**CASE A** : *x*˙5 �= 0

Firstly we propose to express fluxes *x*<sup>3</sup> and *x*4, from equation (27) we obtain:

$$\begin{aligned} \dot{\mathbf{z}}\_3 &= b\mathbf{x}\_3 + c\mathbf{x}\_5\mathbf{x}\_4\\ \dot{\mathbf{z}}\_4 &= b\mathbf{x}\_4 - c\mathbf{x}\_5\mathbf{x}\_3 \end{aligned}$$

We deduce

$$\mathbf{x}\_3 = \frac{\mathbf{z}\_3 - c\mathbf{x}\_5\mathbf{x}\_4}{b} \tag{35}$$

$$\mathbf{x}\_4 = \frac{\mathbf{z}\_4 + c\mathbf{x}\_5\mathbf{x}\_3}{b} \tag{36}$$

By substituting *x*<sup>4</sup> by its expression in (35) and *x*<sup>3</sup> in (36) we have:

$$\begin{aligned} \mathbf{x}\_3 &= \frac{\mathbf{\hat{z}}\_4 + \frac{c}{b} \mathbf{z}\_3 \mathbf{x}\_5}{b + \frac{c^2 \mathbf{x}\_5^2}{b}}\\ \mathbf{x}\_4 &= \frac{\mathbf{\hat{z}}\_3 - \frac{c}{b} \mathbf{z}\_4 \mathbf{x}\_5}{b + \frac{c^2 \mathbf{x}\_5^2}{b}} \end{aligned} \tag{37}$$

Now let us express *x*5. From (2) we know

$$
\dot{\mathbf{x}}\_3 = a\mathbf{x}\_1 - b\mathbf{x}\_3 - c\mathbf{x}\_5\mathbf{x}\_4
$$

$$
\dot{\mathbf{x}}\_4 = a\mathbf{x}\_2 - b\mathbf{x}\_4 + c\mathbf{x}\_5\mathbf{x}\_3\tag{38}
$$

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

$$
\begin{aligned}
\dot{x}\_3 &= az\_1 - \dot{z}\_3 \\
\dot{x}\_4 &= az\_2 - \dot{z}\_4
\end{aligned}
\tag{39}
$$

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

$$\mathbf{z\_3} = b\mathbf{x\_3} + c\mathbf{x\_5}\mathbf{x\_4} \tag{40}$$

$$
\hat{x}\_4 = b\mathbf{x}\_4 - c\mathbf{x}\_5\mathbf{x}\_3 \tag{41}
$$

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

$$
\dot{z}\_3 = \dot{z}\_5 = b\dot{\mathbf{x}}\_3 + c\dot{\mathbf{x}}\_5\mathbf{x}\_4 + c\dot{\mathbf{x}}\_4\mathbf{x}\_5 \tag{42}
$$

$$
\dot{z}\_4 = \dot{z}\_6 = b\dot{x}\_4 - c\dot{x}\_5 x\_3 - c\dot{x}\_3 x\_5 \tag{43}
$$

From (42) we have:

12 Will-be-set-by-IN-TECH

*e*<sup>1</sup> = *e*<sup>2</sup> = *e*3 = *e*<sup>4</sup> = 0

*z*ˆ1 = *z*1, *z*ˆ2 = *z*<sup>2</sup> *z*ˆ3 = *z*˜3, *z*ˆ4 = *z*˜4 *z*˜5 = *z*5, *z*˜6 = *z*<sup>6</sup>

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

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

*<sup>x</sup>*<sup>3</sup> <sup>=</sup> *<sup>z</sup>*ˆ3 <sup>−</sup> *cx*5*x*<sup>4</sup>

*<sup>x</sup>*<sup>4</sup> <sup>=</sup> *<sup>z</sup>*ˆ4 <sup>+</sup> *cx*5*x*<sup>3</sup>

*<sup>x</sup>*<sup>3</sup> <sup>=</sup> *<sup>z</sup>*ˆ4 <sup>+</sup> *<sup>c</sup>*

*<sup>x</sup>*<sup>4</sup> <sup>=</sup> *<sup>z</sup>*ˆ3 <sup>−</sup> *<sup>c</sup>*

*b* +

*b* +

*x*˙3 = *ax*<sup>1</sup> − *bx*<sup>3</sup> − *cx*5*x*<sup>4</sup>

*<sup>b</sup> z*<sup>3</sup> *x*<sup>5</sup>

*<sup>b</sup> z*<sup>4</sup> *x*<sup>5</sup>

*c*<sup>2</sup> *x*<sup>2</sup> 5 *b*

*c*<sup>2</sup> *x*<sup>2</sup> 5 *b*

2

2

*<sup>b</sup>* (35)

*<sup>b</sup>* (36)

*x*˙4 = *ax*<sup>2</sup> − *bx*<sup>4</sup> + *cx*5*x*<sup>3</sup> (38)

(37)

*<sup>α</sup>*<sup>3</sup> <sup>−</sup> *max*(*z*5) (33)

*<sup>α</sup>*<sup>4</sup> <sup>−</sup> *max*(*z*6) (34)

*α*<sup>3</sup> > *max*(*z*5), *λ*<sup>3</sup> > (*max*(*z*5) + *α*3)

*α*<sup>4</sup> > *max*(*z*6), *λ*<sup>4</sup> > (*max*(*z*6) + *α*4)

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

Firstly we propose to express fluxes *x*<sup>3</sup> and *x*4, from equation (27) we obtain:

By substituting *x*<sup>4</sup> by its expression in (35) and *x*<sup>3</sup> in (36) we have:

Now let us express *x*5. From (2) we know

and we get

variables.

We deduce

**CASE A** : *x*˙5 �= 0

i.e.

$$\dot{\mathfrak{x}}\_{5} = \frac{\mathfrak{z}\_{5} - b\dot{\mathfrak{x}}\_{3} - c\mathfrak{x}\_{5}\dot{\mathfrak{x}}\_{4}}{c\mathfrak{x}\_{4}} \tag{44}$$

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

$$\mathfrak{z}\_6 = b\dot{\mathfrak{x}}\_4 - c\mathfrak{x}\_5\dot{\mathfrak{x}}\_3 - \mathfrak{x}\_3\frac{\mathfrak{z}\_5 - b\dot{\mathfrak{x}}\_3 - c\mathfrak{x}\_5\dot{\mathfrak{x}}\_4}{\mathfrak{x}\_4} \tag{45}$$

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

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

$$
\pi\_1 \mathbf{x}\_5^2 + \pi\_2 \mathbf{x}\_5 + \pi\_3 = \mathbf{0} \tag{46}
$$

where

$$\begin{aligned} \pi\_1 &= \frac{c}{b} \left[ (az\_2 - \widehat{z}\_4)\widehat{z}\_3 - (az\_1 - \widehat{z}\_3)\widehat{z}\_4 \right] \\ \pi\_2 &= \frac{c}{b} \left[ b(az\_1 - \widehat{z}\_3)\widehat{z}\_3 - b(az\_2 - \widehat{z}\_4)\widehat{z}\_4 - \widetilde{z}\_5\widehat{z}\_3 + \widehat{z}\_4\widetilde{z}\_6 \right] \\ \pi\_3 &= \widehat{z}\_3 \left[ -\widetilde{z}\_6 + b(az\_2 - \widehat{z}\_4) \right] - \widehat{z}\_4 \left[ \widetilde{z}\_5 - b(az\_1 - \widehat{z}\_3) \right] \end{aligned}$$

**CASE B** : *x*˙5 = 0

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> :

$$\mathbf{x}\_{\mathfrak{F}} = \frac{\mathfrak{z}\_{\mathfrak{F}} - b\mathfrak{x}\_{\mathfrak{3}}}{c\mathfrak{x}\_{\mathfrak{4}}} \tag{47}$$

or

$$\mathbf{x}\_{5} = \frac{b\dot{\mathbf{x}}\_{4} - \tilde{\mathbf{z}}\_{6}}{c\dot{\mathbf{x}}\_{3}} \tag{48}$$

### 14 Will-be-set-by-IN-TECH 372 Induction Motors – Modelling and Control Industrial Application of a Second Order Sliding Mode Observer for Speed and Flux Estimation in Sensorless Induction Motor <sup>15</sup>

we change *x*˙3 by expression (35) and *x*˙4 by expression(36)

$$\chi\_5 = \frac{\mathfrak{z}\_5 - baz\_1 + b\mathfrak{z}\_3}{czz\_2 - c\mathfrak{z}\_4} \tag{49}$$

⎧

*z*ˆ1(*k*) = *z*ˆ1(*k* − 1) + *Te*

*z*ˆ2(*k*) = *z*ˆ2(*k* − 1) + *Te*

*z*ˆ3(*k*) = *z*˜3(*k* − 1) + *Te E*<sup>1</sup> *E*<sup>2</sup>

*z*ˆ4(*k*) = *z*ˆ4(*k* − 1) + *Te E*<sup>1</sup> *E*<sup>2</sup>

�

�

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

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

error of Euler's method, seen in the previous subsection (4.1).

two discrete time systems: *<sup>H</sup>*<sup>1</sup> sampled at frequency *fs*<sup>1</sup> <sup>=</sup> <sup>1</sup>

enough. The discrete time is given by *tH*<sup>1</sup> = *nTe* for *H*<sup>1</sup> and *tH*2 = *k*

� *k Te N* �

Assume a continuous autonomous system of the form:

can be approximated by the explicit Euler's method:

For small values of *Te*, *O*(*T*<sup>3</sup>

*e* .

= *t*<sup>0</sup> and *x*(*nTe*) = *x*

proportional to *T*<sup>2</sup>

*fs*<sup>2</sup> <sup>=</sup> *<sup>N</sup> Te*

*nTe* <sup>=</sup> *<sup>k</sup> <sup>T</sup>*

*Te*

system *H*<sup>1</sup> can be written as:

*z*˜3(*k*) = *z*˜3(*k* − 1) + *Te α*<sup>1</sup> *sign*(*e*1(*k* − 1))

*z*˜4(*k*) = *z*˜4(*k* − 1) + *Te α*<sup>2</sup> *sign*(*e*2(*k* − 1))

+ *λ*<sup>1</sup> |*e*1(*k* − 1)|

+ *λ*<sup>2</sup> |*e*2(*k* − 1)|

�

�

1

1

To achieve good accuracy, a small sample period and fast DSP are needed. In the industrial application, the DSP clock frequency is only 150*MHz*, which does not allow a small enough sample period. So in experimentation an over-sample technique is proposed. In the following paragraphs we show that, under a few low restrictive conditions, it is possible to reduce the

Hereafter we first present the oversampling method in a very simple use, where *<sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*∞.

Assume in addition that the system is discretized at a sampling time *Te*. Then the system ((54))

*<sup>x</sup>*(*tk*<sup>+</sup>1) = *<sup>x</sup>*(*tk*) + *Te <sup>x</sup>*˙(*tk*) + *<sup>O</sup>*(*T*<sup>2</sup>

Suppose now that the system (54) is discretized at two different sample rates resulting in

Assume that the initial times and the initial conditions are the same for both of them, that is:

; and let us compare the truncation error of each one after *Te* seconds for *N* large

*θ z*˜3(*k* − 1) − *γ z*1(*k* − 1) + *ξ v*1(*k* − 1)

<sup>2</sup> *sign*(*e*1(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>))�

*θ z*˜4(*k* − 1) − *γ z*2(*k* − 1) + *ξ v*2(*k* − 1)

<sup>2</sup> *sign*(*e*2(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>))�

*z*˜5(*k* − 1) + *λ*<sup>3</sup> |*e*3(*k* − 1)|

*z*˜6(*k* − 1) + *λ*<sup>4</sup> |*e*4(*k* − 1)|

1

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

1

*x*˙ = *f*(*x*); *x*(*t*0) = *x*<sup>0</sup> (54)

*<sup>e</sup>* ) is neglected and the truncation error is approximately

*Te*

*Te*

= *x*(*t*0) = *x*0, The dynamics of the discrete time

<sup>2</sup> *sign*(*e*3(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>))�

(53)

373

<sup>2</sup> *sign*(*e*4(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>))�

*<sup>e</sup>* ) (55)

and *H*<sup>2</sup> sampled at frequency

*<sup>N</sup>* for *<sup>H</sup>*2, with *<sup>n</sup>*, *<sup>k</sup>* <sup>∈</sup> **<sup>N</sup>**.

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

**4.2. Oversampling**

or

$$\alpha\_5 = \frac{\mathfrak{z}\_6 + baz\_2 - b\mathfrak{z}\_4}{cz\_1 - c\mathfrak{z}\_3} \tag{50}$$

Equations (49) and (50) are true only if :

$$
\circledast az\_2 - c\sharp\_4 \neq 0 \text{ for (49) and } caz\_1 - c\sharp\_3 \neq 0 \text{ for (50)}.
$$

### **Speed estimation**

In order to avoid singularities of speed estimation in (49) and (50), we use the fact that (49) and (50) are in quadrature and thus we get the estimation of speed *x*<sup>5</sup> as follows :

$$\mathbf{x\_5} = \frac{(\mathbf{\tilde{z}\_5} - baz\_1 + b\mathbf{\hat{z}\_3})(czz\_2 - c\mathbf{\hat{z}\_4}) + (\mathbf{\tilde{z}\_6} + baz\_2 - b\mathbf{\hat{z}\_4})(czz\_1 - c\mathbf{\hat{z}\_3})}{(czz\_2 - c\mathbf{\hat{z}\_4})^2 + (czz\_1 - c\mathbf{\hat{z}\_3})^2} \tag{51}$$

### **Flux estimation**

The rotor flux are obtained by replacing the estimation speed (51) in (37)

### **Flux position estimation**

Having the rotor flux estimation (37), we can obtain rotor flux position *ρ*

$$\rho = \operatorname{atan}(\frac{\mathbf{x\_4}}{\mathbf{x\_3}}) \tag{52}$$

### **4. Discrete time implementation**

### **4.1. Explicit Euler method**

For the industrial application in real time, the discrete time observer is designed. The explicit Euler's method is chosen to transform continuous observer to discrete observer. This is due to the simplicity of computation. Considering a differential equation :

$$
\dot{\mathfrak{x}} = f(\mathfrak{x}).
$$

The explicit Euler's method with a sampling time *Te* gives:

$$\mathbf{x}(k) = \mathbf{x}(k-1) + T\_\ell f(\mathbf{x}(k-1))$$

the data acquisition period *Te* is also the computation period.

Applying the explicit Euler's method for the second order sliding mode observer, the discrete observer is obtained:

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *z*ˆ1(*k*) = *z*ˆ1(*k* − 1) + *Te* � *θ z*˜3(*k* − 1) − *γ z*1(*k* − 1) + *ξ v*1(*k* − 1) + *λ*<sup>1</sup> |*e*1(*k* − 1)| 1 <sup>2</sup> *sign*(*e*1(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>))� *z*˜3(*k*) = *z*˜3(*k* − 1) + *Te α*<sup>1</sup> *sign*(*e*1(*k* − 1)) *z*ˆ2(*k*) = *z*ˆ2(*k* − 1) + *Te* � *θ z*˜4(*k* − 1) − *γ z*2(*k* − 1) + *ξ v*2(*k* − 1) + *λ*<sup>2</sup> |*e*2(*k* − 1)| 1 <sup>2</sup> *sign*(*e*2(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>))� *z*˜4(*k*) = *z*˜4(*k* − 1) + *Te α*<sup>2</sup> *sign*(*e*2(*k* − 1)) *z*ˆ3(*k*) = *z*˜3(*k* − 1) + *Te E*<sup>1</sup> *E*<sup>2</sup> � *z*˜5(*k* − 1) + *λ*<sup>3</sup> |*e*3(*k* − 1)| 1 <sup>2</sup> *sign*(*e*3(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>))� *z*˜5(*k*) = *z*˜5(*k* − 1) + *Te E*<sup>1</sup> *E*<sup>2</sup> *α*<sup>3</sup> *sign*(*e*3(*k* − 1)) *z*ˆ4(*k*) = *z*ˆ4(*k* − 1) + *Te E*<sup>1</sup> *E*<sup>2</sup> � *z*˜6(*k* − 1) + *λ*<sup>4</sup> |*e*4(*k* − 1)| 1 <sup>2</sup> *sign*(*e*4(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>))� *z*˜6(*k*) = *z*˜6(*k* − 1) + *Te E*<sup>1</sup> *E*<sup>2</sup> *α*<sup>4</sup> *sign*(*e*4(*k* − 1)) (53)

### **4.2. Oversampling**

14 Will-be-set-by-IN-TECH

*<sup>x</sup>*<sup>5</sup> <sup>=</sup> *<sup>z</sup>*˜5 <sup>−</sup> *baz*<sup>1</sup> <sup>+</sup> *bz*ˆ3 *caz*<sup>2</sup> − *cz*ˆ4

*<sup>x</sup>*<sup>5</sup> <sup>=</sup> *<sup>z</sup>*˜6 <sup>+</sup> *baz*<sup>2</sup> <sup>−</sup> *bz*ˆ4 *caz*<sup>1</sup> − *cz*ˆ3

*caz*<sup>2</sup> − *cz*ˆ4 �= 0 for (49) and *caz*<sup>1</sup> − *cz*ˆ3 �= 0 for (50)

In order to avoid singularities of speed estimation in (49) and (50), we use the fact that (49)

*<sup>x</sup>*<sup>5</sup> <sup>=</sup> (*z*˜5 <sup>−</sup> *baz*<sup>1</sup> <sup>+</sup> *bz*ˆ3)(*caz*<sup>2</sup> <sup>−</sup> *cz*ˆ4)+(*z*˜6 <sup>+</sup> *baz*<sup>2</sup> <sup>−</sup> *bz*ˆ4)(*caz*<sup>1</sup> <sup>−</sup> *cz*ˆ3)

*ρ* = *atan*(

For the industrial application in real time, the discrete time observer is designed. The explicit Euler's method is chosen to transform continuous observer to discrete observer. This is due

*x*˙ = *f*(*x*)

*x*(*k*) = *x*(*k* − 1) + *Te f*(*x*(*k* − 1))

Applying the explicit Euler's method for the second order sliding mode observer, the discrete

*x*4 *x*3

(*caz*<sup>2</sup> <sup>−</sup> *cz*ˆ4)<sup>2</sup> + (*caz*<sup>1</sup> <sup>−</sup> *cz*ˆ3)<sup>2</sup> (51)

) (52)

and (50) are in quadrature and thus we get the estimation of speed *x*<sup>5</sup> as follows :

The rotor flux are obtained by replacing the estimation speed (51) in (37)

Having the rotor flux estimation (37), we can obtain rotor flux position *ρ*

to the simplicity of computation. Considering a differential equation :

The explicit Euler's method with a sampling time *Te* gives:

the data acquisition period *Te* is also the computation period.

(49)

(50)

we change *x*˙3 by expression (35) and *x*˙4 by expression(36)

Equations (49) and (50) are true only if :

or

**Speed estimation**

**Flux estimation**

**Flux position estimation**

**4. Discrete time implementation**

**4.1. Explicit Euler method**

observer is obtained:

To achieve good accuracy, a small sample period and fast DSP are needed. In the industrial application, the DSP clock frequency is only 150*MHz*, which does not allow a small enough sample period. So in experimentation an over-sample technique is proposed. In the following paragraphs we show that, under a few low restrictive conditions, it is possible to reduce the error of Euler's method, seen in the previous subsection (4.1).

Hereafter we first present the oversampling method in a very simple use, where *<sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*∞. Assume a continuous autonomous system of the form:

$$\dot{\mathbf{x}} = f(\mathbf{x}); \quad \mathbf{x}(t\_0) = \mathbf{x}\_0 \tag{54}$$

Assume in addition that the system is discretized at a sampling time *Te*. Then the system ((54)) can be approximated by the explicit Euler's method:

$$\mathbf{x}(t\_{k+1}) = \mathbf{x}(t\_k) + T\_\varepsilon \dot{\mathbf{x}}(t\_k) + \mathcal{O}(T\_\varepsilon^2) \tag{55}$$

For small values of *Te*, *O*(*T*<sup>3</sup> *<sup>e</sup>* ) is neglected and the truncation error is approximately proportional to *T*<sup>2</sup> *e* .

Suppose now that the system (54) is discretized at two different sample rates resulting in two discrete time systems: *<sup>H</sup>*<sup>1</sup> sampled at frequency *fs*<sup>1</sup> <sup>=</sup> <sup>1</sup> *Te* and *H*<sup>2</sup> sampled at frequency *fs*<sup>2</sup> <sup>=</sup> *<sup>N</sup> Te* ; and let us compare the truncation error of each one after *Te* seconds for *N* large enough. The discrete time is given by *tH*<sup>1</sup> = *nTe* for *H*<sup>1</sup> and *tH*2 = *k Te <sup>N</sup>* for *<sup>H</sup>*2, with *<sup>n</sup>*, *<sup>k</sup>* <sup>∈</sup> **<sup>N</sup>**. Assume that the initial times and the initial conditions are the same for both of them, that is: *nTe* <sup>=</sup> *<sup>k</sup> <sup>T</sup> Te* = *t*<sup>0</sup> and *x*(*nTe*) = *x* � *k Te N* � = *x*(*t*0) = *x*0, The dynamics of the discrete time system *H*<sup>1</sup> can be written as:

$$\propto \left( (n+1)T\_{\ell} \right) = \propto \left( nT\_{\ell} \right) + T\_{\ell} \left f \left( \mathfrak{x} \left( nT\_{\ell} \right) \right) + O \left( T\_{\ell}^{2} \right) \tag{56}$$

*<sup>ε</sup>*<sup>2</sup> <sup>≈</sup> *<sup>ε</sup>*<sup>1</sup>

In practice, to achieve the benefits of oversampling, we emulate this technique based on the assumption that between two consecutive samples of an input signal, its derivative is nearly constant. In this way the new" samples are obtained by linear interpolation between consecutive "measured" samples. This technique is shown in figure 5. On the top the

Then the oversampled system *H*<sup>2</sup> reduces the truncation error about *N* times.

(*k* − 1)*Te k Te*

of this technique are exposed and validated by experimental tests.

practice, and in Table (2) main VAR-CNTRL card features are presented.

inequalities (31), 32, 33 and 34 to satisfy convergence conditions.

Δ*T*

**Figure 5.** Comparison between sampling and oversampling implementation.

"classical sampling" is shown and on the bottom, the oversampling technique is depicted. As

technique reduces the truncation error, inherent to Euler's method, three times. The benefits

Table (1) presents all electrical and mechanical parameters of Induction machine used in

The tunning parameters *αi*, *λi*, *i* = 1, ..., 4 of the proposed observer are chosen according to

we can see, the sample period is reduced three times its original value, that is, *Te*

*x*[*n Te*]

(*k* − 1)*Te*

*x*[*n Te*]

*x*[*k Te*]

*x*[(*k* − 1)*Te*]

*x*[*k Te*]

Δ*x*

**5. Experimentations**

**5.1. Test bench**

*x*[(*k* − 1)*Te*]

*<sup>N</sup>* (63)

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

375

*n Te*

*<sup>N</sup>* <sup>=</sup> *Te*

<sup>3</sup> . This

*n Te*

*k Te*

*<sup>H</sup>*2, sampled at *fs*<sup>2</sup> <sup>=</sup> *<sup>N</sup> Te* , evolves as follows:

$$\begin{split} \mathbf{x}((k+1)\frac{T\_{\varepsilon}}{N}) &= \mathbf{x}(k\frac{T\_{\varepsilon}}{N}) + \frac{T\_{\varepsilon}}{N}f\left(\mathbf{x}(k\frac{T\_{\varepsilon}}{N})\right) + O\left(\frac{T\_{\varepsilon}}{N}\right)^{2} \\ \mathbf{x}((k+2)\frac{T\_{\varepsilon}}{N}) &= \mathbf{x}((k+1)\frac{T\_{\varepsilon}}{N}) + \frac{T\_{\varepsilon}}{N}f\left(\mathbf{x}((k+1)\frac{T\_{\varepsilon}}{N})\right) + O\left(\frac{T\_{\varepsilon}}{N}\right)^{2} \\ &= \mathbf{x}(k\frac{T\_{\varepsilon}}{N}) + \frac{T\_{\varepsilon}}{N}f\left(\mathbf{x}(k\frac{T\_{\varepsilon}}{N})\right) \\ &\quad + \frac{T\_{\varepsilon}}{N}f\left(\mathbf{x}(k\frac{T\_{\varepsilon}}{N}) + \frac{T\_{\varepsilon}}{N}f\left(\mathbf{x}(k\frac{T\_{\varepsilon}}{N})\right) + O\left(\frac{T\_{\varepsilon}}{N}\right)^{2}\right) + 2O\left(\frac{T\_{\varepsilon}}{N}\right)^{2} \end{split} \tag{57}$$

For *N* large enough we can consider the influence of the error term *O Te N* <sup>2</sup> over the function *Te <sup>N</sup> <sup>f</sup>*(·) as a term in *<sup>O</sup> Te N* <sup>3</sup> , then:

$$\begin{split} \mathbf{x}((k+2)\frac{T\_{\varepsilon}}{N}) &\approx \mathbf{x}(k\frac{T\_{\varepsilon}}{N}) + \frac{T\_{\varepsilon}}{N}f\left(\mathbf{x}(k\frac{T\_{\varepsilon}}{N})\right) + \frac{T\_{\varepsilon}}{N}f\left\{\mathbf{x}\left(k\frac{T\_{\varepsilon}}{N}\right) + \frac{T\_{\varepsilon}}{N}f\left(\mathbf{x}\left(k\frac{T\_{\varepsilon}}{N}\right)\right)\right\} \\ &+ 2O\left(\frac{T\_{\varepsilon}}{N}\right)^{2} \\ &= \mathbf{x}(k\frac{T\_{\varepsilon}}{N}) + \frac{T\_{\varepsilon}}{N}f\left(\mathbf{x}(k\frac{T\_{\varepsilon}}{N})\right) + \frac{T\_{\varepsilon}}{N}f(\mathbf{x}((k+1)\frac{T\_{\varepsilon}}{N})) + 2O\left(\frac{T\_{\varepsilon}}{N}\right)^{2} \end{split} \tag{58}$$

So, in a general way, we have:

$$\text{tr}((k+N)\frac{T\_\varepsilon}{N}) \approx \text{x}(k\frac{T\_\varepsilon}{N}) + \frac{T\_\varepsilon}{N} \sum\_{i=k}^{k+N-1} f(\text{x}(i\frac{T\_\varepsilon}{N})) + NO\left(\frac{T\_\varepsilon}{N}\right)^2\tag{59}$$

As we can see from (56) and (59), the truncation errors of the discrete systems *H*<sup>1</sup> and *H*<sup>2</sup> are

$$
\varepsilon\_1 = O\left(T\_e^2\right) \tag{60}
$$

and

$$
\varepsilon\_2 = NO\left(\left(\frac{T\_\varepsilon}{N}\right)^2\right) \tag{61}
$$

The truncation errors *ε*<sup>1</sup> and *ε*2, given by (60) and (61) respectively give

$$NO\left(\left(\frac{T\_\ell}{N}\right)^2\right) \approx \frac{O\left(T\_\ell^2\right)}{N} \tag{62}$$

374 Induction Motors – Modelling and Control Industrial Application of a Second Order Sliding Mode Observer for Speed and Flux Estimation in Sensorless Induction Motor <sup>17</sup> 375 Industrial Application of a Second Order Sliding Mode Observer for Speed and Flux Estimation in Sensorless Induction Motor

$$
\varepsilon\_2 \approx \frac{\varepsilon\_1}{N} \tag{63}
$$

Then the oversampled system *H*<sup>2</sup> reduces the truncation error about *N* times.

In practice, to achieve the benefits of oversampling, we emulate this technique based on the assumption that between two consecutive samples of an input signal, its derivative is nearly constant. In this way the new" samples are obtained by linear interpolation between consecutive "measured" samples. This technique is shown in figure 5. On the top the

**Figure 5.** Comparison between sampling and oversampling implementation.

"classical sampling" is shown and on the bottom, the oversampling technique is depicted. As we can see, the sample period is reduced three times its original value, that is, *Te <sup>N</sup>* <sup>=</sup> *Te* <sup>3</sup> . This technique reduces the truncation error, inherent to Euler's method, three times. The benefits of this technique are exposed and validated by experimental tests.

### **5. Experimentations**

### **5.1. Test bench**

16 Will-be-set-by-IN-TECH

*x* ((*n* + 1)*Te*) = *x* (*nTe*) + *Te f* (*x*(*nTe*)) + *O*(*T*<sup>2</sup>

*x*((*k* + 1)

*Te N* ) + *O*( *Te N* )2

*<sup>N</sup> <sup>f</sup>*(*x*((*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)

*f*(*x*(*i Te* *Te*

*<sup>N</sup>* )) + *N O*

*<sup>N</sup>* )) + <sup>2</sup>*<sup>O</sup>*

 *Te N* <sup>2</sup>

*<sup>N</sup>* (62)

 *Te N* <sup>2</sup>

*<sup>H</sup>*2, sampled at *fs*<sup>2</sup> <sup>=</sup> *<sup>N</sup>*

*x*((*k* + 1)

*x*((*k* + 2)

*<sup>N</sup> <sup>f</sup>*(·) as a term in *<sup>O</sup>*

*Te*

So, in a general way, we have:

*x*((*k* + 2)

*Te*

and

*Te*

*Te*

*Te*

*<sup>N</sup>* ) = *<sup>x</sup>*(*<sup>k</sup> Te*

*<sup>N</sup>* ) = *<sup>x</sup>*((*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)

<sup>=</sup> *<sup>x</sup>*(*<sup>k</sup> Te*

+ *Te N f*

 *Te N* <sup>3</sup>

+2*O*

<sup>=</sup> *<sup>x</sup>*(*<sup>k</sup> Te*

*x*((*k* + *N*)

*<sup>N</sup>* ) <sup>≈</sup> *<sup>x</sup>*(*<sup>k</sup> Te*

, evolves as follows:

*Te <sup>N</sup>* ) + *Te N f* 

For *N* large enough we can consider the influence of the error term *O*

, then:

*<sup>N</sup>* ) + *Te N f <sup>x</sup>*(*<sup>k</sup> Te N* ) + *Te N f x k Te N* + *Te N f x k Te N*

 *Te N* <sup>2</sup>

*<sup>N</sup>* ) + *Te N f <sup>x</sup>*(*<sup>k</sup> Te N* ) + *Te*

*<sup>N</sup>* ) <sup>≈</sup> *<sup>x</sup>*(*<sup>k</sup> Te*

*<sup>N</sup>* ) + *Te N*

*ε*<sup>2</sup> = *N O*

 *Te N* <sup>2</sup> 

The truncation errors *ε*<sup>1</sup> and *ε*2, given by (60) and (61) respectively give

*N O*

*k*+*N*−1 ∑ *i*=*k*

As we can see from (56) and (59), the truncation errors of the discrete systems *H*<sup>1</sup> and *H*<sup>2</sup> are

 *T*2 *e* 

 *Te N* <sup>2</sup> 

> <sup>≈</sup> *<sup>O</sup> T*2 *e*

*ε*<sup>1</sup> = *O*

*Te*

*<sup>N</sup>* ) + *Te N f <sup>x</sup>*(*<sup>k</sup> Te N* ) + *O Te N* <sup>2</sup> 

*<sup>N</sup>* ) + *Te N f <sup>x</sup>*(*<sup>k</sup> Te N* ) + *O Te N* <sup>2</sup>

*<sup>N</sup>* ) + *Te N f <sup>x</sup>*(*<sup>k</sup> Te N* ) 

> *<sup>x</sup>*(*<sup>k</sup> Te*

*<sup>e</sup>* ) (56)

+ 2*O*

 *Te N* <sup>2</sup>  *Te N* <sup>2</sup>

over the function

(57)

(58)

(59)

(60)

(61)

Table (1) presents all electrical and mechanical parameters of Induction machine used in practice, and in Table (2) main VAR-CNTRL card features are presented.

The tunning parameters *αi*, *λi*, *i* = 1, ..., 4 of the proposed observer are chosen according to inequalities (31), 32, 33 and 34 to satisfy convergence conditions.


**VAR-CNTRL** is a electronic card designed by **GS Maintenance** and dedicated to motor control ( Synchronous, Induction machine, Brushless, and DC motor). Equipped with a **DSP TMS320F2812** from **Texas Instrument**,this component is a fixed point; data are represented

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377

• **ADC**'s (Analog-to-Digital Coder) of 12 bits provide bus voltage (*VDC*), and phase currents

In addition to the **VAR-CNTRL**, a **MMI** (Man Machine Interface) permits to visualize **DSP** data registers in representation format 8.8 that means possible variations are from [-127.996 to

In this section we propose some experimentation results, that allow the following points:

In section 4.1 we introduced Euler Explicit Sampling Method to discretize a continuous system. Some technical limits about sampling frequency *Fe* lead us to introduce Oversampling strategy (c.f. section 4.2) . At first glance we propose to validate Super Twisting Observer strategy (c.f. system 7), we will take account of subsystem Σ<sup>1</sup> in figure 4 with the following

Practicals tests have been done under the following configurations:

under 32 bits.

128].

• An **IM**.

• A **MMI**.

**5.2. Results**

• Fe , Sampling frequency of 8KHZ. • Fcyc, **DSP** clock frequency of 150MHZ. • 1024 points encoder, as speed sensor.

(*IA*,*IB*) frames under 12 bits.

• A control card , **VAR-CNTRL**.

To summarize our Bed Test description, we have :

• A speed sensor, a voltage sensor, and two current sensors.

**Figure 7.** MMI capture : ( *z*1,*v*1) and ( *z*ˆ1, *z*˜3) on convergence phase.

• Validate Super Twisting Algorithm convergence.

• Evaluate Oversampling method efficiency. • Evaluate Motor variables estimation.

entries : (*v*<sup>1</sup> , *z*1), and outputs : (*z*ˆ1, *z*˜3).

• A two-level **VSI** (Voltage Source Inverter).

**Table 1.** Induction machine and observer parameters.

**Figure 6.** VAR-CNTRL card a product of GS Maintenance.


**Table 2.** VAR-CNTRL card main elements.

**VAR-CNTRL** is a electronic card designed by **GS Maintenance** and dedicated to motor control ( Synchronous, Induction machine, Brushless, and DC motor). Equipped with a **DSP TMS320F2812** from **Texas Instrument**,this component is a fixed point; data are represented under 32 bits.

Practicals tests have been done under the following configurations:


In addition to the **VAR-CNTRL**, a **MMI** (Man Machine Interface) permits to visualize **DSP** data registers in representation format 8.8 that means possible variations are from [-127.996 to 128].

To summarize our Bed Test description, we have :

• An **IM**.

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*PN* rated power 1.5KW *VN* rated voltage 230V *IN* rated current 3.2A *FN* rated frequency 50Hz *NN* rated speed 2998tr/min p number of pair of poles 1 *RS* stator resistance 4.2Ω *RR* rotor resistance 2.8Ω *LS* stator inductance 0.522 H *LR* stator inductance 0.537 H *MSR* mutual inductance 0.502 H f viscous coefficient 1N.s/rad *α*1, *λ*<sup>1</sup> tunning parameters *α*<sup>1</sup> = 1500, *λ*<sup>1</sup> = 2500 *α*2, *λ*<sup>2</sup> tunning parameters *α*<sup>2</sup> = *α*1, *λ*<sup>2</sup> = *λ*<sup>1</sup> *α*3, *λ*<sup>3</sup> tunning parameters *α*<sup>3</sup> = 1500, *λ*<sup>3</sup> = 2000 *α*4, *λ*<sup>4</sup> tunning parameters *α*<sup>4</sup> = *α*3, *λ*<sup>4</sup> = *λ*<sup>3</sup>

**Table 1.** Induction machine and observer parameters.

**Figure 6.** VAR-CNTRL card a product of GS Maintenance.

2 Communication port. (1 *RS*232) 3 Logical Output connector. (6 Outputs )

5 Logical Input connector. (8 Isolated Inputs)

7 Measurements connector. (*VDC*, *IA*, *IB*)

9 PWM connector. (6 Output signals).

6 Supply voltage connector. (3.3V- 5V - (±15V) - 24V)

4 QEP connector. (*A*-*B*-*Z*)

8 DSP *TMS*320*F*2812

**Table 2.** VAR-CNTRL card main elements.

1 Analog Input/Output connectors. (3 Inputs /3 0utputs)


## **5.2. Results**

In this section we propose some experimentation results, that allow the following points:


In section 4.1 we introduced Euler Explicit Sampling Method to discretize a continuous system. Some technical limits about sampling frequency *Fe* lead us to introduce Oversampling strategy (c.f. section 4.2) . At first glance we propose to validate Super Twisting Observer strategy (c.f. system 7), we will take account of subsystem Σ<sup>1</sup> in figure 4 with the following entries : (*v*<sup>1</sup> , *z*1), and outputs : (*z*ˆ1, *z*˜3).

**Figure 7.** MMI capture : ( *z*1,*v*1) and ( *z*ˆ1, *z*˜3) on convergence phase.

### 20 Will-be-set-by-IN-TECH 378 Induction Motors – Modelling and Control Industrial Application of a Second Order Sliding Mode Observer for Speed and Flux Estimation in Sensorless Induction Motor <sup>21</sup>

On figure 7 we validate the convergence of Σ<sup>1</sup> in figure 4, we can see that under some initials values *z*ˆ1 converge to *z*<sup>1</sup> in a finite time.

Figures 8 and 9 permit to assume that oversampling method is efficient, in fact we see that signals estimated by the observer (53) of the subsystem Σ<sup>1</sup> in figure 4 are much more better with an oversampling than without.

**Figure 10.** Flux Estimation: *x*<sup>3</sup> and *x*<sup>4</sup> during static phase.

379

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

**Figure 11.** Estimated of rotor flux position: *ρ*.

**Figure 12.** Measure and estimate of speed during static phase: *x*<sup>5</sup> and *x*ˆ5

**Figure 8.** MMI capture : (*z*1,*v*1) and ( *z*ˆ1, *z*˜3) without oversampling method.

**Figure 9.** MMI capture : (*z*1,*v*1) and (*z*ˆ1, *z*˜3) with oversampling method. ( *N* = 10.)

Thus at the same operating point, we assume that with oversampling method we improve efficiency of the algorithm. With this validated data we can now abort estimation of *IM* magnetic (*x*3, *x*4) and mechanical (*x*5) variables including the rotor position flux *ρ* given in equations 51, (35-36) and (52) respectively.

The main objective of this work is to provide a motor speed estimation without any mechanical sensor, and then drive it. Note that the speed sensor is only used in comparison of estimated speed with its measure. To validate our strategy we propose some tests into different conditions.

Figures 12 and 13 permit to validate accuracy of estimated speed compare to measured speed in high variation range. However, it is admit that at low and very low speed, estimated speed damages more and more, as we can see on figures 14 and 15.

Now we propose some dynamical test results. During acceleration and deceleration phases (c.f. 16) , estimated speed is steel working although there is small delay between *x*<sup>5</sup> and *x*ˆ5.

378 Induction Motors – Modelling and Control Industrial Application of a Second Order Sliding Mode Observer for Speed and Flux Estimation in Sensorless Induction Motor <sup>21</sup> 379 Industrial Application of a Second Order Sliding Mode Observer for Speed and Flux Estimation in Sensorless Induction Motor

**Figure 10.** Flux Estimation: *x*<sup>3</sup> and *x*<sup>4</sup> during static phase.

20 Will-be-set-by-IN-TECH

On figure 7 we validate the convergence of Σ<sup>1</sup> in figure 4, we can see that under some initials

Figures 8 and 9 permit to assume that oversampling method is efficient, in fact we see that signals estimated by the observer (53) of the subsystem Σ<sup>1</sup> in figure 4 are much more better

**Figure 8.** MMI capture : (*z*1,*v*1) and ( *z*ˆ1, *z*˜3) without oversampling method.

**Figure 9.** MMI capture : (*z*1,*v*1) and (*z*ˆ1, *z*˜3) with oversampling method. ( *N* = 10.)

damages more and more, as we can see on figures 14 and 15.

equations 51, (35-36) and (52) respectively.

different conditions.

Thus at the same operating point, we assume that with oversampling method we improve efficiency of the algorithm. With this validated data we can now abort estimation of *IM* magnetic (*x*3, *x*4) and mechanical (*x*5) variables including the rotor position flux *ρ* given in

The main objective of this work is to provide a motor speed estimation without any mechanical sensor, and then drive it. Note that the speed sensor is only used in comparison of estimated speed with its measure. To validate our strategy we propose some tests into

Figures 12 and 13 permit to validate accuracy of estimated speed compare to measured speed in high variation range. However, it is admit that at low and very low speed, estimated speed

Now we propose some dynamical test results. During acceleration and deceleration phases (c.f. 16) , estimated speed is steel working although there is small delay between *x*<sup>5</sup> and *x*ˆ5.

values *z*ˆ1 converge to *z*<sup>1</sup> in a finite time.

with an oversampling than without.

**Figure 11.** Estimated of rotor flux position: *ρ*.

**Figure 12.** Measure and estimate of speed during static phase: *x*<sup>5</sup> and *x*ˆ5

divergence as small as the time to cross it; in fact this phenomenon underlines that speed is

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

381

non observable with low current dynamic.

**Figure 16.** Measure and estimate of speed during variable phase: *x*<sup>5</sup> and *x*ˆ5

**Figure 17.** Measure and estimate of speed during acceleration phase: *x*<sup>5</sup> and *x*ˆ5

**Figure 13.** Measure and estimate of speed during static phase: *x*<sup>5</sup> and *x*ˆ5

**Figure 14.** Measure and estimate of speed during static phase: *x*<sup>5</sup> and *x*ˆ5

**Figure 15.** Measure and estimate of speed during static phase: *x*<sup>5</sup> and *x*ˆ5

During static phase operation we saw that at low and very low speed , speed observation does not work very well. However on figures (17) and (18) we cross 0 speed, we denote a small divergence as small as the time to cross it; in fact this phenomenon underlines that speed is non observable with low current dynamic.

22 Will-be-set-by-IN-TECH

**Figure 13.** Measure and estimate of speed during static phase: *x*<sup>5</sup> and *x*ˆ5

**Figure 14.** Measure and estimate of speed during static phase: *x*<sup>5</sup> and *x*ˆ5

**Figure 15.** Measure and estimate of speed during static phase: *x*<sup>5</sup> and *x*ˆ5

During static phase operation we saw that at low and very low speed , speed observation does not work very well. However on figures (17) and (18) we cross 0 speed, we denote a small

**Figure 16.** Measure and estimate of speed during variable phase: *x*<sup>5</sup> and *x*ˆ5

**Figure 17.** Measure and estimate of speed during acceleration phase: *x*<sup>5</sup> and *x*ˆ5

Thus the practical results permit us to do a first assessment:

• we validate speed estimation during static and dynamic steps.

number could improve approximation of the continuous system .

In next step some tests will done to validate:

• Validation of hardiness to load variation.

*ECS - Lab, ENSEA and GS Maintenance, France*

• Validation in closed loop.

estimator.

**Author details**

**7. References**

Wiley-IEEE Press.

*Science* 38(10): 803–815.

*Conference on*, IEEE, pp. 1240–1245.

• we obtained an image of rotor flux(*x*3, and *x*4), and also rotor position.

• we validate our oversampling method introduced to overcome low speed data acquisition.

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

383

Compared with mechanical sensor the precision provides by the observer on the size speed offer a precision inferior or equal to 5% in the operating speed range from: 25% to 100%.

In term of prospects, it possible to improve the threshold of operation in low mode (25% to 5%) by adaptation oversampling number to stator frequency value, indeed a larger sample

About Observability loose at very low speed a first solution could be to switch with a speed

Sebastien Solvar, Malek Ghanes, Leonardo Amet, Jean-Pierre Barbot and Gaëtan Santomenna

[1] Aurora, C. & Ferrara, A. [2007]. A sliding mode observer for sensorless induction motor

[2] Bartolini, G., Ferrara, A. & Usani, E. [1998]. Chattering avoidance by second-order sliding

[3] Canudas De Wit, C., Youssef, A., Barbot, J., Martin, P. & Malrait, F. [2000]. Observability conditions of induction motors at low frequencies, *Decision and Control, 2000. Proceedings*

[4] Chiasson, J. [2005]. *Modeling and high performance control of electric machines*, Vol. 24,

[5] Davila, J., Fridman, L. & Levant, A. [2005]. Second-order sliding-mode observer for mechanical systems, *IEEE Transactions on Automatic Control* Vol. 50(11): 1785–1789. [6] Dib, A., Farza, M., MŠSaad, M., Dorléans, P. & Massieu, J. [2011]. High gain observer for

[7] Floquet, T. & Barbot, J. [2007]. Super twisting algorithm-based step-by-step sliding mode observers for nonlinear systems with unknown inputs, *International Journal of Systems*

[8] Ghanes, M., Barbot, J., De Leon, J. & Glumineau, A. [2010]. A robust sensorless output feedback controller of the induction motor drives: new design and experimental

[9] Ghanes, M., De Leon, J. & Glumineau, A. [2006]. Observability study and observer-based interconnected form for sensorless induction motor, *Decision and Control, 2006 45th IEEE*

speed regulation, *International Journal of Systems Science* 38(11): 913–929.

mode control, *Automatic control, IEEE Transactions on* 43(2): 241–246.

sensorless induction motor, *World Congress*, Vol. 18, pp. 674–679.

*of the 39th IEEE Conference on*, Vol. 3, IEEE, pp. 2044–2049.

validation, *International Journal of Control* 83(3): 484–497.

**Figure 18.** Measure and estimate of speed during deceleration phase: *x*<sup>5</sup> and *x*ˆ5

Figures 12 and 13 at high speed, show that speed approximation proposed in equation (51) work and permit to obtain magnitude and speed sign. This efficiency is also proved during dynamical phases as we can see on figure 16.

About this bad results, we have 2 arguments:


To overcome all this features, we propose to use an on-line resistor measurement of stator threw temperature.

## **6. Conclusion**

Through this chapter an original method of observation without mechanical sensors for induction machine was introduced.

Designed for a embedded system (VAR-CNTRL) equipped with a fixed point DSP, we carried out various tests of validation.

We used concept of Sliding Mode through Super Twisting Algorithm, and oversampling method being based on the explicit Euler development. The contribution of this paper is mainly based on the applicability of the proposed observer for sensorless induction motor when a basic microprocessors are used in an industrial context.

At the time of the setting works of our strategy some technical constraints brought us to introduce a news strategy.

Thus the practical results permit us to do a first assessment:


Compared with mechanical sensor the precision provides by the observer on the size speed offer a precision inferior or equal to 5% in the operating speed range from: 25% to 100%.

In term of prospects, it possible to improve the threshold of operation in low mode (25% to 5%) by adaptation oversampling number to stator frequency value, indeed a larger sample number could improve approximation of the continuous system .

In next step some tests will done to validate:


About Observability loose at very low speed a first solution could be to switch with a speed estimator.
