**6. Conclusions**

. *N c*

= - (41)

*mm m* = + (42)

= - (43)

<sup>G</sup> = - (44)

(45)

w

 w

q

q

q

*<sup>T</sup> <sup>d</sup> T T* q

, , , *ref ref eq ref d*

, <sup>1</sup> , *ref eq* <sup>1</sup> *<sup>m</sup> <sup>f</sup> <sup>d</sup>*

, sign( ), *d*

According to the methodology shown in previous sections, the position control system is stable

q

*<sup>m</sup> <sup>s</sup> <sup>d</sup>* q

> <sup>2</sup> . *<sup>d</sup> <sup>f</sup>* G > q

 q

The block diagram of the proposed cascade position control structure is shown in the **Figure 10**. Performance of the control structure for nominal load operation is shown in **Figure 11**. First, the simulation study for the ideal case is shown (**Figure 11a**), the sign function is used in the control algorithm directly. It can be seen that the position follows the reference signal with required dynamics. Due to the torque constraint introduced, some small dynamical

q

By analogy to equation (30), the reference speed control signal is as follows:

ww

w

*ref d*

w

*<sup>d</sup>* is a control parameter.

92 Robust Control - Theoretical Models and Case Studies

**Figure 10.** Block diagram of the cascade SM position control.

where *Γθ*

if:

This chapter deals with the SMC of the most important IM variables: torque, speed and position of the shaft, simultaneously ensuring constant value of the stator flux amplitude.

First part of the chapter is connected with the sliding mode DTC for IM. It is proved that the classical DT-SMC approach gives undesirable torque chattering. In order to reduce the chattering, the saturation function is used instead of the sign function. However, the saturation function introduces large steady-state control error. An integral part in the torque switching function is successfully used to eliminate this error. Such designed torque control is then used in the cascade speed and position IM control.

Next part of the chapter is the IM speed control. It is shown that the sliding mode speed control in its classical, direct approach is characterized by a large steady-state error and chattering. It is proved using a specially prepared simulation model and experimental setup. Additionally, the torque value is not constrained in this direct speed control structure. Therefore, the cascade connection of speed and torque regulators is introduced. The equivalent signal-based control method is applied to reduce the chattering. This solution allows to supervise the value of the electromagnetic torque, while reducing the control error effectively.

The last part of the chapter shows the position control of the drive. Conclusions that come from the SM position control analysis are analogical to the ones from the speed control. Direct position control does not ensure the torque and speed supervision, that is solved by the cascade control structure. Simultaneously, the chattering can be reduced using the continuous approximation of the sign function and the equivalent signal-based approach.

All of these control concepts are illustrated using simulation and experimental study. Special attention has been paid to create a simulation model that allows to take the digital realization of modern DSP control applications and measurement delays into account.
