1 Introduction

In industrial applications, nowadays, the three phase induction motor (TIM) is one of the most electric machines that is largely applied, due its robustness and efficiency. Therefore, the research on control system in order to improve the effectiveness of the TIM applications always attracts many attentions from researchers in the world [1,2,3,4,5,6,7]. However, the model of the TIM is a fully complex nonlinear system, with multiple inputs and outputs, the control for TIM is not easy to implement, especially in real applications. As well known, with the results in previous works [1,2,3,4,5,6,7], the vector-oriented control (VOC) method plays an important role in the design of control algorithms. For the TIM control, the VOC technique is a basic derivative to implement the decoupled control of d–q currents [2, 4, 5]. Based on this decoupled control, the sliding mode control (SMC) technique, with its advantages [8,9,10] as the stability and robustness features in the presence of time-varying disturbances, can be considered as the most popular control strategy for the TIM that is applied to be able to achieve high performance [11,12,13,14,15,16,17,18,19,20,21,22]. The authors in [11] combined a backstepping technique into the SMC method to position control for a linear induction motor. In this proposed method [11], to decrease the chattering phenomena, the signum function for the sliding switching surface is replaced by the saturated function. Clearly, in the SMC methods for the TIM control, the chattering phenomena are the important problem that must be solved carefully in order to guarantee the robustness for the SMC control systems [9, 10]. For the TIM control system based the SMC, the chattering phenomena will be increased cause by fast and large amplitude variation of the TIM currents and voltages. Thus, the robustness and stability of the SMC system depend on the selection of the sliding and switching surfaces. To relax these problems for the electric machine SMC-control, Zhang et al. [19] proposed the reaching law (the switching surface) as a nonlinear exponential function. This reaching law can help the proposed control system [19] adapt to the variations of the switching surface and system states [18], suppress the chattering and discontinuous states (for using the signum function as the switching surface). In the other hand, the authors in [20] presented successfully the fast terminal sliding mode scheme to deal with chattering phenomena and improve the control performance for the motor-SMC system. Thus, we can realize that the SMC systems using the sliding nonlinear functions can achieve better the chattering-reducing performances than those of the sliding linear functions [19,20,21,22].

Accordance with the improving of the control effectiveness, research on the position sensorless control for the electric machines have been explored strongly in recent years [23,24,25,26,27,28,29,30,31,32]. Wang et al. in [25] implemented quite well a back electromotive force detection method as a sensorless control for the motor for the high order SMC control without Hall sensors. Based on the previous research works [23,24,25,26,27,28,29,30,31,32], we easy see that observer systems-based approaches can be considered as a useful scheme for the TIM sensorless control methods. By applying, the observers-based models [23,24,25, 27, 30,31,32], these approaches achieved good performances for the TIM control systems. However, because highly dependent on the measured signals of the electric machine control systems, the complexity of system design and numerical calculations will have to increase. In the different approach, the results in [26, 28, 29] proposed the observer based on the soft computing techniques for the TIM sensorless control. These proposed observer [26, 28, 29], have proved the effectiveness of the TIM sensorless control systems in reducing the disadvantages of the observer systems-based models that are the complexity of the control systems and computation burden.

In this research, by applying the merits of the SMC sensorless control methods based soft computing techniques, the decoupled SMC system will be proposed without the requirement of the TIM speed information. The first, the decoupled control method as the VOC technique will create two independent controllers for the flux and rotor speed based the SMC scheme. In order to reduce the chattering phenomena, in the proposed SMC controllers, the sliding mode surface is applied as the nonlinear function that is the hyperbolic tangent sigmoid function (tansig function). The second, the rotor speed of the TIM system will be online estimated by using an observer based the radial basic function (RBF) neural networks (NN). Moreover, the online NN updating algorithm and the SMC laws are designed by using the Lyapunov stability theorem such that the stability and adaptability of the proposed SMC sensorless control system for the TIM are guaranteed.

The paper is organized as follows. Section II describes the dynamics model of the TIM and the problem formulation for the SMC decoupled control strategies. In Section III, the SMC sensorless control algorithms are presented. The comparative simulation and experimental results for the TIM control system are included in Section IV. Finally, conclusion is drawn in Section V.

2 Preliminaries

2.1 System description

We will consider the nonlinear model of the three-phase asynchronous induction motor in the d–q stationary reference frame as the following vector–matrix form [33]:

The stator and rotor voltage equations:

$$ \left[ {\begin{array}{*{20}c} {u_{{{\text{sd}}}} } \\ {u_{{{\text{sq}}}} } \\ \end{array} } \right] {=} \left[ {\begin{array}{*{20}c} {R_{{\text{s}}} } & 0 \\ 0 & {R_{{\text{s}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{{{\text{sd}}}} } \\ {i_{{{\text{sq}}}} } \\ \end{array} } \right] {+} \frac{d}{dt}\left[ {\begin{array}{*{20}c} {\varphi_{{{\text{sd}}}} } \\ {\varphi_{{{\text{sq}}}} } \\ \end{array} } \right] {+} \left[ {\begin{array}{*{20}c} 0 & { - \omega_{{\text{e}}} } \\ {\omega_{{\text{e}}} } & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varphi_{{{\text{sd}}}} } \\ {\varphi_{{{\text{sq}}}} } \\ \end{array} } \right] $$
(1)
$$ \begin{aligned} \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right] = & \left[ {\begin{array}{*{20}c} {R_{{\text{r}}} } & 0 \\ 0 & {R_{{\text{r}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{{{\text{rd}}}} } \\ {i_{{{\text{rq}}}} } \\ \end{array} } \right] + \frac{{\text{d}}}{{{\text{d}}t}}\left[ {\begin{array}{*{20}c} {\varphi _{{{\text{rd}}}} } \\ {\varphi _{{{\text{rq}}}} } \\ \end{array} } \right] \\ &+ \left[ {\begin{array}{*{20}c} 0 & { - (\omega _{{\text{e}}} - p\omega _{r} )} \\ {(\omega _{{\text{e}}} - p\omega _{{\text{r}}} )} & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varphi _{{{\text{rd}}}} } \\ {\varphi _{{{\text{rq}}}} } \\ \end{array} } \right] \\ \end{aligned} $$
(2)

The stator and rotor flux equations:

$$ \left[ {\begin{array}{*{20}c} {\varphi_{{{\text{sd}}}} } \\ {\varphi_{{{\text{sq}}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {L_{{\text{s}}} } & 0 \\ 0 & {L_{{\text{s}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{{{\text{sd}}}} } \\ {i_{{{\text{sq}}}} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {L_{{\text{m}}} } & 0 \\ 0 & {L_{{\text{m}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{{{\text{rd}}}} } \\ {i_{{{\text{rq}}}} } \\ \end{array} } \right] $$
(3)
$$ \left[ {\begin{array}{*{20}c} {\varphi_{{{\text{rd}}}} } \\ {\varphi_{{{\text{rq}}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {L_{{\text{m}}} } & 0 \\ 0 & {L_{{\text{m}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{{{\text{sd}}}} } \\ {i_{{{\text{sq}}}} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {L_{{\text{r}}} } & 0 \\ 0 & {L_{r} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{{{\text{rd}}}} } \\ {i_{{{\text{rq}}}} } \\ \end{array} } \right] $$
(4)

The torque equations:

$$ \left\{ {\begin{array}{*{20}l} {T_{{\text{e}}} = \frac{3}{2}p\left[ {\begin{array}{*{20}c} {\varphi_{{{\text{sd}}}} } & {\varphi_{{{\text{sq}}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {i_{{{\text{sq}}}} } \\ { - i_{{{\text{sd}}}} } \\ \end{array} } \right]} \hfill \\ {J\frac{{\text{d}}}{{{\text{d}}t}}\omega_{r} = (T_{{\text{e}}} - B\omega_{{\text{r}}} - T_{{\text{m}}} )} \hfill \\ \end{array} } \right. $$
(5)

We note that, if the d-axis of the reference frame is aligned with the stator flux vector [33], we have the following results:

$$ \varphi_{{{\text{sq}}}} = \dot{\varphi }_{{{\text{sq}}}} = 0 $$
(6)
$$ u_{{{\text{sq}}}} = R_{{\text{s}}} i_{{{\text{sq}}}} + \omega_{r} \varphi_{{{\text{sd}}}} $$
(7)
$$ u_{{{\text{sd}}}} = R_{{\text{s}}} i_{{{\text{sd}}}} + \dot{\varphi }_{{{\text{sd}}}} $$
(8)
$$ T_{{\text{e}}} = \frac{3}{2}p(\varphi_{{{\text{sd}}}} i_{{{\text{sq}}}} ) $$
(9)
$$ \varphi_{{{\text{sd}}}} = \int {(u_{{{\text{sd}}}} - R_{{\text{s}}} i_{{{\text{sd}}}} ){\text{d}}t} $$
(10)

In this paper, the SMC based on the decoupled VOC method will be consider for the control of the stator flux and the rotor speed. The characteristics of the proposed SMC controllers will be analyzed as follows.

2.2 Stator flux control

In this study, we will apply the SMC technique to control the stator flux. Define the stator flux control error as:

$$ e_{{\text{F}}} = \varphi_{{{\text{sd}}}}^{*} - \varphi_{{{\text{sd}}}} $$
(11)

where \(\varphi_{{{\text{sd}}}}^{*}\) is the desired stator flux that is bounded and uniformly continuous, and it also has derivative up to the second order. The sliding function for this control is defined as the following form:

$$ S_{{\text{F}}} = c_{1} e_{{\text{F}}} + \dot{e}_{{\text{F}}} $$
(12)

where \(c_{1}\) is positive constant. Taking the derivative of \(S_{{\text{F}}}\) with respect to time and applying Eq. (8), yields:

$$ \dot{S}_{{\text{F}}} = - c_{1} (u_{{{\text{sd}}}} - R_{{\text{s}}} i_{{{\text{sd}}}} ) + c_{1} \varphi_{{{\text{sd}}}}^{*} + \ddot{e}_{{\text{F}}} $$
(13)

In order to guarantee the stator flux control, the switching surface function (13) can be chosen as:

$$ \dot{S}_{{\text{F}}} = - \varepsilon_{{\text{F}}} {\text{sign}}(S_{{\text{F}}} ) $$
(14)

where \(\varepsilon_{{\text{F}}} > 0\), the \({\text{sign}}(S_{{\text{F}}} )\) function is defined as:

$$ {\text{sign}}(S_{{\text{F}}} ) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {S_{{\text{F}}} > 0} \hfill \\ {0,} \hfill & {S_{{\text{F}}} = 0} \hfill \\ { - 1} \hfill & {S_{{\text{F}}} < 0} \hfill \\ \end{array} } \right. $$
(15)

Based on the SMC theory [9], from (13) and (14), the SMC input for the stator flux control can be determined as the following form:

$$ u_{{{\text{sd}}}} = \frac{1}{{c_{1} }}(\varepsilon_{{\text{F}}} {\text{sign}}(S_{{\text{F}}} ) + \ddot{e}_{{\text{F}}} ) + \dot{\varphi }_{{{\text{sd}}}}^{*} + R_{{\text{s}}} i_{{{\text{sd}}}} $$
(16)

Individually, this SMC input can be able to guarantee for the tracking control of the stator flux control. However, in the TIM control system, the effectiveness, robustness, and stability of the stator flux control are dependent heavily on the control of the rotor speed performance. In addition, the SMC always has the inevitable chattering phenomena with the signum \({\text{sign}}(S_{{\text{F}}} )\) switching function.

2.3 Rotor speed control

Similar to the stator flux control, the SMC method will be applied to control the rotor speed. Define the rotor speed error as:

$$ e_{{\text{M}}} = \omega_{{\text{r}}}^{*} - \omega_{{\text{r}}} $$
(17)

where \(\omega_{{\text{r}}}^{*}\) is the desired rotor speed that is bounded and uniformly continuous, and it also has derivative up to the second order. The sliding function is defined as the following form:

$$ S_{{\text{M}}} = c_{2} e_{{\text{M}}} $$
(18)

where \(c_{2}\) is positive constant By selecting the switching function as the same of the SMC for the stator flux control as:

$$ \dot{S}_{{\text{M}}} = - \varepsilon_{{\text{M}}} {\text{sign}}(S_{{\text{M}}} ) $$
(19)

where \(\varepsilon_{{\text{M}}} > 0\). And based on the SMC theory [9], in order to guarantee for the rotor speed tracking control, from (7), (10) and (19), with assumption that the good performance of the stator flux control can be obtained, the SMC control input can be designed as the following form:

$$\begin{aligned} u_{{\text{M}}} &=\, \int {\dot{u}_{{{\text{sq}}}} } {\text{d}}t \\& = \int {\left( {\omega_{{\text{r}}} \dot{\varphi }_{{{\text{sd}}}} + R_{{\text{s}}} \dot{i}_{{{\text{sq}}}} + \omega_{{\text{r}}}^{*} \varphi_{{{\text{sd}}}} + \frac{{\varphi_{{{\text{sd}}}} }}{{c_{2} }}\varepsilon_{{\text{M}}} {\text{sign}}(S_{{\text{F}}} )} \right){\text{d}}t} \end{aligned}$$
(20)

However, with this SMC input for the rotor speed control, besides the inevitable chattering phenomena, this controller is only well done when the rotor speed \(\omega_{{\text{r}}}\) can be measured.

In this practical research, the SMC without requirement of rotor speed information will be explored carefully. In the proposed controllers, the chattering phenomena are also relaxed in considering of control system stability. The proposed methods are analyzed and designed as follows.

3 Control algorithms

As we knew, the control goal was to improve the SMC methods for the stator flux and rotor speed tracking control of the TIM system as, \(\varphi_{{\text{s}}}^{{}} (t) \to \varphi_{{{\text{sd}}}}^{*} (t)\) and \(\omega_{{\text{r}}} (t) \to \omega_{{\text{r}}}^{*} (t)\) as the time tends to infinite \(t \to \infty\). The proposed control system strategy is drawn in Fig. 1.

Fig. 1
figure 1

The proposed control system strategy

Full size image

Assumption 3.1

The \(\varphi_{{{\text{sd}}}}^{*}\) and \(\omega_{r}^{*}\) are bounded and uniformly continuous.

Assumption 3.2

The \(\omega_{{\text{r}}}^{{}}\) signal cannot be measured by sensors.

3.1 SMC for the stator flux

As the aforementioned analysis for the stator flux control, the SMC controller has an inevitable drawback when the switching surface is the \({\text{sign}}(S_{{\text{F}}} )\) signum function. The \({\text{sign}}(S_{{\text{F}}} )\) is a linear function is used to smooth nonlinear discontinuous control. Therefore, the chattering phenomenon is likely to occur all the time, especially in cases the control system operates at high frequencies. In this study, in order to deal with this problem, we will apply the \({\text{tansig}}(S_{{\text{F}}} )\) function to replace the \({\text{sign}}(S_{{\text{F}}} )\) function. The \({\text{tansig}}(S_{{\text{F}}} )\) is defined as the following form:

$$ {\text{tansig}}(S_{{\text{F}}} ) = \frac{2}{{(1 + e^{{ - 2S_{F} }} )}} - 1 $$
(21)

Clearly, the \({\text{tansig}}(S_{F} )\) is a smooth nonlinear function, as shown in Fig. 2, it also is the exponential function with rapidly response. Thus, the chattering phenomenon around sliding surface can be decreased when the switching function use this \({\text{tansig}}(S_{{\text{F}}} )\) function [19,20,21,22]. Based on this recommendation, from (21), the proposed SMC input for the stator flux control is designed as:

$$ u_{{{\text{sd}}}} = \frac{1}{{c_{1} }}\left( {\varepsilon_{{\text{F}}} \tan {\text{sig}}(S_{{\text{F}}} ) + \ddot{e}_{{\text{F}}} + K_{{\text{F}}} s} \right) + \dot{\varphi }_{{{\text{sd}}}}^{*} + R_{{\text{s}}} i_{{{\text{sd}}}} $$
(22)

where \(K_{F}\) is the positive constant.

Fig. 2
figure 2

The Tansig function response

Full size image

3.2 SMC for the rotor speed

As shown in Fig. 1, the proposed SMC method for the rotor speed has some improvements so that to decrease the chattering phenomenon, relax the requirement of the rotor speed measurement and increase the time-response of control signal. The detailed design steps of the proposed SMC scheme for the rotor speed are presented clearly as follows.

The first, based on the SMC input for the rotor speed (20) that is considered as an integral control input, we will add a low-pass filter [34] to create the true control input for the rotor speed control. Therefore, we have the following relationship as:

$$ u_{{{\text{sd}}}} =\, \frac{1}{\tau s + 1}u_{{\text{M}}} $$
(23)

where \(\tau\) is an integral time constant, \(s\) is the Laplace operator, and, the following result can be obtained as:

$$ u_{{\text{M}}} = u_{{{\text{sd}}}} + \tau \dot{u}_{{{\text{sd}}}} $$
(24)

By substituting the result in (7) into (24), we have:

$$ \dot{\omega }_{{\text{r}}} = \frac{1}{{\tau \varphi_{{{\text{sd}}}} }}\left( {u_{{\text{M}}} - R_{{\text{s}}} i_{{{\text{sq}}}} - \omega_{{\text{r}}} (\varphi_{{{\text{sd}}}} + \tau \dot{\varphi }_{{{\text{sd}}}} ) - \tau R_{{\text{s}}} \dot{i}_{{{\text{sq}}}} } \right) $$
(25)

From (18) and (25), the sliding switching function of the rotor speed control can be described as the following form:

$$ \begin{aligned}\dot{S}_{{\text{M}}} &= c_{2} \left( {\dot{\omega }_{{\text{r}}}^{*} - \frac{1}{{\tau \varphi_{{{\text{sd}}}} }}\left( {u_{{\text{M}}} - R_{{\text{s}}} i_{{{\text{sq}}}} }\right.}\right. \\ & \quad \left.{\left.{- \omega_{{\text{r}}} (\varphi_{{{\text{sd}}}} + \tau \dot{\varphi }_{{{\text{sd}}}} ) - \tau R_{{\text{s}}} \dot{i}_{{{\text{sq}}}} } \right)\vphantom{\frac{1}{\varphi_1}}} \right)\end{aligned} $$
(26)

here if the sliding switching function in (26) is the \({\text{tansig}}(S_{{\text{M}}} )\) function for chattering phenomenon elimination, the SMC input for the rotor speed control can be designed as:

$$ \begin{aligned} u_{{\text{M}}} = &\, \tau \varphi_{{{\text{sd}}}} \dot{\omega }_{{\text{r}}}^{*} + R_{s} i_{{{\text{sq}}}} + \omega_{r} (\varphi_{{{\text{sd}}}} + \tau \dot{\varphi }_{{{\text{sd}}}} ) \\ & + \tau R_{{\text{s}}} \dot{i}_{{{\text{sq}}}} + \frac{{\tau \varphi_{{{\text{sd}}}} }}{{c_{2} }}(\varepsilon_{{\text{M}}} \tan {\text{sig}}(S_{{\text{M}}} ) + K_{{\text{M}}} S_{M} ) \\ \end{aligned} $$
(27)

where \(K_{{\text{M}}}\) is the positive constant. However, follow to the Assumption 3.2, we have no knowledge of the rotor speed \(\omega_{{\text{r}}}\). Therefore, the SMC control input for the rotor speed control is modified as the following form:

$$ \begin{aligned} u_{{\text{M}}} = &\, \tau \varphi_{{{\text{sd}}}} \dot{\omega }_{{\text{r}}}^{*} + R_{{\text{s}}} i_{{{\text{sq}}}} + \hat{f}(\varphi_{{{\text{sd}}}} + \tau \dot{\varphi }_{{{\text{sd}}}} ) + \tau R_{{\text{s}}} \dot{i}_{{{\text{sq}}}} \\ & + \frac{{\tau \varphi_{{{\text{sd}}}} }}{{c_{2} }}\left( {\varepsilon_{{\text{M}}} \tan {\text{sig}}(S_{{\text{M}}} ) + K_{{\text{M}}} S_{{\text{M}}} } \right) \\ \end{aligned} $$
(28)

where the \(\hat{f}\) is the estimator value of the \(f = \omega_{r}\).

In the SMC method for the rotor speed control, the RBF NN as the observer system, will be applied to approximate the \(f\) function. And from this approximating value, the rotor speed value can be obtained. For simplicity, the NN estimator structure has one hidden layer with activation function is the Gaussian function [35], as shown in Fig. 3. The NN output can be denoted as:

$$ f = H^{T} \theta (x) $$
(29)

where \(H = \left[ {h_{1} , \ldots ,h_{i} , \ldots h_{n} } \right]^{T} ,i = 1 \ldots n\), \(h_{i}\) is the weight between the hidden layer and the output layer, \(n\) is the number of hidden nodes. \(x = \left[ {x_{1} , \ldots ,h_{j} , \ldots h_{m} } \right]^{T} ,j = 1 \ldots m\) is the number of inputs. The \(\theta (x) = \left[ {\theta_{1} , \ldots ,\theta_{i} , \ldots ,\theta_{n} } \right]^{T}\), \(\theta_{i}\) is the Gaussian function:

$$ \theta_{i} (x) = e^{{ - \frac{{\left\| {x - c_{i} } \right\|^{2} }}{{b_{i}^{2} }}}} $$
(30)

where \(c_{i} = \left[ {c_{1i} , \ldots ,c_{{{\text{ji}}}} , \ldots ,c_{{{\text{mi}}}} } \right]^{T}\) is the coordinate value of center point of the Gaussian function, \(b_{i} = \left[ {b_{1i} , \ldots ,b_{{{\text{ji}}}} , \ldots ,b_{{{\text{mi}}}} } \right]^{T}\) is the width value of the Gaussian function. Based on the universal approximation analysis [35], an ideal NN structure with its optimal parameters has following form as:

$$ f = H^{*T} \theta (x) $$
(31)

where \(H^{*}\) is the unknown optimal value. And from (28) and (31), the NN estimator output can be described as:

$$ \hat{f} = \hat{H}^{T} \theta (x) $$
(32)

where \(\hat{H}\) is the approximating value of the \(H^{*}\).

Fig. 3
figure 3

The RBF NN structure

Full size image

By defining \(\tilde{f} = f - \hat{f},\tilde{H} = H^{*} - \hat{H}\), follow to (25, 28, 31, 32), and assuming that we can obtain the following result as:

$$ \begin{aligned} \dot{S}_{{\text{M}}} = & \,\frac{{c_{2} }}{{\tau \varphi_{{{\text{sd}}}} }}(\varphi_{{{\text{sd}}}} + \tau \dot{\varphi }_{{{\text{sd}}}} )\tilde{f} - \varepsilon_{{\text{M}}} \tan {\text{sig}}(S_{{\text{M}}} ) - K_{{\text{M}}} S_{{{\text{ds}}}} \\ = & \frac{{c_{2} }}{{\tau \varphi_{{{\text{sd}}}} }}(\varphi_{{{\text{sd}}}} + \tau \dot{\varphi }_{{{\text{sd}}}} )\tilde{H}^{T} \theta \\ & - \varepsilon_{{\text{M}}} \tan {\text{sig}}(S_{{\text{M}}} ) - K_{{\text{M}}} S_{{\text{M}}} \\ \end{aligned} $$
(33)

Based on the Lyapunov stability theorem [36], the adaptive online learning algorithm for the RBF NN is proposed as:

$$ \dot{\hat{H}} = K_{H} \frac{{c_{2} }}{{\tau \varphi_{{{\text{sd}}}} }}(\varphi_{{{\text{sd}}}} + \tau \dot{\varphi }_{{{\text{sd}}}} )\theta S_{{\text{M}}} $$
(34)

where \(K_{{\text{H}}}\) is the positive and diagonal constant matrix.

3.3 Stability analysis

Theorem

By considering that the TIM dynamics in the Eqs. (1–10), all Assumptions hold. If the SMC law for the stator flux control is designed as (22), the SMC law for the rotor speed control and the online updating algorithm for the RBF NN estimator are designed as (28) and (34), then,

  1. (1)

    The \(S_{F}\) converges to zero.

  2. (2)

    The \(e_{M}\) converges to zero and the RBF NN weights are bounded. The stability of the proposed TAIM control system is guaranteed.

Proof (1)

For the stator flux control, we consider the Lyapunov candidate function as the following form:

$$ V_{1} (S_{F} (t)) = \frac{1}{2}S_{F}^{2} $$
(35)

By differentiating (35) with respect to time, we have:

$$ \dot{V}_{1} = S_{F} \dot{S}_{F} $$
(36)

By substituting (13) into (36), it yields:

$$ \dot{V}_{1} = S_{{\text{F}}} ( - c_{1} (u_{{{\text{sd}}}} - R_{{\text{s}}} i_{{{\text{sd}}}} ) + c_{1} \varphi_{{{\text{sd}}}}^{*} + \ddot{e}_{{\text{F}}} ) $$
(37)

If the SMC law for the stator flux control is chosen as (22), then, (37) can be described as:

$$ \begin{aligned} \dot{V}_{1} = & - \varepsilon_{F} S_{F} \tan {\text{sig}}(S_{F} ) - K_{F} S_{F}^{2} \\ \le & - K_{F} S_{F}^{2} \\ \end{aligned} $$
(38)

Clearly, \(\dot{V}_{1} \le 0\), and based the Lyapunov theorem [36], from this results, we easy to conclude that the stability of the SMC system for stator flux control is guaranteed and \(S_{F}\) will converge to zero as time tends to infinite. Therefore, the stator flux control is guaranteed.

Proof (2)

For the rotor speed control, we consider the Lyapunov candidate function as the following form:

$$ V_{2} (S_{M} (t),\tilde{H}) = \frac{1}{2}S_{M}^{2} + \frac{1}{2}\tilde{H}^{T} K_{H}^{ - 1} \tilde{H} $$
(39)

By differentiating (39) with respect to time, it yields:

$$ \begin{aligned} \dot{V}_{2} = &\, S_{M} \dot{S}_{M} + \tilde{H}^{T} K_{H}^{ - 1} \dot{\tilde{H}} \\ = &\, S_{M} \dot{S}_{M} - \tilde{H}^{T} K_{H}^{ - 1} \dot{\hat{H}} \\ \end{aligned} $$
(40)

By substituting (33) into (40), one can obtain:

$$ \begin{aligned} \dot{V}_{2} & = S_{M} \left( {\frac{{c_{2} }}{{\tau \varphi _{{{\text{sd}}}} }}(\varphi _{{{\text{sd}}}} + \tau \dot{\varphi }_{{{\text{sd}}}} )\tilde{H}^{T} \theta - \varepsilon _{{\text{M}}} \tan {\text{sig}}(S_{{\text{M}}} )} \right. \\ &\quad \left. { - K_{{\text{M}}} S_{{\text{M}}} } \right) - \tilde{H}^{T} K_{{\text{H}}}^{{ - 1}} \dot{\hat{H}} \\ & = - \varepsilon _{{\text{M}}} S_{{\text{M}}} \tan sig(S_{{\text{M}}} ) - K_{{\text{M}}} S_{{\text{M}}}^{2} \\ &\quad + \frac{{c_{2} }}{{\tau \varphi _{{{\text{sd}}}} }}(\varphi _{{{\text{sd}}}} + \tau \dot{\varphi }_{{{\text{sd}}}} )\tilde{H}^{T} \theta S_{{\text{M}}} - \tilde{H}^{T} K_{{\text{H}}}^{{ - 1}} \dot{\hat{H}} \\ \end{aligned} $$
(41)

If the adaptive algorithm of the NN estimator for rotor speed is chosen as (34), it is concluded that:

$$ \begin{aligned} \dot{V}_{2} = & - \varepsilon_{{\text{M}}} S_{{\text{M}}} {\text{tansig}}(S_{{\text{M}}} ) \\ & \quad - K_{{\text{M}}} S_{{\text{M}}}^{2} + \frac{{c_{2} }}{{\tau \varphi_{{{\text{sd}}}} }}(\varphi_{{{\text{sd}}}} + \tau \dot{\varphi }_{{{\text{sd}}}} )\tilde{H}^{T} \theta S_{{\text{M}}} \\ & - \tilde{H}^{T} K_{H}^{ - 1} (K_{H} \frac{{c_{2} }}{{\tau \varphi_{{{\text{sd}}}} }}(\varphi_{{{\text{sd}}}} + \tau \dot{\varphi }_{{{\text{sd}}}} )\theta S_{M} ) \\ = & - \varepsilon_{{\text{M}}} S_{{\text{M}}} {\text{tansig}}(S_{{\text{M}}} ) - K_{{\text{M}}} S_{{\text{M}}}^{2} \le - K_{{\text{M}}} S_{E}^{2} \\ \end{aligned} $$
(42)

Based on the result in (42), \(\dot{V}_{2} (S_{M} (t),\tilde{H})\) is a negative semi-definite function, \(V_{2} (S_{M} (t),\tilde{H}) \le V_{2} (S_{M} (0),\tilde{H})\). Clearly, if \(S_{M} (t),\tilde{H}\) are bounded at the time \(t = 0\), then they will stay at this bounded state as \(t \ge 0\). Therefore, the \(S_{M} (t),\tilde{H}\) are also bounded for \(t \ge 0\). This conclusion is right for the \(S_{F} (t)\) in the Proof (1).

Next, we consider the following definition: \(\Xi (t) \equiv - K_{M} S_{M}^{2}\), and from (42), we have: \(- \dot{V}_{2} \ge \Xi (t)\), by integrating \(\Xi (t)\) with respect to time, we will obtain the following result:

$$ \int\limits_{0}^{t} {\Xi (\vartheta )d\vartheta \le V_{2} (S_{M} (0),\tilde{H}) - V_{2} (S_{M} (t),\tilde{H})} $$
(43)

From the result as (43), based on above analysis, the \(V_{2} (S_{M} (0),\tilde{H})\) is a bounded function, the \(V_{2} (S_{M} (t),\tilde{H})\) also is a bounded non-increasing function, the following result can be achieved:

$$ \mathop {\lim }\limits_{t \to \infty } \int\limits_{0}^{t} {\Xi (\vartheta )d\vartheta < \infty } $$
(44)

Thus, by applying the Barbalat’s Lemma [36] with \(\dot{\Xi }(t)\) is bounded, it shows that \(\mathop {\lim }\limits_{t \to \infty } \int\limits_{0}^{t} {\Xi (\vartheta )d\vartheta = 0}\), and as a result, the \(S_{M} (t)\) will converge to zero as time tends to infinite. As the result, from (18), we can conclude that the \(e_{M}\) also converges to zero as time tends to infinite.

Based the aforementioned results in Proof (1) and Proof (2), the stability and robustness of the proposed SMC controllers, for the stator flux and rotor speed, are guaranteed with all signals in the control system are bounded. The tracking errors of the stator flux and rotor speed will converge to zero as time tends to infinite.

Remark

According to the analysis and designing of the proposed SMC controllers for the stator flux and rotor speed control, the requirement of the rotor speed measurement is relaxed. In addition, the NN estimator for the rotor speed estimation is designed with the online updating algorithm. This advantage has guaranteed the strictly state passive for the proposed control system based on the NN [35], improve the robustness and boundedness features. Moreover, in the SMC controller for the rotor speed control, we can also consider the stator flux is the constant parameter. If the assumption is hold, the control input (28) can be modified as:

$$ \begin{aligned} u_{{\text{M}}} = & \tau \varphi_{{{\text{sd}}}} \dot{\omega }_{{\text{r}}}^{*} + R_{s} i_{{{\text{sq}}}} + \hat{f}(\varphi_{{{\text{sd}}}} ) + \tau R_{s} \dot{i}_{{{\text{sq}}}} \\ & + \frac{{\tau \varphi_{{{\text{sd}}}} }}{{c_{2} }}(\varepsilon_{{\text{M}}} \tan {\text{sig}}(S_{{\text{M}}} ) + K_{{\text{M}}} S_{{\text{M}}} ) \\ \end{aligned} $$
(45)

And the online updating algorithm (34) can be represented as the following form:

$$ \dot{\hat{H}} = K_{H} \frac{{c_{2} }}{\tau }\theta S_{M} $$
(46)

The effectiveness for the tracking control and robustness of the proposed SMC controllers will be addressed by the following numerical and experimental results.

4 Simulation and experimental results

4.1 Simulation results

In order to verify the effectiveness of the proposed SMC methods, we shall consider the simulation model of the proposed control system that is shown in Fig. 4. The simulation process carried out using the Matlab-Simulink package with sample time is \(T_{s} = 2.10^{ - 5} (s)\). The parameters of the TIM model are given as Table 1.

Fig. 4
figure 4

The simulation diagram

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Table 1 The TIM model parameters
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For first simulation case, by observing the results in Fig. 5, we can see the output responses of the three-phase 5-level cascaded inverter that are the common mode voltage (CMV), line voltage, phase current, and fast Fourier transform (FFT) of the phase current and line voltage. Based on the desired constant values of stator flux and rotor speed as \(\varphi_{{{\text{sd}}}}^{*} = 1({\text{Wb}}), \, \omega_{{\text{r}}}^{*} = 150({\text{rad}}/{\text{s}})\), the CMV and THD of line voltage are \(V_{{{\text{cm}}}} = V_{{{\text{dc}}}} /3,{\text{ THD}}_{V} = 18.69(\% )\), the phase current and THD of phase current is \(I_{a} = 4(A),\,{\text{THD}}_{I} = 0.865(\% )\). In Fig. 6, the SMC controllers based on the NN has well guaranteed the tracking control for the stator flux and the rotor speed, with steady time are about 0.15 (s) and 0.25 (s) for the stator flux and the rotor speed, respectively. In addition, based on the result in Fig. 6b, the NN estimator also has good performance when it provides the rotor speed estimating value with good tracking performance in comparing to rotor speed measurement.

Fig. 5
figure 5

The output responses of the three-phase 5-level cascaded inverters. a CMV, line voltage and FFT of line voltage, b phase current and FFT of phase current

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Fig. 6
figure 6

The stator flux and rotor speed responses. a The stator flux, b the rotor speed

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In the second simulation case, the desired rotor speed values are changed with respect to time (as shown on Fig. 7b). The simulation results for this case are provided in Fig. 7. Based on these results, we can draw the following comments. The SMC controllers based on the NN estimator for stator flux and rotor speed still operate well with the changing of the desired rotor speed. In Fig. 7a, the torque of the TIM has occurred overshoot and strong vibration phenomena at the points where the rotor speed was changed quickly. As shown in Fig. 7b, the phase current is about 4 (A) at lowest rotor speed (10 rad/s).

Fig. 7
figure 7

The simulation results with case of changing the rotor speed. a The stator flux and torque Tm, b The rotor speed and phase current

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In the third simulation case, besides the changing of the desired rotor speed, the stator and rotor resistances are increased 1.5 times. In Figs. 8 and 9, the responses of the stator flux, load torque and rotor speed show that the performance of the SMC controllers are still stable. However, the phase current is about 3 (A) at lowest rotor speed (10 rad/s).

Fig. 8
figure 8

The simulation results with case of changing the rotor speed and Rs is increased to 1.5 times. a The stator flux and torque Tm, b the rotor speed and phase current

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Fig. 9
figure 9

The simulation results with case of changing the rotor speed and Rr is increased to 1.5 times. a The stator flux and torque Tm, b the rotor speed and phase current

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In order to verify the robustness of the proposed SMC controllers, in the fourth simulation case, the disturbances will be added in the simulation process. Figures 10 and 11 describe the simulation results with the random and sine wave distribution disturbances, respectively. The disturbances have impact to the responses of the torque, stator flux and rotor speed. However, the tracking control for stator flux and rotor speed is still guaranteed. In addition, the phase current is also up to about 4 (A) at the lowest rotor speed (10 rad/s).

Fig. 10
figure 10

The simulation results with case of adding the random distribution disturbances. a The disturbances and torque Tm, b The rotor speed and phase current

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Fig. 11
figure 11

The simulation results with case of adding the distribution disturbances typed sine wave. a The disturbances and torque Tm, b The rotor speed and phase current

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Based on the simulation results, we easy realize that, with the proposed SMC controllers based on the NN estimator for the TIM 1 HP, 150 rad/s, the adaptive tracking control for the stator flux and rotor speed is well done. The NN estimator has played an important role in providing approximating rotor speed approximate with the rotor speed measured value. Moreover, for rotor speed tracking control, in considering the existence of the disturbances and the increasing of the resistances of stator/rotor (up to 1.5 times), the SMC controller-based NN has proven the robustness feature. This proposed control method cannot only guarantee the tracking control with the changing rotor speed, from lowest speed (10 rad/s) to highest speed (150 rad/s), but also guarantee the stability of phase current, the low CMV and THD.

4.2 Experimental results

Figure 12 shows the experimental TIM control system that is described briefly as follows. The designed SMC controllers-based NN estimator are embedded into the Digital Signal Processing (DSP) 320F28355 card from Simulink/Matlab software (from host computer). The DSP card provide the control signals to control the TAIM via the IGBT inverter and receive the feedback signals from sensors (current, speed, stator flux and moment). The TIM system has technical parameters are same to the TIM model in the simulation process. We will test the effectiveness and robustness of the proposed control method with \(\varphi_{{{\text{sd}}}}^{*} = 1({\text{Wb}}), \, \,\omega_{r}^{*} = 150({\text{rad}}/{\text{s}})\), and the TIM operate with and without the rated load.

Fig. 12
figure 12

The experimental TIM control system

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Figures 13 and 14 present the experimental results of the line voltage, CMV, phase current, THD and FFT of phase current for the cases of no the rated load and adding the rated load, respectively. The CMV value is guaranteed with \(V_{{{\text{CM}}}} = V_{{\text{s}}} /3\), the THD is low with \({\text{THD}} = 2.5 \to 4.062\%\), the phase current is about 4 (A). Figure 15 shows the experimental results of the stator flux and the moment. Based on these results (Fig. 15), the tracking control of stator flux is well done.

Fig. 13
figure 13

The experimental results of the line voltage, CMV, phase current, THD and FFT of phase current (without the rated load)

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Fig. 14
figure 14

The experimental results of the line voltage, CMV, phase current, THD and FFT of phase current (with the rated load)

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Fig. 15
figure 15

The experimental results of the stator flux and moment Tm. a The stator flux, b the torque Tm

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For the tracking control of rotor speed, we can observe based the results in Fig. 16. In Fig. 16a, without the rated load, the rotor speed track well to the desired rotor speed. In the other hand, in Fig. 16b, with the rated load, the tracking control for the rotor speed still guaranteed. However, with the rated load, the rotor speed fluctuates around the desired rotor speed. And this phenomenon shows that the proposed SMC controller based on NN estimator has proven the effectiveness and robustness characteristics. In addition, with and without the rated load, the NN estimator has updated itself and provided the rotor speed information. On the other hand, follow to the experimental result of the TIM torque, we can see that the SMC, with the tansig function for the switching surface, has limited the chattering phenomenon in the control process.

Fig. 16
figure 16

The experimental results of the rotor speed. a Without the rated load, b with the rated load

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Based on the simulation and experimental results, we can confirm that the proposed SMC controllers based on the NN estimator has guaranteed well for the requirement of tracking control and robustness for the TIM control system.

5 Conclusions

The proposed method has successfully implemented the sensorless SMC scheme based on the NN for the TIM in the simulation and real-time practical. The NN estimator as the observer, has provided the rotor speed information for the control system without applying the sensors. By applying the nonlinear function for the sliding switching surface, the proposed SMC method has guaranteed good performance for the chattering reducing and the tracking control of the stator flux and rotor speed. In addition, the stability and robustness of the proposed control system are also well done in the presence of disturbances and the changing of control parameters. The proposed sensorless SMC method based on the NN can be applied as a good alternative in the existing TIM control system.