**2. Mathematical model of induction motor drive**

SMC algorithms are strictly based on the mathematical model of the controlled object, which is the IM in this research. This model will be shown in this section—it is created with commonly known simplifying assumptions [22]. It is written with normalized [per unit (p.u.)] units, in an arbitrary frame, rotating with the angular velocity ωk. Base values, required to the p.u. system transformation, are shown in the appendix.

Stator and rotor voltage equations:

$$\mathbf{u}\_s = r\_s \mathbf{i}\_s + T\_N \frac{\mathbf{d}}{\mathbf{d}t} \Psi\_s + j\phi\_k \Psi\_{s'} \tag{1}$$

$$\mathbf{u}\_{\mathbf{r}} = r\_r \mathbf{i}\_{\mathbf{r}} + T\_N \frac{\mathbf{d}}{\mathbf{d}t} \mathbf{w}\_{\mathbf{r}} + j(o\_k - o\_m) \mathbf{w}\_{\mathbf{r}\text{ }\mathbf{r}}.\tag{2}$$

where **us** = *usα* + *jusβ*, **ur** = *urα* + *jurβ* are stator and rotor voltage vectors, **is** = *isα* + *jisβ*, **ir** = *irα* + *jirβ* are stator and rotor current vectors, are stator and rotor flux vectors, *rs*, *rr* are stator and rotor winding resistances, *T*N = 1/(2*πfsN*) is nominal time constant, appearing after the per unit system is introduced, *fsN* is nominal frequency of the motor and *ωm* is mechanical velocity.

Flux equations:

part in the switching function has also been used to eliminate the dynamical and steady-state

Most of the papers focused on reducing the most negative feature of the SMC, i.e. chattering (large oscillations of controlled variables). Position control with adaptive continuous approx‐ imation of the sign function is proposed in reference [4]. Load torque estimator was introduced in reference [5] to reduce level of the discontinuous part of the control signal. One of the effective solutions to reduce the chattering is the application of higher-order sliding modes. They were introduced for all of the IM variables: torque in reference [6], speed (in a speedsensorless approach) in reference [7] and position in reference [8]. The IM drive control, supplied from a current source inverter with the second-order SMC is introduced in reference [9]. Integral SMC of stator current components is shown in references [10–12] to reduce the

One of the chattering sources is a discretization caused by digital implementations of the drives' control structures; therefore, the discrete SMC methods have been proposed. The IM position discrete control is proposed in references [13] and [14]. The discrete SMC of the IM

Another drawback of the SMC in a direct approach (when the control algorithm defines the transistors' control signals directly) is a variable switching frequency. In order to eliminate this phenomenon, a voltage modulator can be applied. The classical direct torque control (DTC), SMC and space vector modulation (SVM) were combined in references [16] and [17]. Similarly, the indirect field-oriented control (IFOC) method and SMC were combined in reference [18].

In the past years, there have also been the attempts to extend the robustness of the IM control over the reaching phase, not only the sliding phase. The proposed approaches can be divided into two groups. In the first one, the switching line (or a surface) is designed to include the starting point: for speed control in reference [2] and for the position control in reference [19]. The second group consists of the methods with time-varying switching lines. They have been

In this chapter, a comparative analysis of the SMC of all IM state variables is presented. Direct approaches that define the transistor control signals directly are described and illustrated with simulation and experimental results. The cascade connection of sliding mode controllers is proposed for speed and position regulation, presented in a unified manner. The equivalent

This chapter consists of nine numerated sections. The following section presents the mathematical model of IM. Next three sections show the control of IM variables: torque, speed and position, respectively. Sections related to the speed and position control are divided into two subsections that include the direct and the cascade control. After short conclusions section, there is an appendix with experimental setup description and tables with tested IM parameters

signal-based control is used to lower the level of the chattering in regulated variables.

applied mainly in the position control [20], but also for the speed control [21].

and base values, necessary to obtain the normalized unit system.

errors in the torque control [3].

78 Robust Control - Theoretical Models and Case Studies

speed is introduced in reference [15].

chattering.

$$\mathbf{\upmu}\_s = l\_s \mathbf{i}\_s + l\_m \mathbf{i}\_r \tag{3}$$

$$
\Psi \mathbf{\upmu}\_r = l\_r \mathbf{i}\_r + l\_m \mathbf{i}\_s,\tag{4}
$$

where *ls* = *lm* + *lsσ*, *lr* = *lm* + *lrσ* are stator and rotor winding inductances, *lm* is magnetizing inductance and *lsσ*, *lrσ* are stator and rotor leakage inductances.

Electromagnetic torque and the motion equation are as follows:

$$m\_e = \text{Im}\left(\mathbf{\upmu}\_s^\* \mathbf{i}\_s\right) = \boldsymbol{\upmu}\_{s\boldsymbol{\alpha}} \mathbf{i}\_{s\boldsymbol{\beta}} - \boldsymbol{\upmu}\_{s\boldsymbol{\beta}} \mathbf{i}\_{s\boldsymbol{\alpha}},\tag{5}$$

$$\frac{\mathbf{d}\,\alpha\_m}{\mathbf{d}t} = \frac{1}{T\_M} (m\_e - m\_o),\tag{6}$$

where *me* is electromagnetic torque, *mo* is load torque and *TM* is mechanical time constant of the drive.

It is assumed that the IM is supplied by an ideal voltage source inverter (VSI), which can be described by the following matrix equation:

$$\mathbf{u}\_s = \begin{bmatrix} u\_{s\alpha} \\ u\_{s\beta} \end{bmatrix} = \frac{u\_{DC}}{3} \mathbf{T} \mathbf{k}, \quad \mathbf{T} = \begin{bmatrix} 1 & -1/2 & -1/2 \\ 0 & \sqrt{3}/2 & -\sqrt{3}/2 \end{bmatrix}, \tag{7}$$

where **k** = [*kA*, *kB*, *kC*] <sup>T</sup> is the control signals' vector of the VSI transistors and *uDC* is the DC-bus voltage.
