**4. Sliding mode speed control**

### **4.1. Direct sliding mode speed control**

**Figure 2.** Performance of the SM-DTC for induction motor: (a) simulation study: ideal case, digital implementation is not taken into account, (b) simulation study: digital implementation taken into account, (c) experimental study; first row: reference and real (estimated) torque, second row: speed, third row: reference and real (estimated) stator flux,

Results of the integral part introduction in the switching function are shown in **Figure 3b** and **3c** for simulation and experimental tests, respectively. The torque and stator flux are controlled perfectly, without any steady-state or dynamical errors. Additionally, the chattering phenom‐ enon is reduced considerably—level of the oscillations in regulated signals is acceptable now. Sliding mode DTC structure with the modified switching function, shown in this section, will be used to create the cascade speed and position control structures, shown in the following

**Figure 3.** Performance of the SM-DTC for induction motor: (a) simulation study: saturation function used instead of the sign function, (b) simulation study: integral part added in the torque switching function, (c) experimental study; first row: reference and real (estimated) torque, second row: speed, third row: reference and real (estimated) stator

fourth row: phase currents.

flux, fourth row: phase currents.

sections.

where *K*I is positive control parameter.

84 Robust Control - Theoretical Models and Case Studies

The direct IM speed control can be realized very similarly to the DTC. The most significant difference is another switching function that can be expressed as in [1]:

$$\mathbf{s}\_{1} = \mathbf{s}\_{o} = \alpha\_{1} \left( \alpha\_{m}^{\prime \prime} - \alpha\_{n} - T\_{co} \dot{\alpha}\_{n} \right), \tag{22}$$

where *Tcω* is time constant that defines the required dynamics of the speed.

When the switching function is zero, the controlled object acts as first-order inertia with time constant *Tcω*. Settling time of the system (95%) is equal to:

$$T\_s = \mathfrak{Y} T\_{coo}.\tag{23}$$

In this case, the same control algorithm [Eq. (12)] can be applied to regulate motor speed; however, the **D1** matrix in Eq. (14) must be slightly modified (the additional term *Tcω*/*TM* appears):

$$\mathbf{D}\_{1} = \frac{1}{T\_{N}} \begin{bmatrix} \alpha\_{1} \frac{T\_{cs}}{T\_{M}} \left( -\mathbf{i}\_{s\beta} + \frac{1}{\mathbf{x}\_{s}\sigma} \boldsymbol{\nu}\_{s\beta} \right) & \alpha\_{1} \frac{T\_{cs}}{T\_{M}} \left( \mathbf{i}\_{s\alpha} - \frac{1}{\mathbf{x}\_{s}\sigma} \boldsymbol{\nu}\_{s\alpha} \right) \\ -2\alpha\_{2} \boldsymbol{\nu}\_{s\alpha} & -2\alpha\_{2} \boldsymbol{\nu}\_{s\beta} \end{bmatrix} \mathbf{T} \,. \tag{24}$$

The condition of the system stability remains the same as in Eq. (18).

The block diagram of the direct sliding mode speed control is shown in **Figure 4**. It is almost identical as the one shown in **Figure 1**; however, the speed switching function *sω* is provided instead of the torque regulation error. For the clarity of the block diagram, the digital realiza‐ tion and measurement delays will not be presented in the following figures.

**Figure 5a** shows the performance of the direct SM speed control structure in the ideal simu‐ lation case. The reverses of the speed are presented—the speed follows the reference value with the requested dynamics, which is indicated by the *ω<sup>m</sup> ref* ,*dyn* signal. Even in this ideal case, the steady-state error exists—it is shown in the second row of **Figure 5a**. Additionally, electromagnetic torque of the motor is not controlled and supervised. Therefore, it exceeds the maximum value, set at the level 1.0 (it is about 150% of the nominal torque in p.u., see Appendix). If the digital realization of the control structure and measurement delays are taken into account, the steady-state speed error increases significantly—it can be seen in **Figure 5b**. Moreover, the torque and stator flux oscillation levels are much higher, similarly to the DTC algorithm, shown in the previous section. If the saturation function is applied (**Figure 5c**), the regulation error becomes even larger; however, the chattering level is reduced. This simulation study is verified using the experimental tests (**Figure 5d**)—and both of them give almost the same results.

**Figure 4.** Block diagram of the direct SM speed control.

**Figure 5.** Performance of the SM direct speed control: (a) simulation study: control in the ideal case, (b) simulation study: speed control in case of the digital implementation and measurement delays taken into account, (c) simulation study: saturation function used instead of the sign function, (d) experimental tests results for saturation function us‐ age; first row: reference and real speed, second row: speed control error, third row: load, electromagnetic and maxi‐ mum torque, fourth row: reference and real amplitude of stator flux.

In both the previously mentioned cases, direct SM torque and speed control, the torque becomes higher than the acceptable level and can be dangerous for the drive and its mechanical elements. This drawback can be eliminated reducing the desired dynamics, defined by *Tcω* or applying the cascade structure of the SM controllers. The second solution will be now descri‐ bed.

### **4.2. Cascade sliding mode speed control**

In order to create the cascade connection of SM speed and torque regulators, the torque control loop has to be simplified to the first-order inertial element, described by the following transfer function:

$$\frac{m\_e(p)}{m\_e^{ref}(p)} = \frac{1}{T\_{me}p + 1},\tag{25}$$

where *p* is Laplace operator, *Tme* is replacement time constant of the torque control circuit.

The control signals' vector becomes a scalar quantity **k**= *me ref* and the switching functions' vector **s**= *sω* likewise. In this case, the derivative of the switching function can be divided into:

$$\dot{s}\_{ao} = f\_{lao} + f\_{\he{2}o} + d\_{oo} m\_e^{ref} \,\,\,\,\,\,\tag{26}$$

where:

algorithm, shown in the previous section. If the saturation function is applied (**Figure 5c**), the regulation error becomes even larger; however, the chattering level is reduced. This simulation study is verified using the experimental tests (**Figure 5d**)—and both of them give almost the

**Figure 5.** Performance of the SM direct speed control: (a) simulation study: control in the ideal case, (b) simulation study: speed control in case of the digital implementation and measurement delays taken into account, (c) simulation study: saturation function used instead of the sign function, (d) experimental tests results for saturation function us‐ age; first row: reference and real speed, second row: speed control error, third row: load, electromagnetic and maxi‐

same results.

86 Robust Control - Theoretical Models and Case Studies

**Figure 4.** Block diagram of the direct SM speed control.

mum torque, fourth row: reference and real amplitude of stator flux.

$$f\_{1o0} = \dot{\alpha}\_m^{ref} + \frac{T\_{co0} - T\_{me}}{T\_M T\_{me}} m\_e,\tag{27}$$

$$f\_{2\alpha} = \frac{T\_{c\alpha}}{T\_M}\dot{m}\_o + \frac{1}{T\_M}m\_o,\tag{28}$$

$$d\_{o0} = -\frac{T\_{co0}}{T\_M T\_{me}}.\tag{29}$$

In the above equations, *f*1*<sup>ω</sup>* is the part that can be calculated from available variables, *f*2*<sup>ω</sup>* depends on the unknown variables and *dω* stands next to the reference torque.

If the equivalent signal-based control method is applied, then the reference torque signal consists of two parts [26]:

$$m\_{\text{e}}^{ref} = m\_{\text{e}}^{ref, eq} + m\_{\text{e}}^{ref, d},\tag{30}$$

$$m\_e^{ref,eq} = -\frac{1}{d\_{o0}} f\_{1o\nu} \tag{31}$$

$$m\_e^{ref,d} = -\frac{\Gamma\_{a0}^d}{d\_{a0}} \text{sign}(\mathbf{s}\_{a0}),\tag{32}$$

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

Continuous control signal part *me ref* , *eq* is calculated from available signals, and is designed to force the switching function to zero in presence of no load torque and motor parameters changes. The discontinuous part must be included in the SM control system, in order to compensate external disturbances, such as the load torque present in *f*2*<sup>ω</sup>* and the inaccuracy of the simplification from Eq. (25). The switching function derivative becomes:

$$\dot{L} = \mathbf{s}\_{\alpha}\dot{\mathbf{s}}\_{\alpha} = \mathbf{s}\_{\alpha}f\_{2\alpha} - \Gamma^{d}\_{\alpha} \|\mathbf{s}\_{\alpha}\|,\tag{33}$$

while its negative value is ensured if:

$$\left| \Gamma \right|\_{o}^{d} > \left| f\_{2o} \right|. \tag{34}$$

Thus, if the control parameter is chosen properly, the stability of the proposed control system can be guaranteed. The block diagram of the cascade control structure described here is shown in **Figure 6**. Unlike the direct control from **Figure 4**, the speed controller output signal is the reference torque, and it consists of two parts. Furthermore, this signal can be limited at desired value. The reference torque is the input of the SM-DTC structure, described in the previous chapter.

Performance of the cascade SM speed control in presence of the passive load torque is shown in **Figure 7**. The obtained results are shown for the speed reverses. It can be seen that the speed follows the reference signal with required dynamics in all cases. First subfigure shows the relay control—the equivalent signal from Eq. (31) is not taken into account in this case and the control parameter is equal to *Γme <sup>d</sup>* <sup>=</sup>*me max* . Performance of the control structure is presented during simulation tests—despite the ideal conditions, some small dynamic and steady-state error appears. Due to the enormous mechanical vibrations, it is impossible to conduct the experi‐ mental tests. Therefore, the equivalent signal was taken into account and its performance is shown in **Figure 7b** and **7c** for simulation and experimental tests, respectively. The results are almost the same. Electromagnetic torque has acceptable oscillations and is limited on a maximum value. Stator flux amplitude is kept constant at nominal value.

**Figure 6.** Block diagram of the SM speed control in cascade connection.

*ref ref eq ref d* , , *mm m ee e* = + , (30)

= - , (31)


*ref* , *eq* is calculated from available signals, and is designed to

& & (33)

. Performance of the control structure is presented during

(34)

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

, sign( ) *d ref d m s <sup>e</sup> <sup>d</sup>* w

w

where *Γω*

chapter.

parameter is equal to *Γme*

*<sup>d</sup>* <sup>=</sup>*me max*

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

88 Robust Control - Theoretical Models and Case Studies

Continuous control signal part *me*

while its negative value is ensured if:

w

force the switching function to zero in presence of no load torque and motor parameters changes. The discontinuous part must be included in the SM control system, in order to compensate external disturbances, such as the load torque present in *f*2*<sup>ω</sup>* and the inaccuracy of

the simplification from Eq. (25). The switching function derivative becomes:

<sup>2</sup> , *<sup>d</sup> L s s f ss* = = -G ww

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

 w

Thus, if the control parameter is chosen properly, the stability of the proposed control system can be guaranteed. The block diagram of the cascade control structure described here is shown in **Figure 6**. Unlike the direct control from **Figure 4**, the speed controller output signal is the reference torque, and it consists of two parts. Furthermore, this signal can be limited at desired value. The reference torque is the input of the SM-DTC structure, described in the previous

Performance of the cascade SM speed control in presence of the passive load torque is shown in **Figure 7**. The obtained results are shown for the speed reverses. It can be seen that the speed follows the reference signal with required dynamics in all cases. First subfigure shows the relay control—the equivalent signal from Eq. (31) is not taken into account in this case and the control

simulation tests—despite the ideal conditions, some small dynamic and steady-state error appears. Due to the enormous mechanical vibrations, it is impossible to conduct the experi‐ mental tests. Therefore, the equivalent signal was taken into account and its performance is shown in **Figure 7b** and **7c** for simulation and experimental tests, respectively. The results are

w ww w

w

w

**Figure 7.** Performance of the SM cascade speed control: (a) simulation study: relay control, (b) simulation study: equiv‐ alent control, (c) experimental study: equivalent signal-based control; first row: reference and real speed, second row: load, electromagnetic and maximum torques, third row: reference and real amplitude of stator flux.
