**5. Sliding mode position control**

### **5.1. Direct sliding mode position control**

By the analogy to the torque and speed control, presented in previous sections, IM shaft position control can be designed using the direct approach [1]. The control algorithm [Eq. (12)] and the **D1** matrix [Eq. (24)] remain the same as for the direct speed control, *Tcω* is only replaced by *Tcθ*. Switching function for the position control becomes:

$$\mathbf{s}\_{1} = \mathbf{s}\_{\boldsymbol{\theta}} = \alpha\_{1} \left( \boldsymbol{\theta}\_{\boldsymbol{m}}^{\nu \circ \boldsymbol{\ell}} - \boldsymbol{\theta}\_{\boldsymbol{m}} - T\_{\boldsymbol{\theta}} \dot{\boldsymbol{\theta}}\_{\boldsymbol{m}} - T\_{\boldsymbol{c}\boldsymbol{\theta}} \ddot{\boldsymbol{\theta}}\_{\boldsymbol{m}} \right) \tag{35}$$

where *Tθ*, *Tcθ* are time constants that can be selected according to the required settling time (5%) of the position control *Tsθ*, using the following rule [27]:

$$T\_{\theta} = 2T\_{cr},\ T\_{c\theta} = T\_{cr}^2,\ T\_{cr} = \frac{2T\_{s\theta}}{9}.\tag{36}$$

The block diagram of the SM direct position control is presented in **Figure 8**. Performance of the SM direct position control during experimental tests is shown in **Figure 9**. It can be seen that the shaft position has the desired dynamics and tracks the reference value without almost any error. Unfortunately, the chattering visible in the torque is entirely unacceptable. There‐ fore, the saturation function is applied (**Figure 9b**)—the level of the undesired oscillations is greatly reduced. In both cases, similarly to the direct speed control, the electromagnetic torque is not supervised in this type of the control. Simultaneously, the speed does not exceed the accepted value, which is 120% of the nominal value in this research. The flux is kept constant (not shown in the figure).

**Figure 8.** Block diagram of the direct SM position control.

**Figure 9.** Experimental performance of the SM direct position control: (a) with sign function, (b) with saturation func‐ tion.

### **5.2. Cascade sliding mode position control**

**5. Sliding mode position control**

90 Robust Control - Theoretical Models and Case Studies

**5.1. Direct sliding mode position control**

(not shown in the figure).

**Figure 8.** Block diagram of the direct SM position control.

By the analogy to the torque and speed control, presented in previous sections, IM shaft position control can be designed using the direct approach [1]. The control algorithm [Eq. (12)] and the **D1** matrix [Eq. (24)] remain the same as for the direct speed control, *Tcω* is only

> 1 1 ( ) *ref m m m cm s s* == -- -

 q

where *Tθ*, *Tcθ* are time constants that can be selected according to the required settling time

<sup>2</sup> <sup>2</sup> 2, , . <sup>9</sup>

The block diagram of the SM direct position control is presented in **Figure 8**. Performance of the SM direct position control during experimental tests is shown in **Figure 9**. It can be seen that the shaft position has the desired dynamics and tracks the reference value without almost any error. Unfortunately, the chattering visible in the torque is entirely unacceptable. There‐ fore, the saturation function is applied (**Figure 9b**)—the level of the undesired oscillations is greatly reduced. In both cases, similarly to the direct speed control, the electromagnetic torque is not supervised in this type of the control. Simultaneously, the speed does not exceed the accepted value, which is 120% of the nominal value in this research. The flux is kept constant

*cr c cr cr <sup>T</sup> T TT T T*

 q  *T T* q

 q

& && (35)

 q

*s*

q

= == (36)

 q

replaced by *Tcθ*. Switching function for the position control becomes:

aq

q

(5%) of the position control *Tsθ*, using the following rule [27]:

q

According to the previous section, the cascade position control structure will now be analyzed. It is based on the assumption that the SM speed control works perfectly and ensures zero value of the speed switching function from Eq. (22). In such situation, the speed control loop can be described by the following transfer function:

$$\frac{\alpha o\_m(p)}{\alpha o\_m^{ref}(p)} = \frac{1}{T\_{co}p + 1} \tag{37}$$

Similarly, as for the speed control, the control signals vector and switching functions' vector become scalars, **k**= *ω<sup>m</sup> ref* , **s**<sup>=</sup> *<sup>s</sup><sup>θ</sup>* , respectively. In this case, the switching function derivative becomes:

$$\dot{s}\_{\theta} = f\_{1\theta} + f\_{2\theta} + d\_{\theta} o\_{m}^{ref},\tag{38}$$

$$f\_{1\theta} = \dot{\theta}\_m^{ref} - \frac{1}{T\_N} \left(\alpha\_m - \frac{T\_\theta}{T\_{oo}}\right) - \frac{T\_{coo}}{T\_M T\_N} \dot{m}\_\theta,\tag{39}$$

$$f\_{2\theta} = \frac{T\_{c\theta}}{T\_M T\_N} \dot{m}\_o,\tag{40}$$

$$d\_{\theta} = -\frac{T\_{\theta}}{T\_{N}T\_{coo}}.\tag{41}$$

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

$$
\alpha\_m^{ref} = \alpha\_m^{ref,eq} + \alpha\_m^{ref,d},\tag{42}
$$

$$
\rho\_m^{ref, eq} = -\frac{1}{d\_\theta} f\_{1\theta},
\tag{43}
$$

$$
\alpha\_m^{ref, \, d} = -\frac{\Gamma\_\theta^d}{d\_\theta} \text{sign}(\mathbf{s}\_\theta),
\tag{44}
$$

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

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

$$
\Gamma^d\_{\theta} > \left| f\_{2\theta} \right|. \tag{45}
$$

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

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 error of the position control is visible. In order to decrease the level of the chattering of torque and speed, visible especially when the digital operation is taken into account (**Figure 11b**), the saturation function is applied in **Figure 11c**. The oscillations level is reduced successfully, while the position dynamic error is maintained. **Figure 11d** shows the experimental results—because of the digital realization of the control structure and additional, parasitic dynamics (measure‐ ment delays), the controlled variables are characterized by larger chattering; however, its level is acceptable. The position dynamic error is slightly higher than in case of the simulation tests.

**Figure 11.** Performance of the SM equivalent control signal-based position control in a cascade structure: (a) simulation study: ideal case, (b) simulation study: digital operation and delays taken into account, (c) simulation study: saturation function used instead of the sign function in cascade control, (d) experimental results with saturation function used in cascade control; first row: reference and real position, second row: reference, real and maximum speed; third row: ref‐ erence, real (estimated), load and maximum torque; fourth row: reference and real (estimated) stator flux amplitude.
