Electric Motor Control

## **Chapter 4**

## Robust Mechanism for Speed and Position Observers of Electrical Machines

*Marcin Morawiec*

## **Abstract**

In the sensorless control system, the rotor speed or position is not measured but reconstructed in the dedicated observer structure. The observer structure is based on the mathematical model of an electrical machine. This model is often determined in the space vector form by using the stator/rotor flux vector and stator/rotor current vector components. During the machine works, there exist working points in which the observer can be unstable or its accuracy is unsatisfactory. In order to increase the observer system stability, the Lyapunov theorem should be satisfied. Using this, the observer system's proper stabilizing function can be determined. However, in many cases, this procedure is not sufficient and in close to an unstable region properties of the speed observer structure are very poor—the estimation errors have values exceeding 5%, which causes loss of synchronization in case of synchronous machines and errors in the values of electromagnetic torque or stator/rotor fluxes. In order to prevent this undesirable phenomenon, additional laws of estimation should be introduced to the speed or position estimation mechanism, which is proposed in this chapter. This mechanism is named in this chapter "robust" because during the machine works, it increases significantly the properties of the whole sensorless control system, minimizing the speed or position estimation errors almost to zero, close to the unstable region (small rotor speed with the regenerating machine mode or close to synchronous of rotor speed in case of the doubly fed generator). The proposed robust mechanism has been tested by using simulation and experimental investigations prepared for: the squirrel-cage induction machine, permanent magnet synchronous machine, and doubly fed induction generator.

**Keywords:** speed estimation, rotor position, adaptive, non-adaptive, induction machine, permanent magnet synchronous machine, doubly fed induction generator

## **1. Introduction**

In sensorless control of an electrical machine, the rotor speed value or rotor position is not measured but reconstructed by an observer structure. In the literature,

methods of reproducing the rotor speed or rotor position can be divided into three [1]: algorithmic, neural network, and physical methods. The most popular is an algorithmic method in which the observer structure is based on the mathematical model of an electrical machine. This group includes state full and reduced-order observers, [2], the adaptive full-order observer (AFO), [3], Kalman filters, [4], model reference adaptive observers MRAS, [5], sliding mode observers, [6], and backstepping, [6]. The other approach to the estimation of the state variables is to extend the model of a machine with an additional state variable—an auxiliary state, [7]. The rotor speed value in these observers can be reconstructed from the classical adaptation law by using the proportional-integral controller (PI), [1, 7, 8]. The rotor position value can be obtained by using the integration of the rotor speed value in the same integration step, [7, 8]. Other approach to the reconstruction of the rotor speed value is the nonadaptive method. The rotor speed value is obtained by using the suitable algebraic transformation of the estimated state variables, [5, 6].

The main problem in the sensorless control systems is the stability of the observer structure in the wide changes of working points of the machine, [8, 9]: from zero to nominal rotor speed, under load torque injections, and for regenerating mode. Stabilization of the observer structure under regenerating mode and low speed of the induction machine, IM, was studied in many papers, [9–11]. For this case, the frequency of stator voltage is almost zero, and there exist unstable poles of the observer system, [8, 10]. Similarly, the problem occurs for the permanent or interior permanent magnet synchronous machines (PMSM/IPMSM) during the zero rotor speed; while the electromagnetic force (EMF) is not generated, [12–14]. To overcome this problem, a different value of stator current or voltage (high [13] or low [14] frequency) is injected into the stator voltage from an inverter.

A robust mechanism for the rotor speed estimation is proposed in this chapter. The proposed approach is suitable for the speed observer structures, which are based on a mathematical model of an electrical machine (algorithmic) in the space vector form. In Section 2, the mathematical model of an electrical machine is considered in the general form for the nonlinear class of systems. In Section 3, the application to IM is shown. In Section 4, the speed observer of IPMSM with the robust mechanism is proposed. In Section 5, the robust mechanism for the rotor speed and position estimation is adapted to the observer structure of DFIG.

All the theoretical derivations are confirmed by using simulation and experimental investigations.

## **2. Design procedure of the speed and position observer**

One of the most popular design procedures for the speed observer of an electrical machine is based on the second theorem of the Lyapunov of asymptotical stability of the system in the general form

$$
\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u},\tag{1}
$$

$$\mathbf{y} = \mathbf{C}\mathbf{x},\tag{2}$$

where is assumed that *A*, *B,* and *C* are the matrixes that including the system parameters, *x* is the vector of state variables, and *u* is the vector of controls.

Considering only (1) the model can be rewritten to the vector components form in which (αβ) is the stationary reference frame and the index *k* means the number of the state variables in the system defined in (1)–(2)

$$
\dot{\mathfrak{x}}\_{ka} = a\_k \mathfrak{x}\_{ka} + \dots + b\_k \mathfrak{u}\_{ka}, \tag{3}
$$

$$
\dot{\mathfrak{x}}\_{k\beta} = a\_k \mathfrak{x}\_{k\beta} + \dots + b\_k \mathfrak{u}\_{k\beta}.\tag{4}
$$

If the system (3)–(4) will be connected to the rotating reference frame, then the differential equations have the form

$$
\dot{\mathfrak{x}}\_{kd} = a\_k \mathfrak{x}\_{kd} + a\_{dq} \mathfrak{x}\_{kq} + \dots + b\_k \mathfrak{u}\_{kd},\tag{5}
$$

$$
\dot{\mathfrak{X}}\_{kq} = a\_k \mathfrak{x}\_{kq} - a\_{dq} \mathfrak{x}\_{kd} + \dots + b\_k \mathfrak{u}\_{kq},\tag{6}
$$

where *ωdq* is the angular speed of the (d-q) reference frame, and (*ukd*, *ukq*) are the controls defined in (d-q).

It can be assumed that the system model belongs to the operation domain *D* defined by the set of values

$$\mathbf{D} = \left\{ \mathbf{x} \in \mathbb{R}^k, \ |\boldsymbol{x}\_{kd}| \le \boldsymbol{\kappa}\_{kd}^{\max}, \ |\boldsymbol{x}\_{kq}| \le \boldsymbol{\kappa}\_{kq}^{\max}, \ \boldsymbol{\alpha}\_{dq} \le \boldsymbol{\alpha}\_{dq}^{\max} \right\},\tag{7}$$

where

*x* max *kd* , *x* max *kq* , *ω* max *dq* are the maximum values for the state variables, and the parameters in the system *ak*, *bk* have known, constant, and bounded values.

*Assumption 1.* For the system (5)–(6) in which the *ωdq* is treated as the parameter, it is possible to reconstruct its value by using the adaptive and non-adaptive approaches and state of variables *xk*. Moreover, the controls (*ukd*, *ukq*) satisfied the persistent of excitation condition [14].

The first step in the procedure of design of the observer structure is to stabilize the observer for the system (5)–(6). The observer structure has the following form:

$$
\dot{\hat{\mathbf{x}}}\_{kd} = a\_k \hat{\mathbf{x}}\_{kd} + \hat{\boldsymbol{\alpha}}\_{dq} \hat{\mathbf{x}}\_{kq} + \dots + b\_k \boldsymbol{u}\_{kd} + \boldsymbol{v}\_d,\tag{8}
$$

$$
\dot{\hat{\mathbf{x}}}\_{kq} = a\_k \hat{\mathbf{x}}\_{kq} - \hat{\alpha}\_{dq} \hat{\mathbf{x}}\_{kd} + \dots + b\_k \boldsymbol{u}\_{kq} + \boldsymbol{v}\_q,\tag{9}
$$

where the estimated values are marked by "^", and *vd*, *vq* are the inputs to the observer (8)–(9), which stabilize the system.

The estimation errors between estimated (8)–(9) and real/measured values (5)–(6) are expressed by

$$
\tilde{\mathfrak{x}}\_{kd,q} = \hat{\mathfrak{x}}\_{kd,q} - \mathfrak{x}\_{kd,q},
\tag{10}
$$

$$
\tilde{\boldsymbol{\alpha}}\_{dq} = \hat{\boldsymbol{\alpha}}\_{dq} - \boldsymbol{\alpha}\_{dq}. \tag{11}
$$

For the above-defined estimation errors, it is possible to determine the model of estimation errors, which form is as follows:

$$
\dot{\tilde{\mathbf{x}}}\_{kd} = a\_k \tilde{\mathbf{x}}\_{kd} + \hat{\boldsymbol{\alpha}}\_{dq} \tilde{\mathbf{x}}\_{kq} + \tilde{\boldsymbol{\alpha}}\_{dq} \hat{\mathbf{x}}\_{kq} - \tilde{\boldsymbol{\alpha}}\_{dq} \tilde{\mathbf{x}}\_{kq} + \boldsymbol{\upsilon}\_{d},\tag{12}
$$

$$
\dot{\tilde{\mathbf{x}}}\_{kq} = \mathbf{a}\_k \tilde{\mathbf{x}}\_{kq} - \hat{\boldsymbol{\alpha}}\_{dq} \tilde{\mathbf{x}}\_{kd} - \tilde{\boldsymbol{\alpha}}\_{dq} \hat{\mathbf{x}}\_{kd} + \tilde{\boldsymbol{\alpha}}\_{dq} \tilde{\mathbf{x}}\_{kd} + \boldsymbol{\upsilon}\_q. \tag{13}
$$

The next step is to determine the form of stabilizing functions, which stabilize the observer structure (8)–(9). By using the Lyapunov theorem, the observer structure will be stable if the candidate of the Lyapunov function

$$\mathcal{V} = \mathbf{0}.\mathbf{5}\left(\tilde{\boldsymbol{\mathfrak{x}}}\_{kd}^{2} + \tilde{\boldsymbol{\mathfrak{x}}}\_{kq}^{2}\right) + \boldsymbol{V}\_{1} \succeq \mathbf{0},\tag{14}$$

is positively defined, and *V1* > 0 has the form

$$V\_1 = \chi^{-1} \tilde{o}\_{dq}^2. \tag{15}$$

Derivative of Lyapunov function (14) must be negative determined, therefore using (12)–(13), its form is determined

$$\begin{split} \dot{\mathcal{V}} &= \tilde{\mathbf{x}}\_{kd} \big( a\_k \tilde{\mathbf{x}}\_{kd} + \hat{\boldsymbol{\alpha}}\_{dq} \tilde{\mathbf{x}}\_{kq} + \boldsymbol{\nu}\_d \big) + \tilde{\mathbf{x}}\_{kq} \big( a\_k \tilde{\mathbf{x}}\_{kq} - \hat{\boldsymbol{\alpha}}\_{dq} \tilde{\mathbf{x}}\_{kd} + \boldsymbol{\nu}\_q \big) \\ &+ \tilde{\boldsymbol{\alpha}}\_{dq} \big( \mathbf{y}^{-1} \dot{\tilde{\mathbf{o}}}\_{dq} + \hat{\mathbf{x}}\_{kq} \tilde{\mathbf{x}}\_{kd} - \hat{\mathbf{x}}\_{kd} \tilde{\mathbf{x}}\_{kq} \big) \leq \mathbf{0}. \end{split} \tag{16}$$

The observer structure will be asymptotically stable if the Lyapunov theorem is satisfied and the stabilizing functions are chosen

$$
\sigma\_d = -\mathfrak{a}\_k \tilde{\mathfrak{x}}\_{kd},
\tag{17}
$$

$$
\upsilon\_q = -a\_k \tilde{\chi}\_{kq},
\tag{18}
$$

then derivative (16) has the form

$$
\dot{V} = \ddot{a}\_{dq} \left( \hat{\mathbf{x}}\_{kq} \ddot{\mathbf{x}}\_{kd} - \hat{\mathbf{x}}\_{kd} \ddot{\mathbf{x}}\_{kq} \right) \le \mathbf{0}. \tag{19}
$$

To satisfy (19), the value of the parameter *ω*~*dq* should be determined by

$$\dot{\tilde{\phi}}\_{dq} = -\gamma \left( \hat{\mathbf{x}}\_{kq} \tilde{\mathbf{x}}\_{kd} - \hat{\mathbf{x}}\_{kd} \tilde{\mathbf{x}}\_{kq} \right), \tag{20}$$

where

*γ* > 0 is the tuning gain.

For (20), the derivative of the Lyapunov function is always smaller than zero *V*\_ <0, and the Lyapunov condition is satisfied.

## **2.1 Adaptive estimation of parameter** *ωdq*

The estimated value of the parameter *ω*^*dq* can be determined from (20) under assumption that the derivative of the real value is constant in time *ω*\_ *dq* and equal to zero

$$
\dot{\hat{\boldsymbol{\alpha}}}\_{dq} = -\gamma \left( \hat{\boldsymbol{\kappa}}\_{kq} \tilde{\boldsymbol{\kappa}}\_{kd} - \hat{\boldsymbol{\kappa}}\_{kd} \tilde{\boldsymbol{\kappa}}\_{kq} \right). \tag{21}
$$

The above estimation law is named in the literature [15] as the classical adaptation law.

*Remark 1*. The assumption *ω*\_ *dq* ¼ 0 is not desirable for the nonlinear system, in which the highest accuracy of estimation is needed. For *ω*\_ *dq* 6¼ 0 and *ωdq* ¼ *ω*^*dq* � *ω*~*dq*, after substitution (17)–(18) to (16), the derivative of the Lyapunov function has the following form:

*Robust Mechanism for Speed and Position Observers of Electrical Machines DOI: http://dx.doi.org/10.5772/intechopen.107898*

$$\dot{V} = \ddot{\alpha}\_{dq} \left( -\frac{1}{\chi} (\dot{\hat{\alpha}}\_{dq} - \dot{\hat{\alpha}}\_{dq}) \right) \le \mathbf{0}. \tag{22}$$

After substitution (21) to (22), the update form of derivative of the Lyapunov function is achieved

$$\dot{V} = -\ddot{\boldsymbol{\alpha}}\_{dq} \left( -\left( \hat{\boldsymbol{\alpha}}\_{kq} \ddot{\boldsymbol{\alpha}}\_{kd} - \hat{\boldsymbol{\alpha}}\_{kd} \ddot{\boldsymbol{\alpha}}\_{kq} \right) + \frac{1}{\chi} \dot{\ddot{\boldsymbol{\alpha}}}\_{dq} \right) \leq \mathbf{0}. \tag{23}$$

It is easy to check that in (23) the dependence in the internal bracket ð Þ¼ x^*<sup>k</sup>* � x~*<sup>k</sup> x*^*kqx*~*kd* � *x*^*kdx*~*kq* means the cross-product of the pair of two vectors that occur in the observer system. The cross-product for can be determined by using Lagrange's identity [16].

*Assumption 2.* Considering the pair of vectors ð Þ x^*k*, x~*<sup>k</sup>* defined in the observer system (8)–(9) and for the estimation errors (10)–(13), there exists Lagrange's identity [16], which has the following form:

$$\left(\hat{\mathbf{x}}\_{k}\times\bar{\mathbf{x}}\_{k}\right)^{2}\equiv|\hat{\mathbf{x}}\_{k}|^{2}|\tilde{\mathbf{x}}\_{k}|^{2}-\left(\hat{\mathbf{x}}\_{k}\cdot\bar{\mathbf{x}}\_{k}\right)^{2}.\tag{24}$$

Considering the vector components defined in (d-q) reference frame (24) can be rewritten as

$$
\tilde{\mathbf{x}}\_{kd}\hat{\mathbf{x}}\_{kq} - \tilde{\mathbf{x}}\_{kq}\hat{\mathbf{x}}\_{kd} = \sqrt{\left(\hat{\mathbf{x}}\_{kd}^2 + \hat{\mathbf{x}}\_{kq}^2\right)\left(\hat{\mathbf{x}}\_{kd}^2 + \hat{\mathbf{x}}\_{kq}^2\right) - \left(\hat{\mathbf{x}}\_{kd}\tilde{\mathbf{x}}\_{kd} + \hat{\mathbf{x}}\_{kq}\tilde{\mathbf{x}}\_{kq}\right)^2}.\tag{25}
$$

Substituting (25) to (23), the derivative of the Lyapunov function has the following form:

$$\dot{\mathcal{V}} = \ddot{\boldsymbol{\alpha}}\_{dq} \left( \sqrt{\left( \hat{\boldsymbol{\chi}}\_{kd}^{2} + \hat{\boldsymbol{\chi}}\_{dq}^{2} \right) \left( \tilde{\mathbf{x}}\_{kd}^{2} + \tilde{\mathbf{x}}\_{dq}^{2} \right) - \left( \hat{\boldsymbol{\chi}}\_{kd} \tilde{\mathbf{x}}\_{kd} + \hat{\boldsymbol{\chi}}\_{dq} \tilde{\mathbf{x}}\_{dq} \right)^{2}} - \frac{1}{\chi} \dot{\tilde{\boldsymbol{\alpha}}}\_{dq} \right) \le \mathbf{0}. \tag{26}$$

The Lyapunov theorem is satisfied if

$$
\dot{\tilde{\alpha}}\_{dq} = \chi \sqrt{\left(\hat{\mathfrak{x}}\_{kd}^2 + \hat{\mathfrak{x}}\_{kq}^2\right) \left(\tilde{\mathcal{X}}\_{kd}^2 + \tilde{\mathfrak{x}}\_{kq}^2\right) - \left(\hat{\mathfrak{x}}\_{kd}\tilde{\mathfrak{x}}\_{kd} + \hat{\mathfrak{x}}\_{kq}\tilde{\mathfrak{x}}\_{kq}\right)^2},\tag{27}
$$

where *x*^<sup>2</sup> *kd* <sup>þ</sup> *<sup>x</sup>*^<sup>2</sup> *kq* � � *<sup>x</sup>*~<sup>2</sup> *kd* <sup>þ</sup> *<sup>x</sup>*~<sup>2</sup> *kq* � � � *<sup>x</sup>*^*kdx*~*kd* <sup>þ</sup> *<sup>x</sup>*^*kqx*~*kq* � �<sup>2</sup> � � <sup>≥</sup>0.

*Remark 2:* To satisfy the above condition, the negative sign-in (27) must be changed to positive. The form (27) is determined as follows:

$$
\dot{\tilde{\alpha}}\_{dq} = \chi \sqrt{\left(\hat{\mathfrak{x}}\_{kd}^2 + \hat{\mathfrak{x}}\_{kq}^2\right) \left(\tilde{\mathbf{x}}\_{kd}^2 + \tilde{\mathbf{x}}\_{kq}^2\right) + \left(\hat{\mathfrak{x}}\_{kd}\tilde{\mathbf{x}}\_{kd} + \hat{\mathfrak{x}}\_{kq}\tilde{\mathbf{x}}\_{kq}\right)^2}. \tag{28}
$$

Dependence (28) can be used to find the updated form of the classical estimation law (21). It provides an improvement to the stability range of the observer system.

*Assumption 3.* The expression (21) has the form of an open integrator. There is a lack of additional stabilizing function, interconnecting the observer system. To improve the stability range of the observer system, it is proposed to introduce additional input *sω*

$$
\dot{\hat{\alpha}}\_{dq} = -\gamma \left( \hat{\mathbf{x}}\_{kq} \tilde{\mathbf{x}}\_{kd} - \hat{\mathbf{x}}\_{kd} \tilde{\mathbf{x}}\_{kq} \right) + \mathfrak{s}\_{\alpha}.\tag{29}
$$

To stabilize the integrator (29), the stabilization function *<sup>s</sup><sup>ω</sup>* should be *<sup>s</sup><sup>ω</sup>* � *<sup>ω</sup>*~\_ *dq*. The updated estimation law has the following form:

$$\dot{\hat{\boldsymbol{\alpha}}}\_{dq} = -\gamma \left( \hat{\boldsymbol{\alpha}}\_{kq} \ddot{\boldsymbol{\alpha}}\_{kd} - \hat{\boldsymbol{\alpha}}\_{kd} \ddot{\boldsymbol{\alpha}}\_{kq} + \gamma\_1 \mathbf{k}\_f \sqrt{\left( \dot{\hat{\boldsymbol{\alpha}}}\_{kd}^2 + \dot{\hat{\boldsymbol{\alpha}}}\_{kq}^2 \right) \left( \ddot{\hat{\boldsymbol{\alpha}}}\_{kd}^2 + \ddot{\hat{\boldsymbol{\alpha}}}\_{kq}^2 \right) + \left( \hat{\boldsymbol{\alpha}}\_{kd} \ddot{\boldsymbol{\alpha}}\_{kd} + \hat{\boldsymbol{\alpha}}\_{kq} \ddot{\boldsymbol{\alpha}}\_{kq} \right)^2} \right), \tag{30}$$

where *<sup>γ</sup><sup>1</sup>* is the additional gain, and *kf* <sup>¼</sup> *sign <sup>ω</sup>*^*dq* � � is the sign of the estimated parameter.

*Remark 3*. Under the assumption that in

(30) *x*^<sup>2</sup> *kd* <sup>þ</sup> *<sup>x</sup>*^<sup>2</sup> *kq* � � *<sup>x</sup>*~<sup>2</sup> *kd* <sup>þ</sup> *<sup>x</sup>*~<sup>2</sup> *kq* � � <sup>≪</sup> *<sup>x</sup>*^*kdx*~*kd* <sup>þ</sup> *<sup>x</sup>*^*kqx*~*kq* � �<sup>2</sup> the update estimation law can be simplified to the following form

$$\dot{\hat{\alpha}}\_{dq} = -\gamma \left( \hat{\mathbf{x}}\_{kq} \tilde{\mathbf{x}}\_{kd} - \hat{\mathbf{x}}\_{kd} \tilde{\mathbf{x}}\_{kq} + \gamma\_1 \mathbf{k}\_f \mathbf{s}\_{af} \right), \tag{31}$$

where *s<sup>ω</sup>* ¼ *x*^*kdx*~*kd* þ *x*^*kqx*~*kq*, and *sω<sup>f</sup>* is their filtered value by using a low-pass filter LPF (to avoid the algebraic loop).

In (31), there is the cross and scalar product ð Þ¼ x^*<sup>k</sup>* � x~*<sup>k</sup> x*^*kdx*~*kd* þ *x*^*kqx*~*kq* of two vectors. It is worth noticing that for the perpendicular vectors, the scalar product is equal to zero; however, in other cases, it is different from zero and additionally stabilizes the estimation law.

## **2.2 Non-adaptive estimation of parameter** *ωdq*

In the previous section, the parameter *ωdq* was reconstructed from the adaptive law. However, this value can be estimated non-adaptively. Under the assumption of the steady-state for *ak*≈1, *ω*~*dq*≈0 and *vd, q* = 0, from the model of estimation error, the following approximations can be achieved:

$$
\hat{\mathfrak{X}}\_{kd} \approx \hat{\alpha}\_{dq} \hat{\mathfrak{X}}\_{kq},\tag{32}
$$

$$
\tilde{\mathfrak{X}}\_{kq} \approx -\,\hat{\alpha}\_{dq}\hat{\mathfrak{X}}\_{kd},\tag{33}
$$

for whose the following relationships are satisfied

$$
\hat{\boldsymbol{\mathfrak{x}}}\_{kd}^2 + \tilde{\boldsymbol{\mathfrak{x}}}\_{kq}^2 = \hat{\boldsymbol{\alpha}}\_{dq}^2 \left( \hat{\boldsymbol{\mathfrak{x}}}\_{kd}^2 + \hat{\boldsymbol{\mathfrak{x}}}\_{kq}^2 \right), \tag{34}
$$

$$
\hat{\alpha}\_{dq} = \frac{\tilde{\mathcal{X}}\_{kd}\hat{\mathfrak{X}}\_{kq} - \tilde{\mathcal{X}}\_{kq}\hat{\mathfrak{X}}\_{kd}}{\hat{\mathfrak{X}}\_{kd}^2 + \hat{\mathfrak{X}}\_{kq}^2},
\tag{35}
$$

where *x*^<sup>2</sup> *kd* <sup>þ</sup> *<sup>x</sup>*^<sup>2</sup> *kq* 6¼ 0.

Substituting (34)–(35) to (24), the following quadratic function is obtained:

$$f\left(\hat{\boldsymbol{\alpha}}\_{dq}\right) = -\left(\hat{\mathbf{x}}\_{kd}^{2} + \hat{\mathbf{x}}\_{kq}^{2}\right)\left(\hat{\mathbf{x}}\_{kd}^{2} + \hat{\mathbf{x}}\_{kq}^{2}\right)\hat{\boldsymbol{\alpha}}\_{dq}^{2} + \hat{\boldsymbol{\alpha}}\_{dq}\left(\hat{\mathbf{x}}\_{kd}^{2} + \hat{\mathbf{x}}\_{kq}^{2}\right)\left(\bar{\mathbf{x}}\_{kd}\hat{\mathbf{x}}\_{kq} - \bar{\mathbf{x}}\_{kq}\hat{\mathbf{x}}\_{kd}\right) + \tag{36}$$
 
$$\left(\hat{\mathbf{x}}\_{kd}\bar{\mathbf{x}}\_{kd} + \hat{\mathbf{x}}\_{kq}\bar{\mathbf{x}}\_{kq}\right)^{2}.$$

One of the roots of the function *f ω*^*dq* � � can be calculated as follows:

$$\hat{\boldsymbol{\alpha}}\_{dq} = \frac{\tilde{\mathbf{x}}\_{kd}\hat{\mathbf{x}}\_{kq} - \tilde{\mathbf{x}}\_{kq}\hat{\mathbf{x}}\_{kd} + \mathbb{I}\_{\tilde{f}}\sqrt{\left(\left(\tilde{\mathbf{x}}\_{kd}\hat{\mathbf{x}}\_{kq} - \tilde{\mathbf{x}}\_{kq}\hat{\mathbf{x}}\_{kd}\right)^{2} + 4\mathbf{y}\_{1}\left(\hat{\mathbf{x}}\_{kd}\tilde{\mathbf{x}}\_{kd} + \hat{\mathbf{x}}\_{kq}\tilde{\mathbf{x}}\_{kq}\right)^{2}\right)}{2\left(\hat{\mathbf{x}}\_{kd}^{2} + \hat{\mathbf{x}}\_{kq}^{2}\right)}},\tag{37}$$

where *<sup>γ</sup><sup>1</sup>* is the additional tuning gain and *kf* <sup>¼</sup> *sign <sup>ω</sup>*^*dq* � �.

#### **2.3 Practical stability of the observer system**

The practical stability of the observer system was proposed in [17, 18]. Based on the theorem of practical stability and considering that the system belongs to domain *D* defined in (7), the observer structure will be practical stable in the Lyapunov function derivative is

$$
\dot{V} = \delta\_1 |\tilde{\mathbf{x}}\_{kd}| + \delta\_2 |\tilde{\mathbf{x}}\_{kq}| + \delta\_c |\tilde{\boldsymbol{\alpha}}\_{dq}| \le -\mu V + \kappa,\tag{38}
$$

where (*δ*1, *δ*2,*δc*) > 0 and *x*~*kd* ≤*ε*1, *x*~*kd* ≤*ε*2,*ω*~*<sup>r</sup>* ≤*ε*3, *ε*1,2,3 ≪ 1 are sufficient small real numbers *ε*1,2,3 > 0 and where

$$\gamma\_1 = \max \left\{ \frac{\left(\hat{\mathbf{x}}\_{kq}\bar{\mathbf{x}}\_{kd} - \hat{\mathbf{x}}\_{kd}\bar{\mathbf{x}}\_{kq}\right)}{\sqrt{\left(\hat{\mathbf{x}}\_{kd}^2 + \hat{\mathbf{x}}\_{kq}^2\right)\left(\hat{\mathbf{x}}\_{kd}^2 + \hat{\mathbf{x}}\_{kq}^2\right) + \left(\hat{\mathbf{x}}\_{kd}\bar{\mathbf{x}}\_{kd} + \hat{\mathbf{x}}\_{kq}\bar{\mathbf{x}}\_{kq}\right)^2}} + \delta\_\varepsilon \right\},\tag{39}$$

and

$$\mu = \min \left( \delta\_1 - \frac{1}{2\xi\_1^2}, \delta\_2 - \frac{1}{2\xi\_2^2}, \sqrt{2}\delta\_\epsilon \right), \kappa = 0.5(\xi\_1^2 \eta\_1^2 + \xi\_2^2 \eta\_2^2) \ \forall \xi\_i \in (0, 1), i = 1, 2 \tag{40}$$

Hence, (38) implies the convergence of estimated vector values to their real, in finite time, noted as *t1*. The reconstructed parameter *ω*^*dq* converges exponentially to real *ωdq* in finite time *t* > *t2* > *t1*. This condition is satisfied for ideal and constant parameters of the system (3)–(4). According to [17, 18], the tracking errors converge to the ball of radius *κ=μ*. This radius can be decreased by the properly choosing tuning gains of the observer system (8)–(9).

### **2.4 Conclusion**

Presented in Section 2 is the design of the observer structure generalized to the class of system (3)–(4) in the space vector form. The form of the system (3)–(4) was in α-β stationary reference frame. It has been appropriately transformed by using Clark's transformation to the rotational reference frame d-q. In system (5)– (6), there exists the parameter, which is the angular speed of the reference frame in d-q. The system (5)–(6) has been properly written with a separate parameter and has a similar form to an AC electrical machines models presented in the next sections. Therefore, the proposed procedure in Section 2 for designing the observer structure can be directly adapted to the sensorless control system of an AC electrical machine. The proposed solution is based on the classical adaptation law of estimation and non-adaptive. According to the literature, [1, 2, 6–12], the rotor speed whose value is estimated only from the classical law of adaptation and used to tune the observer structure (8)–(9) can lead to instability during the regenerating mode and low speed of the electrical machine. There exist positive poles of the observer structure for which it is unstable. The problem in the classical law of adaptation is the open form of the integrator (21) from which the value of rotor speed is estimated (in the case of an electrical machine). Therefore, in Section 2, the additional stabilization function is introduced also to the classical law of estimation. The proposed stabilization function is based on Lagrange's identity of the pair of vectors in the observer system. The form of additional stabilization law contains the scalar product and the length of the vectors. However, after the simplification shown in Remark 3, it can be assumed that the stabilization function is proportional to the scalar product of the chosen vectors that were presented in [6].

The proposed theoretical issues in Section 2 will be confirmed in the simulation and experimental results for the squirrel-cage induction machine and interior permanent magnet synchronous machine. Also, it can be extended to estimation of the state variables of the doubly fed induction generator (DFIG).

## **3. The speed observer structures of the squirrel-cage induction machine**

The AFO speed observer structure of IM is proposed in this section. The rotor speed will be estimated by using two approaches: from the adaptive estimation law and non-adaptively.

Considering the mathematical model of the induction machine presented in [5, 6], for the pair of vectors ψ*r*, i ð Þ*<sup>s</sup>* according to (8)–(9), the conventional AFO observer structure can be determined in the form

$$\frac{d\hat{\mathbf{i}}\_{sa}}{d\tau} = a\_1 \hat{\mathbf{i}}\_{sa} + a\_2 \hat{\boldsymbol{\upmu}}\_{ra} + a\_3 \hat{\boldsymbol{\upmu}}\_r \hat{\boldsymbol{\upmu}}\_{r\boldsymbol{\upbeta}} + a\_4 \boldsymbol{\upmu}\_{sa} + v\_a,\tag{41}$$

$$\frac{d\hat{\mathbf{i}}\_{t\boldsymbol{\beta}}}{d\tau} = a\_1 \hat{\mathbf{i}}\_{t\boldsymbol{\beta}} + a\_2 \hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} - a\_3 \hat{\boldsymbol{\alpha}}\_r \hat{\boldsymbol{\mu}}\_{ra} + a\_4 \boldsymbol{\mu}\_{t\boldsymbol{\beta}} + v\_{\boldsymbol{\beta}},\tag{42}$$

$$\frac{d\hat{\boldsymbol{\mu}}\_{ra}}{d\tau} = \boldsymbol{a}\_{\mathcal{S}}\hat{\boldsymbol{\mu}}\_{ra} - \hat{\boldsymbol{\alpha}}\_{r}\hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} + \boldsymbol{a}\_{\mathcal{G}}\hat{\boldsymbol{i}}\_{sa} + \boldsymbol{v}\_{\boldsymbol{\mu}a},\tag{43}$$

$$\frac{d\hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}}}{d\boldsymbol{\tau}} = \boldsymbol{a}\_{\mathcal{S}}\hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} + \hat{\boldsymbol{\alpha}}\_{r}\hat{\boldsymbol{\mu}}\_{r\boldsymbol{\alpha}} + \boldsymbol{a}\_{\mathcal{G}}\hat{\boldsymbol{i}}\_{s\boldsymbol{\beta}} + \boldsymbol{v}\_{\boldsymbol{\omega}\boldsymbol{\beta}},\tag{44}$$

where the estimated values are marked by "^".

It is assumed that the stator current vector ^*isα*,*β*, rotor flux vector *ψ*^*r<sup>α</sup>*,*<sup>β</sup>* components, and rotor speed *ω*^*<sup>r</sup>* are estimated in the observer structure (41)–(44), *vα,β,* and *vψα,<sup>β</sup>* are stabilizing functions introduced to the structure. The values *isα*,*<sup>β</sup>* are available in measurement and *usα*,*<sup>β</sup>* are treated as the known variables (from the control system structure of the machine). The machine parameters are included in

$$\mathfrak{a}\_1 = -\frac{R\_r L\_r^2 + R\_r L\_m^2}{L\_r w\_\sigma}, \mathfrak{a}\_2 = \frac{R\_r L\_m}{L\_r w\_\sigma}, \mathfrak{a}\_3 = \frac{L\_m}{w\_\sigma}, \mathfrak{a}\_4 = \frac{L\_r}{w\_\sigma}, \mathfrak{a}\mathfrak{s} = -\frac{R\_r}{L\_r}, \mathfrak{a}\mathfrak{s} = \frac{R\_r L\_m}{L\_r}, \mathfrak{a}\mathfrak{s}$$

*Robust Mechanism for Speed and Position Observers of Electrical Machines DOI: http://dx.doi.org/10.5772/intechopen.107898*

$$
\omega\_{\sigma} = L\_r L\_s - L\_m^2. \tag{45}
$$

The estimation errors for the observer system (41)–(44) are defined.

$$
\tilde{\boldsymbol{\alpha}}\_{\tau} = \hat{\boldsymbol{\alpha}}\_{\tau} - \boldsymbol{\alpha}\_{r}, \tilde{\boldsymbol{\mu}}\_{ra,\theta} = \hat{\boldsymbol{\psi}}\_{ra,\theta} - \boldsymbol{\psi}\_{ra,\theta} \tilde{\mathbf{i}}\_{ra,\theta} = \hat{\mathbf{i}}\_{ra,\theta} - \mathbf{i}\_{sa,\theta} \tag{46}
$$

where it is assumed that components *isα*,*β*, *ψr<sup>α</sup>*,*β*, *ω<sup>r</sup>* are the real values.

The rotor speed value *ω*^*<sup>r</sup>* will be reconstructed adaptively and non-adaptively by using the observer structure (41)–(44) and based on the measurements *isα*,*β*, and *usα*,*β*.

Using the design procedure presented in Section 2, the model of estimation errors is as follows:

$$\frac{d\tilde{\mathbf{i}}\_{sa}}{d\tau} = a\_1 \tilde{\mathbf{i}}\_{sa} + a\_2 \tilde{\boldsymbol{\upmu}}\_{ra} + a\_3 \left( \tilde{\boldsymbol{\upmu}}\_r \hat{\boldsymbol{\upmu}}\_{r\boldsymbol{\upbeta}} + \hat{\boldsymbol{\upmu}}\_r \tilde{\boldsymbol{\upmu}}\_{r\boldsymbol{\upbeta}} - \tilde{\boldsymbol{\upalpha}}\_r \tilde{\boldsymbol{\upmu}}\_{r\boldsymbol{\upbeta}} \right) + \boldsymbol{\upnu}\_a,\tag{47}$$

$$\frac{d\tilde{\mathbf{i}}\_{s\boldsymbol{\beta}}}{d\boldsymbol{\pi}} = \mathbf{a}\_1 \tilde{\mathbf{i}}\_{s\boldsymbol{\beta}} + \mathbf{a}\_2 \tilde{\boldsymbol{\psi}}\_{r\boldsymbol{\beta}} - \mathbf{a}\_3 (\tilde{\boldsymbol{\alpha}}\_r \hat{\boldsymbol{\psi}}\_{ra} + \hat{\boldsymbol{\alpha}}\_r \tilde{\boldsymbol{\psi}}\_{ra} - \tilde{\boldsymbol{\alpha}}\_r \tilde{\boldsymbol{\psi}}\_{ra}) + \boldsymbol{\nu}\_{\boldsymbol{\beta}},\tag{48}$$

$$\frac{d\tilde{\boldsymbol{\mu}}\_{ru}}{d\boldsymbol{\tau}} = \boldsymbol{a}\_{\xi}\tilde{\boldsymbol{\mu}}\_{ru} - \left(\tilde{\boldsymbol{\alpha}}\_{r}\hat{\boldsymbol{\nu}}\_{r\beta} + \hat{\boldsymbol{\alpha}}\_{r}\tilde{\boldsymbol{\nu}}\_{r\beta} - \tilde{\boldsymbol{\alpha}}\_{r}\tilde{\boldsymbol{\nu}}\_{r\beta}\right) + \boldsymbol{a}\_{\xi}\tilde{\boldsymbol{\xi}}\_{ra} + \boldsymbol{\nu}\_{\boldsymbol{\nu}a},\tag{49}$$

$$\frac{d\tilde{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}}}{d\boldsymbol{\tau}} = \boldsymbol{a}\_{\boldsymbol{\delta}}\tilde{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} + \left(\tilde{\boldsymbol{\alpha}}\_{r}\hat{\boldsymbol{\nu}}\_{ra} + \hat{\boldsymbol{\alpha}}\_{r}\tilde{\boldsymbol{\nu}}\_{ra} - \tilde{\boldsymbol{\alpha}}\_{r}\tilde{\boldsymbol{\nu}}\_{ra}\right) + \boldsymbol{a}\_{\boldsymbol{\delta}}\tilde{\boldsymbol{i}}\_{\boldsymbol{\beta}\boldsymbol{\beta}} + \boldsymbol{\nu}\_{\boldsymbol{\nu}\boldsymbol{\beta}}.\tag{50}$$

The Lyapunov function defined for the estimation errors has the form

$$\mathbf{V} = \frac{\mathbf{1}}{2} \left( \hat{\boldsymbol{i}}\_{s\boldsymbol{a}}^{2} + \hat{\boldsymbol{i}}\_{s\boldsymbol{\beta}}^{2} + \tilde{\boldsymbol{\nu}}\_{r\boldsymbol{a}}^{2} + \tilde{\boldsymbol{\nu}}\_{r\boldsymbol{\beta}}^{2} \right) + \mathbf{V}\_{1} > \mathbf{0},\tag{51}$$

where for the non-adaptive speed estimation *V*<sup>1</sup> ¼ 0, and in (47)–(50), *ω*~*<sup>r</sup>* ¼ 0 under the assumption (32)–(33), for the case of adaptive law of estimation *<sup>V</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> *<sup>γ</sup> ω*~<sup>2</sup> *r*.

The derivative of the Lyapunov function will be negatively determined *V*\_ <0 if the stabilizing functions are chosen.

$$
\sigma\_a = -\mathbf{c}\_a \tilde{\mathbf{i}}\_{sa}, \sigma\_\beta = -\mathbf{c}\_a \tilde{\mathbf{i}}\_\beta \tag{52}
$$

$$
\upsilon\_{\varphi a} = -\mathfrak{c}\_{\varphi 1}\tilde{\mathfrak{i}}\_{s\alpha} + \mathfrak{c}\_{\varphi}\hat{\alpha}\_r \tilde{\mathfrak{i}}\_{r\beta}, \\
\upsilon\_{\varphi\beta} = -\mathfrak{c}\_{\varphi 1}\tilde{\mathfrak{i}}\_{s\beta} - \mathfrak{c}\_{\varphi}\hat{\alpha}\_r \tilde{\mathfrak{i}}\_{s\alpha} \tag{53}
$$

where it is assumed (*c<sup>α</sup>* ¼ *f a*ð Þ<sup>1</sup> >0) as well as *c<sup>ψ</sup>* ¼ *f a*ð Þ<sup>3</sup> >0, whereas *c<sup>ψ</sup>*<sup>1</sup> ≥0, *a5* < 0.

The rotor speed value can be estimated directly from the adaptive estimation law (20) presented in Section 2.1, considering the pair of vectors ð Þ� <sup>x</sup>^*k*, <sup>x</sup>~*<sup>k</sup>* <sup>ψ</sup>^*r*, <sup>~</sup>i*<sup>s</sup>* � �

$$\dot{\hat{a}}\_{\hat{r}} = -\gamma \left( \tilde{\mathbf{i}}\_{\text{tar}} \hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} - \tilde{\mathbf{i}}\_{\text{t}\boldsymbol{\beta}} \hat{\boldsymbol{\mu}}\_{ra} + \boldsymbol{\gamma}\_1 \mathbf{k}\_f \hat{\boldsymbol{s}}\_{\boldsymbol{\alpha}} \right), \tag{54}$$

where the robust term ^*s<sup>ω</sup>* is given from

$$
\hat{s}\_{o} = \sqrt{\left(\hat{\boldsymbol{\upmu}}\_{ra}^{2} + \hat{\boldsymbol{\upmu}}\_{r\boldsymbol{\upbeta}}^{2}\right) \left(\hat{\boldsymbol{i}}\_{sa}^{2} + \hat{\boldsymbol{i}}\_{s\boldsymbol{\upbeta}}^{2}\right) + \left(\hat{\boldsymbol{\upmu}}\_{ra}\hat{\boldsymbol{i}}\_{sa} + \hat{\boldsymbol{\upmu}}\_{r\boldsymbol{\upbeta}}\hat{\boldsymbol{i}}\_{s\boldsymbol{\upbeta}}\right)^{2}}{\left(\hat{\boldsymbol{i}}\_{s}\right)^{2} + \left(\hat{\boldsymbol{i}}\_{s}\right)^{2}}},\tag{55}
$$

and *γ<sup>1</sup>* > 0 is the additional gain, *kf* ¼ *sign*ð Þ *ω*^*<sup>r</sup>* .

Considering the non-adaptive scheme for rotor speed estimation presented Section 2.2. For the pair of vectors ð Þ� <sup>x</sup>^*k*, <sup>x</sup>~*<sup>k</sup>* <sup>ψ</sup>^*r*, <sup>~</sup>i*<sup>s</sup>* � �, the rotor speed value can be estimated from

$$
\hat{\boldsymbol{\alpha}}\_r = \frac{\mathbf{\tilde{i}}\_{\rm tr} \hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} - \mathbf{\tilde{i}}\_{r\boldsymbol{\beta}} \hat{\boldsymbol{\mu}}\_{r\boldsymbol{\alpha}} + k\_f \hat{\boldsymbol{s}}\_{\boldsymbol{\alpha}}}{2 \left( \hat{\boldsymbol{\mu}}\_{r\boldsymbol{\alpha}}^2 + \hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}}^2 \right)}, \tag{56}
$$

where

$$
\hat{\mathbf{s}}\_{av} = \sqrt{\left(\hat{\mathbf{i}}\_{s\mathbf{r}}\hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} - \hat{\mathbf{i}}\_{s\boldsymbol{\beta}}\hat{\boldsymbol{\mu}}\_{ra}\right)^2 + 4\gamma\_1 \left(\hat{\boldsymbol{\mu}}\_{ra}\hat{\mathbf{i}}\_{s\mathbf{a}} + \hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}}\hat{\boldsymbol{\mathbf{i}}}\_{s\boldsymbol{\beta}}\right)^2}.\tag{57}$$

The proposed AFO speed observer was tested on the 5.5 kW induction machine, which was clutched to DC motor. The sensorless control system structure was based on feedback control with the multi-scalar variables shown in [5, 6]. The control system contains four PI controllers of the rotor speed, electromagnetic torque *Te*, square of rotor flux *ψ*<sup>2</sup> *<sup>r</sup>* <sup>¼</sup> *<sup>ψ</sup>*<sup>2</sup> *<sup>r</sup><sup>α</sup>* <sup>þ</sup> *<sup>ψ</sup>*<sup>2</sup> *<sup>r</sup><sup>β</sup>*, and the variables *x*<sup>22</sup> ¼ *ψ*^*r<sup>α</sup>* ^*is<sup>α</sup>* <sup>þ</sup> *<sup>ψ</sup>*^*r<sup>β</sup>* ^*isβ*.

The control system was implemented in the interface with a DSP Sharc ADSP21363 floating-point signal processor with Altera Cyclone 2 FPGA. The interrupt time was 6.6 kHz, and the transistor switching frequency was 3.3 kHz. The rotor speed and position were measured by the incremental encoder (11-bits)—only to the accuracy verification of observer structure. The stator current was measured by the current transducers LA 25-NP—in the phases "a" and "b" and transformed to the (αβ) reference frame by using the Park transformation. The nominal parameters of the IM are presented in **Table 1**.

In **Figures 1**–**3**, the following variables are presented:

*ω*^*<sup>r</sup>* - estimated rotor speed, *ωrM* - measured rotor speed, *ω*~*<sup>r</sup>* - rotor speed error,^*s<sup>ω</sup>* additional variables, *ψ*^<sup>2</sup> *<sup>r</sup>* <sup>¼</sup> *<sup>ψ</sup>*^<sup>2</sup> *<sup>r</sup><sup>α</sup>* <sup>þ</sup> *<sup>ψ</sup>*^<sup>2</sup> *<sup>r</sup><sup>β</sup>*, *<sup>T</sup>*^*<sup>e</sup>* <sup>¼</sup> *<sup>ψ</sup>*^*r<sup>α</sup>* ^*is<sup>β</sup>* � *<sup>ψ</sup>*^*r<sup>β</sup>* ^*isα*.

In **Figure 1**, the IM is starting up from 0.1 to 1.0 p.u. The waveforms of the estimated value of rotor speed, measured rotor speed, square of rotor flux


#### **Table 1.**

*Parameters of the IM and references unit.*

*Robust Mechanism for Speed and Position Observers of Electrical Machines DOI: http://dx.doi.org/10.5772/intechopen.107898*

**Figure 1.** *The IM is starting up to 1.0 p.u., non-loaded and the rotor speed value is estimated from a) non-adaptive law (56), b) adaptively (54) – Experimental results.*

components, electromagnetic torque, and reference rotor speed are presented. The reference value of electromagnetic torque is limited to 0.75 p.u. The reference value of the square of the rotor flux vector components is set to 0.9 p.u. The estimated rotor speed error during the dynamic states is about 0.015 p.u., and for the steady state is smaller than 0.01 p.u. In **Figure 1a**, the rotor speed is estimated non-adaptively. In **Figure 1b**, the rotor speed value is estimated from adaptive law.

In **Figure 2**, the rotor speed reversed from a nominal speed 1.0 p.u. to �1.0 p.u. The IM during this test is loaded at about *TL* = 0.08 p.u. The square of rotor flux was set to 0.9 p.u. The value ^*sω*is determined from (57). The value of the rotor speed error for the case presented in **Figure 2a** is smaller than 0.02 in the dynamic states. For the case presented in **Figure 2b**, the value of rotor speed error is almost the same and smaller than 0.02 p.u.

In **Figure 3**, the motoring and regenerating modes of the IM are presented. In the AFO speed observer in which the rotor speed is estimated from the classical law of adaptation, for *γ<sup>1</sup>* = 0 in (54) (for this case, the stabilizing function is omitted), the observer structure is unstable in the regenerating machine mode, what was signaled in [6]. For the adaptive case (**Figure 3b**), if *γ<sup>1</sup>* 6¼ 0 and the value ^*s<sup>ω</sup>* is estimated from (55). The observer structure is stable during the load torque value change from 0.7 to �0.7 p.u. The rotor speed error is smaller than 0.015 in the dynamic states. The value of estimated electromagnetic torque for the regenerating case is about �0.65 p.u. It means that the electromagnetic torque value is estimated with a small value of the error of about 0.05 p.u. in the stationary state, but the observer structure is stable.

For the non-adaptive case presented in **Figure 3a**, the estimated rotor speed value has more oscillations than in **Figure 3a**. It is because the rotor speed is not filtered as in

*The IM is reversing from 1.0 to 1.0 p.u., non-loaded and the rotor speed value is estimated from a) non-adaptive law (56), b) adaptively (54) – Experimental results.*

#### **Figure 3.**

*Motoring and regenerating mode of IM for the rotor speed estimated a) from non-adaptive law (56), b) adaptively (54) – Experimental results.*

the case of adaptive estimation law. However, the estimated value of electromagnetic torque is 0.7 p.u. (the same as the load torque). Hence, the electromagnetic torque is estimated more accurately than in the case of the adaptive law of rotor speed estimation.

#### **3.1 Extended speed observer of the squirrel-cage induction machine**

In [19], the speed observer was proposed, which is based on the extended model of the IM. In the model of the observer structure, an auxiliary variable marked in [5] as "Z" was introduced and defined as follows:

$$
\hat{Z}\_a = \hat{a}\_r \hat{\boldsymbol{\mu}}\_{ra},\tag{58}
$$

$$
\hat{Z}\_{\beta} = \hat{a}\_r \hat{\boldsymbol{\mu}}\_{r\beta}.\tag{59}
$$

Based on the introduced auxiliary variables, the observer model can be determined

$$\frac{d\hat{\mathbf{i}}\_{sa}}{d\tau} = a\_1 \hat{\mathbf{i}}\_{sa} + a\_2 \hat{\boldsymbol{\upmu}}\_{ra} + a\_3 \hat{\mathbf{Z}}\_{\beta} + a\_4 \boldsymbol{u}\_{sa} + k\_1 (\hat{\mathbf{i}}\_{sa} - \mathbf{i}\_{sa}),\tag{60}$$

$$\frac{d\hat{\mathbf{i}}\_{s\boldsymbol{\beta}}}{d\boldsymbol{\tau}} = a\_1 \hat{\mathbf{i}}\_{s\boldsymbol{\beta}} + a\_2 \hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} - a\_3 \hat{\mathbf{Z}}\_a + a\_4 \boldsymbol{\mu}\_{s\boldsymbol{\beta}} + k\_1 (\hat{\mathbf{i}}\_{s\boldsymbol{\beta}} - \mathbf{i}\_{s\boldsymbol{\beta}}),\tag{61}$$

$$\frac{d\hat{\boldsymbol{\mu}}\_{ra}}{d\boldsymbol{\pi}} = \mathbf{a}\_{5}\hat{\boldsymbol{\mu}}\_{ra} - \hat{\mathbf{Z}}\_{\beta} + \mathbf{a}\_{6}\hat{\mathbf{i}}\_{ra} + \mathbf{k}\_{2}(\hat{\mathbf{Z}}\_{\beta} - \mathbf{Z}\_{\beta}),\tag{62}$$

$$\frac{d\hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}}}{d\boldsymbol{\pi}} = \boldsymbol{a}\_{\boldsymbol{\xi}}\hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} + \hat{\boldsymbol{Z}}\_{a} + \boldsymbol{a}\_{\boldsymbol{6}}\hat{\boldsymbol{i}}\_{s\boldsymbol{\beta}} - \boldsymbol{k}\_{2}(\hat{\boldsymbol{Z}}\_{a} - \boldsymbol{Z}\_{a}),\tag{63}$$

$$\frac{d\hat{Z}\_a}{d\tau} = -\hat{\alpha}\_r(\hat{Z}\_\beta - a\_6 \mathbf{i}\_{sa}) - a\_5 \hat{Z}\_a + k\_3(\hat{\mathbf{i}}\_{sa} - \mathbf{i}\_{sa}),\tag{64}$$

$$\frac{d\hat{Z}\_{\beta}}{d\tau} = \left\|{\hat{Z}\_{a} + \mathbf{a}\_{\theta}\mathbf{i}\_{\vartheta}}\right\| - \mathbf{a}\_{\theta}\hat{Z}\_{\beta} + k\_{3}(\hat{\mathbf{i}}\_{\vartheta} - \mathbf{i}\_{\vartheta}),\tag{65}$$

where the derivative of estimated rotor speed can be approximated *<sup>d</sup>ω*^*<sup>r</sup> <sup>d</sup><sup>τ</sup>* <sup>≈</sup> *Δω*^*<sup>r</sup> <sup>Δ</sup><sup>T</sup>* ≈0 in the small interval time *ΔT*, and coefficients *a1* … *a6* are defined in (45).

The rotor speed can be determined non-adaptively from the dependence [19]:

$$
\hat{\boldsymbol{\alpha}}\_r = \frac{\hat{\mathbf{Z}}\_a \hat{\boldsymbol{\mu}}\_{ra} + \hat{\mathbf{Z}}\_\beta \hat{\boldsymbol{\mu}}\_{r\beta}}{\hat{\boldsymbol{\nu}}\_{ra}^2 + \hat{\boldsymbol{\nu}}\_{r\beta}^2}. \tag{66}
$$

The experimental results in this section are limited only to the regenerating mode of the IM, in which the observer structure can be unstable. The reference rotor speed is set to 0.1 p.u.

In the first case presented in **Figure 4a**, (in which the rotor speed is estimated from (66)) after 0.5 s machine is loaded *TL* = �0.6 p.u. For the motoring mode (*ω*^*<sup>r</sup>* > 0, *T*^*<sup>e</sup>* ≥0), the speed observer (60)–(65) is stable. After 1.5 s, when the load torque value is decreased up to �0.2 p.u., the observer system estimates the state variables incorrectly, the error of estimated rotor speed increases up to 0.05 p.u., and after 1.8 s, the electromagnetic torque value achieves its limitation (0.75 p.u.). After 1.9 s, the rotor speed error is higher than 0.05 p.u., and the IM is braking. The observer structure achieves unstable points of operation in which all the estimates do not converge to their real values.

The rotor speed value is estimated from (66), which is suitable only for the motoring mode of the machine. This is the same case as for the AFO speed observer structure. The speed estimation law is based on the algebraic eq. (66), which does not

**Figure 4.** *Regenerating mode of IM for the rotor speed estimated: a) the rotor speed value is estimated from (66), b) the rotor speed is estimated by using the proposed robust law (additional stabilization function) – Experimental results.*

guarantee the stability of the observer structure during the regenerating mode of the machine. Some poles of the observer move to an unstable zone (are zero or positive). The reason for this is the form of dependence (66) in which the additional stabilization function does not exist. The stabilization function, proposed in Section 2.2, which is based on Lagrange's identity, cannot be directly used, because the vectors: ψ^*<sup>r</sup>* and Zhave the same position and different amplitude only. This is the result of the ^ definition (58)–(59). In this case, it is better to use from (58)–(59), and after few simple transformations, one can be obtain

$$
\hat{Z}\_a \hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} - \hat{Z}\_\beta \hat{\boldsymbol{\mu}}\_{ra} = \hat{\boldsymbol{\alpha}}\_r \hat{\boldsymbol{\mu}}\_{ra} \hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} - \hat{\boldsymbol{\alpha}}\_r \hat{\boldsymbol{\mu}}\_{ra} \hat{\boldsymbol{\mu}}\_{r\boldsymbol{\beta}} = \mathbf{0}.\tag{67}
$$

This is satisfied for the ideal case in which all estimation errors are equal to zero. In the other case, taking the left side of (67) as

$$
\hat{\mathbf{s}}\_{\alpha} = \hat{Z}\_{a}\hat{\boldsymbol{\mu}}\_{r\beta} - \hat{Z}\_{\beta}\hat{\boldsymbol{\mu}}\_{ra},\tag{68}
$$

and using (66) the update form of non-adaptive speed estimation with the stabilization function (68) can be determined

$$
\hat{\alpha}\_r = \frac{\hat{Z}\_a \hat{\mu}\_{ra} + \hat{Z}\_\beta \hat{\nu}\_{r\beta} + k\_f \gamma\_1 \hat{s}\_a}{\hat{\nu}\_{ra}^2 + \hat{\nu}\_{r\beta}^2},\tag{69}
$$

where *kf* ¼ *sign*ð Þ *ω*^*<sup>r</sup>* , *γ*<sup>1</sup> ≥0.

The experimental results of the proposed non-adaptive speed estimation with the stabilization function (68) are presented in **Figure 4b**. The reference of rotor speed is set to 0.1 p.u. After 1.5 s, the load torque slowly changes from (the motoring mode)

0.7 to �0.7 p.u. (the regenerating mode). The speed observer correctly estimates the electromagnetic torque value with the rotor speed error smaller than 0.015 p.u. during the dynamic states. The square of rotor flux vector components is stabilized on the almost constant value equal to 0.9 p.u. The proposed stabilized function (68) improves the speed observer properties making the speed observer structure more robust in the regenerating mode, which was confirmed in **Figure 4b**.

## **4. Speed and position observer of interior permanent magnet machine**

This section is concerned with the observer system of interior permanent magnet machines IPMSM and their problems during a sensorless application under disturbances.

#### **4.1 Mathematical models of the IPMSM machines**

The mathematical model of IPMSM is often determined in the rotating reference frame (d-q), which is connected to the position of the rotor. The model in (d-q) was presented in [12–14]. However, sometimes the mathematical model is better considered in the stationary (α-β) reference frame connected to the stator. The model of IPMSM in (α-β) has the following form [12]:

$$\frac{di\_{s\alpha}}{d\pi} = \frac{\alpha\_r}{L\_d}\lambda\_\beta + (-R\_i i\_{s\alpha} + \mu\_{s\alpha})L\_1 + (-R\_i i\_{s\beta} + \mu\_{s\beta})L\_3,\tag{70}$$

$$\frac{di\_{s\beta}}{d\pi} = -\frac{\alpha\_r}{L\_d}\lambda\_a + (-R\_i i\_{sa} + \mu\_{sa})L\_3 + \left(-R\_i i\_{s\beta} + \mu\_{s\beta}\right)L\_4,\tag{71}$$

$$\frac{d\alpha\_r}{d\tau} = \frac{1}{J} \left( \psi\_{fa}\dot{\imath}\_{s\beta} - \psi\_{f\beta}\dot{\imath}\_{sa} + \left(L\_d - L\_q\right)\dot{\imath}\_{sa}\dot{\imath}\_{s\beta} - T\_L\right),\tag{72}$$

$$\frac{d\theta\_r}{d\tau} = a\_r,\tag{73}$$

where

$$
\lambda\_a = L\_d L\_q^{-1} \nu\_{fa} - \left(\mathbf{1} - L\_d L\_q^{-1}\right) (L\_0 i\_{a2} + L\_2 i\_{sa}),
\tag{74}
$$

$$
\lambda\_{\beta} = \mathbf{L}\_d \mathbf{L}\_q^{-1} \boldsymbol{\Psi}\_{f\beta} + \left(\mathbf{1} - \mathbf{L}\_d \mathbf{L}\_q^{-1}\right) \left(\mathbf{L}\_0 \mathbf{i}\_{\beta 2} - \mathbf{L}\_2 \mathbf{i}\_{\beta}\right), \tag{75}
$$

$$L\_0 = 0.5(L\_d + L\_q), \\ L\_1 = L\_d^{-1} \cos^2 \theta\_r + L\_q^{-1} \sin^2 \theta\_r \tag{76}$$

$$L\_2 = 0.5(L\_d - L\_q), \\ L\_3 = 0.5 \left(\frac{1}{L\_d} - \frac{1}{L\_q}\right) \sin\left(2\theta\_r\right) \tag{77}$$

$$L\_4 = L\_d^{-1} \sin^2 \theta\_r + L\_q^{-1} \cos^2 \theta\_r,\tag{78}$$

$$i\_{a2} = i\_{\ast a} \cos 2\theta\_r + i\_{s\beta} \sin 2\theta\_r \tag{79}$$

$$i\_{\beta 2} = -i\_{\alpha} \sin 2\theta\_r + i\_{\beta} \cos 2\theta\_r,\tag{80}$$

$$
\psi\_{fa} = \psi\_f \cos \theta\_r, \psi\_{f\beta} = \psi\_f \sin \theta\_r \tag{81}
$$

**Figure 5.** *The phase "A" back-EMF voltage.*

where *Rs* is the stator resistance, *Ld*, *Lq* are the winding inductances, *isα*,*<sup>β</sup>* are the stator currents, *usα*,*<sup>β</sup>* are the stator voltages, *ψ<sup>f</sup>* is the permanent magnet flux linkage, *ω<sup>r</sup>* is the rotor speed, *θ<sup>r</sup>* is the rotor position, *J* is the rotor inertia,*TL* is the load torque.

The design of the IPMSM machine has a significant impact on the properties of the whole drive system. In IPMSM without the skews in the slots, there occur the rotor slot's harmonics, [20]. The slot's harmonics cause the non-sinusoidal distribution of the electromagnetic force (EMF) generated in the machine, [21]. These have a negative influence on the quality of the control system of the machine, and in particular, on the speed and position observer. In the literature [12–14], these negative effects are named by the disturbances in the IPMSM, which have bounded values. These disturbances have a significant impact on the machine rotor speed smaller than 15% of the nominal value and the idling mode of the IPMSM. For the low-speed range, the stator voltage has a small value, and similarly is the stator currents; because of this, the back-EMF value is significant. The example waveform of back-EMF voltage in 3.5 kW IPMSM machine for 10% of nominal rotor speed, registered by using the digital oscilloscope is presented in **Figure 5**.

In **Figure 5**, there are visible 18 slot's harmonics in the waveform. The IPMSM nominal parameters are shown in **Table 2**.


#### **Table 2.**

*IPMSM nominal parameters and reference units.*

In the next section, the speed and position observer is proposed for the IPMSM in which the disturbances have occurred and the back-EMF voltage has an almost trapezoidal distribution (**Figure 5**).

#### **4.2 Adaptive speed and position observer of IPMSM**

As was mentioned in 4.1, the IPMSM has disturbances in the form of trapezoidal back-EMF voltage with slot's harmonics. The classical structures of the observer in (dq) are not stable if the rotor speed is smaller than 15–20% of the nominal speed. One of the solutions to overcome this problem is to implement a dedicated algorithm in which additional high or low-frequency signals are injected into the stator voltage or current [13, 14]. However, the sensorless control system is then more complicated than classical FOC, and the observer structure contains a few low-passes or bandwidth filters [22]. The procedure of selecting the settings of the observer and the PI controllers in the control system is difficult. Therefore in this section, a new form of the speed and position observer is proposed, which is based on the mathematical model in the stationary reference frame presented in Section 4.1. Considering the procedure of design of the observer stabilization function from Section 2, the AFO speed observer of IPMSM can be determined

$$\frac{d\hat{\mathbf{i}}\_{sa}}{d\tau} = \frac{\hat{\alpha}\_r}{L\_d}\hat{\lambda}\_\beta + \left(-R\_i\hat{\mathbf{i}}\_{sa} + \mathfrak{u}\_{sa}\right)L\_1 + \left(-R\_i\hat{\mathbf{i}}\_{s\beta} + \mathfrak{u}\_{s\beta}\right)L\_3 + \upsilon\_a,\tag{82}$$

$$\frac{d\hat{\mathbf{i}}\_{\imath\beta}}{d\pi} = -\frac{\hat{\alpha}\_r}{L\_d}\hat{\lambda}\_a + \left(-R\_i\hat{i}\_{\imath a} + \mathfrak{u}\_{\imath a}\right)L\_3 + \left(-R\_i\hat{i}\_{\imath\beta} + \mathfrak{u}\_{\imath\beta}\right)L\_4 + \upsilon\_\beta,\tag{83}$$

$$\frac{d\hat{\theta}\_r}{d\tau} = \hat{\alpha}\_r + v\_\theta,\tag{84}$$

where "^" denotes estimated values; *vα,β,* and *v<sup>θ</sup>* are stabilizing functions introduced to (82)–(83).

The values of the rotor flux vector components can be obtained by using (74)–(75) as follows:

$$
\hat{\lambda}\_a = L\_d L\_q^{-1} \hat{\psi}\_{fa} - \left(\mathbf{1} - L\_d L\_q^{-1}\right) \left(L\_0 \hat{i}\_{a2} + L\_2 \hat{i}\_{sa}\right), \tag{85}
$$

$$
\hat{\lambda}\_{\beta} = \mathbf{L}\_{d} \mathbf{L}\_{q}^{-1} \hat{\boldsymbol{\upmu}}\_{f\beta} + \left(\mathbf{1} - \mathbf{L}\_{d} \mathbf{L}\_{q}^{-1}\right) \left(\mathbf{L}\_{0} \hat{\mathbf{i}}\_{\beta 2} - \mathbf{L}\_{2} \hat{\mathbf{i}}\_{\beta \beta}\right), \tag{86}
$$

where

$$L\_0 = 0.5(L\_d + L\_q), \omega\_1 = L\_d^{-1} \cos^2 \hat{\theta}\_r + L\_q^{-1} \sin^2 \hat{\theta}\_r \tag{87}$$

$$L\_2 = 0.5(L\_d - L\_q), \\ L\_3 = 0.5 \left(\frac{1}{L\_d} - \frac{1}{L\_q}\right) \sin\left(2\hat{\theta}\_r\right) \tag{88}$$

$$L\_4 = L\_d^{-1} \sin^2 \hat{\theta}\_r + L\_q^{-1} \cos^2 \hat{\theta}\_r,\tag{89}$$

$$
\hat{i}\_{a2} = \hat{i}\_{sa}\cos 2\hat{\theta}\_r + \hat{i}\_{s\beta}\sin 2\hat{\theta}\_r \tag{90}
$$

$$
\hat{i}\_{\beta 2} = -\hat{i}\_{\alpha} \sin 2\hat{\theta}\_r + \hat{i}\_{\beta} \cos 2\hat{\theta}\_r,\tag{91}
$$

$$
\hat{\boldsymbol{\mu}}\_{fa} = \boldsymbol{\mu}\_f \cos \hat{\boldsymbol{\theta}}\_r,\\
\hat{\boldsymbol{\mu}}\_{f\beta} = \boldsymbol{\mu}\_f \sin \hat{\boldsymbol{\theta}}\_r \tag{92}
$$

To stabilize the observer structure (82)–(84), the appropriate form of the stabilization functions *vα,<sup>β</sup>* and *v<sup>θ</sup>* should be determined to satisfy the Lyapunov theorem. The Lyapunov candidate function has the form

$$\mathbf{V} = \mathbf{0}. \mathbf{5} \left( \left( \tilde{\mathbf{i}}\_{s\alpha}^{2} + \tilde{\mathbf{i}}\_{s\beta}^{2} \right) + \tilde{\theta}\_{r}^{2} + \chi^{-1} \tilde{\phi}\_{r}^{2} \right), \tag{93}$$

where

$$
\tilde{\mathbf{i}}\_{ta,\theta} = \hat{\mathbf{i}}\_{sa,\theta} - \mathbf{i}\_{ta,\theta}, \tilde{\boldsymbol{\alpha}}\_r = \hat{\boldsymbol{\alpha}}\_r - \boldsymbol{\alpha}\_r \tilde{\boldsymbol{\theta}}\_r = \hat{\boldsymbol{\theta}}\_r - \boldsymbol{\theta}\_r \tag{94}
$$

The derivative of the Lyapunov function can be determined by using the estimation errors (94) and the proposed observer structure as

$$\dot{V} = -\left(c\_a \hat{\vec{t}}\_{sa}^2 + c\_a \hat{\vec{t}}\_{s\theta}^2\right) + \ddot{o}\_r \left(-\frac{1}{L\_d} \hat{\lambda}\_\theta \ddot{\vec{t}}\_{sa} + \frac{1}{L\_d} \hat{\lambda}\_a \ddot{\vec{t}}\_{s\theta} + \frac{1}{\chi} \dot{\vec{o}}\_r\right) + \ddot{\theta}\_r (\ddot{o}\_r + v\_\theta) \le 0. \tag{95}$$

The derivative of the Lyapunov function (95) is negative if the stabilizing functions are chosen

$$
\upsilon\_a = -c\_a R\_s L\_1 \tilde{i}\_{sa} + c\_\lambda L\_d^{-1} \hat{\alpha}\_r \hat{\lambda}\_\beta \tilde{i}\_{sa},\tag{96}
$$

$$
\omega v\_{\beta} = -\mathcal{c}\_{a} \mathcal{R}\_{\sharp} L\_{\mathfrak{A}} \tilde{\mathfrak{i}}\_{s\beta} - \mathcal{c}\_{\lambda} L\_{d}^{-1} \hat{\alpha}\_{r} \hat{\lambda}\_{a} \tilde{\mathfrak{i}}\_{s\beta}, \tag{97}
$$

$$
v\_{\theta} = -c\_{\theta}\tilde{\theta}\_{r},\tag{98}$$

where (*cα*, *cθ*) > 0 and *c<sup>λ</sup>* ≤ *RsL*<sup>1</sup> ^*λβ*�*RsL*4^*λα L*�<sup>1</sup> *<sup>d</sup>* <sup>j</sup>*ω*^*r*<sup>j</sup> ^*<sup>λ</sup>* 2 *<sup>α</sup>*þ^*<sup>λ</sup>* 2 *β* � �.

The rotor speed value can be estimated by using the classical adaptation law

$$
\dot{\hat{\alpha}}\_r = \gamma L\_d^{-1} (\hat{\lambda}\_\beta \tilde{\mathbf{i}}\_{sa} - \hat{\lambda}\_a \tilde{\mathbf{i}}\_{\beta}),
\tag{99}
$$

where *γ* > 0.

However, under *Assumption 3* from Section 2.1 in order to improve the quality of reconstruction of the rotor speed value, it is better to introduce the additional stabilization function to the open-integrator (99)

$$
\dot{\hat{\alpha}}\_r = \chi L\_d^{-1} \left( \hat{\lambda}\_\beta \tilde{\mathbf{i}}\_{sa} - \hat{\lambda}\_a \tilde{\mathbf{i}}\_{s\beta} + k\_f \hat{\mathbf{s}}\_{av} \right), \tag{100}
$$

where the value of the stabilizing function can be obtained by using the approach presented in Section 2.1:

$$
\hat{\mathbf{s}}\_{ab} = \sqrt{\left(\hat{\boldsymbol{\lambda}}\_{a}^{2} + \hat{\boldsymbol{\lambda}}\_{\beta}^{2}\right)\left(\hat{\mathbf{i}}\_{sa}^{2} + \hat{\mathbf{i}}\_{s\beta}^{2}\right) + \left(\hat{\boldsymbol{\lambda}}\_{a}\hat{\mathbf{i}}\_{sa} + \hat{\boldsymbol{\lambda}}\_{\beta}\hat{\mathbf{i}}\_{s\beta}\right)^{2}}.\tag{101}
$$

For ^*λ* 2 *<sup>α</sup>* <sup>þ</sup> ^*<sup>λ</sup>* 2 *β* � � <sup>~</sup>*<sup>i</sup>* 2 *<sup>s</sup><sup>α</sup>* <sup>þ</sup>~*<sup>i</sup>* 2 *sβ* � � <sup>≪</sup> ^*λα* <sup>~</sup>*is<sup>α</sup>* <sup>þ</sup> ^*λβ* ~*isβ* � �<sup>2</sup> , the value of (101) can be determined from the simplified form

*Robust Mechanism for Speed and Position Observers of Electrical Machines DOI: http://dx.doi.org/10.5772/intechopen.107898*

$$
\hat{\mathbf{s}}\_{\alpha} = \hat{\lambda}\_{a}\tilde{\mathbf{i}}\_{\alpha} + \hat{\lambda}\_{\beta}\tilde{\mathbf{i}}\_{\varepsilon\beta}. \tag{102}
$$

The rotor position can be obtained directly from (84), and the stabilizing function *vθ* from (98).

In (98) there is the rotor position error, which is defined <sup>~</sup>*θ<sup>r</sup>* <sup>¼</sup> ^*θ<sup>r</sup>* � *<sup>θ</sup>r*, where *<sup>θ</sup><sup>r</sup>* means the real (measured) value of rotor speed. However, in the speed observer structure, the rotor speed is not measured but only estimated. Therefore, it is proposed to replace the deviation ~*θ<sup>r</sup>* by ~*θλ* and (98) is rewritten as

$$
v\_{\theta} = -c\_{\theta}\tilde{\theta}\_{\lambda},\tag{103}$$

where ~*θλ* can be defined as the angle between the rotor flux vector components *λα, <sup>β</sup>* and their estimated values ^*λα*,*β*. Values of deviation ~*θλ* can be determined as

$$\tilde{\theta}\_{\dot{k}} = \tan^{-1}(\varphi),\tag{104}$$

where *<sup>φ</sup>* <sup>¼</sup> *λα*^*λβ* � *λβ* ^*λα* � � *λα*^*λα* <sup>þ</sup> *λβ* ^*λβ* � ��<sup>1</sup> . The rotor flux vector components, *λα, <sup>β</sup>*, can be determined from (74)–(75) in which it is assumed *θr*≈^*θ<sup>r</sup>* and the measured values of *isα*,*<sup>β</sup>* are used; also, *λα*^*λα* <sup>þ</sup> *λβ* ^*λβ* � � 6¼ 0.

Value of ~*θλ* should be projected using

$$\tilde{\theta}\_{\dot{\lambda}} = \begin{cases} \tilde{\theta}\_{\dot{\lambda}} - \pi/2, \text{ if } \rho > 0 \\ \tilde{\theta}\_{\dot{\lambda}} + \pi/2 \text{ if } \rho < 0 \end{cases} \right), \tag{105}$$

It gives the values ~*θλ* in a steady state close to zero, and it can be assumed that ~*θλ*≈~*θr*. The proposed stabilizing function improves the estimated value of the rotor position, particularly in the dynamic states of the IPMSM. The stabilizing function is necessary in the case of IPMSM with the described above disturbances.

Remark 4. The value of rotor flux vector components must be estimated from (74)–(75), however, by using the estimated rotor speed position ^*θ<sup>r</sup>* in (76)–(81).

### **4.3 Non-adaptive speed estimation of the IPMSM**

The rotor speed value can be estimated by using the non-adaptive estimation scheme. Consider the non-adaptive scheme for the rotor speed estimation presented in Section 2.2 for the pair of vectors, ð Þ� <sup>x</sup>^*k*, <sup>x</sup>~*<sup>k</sup>* ^*λr*, <sup>~</sup>i*<sup>s</sup>* � � the rotor speed value can be estimated from

$$
\hat{\boldsymbol{\alpha}}\_r = \frac{\mathbf{\hat{i}}\_{\alpha}\hat{\boldsymbol{\lambda}}\_{\beta} - \mathbf{\tilde{i}}\_{\alpha}\hat{\boldsymbol{\lambda}}\_{\alpha} + k\_f \hat{\boldsymbol{s}}\_{\alpha}}{2\left(\hat{\boldsymbol{\lambda}}\_{\alpha}^2 + \hat{\boldsymbol{\lambda}}\_{\beta}^2\right)},\tag{106}
$$

where

$$
\hat{s}\_{ab} = \sqrt{\left(\hat{\mathbf{i}}\_{sa}\hat{\mathbf{i}}\_{\beta} - \hat{\mathbf{i}}\_{s\beta}\hat{\mathbf{i}}\_{a}\right)^{2} + 4\gamma\_1 \left(\hat{\lambda}\_a \hat{\mathbf{i}}\_{sa} + \hat{\lambda}\_\beta \hat{\mathbf{i}}\_{s\beta}\right)^{2}},\tag{107}
$$

and *kf* ¼ *sign*ð Þ *ω*^*<sup>r</sup>* , *γ*<sup>1</sup> ≥0.

## **4.4 Simulation and experimental results of the speed and position observer of IPMSM**

In this section, the chosen waveforms from the simulation and the experiment setup are shown. The nominal parameters of the IPMSM are shown in **Table 2**. The experimental validations were carried out on 3.5 kW IPMSM. The stator of IPMSM has 18 slots, which are visible in the waveform of EMF from **Figure 5**. The machine is controlled by using the classical FOC control presented in [12, 13, 22]. There are three PI controllers for the rotor speed, isq, and isd stator vector components. Additionally, the MTPA algorithm [12] was applied.

In **Figure 6,** the waveform of the simulation results is shown. The estimated rotor speed *ω*^*r*, stator current vector components ^*isd*,*<sup>q</sup>* estimated rotor speed error *ω*~*r*, and the estimated rotor position ^*θ<sup>r</sup>* are presented. In **Figure 6a**, the machine is starting up to 1.0 p.u. and after 600 ms loaded to about 0.6 p.u. The error of the rotor position is smaller than 0.05 p.u. during the dynamic states, the rotor speed error is smaller than 0.01 p.u. In **Figure 6b**, the machine is reversing to 1.0 p.u. The position error is smaller than 0.1 p.u. during the dynamic states, and the error of rotor speed is smaller than 0.05 p.u. In **Figure 6b**, the measured value of rotor position *θrM* is shown.

In **Figure 7a**, after 100 ms the machine is loaded *TL* = 1.0 p.u. and after 600 ms the regenerating mode is applied and *TL* = 1.0 p.u. The rotor reference speed is equal to 0.1 p.u. The estimated electromagnetic torque *T*^*e*, rotor position error ~*θr*, ~*θλ* defined in (104), *s<sup>ω</sup>* and rotor speed error, and the estimated ^*isd* stator current component are presented. It is worth noticing that the waveforms of ~*θ<sup>r</sup>* as well as ^*s<sup>ω</sup>* and *ω*~*<sup>r</sup>* are converged on each other.

The experimental waveforms are presented in **Figure 8**. The machine's reversal from 1.0 to 1.0 p.u. is shown in **Figure 8a**. The estimated rotor speed *ω*^*r*, stator current vector components ^*isd*,*<sup>q</sup>* estimated rotor speed error *ω*~*r*, and the estimated

**Figure 6.** *a) Machine is starting up to 1.0 p.u and b) reversing to 1.0 p.u. – Simulation results.* *Robust Mechanism for Speed and Position Observers of Electrical Machines DOI: http://dx.doi.org/10.5772/intechopen.107898*

**Figure 7.**

*a) the load torque* TL *is changed from 1.0 to – 1.0 p.u., b) parameters of the machine are changed in the sensorless control system (parameters uncertainties test) – Simulation.*

**Figure 8.**

*Machine is reversing from 1.0 to 1.0 p.u. for a) the sensorless control system with the proposed observer, b) the control system with measured rotor speed and position values – Experimental results.*

rotor position ^*θ<sup>r</sup>* and the stator current module *im* are presented. In **Figure 8b**, the same waveforms but for the measured value from the encoder of the rotor speed and rotor position are presented (for comparison). During the machine reverse, the *isd* value is about 0.25 p.u. It results from the MTPA algorithm [12].

**Figure 9.**

*Regenerating mode of IPMSM machine for the low reference speed 0.025 p.u.,* TL *=* �*0.5 p.u. for the cases a) in (106)* kf *= 0.5, b) in (106)* kf *= 2.5 p.u. – Experimental results.*

In **Figure 9**, the regenerating mode of IPMSM is shown. The rotor speed was set to 0.025 p.u., and the machine was loaded at about �0.5 p.u. In **Figure 9a**, the stabilizing function is *kf* = 0.5 in (106), and the value of the error of estimated rotor position is increased to 0.075 p.u., which is visible in **Figure 9a**. The value of the stator current component ^*isq*was incorrectly estimated (due to rotor position error). In **Figure 9b**, the same case is shown, however for *kf* ¼ 2*:*5 p.u. The rotor position error is almost minimized to zero, and the ^*isq* value is about �0. 5 p.u. (the same as the referenced).

In this section, the presented simulation and experimental results confirmed that the introduced stabilization function into the speed adaptation scheme leads to the improvement of the properties of the observer system and robustness of the occurred disturbances. In this case, these are the non-sinusoidal EMF and slot's harmonics.

## **5. Speed and position observer of doubly fed induction generator**

In this section, the speed and position observer of the doubly fed induction generator DFIG is considered. The rotor is connected to a voltage source converter, and the stator is directly connected to three phases AC-grid. The field-oriented control FOC is used to control the active and reactive stator powers presented in [23]. The rotor speed will be estimated by using a non-adaptive estimation scheme only.

#### **5.1 Mathematical models of the DFIG**

The mathematical model of DFIG can be determined in rotating or stationary reference frames (x-y). Considering the rotor and stator current vector components, the differential equations have the form [24]:

*Robust Mechanism for Speed and Position Observers of Electrical Machines DOI: http://dx.doi.org/10.5772/intechopen.107898*

$$\frac{d\dot{\mathbf{u}}\_{\rm{rx}}}{d\tau} = -\frac{\mathbf{L}\_{\rm{s}}}{\dot{\mathbf{w}}\_{\sigma}}(R\_{\rm{s}}\dot{\mathbf{i}}\_{\rm{rx}} - \mathbf{u}\_{\rm{rx}}) + \frac{\mathbf{L}\_{\rm{m}}}{\dot{\mathbf{w}}\_{\sigma}}\left(\alpha\_{r}\Big(L\_{\rm{m}}\dot{\mathbf{i}}\_{\rm{ry}} + L\_{r}\dot{\mathbf{i}}\_{\rm{ry}}\Big) + R\_{r}\dot{\mathbf{i}}\_{\rm{rx}} - \mathbf{u}\_{\rm{rx}}\Big)\right),\tag{108}$$

$$\frac{d\dot{u}\_{\rm ry}}{d\tau} = -\frac{L\_r}{w\_\sigma} \left( R\_r \dot{i}\_\rm \eta - u\_\rm \eta \right) - \frac{L\_m}{w\_\sigma} \left( \alpha\_r (L\_m \dot{i}\_\rm x} + L\_R \dot{i}\_\rm x \right) - R\_r \dot{i}\_\rm \eta + u\_\rm \eta \right), \tag{109}$$

$$\frac{d\dot{u}\_{\rm rx}}{d\tau} = \frac{L\_{\rm s}}{w\_{\sigma}} \left( -\alpha\_{r} \left( L\_{r} \dot{i}\_{r\gamma} + L\_{m} \dot{i}\_{\gamma\gamma} \right) - R\_{r} \dot{i}\_{r\infty} + u\_{\rm rx} \right) + \frac{L\_{m}}{w\_{\sigma}} (R\_{r} \dot{i}\_{\rm rx} - u\_{\rm rx}), \tag{110}$$

$$\frac{d\dot{\mathbf{u}}\_{r\mathbf{y}}}{d\tau} = \frac{\mathbf{L}\_{\mathbf{r}}}{w\_{\sigma}} \left( a\nu\_{r} (\mathbf{L}\_{r}\dot{\mathbf{i}}\_{r\mathbf{x}} + \mathbf{L}\_{m}\dot{\mathbf{i}}\_{\mathbf{x}\mathbf{}}) - R\_{r}\dot{\mathbf{i}}\_{r\mathbf{y}} + u\_{r\mathbf{y}} \right) + \frac{L\_{m}}{w\_{\sigma}} \left( R\_{r}\dot{\mathbf{i}}\_{\mathbf{y}} - u\_{\mathbf{y}} \right), \tag{111}$$

$$\frac{d\rho\_r}{d\tau} = \frac{L\_m}{fL\_r} \left( i\_{r\mathbf{x}} i\_{r\mathbf{y}} - i\_{r\mathbf{y}} i\_{\mathbf{x}\mathbf{y}} \right) - \frac{1}{J} \left( T\_L + f\_r \rho\_r \right). \tag{112}$$

where the (x-y) coordinate system is associated with any angular speed, and it is assumed that (108)–(109) are connected to the stationary stator windings so the angular speed of the (x,y) system is *ω<sup>a</sup>* ¼ 0, *fr* is the friction.

It is assumed that all the DFIG parameters are known and constant. The components *urx*, *ury* are treated as the control vector variables, and *usx*, *usy*, *isx*, *isy* and *irx*, *iry* components are treated as measured and transformed to the adequate (x-y) reference frame.

To design the observer structure, it is proposed to introduce new auxiliary variables, which are defined

$$H\_{\mathbf{x}} = a o\_r(L\_m i\_{\mathbf{x}} + L\_r i\_{r\mathbf{x}}),\tag{113}$$

$$H\_{\mathcal{Y}} = \alpha\_r \left( L\_m i\_{\mathcal{Y}} + L\_r i\_{r\mathcal{Y}} \right). \tag{114}$$

Considering (113)–(114) and (108)–(111), the observer structure can be determined

$$\frac{d\hat{\mathbf{l}}\_{\mathbf{rx}}}{d\tau} = -\frac{L\_{\mathbf{t}}}{w\_{\sigma}} \left( \hat{\mathbf{H}}\_{\mathbf{y}} + \mathbf{R}\_{r} \dot{\mathbf{r}}\_{\mathbf{rx}} + \boldsymbol{\mu}\_{\mathbf{rx}} \right) + \frac{L\_{m}}{w\_{\sigma}} \left( \mathbf{R}\_{i} \dot{\mathbf{x}}\_{\mathbf{x}} - \boldsymbol{\mu}\_{\mathbf{x}} \right) + \boldsymbol{\upsilon}\_{\mathbf{rx}},\tag{115}$$

$$\frac{d\hat{\mathbf{i}}\_{\tau\mathbf{y}}}{d\tau} = \frac{L\_{\mathbf{s}}}{w\_{\sigma}} \left(\hat{\mathbf{H}}\_{\mathbf{x}} - R\_{r}\dot{\mathbf{i}}\_{\tau\mathbf{y}} + \boldsymbol{\mu}\_{\tau\mathbf{y}}\right) + \frac{L\_{m}}{w\_{\sigma}} \left(\mathbf{R}\_{r}\dot{\mathbf{i}}\_{\tau\mathbf{y}} - \boldsymbol{\mu}\_{\mathbf{s}\mathbf{y}}\right) + \boldsymbol{\nu}\_{\tau\mathbf{y}},\tag{116}$$

$$\frac{d\hat{H}\_{\text{x}}}{d\tau} = \hat{\alpha}\_{r} \left( -\hat{H}\_{\text{y}} - R\_{r}\dot{\imath}\_{\text{rx}} + \boldsymbol{\mu}\_{\text{rx}} \right) + \boldsymbol{v}\_{\text{Hx}},\tag{117}$$

$$\frac{d\hat{H}\_{\text{y}}}{d\tau} = \left\|{\hat{H}\_{\text{x}} - R\_{r}i\_{r\text{y}} + u\_{r\text{y}}}\right\| + \nu\_{\text{Hy}},\tag{118}$$

$$\frac{d\hat{\theta}\_r}{d\tau} = \hat{\alpha}\_r + v\_\theta,\tag{119}$$

where estimated state variables are marked by "^" and *<sup>d</sup>ω*^*<sup>r</sup> <sup>d</sup><sup>τ</sup>* <sup>≈</sup> <sup>Δ</sup>*ω*^*<sup>r</sup>* <sup>Δ</sup>*<sup>T</sup>* , the observer contains the stabilization functions *vrx*, *vry* and *vHx*, *vHy*, *v<sup>θ</sup>* .

According to the design procedure of the observer presented in Section 2, in order to stabilize the observer structure (115)–(119), appropriate form of the stabilization functions *vrx*, *vry* and *vHx*, *vHy*, *v<sup>θ</sup>* should be determined to satisfy the Lyapunov theorem. The Lyapunov function has the form

*New Trends in Electric Machines - Technology and Applications*

$$\mathbf{V} = \frac{1}{2} \left( \tilde{\mathbf{i}}\_{r\mathbf{x}}^{2} + \tilde{\mathbf{i}}\_{r\mathbf{y}}^{2} + \tilde{\mathbf{H}}\_{\mathbf{x}}^{2} + \tilde{\mathbf{H}}\_{\mathbf{y}}^{2} + \tilde{\theta}\_{r}^{2} \right) > \mathbf{0},\tag{120}$$

where.

$$\ddot{\mathbf{i}}\_{\mathbf{rx}} = \hat{\mathbf{i}}\_{\mathbf{rx}} - \mathbf{i}\_{\mathbf{rx}} \ddot{\mathbf{i}}\_{\mathbf{ry}} = \hat{\mathbf{i}}\_{\mathbf{ry}} - \mathbf{i}\_{\mathbf{ry}} \ddot{H}\_{\mathbf{x}} = \hat{H}\_{\mathbf{x}} - H\_{\mathbf{x}} \ddot{H}\_{\mathbf{y}} = \hat{H}\_{\mathbf{y}} - H\_{\mathbf{y}} \text{ and } \ddot{\boldsymbol{\theta}}\_{\mathbf{r}} = \hat{\boldsymbol{\theta}}\_{\mathbf{r}} - \boldsymbol{\theta}\_{\mathbf{r}}.\tag{121}$$

The proposed observer structure will be asymptotically stable if *V*\_ <sup>2</sup> < 0 and if the stabilizing functions introduced to the structure are determined as

$$
\boldsymbol{v}\_{\rm rx} = -\boldsymbol{c}\_{\rm x} \boldsymbol{\tilde{i}}\_{\rm rx},\tag{122}
$$

$$
v\_{\gamma} = -c\_{\gamma}\ddot{i}\_{\gamma},\tag{123}$$

$$v\_{Hx} = c\_{Hx} \left( -\frac{L\_t}{w\_\sigma} \tilde{i}\_{r\gamma} - \hat{\alpha}\_r R\_r \tilde{i}\_{r\chi} \right),\tag{124}$$

$$
\sigma\_{\rm Hy} = \sigma\_{\rm Hy} \left( \frac{L\_{\rm s}}{w\_{\sigma}} \tilde{\dot{i}}\_{\rm rx} + \hat{\alpha}\_{r} R\_{r} \tilde{\dot{i}}\_{\gamma \gamma} \right), \tag{125}
$$

where (*cx*, *cy*, *cHx*, *cHy*, *cθ*) > 0 are the observer tuning gains and

$$
\sigma\_{\theta} = -\varepsilon\_{\theta} \tilde{\theta}\_{r}. \tag{126}
$$

The speed observer structure will be asymptotically stable if (122)–(126) is satisfied. In the sensorless control, the rotor speed is not measured, therefore the deviation ~*θ<sup>r</sup>* in (126) should be replaced by ~*θH*. This means that the deviation between the estimated values of *Hx* and *Hy*, calculated from (113)–(114) and estimated from the observer structure in (117)–(118) is as follows:

$$\tilde{\theta}\_H = \tan^{-1}(\theta),\tag{127}$$

where

$$\theta = \frac{H\_{\text{x}}\hat{H}\_{\text{y}} - H\_{\text{y}}\hat{H}\_{\text{x}}}{H\_{\text{x}}\hat{H}\_{\text{x}} + H\_{\text{y}}\hat{H}\_{\text{y}}} \text{ and } \left(H\_{\text{x}}\hat{H}\_{\text{x}} + H\_{\text{y}}\hat{H}\_{\text{y}}\right) \neq \mathbf{0}.\tag{128}$$

The value *H*^ *<sup>x</sup>*,*H*^ *<sup>y</sup>* can be estimated from

$$
\hat{H}\_{\text{x}} = \hat{a}\_{r} \left( L\_{m} \mathbf{i}\_{\text{xx}} + L\_{r} \hat{\mathbf{i}}\_{r\text{x}} \right),
\tag{129}
$$

$$
\hat{H}\_{\text{\textquotedblleft}} = \hat{\alpha}\_r \left( L\_m \dot{\imath}\_{\text{\textquotedblleft}} + L\_r \hat{\dot{\imath}}\_{r\text{\textquotedblright}} \right). \tag{130}
$$

In the observer structure, the rotor speed is not estimated adaptively, therefore the rotor speed error in (120) is not considered. The rotor speed value can be estimated from the non-adaptive scheme. From (129)–(130), after some calculation, the rotor speed value can be determined from

$$
\hat{\rho}\_r = \frac{\hat{H}\_x \hat{\boldsymbol{\mu}}\_{rx} + \hat{H}\_y \hat{\boldsymbol{\mu}}\_{ry} - \mathbf{c}\_f \mathbf{s}\_w}{\hat{\boldsymbol{\mu}}\_{rx}^2 + \hat{\boldsymbol{\mu}}\_{ry}^2},
\tag{131}
$$

*Robust Mechanism for Speed and Position Observers of Electrical Machines DOI: http://dx.doi.org/10.5772/intechopen.107898*

where

$$
\sigma\_{\rm av} = \hat{H}\_{\rm x} \hat{\boldsymbol{\mu}}\_{\rm ry} - \hat{H}\_{\rm y} \hat{\boldsymbol{\mu}}\_{\rm rx}, \tag{132}
$$

$$
\hat{\Psi}\_{r\mathbf{x}} = L\_m \mathbf{i}\_{\mathbf{x}} + L\_r \hat{\mathbf{i}}\_{r\mathbf{x}},\tag{133}
$$

$$
\hat{\Psi}\_{ry} = L\_m \dot{\mathbf{i}}\_{ry} + L\_r \hat{\mathbf{i}}\_{ry},\tag{134}
$$

*cf* ≥ 0 and *ψ*^<sup>2</sup> *rx* <sup>þ</sup> *<sup>ψ</sup>*^<sup>2</sup> *ry* 6¼ 0.

In **Figure 10a**, the responses of DFIG on the active and reactive power changes are shown. After 0.1 s, the active power *sp* value is changed from �0.1 to �0.35, and reactive power is set to �0.6 p.u. and changed at the same time. After 0.4 s, the active and reactive powers are changed *sp* to 0.35 and *sq* to 0.2 p.u. The rotor speed estimation error is smaller than 0.015 in the dynamic states, the same as the rotor position error.

In **Figure 10b**, the active power *sp* is set to 0.02 p.u. and reactive power *sq* is set to �0.6 p.u. The rotor speed of the DFIG is changed from the sub-synchronous to supersynchronous mode. Close to synchronous rotor speed (1.0 p.u.), the estimated speed error was smaller than 0.01 p.u., and it is increasing when the speed is growing. The rotor position error has almost the same value.

In **Figure 11a**, the active power is changed from �0.1 to �0.35 p.u. The rotor speed estimation error is smaller than 0.05 p.u. and the rotor position is smaller than 0.1 p.u. during these changes (in the experimental results). In **Figure 11b**, the reactive power is changed from �0.7 to �0.4 p.u., the rotor speed error is smaller than 0.035 p.u., and the rotor position error value is changed from 0.05 to �0.05 p.u.

In **Figure 12a**, the rotor speed crosses from �1.1 (super-synchronous mode) to �0.7 p.u. (sub-synchronous mode). The estimated value of rotor position is growing and close to synchronous speed (�1.0 p.u.) achieving the value of about 0.12. The reactive power value was set to �0.6 p.u. With LPF, the filtered value of stabilizing

**Figure 11.**

*The changes: a) the active power from 0.1 to 0.35 p.u., b) reactive power from 0.7 to 0.4 p.u. – Experimental results.*

#### **Figure 12.**

*The waveforms of chosen variables in a) the rotor speed are changed from the super-synchronous to sub-synchronous mode, b) the steady state – Experimental results.*

function ^*s<sup>ω</sup><sup>f</sup>* is presented. The value of this function is about 0.005 for the supersynchronous mode and about 0.001 p.u. for the sub-synchronous mode. The estimated rotor speed error is about 0.05 p.u. during the crossing through the synchronous speed.

## **6. Conclusions**

In this chapter, robust mechanism for different structures of speed observers or rotor position was presented. The solution was tagged "robust mechanism" because of the introduction of stabilizing function in the speed or rotor position estimation schemes. The additional stabilization law prevents the unstable working range of the speed observer structure (positive poles of the observer). In this chapter, the stability analysis based on the Lyapunov function was presented. The introduced additional stabilizing function to the observer structure is based on Lagrange's identity, which is the main contribution of this chapter. The form of the proposed robust mechanism is based mainly on the vector and scalar product of the two chosen vectors in the observer system. The mutual position of these vectors directly influences the position of the estimated vectors of the observer and also influences the estimated rotor speed value or the rotor position. The mutual position of a vector influences the value of the estimated electromagnetic torque of the machine. The proposed solution has significant meaning during the low speed of the IM and IPMSM (due to the unstable working points of the observer structure), as well as during the synchronous rotor speed of the DFIG system. The proposed robust mechanism for the speed estimation scheme can be applied to each observer structure, which is based on the space vector form of the mathematical model of an observer system.

## **Nomenclature**



## **Author details**

Marcin Morawiec Department of Electric Drives and Energy Conversion, Gdańsk University of Technology, Gdańsk, Poland

\*Address all correspondence to: marcin.morawiec@pg.edu.pl

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Robust Mechanism for Speed and Position Observers of Electrical Machines DOI: http://dx.doi.org/10.5772/intechopen.107898*

## **References**

[1] Orlowska-Kowalska T, Korzonek M, Tarchała G. Stability analysis of selected speed estimators for induction motor drive in regenerating mode—A comparative study. IEEE Transactions on Industrial Electronics. 2017;**62**(10)

[2] Kubota H, Matsuse K, Nakano T. "DSP-based speed adaptive flux observer of induction motor". IEEE Transactions on Industry Applications. Mar./Apr. 1993;**29**:344-348

[3] Yin Z, Li G, Zhang Y, Liu J. Symmetric-strong-trackingextended-Kalman-filter-based Sensorless control of IM drives for Modeling error reduction. IEEE Transactions on Industrial Informatics. 2019;**15**:650-662

[4] Comanescu M. Design and implementation of a highly robust sensorless sliding mode observer for the flux magnitude of the induction motor. IEEE Transactions on Energy Conversion. 2016;**2**:649-657

[5] Morawiec M. Z type observer backstepping for induction machines. IEEE Transactions on Industrial Electronics. 2015;**62**(4):2090-2103

[6] Morawiec M, Kroplewski P, Odeh C. Nonadaptive rotor speed estimation of induction machine in an adaptive fullorder observer. IEEE Transactions on Industrial Electronics. 2022;**69**:2333-2344

[7] Kubota H, Matsuse K. Speed Sensorless field-oriented control of induction motor with rotor resistance adaptation. IEEE Transactions on Industry Applications. 1994;**30**: 1219-1224

[8] Sunwankawin S, Sangwongwanich S. Design strategy of an adaptive full-order observer for speed-sensorless induction

motor drive – Tracking performance and stabilization. IEEE Transactions on Industrial Electronics. 2006;**53**(1): 96-119

[9] Sun W, Liu K, Jiang D, Ou R. Zero synchronous speed stable operation strategy for speed Sensorless induction motor drive with virtual voltage injection. IEEE Energy Conversion Congress and Exposition (ECCE). 2018; **2018**:337-343. DOI: 10.1109/ ECCE.2018.8557756

[10] Chen J, Huang J. Globally stable speed-adaptive observer with auxiliary states for Sensorless induction motor drives. IEEE Transactions on Power Electronics. 2019;**34**(1):33-39

[11] Hinkkanen M, Luomi J. "Stabilization of regenerating-mode operation in sensorless induction motor drives by full-order flux observer design". IEEE Transactions on Industrial Electronics. Dec. 2004;**51**(6): 1318-1328

[12] Glumineau A, Morales JL. Sensorless AC Electric Motor Control Robust Advanced Design Techniques and Applications. Cham: Springer; 2015

[13] Gou L, Wang C, You X, Zhou M, Dong S. IPMSM sensorless control for zero- and low-speed regions under low switching frequency condition based on fundamental model. IEEE Transactions on Industrial Electronics. 2021;**2021**

[14] Kim S, Im J, Song E, Kim R. A new rotor position estimation method of IPMSM using all-pass filter on highfrequency rotating voltage signal injection. IEEE Transactions on Industrial Electronics. 2016;**63**: 6499-6509

[15] Ioannou P, Sun J. Robust Adaptive Control. Englewood cliffs, NJ: Prentice Hall; 1996

[16] Howard A, Rorres C. Relationships between dot and cross products. In: Elementary Linear Algebra: Applications Version. 10th ed. Hoboken, NJ: John Wiley & Sons; 2010. p. 162

[17] Laskhmikanthan V, Leela S, Martynyuk A. Practical Stability of Nonlinear Systems. Singapore: Word Scientific; 1990

[18] Traoré D, Plestan F, Glumineau A, Leon J. Sensorless induction motor: High-order sliding-mode controller and adaptive interconnected observers. IEEE Transactions on Industrial Electronics. 2008;**55**(11):3818-3827

[19] Krzeminski Z. A new speed observer for control system of induction motor. In: IEEE Int. Conference on Power Electronics and Drive Systems, PESC'99. Hong Kong; 1999

[20] Xiaoyuan W, Pingxin W, Jiahong Y, Gengji W. Study the Effects of Slotted Rotor on q-axis Direction on Permanent Magnet Synchronous Motor. In: 15th International Conference on Electrical Machines and Systems (ICEMS). Sapporo, Japan: Hokkaido Citizens Actives Center; 2012

[21] Jung S, Kobayashi H, Doki S, Okuma S. An improvement of sensorless control performance by a mathematical modelling method of spatial harmonics for a SynRM. In: International Power Electronics Conference - ECCE ASIA. 2010. pp. 2010-2015

[22] Agrawal J, Bodkhe S. Experimental study of low speed Sensorless control of PMSM drive using high frequency signal injection. Power Engineering and

Electrical Engineering. 2016;**14**(1). DOI: 10.15598/aeee.v14i1.1564

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[24] Morawiec M, Blecharz K, Lewicki A. Sensorless rotor position estimation of doubly-fed induction generator based on backstepping technique. IEEE Transactions On Industrial Electronics. 2020;**67**(7):5889-5899

## **Chapter 5**

## Classical Direct Torque Control for Switched Reluctance Motor Drive

*V. Pushparajesh and B.M. Nandish*

## **Abstract**

The modern electrical machines require higher efficiency in concern with pollution of the environment. Industries are focusing on bringing out new avenues in controlling the electric motors to adjust the speed and torque without compromising. The Direct Torque Control technique is suggested in this study. Slip control, which exploits a peculiar link between slip and torque, is the basic concept underlying this regulation. Direct torque control provides various benefits over field-oriented control, including reduced sensitivity to machine parameters, easy assembly, and quick dynamic torque response. As the voltage space vector is chosen in response to the inaccuracy in the flux linkage and torque, a current controller is unnecessary in this design. Low torque ripple, reduced noise, and reduced mechanical vibration are all attainable through proper torque management in the switching reluctance motor.

**Keywords:** direct torque control (DTC), field-oriented control, dynamic torque response, flux linkage, slip control, torque control

## **1. Introduction**

Because of its simple mechanical structure, low cost, efficiency, The Switched Reluctance Motor (SRM) has the potential to become one of the most widely used low-cost electromechanical energy converters due to its advantageous torque/speed characteristic and very minimal requirement for maintenance. However, this drive's non-uniform torque output characteristics and doubly salient construction mean it generates more noise and torque ripples, limiting its usefulness. As a result, various methods have been developed for reducing torque ripple in switching reluctance motors. Several torque control techniques are studied with the goal of enhancing the drive's efficiency through reduced torque ripple and faster response times.

The early 1980s saw the development of Direct Torque Control for use with AC drives. In 2012, Yong Chang Zhang, et al., proposed a new direct torque control for three-level inverter supplied AC drives [1]. By adjusting the state voltage vectors in relation to the torque and flux errors, we are able to exert direct control over the torque. Direct torque control for a switching reluctance motor was developed by Moron et al. using the lyapunov function [2]. In order to precisely control the torque applied to a switching reluctance motor, Sahoo et al. presented a lyapunov function [3]. It was proposed by Yong Chang Zhang et al. in 2012 that a sensorless drive for a

three-level inverter-fed induction motor could benefit from enhanced direct torque control [1]. A low-ripple torque control at high speeds was implemented by Jin Ye et al. for switching reluctance motor drives [4].

Improvements in switching reluctance motor performance were studied by Qingguo Sun et al., who looked into the role of direct torque control and torque sharing function [5]. Conventional direct torque control of the four-phase switching reluctance motor was created by Srinivas Pratapgiri and Prasad Polaki Venkata Narsimha in 2012 [6]. As the bandwidth of the hysteresis controller is restricted in this control, the decrease of torque ripple is minimal at best. For switching reluctance motors, proposed a method of shared control of current and flux linkage [7]. For switching reluctance motors, Jipun and Luk achieved sensorless direct torque control [8]. This article discusses a machine with a shorter flux path and modifies the electrical and mechanical phases so that they both add up to 45 degrees. When it comes to the direct torque management of a switching reluctance motor, Bosra et al. proposed a four-level converter [9].

Due to the odd number of phases, there is a paucity of material on direct torque control with four phase SRM. Compared to field orientation control, the DTC's many benefits include reduced reliance on machine parameters, a quicker dynamic torque response, and a more straightforward design. Any current controller is unnecessary for the DTC as the voltage space vectors are chosen in response to flux linkage and torque faults. Direct torque control, or DTC, is a method whereby a motor's torque and speed are adjusted in response to changes in the motor's electromagnetic field.

## **2. Control strategy**

When the converter switches, the motor's flux and torque are directly controlled, making direct torque control an optimal AC variable frequency control concept. **Figure 1** depicts the rudimentary DTC block diagram. Based on the estimated flux and torque and the reference flux and torque, the stator voltage and resistance may be calculated. Based on the output levels of this hysteresis comparator, the error is sent to the hysteresis comparator, and the switching table is used to determine the voltage vector that will be supplied to the voltage source inverter in order to obtain the reference torque. Direct torque control for a four-phase switching reluctance motor is

**Figure 1.** *Block diagram of classical DTC.*

#### *Classical Direct Torque Control for Switched Reluctance Motor Drive DOI: http://dx.doi.org/10.5772/intechopen.107876*

outlined, together with its underlying principles, the specific steps involved, and the means for putting it into practise [10].

Because of their unique four-phase, eight-by-six-polar configuration, synchronous motors cannot be controlled using the same direct torque technique as inductance motors. SRM employs the reluctance principle for producing torque, with each phase functioning separately and sequentially. Torque is generated in either a positive or negative direction depending on the magnitude of the change in stator flux amplitude in relation to the rotor's location. We call the former "flux acceleration" and the latter "flux deceleration," respectively, when the value is positive or negative. Therefore, the following constitutes a definition of a novel approach to SRM regulation [11].


The control goal is accomplished by varying the voltage vector and speeding up or slowing down the stator flux vector in relation to the rotor rotation [12]. The magnitude of the torque is also a function of the instantaneous current, which is different from the way things are handled with traditional control. It is also shown that the stator current in this drive control system exhibits a first order delay with respect to the variation in stator flux. In this way, it is safe to assume that the current remains stable even if the flux is sped up or slowed down under control [13]. This permits the control method to regulate torque solely with respect to the value of the flux acceleration and deceleration, and independently of the current change. This is similar to the traditional control scheme, which assumes that the rotor flux remains unchanged despite variations in the stator flux and modifies the motor's torque via regulation of the stator flux acceleration [14].

## **3. Impact of voltage vectors**

The Direct Torque Control loop has a torque hysteresis controller with three levels (1, 0, and 1) and a flux hysteresis controller with two levels (1 & 1). The SR motor has a distinctive pole structure, hence the voltage space vector for each phase is said to be perpendicular to the pole's central axis. Keep in mind that the physical winding topology of the motor has not been altered from its standard setting.

Each motor phase can be in one of three voltage states, determined on the drive's circuit topology configuration. A zero-voltage loop arises and is defined as the condition "0" when current is flowing with one device off (V1 or V2). Similar to how the condition is described as "1" when both devices (V1&V2) are switched off and "1" when both devices (V1&V2) are turned on, the motor phase experiences a negative voltage when both are off and a positive value when both are on. **Table 1** displays the three voltage states that can exist in a four-phase SRM drive, which is a result of the switching function.

As a result, there are (mn, where m represents no of voltage states and n represents no of phases) 81 distinct permutations, as opposed to just two for the classical DTC of the AC motor. **Figure 2** depicts the eight alternative spatial locations of switching voltage vectors for defining the voltage states, which are defined in the same way as in the standard direct torque control algorithm but with equal amplitude voltage vectors that are separated by radians. The mathematical model of IM has typically been


#### **Table 1.**

*Switching function table.*

**Figure 2.** *Spatial location of switching voltage vectors.*

analysed using the dq coordinate system. Within a two-axis rotating reference frame, the abc to dq0 transformation determines the direct axis, quadratic axis, and zero sequence qualities of a three-phase sinusoidal signal. The park transformation is a standard tool for modelling three-phase electric machines. Because the stator and rotor values can be referred to a stationary or rotating reference frame, time-varying inductances can be eliminated.

Each stator winding's flux linkage is assumed to be at the magnetic pole's centreline for the sake of convenience. **Figure 3** shows the stable α-β coordinate as a result. In Eqs. (1) to (4), and Ψα and Ψβ stand for the two-flux linkage that flows in the two equivalent rotors, generating the same flux as the stator ψ1, ψ2, ψ<sup>3</sup> and ψ<sup>4</sup> currents (1.4).

$$\Psi \mathfrak{a} = \frac{1}{\sqrt{2}} \left[ \Psi \mathfrak{1} + \Psi \mathfrak{2} \cos 4\mathfrak{5} - \Psi \mathfrak{4} \cos 4\mathfrak{5} \right] \tag{1}$$

$$
\Psi\mathfrak{P} = \frac{1}{\sqrt{2}} \left[ \Psi\mathfrak{J} + \Psi\mathfrak{L}\sin 4\mathfrak{S} + \Psi\mathfrak{A}\sin 4\mathfrak{S} \right] \tag{2}
$$

$$
\Psi \mathbf{\dot{s}} = \sqrt{\Psi \mathbf{a}^2 + \Psi \mathbf{\dot{\beta}}^2} \tag{3}
$$

$$\delta = \text{arrt}(\Psi\_{\emptyset}, \Psi\_a) \tag{4}$$

where, Ψ<sup>s</sup> is instantaneous composition flux linkage, is spatial position angle of composition flux linkage.

*Classical Direct Torque Control for Switched Reluctance Motor Drive DOI: http://dx.doi.org/10.5772/intechopen.107876*


#### **Table 2.**

*Switching voltage vector for converter.*

**Figure 3.** *Composition flux linkage vector of switched reluctance motor.*

Coordinate decomposition concept yields composition flow linkage height that is 1.4% greater than energy conservation approach, as seen in the aforementioned formulae. As a result, that SRM will spend a lot of time operating at the magnetic saturation point. While steady error is still ensured, motor efficiency drops dramatically towards magnetic saturation. Using the α-β vector block, we can determine which part of the plane the flow vector occupies. Sector of the plane in which the flow vector lies is one of eight, with each sector separated from the others by 45 degrees. If the stator flux linkage is in the kth region, increasing the flux with the switching vectors U k+1 and U k1, or decreasing it with U k+3 and U k3, is possible. Switching vectors U k +1 and U k +3 can be used to enhance the torque, whereas U k1 and U k3 can be used to decrease it. **Table 2** illustrates the converter's voltage switching vector selection. There are eight "active" voltage vectors labelled U1 through U8 and two "null" vectors labelled U0 and U9.

#### **3.1 Classical direct torque control results**

The aforementioned technique is used to model the direct torque control scheme implemented in a MATLAB/Simulink simulation of a four-phase switching reluctance motor drive. One torque hysteresis controller with three levels and another flux hysteresis controller with two levels are used to create the simulation model. With reference to equations 1.2–1.4, the, α, β flux transformation is performed. The phase count and rotor-position angle inform the design of the flux sector. The voltage switching vectors form the basis for the lookup table. In the virtual experiment, a 4-phase motor is used. Each zone's switching vector is chosen from the vector table based on the output signals from the two-hysteresis comparator, which are in turn determined by the position sensor. The stator flux hysteresis band in this control approach is set to 0.01Wb, and the torque hysteresis band is set to 0.3Nm; both values are held within these bounds throughout all simulations. Within 50 milliseconds, the motor will have reached the desired speed, having drawn a starting current that was capped by the converter's components' maximum ratings. The tests are performed at a constant stator flux of 0.3Wb under a wide range of load torques.

Using a proportional integral controller with kp and ki set to 0.10 and 0.01 respectively, the speed error is transformed into the reference torque. The simulation was run to examine how well the hysteresis controller worked under varying speeds and loads. The 1.5 kilowatt, 3800 revolution per minute, 4 Newton-meter reluctance drive is put through its paces with a dc voltage of 120 volts. It has been decided that 10mH will be the aligned Inductance and 49mH will be the unaligned Inductance. We've decided on a value of 0.008kg.m.m for the moment of inertia and 0.01N.m.s for the coefficient of friction. Below are examples of switching reluctance motor drive performance under a variety of load and speed scenarios.

## **4. Performance at rated load condition**

When first activated, the switching reluctance drive is subjected to the rated load torque and speed. In **Figure 4**, we see the current and torque response in this situation. The phase current and total torque time scale variation is assumed to be 0.71– 0.76 seconds for simplicity in interpreting the responses. It can be seen from the curve that the rated condition yields a maximum current of 8A.

The hysteresis controller minimises torque inaccuracy by choosing the best switching vector, which in turn minimises torque ripple as depicted in the image. Using the following Equation, one can get both the total torque production and the torque ripple in percentage terms (5).

$$\mathbf{T\_{ripple}} = \left[ (\mathbf{T\_{max}} - \mathbf{T\_{min}}) / \mathbf{T\_{avg}} \right]^\* \mathbf{100\%}\tag{5}$$

The figure is further enlarged to specify the variation of the torque ripple accurately at rated torque over a wide range of speed and is shown in **Figure 5**.

The response curve (a) indicates that the torque achieves a minimum of 3.94 Nm and a maximum of 4.15 Nm, with an average of 4.0 Nm reached at 0.09 s. Torque is 5.2% of the total estimated force. According to the torque response curve(b), the specified torque is reached in 0.09 seconds, after which the output torque ranges between a maximum of 4.18 Nm and a minimum of 3.92Nm. An average torque value of 4.01Nm was measured. The percentage of torque ripple is 6.47 percent. In the torque response curve (c), the rated torque is reached in 0.09 s, following which the output torque ranges between a maximum of 4.2 Nm and a minimum of 3.9 Nm. Torque production (Tavg) is measured to be 4.05 Nm on average. Torque ripple is 7.4 percent, according to the calculations.

**Figure 4.** *Current and torque response of the motor at rated load condition.*

**Figure 5.** *Torque ripple waveforms at rated torque (a) Rated speed (b) Half rated speed (c) 10% of rated speed.*

## **5. Performance at 75% of the rated load**

At first, the SRM drive is stimulated at 75% load torque at a variety of rated speeds. **Figure 6** displays the resulting current and torque response. According to the graph, the rated condition produces a maximum current of 6A.

**Figure 6.** *Phase current and torque response at 75% of the rated load with rated speed.*

The suggested controller outperforms the alternative proposed controller in terms of torque response and current response under the aforementioned conditions. **Figure 7** enlarges the torque response curve displayed in **Figure 6** to show the precise fluctuation of torque under 75% of the rated load situation with respect to the extensive range of speed variation.

In 0.08 seconds, the output torque (Tout) reaches 75% of the specified torque, and from there it varies between a maximum of 3.13 Nm and a minimum of 2.98 Nm, as

**Figure 7.**

*Torque response at 75% of the rated load (a) Rated speed (b) half of rated speed (c) 10% of rated speed.*

shown in the response curve (a). Generally speaking, 3.0Nm is the average torque production (Tavg) that was measured. Torque ripple is 5.0% as determined by the calculation.

Torque output (Tout) reaches 75% of rated torque in 0.08 seconds, as shown by the response curve (b), and then ranges between a maximum of 3.18 Nm and a minimum of 2.98 Nm. Torque production (Tavg) is measured to be 3.1 Nm on average. 6.4% is the percentage value derived for the torque ripple. Torque output (Tout) reaches 75% rated torque in 0.08 seconds, and after that, it ranges between a maximum of 3.12 Nm and a minimum of 2.90 Nm, as shown by the torque response curve (c). It has been determined that the average torque output (Tavg) is 3.05 Nm, and that the torque ripple is 7.2% of that value. The output torque (Tout) achieves 75% of rated torque in 0.08 s, as shown by the response curve (a), and then ranges between a maximum (Tmax) of 3.13 Nm and a minimum (Tmin) of 2.98 Nm.

Generally speaking, 3.0Nm is the average torque production (Tavg) that was measured. Torque ripple is 5.0% as determined by the calculation. Torque output (Tout) reaches 75% of rated torque in 0.08 seconds, as shown by the response curve (b), and then ranges between a maximum of 3.18 Nm and a minimum of 2.98 Nm. Torque production (Tavg) is measured to be 3.1 Nm on average. 6.4% is the percentage value derived for the torque ripple. Torque output (Tout) reaches 75% rated torque in 0.08 seconds, and after that, it ranges between a maximum of 3.12 Nm and a minimum of 2.90 Nm, as shown by the torque response curve (c). It has been determined that the average torque output (Tavg) is 3.05 Nm, and that the torque ripple is 7.2% of that value (**Figure 8**).

In order to clearly demonstrate the fluctuation in torque under 50% of the rated load condition with respect to the different speed conditions, the above depicted torque response curve has been extended. **Figure 9** demonstrates the observable range of values.

**Figure 8.** *Current and torque response at 50% of the load torque with rated speed.*

#### **Figure 9.** *Torque response at 50% of the rated load (a) Rated speed (b) half rated speed (c) 10% of rated speed.*

In 0.08 seconds, the output torque (Tout) reaches 50% of its rated value, and from there it varies between a maximum (Tmax) of 2.1 Nm and a minimum (Tmin) of 2.0 Nm, as shown in the response curve (a). Torque production (Tavg) is calculated to be 2.02 Nm on average. 4.9% is the percentage value determined to be the torque ripple output. The torque output (Tout) reaches 50% of the specified torque in 0.08 s, and then it ranges between 2.11 Nm and 1.99 Nm (Tmax and Tmin, respectively) as shown in the torque response curve (b). Torque production (Tavg) was measured to be 2.0Nm on average. 6 percent is the calculated ripple percentage of torque. The torque output (Tout) reaches 50% of the specified torque in 0.08 seconds, as shown by the curve (c), and then ranges between a maximum (Tmax) of 2.13 Nm and a minimum (Tmin) of 1.99 Nm.

Generally speaking, we can say that the torque output (Tavg) is 2.02 Nm. Torque ripple is determined to be 6.9% of total output.

## **6. Performance at 25% of the rated load torque**

In order to trigger the SRM dive, a 25% load torque is applied while the speed is varied. We evaluate the performance of the proposed controller by measuring and tabulating the torque and current responses. **Figure 10** depicts this rated-speed torque and current response. This curve shows that at the rated condition, the maximum current is 2A.

The aforementioned torque response curve is simplified to clearly demonstrate the accurate fluctuation of torque under 25% of the rated load situation with respect to the broad range of speed variation. **Figure 11** displays this variance.

It can be seen in the torque output profile (a) that the output torque (Tout) reaches 25% of the specified torque in 0.07 seconds, and then ranges between a maximum of 1.01 Nm and a minimum of 0.99Nm. The computed proportion of torque ripple output is 2.0%, and the average torque output (Tavg) is 1.00Nm.

Torque output (Tout) achieves 25% rated torque in 0.07 s, as shown by the torque response curve (b), and then ranges between a maximum of 1.015 Nm and a minimum of 0.99 Nm. To be precise, Tavg is 1.0Nm, which is the average torque production. According to the numbers, the torque ripple is 2.5%.

*Classical Direct Torque Control for Switched Reluctance Motor Drive DOI: http://dx.doi.org/10.5772/intechopen.107876*

**Figure 10.** *Current and torque profile for 25% of the load torque at rated speed.*

**Figure 11.** *Torque profile for 25% of the load torque (a) Rated speed (b) half rated speed (c) 10% of rated speed.*

Torque output (Tout) achieves 25% rated torque in 0.07 s, as suggested by the response curve (c), and then ranges between a maximum of 1.02 Nm and a minimum of 0.99 Nm. Torque production (Tavg) averaged out to be 1.01 Nm. The percentage of torque output is 3.0%.

## **7. Performance under variable load condition**

In the dynamic simulation, the speed was maintained at a constant rated condition throughout. There were two potential scenarios: It took 0.34 seconds to double the

**Figure 12.** *Torque profile under external load variation.*

command torque Tcom from the first case shown in **Figure 12** to the second case whereas it took 0.65 seconds to halve the torque back down to 2 N.m from 4 N.m. According to the figure, the controller has excellent dynamic performance, with Tout increasing to 2 Nm in just 0.06s.

The most significant drawback of the conventional direct torque control is its slow response to initial torque and flux changes. Changes between steady state and step-up states use the same vectors, making it impossible to differentiate between large and tiny errors in flux and torque. Both AI and conventional techniques of control over AI can alleviate these issues. To smooth out the driving torque at any rated speed and torque, the next few chapters will focus on such smart controllers.

## **8. Comparative study**

An analysis is made on Neural Network Controller and the revealed observations are made; **Table 3** shows the performance comparisons between different controllers.


**Table 3.**

*Performance comparison between different controllers.*

## **9. Conclusion**

The direct torque control technique is tested for switched reluctance motor, Direct torque Control technique is able to minimize the ripple content in the motor torque output at different operating conditions. the torque ripple is almost minimized in the range of about 1.5% to 2% for a fixed speed with variable torque. The settling time of the torque and the response time of the speed is also reduced, which in turn increases the efficiency of the machine. The major drawback with the proposed controller is the fixation of weights during the real time application which reduce the flexibility and adaptability of the system. This drawback or limitation can be overcome by using hybrid intelligent controller.

## **Author details**

V. Pushparajesh\* and B.M. Nandish Department of Electrical and Electronics, FET, Jain Deemed to be University, Karnataka, India

\*Address all correspondence to: pushparajesh.v@gmail.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

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## *Edited by Miguel Delgado-Prieto, José A. Antonino Daviu and Roque A. Osornio Rios*

Electric machines are widely used in industry, energy, and transport, among other sectors. They are also heavily researched with scientific and technological efforts focusing on three main areas: electric motor technologies, supervision and maintenance strategies, and control schemes. This book provides a comprehensive overview of the technology and applications of electric machines. It is organized into two sections on maintenance and control. Chapters address such topics as signal processing and artificial intelligence, fault detection and identification schemes, predictive maintenance strategies, direct torque control and Lyapunov-based sensorless control, and more.

Published in London, UK © 2023 IntechOpen © Kiruba Arun / iStock

New Trends in Electric Machines - Technology and Applications

New Trends in Electric

Machines

Technology and Applications

*Edited by Miguel Delgado-Prieto,* 

*José A. Antonino Daviu and Roque A. Osornio Rios*