**2. The asymmetrical dual three-phase AC drive**

The asymmetrical dual three-phase induction machine is supplied by a 6-phase voltage source inverter (VSI) and a Dc Link, as shown in Figure 1. This six-phase machine is a continuous system which can be described by a set of differential equations. A methodology that simplifies the modeling is based on the vector space decomposition (VSD) theory introduced in [3] to transform the original six-dimensional space of the machine into three

**Figure 1.** A general scheme of an asymmetrical dual three-phase drive

two-dimensional orthogonal subspaces in stationary reference frame (*α* − *β*), (*x* − *y*) and (*z*<sup>1</sup> − *z*2), by means of a 6 × 6 transformation matrix using an amplitude invariant criterion:

2 Induction Motor

improving the drive performance [9]. This chapter wide the concept of the MBPC techniques to the speed control of a dual three-phase induction machine, by using an Kalman Filter (KF) to improve the estimation of states through an optimal estimation of the rotor current. The KF is an efficient recursive filter that estimates the internal state of a dynamic system from a series of noisy measurements. Its purpose is to use measurements that are observed over time that contain noise (random variations) and other inaccuracies (including modeling errors), and produce values that tend to be closer to the true values of the measurements and their associated calculated values. This feature is an attractive solution in the predictive control of induction machines based on the model, mainly if not precisely known internal parameters of the drive, and the measurement of the state variables are perturbed by gaussian noise.

The chapter includes simulation results of the current control based on a predictive model of the asymmetrical dual three-phase induction machine and proposes a new approach to speed control based on MBPC technique. The results provided confirm the feasibility of the speed control scheme for multi-phase machines. The rest of the chapter is organized as follows. Section 2 introduces an asymmetrical dual three-phase AC drive used for simulations. Section 3 details the general principles of the predictive current control method for AC drives. Section 4 shows the simulation results obtained from the inner loop of predictive current control and proposed a new approach to speed control for the dual three-phase induction machine, on the other hand presents a discussion of the obtained results from the proposed approach. The

The asymmetrical dual three-phase induction machine is supplied by a 6-phase voltage source inverter (VSI) and a Dc Link, as shown in Figure 1. This six-phase machine is a continuous system which can be described by a set of differential equations. A methodology that simplifies the modeling is based on the vector space decomposition (VSD) theory introduced in [3] to transform the original six-dimensional space of the machine into three

*a d be c f*

*a d b e c f*

r

*Sa Sd Sb Se Sc Sf*

*<sup>a</sup> Sd Sb Se Sc Sf*

*is*

*f*

*N N'*

6-phase VSI

chapter ends with Section 5 where the conclusions are presented.

**2. The asymmetrical dual three-phase AC drive**

*a*

*r*

*S***S**

*d*

d

q

*e*

*<sup>b</sup> <sup>c</sup>*

*N*

**Figure 1.** A general scheme of an asymmetrical dual three-phase drive

*N'*

Vdc

Dc Link

$$\mathbf{T} = \frac{1}{3} \begin{bmatrix} 1 & \frac{\sqrt{3}}{2} & -\frac{1}{2} & -\frac{\sqrt{3}}{2} & -\frac{1}{2} & 0\\ 0 & \frac{1}{2} & \frac{\sqrt{3}}{2} & \frac{1}{2} & -\frac{\sqrt{3}}{2} & -1\\ 1 & -\frac{\sqrt{3}}{2} & -\frac{1}{2} & \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0\\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} & \frac{1}{2} & \frac{\sqrt{3}}{2} & -1\\ 1 & 0 & 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 & 0 & 1 \end{bmatrix} \tag{1}$$

The VSI has a discrete nature and has a total number of 26 = 64 different switching states defined by six switching functions corresponding to the six inverter legs [*Sa*, *Sd*, *Sb*, *Se*, *Sc*, *Sf* ], where **S***<sup>i</sup>* ∈ {0, 1}. The different switching states and the voltage of the DC link (Vdc) define the phase voltages which can in turn be mapped to the (*α* − *β*) − (*x* − *y*) space according to the VSD approach. Consequently, the 64 different on/off combinations of the six VSI legs lead to 64 space vectors in the (*α* − *β*) and (*x* − *y*) subspaces. Figure 2 shows the active vectors in the (*α* − *β*) and (*x* − *y*) subspaces, where each vector switching state is identified using the switching function by two octal numbers corresponding to the binary numbers [*SaSbSc*] and [*SdSeSf* ], respectively. For the sake of conciseness, the 64 VSI switching vectors will be usually referred as voltage vectors, or just vectors, in what follows. It must be noted that the 64 possibilities imply only 49 different vectors in the (*α* − *β*) − (*x* − *y*) space. Nevertheless, redundant vectors should be considered as different vectors because they have a different impact on the switching frequency even though they generate identical torque and losses in the six-phase machine.

**Figure 2.** Voltage vectors and switching states in the (*α* − *β*) and (*x* − *y*) subspaces for a 6-phase asymmetrical VSI

To represent the stationary reference frame (*α* − *β*) in dynamic reference (*d* − *q*), a rotation transformation must be used. This transformation is given by:

$$\mathbf{T}\_{dq} = \begin{bmatrix} \cos\left(\delta\_r\right) - \sin\left(\delta\_r\right) \\ \sin\left(\delta\_r\right) \cos\left(\delta\_r\right) \end{bmatrix} \tag{2}$$

where *δ<sup>r</sup>* is the rotor angular position referred to the stator as shown in Figure 1.
