**Appendix A. Derivation of Eq. (9) in detail**

As stated in Section 3, the Lyapunov function is defined in (8) as

$$\dot{V} = \frac{1}{2}\mathbf{S}^2 + \frac{1}{2\rho}\ddot{P}^2 \rightarrow \dot{V} = \mathbf{S}\dot{\mathbf{S}} + \frac{1}{\rho}\ddot{P}\dot{\tilde{P}} \tag{A.1}$$

where *<sup>P</sup>*<sup>~</sup> <sup>¼</sup> *<sup>P</sup>* � *<sup>P</sup>*^ and *<sup>P</sup>* <sup>¼</sup> <sup>1</sup> *<sup>b</sup> <sup>ω</sup>*\_ <sup>Δ</sup>*<sup>a</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>b</sup> <sup>T</sup>*\_ *<sup>e</sup>* � *<sup>T</sup>*\_ *<sup>L</sup>* defined from (5) are assumed to be constant during the sampling period. Hence, it can be concluded that ~\_ *<sup>P</sup>* ¼ � \_ *P*^. By substituting (7) into (A.1), and adding and subtracting *K*1*S*<sup>2</sup> , the derivative of Lyapunov function can be written as

$$\dot{V} = \mathbf{S}\dot{\mathbf{S}} - \frac{\mathbf{1}}{\rho}\dot{\tilde{P}}\dot{\tilde{P}} = -K\_1\mathbf{S}^2 + K\_1\mathbf{S}^2 + \mathbf{S}\left[-(\sigma + a)\dot{\phi} - bu + b\dot{T}\_\perp - b\left(\ddot{P} + \dot{P}\right)\right] - \frac{\mathbf{1}}{\rho}\dot{\tilde{P}}\dot{\tilde{P}} \tag{A.2}$$

Finally it can be simplified as

$$\dot{V} = -K\_1 \mathbf{S}^2 + \mathbf{S} \left[ -(\sigma + a)\dot{a} - b \left( u + \dot{P} \right) + b \dot{T}\_\perp + K\_1 \mathbf{S} \right] - \tilde{P} \left( \frac{\mathbf{1}}{\rho} \dot{\tilde{P}} + b \mathbf{S} \right) \tag{A.3}$$

### **B. Numerical values of the SRM parameters**


**7. Conclusion**

*Control Theory in Engineering*

**Figure 16.**

perturbation.

**138**

dynamic model is considered.

Two nonlinear adaptive sliding mode control algorithms are developed for speed control of SRM in the application of EVs, especially in urban areas that the speed is low and the driving torque is highly fluctuated. This condition is emulated by applying a Gaussian noise to perturb the load torque. First, a conventional adaptive sliding mode control is designed to yield a chattering-free control algorithm. However, linearized SRM model is used, and only mechanical uncertainty of SRM

To implement a high-performance control algorithm for removing these drawbacks, a robust adaptive sliding mode control, namely, RASMC, is proposed. An augmented time-varying uncertainty is defined taking into account the SRM full nonlinear dynamic model and the system uncertainties as well as time-varying load

The upper bound of defined augmented uncertainty is not required to be known

to make the control gain small enough in order to decrease the chattering and efforts of the controller. Moreover, the exponential terms in sliding function are used to prevent chattering in the control signal. The proposed controller is capable of achieving torque ripple minimization with reduced smooth control effort. The simulation and experimental results also confirm the robustness properties against model uncertainties and time-varying load torque disturbances with significant

improvement in speed control loop transient response.

*Two phase currents and voltages for (a) CASMC and (b) RASMC.*

*Control Theory in Engineering*
