Speed Control of Induction Motor Servo Drives Using Terminal Sliding-Mode Controller

Abstract

The induction motor (IM) is one of the most common electrical motor used in most applications. This motor runs at a speed less than its synchronous speed, therefore it is also called as asynchronous motor. The synchronous speed is the speed of rotation of the magnetic field in a rotary machine and it depends upon the frequency and number poles of the IM. The IM has been extensively used in many practical applications due to its simply construction, lower repair and maintenance costs, high reliability and relatively low manufacturing cost. With the development of power electronics, electrical technique and control theories, IMs have been able to be used in high-performance servo systems, such as speed servo systems, even position servo systems. This chapter describes the speed control method and technique of IM servo drives using terminal sliding-mode controller.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

The induction motor (IM) is one of the most common electrical motor used in most applications. This motor runs at a speed less than its synchronous speed, therefore it is also called as asynchronous motor. The synchronous speed is the speed of rotation of the magnetic field in a rotary machine and it depends upon the frequency and number poles of the IM. The IM has been extensively used in many practical applications due to its simply construction, lower repair and maintenance costs, high reliability and relatively low manufacturing cost, etc [1]. With the development of power electronics, electrical technique and control theories, IMs have been able to be used in high-performance servo systems, such as speed servo systems, even position servo systems.

Three methods can be used for the control of IMs: the scalar control, the direct torque control (DTC) and the field oriented control (FOC). The former method is very simple method for controlling the speed of IM compared to the vector control which is more complex. The latter two methods can be utilized to implement the high-performance IM servo systems.

In DTC-based IM servo systems both the stator flux and the torque are regulated respectively using the bang-bang control strategies. This control method may lead to the torque ripple. If the IM runs at low speed, its performances will become poorer, and the speed range will be limited.

FOC is widely used in high performance control of IM servo systems. Since the torque and flux of an IM are decoupled using FOC, the IM systems can yield faster dynamic response and lower steady-state error. The mathematical model of an IM in a three-dimensional stationary reference frame (abc) can be converted into a model in a two-dimensional rotating reference frame (dq) using the Clarke-Park transformation. The d-axis current in the stator represents the rotor flux and the q-axis current represents the torque. Therefore, the decoupled rotor flux and the torque of an IM can be separately controlled like as a decoupled excited DC motor. Consequently it is possible to achieve good steady-state and dynamic performances of IMs [2, 3].

However, an accurate information on both the magnitude and the angular position of the rotor flux are needed by FOC for the transformation between the rotating and the stationary reference frames. There are two main methods for obtaining the magnitude and the angular position of the rotor flux, the direct measurements or the indirect estimation. The former needs special sensors, therefore it is difficult in practical applications. The latter is popular and widely used [4]. It applies the measurements of the stator currents, stator voltages and the motor speed into some estimation algorithms to estimate the magnitude and the angular position of the rotor flux. A lot of estimation methods for the rotor flux have been proposed, such as Luenberger observer-based methods [5, 6], model reference methods [7, 8], Kalman filter-based methods [9], and neural networks [10].

The high performance control of IMs is a challenge due to multi-variable, strong coupling, and nonlinearities in the model of IMs [11, 12]. So far, a lot of control methods have been proposed to improve the robustness and dynamical performances of IMs, such as neural network control, fuzzy control, optimal control, adaptive control and sliding-mode control [13, 14]. Sliding-mode control has attractive advantages compared to other control methods, such as low sensitivity to the system parameter variations and strong robustness to external disturbances [15, 16]. However, the chattering phenomena limit the practical applications of conventional sliding-mode control [17]. In this chapter, a nonsingular terminal sliding-mode control (NTSM) method is applied for IM velocity servo systems. To implement the FOC of IMs, an NTSM observer is designed in the chapter to estimate the rotor flux of IMs with equivalent smooth control signals [18, 19]. Additionally, the speed sensorless technology is also utilized in this chapter, afterwards an NTSM observer is utilized in the FOC system of IMs to estimate the speed instead of practical sensors.The simulations have been carried out to validate the applied method.

2 Mathematical Model of Induction Motor

An accurate mathematical model of IMs is the basic factor for the implementation of high-performance servo systems of IMs, especially for the FOC algorithms. For simplicity of the analysis some assumptions for IMs can be described as follows:

(1) the effect of magnetic saturation is neglected.

(2) the three-phase windings have the same structure and the fringe effect is neglected.

(3) the slots effect is ignored.

(4) the iron core loss is not taken into account.

2.1 Mathematical Model of IM in Three-Dimensional Stationary Coordinate (abc)

The mathematical model of IMs is usually composed of the voltage, flux, and motion equations. Based on the assumptions above, the voltage equations of the IMs in three-dimensional stationary coordinate (abc) are below:

$$\begin{aligned} \left[ \begin{array}{l} {u_{sa}}\\ {u_{sb}}\\ {u_{sc}}\\ {u_{ra}}\\ {u_{rb}}\\ {u_{rc}} \end{array} \right] = \left[ {\begin{array}{*{20}{c}} {{R_s}}&{}0&{}0&{}0&{}0&{}0\\ 0&{}{{R_s}}&{}0&{}0&{}0&{}0\\ 0&{}0&{}{{R_s}}&{}0&{}0&{}0\\ 0&{}0&{}0&{}{{R_r}}&{}0&{}0\\ 0&{}0&{}0&{}0&{}{{R_r}}&{}0\\ 0&{}0&{}0&{}0&{}0&{}{{R_r}} \end{array}} \right] \left[ \begin{array}{l} {i_{sa}}\\ {i_{sb}}\\ {i_{sc}}\\ {i_{ra}}\\ {i_{rb}}\\ {i_{rc}} \end{array} \right] + \left[ \begin{array}{l} {{\dot{\psi }}_{sa}}\\ {{\dot{\psi }}_{sb}}\\ {{\dot{\psi }}_{sc}}\\ {{\dot{\psi }}_{ra}}\\ {{\dot{\psi }}_{rb}}\\ {{\dot{\psi }}_{rc}} \end{array} \right] \end{aligned}$$

(14.1)

where $u_{sa}$, $u_{sb}$ and $u_{sc}$ are three stator voltages in abc axes;

     $u_{ra}$, $u_{rb}$ and $u_{rc}$ are three rotor voltages in abc axes;

     $i_{sa}$, $i_{sb}$ and $i_{sc}$ are three stator currents in abc axes;

     $i_{ra}$, $i_{rb}$ and $i_{rc}$ are three rotor currents in abc axes;

     $\psi _{sa}$, $\psi _{sb}$ and $\psi _{sc}$ are three stator fluxes in abc axes;

     $\psi _{ra}$, $\psi _{rb}$ and $\psi _{rc}$ are three rotor fluxes in abc axes;

     $R_s$ is the stator resistance;

     $R_r$ is the rotor resistance.

Therefore the flux equations of an IM in the three-dimensional stationary coordinate (abc) include both the stator and rotor flux equations and describe the relationships between the currents and the fluxes in the stator and rotor. They are given by the following equations:

$$\begin{aligned} \left[ \begin{array}{l} {\psi _s}\\ {\psi _r} \end{array} \right] = \left[ {\begin{array}{*{20}{c}} {{L_{ss}}}&{}{{L_{sr}}}\\ {{L_{rs}}}&{}{{L_{rr}}} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {{i_s}}\\ {{i_r}} \end{array}} \right] \end{aligned}$$

(14.2)

with

                                                                        ${L_{ss}} = \left[ {\begin{array}{*{20}{c}} {{L_{sm}} + {L_{sl}}}&{}{ - \frac{1}{2}{L_{sm}}}&{}{ - \frac{1}{2}{L_{sm}}}\\ { - \frac{1}{2}{L_{sm}}}&{}{{L_{sm}} + {L_{sl}}}&{}{ - \frac{1}{2}{L_{sm}}}\\ { - \frac{1}{2}{L_{sm}}}&{}{ - \frac{1}{2}{L_{sm}}}&{}{{L_{sm}} + {L_{sl}}} \end{array}} \right]$, ${L_{rr}} = \left[ {\begin{array}{*{20}{c}} {{L_{sm}} + {L_{rl}}}&{}{ - \frac{1}{2}{L_{sm}}}&{}{ - \frac{1}{2}{L_{sm}}}\\ { - \frac{1}{2}{L_{sm}}}&{}{{L_{sm}} + {L_{rl}}}&{}{ - \frac{1}{2}{L_{sm}}}\\ { - \frac{1}{2}{L_{sm}}}&{}{ - \frac{1}{2}{L_{sm}}}&{}{{L_{sm}} + {L_{rl}}} \end{array}} \right]$,

$${L_{sr}} = L_{rs}^T = {L_{sm}}\left[ {\begin{array}{*{20}{c}} {\cos \theta }&{}{\cos (\theta - {{120}^\circ })}&{}{\cos (\theta + {{120}^\circ })}\\ {\cos (\theta + {{120}^\circ })}&{}{\cos \theta }&{}{\cos (\theta - {{120}^\circ })}\\ {\cos (\theta - {{120}^\circ })}&{}{\cos (\theta + {{120}^\circ })}&{}{\cos \theta } \end{array}} \right] $$

where $\psi _s$ and $\psi _r$ are the stator flux phase and the rotor flux phase in abc axes respectively, by $\psi _s = [\psi _{sa}, \psi _{sb}, \psi _{sc}]^T$, $\psi _r = [\psi _{ra}, \psi _{rb}, \psi _{rc}]^T$

${{\varvec{i}}}_s$ and ${{\varvec{i}}}_r$ are the stator current vector and the rotor current vector in abc axes respectively, by ${{\varvec{i}}}_s = [{{\varvec{i}}}_{sa}, {{\varvec{i}}}_{sb}, {{\varvec{i}}}_{sc}]^T$, ${{\varvec{i}}}_r = [{{\varvec{i}}}_{ra}, {{\varvec{i}}}_{rb}, {{\varvec{i}}}_{rc}]^T$;

$L_{sl}$ and $L_{rl}$ are the stator and rotor leakage inductance respectively;

$L_{sm}$ and $L_{rm}$ are the mutual inductance between the stator and rotor windings;

$\theta$ is the electric angle between the stator and rotor windings.

The torque equation of IMs can be given as follows:

$$\begin{aligned} \begin{array}{l} {T_{e}} = {n_p}{L_{sm}}[({i_{sa}}{i_{ra}} + {i_{sb}}{i_{rb}} + {i_{sc}}{i_{rc}})\sin \theta + ({i_{sa}}{i_{ra}} + {i_{sb}}{i_{rb}} + {i_{sc}}{i_{rc}})\sin (\theta + {120^\circ })\\ {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} + ({i_{sa}}{i_{ra}} + {i_{sb}}{i_{rb}} + {i_{sc}}{i_{rc}})\sin (\theta - {120^\circ })] \end{array} \end{aligned}$$

(14.3)

where $n_p$ is the number of pole pairs.

Finally the mechanical equation of an IM can be described as follows:

$$\begin{aligned} {T_{e}} = {T_{L}} + \frac{J}{{{n_p}}}\dot{\omega }\end{aligned}$$

(14.4)

where $\omega$ is the electric angular velocity of the rotor;

       J is the inertia of the motor;

       $T_L$ is the load torque.

2.2 Mathematical Model of IM in Two-Dimensional Stationary Coordinate ($\alpha \beta$)

Based on Clarke transformation, the mathematical model of IMs in a three-dimensional stationary coordinate (abc) can be converted into a two-dimensional stationary coordinate ($\alpha \beta$). The Clarke transformation is given by the following equation:

$$\begin{aligned} {{\varvec{C}}}_{3/2} = {\sqrt{\frac{2}{3}}} \left[ {\begin{array}{*{20}{c}} 1&{}{ - \frac{\mathrm{{1}}}{\mathrm{{2}}}}&{}{ - \frac{\mathrm{{1}}}{\mathrm{{2}}}}\\ 0&{}{\frac{{\sqrt{\mathrm{{3}}} }}{\mathrm{{2}}}}&{}{\frac{{\sqrt{\mathrm{{3}}}}}{\mathrm{{2}}}} \end{array}} \right] \end{aligned}$$

(14.5)

Subsequently, the voltage equations of the IMs in a two-dimensional stationary coordinate ($\alpha \beta$) can be described as follow:

$$\begin{aligned} \left\{ \begin{array}{l} {u_{s\alpha }} = {R_s}{i_{s\alpha }} + {L_s}{{\dot{i}}_{s\alpha }} + {L_m}{{\dot{i}}_{r\alpha }}\\ {u_{s\beta }} = {R_s}{i_{s\beta }} + {L_s}{{\dot{i}}_{s\beta }} + {L_m}{{\dot{i}}_{rb}}\\ {u_{r\alpha }} = {R_r}{i_{r\alpha }} + {L_r}{{\dot{i}}_{r\alpha }} + {L_m}{{\dot{i}}_{r\alpha }} + \omega \mathrm{{(}}{L_m}{i_{s\beta }} + {L_r}{i_{r\beta }}\mathrm{{)}}\\ {u_{r\beta }} = {R_r}{i_{r\beta }} + {L_r}{{\dot{i}}_{r\beta }} + {L_m}{{\dot{i}}_{s\beta }} + \omega \mathrm{{(}}{\mathrm{{L}}_m}{i_{s\alpha }} + {L_r}{i_{r\alpha }}\mathrm{{)}} \end{array} \right. \end{aligned}$$

(14.6)

where $u_{s \alpha }$ and $u_{s \beta }$ are two stator voltages in $\alpha \beta$ axes;

     $u_{r \alpha }$ and $u_{r \beta }$ are two rotor voltages in $\alpha \beta$ axes;

     $i_{s \alpha }$ and $i_{s \beta }$ are two stator currents in $\alpha \beta$ axes;

     $i_{r \alpha }$ and $i_{r \beta }$ are two rotor currents in $\alpha \beta$ axes;

     $L_s$, $L_r$, and $L_m$ the stator inductance, the rotor inductance and the mutual inductance between the stator and rotor windings, which are described by

$$\begin{aligned} {L_s} = {L_m} + {L_{ls}}, \, {L_r} = {L_m} + {L_{lr}}, \, {L_m} = \frac{3}{2}{L_{ms}} \end{aligned}$$

Then the flux equations of IMs in a two-dimensional stationary coordinate ($\alpha \beta$) can be described as follow:

$$\begin{aligned} \left\{ \begin{array}{l} {\psi _{s\alpha }} = {L_s}{i_{s\alpha }} + {L_m}{i_{r\alpha }}\\ {\psi _{s\beta }} = {L_s}{i_{s\beta }} + {L_m}{i_{r\beta }}\\ {\psi _{r\alpha }} = {L_m}{i_{s\alpha }} + {L_r}{i_{r\alpha }}\\ {\psi _{r\beta }} = {L_m}{i_{s\beta }} + {L_r}{i_{r\beta }} \end{array} \right. \end{aligned}$$

(14.7)

where ${\psi _{s\alpha }}$ and ${\psi _{s\beta }}$ are the stator fluxes in $\alpha \beta$ axes respectively;

     ${\psi _{r\alpha }}$ and ${\psi _{r\beta }}$ are the rotor fluxes in axes respectively.

The torque equation of IMs in a two-dimensional stationary coordinate ($\alpha \beta$) is described as follows:

$$\begin{aligned} {T_e} = {n_p}{L_m}({i_{s\beta }}{i_{r\alpha }} - {i_{s\alpha }}{i_{r\beta }}) \end{aligned}$$

(14.8)

2.3 Mathematical Model of IMs in Two-Dimensional Rotating Coordinate (dq)

The FOC strategy can decouple the d-axis and q-axis currents in the stator of an IM applying the Park transformation, and make the rotor flux and torque of the IM controlled separately like as decoupled excited DC motors. The relationship between the two-dimensional stationary and rotating coordinates can be shown in Fig. 14.1, where $\theta _e$ represents the electric angle between the d-axis and the $\alpha$-axis. The speed of the two-dimensional rotating coordinate equals to $\omega _1$ which is the synchronous electric angular velocity of the IM.

Fig. 14.1

$\alpha \beta$ and dq coordinates

The Park transformation is given by

$$\begin{aligned} {{\varvec{C}}}_{2{s}/2{r}} = \left[ {\begin{array}{*{20}{c}} {\cos {\theta _{e}}}&{}{\sin {\theta _{e}}}\\ { - \sin {\theta _{e}}}&{}{\cos {\theta _{e}}} \end{array}} \right] \end{aligned}$$

(14.9)

Then the voltage equations of the IMs in a two-dimensional rotating coordinate (dq) can be described as follow:

$$\begin{aligned} \left\{ \begin{array}{l} {u_{sd}} = {R_s}{i_{sd}} + {{\dot{\psi }}_{sd}} - {\omega _1}{\psi _{sq}}\\ {u_{sq}} = {R_s}{i_{sq}} + {{\dot{\psi }}_{sq}} + {\omega _1}{\psi _{sd}}\\ {u_{rd}} = {R_r}{i_{rd}} + {{\dot{\psi }}_{rd}} - {\omega _s}{\psi _{rq}}\\ {u_{rq}} = {R_r}{i_{rq}} + {{\dot{\psi }}_{rq}} - {\omega _s}{\psi _{rd}} \end{array} \right. \end{aligned}$$

(14.10)

where $u_{sd}$ and $u_{sq}$ are two stator voltages in dq axes;

     $u_{rd}$ and $u_{rq}$ are two rotor voltages in dq axes;

     $i_{sd}$ and $i_{sq}$ are the stator currents in d- and q- axes;

     $i_{rd}$ and $i_{rq}$ are the rotor currents in d- and q- axes;

     ${{\psi }_{sd}}$ and ${{\psi }_{sq}}$ are the stator fluxes in d- and q- axes;

     ${{\psi }_{rd}}$ and ${{\psi }_{rq}}$ are the rotor fluxes in d- and q- axes;

     ${\omega _s}$ is the slip angle velocity;

     ${\omega _1}$ is the synchronous angular velocity.

The flux equations of the IMs in a two-dimensional rotating coordinate system (dq) can be described as follow:

$$\begin{aligned} \left\{ \begin{array}{l} {\psi _{sd}} = {L_s}{i_{sd}} + {L_m}{i_{rd}}\\ {\psi _{sq}} = {L_s}{i_{sq}} + {L_m}{i_{rq}}\\ {\psi _{rd}} = {L_m}{i_{sd}} + {L_r}{i_{rq}}\\ {\psi _{rq}} = {L_m}{i_{sq}} + {L_r}{i_{rq}} \end{array} \right. \end{aligned}$$

(14.11)

The torque equation of IMs in a two-dimensional rotating coordinate (dq) is given by

$$\begin{aligned} {T_e} = \frac{{{n_p}{L_m}}}{{{L_r}}}({i_{sq}}{\psi _{rd}} - {i_{sd}}{\psi _{rq}}) \end{aligned}$$

(14.12)

Summarizing, the mathematical model of IMs in a three-dimensional stationary coordinate (abc) is described by Eqs.(14.1)–(14.4), which is further transferred to the model in a two-dimensional stationary coordinate ($\alpha \beta$), as shown as in Eqs. (14.6), (14.7) and (14.8), by the Clark transformation (14.5). Finally, the model of IMs in a two-dimensional rotating coordinate (dq) is obtained in Eqs. (14.4), (14.10), (14.11) and (14.12) by the Park transformation (14.9).

3 Field Oriented Control System

For squirrel cage IMs, the rotor voltages are

$$\begin{aligned} {u_{rd}} = {u_{rq}} = \mathrm{{0}} \end{aligned}$$

(14.13)

The mathematical model of the IMs in the FOC systems can be finally described in dq axes as follows by using Eqs. (14.4), and (14.10)–(14.13):

$$\begin{aligned} \left\{ \begin{array}{l} {{\dot{i}}_{sd}} = \xi \frac{1}{{{T_r}}}{\psi _{rd}} + \xi \omega {\psi _{rq}} - \lambda {i_{sd}} + {\omega _1}{i_{sq}} + K{u_{sd}}\\ {{\dot{i}}_{sq}} = - \xi \omega {\psi _{rd}} + \xi \frac{1}{{{T_r}}}{\psi _{rq}} - {\omega _1}{i_{sd}} - \lambda {i_{sq}} + K{u_{sq}}\\ {{\dot{\psi }}_{rd}} = - \frac{1}{{{T_r}}}{\psi _{rd}} + ({\omega _1} - \omega ){\psi _{rq}} + \frac{{{L_m}}}{{{T_r}}}{i_{sd}}\\ {{\dot{\psi }}_{rq}} = - ({\omega _1} - \omega ){\psi _{rd}} - \frac{1}{{{T_r}}}{\psi _{rq}} + \frac{{{L_m}}}{{{T_r}}}{i_{sq}}\\ \dot{\omega }= \frac{{n_p^2{L_m}}}{{J{L_r}}}\left( {{i_{sq}}{\psi _{rd}} - {i_{sd}}{\psi _{rq}}} \right) - \frac{{{n_p}}}{J}{T_L} \end{array} \right. \end{aligned}$$

(14.14)

where $\sigma = 1- {{L_m^2} / {L_sL_r}}$ is the leakage coefficient; ${T_r} = {L_r}/{R_r}$ is the rotor time constant; and $K = {{1} /{\sigma L_s}}$, $\xi = K({L_m}/{L_r})$, $\lambda = K({R_s} + L_m^2/{T_r}{L_r})$.

Based on the FOC strategy, the d-axis is orientated in accord with the axis of the rotor flux. Therefore d- and q- axis rotor flux can be determined as follows

$$\begin{aligned} \left\{ {\begin{array}{*{20}{c}} {{\psi _{rd}} = {\psi _{r}}}\\ {{\psi _{rq}} = 0} \end{array}} \right. \end{aligned}$$

(14.15)

where ${\psi _{r}}$ is the rotor flux.

The FOC of IMs is based on their mathematical model (14.14) and the field orientated principle (14.15). It can be seen that the control of IMs is decoupled into the rotor flux control and torque control. Then the FOC-based speed servo system of IMs can be built using four nonsingular terminal sliding-mode controllers in the speed-, flux-, d-axis current- and q-axis current-loops respectively, as shown in Fig. 14.2.

Fig. 14.2

An FOC-based speed closed-loop control system of IMs

In Fig. 14.2, ${\omega _{{m_{ref}}}}$ is the required speed of the IM. The outputs of the speed and flux controllers are the references of the stator currents in d- and q-axes respectively. To implement high-performance control of the FOC position servo system of IMs, four full-order sliding-mode controllers will be designed in the next section.

4 NTSM Controllers for IM Servo System

4.1 Speed Controller

The motion equation of the IMs can be obtained from the model of the IMs (14.14) as follows:

$$\begin{aligned} \dot{\omega }_m = \frac{{{n_p}{L_m}}}{{J{L_r}}}{i_{sq}}{\psi _r} - \frac{1}{J}{T_L} \end{aligned}$$

(14.16)

where $\omega _m = \omega /n_p$ is the mechanical angular velocity of the rotor.

Defining the desired mechanical velocity of the motor as ${\omega _{{m_{ref}}}}$, which should be smooth enough up to the second order time derivative, the error between the actual velocity and the given velocity $e_{\omega }$ is:

$$\begin{aligned} {e_\omega } = {\omega _{{m_{ref}}}} - {\omega _m} \end{aligned}$$

(14.17)

Then the speed error dynamics can be obtained as follows using Eqs.(14.16) and (14.17):

$$\begin{aligned} {\dot{e}_\omega } = {\dot{\omega }_{{m_{ref}}}} - {\dot{\omega }_m} = {\dot{\omega }_{{m_{ref}}}} - \frac{{{n_p}{L_m}}}{{J{L_r}}}{\psi _r}{i_{sq}} + \frac{1}{J}{T_L} \end{aligned}$$

(14.18)

A NTSM manifold [7] is designed as follows:

$$\begin{aligned} {s_\omega } = {e_\omega } + {\gamma _1}{\dot{e}_\omega }^{{p_1}/{q_1}} \end{aligned}$$

(14.19)

where ${\gamma _1}>0$; $p_1$ and $q_1$ are odd, and $1<p_1/q_1<2$.

Theorem 14.1

The NTSM surface (14.19) and the following control assure that finite-time convergence of the speed error dynamics (14.18):

$$\begin{aligned} {i_{s{q_{ref}}}} = {i_{s{q_{eq}}}} + {i_{s{q_n}}} \end{aligned}$$

(14.20)

$$\begin{aligned} {i_{s{q_{eq}}}} = \frac{{J{L_r}}}{{{n_p}{L_m}{\psi _r}}}{\dot{\omega }_{{m_{ref}}}} \end{aligned}$$

(14.21)

$$\begin{aligned} {\dot{i}_{s{q_n}}} + T{i_{s{q_n}}} = {v_\omega } \end{aligned}$$

(14.22)

$$\begin{aligned} {v_\omega } = \frac{{J{L_r}}}{{{n_p}{L_m}{\psi _r}}}\left( {\frac{{{s_\omega }{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}}}{{{{\left| {{s_\omega }{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}} \right| }^2}}}\left| {{s_\omega }} \right| \left| {{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}} \right| ({k_1} + {\eta _1}) + \frac{{{q_1}}}{{{\gamma _1}{p_1}}}{{\dot{e}}_\omega }^{2 - {p_1}/{q_1}}} \right) \end{aligned}$$

(14.23)

where $k_1>0$, $\eta _1>0$ are design parameters, and ${k_1} > \left( {\left| {\frac{1}{J}{{\dot{T}}_L}} \right| + \left| {\frac{{{n_p}{L_m}{\psi _r}}}{{J{L_r}}}T{i_{s{q_n}}}} \right| } \right)$.

Proof

The following Lyapunov function is considered ${V_\omega }(t) = 0.5{s_\omega }^2(t)$. We have:

$$\begin{aligned} {\dot{V}_\omega }(t) = {s_\omega }(t){\dot{s}_\omega }(t) = {s_\omega }\left[ {{{\dot{e}}_\omega } + \frac{{{\gamma _1}{p_1}}}{{{q_1}}}{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}{{\ddot{e}}_\omega }} \right] \end{aligned}$$

$$\begin{aligned} = \frac{{{s_\omega }{\gamma _1}{p_1}}}{{{q_1}}}{\dot{e}_\omega }^{{p_1}/{q_1} - 1}\left[ { - \frac{{{n_p}{L_m}}}{{J{L_r}}}{\psi _r}{{\dot{i}}_{s{q_n}}} + \frac{1}{J}{{\dot{T}}_L} + \frac{{{q_1}}}{{{\gamma _1}{p_1}}}{{\dot{e}}_\omega }^{2 - {p_1}/{q_1}}} \right] \end{aligned}$$

$$\begin{aligned} = \frac{{{s_\omega }{\gamma _1}{p_1}}}{{{q_1}}}{\dot{e}_\omega }^{{p_1}/{q_1} - 1}\left[ { - \frac{{{n_p}{L_m}{\psi _r}}}{{J{L_r}}}{v_\omega } + \frac{{{n_p}{L_m}{\psi _r}}}{{J{L_r}}}T{i_{s{q_n}}} + \frac{1}{J}{{\dot{T}}_L} + \frac{{{q_1}}}{{{\gamma _1}{p_1}}}{{\dot{e}}_\omega }^{2 - {p_1}/{q_1}}} \right] \end{aligned}$$

$$\begin{aligned} = \frac{{{s_\omega }{\gamma _1}{p_1}}}{{{q_1}}}{\dot{e}_\omega }^{{p_1}/{q_1} - 1}\left[ { - \frac{{{s_\omega }{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}}}{{{{\left| {{s_\omega }{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}} \right| }^2}}}\left| {{s_\omega }} \right| \left| {{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}} \right| ({k_1} + {\eta _1}) + \frac{{{n_p}{L_m}{\psi _r}}}{{J{L_r}}}T{i_{s{q_n}}} + \frac{1}{J}{{\dot{T}}_L}} \right] \end{aligned}$$

$$\begin{aligned} = \frac{{{\gamma _1}{p_1}}}{{{q_1}}}\left[ { - \left| {{s_\omega }} \right| \left| {{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}} \right| ({k_1} + {\eta _1}) + {s_\omega }{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}\left( {\frac{1}{J}{{\dot{T}}_L} + \frac{{{n_p}{L_m}{\psi _r}}}{{J{L_r}}}T{i_{s{q_{ref}}}}} \right) } \right] \end{aligned}$$

$$\begin{aligned} \le \frac{{{\gamma _1}{p_1}}}{{{q_1}}}\left| {{s_\omega }} \right| \left| {{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}} \right| \left[ { - ({k_1} + {\eta _1}) + \left| {\frac{1}{J}{{\dot{T}}_L}} \right| + \left| {\frac{{{n_p}{L_m}{\psi _r}}}{{J{L_r}}}T{i_{s{q_{ref}}}}} \right| } \right] \end{aligned}$$

$$\begin{aligned} \le \frac{{{\gamma _1}{p_1}}}{{{q_1}}}{\eta _1}\left| {{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}} \right| \left| {{s_\omega }} \right| \end{aligned}$$

since ${k_1} > \left( {\left| {{{\dot{T}}_L}/J} \right| + \left| {({n_p}{L_m}{\psi _r}/J{L_r})T{i_{s{q_{ref}}}}} \right| } \right)$, therefore

$$\begin{aligned} {\dot{V}_\omega }(t){} {} \le - \gamma {}_1({p_1}/{q_1}){\eta _1}\left| {{{\dot{e}}_\omega }^{{p_1}/{q_1} - 1}} \right| \left| {{s_\omega }} \right| | < 0 \quad \mathrm{{for}} \, |{s_\omega }| \ne 0 \end{aligned}$$

which means that the speed error dynamics (14.18) can reach the sliding-mode surface in finite time, and then both ${e_\omega }$ and ${\dot{e}_\omega }$ can converge to zero within infinite time. This completes the proof.

4.2 Rotor Flux Controller Design

Define the desired rotor flux as ${\psi _{{r_{ref}}}}=$ sconst, the error between the actual rotor flux and the desired flux is $e_{\psi }$:

$$\begin{aligned} {e_\psi } = {\psi _{{r_{ref}}}} - {\psi _r} \end{aligned}$$

(14.24)

The rotor flux error system can be obtained as follows according to the mathematical model (14.14):

$$\begin{aligned} {\dot{e}_\psi } = - {\dot{\psi }_r} = \frac{1}{{{T_r}}}{\psi _r} - \frac{{{L_m}}}{{{T_r}}}{i_{sd}} \end{aligned}$$

(14.25)

The sliding-mode surface $s_{\psi }$ is designed as the follows:

$$\begin{aligned} {s_\psi } = {e_\psi } + {\gamma _2}{\dot{e}_\psi }^{{p_2}/{q_2}} \end{aligned}$$

(14.26)

where $\gamma _2 > 0$, $p_2$, $q_2$ are odd, and $1<p_2/q_2<2$.

Theorem 14.2

The NTSM surface (14.26) and the following control assure the finite-time convergence of the rotor flux error system (14.25):

$$\begin{aligned} {i_{s{d_{ref}}}} = {i_{s{d_{eq}}}} + {i_{s{d_n}}} \end{aligned}$$

(14.27)

$$\begin{aligned} {i_{s{d_{eq}}}} = \frac{{{\psi _r}}}{{{L_m}}} \end{aligned}$$

(14.28)

$$\begin{aligned} {\dot{i}_{s{d_n}}} + T{i_{s{d_n}}} = {v_\psi } \end{aligned}$$

(14.29)

$$\begin{aligned} {v_\psi } = \frac{{J{L_r}}}{{{n_p}{L_m}{\psi _r}}}\left( {\frac{{{s_\psi }{{\dot{e}}_\psi }^{{p_2}/{q_2} - 1}}}{{{{\left| {{s_\psi }{{\dot{e}}_\psi }^{{p_2}/{q_2} - 1}} \right| }^2}}}\left| {{s_\psi }} \right| \left| {{{\dot{e}}_\psi }^{{p_2}/{q_2} - 1}} \right| ({k_2} + {\eta _2}) + \frac{{{q_2}}}{{{\gamma _2}{p_2}}}{{\dot{e}}_\psi }^{2 - {p_2}/{q_2}}} \right) \end{aligned}$$

(14.30)

where $k_2>0$, $\eta _2 > 0$ is the design parameter.

Proof

This follows straightforwardly from Theorem 14.1.

4.3 q-axis Current Controller Design

Define the error between the desired current in q-axis (${i_{s{q_{ref}}}}$) and the actual current in q-axis ($i_{sq}$) as follows:

$$\begin{aligned} {e_{sq}} = {i_{s{q_{ref}}}} - {i_{sq}} \end{aligned}$$

(14.31)

The q-axis current error system can be obtained as the follow according to the mathematical model (14.14):

$$\begin{aligned} {\dot{e}_{sq}} = {\dot{i}_{s{q_{ref}}}} - {\dot{i}_{sq}} = {\dot{i}_{s{q_{ref}}}} + \xi \omega {\psi _r} + {\omega _1}{i_{sd}} + \lambda {i_{sq}} - K{u_{sq}} \end{aligned}$$

(14.32)

A NTSM manifold $s_{sq}$ is designed as the follow:

$$\begin{aligned} {s_{sq}} = {e_{sq}} + {\gamma _3}{\dot{e}_{sq}}^{{p_3}/{q_3}} \end{aligned}$$

(14.33)

where $\gamma _3>0$, $p_3$, $q_3$ are odd, and $1<p_3/q_3<2$.

Theorem 14.3

The NTSM surface (14.33) and the following control assure the finite-time convergence of the q-axis current error system (14.32):

$$\begin{aligned} {u_{sq}} = {u_{s{q_{eq}}}} + {u_{s{q_n}}} \end{aligned}$$

(14.34)

$$\begin{aligned} {u_{s{q_{eq}}}} = {{\left( {{{\dot{i}}_{s{q_{ref}}}} + \xi \omega {\psi _r} + {\omega _1}{i_{sd}} + \lambda {i_{sq}}} \right) } / K} \end{aligned}$$

(14.35)

$$\begin{aligned} {\dot{u}_{s{q_n}}} + T{u_{s{q_n}}} = {v_{sq}} \end{aligned}$$

(14.36)

$$\begin{aligned} {v_{sq}} = \frac{1}{K}\left( {\frac{{{s_{sq}}{{\dot{e}}_{sq}}^{{p_3}/{q_3} - 1}}}{{{{\left| {{s_{sq}}{{\dot{e}}_{sq}}^{{p_3}/{q_3} - 1}} \right| }^2}}}\left| {{s_{sq}}} \right| \left| {{{\dot{e}}_{sq}}^{{p_3}/{q_3} - 1}} \right| ({k_3} + {\eta _3}) + \frac{{{q_3}}}{{{\gamma _3}{p_3}}}{{\dot{e}}_{sq}}^{2 - {p_3}/{q_3}}} \right) \end{aligned}$$

(14.37)

where $k_3>0$, $\eta _3 > 0$ are design parameters.

Proof

This follows straightforwardly from Theorem 14.1 as well.

4.4 d-axis Current Controller Design

Define the error between the desired and the actual d-axis current as the follow:

$$\begin{aligned} {e_{sd}} = {i_{s{d_{ref}}}} - {i_{sd}} \end{aligned}$$

(14.38)

The d-axis current error system can be obtained as follows according to the mathematical model (14.14):

$$\begin{aligned} {\dot{e}_{sd}} = {i_{s{d_{ref}}}} - \xi \frac{1}{{{T_r}}}{\psi _r} + \lambda {i_{sd}} - {\omega _1}{i_{sq}} - K{u_{sd}} \end{aligned}$$

(14.39)

A NTSM manifold is designed as the follow:

$$\begin{aligned} {s_{sd}} = {e_{sd}} + {\gamma _4}{\dot{e}_{sd}}^{{p_4}/{q_4}} \end{aligned}$$

(14.40)

where $\gamma _4>0$, $p_4$ and $q_4$ are odd, and $1<p_4/q_4<2$.

Theorem 14.4

The NTSM surface (14.40) and the following control assure the finite-time convergence of the d-axis current error system (14.39):

$$\begin{aligned} {u_{sd}} = {u_{s{d_{eq}}}} + {u_{s{d_n}}} \end{aligned}$$

(14.41)

$$\begin{aligned} {u_{sdeq}} = \left( {{{\dot{i}}_{s{d_{ref}}}} - \xi \frac{1}{{{T_r}}}{\psi _r} + \lambda {i_{sd}} - {\omega _1}{i_{sq}}} \right) /K \end{aligned}$$

(14.42)

$$\begin{aligned} {\dot{u}_{s{d_n}}} + T{u_{s{d_n}}} = {v_{sd}} \end{aligned}$$

(14.43)

$$\begin{aligned} {v_{sd}} = \frac{1}{K}\left( {\frac{{{s_{sd}}{{\dot{e}}_{sd}}^{{p_4}/{q_4} - 1}}}{{{{\left| {{s_{sd}}{{\dot{e}}_{sd}}^{{p_4}/{q_4} - 1}} \right| }^2}}}\left| {{s_{sd}}} \right| \left| {{{\dot{e}}_{sd}}^{{p_4}/{q_4} - 1}} \right| ({k_4} + {\eta _4}) + \frac{{{q_4}}}{{{\gamma _4}{p_4}}}{{\dot{e}}_{sd}}^{2 - {p_4}/{q_4}}} \right) \end{aligned}$$

(14.44)

where $k_4>0$, $\eta _4 > 0$ are parameters to design.

Proof

This follows straightforwardly from Theorem 14.1.

5 Numerical Simulation Test

Some simulations are carried out for an IM control system to validate the applied NTSM controllers in MATLAB-Simulink. The parameters of the IM are give as follows:

$P_N{=}\,1.1\,kW$, $I_N{=}\,2.8\,A$, $U_N{=}\,380\,V$, $f_N{=}\,50\,Hz$, $n_p=2$, $R_s=5.9 \,\Omega$, $R_r=5.6\,\Omega$, $L_r=37.94\,mH$, $L_s=0.58\,H$, $L_m=0.55\,H$, $J=0.021\,kg\cdot m^2$, $\psi _{ref}=0.7\,Wb$.

And the NTSM controllers are designed with the following parameters:

$p_1=5$, $q_1=3$, $\gamma _1=50$, $k_1=5000$, $T_1=10$; $p_2=5$, $q_2=3$, $\gamma _2=30$, $k_2=3000$, $T_2=10$; $p_3=7$, $q_3=5$, $\gamma _3=20$, $k_3=6000$, $T_3=5$; $p_4=13$, $q_4=11$, $\gamma _4=10$, $k_4=3000$, $T_4=5$.

The desired speed is 20sin(4t) rpm and the desired rotor flux is 0.7Wb.

The motor speed and the its tracking error are displayed in Fig. 14.3. For clear observation, the desired speed is shifted a little bit manually. It can be seen that the motor speed can track its reference fast and accurately. The rotor flux and its tracking error are shown in Fig. 14.4. It is clear that the rotor flux can track its reference value reference. The desired flux is shifted a little as well. The stator currents are displayed in Fig. 14.5. Both of the two currents track their references quickly and accurately. The control signals of the NTSM current controllers are shown in Fig. 14.6. It can be seen that the two control signals are smooth, which means that the chattering is attenuated by the applied new high-order NTSM method, and better performances of IM systems can be obtained.

Fig. 14.3

The motor speed and its tracking error

Fig. 14.4

The rotor flux and its tracking error

Fig. 14.5

The stator currents in dq

Fig. 14.6

The control signals of the NTSM current controllers

6 Conclusion

This chapter has introduced a nonsingular terminal sliding-mode control method for IM velocity servo systems. The NTSM controllers are applied into the speed, flux and current closed loop of the FOC-based IM velocity servo systems. The designed NTSM control law can suppress the chattering which exists in conventional sliding-mode control. The results of simulation have proved that the applied method is corrective and effective.