup>≥</sup> 0. The rising trajectory is given by *<sup>e</sup>*¨1 <sup>=</sup> <sup>−</sup>(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +).

By choosing

6 Will-be-set-by-IN-TECH

*Proof*: Consider system (6). To show the convergence of (*x*ˆ1, *x*ˆ2) to (*x*1, *x*2) (ie., (*e*1,*e*2) →

Figure 2 illustrates the finite time convergence behavior of the proposed observer for system 6. In what follows we will give the error trajectory for each quadrant in the worst cases.

<sup>−</sup> *<sup>α</sup>sign*(*e*1) <sup>−</sup> *<sup>λ</sup>*

*f* <sup>+</sup> = *max*(*f*(*t*, *x*)),

<sup>2</sup> <sup>|</sup>*e*1<sup>|</sup> − 1

> <sup>2</sup> <sup>|</sup>*e*1<sup>|</sup> − 1

<sup>|</sup>*e*˙1(0)<sup>|</sup> <sup>&</sup>lt; <sup>1</sup> (9)

<sup>2</sup> *e*˙1 (10)

<sup>2</sup> *e*˙1 (11)


*<sup>e</sup>*¨1 <sup>=</sup> *<sup>f</sup>*(*x*, *<sup>t</sup>*) <sup>−</sup> *<sup>α</sup>sign*(*e*1) <sup>−</sup> *<sup>λ</sup>*

**Figure 2.** Upper bound of finite time convergence curve.

of figure 2, where *e*˙1(*T*2) = *C* and *e*˙1(0) = *A*.

*<sup>e</sup>*¨1 <sup>∈</sup>

<sup>−</sup>*<sup>f</sup>* <sup>+</sup>, *<sup>f</sup>* <sup>+</sup>

Let consider the system's dynamic *e*¨1

*dt* <sup>=</sup> *xsign* ˙ (*x*).

Equation (10) leads to

with *<sup>d</sup>* <sup>|</sup>*x*<sup>|</sup>

where

(0, 0)), we need to show that

$$
\mathfrak{a} > f^+ \tag{12}
$$

we ensure that *e*¨1 < 0 and hence *e*˙1 decreases and tends towards the y-axis, corresponding to *e*˙1 = 0 (point B in figure 2).

### **Computing of** *e*1(*T*1)

From (10), we have

$$
\vec{e}\_1 = -(\mathfrak{a} - f^+)
$$

Which implies that

$$
\dot{e}\_1(t) = -(\mathfrak{a} - f^+)t + \dot{e}\_1(0) \tag{13}
$$

And

$$e\_1(t) = -(\mathfrak{a} - f^+)\frac{t^2}{2} + \dot{e}\_1(0)t$$

From (13), since *e*˙1(*T*1) = 0, we obtain the necessary time for going from *A* to *B* with *B* = *e*1(*T*1).

$$T\_1 = \frac{\dot{e}\_1(0)}{(\alpha - f^+)}\tag{14}$$

Then we can compute *e*(*T*1) as follows

$$e\_1(T\_1) = -(\mathfrak{a} - f^+) \frac{\dot{e}\_1^2(0)}{2(\mathfrak{a} - f^+)} + \frac{\ddot{e}\_1^2(0)}{(\mathfrak{a} - f^+)}$$

$$= \frac{\dot{e}\_1^2(0)}{2(\mathfrak{a} - f^+)}\tag{15}$$

**Second quadrant**: *e*<sup>1</sup> > 0 and *e*˙1 < 0

In this case, *<sup>e</sup>*¨1 <sup>=</sup> <sup>−</sup>*<sup>f</sup>* <sup>+</sup> <sup>−</sup> *<sup>α</sup>sign*(*e*1) <sup>−</sup> *<sup>λ</sup>* <sup>2</sup> <sup>|</sup>*e*1<sup>|</sup> − 1 <sup>2</sup> *e*˙1

becomes negative (*e*¨1 < 0) on making a good choice of *α* which leads to

$$|(a + f\_1^+) > -\frac{\lambda\_1}{2}|e\_1|^{-\frac{1}{2}}\dot{e}\_1\tag{16}$$

Since *e*˙1 is negative, then

$$|\dot{e}\_1(t)| \le \frac{2(\varkappa + f\_1^+)}{\lambda} |e\_1(t)|^{\frac{1}{2}} \tag{17}$$

Considering by the sake of simplicity (17), *e*<sup>1</sup> > 0 and *e*˙1 < 0.

Integrating (17) with *e*1(0) = 0 gives

$$
\sqrt{e\_1(t)} = \frac{(\alpha + f^+)}{\lambda} t \tag{18}
$$

we obtain a bounded limit

The associated observer is:

*<sup>T</sup>*<sup>∞</sup> <sup>≤</sup> ( <sup>1</sup> 1 −

equal to *sin*(*t*) with *<sup>f</sup>* <sup>+</sup> <sup>=</sup> *max*{*sin*(*t*)} <sup>=</sup> 1. We get

*x*1, and then slides along *x*<sup>1</sup> path, and equal to *x*˜2.

*IM* model (2) (without *x*˙5 equation) into a form 6:

**3.2. Application to** *Induction Motor*

<sup>√</sup>2(*α*+*<sup>f</sup>* +) *λ* <sup>√</sup>*α*−*<sup>f</sup>* <sup>+</sup>

> ⎧ ⎪⎪⎨

> ⎪⎪⎩

*x*ˆ1 = *x*˜2 + *λ*|*e*1|

˙*x*˜2 = *αsign*(*e*1)

� ˙

⎧

*z*<sup>1</sup> = *x*<sup>1</sup> *z*<sup>2</sup> = *x*<sup>2</sup>

*z*<sup>5</sup> = *z*˙3 *z*<sup>6</sup> = *z*˙4

From the *IM* model (2) and (27), we obtain a new dynamical system as following:

*z*˙3 = *z*<sup>5</sup> *z*˙4 = *z*<sup>6</sup>

⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧ ⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

)( *<sup>λ</sup>* <sup>√</sup>2(*<sup>α</sup>* <sup>+</sup> *<sup>f</sup>* +)

Here we give simulations of a very simple example. The function *f*(*t*, *x*) in system (6) is set

1 <sup>2</sup> *sign*(*e*1)

The simulation results are shown in figure 3. It can be seen that figure (3) spotlight *two* steps into Super Twisting Algorithm, which are convergence step in finite time, and sliding mode. Indeed observer is working on *t* = 1*s* with *x*ˆ1(0) = 1 and *x*˜2(0) = 1. *x*ˆ1 converges under 1*s* to

At first, due to the nonlinearity of flux and speed product, the *IM* model (2) is not written in a suitable form allowing to apply the super twisting algorithm presented in previous section. To overcome this difficulty, we make the following change of variables in order to rewrite the

> *z*<sup>3</sup> = *b x*<sup>3</sup> + *c x*5*x*<sup>4</sup> *z*<sup>4</sup> = *b x*<sup>4</sup> − *c x*5*x*<sup>3</sup>

Equation 27 is not a diffeomorphism, not an homeomorphism but only an immersion, because the dimension of *x* is 5 and the dimension of *z* is 6. Nevertheless, this immersion is used in order to avoid some singularities in a speed estimation as this will be pointed out in the next.

> *z*˙1 = −*z*<sup>1</sup> + *θz*<sup>3</sup> + *ξv*<sup>1</sup> *z*˙2 = −*γz*<sup>2</sup> + *θz*<sup>4</sup> + *ξv*<sup>2</sup>

*x*˙1 = *x*<sup>2</sup> *x*˙2 = *sin*(*t*) *y* = *x*<sup>1</sup>

<sup>+</sup> <sup>1</sup>) *<sup>e</sup>*˙1(0) (*<sup>α</sup>* − *<sup>f</sup>* +)

Industrial Application of a Second Order Sliding Mode Observer for Speed and Flux Estimation in Sensorless Induction Motor

(25)

367

(26)

(27)

(28)

At *t* = *T*2, we should make the inverse of function (18) from point *B* to *C* in figure 2. This leads to

$$e\_1(T\_2) = e\_1(T\_1).\tag{19}$$

Then (18) becomes

$$
\sqrt{e\_1(T\_1)} = \frac{(\alpha + f^+)}{\lambda} T\_2 \tag{20}
$$

By replacing *e*1(*T*1) coming from (15) in equation (20), we get the necessary time for going from *B* to *C*

$$T\_2 = \frac{\lambda}{\left(\mathfrak{a} + f^+\right)} \frac{\dot{\mathfrak{e}}\_1(0)}{\sqrt{2}(\mathfrak{a} - f^+)}\tag{21}$$

After that, by using the argument of (19) in equation (17) evaluated at *t* = *T*<sup>2</sup> in the worth case, we get

$$|\dot{e}\_1(T\_2)| = \frac{2(a + f\_1^+)}{\lambda} |e\_1(T\_1)|^{\frac{1}{2}} \tag{22}$$

By replacing *e*1(*T*1) by its expression given by (15) in (22), we get

$$|\dot{e}\_1(T\_2)| = \frac{2(\alpha + f\_1^+)}{\lambda} \frac{|\dot{e}\_1(0)|}{\sqrt{2}\sqrt{(\alpha - f^+)}}\tag{23}$$

Thus, by satisfying inequality (9) in equation (23) *λ* should be chosen as

$$
\lambda > (\mathfrak{a} + f\_1^+) \sqrt{\frac{2}{(\mathfrak{a} - f^+)}} \tag{24}
$$

Finally, conditions (12) and (24) of the observer parameters are sufficient conditions guaranteeing the state convergence (i.e. the states (*e*1,*e*˙1) tend towards *e*<sup>1</sup> = *e*˙1 = 0 (Figure 2).

This ends the proof.

Moreover the convergence is in finite time, because from (14) and 21 we obtain

$$T\_{\infty} \le (\sum\_{i=0}^{+\infty} (\frac{\sqrt{2}(\alpha + f^{+})}{\lambda \sqrt{\alpha - f^{+}}})^i)(\frac{\lambda}{\sqrt{2}(\alpha + f^{+})} + 1)\frac{\dot{e}\_1(0)}{(\alpha - f^{+})} $$

as

$$|
\frac{
\sqrt{2}(a+f^+)
}{
\lambda
\sqrt{a-f^+}
}| < 1$$

we obtain a bounded limit

8 Will-be-set-by-IN-TECH

*<sup>e</sup>*1(*t*) = (*<sup>α</sup>* <sup>+</sup> *<sup>f</sup>* +)

At *t* = *T*2, we should make the inverse of function (18) from point *B* to *C* in figure 2. This

*<sup>e</sup>*1(*T*1) = (*<sup>α</sup>* <sup>+</sup> *<sup>f</sup>* +)

By replacing *e*1(*T*1) coming from (15) in equation (20), we get the necessary time for going

After that, by using the argument of (19) in equation (17) evaluated at *t* = *T*<sup>2</sup> in the worth

1 ) *λ*

> 1 )

Finally, conditions (12) and (24) of the observer parameters are sufficient conditions guaranteeing the state convergence (i.e. the states (*e*1,*e*˙1) tend towards *e*<sup>1</sup> = *e*˙1 = 0 (Figure 2).

> )( *<sup>λ</sup>* <sup>√</sup>2(*<sup>α</sup>* <sup>+</sup> *<sup>f</sup>* +)

*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* <sup>+</sup> <sup>|</sup> <sup>&</sup>lt; <sup>1</sup>

<sup>√</sup>2(*<sup>α</sup>* <sup>+</sup> *<sup>f</sup>* +)

1 ) *<sup>λ</sup>* <sup>|</sup>*e*1(*T*1)<sup>|</sup>

√2

2

1


*e*˙1(0)

*<sup>λ</sup> <sup>t</sup>* (18)

*<sup>λ</sup> <sup>T</sup>*<sup>2</sup> (20)

<sup>√</sup>2(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +) (21)

<sup>2</sup> (22)

(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +) (23)

(*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* +) (24)

<sup>+</sup> <sup>1</sup>) *<sup>e</sup>*˙1(0) (*<sup>α</sup>* − *<sup>f</sup>* +)

*e*1(*T*2) = *e*1(*T*1). (19)

Considering by the sake of simplicity (17), *e*<sup>1</sup> > 0 and *e*˙1 < 0.

*<sup>T</sup>*<sup>2</sup> <sup>=</sup> *<sup>λ</sup>*

(*α* + *f* +)

<sup>|</sup>*e*˙1(*T*2)<sup>|</sup> <sup>=</sup> <sup>2</sup>(*<sup>α</sup>* <sup>+</sup> *<sup>f</sup>* <sup>+</sup>

<sup>|</sup>*e*˙1(*T*2)<sup>|</sup> <sup>=</sup> <sup>2</sup>(*<sup>α</sup>* <sup>+</sup> *<sup>f</sup>* <sup>+</sup>

*λ* > (*α* + *f* <sup>+</sup>

Moreover the convergence is in finite time, because from (14) and 21 we obtain

<sup>√</sup>2(*<sup>α</sup>* <sup>+</sup> *<sup>f</sup>* +)

*<sup>α</sup>* <sup>−</sup> *<sup>f</sup>* <sup>+</sup> )*<sup>i</sup>*


*λ*

Thus, by satisfying inequality (9) in equation (23) *λ* should be chosen as

By replacing *e*1(*T*1) by its expression given by (15) in (22), we get

Integrating (17) with *e*1(0) = 0 gives

leads to

Then (18) becomes

from *B* to *C*

case, we get

This ends the proof.

as

*T*<sup>∞</sup> ≤ (

+∞ ∑ *i*=0 (

*λ*

$$T\_{\infty} \le (\frac{1}{1 - \frac{\sqrt{2}(\alpha + f^{+})}{\lambda \sqrt{\alpha - f^{+}}}}) (\frac{\lambda}{\sqrt{2}(\alpha + f^{+})} + 1) \frac{\dot{e}\_{1}(0)}{(\alpha - f^{+})} $$

Here we give simulations of a very simple example. The function *f*(*t*, *x*) in system (6) is set equal to *sin*(*t*) with *<sup>f</sup>* <sup>+</sup> <sup>=</sup> *max*{*sin*(*t*)} <sup>=</sup> 1. We get

$$\begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{x}\_2 \\ \dot{\mathbf{x}}\_2 = \sin(t) \\ y\_1 = \mathbf{x}\_1 \end{cases} \tag{25}$$

The associated observer is:

$$\begin{cases} \dot{\mathfrak{X}}\_1 = \mathfrak{x}\_2 + \lambda |e\_1|^{\frac{1}{2}} \text{sign}(e\_1) \\\\ \dot{\mathfrak{X}}\_2 = \operatorname{sign}(e\_1) \end{cases} \tag{26}$$

The simulation results are shown in figure 3. It can be seen that figure (3) spotlight *two* steps into Super Twisting Algorithm, which are convergence step in finite time, and sliding mode. Indeed observer is working on *t* = 1*s* with *x*ˆ1(0) = 1 and *x*˜2(0) = 1. *x*ˆ1 converges under 1*s* to *x*1, and then slides along *x*<sup>1</sup> path, and equal to *x*˜2.

### **3.2. Application to** *Induction Motor*

At first, due to the nonlinearity of flux and speed product, the *IM* model (2) is not written in a suitable form allowing to apply the super twisting algorithm presented in previous section. To overcome this difficulty, we make the following change of variables in order to rewrite the *IM* model (2) (without *x*˙5 equation) into a form 6:

$$\begin{cases} z\_1 = x\_1 \\ z\_2 = x\_2 \\ z\_3 = b \ x\_3 + c \ x\_5 x\_4 \\ z\_4 = b \ x\_4 - c \ x\_5 x\_3 \\ z\_5 = \dot{z}\_3 \\ z\_6 = \dot{z}\_4 \end{cases} \tag{27}$$

Equation 27 is not a diffeomorphism, not an homeomorphism but only an immersion, because the dimension of *x* is 5 and the dimension of *z* is 6. Nevertheless, this immersion is used in order to avoid some singularities in a speed estimation as this will be pointed out in the next.

From the *IM* model (2) and (27), we obtain a new dynamical system as following:

$$\begin{cases} \dot{z}\_1 = -z\_1 + \theta z\_3 + \xi v\_1 \\ \dot{z}\_2 = -\gamma z\_2 + \theta z\_4 + \xi v\_2 \\ \dot{z}\_3 = z\_5 \\ \dot{z}\_4 = z\_6 \end{cases} \tag{28}$$

**Figure 3.** Super Twisting Algorithm example.

Thus, we can propose a new observer structure for dynamical system (28):

$$\begin{cases} \dot{\tilde{z}}\_{1} = \theta \, \tilde{z}\_{3} - \gamma \, z\_{1} + \tilde{\xi} \, v\_{1} + \lambda\_{1} \, |e\_{1}|^{\frac{1}{2}} \operatorname{sign}(e\_{1}) \\ \dot{\tilde{z}}\_{3} = \alpha\_{1} \operatorname{sign}(e\_{1}) \\ \dot{\tilde{z}}\_{2} = \theta \, \tilde{z}\_{4} - \gamma \, z\_{2} + \tilde{\xi} \, v\_{2} + \lambda\_{2} \, |e\_{2}|^{\frac{1}{2}} \operatorname{sign}(e\_{2}) \\ \dot{\tilde{z}}\_{4} = \alpha\_{2} \operatorname{sign}(e\_{2}) \\ \dot{\tilde{z}}\_{3} = E\_{1} \, E\_{2} \left( \tilde{z}\_{5} + \lambda\_{3} \, |e\_{3}|^{\frac{1}{2}} \operatorname{sign}(e\_{3}) \right) \\ \dot{\tilde{z}}\_{5} = E\_{1} \, E\_{2} \, a\_{3} \operatorname{sign}(e\_{3}) \\ \dot{\tilde{z}}\_{4} = E\_{1} \, E\_{2} \left( \tilde{z}\_{6} + \lambda\_{4} \, |e\_{4}|^{\frac{1}{2}} \operatorname{sign}(e\_{4}) \right) \\ \dot{\tilde{z}}\_{6} = E\_{1} \, E\_{2} \, a\_{4} \operatorname{sign}(e\_{4}) \end{cases} \tag{29}$$

This observer structure depends on Super Twisting Algorithm presented in previous section and Step by Step proficiencies [7]. We propose to put in multiples-series observers with

Industrial Application of a Second Order Sliding Mode Observer for Speed and Flux Estimation in Sensorless Induction Motor

The functions *Ei* ensure that the next steps errors do not escape too far before one has the

The gains *αi*, *λ<sup>i</sup>* are chosen which respect to the reachability condition of the Super Twisting

2

2

*<sup>α</sup>*<sup>1</sup> <sup>−</sup> *max*(*θz*3) (31)

369

*<sup>α</sup>*<sup>2</sup> <sup>−</sup> *max*(*θz*4) (32)

algorithm as stated in inequalities (12) and (24) of previous section. By choosing

*α*<sup>1</sup> > *max*(*θz*3), *λ*<sup>1</sup> > (*max*(*θz*3) + *α*1)

*α*<sup>2</sup> > *max*(*θz*4), *λ*<sup>2</sup> > (*max*(*θz*4) + *α*2)

functions (*Ei*).

**Figure 4.** General *IM* Observer Structure.

convergent of the last step error.

$$\text{with } E\_i \begin{cases} 1 \text{ if } e\_i = z\_i - 2\_i = 0, i = 1, 2 \\ 0 \text{ if } \text{not} \end{cases} \tag{30}$$

This observer structure depends on Super Twisting Algorithm presented in previous section and Step by Step proficiencies [7]. We propose to put in multiples-series observers with functions (*Ei*).

**Figure 4.** General *IM* Observer Structure.

10 Will-be-set-by-IN-TECH

**Figure 3.** Super Twisting Algorithm example.

⎧

˙

˙

˙

˙

with *Ei*

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Thus, we can propose a new observer structure for dynamical system (28):

*z*˜3 = *α*<sup>1</sup> *sign*(*e*1)

*z*˜4 = *α*<sup>2</sup> *sign*(*e*2)

*z*ˆ3 = *E*<sup>1</sup> *E*<sup>2</sup> (*z*˜5 + *λ*<sup>3</sup> |*e*3|

*z*ˆ4 = *E*<sup>1</sup> *E*<sup>2</sup> (*z*˜6 + *λ*<sup>4</sup> |*e*4|

*z*˜5 = *E*<sup>1</sup> *E*<sup>2</sup> *α*<sup>3</sup> *sign*(*e*3)

*z*˜6 = *E*<sup>1</sup> *E*<sup>2</sup> *α*<sup>4</sup> *sign*(*e*4)

*z*ˆ1 = *θ z*˜3 − *γ z*<sup>1</sup> + *ξ v*<sup>1</sup> + *λ*<sup>1</sup> |*e*1|

*z*ˆ2 = *θ z*˜4 − *γ z*<sup>2</sup> + *ξ v*<sup>2</sup> + *λ*<sup>2</sup> |*e*2|

<sup>2</sup> *sign*(*e*3)) ˙

<sup>2</sup> *sign*(*e*4)) ˙

<sup>2</sup> *sign*(*e*1) ˙

<sup>2</sup> *sign*(*e*2) ˙

1

1

� 1 if *ei* <sup>=</sup> *zi* <sup>−</sup> *<sup>z</sup>*ˆ*<sup>i</sup>* <sup>=</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, 2

1

1

0 if not (30)

(29)

The functions *Ei* ensure that the next steps errors do not escape too far before one has the convergent of the last step error.

The gains *αi*, *λ<sup>i</sup>* are chosen which respect to the reachability condition of the Super Twisting algorithm as stated in inequalities (12) and (24) of previous section. By choosing

$$
\mu\_1 > \max(\theta z\_3), \quad \lambda\_1 > (\max(\theta z\_3) + a\_1)\sqrt{\frac{2}{\alpha\_1 - \max(\theta z\_3)}}\tag{31}
$$

$$
\alpha\_2 > \max(\theta z\_4), \quad \lambda\_2 > (\max(\theta z\_4) + \alpha\_2)\sqrt{\frac{2}{\alpha\_2 - \max(\theta z\_4)}}\tag{32}
$$

$$
\alpha\_3 > \max(z\_5), \quad \lambda\_3 > (\max(z\_5) + \alpha\_3) \sqrt{\frac{2}{\alpha\_3 - \max(z\_5)}}\tag{33}
$$

Firstly we propose to write *x*˙3 and *x*˙4 as a function of variables *z*. By using (27) in (38), we get:

*x*˙3 = *az*<sup>1</sup> − *z*ˆ3

By Replacing (39) in (38) and using the two first equations in (27), it follows

*z*˜6 = *bx*˙4 − *cx*5*x*˙3 − *x*<sup>3</sup>

*π*1*x*<sup>2</sup>

−*z*˜6 + *b*(*az*<sup>2</sup> − *z*ˆ4)

After a straightforward computations, we obtain a second order expression of *x*5:

(*az*<sup>2</sup> − *z*ˆ4)*z*ˆ3 − (*az*<sup>1</sup> − *z*ˆ3)*z*ˆ4

From (42) we have:

where

or

**CASE B** : *x*˙5 = 0

By substituting (44) in (43), we get

coming from (37) and (39), respectively.

*<sup>π</sup>*<sup>1</sup> <sup>=</sup> *<sup>c</sup> b* 

*<sup>π</sup>*<sup>2</sup> <sup>=</sup> *<sup>c</sup> b* 

*π*<sup>3</sup> = *z*ˆ3

Taking the time derivative of (41) and using third-fourth equations in (27) yields to

*<sup>x</sup>*˙5 <sup>=</sup> *<sup>z</sup>*˜5 <sup>−</sup> *bx*˙3 <sup>−</sup> *cx*5*x*˙4 *cx*<sup>4</sup>

Then we can deduce the motor speed *x*<sup>5</sup> by replacing in (45) expressions of *x*3-*x*<sup>4</sup> and *x*˙3-*x*˙4

*x*˙4 = *az*<sup>2</sup> − *z*ˆ4 (39)

Industrial Application of a Second Order Sliding Mode Observer for Speed and Flux Estimation in Sensorless Induction Motor

*z*ˆ3 = *bx*<sup>3</sup> + *cx*5*x*<sup>4</sup> (40) *z*ˆ4 = *bx*<sup>4</sup> − *cx*5*x*<sup>3</sup> (41)

<sup>5</sup> + *π*2*x*<sup>5</sup> + *π*<sup>3</sup> = 0 (46)

*z*˜5 − *b*(*az*<sup>1</sup> − *z*ˆ3)

(44)

371

(45)

(47)

(48)

*z*˙3 = *z*˜5 = *bx*˙3 + *cx*˙5*x*<sup>4</sup> + *cx*˙4*x*<sup>5</sup> (42) *z*˙4 = *z*˜6 = *bx*˙4 − *cx*˙5*x*<sup>3</sup> − *cx*˙3*x*<sup>5</sup> (43)

> *z*˜5 − *bx*˙3 − *cx*5*x*˙4 *x*4

*b*(*az*<sup>1</sup> − *z*ˆ3)*z*ˆ3 − *b*(*az*<sup>2</sup> − *z*ˆ4)*z*ˆ4 − *z*˜5*z*ˆ3 + *z*ˆ4*z*˜6

 − *z*ˆ4 

We propose this hypothesis because of dynamical gap evolution between electrical and mechanical variables, in fact speed evolves much more slowly than currents or fluxes. Thus with this hypothesis we simplify (42) and (43), and obtain two expressions of *x*<sup>5</sup> :

> *<sup>x</sup>*<sup>5</sup> <sup>=</sup> *<sup>z</sup>*˜5 <sup>−</sup> *bx*˙3 *cx*˙4

> *<sup>x</sup>*<sup>5</sup> <sup>=</sup> *bx*˙4 <sup>−</sup> *<sup>z</sup>*˜6 *cx*˙3

$$
\alpha\_4 > \max(z\_6), \quad \lambda\_4 > (\max(z\_6) + \alpha\_4)\sqrt{\frac{2}{\alpha\_4 - \max(z\_6)}}\tag{34}
$$

and we get

$$e\_1 = e\_2 = e \\ \mathfrak{J} = e\_4 = 0$$

i.e.

$$\begin{aligned} \label{eq:1} \hat{z}\_1 &= z\_{1\prime} & \hat{z}\_2 &= z\_2\\ \hat{z}\_3 &= \tilde{z}\_{3\prime} & \hat{z}\_4 &= \tilde{z}\_4\\ \tilde{z}\_5 &= z\_{5\prime} & \tilde{z}\_6 &= z\_6 \end{aligned}$$

Consequently all variables *z*1, *z*2, *z*ˆ3, *z*ˆ4, *z*˜5, *z*˜6 are available and then we can deduce *IM* variables.

We propose to treat this problem in two different cases: *x*˙5 �= 0, and *x*˙5 = 0
