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

Authors: Xiaoqi Song; Dezhi Xu; Weilin Yang; Yan Xia; Bin Jiang
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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: (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 [7–9]. 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: (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. Section2 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 Figure1. 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 Q to express this phenomenon as(1)Q=l·Rrv·Lr,where l denotes the primary length, v denotes the speed of a LIM, and Lr and Rr denote the secondary inductance and resistance, respectively.Figure 1
Structure of a LIM.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]:(2)Vsd=Rsisd+pϕsd-ωeϕsqVsq=Rsisq+pϕsq-ωeϕsdVrd=Rrird+pϕrd-ωe-ωrϕrqVrq=Rrirq+pϕrq+ωe-ωrϕrd,where Vsd,Vrd, isd,ird, and ϕsd,ϕrd denote the primary and secondary voltage, current, and flux linkage in d-axis; Vsq,Vrq, isq,irq, and ϕsq,ϕrq denote the corresponding parameters in q-axis; Rs denotes the primary resistance; ωe and ωr denote the angular frequency of stator and rotor; and p denotes the differential operator.According to [8, 11], the flux linkage in dq-axis can be expressed as follows:(3)ϕsd=Lslisd+Lm1-fQisd+irdϕrq=Lslisq+Lm1-fQisq+irqϕrd=Lrlird+Lm1-fQisd+irdϕrq=Lrlirq+Lm1-fQisq+irq,where f(Q)=1-e-Q/Q is an important parameter during the process of modeling for a LIM, Lm is the magnetic inductance, and Lsl and Lrl are the primary and secondary leakage inductance. Meanwhile, the electromagnetic thrust force can be expressed as(4)Fet=Kfϕrd·isq-ϕrq·isd,where Kf=3πPLm/(2hLr), P 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 thed-axis, and we get(5)ϕrq=ϕ˙rq=0Vrd=Vrq=0,where ϕ˙rq denotes the differential of ϕrq.According to (2)–(5), the dynamic model of LIM considering end effects under IVC can be described as(6)disddt=-RsLQisd+VsdLQ+ωeisqdisqdt=-ωeisd+Lm1-fQLQLr-LmfQϕdr-RsLQisq+VsqLQdϕrddt=Lm1-fQisd-ϕrdTr-LmfQ/Rrωsl=ωe-ωr=Lm1-fQTr-LmfQ/RrisqϕrdFet=KTisq=M·dvdt+D·v+FLoad,where M denotes the total mass of the moving object, D denotes the viscosity coefficient, FLoad denotes the external force disturbance, ωsl denotes the slip frequency, and(7)LQ=Ls-LmfQ-Lm1-fQ2Lr-LmfQKT=32PπhLm1-fQLr-LmfQϕdrTr=LrRrBesides, according to (6), the acceleration of LIM can be redescribed as(8)dvdt=KTMisq+A·v+B,where A=-D/M;B=-FLoad/M.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(9)uqsmin⩽uqs⩽uqsmax;u˙qsmin⩽u˙qs⩽u˙qsmax,where u˙qs denotes the differential of uqs and (uqsmin,uqsmax) and (u˙qsmin,u˙qsmax) denote the lower and upper bound of uqs and u˙qs.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(10)xt+1=gxt,…,xt-tx,ut,…,ut-tu,dt,…,dt-td,where system output x denotes the speed of the LIM v, input u denotes the primary voltage in q-axis uqs, and disturbance d denotes the external force disturbance FLoad. And tx, tu, and td mean the unknown orders, and g(·) is the unknown function. Apparently, the LIM satisfies the following two basic assumptions.Assumption 2.
The partial derivatives ofg(·) for u(t) and d(t) are continuous.Assumption 3.
The plant (10) satisfies the condition of generalised Lipschitz, that is to say, ∀t, Δu(t-1)≠0 and Δd(t-1)≠0, satisfying Δx(t)⩽Λ1Δu(t-1) and Δx(t)⩽Λ2Δu(t-1), where Δx(t)=x(t)-x(t-1), Δu(t)=u(t)-u(t-1), and Δd(t)=d(t)-d(t-1), and Λ1, Λ2 are unknown constants.Remark 4.
For general nonlinear systems, Assumption2 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 Δu(t)≠0, we can obtain the data-driven model as(11)Δxt+1=θ1Δut+θ2Δdt,where θ1⩽Λ1 and θ2⩽Λ2 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:(12)xt+1=xt+ZTtΦt,where Zt=Δu(t),Δd(t)T and Φ(t)=θ1(t),θ2(t)T.The system output identification observer can be designed as(13)x^t+1=x^t+ZTtΦ^t+Meet,where x^(t) and Φ^(t) mean the estimated value of output and PPDs of the system at time t, ee(t)=x(t)-x^(t) denotes the estimation error of the system output, and the gain K is chosen in the unit cycle. According to (12) and (13), the dynamic of the estimation error ee(t) can be described as(14)eet+1=ZTtΦ~t+Neet,where N=1-M and Φ~(t)=Φ(t)-Φ^(t) means the estimation error of the PPDs. The adaptive update PPD algorithm is given by(15)Φ^t+1=Φ^t+ZtΓteet+1-Neet,where the gain function is chosen as(16)Γt=2Zt2+∂Due to the fact that ∂>0 is a chosen positive constant, it is for sure that Γ(t) is positive. Besides, according to the practical assumption Z(t)⩽Ω, Γ(t) can be limited as(17)Γt⩾2Ω2+∂=υ>0In view of (14) and (15), the error dynamics of the system can be obtained as(18)eet+1=ZTtΦ~t+NeetΦ~t+1=HΦ~t,where H=I2×2-Z(t)Γ(t)ZT(t) and I2×2 means the two-order unit matrix.Theorem 5.
The equivalent ofee,Φ~ is globally uniformly stable. Furthermore, the estimation error of output ee(t) converges to 0; that is to say, limt→∞ee(t)=0Proof.
Consider the Lyapunov function as(19)VAt=PAee2t+λAΦ~TtΦ~t,where λA is a positive constant and PA is also a positive constant figured by PA-FA2PA=QA with QA being a positive constant. Then, the difference of VA(t) can be written as(20)ΔVAt=VAt+1-VAt=PAZTtΦ~tΦ~TtZt-2PAFAZTtΦ~teet+PAFA2ee2t+Φ~TtλAHTH-λAΦ~t+PAee2t=-ΘATtλAμAΓTtΓt-PAΘAt-QAee2t+2PAFAeetΘAt≤-λAμAΓTtΓt-PAΘAt2+2PAFAeetΘAt-QAeet2≤-a1eet2-a2ΘAt2,where ΘA(t)=ZT(t)Φ~(t), a2=λAμAυ2-PA-ιPA2FA2, and a1=QA-1/ι. Thus, ΔVA(t)≤0 can confirm that ι, QA, and λA satisfy the following inequalities:(21)QA>1ι,λAμAυ2-PA-ιPA2FA2>0Since VA(t) is a nonnegative function and ΔVA(t) is negative for sure, we can get the conclusion that when t→∞, VA(t)→0. It is a signal where, for all k, ee(t) and Φ~(t) are bounded, and limt→∞ee(t)=0.
From (13), we get the true value of the system output as follows:(22)xt+1=x^t+ZTtΦ^t+Meet+eet+1It is worth noting that ee(t+1) is unknown in time t. So, we transfer ee(t+1) into the following expression by two-step estimation technique: (23)eet+1≈2eet-eet-1Therefore, (22) can be rewritten as(24)xt+1=x^t+ZTtΦ^t+2-Meet-eet-1Remark 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]:(25)Φ^t=Φ^1,ifΦ^TtΦ^t≤ϑorθ^1t≤ϑ,where ϑ is a tiny positive constant and Φ^(1) is the original value of Φ^(t).


### 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 as(26)et=v∗t-vt-ξt,where v∗(t) means the given velocity reference value and ξ(t) is the compensator signal which will be given later. To design the SMC, we choose an integral sliding surface as(27)st=et+ψ∑i=1tTsei,where ψ>0 and Ts 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, fors(t)≤Ω, the tracking error is bounded to a region as limt→∞e(t)≤2Ω/ψTs.Proof.
According to (27), we get(28)et+1=et1+ψTs+st+1-st1+ψTs≤11+ψTset+11+ψTsst+1+st≤11+ψTset+2Ω1+ψTsDue to the fact that 0<1/(1+ψTs)<1 and 2Ω/(1+ψTs) is bounded, according to the stability criteria in [29], the tracking error can be bounded as(29)limt→∞⁡et≤2ΩψTsThe SMC law of the LIM can be designed based on observer (24) as(30)ust=ut-1+uft+uetut=Satut-1+Satust-ut-1,Tsu˙sqmin,Tsu˙sqmax,Tsusqmin,Tsusqmax,where uf(t) and ue(t) denote the feedback and equivalent laws and us(t) and u(t) denote the primary and actual control input signals, respectively. And Sat(·) function is defined as(31)Sath,hmin,hmax=hmaxh≥hmaxhhmax>h>hminhminhmin≥h,where hmax and hmin 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:(32)ξt=γξt-1+θ^1tust-ut-1,where γ is chosen in the unit disk.Remark 8.
Sinceγ lies in the unit disk and assuming us(t)-u(t-1) is bounded, the signal ξ(t) is uniformly ultimately bounded (UUB) for all t according to [28].Moreover, we concretely give the expressions ofuf(t) and ue(t) as(33)uft=-θ^2tKsstθ^22t+κ1+ψTsuet=θ^2tθ^22t+κx∗t+1-θ^1tΔdt-x^t-2-Meet+eet-1-et1+ψTs-γξt,where κ is also chosen in the unit disk, Ks is a negative constant chosen by Ks2/2+Ks<0, and x∗(t+1) means the reference signal value in time t+1.Theorem 9.
For givenΔx∗(t)-Δx∗(t-1)≤Δx∗, using control laws (31)–(33), the velocity tracking error of the LIM is UUB for all t with ultimate bound as limt→∞e(t)≤q1(t)+q12(t)+4q0(t)q2(t)/q0(t)ψTs.Here,q0 is a constant given by q0(t)≤-Ks2/2+Ks, and (34)q1t=κKs+11+ψTsqtθ^12t+κq2t=κ21+ψTs2q2t2θ^12t+κqt=-Ksst1+ψTs+x∗t+1-θ^2tΔdt-x^t-2-Meet+eet-1-et1+ψTs-γξtProof.
Define the Lyapunov functionVB(t)=(1/2)s2(t); then, the difference of VB(t) can be figured by(35)ΔVBt+1=VBt+1-VBt=Δst+112Δst+1+st,where Δs(t+1) is figured by(36)Δst+1=st+1-st=1+ψTset+1-et=1+ψTsx∗t+1-x^t-2-Meet-Φ^tZt+eet-1-ξt+1-et=1+ψTsx∗t+1-x^t-2-Meet+eet-1-γξt-θ^2tΔdt-θ^1tuft+uet-et=Ksst+κ1+ψTsθ^12t+κ-Ksst1+ψTs-x^t-θ^2tΔdt+x∗t+1-2-Meet+eet-1-et1+ψTs-γξt=Ksst+κ1+ψTsθ^12t+κqtBesides, referring to (33), then we get(37)ΔVt+1=12Ks2+Kss2t+κ21+ψTs2q2t2θ^12t+κ+κKs+11+ψTsqtθ^12t+κst≤-q0ts2t+q1tst+q2t,where Ks is chosen to make Ks2/2+Ks<0, and then q0(t)>0 is for sure. And when s(t)>q1+q12+4q0q2/2q0, ΔV(t+1)<0 can be guaranteed. Hence, the sliding surface is bounded as limt→∞s(t)≤q1+q12+4q0q2/2q0. Finally, according to Theorem 9, we can get the conclusion that(38)limt→∞⁡et≤q1+q12+4q0q2q0ψTsRemark 10.
Becauseψ and κ are tiny positive constants, respectively, the ultimate bound of tracking error is 0; i.e., limt→∞e(t)=0.


## 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 Δu(t)≠0, we can obtain the data-driven model as(11)Δxt+1=θ1Δut+θ2Δdt,where θ1⩽Λ1 and θ2⩽Λ2 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:(12)xt+1=xt+ZTtΦt,where Zt=Δu(t),Δd(t)T and Φ(t)=θ1(t),θ2(t)T.The system output identification observer can be designed as(13)x^t+1=x^t+ZTtΦ^t+Meet,where x^(t) and Φ^(t) mean the estimated value of output and PPDs of the system at time t, ee(t)=x(t)-x^(t) denotes the estimation error of the system output, and the gain K is chosen in the unit cycle. According to (12) and (13), the dynamic of the estimation error ee(t) can be described as(14)eet+1=ZTtΦ~t+Neet,where N=1-M and Φ~(t)=Φ(t)-Φ^(t) means the estimation error of the PPDs. The adaptive update PPD algorithm is given by(15)Φ^t+1=Φ^t+ZtΓteet+1-Neet,where the gain function is chosen as(16)Γt=2Zt2+∂Due to the fact that ∂>0 is a chosen positive constant, it is for sure that Γ(t) is positive. Besides, according to the practical assumption Z(t)⩽Ω, Γ(t) can be limited as(17)Γt⩾2Ω2+∂=υ>0In view of (14) and (15), the error dynamics of the system can be obtained as(18)eet+1=ZTtΦ~t+NeetΦ~t+1=HΦ~t,where H=I2×2-Z(t)Γ(t)ZT(t) and I2×2 means the two-order unit matrix.Theorem 5.
The equivalent ofee,Φ~ is globally uniformly stable. Furthermore, the estimation error of output ee(t) converges to 0; that is to say, limt→∞ee(t)=0Proof.
Consider the Lyapunov function as(19)VAt=PAee2t+λAΦ~TtΦ~t,where λA is a positive constant and PA is also a positive constant figured by PA-FA2PA=QA with QA being a positive constant. Then, the difference of VA(t) can be written as(20)ΔVAt=VAt+1-VAt=PAZTtΦ~tΦ~TtZt-2PAFAZTtΦ~teet+PAFA2ee2t+Φ~TtλAHTH-λAΦ~t+PAee2t=-ΘATtλAμAΓTtΓt-PAΘAt-QAee2t+2PAFAeetΘAt≤-λAμAΓTtΓt-PAΘAt2+2PAFAeetΘAt-QAeet2≤-a1eet2-a2ΘAt2,where ΘA(t)=ZT(t)Φ~(t), a2=λAμAυ2-PA-ιPA2FA2, and a1=QA-1/ι. Thus, ΔVA(t)≤0 can confirm that ι, QA, and λA satisfy the following inequalities:(21)QA>1ι,λAμAυ2-PA-ιPA2FA2>0Since VA(t) is a nonnegative function and ΔVA(t) is negative for sure, we can get the conclusion that when t→∞, VA(t)→0. It is a signal where, for all k, ee(t) and Φ~(t) are bounded, and limt→∞ee(t)=0.
From (13), we get the true value of the system output as follows:(22)xt+1=x^t+ZTtΦ^t+Meet+eet+1It is worth noting that ee(t+1) is unknown in time t. So, we transfer ee(t+1) into the following expression by two-step estimation technique: (23)eet+1≈2eet-eet-1Therefore, (22) can be rewritten as(24)xt+1=x^t+ZTtΦ^t+2-Meet-eet-1Remark 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]:(25)Φ^t=Φ^1,ifΦ^TtΦ^t≤ϑorθ^1t≤ϑ,where ϑ is a tiny positive constant and Φ^(1) is the original value of Φ^(t).


## 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 as(26)et=v∗t-vt-ξt,where v∗(t) means the given velocity reference value and ξ(t) is the compensator signal which will be given later. To design the SMC, we choose an integral sliding surface as(27)st=et+ψ∑i=1tTsei,where ψ>0 and Ts 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, fors(t)≤Ω, the tracking error is bounded to a region as limt→∞e(t)≤2Ω/ψTs.Proof.
According to (27), we get(28)et+1=et1+ψTs+st+1-st1+ψTs≤11+ψTset+11+ψTsst+1+st≤11+ψTset+2Ω1+ψTsDue to the fact that 0<1/(1+ψTs)<1 and 2Ω/(1+ψTs) is bounded, according to the stability criteria in [29], the tracking error can be bounded as(29)limt→∞⁡et≤2ΩψTsThe SMC law of the LIM can be designed based on observer (24) as(30)ust=ut-1+uft+uetut=Satut-1+Satust-ut-1,Tsu˙sqmin,Tsu˙sqmax,Tsusqmin,Tsusqmax,where uf(t) and ue(t) denote the feedback and equivalent laws and us(t) and u(t) denote the primary and actual control input signals, respectively. And Sat(·) function is defined as(31)Sath,hmin,hmax=hmaxh≥hmaxhhmax>h>hminhminhmin≥h,where hmax and hmin 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:(32)ξt=γξt-1+θ^1tust-ut-1,where γ is chosen in the unit disk.Remark 8.
Sinceγ lies in the unit disk and assuming us(t)-u(t-1) is bounded, the signal ξ(t) is uniformly ultimately bounded (UUB) for all t according to [28].Moreover, we concretely give the expressions ofuf(t) and ue(t) as(33)uft=-θ^2tKsstθ^22t+κ1+ψTsuet=θ^2tθ^22t+κx∗t+1-θ^1tΔdt-x^t-2-Meet+eet-1-et1+ψTs-γξt,where κ is also chosen in the unit disk, Ks is a negative constant chosen by Ks2/2+Ks<0, and x∗(t+1) means the reference signal value in time t+1.Theorem 9.
For givenΔx∗(t)-Δx∗(t-1)≤Δx∗, using control laws (31)–(33), the velocity tracking error of the LIM is UUB for all t with ultimate bound as limt→∞e(t)≤q1(t)+q12(t)+4q0(t)q2(t)/q0(t)ψTs.Here,q0 is a constant given by q0(t)≤-Ks2/2+Ks, and (34)q1t=κKs+11+ψTsqtθ^12t+κq2t=κ21+ψTs2q2t2θ^12t+κqt=-Ksst1+ψTs+x∗t+1-θ^2tΔdt-x^t-2-Meet+eet-1-et1+ψTs-γξtProof.
Define the Lyapunov functionVB(t)=(1/2)s2(t); then, the difference of VB(t) can be figured by(35)ΔVBt+1=VBt+1-VBt=Δst+112Δst+1+st,where Δs(t+1) is figured by(36)Δst+1=st+1-st=1+ψTset+1-et=1+ψTsx∗t+1-x^t-2-Meet-Φ^tZt+eet-1-ξt+1-et=1+ψTsx∗t+1-x^t-2-Meet+eet-1-γξt-θ^2tΔdt-θ^1tuft+uet-et=Ksst+κ1+ψTsθ^12t+κ-Ksst1+ψTs-x^t-θ^2tΔdt+x∗t+1-2-Meet+eet-1-et1+ψTs-γξt=Ksst+κ1+ψTsθ^12t+κqtBesides, referring to (33), then we get(37)ΔVt+1=12Ks2+Kss2t+κ21+ψTs2q2t2θ^12t+κ+κKs+11+ψTsqtθ^12t+κst≤-q0ts2t+q1tst+q2t,where Ks is chosen to make Ks2/2+Ks<0, and then q0(t)>0 is for sure. And when s(t)>q1+q12+4q0q2/2q0, ΔV(t+1)<0 can be guaranteed. Hence, the sliding surface is bounded as limt→∞s(t)≤q1+q12+4q0q2/2q0. Finally, according to Theorem 9, we can get the conclusion that(38)limt→∞⁡et≤q1+q12+4q0q2q0ψTsRemark 10.
Becauseψ and κ are tiny positive constants, respectively, the ultimate bound of tracking error is 0; i.e., limt→∞e(t)=0.


## 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 Figure2. Meanwhile, the parameters of the LIM are given in Table 1.Table 1
Parameters of LIM.
Parameters Representation Value Rs(Ω) Primary phase resistance 6.2689 Rr(Ω) Secondary phase resistance 3.784 Lm(H) Mutual inductance 0.0825 Lr(H) Secondary phase inductance 0.1021 Ls(H) Primary phase inductance 0.1021 M (kg) Total mass of the object 3.25 D (kg/s) Viscosity coefficient 40.95 h (m) Polar distance 0.057Figure 2
Diagram of LIM control systems.In order to obtain a satisfactory control effect, we choose the parameters of the controller asN=0.99, ϑ=300, Φ(1)=0.012,0.005T, Ks=-1.5, κ=0.0001, ψ=10000, Ts=0.001, and γ=0.3. Meanwhile, the parameters of the PID controller are P=300, I=10000, and D=0.1. 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 Figure3. 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 Figure8. 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.Figure 3
The reference tracking and tracking error curve of the proposed controller and PID controller (step signal).Figure 4
The reference tracking and tracking error curve of the proposed controller and PID controller (periodic signal).Figure 5
The value of input signals (periodic signal).Figure 6
The value of compensator signals (periodic signal).Figure 7
The value of PPDs (periodic signal).Figure 8
Velocity tracking curve under different mover mass (periodic signal).


## 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.



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