**2. Theory and modeling**

In this chapter, a combination between the SVM technique, the DTC strategy and the IOFL technique is put forward. The SVM is suggested in order to prevent ripples and distortions, and it provides an operation with a fixed switching frequency. IOFL is used in order to achieve decoupled control between the torque and flux quantities. The principle of these techniques is detailed in the following subsections.

### **2.1 Model presentation**

The IM model is presented as follows, which will be used to design the proposed IOFL approach.

$$
\dot{\boldsymbol{\omega}} = \boldsymbol{f}(\boldsymbol{\omega}) + \mathbf{g} \ \boldsymbol{v}\_{s\boldsymbol{a}\boldsymbol{\beta}} \tag{1}
$$

with:

$$\mathbf{x} = \begin{bmatrix} \mathbf{i}\_{sa} & \mathbf{i}\_{s\beta} & \boldsymbol{\phi}\_{sa} & \boldsymbol{\phi}\_{s\beta} \end{bmatrix}^{T} \tag{2}$$

$$f(\mathbf{x}) = \begin{bmatrix} -\frac{1}{\sigma} \left( \frac{1}{T\_r} + \frac{1}{T\_s} \right) i\_{s\alpha} - \alpha m\_i i\_{s\alpha} + \frac{1}{\sigma L\_s T\_r} \phi\_{s\alpha} + \frac{\alpha\_m}{\sigma L\_s} \phi\_{s\beta} \\\\ \alpha\_m i\_{s\alpha} - \frac{1}{\sigma} \left( \frac{1}{T\_r} + \frac{1}{T\_s} \right) i\_{s\beta} - \frac{\alpha\_m}{\sigma L\_s} \phi\_{s\alpha} + \frac{1}{\sigma L\_s T\_r} \phi\_{s\beta} \\\\ -R\_s i\_{s\alpha} \\\\ -R\_s i\_{s\beta} \end{bmatrix} \tag{3}$$

$$\mathbf{g} = \begin{bmatrix} \frac{1}{\sigma L\_s} & \mathbf{0} & \mathbf{1} & \mathbf{0} \\\\ \mathbf{0} & \frac{1}{\sigma L\_s} & \mathbf{0} & \mathbf{1} \end{bmatrix} \tag{4}$$

### where:

*isα*, *is<sup>β</sup>* � � :the stator current components, *vsα*, *vs<sup>β</sup>* � � :the voltage vectors components, *ϕs<sup>α</sup>*, *ϕs<sup>β</sup>* � � :the stator flux vector components, *Rr* ð Þ , *Rs* : the rotor and stator resistance respectively, *Lr* ð Þ , *Ls* : the rotor and stator inductance respectively, *Tr* ð Þ , *Ts* : the rotor and stator time constants, *ωm*(red/sec): the electric rotor speed. *<sup>σ</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>M</sup>*<sup>2</sup> *sr LrLs* : The Blondel coefficient, where Msr presents the mutual inductance.

### **2.2 Space vector modulation**

The classical DTC based on fixed-bandwidth hysteresis controllers produces high ripples and distortions. Indeed, if a larger hysteresis-band of the torque is chosen, the torque ripples increase. For a smaller hysteresis band, the torque ripples are reduced and the switching frequency goes up, which consequently increases the commutation losses in the inverter IGBT transistors [36]. Thus, the SVM technique is proposed in this chapter in order to maintain a fixed switching frequency and reduce the ripples [37, 38]. The SVM principle consists in modulating reference voltage vector components in order to generate the more appropriate voltage vector that characterizes

**Figure 1.** *Voltage vectors.*

*Robust Control Based on Input-Output Feedback Linearization for Induction Motor Drive… DOI: http://dx.doi.org/10.5772/intechopen.104645*

inverter control signals. As shown in **Figure 1**, the reference voltage vector can be determined by projecting it on the two vectors that bound the sector, using Eq. (5).

The time allowed for each voltage vector application can be determined by vector calculations. The rest of the sampling period can be filled by applying the null vector in order to grantee a fixed switching frequency [39, 40]. An example for the first section, by projection on vectors V1 and V2, the voltage vector application times T1 and T2 are given by Eq. (5):

$$\begin{cases} \vec{V}\_S = V\_{sa}^\* + jV\_{s\theta}^\* = \frac{T\_1}{T\_m}\vec{V}\_1 + \frac{T\_2}{T\_m}\vec{V}\_2\\\\ \vec{V}\_1 = \sqrt{\frac{2}{3}}U\_{dc}(\cos\left(0\right) + j\sin\left(0\right)) = \sqrt{\frac{2}{3}}U\_{dc} \\\\ \vec{V}\_2 = \sqrt{\frac{2}{3}}U\_{dc}\left(\cos\left(\frac{\pi}{3}\right) + j\sin\left(\frac{\pi}{3}\right)\right) \\\\ T\_m = T\_1 + T\_2 + T\_0 \\\\ T\_1 = \left(\sqrt{\frac{3}{2}}V\_{Sx} - \frac{1}{\sqrt{2}}V\_{S\theta}\right)\frac{T\_m}{U\_{dc}} \\\\ T\_2 = \sqrt{2}V\_{Sx}\frac{T\_{mod}}{U\_{dc}} \end{cases} (5)$$

where *v* <sup>∗</sup> *<sup>s</sup><sup>α</sup>*, *v* <sup>∗</sup> *sβ* � � represents the components of the reference voltage vector, T1 and T2 denote the commutation time, Tm is the sampling time, and Udc is the DC voltage.

### **2.3 IOFL theory**

This section illustrates the Feedback Linearization (FL) based DTC for an IM drive. The FL technique utilizes an inverse mathematical transformation in order to determine the desired control law for controlling the nonlinear system such as the IM. Furthermore, the FL technique is utilized to obtain decoupled control between the torque and flux. In this study, the suggested system outputs are the electromagnetic torque and the square root of the stator flux norm. Referring to the IOFL theory, the output variables are expressed as:

$$\begin{cases} h\_1(\mathbf{x}) = T\_{em} = \frac{3}{2} N\_p \left( i\_{s\beta} \phi\_{sa} - i\_{sa} \phi\_{s\beta} \right) \\\ h\_2(\mathbf{x}) = \left| \phi\_s \right|^2 = \phi\_{sa}^2 + \phi\_{s\beta}^2 \end{cases} \tag{6}$$

where *Tem* is the estimated electromagnetic torque, and *ϕ<sup>s</sup>* j j is the norm of the stator flux. Assuming the controller objectives y1 and y2 as, we get:

$$\begin{cases} \mathcal{y}\_1 = h\_1(\mathbf{x}) - T\_{em}^\* = T\_{em} - T\_{em}^\* \\ \mathcal{y}\_2 = h\_2(\mathbf{x}) - \left| \phi\_\varepsilon^\* \right|^2 = \left| \phi\_\varepsilon \right|^2 - \left| \phi\_\varepsilon^\* \right|^2 \end{cases} \tag{7}$$

where *T*<sup>∗</sup> *em* and *ϕ*<sup>∗</sup> *s* � � � � are the torque and flux references, respectively. Utilizing the presented equations, the time derivative of the controller objectives can be written as:

$$
\begin{bmatrix}
\dot{\mathcal{V}}\_1 \\
\dot{\mathcal{y}}\_2
\end{bmatrix} = \begin{bmatrix}
\mathbf{g}\_1(\mathbf{x}) \\
\mathbf{g}\_2(\mathbf{x})
\end{bmatrix} + G(\mathbf{x}) \begin{bmatrix}
\boldsymbol{v}\_{s\boldsymbol{a}} \\
\boldsymbol{v}\_{s\boldsymbol{\theta}}
\end{bmatrix} \tag{8}
$$

with:

$$\begin{cases} \begin{aligned} \mathbf{g}\_{1}(\mathbf{x}) &= \frac{3}{2} N\_{p} \left[ -\frac{1}{\sigma} \left( \frac{1}{T\_{r}} + \frac{1}{T\_{s}} \right) \phi\_{sa} \, i\_{sa} + \alpha\_{m} \phi\_{sa} \, i\_{sa} - \frac{\alpha\_{m}}{\sigma L\_{s}} \phi\_{sa}^{2} \right. \\ &\left. + \frac{1}{\sigma} \left( \frac{1}{T\_{r}} + \frac{1}{T\_{s}} \right) \phi\_{s\beta} \, i\_{sa} + \alpha\_{m} \phi\_{s\beta} \, i\_{s\beta} - \frac{\alpha\_{m}}{\sigma L\_{s}} \phi\_{s\beta}^{2} \right] - \mathbf{T}\_{cm}^{\*} \\\\ \mathbf{g}\_{2}(\mathbf{x}) &= -2R\_{r} \phi\_{sa} \, i\_{sa} - 2R\_{r} \phi\_{s\beta} \, i\_{s\beta} - \left| \phi\_{s}^{\bullet} \right| \\\\ \mathbf{G}(\mathbf{x}) &= \begin{bmatrix} 2\phi\_{sa} & 2\phi\_{s\beta} \\\\ \frac{3}{2} N\_{p} \left( i\_{s\beta} - \frac{1}{\sigma L\_{s}} \phi\_{s\beta} \right) & \frac{3}{2} N\_{p} \left( i\_{sa} - \frac{1}{\sigma L\_{s}} \phi\_{sa} \right) \end{bmatrix} \end{aligned} \tag{9}$$

Based on the IOFL technique, the control inputs can be expressed as follows [41].

$$
\begin{bmatrix} v\_{sa} \\ v\_{t\beta} \end{bmatrix} = \mathbf{G}^{-1}(\mathbf{x}) \begin{bmatrix} -\mathbf{g}\_1(\mathbf{x}) + v\_1 \\ -\mathbf{g}\_2(\mathbf{x}) + v\_2 \end{bmatrix} \tag{10}
$$

where *v*<sup>1</sup> and *v*<sup>2</sup> are assumed to be two auxiliary inputs with the purpose of ensuring more desired behavior and tracking accuracy for the torque and the stator flux, with:

$$\begin{cases} v\_1 = -k\_1 y\_1 \\ v\_2 = -k\_2 y\_2 \end{cases} \tag{11}$$

where k1 and k2 are positive constants. The SVM-DTC-IOFL performance strongly depends on the suitable choice of parameters k1 and k2. In fact, the high values of such parameters are able to cause the system instability. On the other hand, the small values will lead to a poor robustness and slow convergence. Finally, it is necessary to better choose such parameters for guarantying high control technique performance [13]. The combination between (8), (10) and (11) gives the following expression:

$$
\begin{bmatrix}
\dot{\mathcal{Y}}\_1 \\
\dot{\mathcal{Y}}\_2
\end{bmatrix} = \begin{bmatrix}
\mathbf{0} & -k\_2
\end{bmatrix} \begin{bmatrix}
\mathcal{Y}\_1 \\
\mathcal{Y}\_2
\end{bmatrix} \tag{12}
$$

Utilizing the IM model, the relation between the rotor and the stator fluxes is given below:

$$\begin{cases} \phi\_{ra} = \frac{\sigma L\_s L\_r}{M\_{sr}} \left( \frac{1}{\sigma L\_s} \phi\_{sa} - i\_{sa} \right) \\\\ \phi\_{r\beta} = \frac{\sigma L\_s L\_r}{M\_{sr}} \left( \frac{1}{\sigma L\_s} \phi\_{s\beta} - i\_{s\beta} \right) \end{cases} \tag{13}$$

Utilizing matrix G(x), defined in (9) and Eq. (13), the determinant of G(x) is given as follows:

$$\mathbf{G}(\mathbf{x}) = \frac{\mathbf{3}\mathbf{M}\_{sr}}{\sigma L\_{s} L\_{r}} \mathbf{N}\_{p} \left(\phi\_{ra}\phi\_{sa} + \phi\_{r\beta}\phi\_{s\beta}\right) \tag{14}$$

*Robust Control Based on Input-Output Feedback Linearization for Induction Motor Drive… DOI: http://dx.doi.org/10.5772/intechopen.104645*

**Figure 2.** *Global diagram of the proposed SVM-DTC-IOFL.*

Referring to Eq. (14), it can be noticed that the product between the rotor flux and the stator flux cannot be zero, and matrix G(x) is nonsingular [42].

The FL control law is used in order to satisfy the stability condition defined by the Lyapunov approach. To study the stability of the control law, the Lyapunov function is given as:

$$\mathbf{V} = \frac{\mathbf{1}}{2} \mathbf{y}^T \mathbf{y} \tag{15}$$

The time derivative of (15) is given as follows:

$$\dot{V} = \mathbf{y}^T \dot{\mathbf{y}} = \begin{bmatrix} y\_1 & y\_2 \end{bmatrix} \begin{bmatrix} -k\_1 & \mathbf{0} \\ \mathbf{0} & -k\_2 \end{bmatrix} \begin{bmatrix} y\_1 \\ y\_2 \end{bmatrix} = -k\_1 \mathbf{y}\_1^2 - k\_2 \mathbf{y}\_2^2 < \mathbf{0} \tag{16}$$

Parameters k1 and k2 are positive, so derivative *V*\_ is negative, which demonstrates the stability of the control system. The global diagram of the proposed SVM-DTC-IOFL is given by **Figure 2**.
