**3. Sliding mode direct torque control**

In order to create a cascade connection of sliding mode controllers (for example torque and speed controllers), it is necessary to design first the sliding mode DTC. This method of control utilizes the IM mathematical model and its equations, shown in the previous chapter.

The first step in the designing is to define the so-called switching functions. The classical approach is first taken into account [1]:

$$\mathbf{s} = \begin{bmatrix} \mathbf{s}\_1 & \mathbf{s}\_2 & \mathbf{s}\_3 \end{bmatrix}^\mathrm{T},\tag{8}$$

where the components of **s** vector allow to control the motor torque, stator flux amplitude and to ensure the three-phase balance of the system, respectively:

$$\mathbf{s}\_{\mathbf{i}} = \alpha\_{\mathbf{i}} \left( m\_{\mathbf{i}}^{\prime \eta^{\prime}} - m\_{\mathbf{i}} \right), \tag{9}$$

$$\mathbf{s}\_2 = \alpha\_2 \left( (\boldsymbol{\nu}\_s^{\mathrm{ref}})^2 - \boldsymbol{\nu}\_s^2 \right), \tag{10}$$

$$\mathbf{a}\_{3} = \alpha\_{3} \left| \left( k\_{\mathcal{A}} + k\_{\mathcal{B}} + k\_{\mathcal{C}} \right) \mathbf{d}t, \tag{11}$$

where *α*1, *α*2, *α*3 are control parameters, that need to be chosen.

The goal of the sliding mode controller will be to force the switching functions from Eqs. (9) to (11) to zero, which means that the real values will follow the reference ones. This goal can be achieved using the classical sliding-mode control formula, expressed as:

Sliding Mode Speed and Position Control of Induction Motor Drive in Cascade Connection 81 http://dx.doi.org/10.5772/63407

$$\begin{aligned} \mathbf{k} &= -\text{sign}(\mathbf{s}^\ast)^T, \\ \mathbf{s}^\ast &= \mathbf{s}^T \mathbf{D}, \end{aligned} \tag{12}$$

where the D matrix comes from the division of the switching function derivative into:

$$
\dot{\mathbf{s}} = \mathbf{f} + \mathbf{D}\mathbf{k} \tag{13}
$$

and can be calculated as follows:

$$\mathbf{D} = \begin{bmatrix} \mathbf{D}\_1 \\ \alpha\_3 & \alpha\_3 & \alpha\_3 \end{bmatrix}' \tag{14}$$

$$\mathbf{D}\_{1} = \frac{1}{T\_{N}} \begin{bmatrix} \alpha\_{1} \left( -i\_{s\beta} + \frac{1}{\mathbf{x}\_{s}\sigma} \boldsymbol{\nu}\_{s\beta} \right) & \alpha\_{1} \left( i\_{s\alpha} - \frac{1}{\mathbf{x}\_{s}\sigma} \boldsymbol{\nu}\_{s\alpha} \right) \\ -2\alpha\_{2} \boldsymbol{\nu}\_{s\alpha} & -2\alpha\_{2} \boldsymbol{\nu}\_{s\beta} \end{bmatrix} \text{T}. \tag{15}$$

In order to check the usefulness of the proposed control algorithm and to verify the stability of the proposed control system, the Lyapunov function method is applied. A positive defined Lyapunov function is proposed as follows:

$$L = \frac{1}{2} \mathbf{s}^{\mathsf{T}} \mathbf{s} = \frac{1}{2} \left( \mathbf{s}\_1^2 + \mathbf{s}\_s^2 + \dots + \mathbf{s}\_n^2 \right) > 0. \tag{16}$$

Its derivative can be calculated as:

$$\begin{aligned} \dot{L} &= \mathbf{s}^{\mathrm{T}} \dot{\mathbf{s}} = \\ &= \mathbf{s}^{\mathrm{T}} \left( \mathbf{f} - \mathbf{D} \text{sign}(\mathbf{s}^{\mathrm{\*}})^{\mathrm{T}} \right) = \\ &= \mathbf{s}^{\mathrm{T}} \mathbf{f} - \mathbf{I} \left| \mathbf{s}^{\mathrm{T}} \mathbf{D} \right|^{\mathrm{T}} \end{aligned} \tag{17}$$

where s = 1 =[ | s = | | s = | ] , I=[1 1 1].

The stability inequality [negative value of Eq. (17)] is defined as:

$$\|\mathbf{f}\| < \|\mathbf{D}\|\mathbf{I}^{\top}.\tag{18}$$

If the control parameters *α*1, *α*2, *α*3, included in the **D** matrix in Eq. (14) are high enough to fulfill the condition [Eq. (18)], the system is stable and the real values follow their reference values.

Full SM-DTC block diagram is shown in **Figure 1**. The control structure defines the control signals *kA*, *kB*, *kC* directly, to control the switches of the VSI without any voltage modulator. The input values are the reference values of stator flux amplitude and electromagnetic torque. If the speed exceeds the nominal value, the amplitude of the flux must be weakened in order to ensure the constant power operation of the induction machine.

It is also necessary to provide the measurement of the DC-bus voltage [this value is present in the **T** matrix in Eq. (14)] and stator phase currents (transformed to the stationary *α*-*β* frame from two-phase currents, when the three-phase symmetry is assumed). The control structure also needs estimated values, such as stator flux vector components (or its magnitude and angle) and electromagnetic torque (the hat "∧" indicates the estimated value). They must be deter‐ mined by a proper estimator—this problem will not be addressed in this chapter. If the estimator requires the stator voltage vector knowledge, its components can be transformed from measured signals or calculated using the Eq. (7), taking into account the inverter deadtime [23].

The block diagram, shown in **Figure 1**, also emphasizes the digital implementation of the SM-DTC, together with the measurement delays (*τ<sup>d</sup>* for current and voltage measurement and *τdω* for speed measurement). Nowadays, the continuous algorithms are realized in a discrete form using the digital signal processors (DSPs). The influence of the digital implementation will be shown in the following part of the chapter.

**Figure 1.** Block diagram of the SM-DTC (digital realization).

**Figure 2** shows a comparative study of performance of the SM-DTC structure for three different cases: ideal simulation study (**Figure 2a**), simulation study with the DSP discretiza‐ tion taken into account (**Figure 2b**) and experimental results (**Figure 2c**).

**Figure 2a** shows the ideal operation of the SM-DTC and proves its perfect dynamical features. The torque and stator flux amplitudes follow their reference values almost immediately and without any oscillations. The speed is a result of the motor and load torque difference [ac‐ cording to Eq. (6)] and therefore is changing in a triangular way. Three-phase currents are smooth and sinusoidal—their frequency is changed automatically by the control structure.

If the control parameters *α*1, *α*2, *α*3, included in the **D** matrix in Eq. (14) are high enough to fulfill the condition [Eq. (18)], the system is stable and the real values follow their reference values. Full SM-DTC block diagram is shown in **Figure 1**. The control structure defines the control signals *kA*, *kB*, *kC* directly, to control the switches of the VSI without any voltage modulator. The input values are the reference values of stator flux amplitude and electromagnetic torque. If the speed exceeds the nominal value, the amplitude of the flux must be weakened in order to

It is also necessary to provide the measurement of the DC-bus voltage [this value is present in the **T** matrix in Eq. (14)] and stator phase currents (transformed to the stationary *α*-*β* frame from two-phase currents, when the three-phase symmetry is assumed). The control structure also needs estimated values, such as stator flux vector components (or its magnitude and angle) and electromagnetic torque (the hat "∧" indicates the estimated value). They must be deter‐ mined by a proper estimator—this problem will not be addressed in this chapter. If the estimator requires the stator voltage vector knowledge, its components can be transformed from measured signals or calculated using the Eq. (7), taking into account the inverter dead-

The block diagram, shown in **Figure 1**, also emphasizes the digital implementation of the SM-DTC, together with the measurement delays (*τ<sup>d</sup>* for current and voltage measurement and *τdω* for speed measurement). Nowadays, the continuous algorithms are realized in a discrete form using the digital signal processors (DSPs). The influence of the digital implementation will be

**Figure 2** shows a comparative study of performance of the SM-DTC structure for three different cases: ideal simulation study (**Figure 2a**), simulation study with the DSP discretiza‐

tion taken into account (**Figure 2b**) and experimental results (**Figure 2c**).

ensure the constant power operation of the induction machine.

shown in the following part of the chapter.

82 Robust Control - Theoretical Models and Case Studies

**Figure 1.** Block diagram of the SM-DTC (digital realization).

time [23].

Unfortunately, one of the negative properties of the SMC structures is the phenomenon called chattering [24]. There are many sources of the chattering—one of them is the discretization, connected with limited sampling rate of modern processors [25]. In order to check the influence of this phenomenon, special simulation model has been built. Suitable results are shown in **Figure 2b**. Large chattering (sometimes called the discretization chattering) can be seen in the controlled variables. It is also visible in phase currents. Due to the moment of inertia of the drive system, speed signal is still smooth.

Simulation test results have been validated using an experimental setup (see Appendix). Obtained results illustrate the same situation in a very similar way—the chattering can be seen in torque, flux and currents. The level of the obtained oscillations is even higher than during the simulation tests—it causes mechanical stress, dangerous for the drive, and acoustic noise.

One of the efficient solutions to avoid the chattering, visible in **Figure 2**, is to use the continuous approximation of the sign function. One of them is a saturation function:

$$\mathbf{d} = -0.5 \Big( \text{sat}(\mathbf{s}^\ast, \boldsymbol{\varepsilon}\_{me})^\mathsf{T} + \mathsf{l} \Big), \tag{19}$$

where *εme* is positive control parameter to be chosen and column vector **d** = [*dA*, *dB*, *dC*] T is duty cycles' vector and the saturation function:

$$\text{sat}(\mathbf{s}) = \begin{cases} \mathbf{s}/\boldsymbol{\varepsilon} & \text{if } \|\mathbf{s}\| \le \boldsymbol{\varepsilon} \\ \text{sign}(\mathbf{s}) & \text{if } \|\mathbf{s}\| > \boldsymbol{\varepsilon} \end{cases} \tag{20}$$

In this case, the control structure defines not the transistor control signals directly, but the duty cycle functions for each phase (relation of the switching-on time to the whole sampling period). Specific form of the Eq. (19) is imposed by the duty cycle feature—its values can vary between 0 and 1 (0% and 100%).

Effects of the saturation function usage are shown in **Figure 3a**. It can be seen that the oscilla‐ tions level is greatly reduced. However, a significant and changing in time, regulation error can be seen in the electromagnetic torque transient. It can be eliminated using simple modifi‐ cation of the switching function [Eq. (9)], to obtain the following formula [3]:

$$\mathbf{S}\_{1} = \alpha\_{1} \left( m\_{\boldsymbol{e}}^{\prime \text{v}\prime} - m\_{\boldsymbol{e}} \right) + K\_{\boldsymbol{\iota}} \int \left( m\_{\boldsymbol{e}}^{\prime \text{v}\prime} - m\_{\boldsymbol{e}} \right) \mathbf{d}t \tag{21}$$

**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, fourth row: phase currents.

where *K*I is positive control parameter.

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 sections.

**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 flux, fourth row: phase currents.
