**2.1. Machine model in the** (*α* − *β*) **subspace**

The asymmetrical dual three-phase machine model can be obtained using a specific and convenient choice of state-space variables, for example, stator and rotor currents. Thus the six-phase machine can be modelled in a stationary reference frame according to the VSD approach as:

$$[\mathbf{u}]\_{\alpha\beta} = [\mathbf{G}] \frac{d}{dt} \begin{bmatrix} \mathbf{x} \end{bmatrix}\_{\alpha\beta} + [\mathbf{F}] \begin{bmatrix} \mathbf{x} \end{bmatrix}\_{\alpha\beta} \tag{3}$$

**3. Predictive model**

equations can be written as:

*Sa*, *Sd*, *Sb*, *Se*, *Sc*, *Sf*

*x*˙

*x*˙

*x*˙

*x*˙

where *c*1-*c*<sup>4</sup> are constant coeficients defined as:

�

*<sup>c</sup>*<sup>1</sup> <sup>=</sup> *Ls* · *Lr* <sup>−</sup> *<sup>L</sup>*<sup>2</sup>

**<sup>M</sup>** <sup>=</sup> <sup>1</sup> 3

(*x* − *y*) axes and gathered in a row vector **U***αβxys* computed as:

�

**X** ˙

**U***αβxys* =

with state vector **X**(*t*) = [*x*1, *x*2, *x*3, *x*4]

**Y**(*t*) = [*x*1, *x*2]

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

The machine model must be discretized in order to be of use as a predictive model. Taking into account that the electromechanical energy conversion involves only quantities in the (*α* − *β*) subspace, the predictive model could be simplified, discarding the (*x* − *y*) subspace. Assuming the asymmetrical dual three-phase induction machine model (see Equation 3) and using the following state components (*x*<sup>1</sup> = *iαs*, *x*<sup>2</sup> = *iβs*, *x*<sup>3</sup> = *iαr*, *x*<sup>4</sup> = *iβr*), the resulting

<sup>1</sup> = *c*<sup>3</sup> (*Rrx*<sup>3</sup> + *ωrx*4*Lr* + *ωrx*2*Lm*) + *c*<sup>2</sup> (*uα<sup>s</sup>* − *Rsx*1)

*<sup>m</sup>*, *<sup>c</sup>*<sup>2</sup> <sup>=</sup> *Lr*

<sup>3</sup> = *c*<sup>4</sup> (−*Rrx*<sup>3</sup> − *ωrx*4*Lr* − *ωrx*2*Lm*) + *c*<sup>3</sup> (−*uα<sup>s</sup>* + *Rsx*1)

*c*1

2 0 −1 0 −1 0 020 −1 0 −1 −10 2 0 −1 0 0 −10 2 0 −1 −1 0 −10 2 0 0 −1 0 −10 2

An ideal inverter converts gating signals to stator voltages that can be projected to (*α* − *β*) and

being *Vdc* the Dc Link voltage and superscript (*T*) indicates the transposed matrix. Combining Equations 9-12 a nonlinear set of equations arises that can be written in state space form:

(*t*) = *f* (**X**(*t*), **U**(*t*))

straightforward manner from Equation 9 and the definitions of state and output vector.

*<sup>T</sup>*, input vector **<sup>U</sup>**(*t*) = �

*<sup>T</sup>*. The components of vectorial function *f* and matrix **C** are obtained in a

*<sup>u</sup>αs*, *<sup>u</sup>βs*, *uxs*, *uys*, 0, 0�*<sup>T</sup>*

Stator voltages are related to the control input signals through the inverter model. The simplest model has been selected for this case study for the sake of speeding up the optimization process. Then if the gating signals are arranged in vector **<sup>S</sup>** <sup>=</sup> �

, *<sup>c</sup>*<sup>3</sup> <sup>=</sup> *Lm c*1

<sup>∈</sup> **<sup>R</sup>**6, with **<sup>R</sup>** <sup>=</sup> {0, 1} the stator voltages are obtained from:

�

The Asymmetrical Dual Three-Phase Induction Machine and the MBPC in the Speed Control 389

�

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

**Y**(*t*) = **CX**(*t*) (13)

*uαs*, *uβ<sup>s</sup>* �

*uβ<sup>s</sup>* − *Rsx*<sup>2</sup>

, *<sup>c</sup>*<sup>4</sup> <sup>=</sup> *Ls c*1

−*uβ<sup>s</sup>* + *Rsx*<sup>2</sup>

�

�

· **<sup>S</sup>***<sup>T</sup>* (11)

= *Vdc* · **T** · **M** (12)

, and output vector

(9)

(10)

<sup>2</sup> = *c*<sup>3</sup> (*Rrx*<sup>4</sup> − *ωrx*3*Lr* − *ωrx*1*Lm*) + *c*<sup>2</sup>

<sup>4</sup> = *c*<sup>4</sup> (−*Rrx*<sup>4</sup> + *ωrx*3*Lr* + *ωrx*1*Lm*) + *c*<sup>3</sup>

$$\mathbf{u}[\mathbf{u}]\_{a\not\rhd} = \begin{bmatrix} \boldsymbol{u}\_{a\mathbf{s}} \ \boldsymbol{u}\_{\beta\mathbf{s}} \ \mathbf{0} \ \mathbf{0} \end{bmatrix}^T ; \begin{bmatrix} \mathbf{x} \end{bmatrix}\_{a\not\rhd} = \begin{bmatrix} \mathbf{i}\_{a\mathbf{s}} \ \mathbf{i}\_{\beta\mathbf{s}} \ \mathbf{i}\_{a\mathbf{r}} \ \mathbf{i}\_{\beta\mathbf{r}} \end{bmatrix}^T \tag{4}$$

where [**u**]*αβ* is the input vector, [**x**]*αβ* is the state vector and [**F**] and [**G**] are matrices that define the dynamics of the electrical drive that for this set of state variables are:

$$\begin{aligned} \mathbf{[F]} = \begin{bmatrix} R\_s & 0 & 0 & 0 \\ 0 & R\_s & 0 & 0 \\ 0 & \omega\_r \cdot L\_m & R\_r & \omega\_r \cdot L\_r \\ -\omega\_r \cdot L\_m & 0 & -\omega\_r \cdot L\_r & R\_r \end{bmatrix}; [\mathbf{G}] = \begin{bmatrix} L\_s & 0 & L\_m & 0 \\ 0 & L\_s & 0 & L\_m \\ L\_m & 0 & L\_r & 0 \\ 0 & L\_m & 0 & L\_r \end{bmatrix} \end{aligned} \tag{5}$$

where *ω<sup>r</sup>* is the rotor angular speed, and the electrical parameters of the machine are the stator and rotor resistances *Rs*, *Rr*, the stator and rotor inductances *Ls* = *Lls* + *Lm*, *Lr* = *Llr* + *Lm*, the stator and rotor leakage inductances *Lls*, *Llr* and the magnetization inductance *Lm*. Using selected state-space variables and amplitude invariant criterion in the transformation, the mechanical part of the drive is given by the following equations:

$$T\_{\ell} = \Im P \left( \psi\_{\beta r} i\_{ar} - \psi\_{ar} i\_{\beta r} \right) \tag{6}$$

$$J\_i \frac{d}{dt} \omega\_r + B\_i \omega\_r = P \left( T\_\varepsilon - T\_L \right) \tag{7}$$

where *Te* is the generated torque, *TL* the load torque, *P* the number of pair of poles, *Ji* the inertia coefficient, *Bi* the friction coefficient and *ψαβ<sup>r</sup>* the rotor flux.

The (*<sup>α</sup>* <sup>−</sup> *<sup>β</sup>*)<sup>2</sup> axes are selected in such a manner that they coincide with the plane of rotation of the airgap flux. Therefore, these variables will are associated with the production of the airgap flux in the machine and with the electromechanical energy conversion related [3].
