Improved Model-Free Adaptive Sliding-Mode-Constrained Control for Linear Induction Motor considering End Effects

Song, Xiaoqi; Xu, Dezhi; Yang, Weilin; Xia, Yan; Jiang, Bin

doi:https://doi.org/10.1155/2018/4341825

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2018 | Article ID 4341825 | https://doi.org/10.1155/2018/4341825

Improved Model-Free Adaptive Sliding-Mode-Constrained Control for Linear Induction Motor considering End Effects

Xiaoqi Song,¹Dezhi Xu,¹Weilin Yang,¹Yan Xia,¹and Bin Jiang²

Academic Editor: Tarek Ahmed-Ali

Received05 Feb 2018

Accepted29 Apr 2018

Published29 May 2018

Abstract

As a kind of special motors, linear induction motors (LIM) have been an important research field for researchers. However, it gives a great velocity control challenge due to the complex nonlinearity, high coupling, and unique end effects. In this article, an improved model-free adaptive sliding-mode-constrained control method is proposed to deal with this problem dispensing with internal parameters of the LIM. Firstly, an improved compact form dynamic linearization (CFDL) technique is used to simplify the LIM plant. Besides, an antiwindup compensator is applied to handle the problem of the actuator under saturations in case during the controller design. Furthermore, the stability of the closed system is proved by Lyapunov stability method theoretically. Finally, simulation results are given to demonstrate that the proposed controller has excellent dynamic performance and stronger robustness compared with traditional PID controller.

1. Introduction

In the past few decades, the LIM has been widely used in many fields, such as military, household appliances, industrial automation, and transportation [1–4]. Compared with the conventional rotary induction motors (RIM), the main advantages of LIM are as follows: it does not have any converter, gear, or other intermediate conversion mechanism which can reduce mechanical loss; it is only driven by magnetic force which makes the LIM have the features of high speed and low noise [5, 6]. Even though the driving principle of a LIM is similar to that of a RIM, the parameters of LIM are time-varying, such as end effects, slip frequency, dynamic air gap, three-phase imbalance, and track structure [7–9]. Among them, the end effects greatly affect the LIM control performance, and the faster the speed, the more significant the impact. Therefore, during the modeling of the LIM, the end effects must be considered.

With the quick development of science and technology, many model-based control methods are proposed to handle LIM control problems. In [10], an adaptive backstepping method is proposed to deal with the position tracking problem of the LIM. In [11], an optimized adaptive tracking control is applied for a LIM considering the uncertainties. In [12], the authors use input-output feedback linearization control technique with online model reference adaptive system (MRAS) method suiting the induction resistance to realize the velocity following goal, whereas the three mentioned methods are highly dependent on the accuracy of the model. Once the model is improperly defined or the system parameters cannot be accurately obtained, the dynamic response of the system will hardly be satisfied. Besides, some non-model-based control methods are also proposed for LIM control problems. In [13], the researchers present a real-time discrete neural control scheme based on a recurrent high order neural network trained online to a LIM. In [14, 15], some methods based on fuzzy control are also used to have the problem solved. However, even if we neglect the complexity of the selection of fuzzy rules and the uncertainty of the neural network nodes, these methods have not considered the input saturation problems which may result in system instability.

Model-free adaptive control (MFAC) was first proposed in 1994 and is a hot topic in the field of data-driven modeling [16–19]. It is a method that only relies on input/output (I/O) data and does not need any internal information of the plant. The main design steps of the MFAC are divided into three categories: using CFDL technique to transfer the nonlinear system into self-designed linear model based on a parameter called pseudo-partial-derivative (PPD), estimating the value of the PPD through a variety of methods, and devising the controller based on self-designed linear model. For now, MFAC has been widely applied in all kinds of fields, such as multiagent systems [20], chemical process [19, 21], and intelligent transportation [22]. Moreover, due to the fact that sliding-mode control (SMC) is designed without object parameters and disturbance, it gets the merits of quick response and high fitness. SMC is also a hot topic and is applied in a variety of fields [23, 24] and has been used in combination with MFAC firstly in [25].

In this paper, an improved CFDL technique is used to linearize the LIM model considering end effects based on PPD estimation algorithm. And we design a model-free adaptive constrained sliding-mode control for the system considering input saturations. So as to avoid the instability caused by saturations, we design an antiwindup compensator to make the output continue to follow the given reference.

The rest of this paper is organized as follows. Section 2 briefly introduces the model of the LIM considering end effects. In Section 3, the main results of the proposed control strategy are given. The simulation results are shown in Section 4 to verify the effectiveness and robustness of the method. Finally, some conclusions are drawn in Section 5.

2. Problem Formulation for LIM

Similar to a RIM, a LIM is made up of primary and secondary components as shown in Figure 1. Besides, a LIM is obtained by a RIM that is opened longitudinally in a transverse direction. However, the biggest difference between a LIM and a RIM is that the LIM contains end effects which are caused by its structure. The end effects can be explained as follows: when the primary moves, eddy current occurs in the corresponding secondary conductor plate at the outlet and inlet terminals, and the direction of flow is opposite to the primary current, so that the air gap magnetic field will be distorted [9, 11]. Researchers generally use a parameter to express this phenomenon aswhere denotes the primary length, denotes the speed of a LIM, and and denote the secondary inductance and resistance, respectively.

When the LIM is in a stationary state, we can consider its equivalent circuit as a RIM. Nevertheless, when the LIM is in a motion state, the model of a LIM in synchronously rotating reference frame should be improved as follows [8, 11]:where , , and denote the primary and secondary voltage, current, and flux linkage in -axis; , , and denote the corresponding parameters in -axis; denotes the primary resistance; and denote the angular frequency of stator and rotor; and denotes the differential operator.

According to [8, 11], the flux linkage in -axis can be expressed as follows:where is an important parameter during the process of modeling for a LIM, is the magnetic inductance, and and are the primary and secondary leakage inductance. Meanwhile, the electromagnetic thrust force can be expressed aswhere , means the pole numbers, and is the pole pitch.

By using the indirect vector control (IVC) technology, we can convert the linear induction motor model into a DC motor model which brings about great convenience to the control of the LIM. Thus, with IVC technology, orientate the rotor flux to the -axis, and we getwhere denotes the differential of .

According to (2)–(5), the dynamic model of LIM considering end effects under IVC can be described aswhere denotes the total mass of the moving object, denotes the viscosity coefficient, denotes the external force disturbance, denotes the slip frequency, andBesides, according to (6), the acceleration of LIM can be redescribed aswhere .

Remark 1. Taking into account the physical characteristics of the inverter structure and the safety of the system, the input saturation conditions must be considered. The control inputs are limited towhere denotes the differential of and () and () denote the lower and upper bound of and .

As speed is the most important performance parameter of motor control, we choose the velocity as our main control objective. Then, the model of a LIM considering end effects can be described in the following discrete-time unknown Nonlinear AutoRegressive with eXogenous input (NARX) modelwhere system output denotes the speed of the LIM , input denotes the primary voltage in -axis , and disturbance denotes the external force disturbance . And , , and mean the unknown orders, and is the unknown function. Apparently, the LIM satisfies the following two basic assumptions.

Assumption 2. The partial derivatives of for and are continuous.

Assumption 3. The plant (10) satisfies the condition of generalised Lipschitz, that is to say, , and , satisfying and , where , , and , and , are unknown constants.

Remark 4. For general nonlinear systems, Assumption 2 is a common condition in the process of controller design. And Assumption 3 is a constrained condition that limits the changes of the outputs of the plant caused by system inputs and disturbance.

3. Main Results

In this section, an improved model-free adaptive SMC scheme is proposed for the LIM through the CFDL technology. The main contributions of this section are as follows:(1)Transferring the LIM system into a data-based CFDL model considering the disturbance.(2)Proposing the PPD estimation algorithm based on observers.(3)Designing the model-free adaptive integral sliding-model controller via an antiwindup compensator.(4)Proving the stability of the closed-loop system by Lyapunov stability theory.

3.1. Data-Driven Modeling for LIM and PPD Estimation Algorithm

Data-driven modeling method was originally proposed by HOU [17, 18, 26], and it is totally divided into three forms: CFDL, partial form dynamic linearization (PFDL), and full-form dynamic linearization (FFDL). In this paper, the CFDL technique is used to linearize the LIM system. When , we can obtain the data-driven model aswhere and are the PPDs of the system.

The process of the proof is the same as that of [27].

To describe the system more conveniently, model (11) can be rewritten as follows:where and .

The system output identification observer can be designed aswhere and mean the estimated value of output and PPDs of the system at time , denotes the estimation error of the system output, and the gain is chosen in the unit cycle. According to (12) and (13), the dynamic of the estimation error can be described aswhere and means the estimation error of the PPDs. The adaptive update PPD algorithm is given bywhere the gain function is chosen asDue to the fact that is a chosen positive constant, it is for sure that is positive. Besides, according to the practical assumption , can be limited asIn view of (14) and (15), the error dynamics of the system can be obtained aswhere and means the two-order unit matrix.

Theorem 5. The equivalent of is globally uniformly stable. Furthermore, the estimation error of output converges to 0; that is to say,

Proof. Consider the Lyapunov function aswhere is a positive constant and is also a positive constant figured by with being a positive constant. Then, the difference of can be written aswhere , , and . Thus, can confirm that , , and satisfy the following inequalities:Since is a nonnegative function and is negative for sure, we can get the conclusion that when , . It is a signal where, for all , and are bounded, and .
From (13), we get the true value of the system output as follows:It is worth noting that is unknown in time . So, we transfer into the following expression by two-step estimation technique: Therefore, (22) can be rewritten as

Remark 6. In order to make the parameter estimation law (15) have a strong capability in tracing time-varying parameters, a reset scheme should be considered as follows [17]:where is a tiny positive constant and is the original value of .

3.2. Model-Free Adaptive SMC Design and Stability Analysis

In order to eliminate the output non-following problem produced by the actuator saturation, an integral SMC based on antiwindup compensator is proposed [28]. Define the velocity tracking error aswhere means the given velocity reference value and is the compensator signal which will be given later. To design the SMC, we choose an integral sliding surface aswhere and denotes the sampling time of the control system. The closed-loop system stability can be guaranteed according to the following theorem.

Theorem 7. When the integral sliding-mode surface is bounded, the tracking error of the control system is bounded, too. More specifically, for , the tracking error is bounded to a region as .

Proof. According to (27), we getDue to the fact that and is bounded, according to the stability criteria in [29], the tracking error can be bounded asThe SMC law of the LIM can be designed based on observer (24) aswhere and denote the feedback and equivalent laws and and denote the primary and actual control input signals, respectively. And function is defined aswhere and mean the upper and lower bound of . One important thing is that when the input signal is within saturation, the tracking performance cannot be guaranteed. Thus, we design an antiwindup compensator signal as follows:where is chosen in the unit disk.

Remark 8. Since lies in the unit disk and assuming is bounded, the signal is uniformly ultimately bounded (UUB) for all according to [28].

Moreover, we concretely give the expressions of and aswhere is also chosen in the unit disk, is a negative constant chosen by , and means the reference signal value in time .

Theorem 9. For given , using control laws (31)–(33), the velocity tracking error of the LIM is UUB for all with ultimate bound as .

Here, is a constant given by , and

Proof. Define the Lyapunov function ; then, the difference of can be figured bywhere is figured byBesides, referring to (33), then we getwhere is chosen to make , and then is for sure. And when , can be guaranteed. Hence, the sliding surface is bounded as . Finally, according to Theorem 9, we can get the conclusion that

Remark 10. Because and are tiny positive constants, respectively, the ultimate bound of tracking error is 0; i.e., .

4. Simulation Results

In this section, a few simulation examples are given to testify the effectiveness of the designed controller compared to the classical PID controller. First of all, to clearly understand the control process of the LIM, a block diagram is given in Figure 2. Meanwhile, the parameters of the LIM are given in Table 1.

In order to obtain a satisfactory control effect, we choose the parameters of the controller as , , , , , , , and . Meanwhile, the parameters of the PID controller are , , and . Two kinds of simulation experiments below are designed to prove the effectiveness of the proposed controller in this paper. By comparing the proposed controller with the PID controller, we will analyze the control performance from the following aspects: dynamic performance, static performance, anti-interference, and robustness.(1)To test the tracking performance and anti-interference, we select the step signal and time-varying periodic signal as our given velocity reference, respectively. Meanwhile, the load torque changes as shown in Figure 3. The velocity tracking performance and tracking error are also shown in Figures 3 and 4. As the figures show, it can be clearly known that both controllers can ensure that there is no steady-state error at steady state for step signal. However, the proposed control method enables the control system to enter steady state faster within 0.12 s (within is 0.3 s for the PID controller). Besides, when the load torque changes at 1.5 s and 3.5 s, the speed of the LIM under the proposed controller is still able to track the given signal quickly within 0.1 s after a small fluctuation (within 0.32 s for the PID controller). It can be seen more prominently in Figure 4 that the proposed controller can make the system output track the time-varying periodic signal perfectly with less than 0.05 m/s error. The input signal under time-varying periodic signal is shown in Figure 5. The compensator signal under time-varying periodic signal is shown in Figure 6. From Figures 5 and 6, we can get the information that, by adding the antiwindup compensator, the control system can quickly exit from saturation but still trace the reference quite well. The values of the PPDs are shown in Figure 7.(2)To test the robustness of the proposed controller, we increase the mover mass to three times and five times the original, and this simulation is also under time-varying periodic signal. The tracking performance is shown in Figure 8. From Figure 8, we know that no matter how the mover mass changes, the speed of the LIM can always follow the given reference satisfactorily, and that is another merit of the model-free adaptive sliding-mode-constrained controller. Therefore, this simulation verifies the robustness of the proposed controller.

5. Conclusion

In this paper, a model-free adaptive sliding-mode controller is proposed to deal with the problem of the speed tracking of the LIM considering end effects. First of all, the CFDL technique is applied to linearize the LIM model which has been transferred into a NARX form. Then, the controller is designed based on PPD estimation algorithm. Through the process of designing, an antiwindup compensator is designed to handle the problem of input saturation. Lyapunov stability theory proves the stability of the closed-loop system theoretically, and the simulation results verify the effectiveness of the proposed method to the LIM system.

Data Availability

All the underlying data related to this article are available upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (61503156, 61403161, and 61473250) and the National Key Research and Development Program (2016YFD0400300).

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Copyright © 2018 Xiaoqi Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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