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

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.


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][2][3][4].Compared with the conventional rotary induction motors (RIM), the main advantages of LIM are as follows: (1) it does not have any converter, gear, or other intermediate conversion mechanism which can reduce mechanical loss; (2) 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][8][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][17][18][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: (1) using CFDL technique to transfer the nonlinear system into self-designed linear model based on a parameter called pseudo-partial-derivative (PPD), (2) estimating the value of the PPD through a variety of methods, and (3) 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.

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 as where  denotes the primary length, V 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 () = (1 −  − )/ 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 as where   = 3  /(2ℎ  ),  means the pole numbers, and ℎ 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 get where φ  denotes the differential of   .According to (2)-( 5), the dynamic model of LIM considering end effects under IVC can be described as where  denotes the total mass of the moving object,  denotes the viscosity coefficient,  Load denotes the external force disturbance,   denotes the slip frequency, and Besides, according to (6), the acceleration of LIM can be redescribed as where  = −/;  = − Load /.
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 to where u  denotes the differential of   and (  min ,   max ) and ( u  min , u  max ) denote the lower and upper bound of   and u  .
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) model  ( + 1) =  ( () , . . .,  ( −   ) ,  () , . . .,  ( −   ) ,  () , . . .,  ( −   )) , (10) where system output  denotes the speed of the LIM V, input  denotes the primary voltage in -axis   , and disturbance  denotes the external force disturbance  Load .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.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.

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.
(4) Proving the stability of the closed-loop system by Lyapunov stability theory.

Mathematical Problems in Engineering
The system output identification observer can be designed as x ( + 1) = x () +   () Φ () +   () , (13) where x() and Φ() mean the estimated value of output and PPDs of the system at time ,   () = () − x() 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 as where  = 1− and Φ() = Φ()− Φ() means the estimation error of the PPDs.The adaptive update PPD algorithm is given by where the gain function is chosen as Due to the fact that  > 0 is a chosen positive constant, it is for sure that Γ() is positive.Besides, according to the practical assumption ‖()‖ ⩽ Ω, Γ() can be limited as In view of ( 14) and ( 15), the error dynamics of the system can be obtained as where  =  2×2 −()Γ()  () and  2×2 means the two-order unit matrix.
Theorem 5.The equivalent of [  , Φ] is globally uniformly stable.Furthermore, the estimation error of output   () converges to 0; that is to say, lim →∞ |  ()| = 0 Proof.Consider the Lyapunov function as where   is a positive constant and   is also a positive constant figured by   − where , and  1 =   − (1/).Thus, Δ  () ≤ 0 can confirm that ,   , and   satisfy the following inequalities: Since   () is a nonnegative function and Δ  () is negative for sure, we can get the conclusion that when  → ∞,   () → 0. It is a signal where, for all ,   () and Φ() are bounded, and lim →∞   () = 0. From ( 13), we get the true value of the system output as follows: It is worth noting that   ( + 1) is unknown in time .So, we transfer   ( + 1) 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 Φ(1) is the original value of Φ().

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 as where V * () 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 as where  > 0 and   denotes the sampling time of the control system.The closed-loop system stability can be guaranteed according to the following theorem.Proof.According to (27), we get Due to the fact that 0 < 1/(1 +   ) < 1 and 2Ω/(1 +   ) is bounded, according to the stability criteria in [29], the tracking error can be bounded as The SMC law of the LIM can be designed based on observer (24) as where   () and   () denote the feedback and equivalent laws and   () and () denote the primary and actual control input signals, respectively.And Sat(⋅) function is defined as where ℎ max and ℎ min mean the upper and lower bound of Sat(⋅).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.

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  = 0.99,  = 300, Φ( 1 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 timevarying 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  (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 timevarying 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 modelfree adaptive sliding-mode-constrained controller.Therefore, this simulation verifies the robustness of the proposed controller.

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.

Figure 1 :
Figure 1: Structure of a LIM.

Theorem 7 .
When the integral sliding-mode surface is bounded, the tracking error of the control system is bounded, too.More specifically, for |()| ≤ Ω, the tracking error is bounded to a region as lim →∞ |()| ≤ 2Ω/  .

Figure 2 :
Figure 2: Diagram of LIM control systems.

Figure 3 :
Figure 3: The reference tracking and tracking error curve of the proposed controller and PID controller (step signal).

Figure 4 :
Figure 4: The reference tracking and tracking error curve of the proposed controller and PID controller (periodic signal).