**1. Introduction**

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Newnes.

–827.

Recently, considerable research efforts are focused on the sensorless Induction Motors (*IM*) control problem. We refer the reader to [12] for a tutorial account on the topic. Indeed, industries concerned by sensorless *IM* drives are continuously seeking for cost reductions in their products. The main drawback of *IM* is the mechanical sensor. The use of such direct speed sensor induces additional electronics, extra wiring, extra space, frequent maintenance, careful mounting and default probability. Moreover, the sensor is vulnerable for electromagnetic noise in hostile environments and has a limited temperature range.

To avoid mechanical sensor (speed, position and load torque) of *IM*, several approaches for the so-called "sensorless control" have attracted a great deal of attention recently (see for example [21], [15], [22], [16], [11], [14], [6], [10], [1], [8], [19]. These methods can be classified into three main strategies.


This chapter belongs to the third strategy using mainly observer methods.

First and second strategies have been a subject of growing interest in recent years. For example the second strategy based on IM spatial saliency with extra converters is a robust and physical method. But artificial intelligence and spatial saliency algorithms are quite heavy for basic microprocessors.

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

As the mechanical position and magnetic variables are unknown, *d* − *q* frame is well

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

In order to construct the proposed observer for an industrial application, we work with a per

*x*˙1 = −*γ x*<sup>1</sup> + *θ* (*b x*<sup>3</sup> + *c x*5*x*4) + *ξ v*<sup>1</sup> *x*˙2 = −*γ x*<sup>2</sup> + *θ* (*b x*<sup>4</sup> − *c x*5*x*3) + *ξ v*<sup>2</sup>

(2)

361

*<sup>x</sup>*<sup>5</sup> <sup>=</sup> *<sup>p</sup>* <sup>Ω</sup> *ωref*

*x*˙5 = *h* (*x*3*x*<sup>2</sup> − *x*4*x*1) − *d x*<sup>5</sup> − *e Tl*

*<sup>x</sup>*<sup>3</sup> <sup>=</sup> *<sup>ω</sup>ref <sup>φ</sup>r<sup>α</sup> Vref*

*<sup>θ</sup>* <sup>=</sup> *KVref Iref ωref*

*<sup>τ</sup><sup>r</sup>* <sup>=</sup> *Lr Rr*

Thus for the sake of homogeneity, hereafter experimental results will be given in per-unit

*<sup>c</sup>* <sup>=</sup> *<sup>ω</sup>ref <sup>d</sup>* <sup>=</sup> *fv*

*<sup>x</sup>*<sup>4</sup> <sup>=</sup> *<sup>ω</sup>ref <sup>φ</sup>r<sup>β</sup> Vref*

*J*

*<sup>ξ</sup>* <sup>=</sup> *Vref σ LsIref*

*<sup>K</sup>* <sup>=</sup> *Msr σLsLr*

*x*˙3 = *a x*<sup>1</sup> − *b x*<sup>3</sup> − *c x*5*x*<sup>4</sup> *x*˙4 = *a x*<sup>2</sup> − *b x*<sup>4</sup> + *c x*5*x*<sup>3</sup>

appropriate for sensorless observer based control design.

*IM* **parameters:**

• *J*: inertia (*Kg*.*m*2).

• *vsα*,*β*: Stator voltage (V). • *isα*,*β*: Stator current (A). • *φrα*,*β*: Rotor flux (Wb).

• *Tl*: Load torque (Nm).

• Ω: Mechanical speed (*rad*/*s*).

With the following parameters:

*<sup>x</sup>*<sup>1</sup> <sup>=</sup> *is<sup>α</sup> Iref*

*<sup>e</sup>* <sup>=</sup> *<sup>p</sup> J ωref*

(p.u.).

*<sup>σ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>M</sup>*<sup>2</sup>

*sr LsLr*

*<sup>a</sup>* <sup>=</sup> *MsrIref <sup>ω</sup>ref TrVref*

unit model, under the following equations :

⎧ ⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

*<sup>x</sup>*<sup>2</sup> <sup>=</sup> *is<sup>β</sup> Iref*

*<sup>b</sup>* <sup>=</sup> <sup>1</sup> *τr*

*<sup>γ</sup>* <sup>=</sup> *RsL*<sup>2</sup>

*<sup>h</sup>* <sup>=</sup> *<sup>p</sup>*<sup>2</sup> *MsrIref Vref Jω*<sup>2</sup> *ref Lr*

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

*σLsL*<sup>2</sup> *r*

*IM* **variables :**

• *RS*: Stator resistance (*Ohms*). • *RR*: Rotor resistance (*Ohms*). • *LS*: Stator inductance (*Ohms*). • *LR*: Rotor inductance (*H*). • *LM*: Mutual inductance (*H*). • *p*: number of pole pairs.

• *f* : viscous friction coefficient (*Nm*.*s*/*rad*).

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

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 other methods.

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, an industrial application is proposed.

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 conclusions and remarks are drawn in section VII.
