*3.4. Torque Estimation*

In this analysis, the proposed AFOO can also be used to determine the electromagnetic torque to reduce the error between the reference and desired motor output torque based on estimated stator current and flux. Delay between acquisition time and application time is important to be considered.

$$
\hat{\mathbf{T}}\_{\theta} = \frac{3}{2} \text{ P Im} \left( \stackrel{\rightarrow}{\hat{\Psi}}\_{s}^{\*} \stackrel{\rightarrow}{\hat{\mathbf{1}}}\_{s,k-1} \right) \tag{23}
$$

where k is the sampling time index.

> As indicated in Figure 4, our proposed observer can easily estimate stator flux angle as:

$$\Theta\_s = \tan^{-1} \left( \frac{\Psi\_{\beta s}}{\Psi\_{\alpha s}} \right) \tag{24}$$

**Figure 4.** Stator and rotor flux vectors relationship in stationary reference frame.

The angular speed of stator flux linkage relative to rotor flux linkage may also be computed as:

$$\delta = \tan^{-1} \left( \frac{\hat{\Psi}\_{\beta r}}{\Psi\_{\alpha r}} \right) - \Theta\_{\theta} \tag{25}$$

To incorporate the system (17) into the digital processor, it is crucial to acquire a discretetime state-space representation of proposed system. It should be noted that matrix Aˆ depending on instantaneous calculated rotor speed value ω<sup>ˆ</sup> r making A ˆ = A ˆ (ω<sup>ˆ</sup> r(t)) as a linear time-varying system. Time dependence implies that variation of ωr, the offline numerical estimate of discrete-time equivalent system cannot be obtained. The solution would be to achieve a discrete-time varying method that can then be modified with the new estimated value of ω<sup>ˆ</sup> r at every sampling period. A direct computation or Euler approximation (first-order sequence expansion) [46] are the more popular methods of obtaining the representation of sampled-data for the model. The approximation of Euler is an easy way of obtaining the discrete-time model with a similar response of dynamic behavior. The direct computation of state trajectory in (17) may not be simple as Euler approximation of the first order but gives a more precise representation in the discrete-time (for example, see [47] and references therein).

## *3.5. Conventional MPTC*

For conventional MPTC, just one voltage vector is chosen and not applied till the next control time owing to the updated mechanism for modern microprocessors. The conventional MPTC diagram shown is in Figure 5. Using a PI controller, torque reference would be created via an outer speed control loop and stator flux reference is kept constant at asset value since field weakening and efficiency optimization process are not taken into account in this procedure [29]. In MPTC technique, torque and stator flux are being predicted for all possible voltage vectors supplied by the inverter based on the system model. Then, the best one is decided by minimizing an objective cost function consisting of a torque-flux tracking error.

**Figure 5.** Conventional MPTC scheme.

## *3.6. Conventional MPFC*

Torque and stator flux references are equivalently transformed into single reference for stator flux vector in this process. The MPFC diagram can be seen in Figure 6.
