Stator flux oriented multiple sliding-mode speed control design of induction motor drives

Abstract

Due to superior robustness characteristic of sliding-mode control techniques, this study proposes a multiple sliding-mode control (MSMC) strategy based on the stator flux oriented vector scheme for speed control of three-phase AC induction motor (IM) drives in the presence of an external disturbance and uncertainties. At first, the dynamic model of a three-phase IM drive is transformed into two-axe orthogonal model (i.e. d and q axes) in the synchronously rotating frame so that vector control can be applied. Then, based on the stator flux oriented scheme (i.e. zero stator flux at q-axis and constant at d-axis), the proposed MSMC causes mechanical angular speed and stator current at q-axis reach toward predefined sliding surfaces. Moreover, stator flux and current at d-axis are respectively indirect and direct controlled such that tracking errors approach toward designed sliding surfaces. The closed-loop stability of the proposed MSMC is proved to possess uniformly ultimately bounded (UUB) performance by Lyapunov stability criteria. Furthermore, the simulation results reveal that the proposed MSMC strategy has a high level of robustness despite addition of an external load and random uncertainties on system parameters. In the meantime, the simulations for comparing the baseline controller (i.e. conventional PI control) are also conducted to verify the superiority of the proposed control scheme.

Introduction

IM drives have widely been applied to numerous kinds of fields such as industries, traffic transportation and medical equipment etc. Many researchers and engineers have been devoting to control of AC IM drives in order to improve control performance and reduce cost. Over past two decades, vector control has become a popular technique to effectively manipulate AC motor drives in terms of control performance.^1,2 However, as ones know, control of motor drives with excellent performance is a challenging problem as operated at low-speed in the presence of system uncertainties. Moreover, an external load, that is, disturbance, to motor drives is another big challenge in fast response and precise tracking. Based on these concerns, developing a speed controller with a high-level of robustness and a superior performance for AC motor drives is strongly in need.

There have been some research findings in the published literatures about the control of IM drives and some representative studies are reviewed here. Field-oriented control (FOC) and direct torque control (DTC) schemes are two most popular approaches to control of AC motor drives. Like in Wu et al., ³ the FOC and DTC approaches are both applied to a five-phase permanent magnet motor. The results reveal that FOC scheme has low torque ripples and DTC scheme has fast dynamic response. Awan et al. ⁴ presented an exact input-output feedback linearization structure based on stator-flux-oriented control method for the performance improvement of permanent-magnet synchronous motors and reluctance motors. In addition, Lin et al. ⁵ proposed an advanced deadbeat direct torque and flux control (DB-DTFC) in permanent magnet synchronous motor (PMSM). The algorithm model based on stator flux oriented coordinate system is deduced and a general solution method is presented to solve the reference stator voltage. The results show that the performance in the aspects of electromagnetic torque and stator current ripples reduction, dynamic response, computation burden, and robustness to motor parameters variation are all improved. With regard to control of induction motors, a model predictive control (MPC) based filed-weakening algorithm was proposed in Su et al. ⁶ for a traction electric vehicle using a low-voltage induction motor. The augmented prediction state relationship between stator voltage and flux-producing current is established for motor current control. The system eigenvalues are adjusted in real time for various speed regions and the controller performance is evaluated with the amplitude of the eigenvalues. The simulation and experimental results are provided to verify its feasibility and effectiveness. Konstantopoulos et al. ⁷ has developed a new nonlinear controller for speed regulation of voltage source inverter (VSI)-driven induction motors. The proposed controller directly provides the duty-ratio input of the VSI in the permitted range to ensure linear modulation and is fully independent to the system parameters and suitably regulates the motor speed and the stator flux to the desired values. The simulation and experimental results show that the system responses have fast convergence to the equilibriums after limited transients. Furthermore, a neural network (NN) vector control was proposed by Fu and Li ⁸ for a three-phase induction motor. The role of the NN controller is to substitute the two decoupled current-loop PI controllers in conventional vector control techniques. The experimental results demonstrated that the NN vector control can succeed in driving the induction motor without audible noise using relatively lower switching frequency or lower sampling rate.

One of well-known robust controllers, that is, sliding-mode control, has been proven to the most effective and feasible technique for any kinds of control systems. Here, some typical works with regards to induction motors are surveyed and reviewed. An adaptive sliding-mode control based model predictive torque control ⁹ was proposed to enhance the robustness and improve the performance by adjusting the switching gain adaptively. In Chen and Yu ¹⁰ presented a backstepping sliding mode control combining with a disturbance observer to regulate the speed of an induction motor. Experimental results verify its effectiveness and practicability. Furthermore, a speed observer design based on backstepping and sliding-mode strategies is used to perform speed operation, especially, at low-speed operation. From experimental tests, the control system is indeed robust on machine parameters and under load torque injection even operating at low-speed. ¹¹ Obviously, it reveals that the sliding-mode control strategy may be a suitable solution for control of induction motor drives.

Based on the aforementioned literatures review, FOC and DTC based approaches are usually used to tackle three-phase AC motor drives. To the best of our knowledge, a multiple sliding-mode control (MSMC) strategy^12,13 based on the stator flux oriented vector scheme is firstly proposed for control of IM drives and can effectively deal with the IM drives in spite of suffering an external load and random uncertainties. The contributions of this study can be summarized as follows: (1) A viable solution that is, the proposed MSMC, is presented for the speed control of three-phase AC IM drives (2) The proposed MSMC scheme which is hierarchically designed based on stator flux oriented vector strategy can become a standard design procedure applied to three-phase AC IM drives (3) The proposed control demonstrates the superior performance to a conventional PI control and possesses a high level of robustness despite addition of an external load and random uncertainties on system parameters.

The rest of the paper is organized as follows: In section 2, the dynamic modeling of the IM drive, decoupled d-q axes transformation and the problem formulation are given. The stator flux oriented multiple sliding-mode control design are described in section 3. The simulation results and discussions are presented in section 4. Finally, in section 5 the conclusions are made.

Dynamic model of an IM drive, decoupled d-q axes transformation and problem formulation

Dynamic model of the IM drive

First, an AC IM drive is three-phase balanced, hence, its dynamic model of the stator, rotor electrical circuits and electromechanical part in space vector form can be expressed as follows:

$$R_s{\mathbf{i}}_{\mathbf{s}}+L_s{\stackrel{·}{\mathbf{i}}}_{\mathbf{s}}+L_m{\stackrel{·}{\mathbf{i}}}_{\mathbf{r}}{e}^{j{\theta }_r}+j{\omega }_r{L}_m{\mathbf{i}}_{\mathbf{r}}{e}^{j{\theta }_r}={\mathbf{v}}_{\mathbf{s}}$$

(1a)

$$R_r{\mathbf{i}}_{\mathbf{r}}+L_r{\stackrel{·}{\mathbf{i}}}_{\mathbf{s}}+L_m{\stackrel{·}{\mathbf{i}}}_{\mathbf{r}}{e}^{-j{\theta }_r}-j{\omega }_r{L}_m{\mathbf{i}}_{\mathbf{s}}{e}^{-j{\theta }_r}=0$$

(1b)

$$J_m{\stackrel{·}{\omega }}_{\mathrm{rm}}+B_m{\omega }_{\mathrm{rm}}+T_L=T_e$$

(1c)

where ${\mathbf{i}}_{\mathbf{s}}=i_{\mathrm{as}}+i_{\mathrm{bs}}{e}^{j\frac{2\pi }{3}}+i_{\mathrm{cs}}{e}^{j\frac{4\pi }{3}};$ ${\mathbf{i}}_{\mathbf{r}}=i_{\mathrm{ar}}+i_{\mathrm{br}}{e}^{j\frac{2\pi }{3}}+i_{\mathrm{cr}}{e}^{j\frac{4\pi }{3}};$ ${\mathbf{v}}_{\mathbf{s}}=v_{\mathrm{as}}+v_{\mathrm{bs}}{e}^{j\frac{2\pi }{3}}+v_{\mathrm{cs}}{e}^{j\frac{4\pi }{3}};$ the superscript * indicates complex conjugate; $R_s={\overline{R}}_s+\mathrm{\Delta }{R}_s$ denotes the stator winding resistor of each phase; $L_s={\overline{L}}_s+\mathrm{\Delta }{L}_s$ is the stator inductance of each phase; $L_m={\overline{L}}_m+\mathrm{\Delta }{L}_m$ is the mutual inductance; $R_r={\overline{R}}_r+\mathrm{\Delta }{R}_r$ represents the rotor winding resistor of each phase; $L_r={\overline{L}}_r+\mathrm{\Delta }{L}_r$ is the rotor inductance of each phase; ${\omega }_{\mathrm{rm}}$ is mechanical angular speed. If the pole pairs of the IM is $P,$ it has relationship ${\omega }_{\mathrm{rm}}=\frac{2}{P}{\omega }_r$ where ${\omega }_r$ denotes rotor angular speed, $J_m={\overline{J}}_m+\mathrm{\Delta }{J}_m,$ $B_m={\overline{B}}_m+\mathrm{\Delta }{B}_m,$ ${\theta }_r,$ $T_L$ and $T_e$ respectively represent as moment of inertia, friction coefficient, rotor angle, external load and electromagnetic torque. To reflect real situation, the above system parameters include nominal denoted $\overline{X}$ and uncertain terms indicated $\mathrm{\Delta }X$ ; $\mathrm{Im}[•]$ represents as an imaginary part. The above equation (1) denotes the complete dynamic model of a three-phase IM drive with 7 state variables (i.e. ${\mathbf{i}}_{\mathbf{s}},$ ${\mathbf{i}}_{\mathbf{r}},$ and ${\omega }_{\mathrm{rm}}$ ). In terms of position control, one more state variable (i.e. ${\theta }_r$ ) is added. The dynamic model is based on a three-phase system, that is, a three-axis coordinate system respectively with 120° phase difference in two-dimensional space. The mathematical expression is quite complicated compared with conventional orthogonal coordinate system. Therefore, the dynamic model (1) is not suitable to be directly used for controller design. The suitable coordinate transformation for decoupling is required. In the next subsection, the often used d-q axes transformation is addressed for controller design.

Decoupled d-q axes transformation

Consider a coordinate transformation from an original three-phase coordinate system to an orthogonal coordinate frame rotating with an angular speed of power source ${\omega }_e,$ the original dynamic model of the IM drive (1) expressed in the synchronously rotating frame is given as Kocabas et al. ¹⁴ and Comanescu: ¹⁵

$$(R_s+j{\omega }_e{L}_s+L_s\nu ){\mathbf{i}}_{\mathbf{s}}^{\mathbf{e}}+(j{\omega }_e{L}_m+L_m\nu ){\mathbf{i}}_{\mathbf{r}}^{\mathbf{e}}={\mathbf{v}}_{\mathbf{s}}^{\mathbf{e}}$$

(2a)

$$(R_r+j{\omega }_{\mathrm{sl}}{L}_r+L_r\nu ){\mathbf{i}}_{\mathbf{r}}^{\mathbf{e}}+(j{\omega }_{\mathrm{sl}}{L}_m+L_m\nu ){\mathbf{i}}_{\mathbf{s}}^{\mathbf{e}}=0$$

(2b)

$$J_m\nu {\omega }_{\mathrm{rm}}+B_m{\omega }_{\mathrm{rm}}+T_L=T_e$$

(2c)

where $\nu =\frac{d}{\mathrm{dt}}(·)$ is a differential operator; ${\mathbf{i}}_{\mathbf{s}}^{\mathbf{e}}=i_{\mathrm{ds}}^e+{\mathrm{ji}}_{\mathrm{qs}}^e;$ ${\mathbf{i}}_{\mathbf{r}}^{\mathbf{e}}=i_{\mathrm{dr}}^e+{\mathrm{ji}}_{\mathrm{qr}}^e;$ ${\omega }_e$ is also called as synchronous angular speed and ${\omega }_{\mathrm{sl}}$ represents slip angular speed with the relation ${\omega }_{\mathrm{sl}}={\omega }_e-{\omega }_r.$ In this expression, there are only five state variables, namely ${\mathbf{i}}_{\mathbf{s}}^{\mathbf{e}},$ ${\mathbf{i}}_{\mathbf{r}}^{\mathbf{e}}$ and ${\omega }_{\mathrm{rm}}$ . For the stator flux oriented control, the rotor current ${\mathbf{i}}_{\mathbf{r}}^{\mathbf{e}}$ must be replaced with the stator flux and stator current as

$${\mathbf{i}}_{\mathbf{r}}^{\mathbf{e}}=\frac{1}{L_m}({\mathbf{\Phi }}_{\mathbf{s}}^{\mathbf{e}}-L_s{\mathbf{i}}_{\mathbf{s}}^{\mathbf{e}})$$

(3)

Through the coordinate transformation, equations (2) can be rewritten as

$$R_s{\mathbf{i}}_{\mathbf{s}}^{\mathbf{e}}+(j{\omega }_e+\nu ){\mathbf{\Phi }}_{\mathbf{s}}^{\mathbf{e}}={\mathbf{v}}_{\mathbf{s}}^{\mathbf{e}}$$

(4a)

$$\begin{array}{c}(\frac{R_r}{L_m}+j\frac{{\omega }_{\mathrm{sl}}{L}_r}{L_m}+\frac{L_r}{L_m}\nu ){\mathbf{\Phi }}_{\mathbf{s}}^{\mathbf{e}}\hfill \\ -(\frac{L_s{R}_r}{L_m}+j{\omega }_{\mathrm{sl}}\frac{\sigma L_s{L}_r}{L_m}+\frac{\sigma L_s{L}_r}{L_m}\nu ){\mathbf{i}}_{\mathbf{s}}^{\mathbf{e}}=0\hfill \end{array}$$

(4b)

$$J_m\nu {\omega }_{\mathrm{rm}}+B_m{\omega }_{\mathrm{rm}}+T_L=T_e$$

(4c)

where complex vectors ${\mathbf{\Phi }}_{\mathbf{s}}^{\mathbf{e}}={\varphi }_{\mathrm{ds}}^e+j{\varphi }_{\mathrm{qs}}^e$ and ${\mathbf{v}}_{\mathbf{s}}^{\mathbf{e}}=v_{\mathrm{ds}}^e+{\mathrm{jv}}_{\mathrm{qs}}^e;$ $\sigma =1-\frac{L_m^2}{L_s{L}_r}=\overline{\sigma }+\mathrm{\Delta }\sigma .$

Problem formulation

After the above coordinate transformation, the two-axe dynamic model with orthogonal features along with the electro-mechanical model is obtained in the equations (4) so that the individual axis can be controlled via a feedback decoupled approach. To achieve the stator flux oriented vector control, the stator flux must be controlled to zero at q-axis, that is, all stator flux is located at d-axis. In the meantime, the stator flux at d-axis must be maintained as a constant value. In this way, the electromagnetic torque $T_e$ generated by the IM drives is proportional to the stator current at q-axis (i.e. $i_{\mathrm{qs}}^e$ ). In other words, the decoupled stator currents (i.e. $i_{\mathrm{ds}}^e$ and $i_{\mathrm{qs}}^e$ ) at d and q-axes can be separately controlled. The control problem is to design ${\mathbf{V}}_{\mathbf{s}}^{\mathbf{e}}$ (i.e. $v_{\mathrm{ds}}^e$ and $v_{\mathrm{qs}}^e$ ) using the proposed SMC strategies such that the reference stator current ${\mathbf{i}}_{\mathbf{s}}^{\mathbf{e}\mathbf{*}}$ can be obtained where the reference stator current $i_{\mathrm{ds}}^{e^{*}}$ at d-axis and $i_{\mathrm{qs}}^{e^{*}}$ at q-axis are respectively generated using the proposed SMC strategies. Subsequently, the reference mechanical angular speed ${\omega }_{\mathrm{rm}}^{*}$ and reference stator flux ${\varphi }_s^{*}$ can be achieved using the proposed SMC strategies as well. The overall control block diagram called MSMC based on stator flux oriented vector control scheme is illustrated in the following Figure 1.

Figure 1.

Overall control block diagram of the IM drive system.

Stator flux oriented multiple sliding-mode control design

As the foregoing, the stator current needs to be controlled where the reference stator current is generated via the desired stator flux and its dynamics. In addition, the desired torque is obtained based on the dynamics of the mechanical angular speed error. Consequently, MSMC is proposed to control the stator current ${\mathbf{i}}_{\mathbf{s}}^{\mathbf{e}}$ and mechanical angular speed ${\omega }_{\mathrm{rm}}.$ Before the proposed MSMC is described, the following errors are defined.

$$e_{{\omega }_{\mathrm{rm}}}={\omega }_{\mathrm{rm}}^{*}-{\omega }_{\mathrm{rm}}$$

(5a)

$$e_{i_{\mathrm{qs}}}=i_{\mathrm{qs}}^{e*}-i_{\mathrm{qs}}^e$$

(5b)

$$e_{i_{\mathrm{ds}}}=i_{\mathrm{ds}}^{e*}-i_{\mathrm{ds}}^e$$

(5c)

$$e_{{\varphi }_s}={\varphi }_s^{*}-{\hat{\varphi }}_s$$

(5d)

In the beginning, the first sliding surface is defined as

$$s_1=k_{\mathrm{Ps}}{e}_{{\omega }_{\mathrm{rm}}}+k_{\mathrm{Is}}\int e_{{\omega }_{\mathrm{rm}}}\mathrm{dt}$$

(6)

where $k_{\mathrm{Ps}}\in \Re$ and $k_{\mathrm{Is}}\in \Re$ are the selected proportional and integral gains such that the first sliding surface is stable and determine its convergent rate. To make the mechanical angular speed error converge into the designed sliding surface and then asymptotically approach to zero, the reference electromagnetic torque $T_e^{*}$ is designed and contains two parts: ones is equivalent control $T_{e,\mathrm{eq}}^{*}$ that deals with a nominal system and the other one is switching control $T_{e,\mathrm{sw}}^{*}$ which is used to cope with external disturbances or uncertainties as

$$T_{e.\mathrm{eq}}^{*}={\overline{B}}_m{\omega }_{\mathrm{rm}}+\frac{{\overline{J}}_m}{k_{\mathrm{Ps}}}(k_{\mathrm{ps}}{\stackrel{·}{\omega }}_{\mathrm{rm}}^{*}+k_{\mathrm{Is}}{e}_{{\omega }_{\mathrm{rm}}})$$

(7a)

$$T_{e,\mathrm{sw}}^{*}=\frac{{\overline{J}}_m}{k_{\mathrm{Ps}}}[{\alpha }_1{s}_1+{\alpha }_2{s}_1/(\left|s_1\right|+{\epsilon }_1)]/(1-{\lambda }_1)$$

(7b)

$$T_e^{*}=T_{e,\mathrm{eq}}^{*}+T_{e,\mathrm{sw}}^{*}$$

(7c)

where ${\alpha }_1>{\delta }_1/2>0$ and ${\delta }_1$ represents the exponential convergent rate to the sliding surface; ${\alpha }_2>0$ is the switching gain; ${\epsilon }_1\ge 0$ is a boundary layer of the first sliding surface; ${\lambda }_1>0$ denotes the upper bound of the uncertain control gain and satisfies the following inequality

$$\left|{\overline{J}}_m{\mathrm{\Delta }}_J\right|\le {\lambda }_1<1$$

(8a)

where ${\mathrm{\Delta }}_J=-{({\overline{J}}_m+\mathrm{\Delta }{J}_m)}^{-1}{\overline{J}}_m^{-1}\mathrm{\Delta }{J}_m$ . In addition, it is assumed that the uncertainty function has an upper bound and satisfies the following inequality

$$\begin{array}{c}|-k_{\mathrm{Ps}}{T}_{e,\mathrm{eq}}^{*}{\mathrm{\Delta }}_J+k_{\mathrm{Ps}}(1+{\mathrm{\Delta }}_J){\overline{J}}_m^{-1}(T_L+\mathrm{\Delta }{B}_m{\omega }_{\mathrm{rm}})|\hfill \\ \le {\mu }_{10}+{\mu }_{11}{p}_1\hfill \end{array}$$

(8b)

where $p_1$ is an known positive function and ${\mu }_{10},{\mu }_{11}\ge 0$ are constants constrained by the stability of the closed-loop function.

Before discussing the result of the proposed controller, the following property about “uniformly ultimately bounded (UUB)” is given.

Definition 1: Wu and Karkoub ¹⁶ The solutions of a dynamic system are said to be UUB if there exist positive constants $\upsilon$ and $\vartheta ,$ and for every $\delta \in (0,\vartheta ),$ there is a positive constant $T=T\left(\delta \right),$ such that $\left|x\left(t_0\right)\right|<\delta \Rightarrow \left|x\left(t\right)\right|\le \upsilon ,\forall t\ge t_0+T.$

Theorem 1: Consider the transformed two-axes dynamic model of an IM drive with the known upper bound of an external load in (4), applying the proposed sliding-mode control results in finite time to reach the predefined sliding surface and satisfy the following convex set $D_{\sigma }$

$$D_{\sigma }=\{{\sigma }_n\in \Re |\left|{\sigma }_n\right|\le s_{\sigma }\}$$

(9a)

where

$$s_{\sigma }\ge (-v_{{\sigma }_1}+\sqrt{v_{{\sigma }_1}^2+4v_{{\sigma }_2}{v}_{{\sigma }_0}})/2v_{{\sigma }_2}$$

(9b)

$$\begin{array}{c}{v}_{{\sigma }_2}=|{\alpha }_1-{\delta }_1/2|\hfill \\ v_{{\sigma }_1}=|{\epsilon }_1{\alpha }_1+{\alpha }_2|-({\mu }_{10}+{\mu }_{11}{p}_1)-{\epsilon }_1{\delta }_1/2\hfill \\ v_{{\sigma }_0}=-{\epsilon }_1({\mu }_{10}+{\mu }_{11}{p}_1)\hfill \end{array}$$

(9c)

Then, $e_{{\omega }_{\mathrm{rm}}}$ is UUB as $t\to \infty .$

Proof: See Appendix A.

Moreover, as the mechanical angular speed is equal or less than rated speed, the stator flux is a constant value under constant power operation. Hence, based on the result of stator flux oriented control, the reference $i_{\mathrm{qs}}^{e*}$ can be given as

$$i_{\mathrm{qs}}^{e*}=\frac{4T_e^{*}}{3P{\varphi }_s^{*}}$$

(10)

To achieve the tracking of the reference $i_{\mathrm{qs}}^{e*}$ , the second sliding-mode control is proposed and then the second sliding surface is defined as follows:

$$s_2=k_{\mathrm{Pq}}{e}_{i_{\mathrm{qs}}}+k_{\mathrm{Iq}}\int e_{i_{\mathrm{qs}}}\mathrm{dt}$$

(11)

where $k_{\mathrm{Pq}}$ and $k_{\mathrm{Iq}}$ are the chosen proportional and integral gains such that the second sliding surface is stable. Based on the dynamic model of IM drives, the equivalent and switching control of $v_{\mathrm{qs}}^e$ are designed as

$$\begin{array}{c}\multicolumn{1}{c}{v_{\mathrm{qs},\mathrm{eq}}^e={\overline{L}}_{\sigma }[(\frac{{\overline{R}}_s}{{\overline{L}}_{\sigma }}+\frac{1}{\overline{\sigma }{\tau }_r})i_{\mathrm{qs}}^e+{\omega }_{\mathrm{sl}}{i}_{\mathrm{ds}}^e+\frac{{\omega }_r}{{\overline{L}}_{\sigma }}{\varphi }_{\mathrm{ds}}^e]}\\ +{\stackrel{·}{i}}_{\mathrm{qs}}^{e*}+\frac{k_{\mathrm{Iq}}}{k_{\mathrm{Pq}}}{e}_{i_{\mathrm{qs}}}\hfill \end{array}$$

(12a)

$$v_{\mathrm{qs},\mathrm{sw}}^e=\frac{{\overline{L}}_{\sigma }}{k_{\mathrm{Pq}}}[{\beta }_1{s}_2+{\beta }_2{s}_2/(\left|s_2\right|+{\epsilon }_2)]/(1-{\lambda }_2)$$

(12b)

$$v_{\mathrm{qs}}^e=v_{\mathrm{qs},\mathrm{eq}}^e+v_{\mathrm{qs},\mathrm{sw}}^e$$

(12c)

where $L_{\sigma }=\sigma L_s={\overline{L}}_{\sigma }+\mathrm{\Delta }{L}_{\sigma };$ ${\beta }_1>{\delta }_2/2>0$ and ${\delta }_2$ is an exponentially convergent rate to the sliding surface; ${\beta }_2>0$ is the switching gain; ${\epsilon }_2\ge 0$ is the boundary layer of the second sliding surface; ${\lambda }_2>0$ is the upper bound of the uncertain control gain and satisfies the following inequality

$$|\mathrm{\Delta }{L}_{\sigma }/{\overline{L}}_{\sigma }|\le {\lambda }_2<1$$

(13)

Remark: In equation (12a), ${\stackrel{·}{i}}_{\mathrm{qs}}^{e*}$ is substituted with ${\stackrel{·}{i}}_{\mathrm{qs}}^{e*}=[i_{\mathrm{qs}}^{e*}\left(\mathrm{kT}\right)-i_{\mathrm{qs}}^{e*}\left((k-1)T\right)]/T$ where $T$ denotes the sampling time of the control cycle. In general, the smaller the sampling time $T$ is, the more accurate the time derivatives is. The approximation error between them is regarded as part of uncertainties or disturbances. The proof is analogous to the proof of Theorem 1. For brevity, it is omitted.

Since the stator flux ${\varphi }_s$ and stator current at d-axis $i_{\mathrm{ds}}^e$ have coupling relationship with $v_{\mathrm{ds}}^e,$ direct control $i_{\mathrm{ds}}^e$ and indirect control ${\varphi }_s$ via $v_{\mathrm{ds}}^e$ are designed. Hence, the reference $i_{\mathrm{ds}}^{e*}$ can be given via SMC. The third sliding surface is then defined as

$$s_3=k_{\mathrm{Pf}}{e}_{{\varphi }_s}+k_{\mathrm{If}}\int e_{{\varphi }_s}\mathrm{dt}$$

(14)

where the selected gains, $k_{\mathrm{Pf}}$ and $k_{\mathrm{If}}$ , make the sliding surface stable. Then, the reference $i_{\mathrm{ds}}^{e*}$ is designed as

$$i_{\mathrm{ds},\mathrm{eq}}^{e*}={\left(k_{\mathrm{Pf}}{R}_s\right)}^{-1}[k_{\mathrm{Pf}}{v}_{\mathrm{ds}}^e-k_{\mathrm{If}}{e}_{\varphi s}]$$

(15a)

$$i_{\mathrm{ds},\mathrm{sw}}^{e*}={\left(k_{\mathrm{Pf}}{R}_s\right)}^{-1}[{\gamma }_1{s}_3+{\gamma }_2{s}_3/(\left|s_3\right|+{\epsilon }_3)]/(1-{\lambda }_3)$$

(15b)

$$i_{\mathrm{ds}}^{e*}=i_{\mathrm{ds},\mathrm{eq}}^{e*}+i_{\mathrm{ds},\mathrm{sw}}^{e*}$$

(15c)

where ${\gamma }_1>{\delta }_2/2>0$ and ${\delta }_2$ is an exponentially convergent rate to the sliding surface; ${\gamma }_2>0$ is the switching gain; ${\epsilon }_3\ge 0$ is the boundary layer of the third sliding surface; ${\lambda }_3>0$ is the upper bound of uncertain control gain and satisfies the following inequality. The proof is analogous to the proof of Theorem 1 and omitted for brevity.

Then, the fourth sliding surface is defined as

$$s_4=k_{\mathrm{Pd}}{e}_{i_{\mathrm{ds}}}+k_{\mathrm{Id}}\int e_{i_{\mathrm{ds}}}\mathrm{dt}$$

(16)

where $k_{\mathrm{Pd}}$ and $k_{\mathrm{Id}}$ are the selected proportional and integral gains such that the fourth sliding surface is stable. The equivalent and switching control for the $v_{\mathrm{ds}}^e$ can be derived as follows:

$$\begin{array}{c}\multicolumn{1}{c}{v_{\mathrm{ds},\mathrm{eq}}^e={(k_{\mathrm{Pd}}{k}_{\mathrm{Pf}}+k_{\mathrm{Pd}}/{\overline{L}}_{\sigma })}^{-1}}\\ \{k_{\mathrm{Pd}}{k}_{\mathrm{Pf}}{\stackrel{·}{\varphi }}_s^{*}+k_{\mathrm{Pd}}{k}_{\mathrm{Pf}}{R}_s{i}_{\mathrm{ds}}^e+k_{\mathrm{Pd}}{k}_{\mathrm{If}}{e}_{{\varphi }_s}+k_{\mathrm{Id}}{e}_{i_{\mathrm{ds}}}\hfill \\ -k_{\mathrm{Pd}}[-(\frac{{\overline{R}}_s}{{\overline{L}}_{\sigma }}+\frac{1}{\overline{\sigma }{\overline{\tau }}_r})i_{\mathrm{ds}}^e+{\omega }_{\mathrm{sl}}{i}_{\mathrm{qs}}^e+\frac{1}{{\overline{L}}_{\sigma }{\overline{\tau }}_r}{\varphi }_{\mathrm{ds}}^e]\}\hfill \end{array}$$

(17a)

$$\begin{array}{c}\multicolumn{1}{c}{v_{\mathrm{ds},\mathrm{sw}}^e={(k_{\mathrm{Pd}}{k}_{\mathrm{Pf}}+k_{\mathrm{Pd}}/{\overline{L}}_{\sigma })}^{-1}}\\ [{\mu }_1{s}_4+{\mu }_2{s}_4/(\left|s_4\right|+{\epsilon }_4)]/(1-{\lambda }_4)\hfill \end{array}$$

(17b)

$$v_{\mathrm{ds}}^e=v_{\mathrm{ds},\mathrm{eq}}^e+v_{\mathrm{ds},\mathrm{sw}}^e$$

(17c)

where $k_{\mathrm{Pd}}{k}_{\mathrm{Pf}}+k_{\mathrm{Pd}}/{\overline{L}}_{\sigma }$ is nonsingular; ${\overline{\tau }}_r$ is the nominal term of ${\tau }_r=\frac{L_r}{R_r}={\overline{\tau }}_r+\mathrm{\Delta }{\tau }_r$ where $\mathrm{\Delta }{\tau }_r$ is uncertain terms; ${\mu }_1>{\delta }_4/2>0$ where ${\delta }_4$ is the exponentially convergent rate to the sliding surface; ${\mu }_2>0$ is the switching gain; ${\epsilon }_4\ge 0$ is the boundary layer of the fourth sliding surface; ${\lambda }_4>0$ is the upper bound of the uncertain control gain and satisfies the following inequality

$$|(k_{\mathrm{Pd}}{k}_{\mathrm{Pf}}+k_{\mathrm{Pd}}/\mathrm{\Delta }{L}_{\sigma })/(k_{\mathrm{Pd}}{k}_{\mathrm{Pf}}+k_{\mathrm{Pd}}/{\overline{L}}_{\sigma })|\le {\lambda }_4<1$$

(18)

Finally, the proposed MSMC in equations (12) and (17) generate three-phase voltage commands via coordinate transformation fed into a inverter to achieve the speed control. The proof is analogous to the proof of Theorem 1 and omitted for brevity.

Simulations and discussions

To evaluate the performance and validate feasibility of the proposed control, two cases of simulations are performed for the tracking of reference speeds that are respectively constant and sinusoidal speeds. The chosen system and control parameters in the simulation are shown in Table B1 of Appendix B. First, the simulations of the IM drive operated with the constant speed 2000 $r/min$ under the external disturbance $T_L=2\mathrm{N}-\mathrm{m}$ and a maximum of 10% error of random variance on the system parameters are conducted. The tracking of reference mechanical angular speed shown in Figure 2(a) is excellent despite suffering the external disturbance $T_L=2\mathrm{N}-\mathrm{m}$ after 1 sec and random uncertainties. Figure 2(b) represents the response of the stator flux at d-axis and achieves the constant value 0.3019 as desired. In addition, the responses of the stator currents at d and q-axes are respectively shown in Figure 2(c) and (d). Obviously, the stator current at q-axis becomes larger after addition of an external load since 1 s and the stator current at d-axis is almost constant. Simultaneously, the value of electromagnetic torque $T_e$ becomes bigger shown as Figure 2(e) since addition of the external load. It is quite reasonable that the IM increases $T_e$ as it tackles the external load $T_L$ under the steady state of mechanical angular speed ${\omega }_{\mathrm{rm}}$ (obtained from equation (2c)). Further, to demonstrate the superiority of the proposed control scheme, the tracking of the IM drive operated with a low reference sinusoidal mechanical angular speed ${\omega }_{\mathrm{rm}}^{*}=200\sin (\pi t/3)$ is investigated under addition of the external load. Figure 3(a) demonstrated the tracking response of the reference ${\omega }_{\mathrm{rm}}^{*}=200\sin (\pi t/3)$ is quite satisfactory in spite of suffering the external load and random uncertainties on system parameters. Other responses consisting of the stator flux at d-axis ${\varphi }_{\mathrm{ds}}$ , stator currents at d and q-axes, and the electromagnetic torque $T_e$ are exhibited in Figure 3(b) to (e), respectively. Furthermore, the control performance of the proposed MSMC is comparative with a baseline controller (i.e. conventional PI controller) under the same conditions is shown in Figure 4 where the proportional and integral gains are the identical to the gains of multiple sliding surfaces. Clearly, it can be seen that tracking performance of the mechanical angular speed applying the proposed MSMC is superior to that of using PI controllers (cf. Figures 3(a) and 4) while the IM drive is subject to an external load. The above simulation results specifically indicate that the proposed MSMC scheme possess effectiveness and a high level of robustness.

Figures 2.

Responses of the IM drive with a constant angular speed under addition of disturbance after 1 s. and a maximum of 10% error of random variance on the system parameters: (a) tracking of real and reference mechanical angular speeds with addition of disturbance after 1 s, (b) stator flux at d-axis, ${\varphi }_{\mathrm{ds}}$ , (c) stator current at d-axis, $i_{\mathrm{ds}}^e$ , (d) stator current at q-axis, $i_{\mathrm{qs}}^e$ , and (e) electromagnetic torque, $T_e$

Figures 3.

Responses of the IM drive operated with a sinusoidal angular speed under addition of disturbance after 1 s. and a maximum of 10% error of random variance on the system parameters: (a) low-speed operating, (b) stator flux at d-axis, ${\varphi }_{\mathrm{ds}}$ , (c) stator current at d-axis, $i_{\mathrm{ds}}^e$ , (d) stator current at q-axis, $i_{\mathrm{qs}}^e$ , (e) electromagnetic torque, $T_e$

Figure 4.

Tracking of the sinusoidal mechanical angular speed using conventional PI controllers under the same conditions with Figure 3.

Conclusions

The dynamic model of a three-phase IM is decoupled and represented via the d-q axes transformation such that the corresponding controller can be designed with ease. Then, the speed control of an IM drive based on the stator flux oriented MSMC strategy was proposed to resolve the typical and potential problems for example, low-speed tracking performance and a high-level of robustness. The proposed MSMC not only had outstanding tracking performance but also possessed a high level of robustness in the presence of an external load and random uncertainties on system parameters. The overall system stability with the UUB performance is assured via Lyapunov stability criteria. Finally, the simulation results also validate the effectiveness and feasibility of the proposed MSMC scheme. Additionally, the comparative simulations with a baseline controller (i.e. conventional PI) exhibit the superior performance of the proposed MSMC under different operating conditions. Due to the necessity of feedback states measured by sensors for the proposed controller, it will cause the increase of cost. Thus, the direction of next research will be sensorless speed control of IM drives without sensors using our proposed control scheme.