**4. Induction motor speed control using fractional order PI controllers**

In this section we present a field oriented control scheme like the one shown in Figure 2, where the PI controller used in the speed loop is changed to a PI controller in which the integral order is not unity (fractional integral effect). The main idea to explore is that fractional order integrals are of benefit in this kind of IM controllers.

## **4.1. Fractional order PI controllers**

The FOPI controller is based on the same principles as the classical PI controller, with the difference that in this case the control action is calculated by means of fractional order integrals. The transfer function of a FOPI controller is given by

$$H\_{FOPI}(\mathbf{s}) = k\_{\,\,p} + k\_{\,\,i} \,/\,\,\text{s}^{\nu} \tag{25}$$

Advanced Control Techniques for Induction Motors 317

(27)

 <sup>1</sup> <sup>1</sup> 0 0

*k*

*sFs s D f*

Other definitions commonly used in fractional calculus can be found in Oldham & Spanier

In this study the FOC scheme shown in Figure 2 (changing the PI controllers by FOPI controllers in the spped control block) is used to control the speed of an IM. The "Speed Controller" block shown in Figure 2 corresponds to a FOPI controller, in which the parameter will be modified to analyze its effects on the controlled system. This strategy will be denoted as FOC-FOPI and will be compared with the classical strategy using a standard PI controller (BCS), which will be denoted as FOC-PI. Notice that this case corresponds to the FOC-FOPI strategy when *ν* = 1*.* "Current Controller" block and "Flux Controller" block in Figure 2 correspond to proportional controllers. All these controller parameters (proportional constants) will be kept constant at values indicated in Section 4.2.

The controlled system corresponds to the Siemens 3-phase IM, model 1LA7080 descrbed in Section 3.3. All the simulations shown in this study were performed using MATLAB/Simulink. The following describes the tests performed on the IM, in simulation analysis. These tests were used to determine the general features of FOC-FOPI scheme, and will be compared with FOC-PI scheme. For the sake of space only results concerning the regulation of the controlled system (capacity of the system reject external perturbations at different levels of load) are shown. Although the tracking study (capacity of the system to reach and follow a pre-specified speed reference at different levels of load) was also done (Mira, 2008; Mira & Duarte-Mermoud, 2009; Duarte-Mermoud et al, 2009 ) the simulated results are not shown here.The simulation results will be analyzed and discussed including

**Test 1(Regulation):** The speed reference increases from zero at a rate of 16 [��� �� ⁄ ]) until nominal speed (146.08[��� �⁄ ] � 1���[���]) in �.��[�]. Then the reference is kept constant at the nominal speed until the end of the test (�1�[�]). (See Figure 4). The mechanical load

The FOPI controller has the transfer function shown in (7). In all tests, parameters *kp* and *ki* were kept fixed at 0.5 and 0.05 respectively. These values were chosen after performing a series of preliminary simulation tests, analyzing the stabilization time and the control effort for different values. The values of the proportional constant used in current and flux proportional controllers were chosen to be 45. This value was also determined after a series of preliminary tests. The integration order was changed to explore the system's sensitivity with respect to parameter �. The results shown in the next section include orders 0.7, 1.0 (Classical PI), 1.7 and 2.0. Theoretically the limit of stability is at = 2, a fact verified at the

*sFs*

(1974), Valério (2005), Kilbas et al (2006) and Sabatier et al (2007).

stabilization time, rise time and control effort, among other aspects.

varies during the test as shown in Table 2 and Figure 17.

( ) (0), 0 *<sup>n</sup> kx k*

 

( )

*<sup>d</sup> L fx dx*

**4.2. Simulation set up** 

( ), 0

where *ν* denotes the integration order, *kp* is the proportional constant and *ki* is the integral constant. The detailed computation of fractional integrals is shown (Valério, 2005). Expression (8) allows computing fractional integrals *ν < 0* and fractional derivatives *ν > 0*, corresponding to Caputo's definition (Oldham & Spanier, 1974; Kilbas et al., 2006; Sabatier et al., 2007).

$$\frac{d^{\nu}}{d\boldsymbol{x}^{\nu}} = \begin{cases} \frac{1}{\Gamma(-\nu)} \int\_{\boldsymbol{x}\_{0}}^{\boldsymbol{x}} (\boldsymbol{x} - t)^{-\nu - 1} f(t) dt, & \nu < 0 \\\ f(\boldsymbol{x}), & \nu = 0 \\\ \frac{1}{\Gamma(-\nu)} \int\_{\boldsymbol{x}\_{0}}^{\boldsymbol{x}} (\boldsymbol{x} - t)^{-\nu - 1} \Big[ \boldsymbol{D}^{\boldsymbol{m}} f(t) \Big] dt, & \nu > 0 \end{cases} \tag{26}$$

In this equation *x* corresponds to the integration variable, corresponds to Gamma function, *m* denotes the integer immediately greater than , and *D* denotes the integer derivative with respect to *x*. The Laplace Transform of fractional order derivatives and integrals (according to Caputo's definition) is shown in (9), where ���� = ℒ������ .

$$L\left\{\frac{d^{\nu}}{dx^{\nu}}f(\mathbf{x})\right\} = \begin{cases} \mathbf{s}^{\nu}F(\mathbf{s}), & \nu \le 0 \\\\ \mathbf{s}^{\nu}F(\mathbf{s}) - \sum\_{k=0}^{n-1} \mathbf{s}^{k} \left(\,\_{0}^{\nu}D^{\nu-k-1}\right)f(\mathbf{0}), & \nu > 0 \end{cases} \tag{27}$$

Other definitions commonly used in fractional calculus can be found in Oldham & Spanier (1974), Valério (2005), Kilbas et al (2006) and Sabatier et al (2007).

In this study the FOC scheme shown in Figure 2 (changing the PI controllers by FOPI controllers in the spped control block) is used to control the speed of an IM. The "Speed Controller" block shown in Figure 2 corresponds to a FOPI controller, in which the parameter will be modified to analyze its effects on the controlled system. This strategy will be denoted as FOC-FOPI and will be compared with the classical strategy using a standard PI controller (BCS), which will be denoted as FOC-PI. Notice that this case corresponds to the FOC-FOPI strategy when *ν* = 1*.* "Current Controller" block and "Flux Controller" block in Figure 2 correspond to proportional controllers. All these controller parameters (proportional constants) will be kept constant at values indicated in Section 4.2.

### **4.2. Simulation set up**

316 Induction Motors – Modelling and Control

Annaswamy, 1989) could be used.

**4.1. Fractional order PI controllers** 

et al., 2007).

PBC strategies (Travieso, 2002; Pelisssier, 2006). In particular the effects of initial conditions on APBC strategies, as well as the effects of using fixed and time-varying adaptive gains were analyzed. It was observed, in general, that time-varying gains improve transient behavior and diminish initial control error. In simulations, a small noise was added on these signals and the performance of the method were not affected significantly (González, 2005). At the experimental level, the influence of the normal noise present in the measurement of current signals during the test did not affect the behavior of the APBC. For higher noise levels some deterioration of the control system behavior was observed. In this case, more robust adaptive laws should be used. For instance the -modification (Narendra &

**4. Induction motor speed control using fractional order PI controllers** 

fractional order integrals are of benefit in this kind of IM controllers.

integrals. The transfer function of a FOPI controller is given by

In this section we present a field oriented control scheme like the one shown in Figure 2, where the PI controller used in the speed loop is changed to a PI controller in which the integral order is not unity (fractional integral effect). The main idea to explore is that

The FOPI controller is based on the same principles as the classical PI controller, with the difference that in this case the control action is calculated by means of fractional order

where *ν* denotes the integration order, *kp* is the proportional constant and *ki* is the integral constant. The detailed computation of fractional integrals is shown (Valério, 2005). Expression (8) allows computing fractional integrals *ν < 0* and fractional derivatives *ν > 0*, corresponding to Caputo's definition (Oldham & Spanier, 1974; Kilbas et al., 2006; Sabatier

1

*x t f t dt*

<sup>1</sup> ( ) () , 0 ( )

( ), 0

<sup>1</sup> ( ) () , 0 ( )

*x t D f t dt*

*m*

1

In this equation *x* corresponds to the integration variable, corresponds to Gamma function, *m* denotes the integer immediately greater than , and *D* denotes the integer derivative with respect to *x*. The Laplace Transform of fractional order derivatives and

0

*x*

*x*

0

integrals (according to Caputo's definition) is shown in (9), where ���� = ℒ������ .

*x*

*x*

*<sup>d</sup> f x dx*

/ *p i k ks*

*H (s) FOPI* (25)

(26)

The controlled system corresponds to the Siemens 3-phase IM, model 1LA7080 descrbed in Section 3.3. All the simulations shown in this study were performed using MATLAB/Simulink. The following describes the tests performed on the IM, in simulation analysis. These tests were used to determine the general features of FOC-FOPI scheme, and will be compared with FOC-PI scheme. For the sake of space only results concerning the regulation of the controlled system (capacity of the system reject external perturbations at different levels of load) are shown. Although the tracking study (capacity of the system to reach and follow a pre-specified speed reference at different levels of load) was also done (Mira, 2008; Mira & Duarte-Mermoud, 2009; Duarte-Mermoud et al, 2009 ) the simulated results are not shown here.The simulation results will be analyzed and discussed including stabilization time, rise time and control effort, among other aspects.

**Test 1(Regulation):** The speed reference increases from zero at a rate of 16 [��� �� ⁄ ]) until nominal speed (146.08[��� �⁄ ] � 1���[���]) in �.��[�]. Then the reference is kept constant at the nominal speed until the end of the test (�1�[�]). (See Figure 4). The mechanical load varies during the test as shown in Table 2 and Figure 17.

The FOPI controller has the transfer function shown in (7). In all tests, parameters *kp* and *ki* were kept fixed at 0.5 and 0.05 respectively. These values were chosen after performing a series of preliminary simulation tests, analyzing the stabilization time and the control effort for different values. The values of the proportional constant used in current and flux proportional controllers were chosen to be 45. This value was also determined after a series of preliminary tests. The integration order was changed to explore the system's sensitivity with respect to parameter �. The results shown in the next section include orders 0.7, 1.0 (Classical PI), 1.7 and 2.0. Theoretically the limit of stability is at = 2, a fact verified at the simulation level (Mira & Duarte-Mermoud, 2009). See also Figure 21. Many other integration orders have been analyzed at the simulation level but they are not shown here for the sake of space. The reader is referred to Mira (2008) for more details.

Advanced Control Techniques for Induction Motors 319

**Speed [rad/s]**

**Speed [rad/s]**

**Figure 18.** Simulation results for ߥ ൌ ͲǤͲ

**Figure 19.** Simulation results for ߥ ൌ ͳǤͲͲ

**Figure 20.** Simulation results for ߥ ൌ ʹǤͲͲ

**Figure 17.** Description of Test 1 (Regulation)


**Table 1.** Variation of load torque
