*2.2. Discrete-Time Model*

The first-order approximation, as shown in Equation (8), is usually used to transfer the continuous-time model to the discrete-time model.

$$\frac{dx}{dt} = \frac{x(k) - x(k-1)}{T\_s} \tag{8}$$

where *Ts* is the sample period. By substituting Equation (8) into Equations (6) and (7), the discrete-time model of the induction machine control system can be obtained as

 $v\_{qs}(k) = (R\_s + \frac{\delta L\_s}{T\_s})i\_{qs}(k) + \omega\_d L\_s i\_{ds}(k) - \delta$  $\frac{\delta L\_s}{T\_s}i\_{qs}(k-1) + \frac{\omega\_r L\_m}{R\_r T\_s}(\psi\_r(k) - \psi\_r(k-1))$ 

$$\upsilon\_{ds}(k) = (R\_s + \frac{\delta L\_s}{T\_s})i\_{ds}(k) - \frac{\delta L\_s}{T\_s}i\_{ds}(k-1) - \omega\_c \delta L\_s i\_{qs}(k) + \frac{L\_m}{L\_r T\_s}(\psi\_r(k) - \psi\_r(k-1))$$

The rotor flux varies slowly compared to the variation in the current and voltage. Therefore, Equations (9) and (10) can be simplified as

$$w\_{\mathfrak{g}^{\mathfrak{g}}}(k) = (R\_{\mathfrak{s}} + \frac{\delta L\_{\mathfrak{s}}}{T\_{\mathfrak{s}}})i\_{\mathfrak{g}^{\mathfrak{g}}}(k) - \frac{\delta L\_{\mathfrak{s}}}{T\_{\mathfrak{s}}}i\_{\mathfrak{g}^{\mathfrak{g}}}(k-1) + \omega\_{\mathfrak{e}}L\_{\mathfrak{e}}i\_{\mathrm{ds}}(k) \tag{11}$$

$$w\_{ds}(k) = (R\_s + \frac{\delta L\_s}{T\_s})i\_{ds}(k) - \frac{\delta L\_s}{T\_s}i\_{ds}(k-1) - \omega\_c \delta L\_s i\_{qs}(k) \tag{12}$$
