# A Novel Compound Nonlinear State Error Feedback Super-Twisting Fractional-Order Sliding Mode Control of PMSM Speed Regulation System Based on Extended State Observer

Authors: Peng Gao; Guangming Zhang; Xiaodong Lv
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this article, a novel compound nonlinear state error feedback super-twisting fractional-order sliding mode control (NLSEF-STFOSMC) is proposed for the control of the permanent magnet synchronous motor (PMSM) speed regulation system. Firstly, a novel fractional-order proportion integration differentiation (FOPID) switching manifold is designed. A modified sliding mode control (SMC) is constructed by a super-twisting reaching law and the novel FOPID sliding surface. Secondly, the nonlinear state error feedback control law (NLSEF) has been widely used because of high control accuracy, fast convergence, and flexible operation. Therefore, combining the modified SMC with the NLSEF, the compound NLSEF-STFOSMC is proposed, which has an excellent performance. At the same time, the external disturbance of the system is observed by a novel extended state observer. Finally, the performance of the corresponding control law to the speed operation of the PMSM is fully investigated compared with other related algorithms to demonstrate the effectiveness. The comparison results show that the proposed compound control strategy has excellent dynamic and static performance and strong robustness.


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## Body

## 1. Introduction

The permanent magnet synchronous motor (PMSM) with the advantages of low rotor inertia, simple structure, high efficiency, and high power density has been widely used in many fields [1, 2]. At present, the traditional linear control scheme still occupies the active position in the control of PMSM in the industrial field, but because of the disturbance and time-varying and strong coupling of PMSM, it is difficult to obtain the satisfactory speed regulation performance of the traditional linear control scheme [3, 4]. Nowadays, many scholars have proposed many different nonlinear control strategies to solve the problems in traditional linear control scheme, such as backstepping control [5, 6], finite time control [7, 8], model predictive control [9], fuzzy logic control [10], iterative learning control [11], and active disturbance rejection control [12]. The sliding mode control (SMC) has been widely used for many decades, because of its simple implementation, disturbance rejection, strong robustness, and fast dynamic responses [13, 14]. Till now, approaches proposed by many researchers were the combination of SMC with the different methods for the control of the PMSM speed regulation system [15–19].Combined with the SMC of classical proportion integration differentiation (PID) control, the control performance can be improved by adjusting the parameters of PID control. Fractional theory has been studied by scholars as pure mathematics theory for nearly 300 years. The traditional PID control theory is integrated with the fractional-order theory to come out the fractional-order PID. Compared with the integer-order controller, the extra degree of freedom of the fractional-order integrator and differentiator can further improve the control performance of the system [20]. Recently, SMC strategies based on the fractional-order PID (FOPID) sliding surface have been widely applied to various systems [21–23]. The SMC strategies based on the fractional-order PID (FOPID) sliding surface also have been designed for the speed control of PMSM [24]. However, the conventional FOPID also has the disadvantages of simple and rough signal processing [25–27]. In this study, a nonlinear saturation function is introduced into the integral term and differential term of the traditional FOPID sliding surface, thus forming a novel FOPID sliding surface, which can effectively settle the abovementioned issues.The derivative information of the super-twisting (ST) scheme is not required as it is a high-order SMC scheme with many advantages [28]. The ST scheme has been widely used in various control systems [13, 29–34]. In this paper, a super-twisting fractional-order sliding mode control (STFOSMC) is constructed by the ST scheme and the novel FOPID sliding surface. Han proposed a nonlinear state error feedback control law (NLSEF) in [35]. The NLSEF has been widely used because of high control accuracy, fast convergence, and performance of flexibility [36–39]. Considering the robustness of SMC and the high precision and fast convergence of NLSEF and combining the STFOSMC and the NLSEF, a novel compound nonlinear state error feedback super-twisting fractional-order sliding mode control (NLSEF-STFOSMC) is proposed.Since the extended state observer (ESO) rarely relies on the mathematical model of the system, it is widely used to estimate the total disturbance of the system [35, 40–42]. However, the nonsmooth and nonderivable characteristics of the conventional ESO can easily cause flutter phenomenon in the response of the system. In this paper, a smooth and derivable ESO which reduces the flutter phenomenon is built.Motivated by the above discussion and our previous research [43], the main contributions of the proposed control method, which are novel in comparison to previous studies, are listed as follows:(1)
A novel compound NLSEF-STFOSMC combining the advantages of the modified SMC and the NLSEF is proposed. The modified SMC uses a novel FOPID sliding surface to improve the control performance of the sliding mode stage. The proposed compound controller is analyzed and used for the PMSM speed regulation system.(2)
The external disturbance of the system is observed by one smooth and derivable ESO. The modified ESO based on a smooth and derivable nonlinear saturation function is introduced to the compound NLSEF-STFOSMC to estimate the external disturbance.(3)
Lyapunov approach and fractional calculus are used to show the stability of the novel compound NLSEF-STFOSMC.Based on our previous research [43], the rest of this paper is organized as follows: the model of PMSM and some definitions of fractional-order calculus are given in Section 2. In Section 3, the control strategies and the stability analysis are given. The comparative simulations and the analysis of simulation results are given in Section 4, which demonstrate that the proposed compound control strategy has excellent dynamic, static performance, and strong robustness. Main conclusions and future work are given in Section 5.


## 2. Mathematical Preliminaries

The model of PMSM and some definitions of fractional-order calculus are given in this section.For surface-mount PMSM, under a mechanical load torque, the equation of mechanical torque is expressed in (1) and the dynamics of PMSM is expressed in (2):(1)Te−TL=Jω˙t+Bωt,Te=1.5pnφfiqt,(2)ω˙t=3pnϕf2Jiqt−BJωt−1JTL,where Te, ϕf, pn, TL, J, B, ωt, and iqt represent the electromagnetic torque, flux linkage, number of pole pairs, load torque, rotational inertia, friction coefficient, mechanical rotor angular speed, and armature currents, respectively.Definition 1.
(see [44, 45]). The Riemann–Liouville fractional derivative and integral of function ft are given by(3)Dtεft=1Γ1−εddt∫t0tfτt−τεdτ,Dt−uft=1Γu∫t0tfτt−τ1−udτ,Γz=∫0∞e−ttz−1dt,where z∈R; Γ⋅ is Euler gamma function; and ε and u are the fractional derivative order and integral order, respectively; 0<ε,u<1.Lemma 1.
(see [44, 45]). The fractional-order integration and differentiation have the following properties:(4)dndtnDtεft=Dtε+nft,(5)dndtnDt−uft=Dtn−uft.Lemma 2.
(see [44, 45]). The fractional-order differentiation has the property of linear operation:(6)Dtεaft+bft=aDtεft+bDtεft.


## 3. Control Strategies and Stability Analysis

In this section, the control strategies and the stability analysis are given.


### 3.1. Design of Controllers

Definition 2.
(see [21–24]). The conventional FOPID sliding surface was defined as(7)st=Kpet+KiDt−uet+KdDtεet,where et=ωrt−ωt, ωrt is desired speed, and Kp∈R+, Ki∈R+, and Kd∈R+.
According to Lemmas1 and 2 and the exponential reaching law (ERL), it can be obtained that(8)Kpe˙t+KiDt1−uet+KdDtε+1et=−kst−ηsignst,where(9)k∈R+,η∈R+,signst=1,st>0,0,st=0,−1,st<0.
Substituting (2) into (8), the following can be derived:(10)Kpω˙rt+BJωt−3pnφf2Jiqt+TLJ+KiDt1−uet+KdDt1+εet=−kst−ηsignst.
Afterwards, the conventional SMC can be given as follows:(11)iqt=2J3pnφfKpKpω˙rt+TLJ+BJωt+KiDt1−uet+KdDt1+εet+kst+ηsignst.Definition 3.
The proposed FOPID sliding surface is given as(12)st=Kpet+KiDt−uΦet,τ+KdDtεΦet,τ,(13)Φet,τ=τ,et>τ,−1τet2+2et,τ≥et≥0,1τet2+2et,0≥et≥−τ,−τ,et<−τ,where Kp>0,Ki>0,andKd>0 and τ is a positive number.
Figure1 shows the corresponding function curve of the Φ⋅ with τ=1. Φ⋅ has the following characteristics [46]: it can be seen that, when et>τ and et<−τ, the error is limited to the value of ±τ and the function has the characteristic of the saturated error; when τ≥et≥0, the minimum value of the function y=Φet,τ−et is zero, that is, Φet,τ≥et is always satisfied; when 0≥et≥−τ, the maximum value of the function y=Φet,τ−et is zero, that is, Φet,τ≤et is always satisfied and the nonlinear function can amplify the error and make the system have better performance even if there is a small error.
The structure diagram of the sliding surface (12) is shown in Figure 2.Figure 1
Characteristic curves ofΦ⋅ with τ=1.Figure 2
Structure diagram of the proposed FOPID sliding surface.Remark 1.
Compared with conventional FOPID sliding surface, the proposed FOPID sliding mode surface with a nonlinear integral term and a nonlinear differential term makes it to effectively meet the control requirements of high quality.
According to Lemmas1 and 2, equations(2) and (12), and ERL, it can be obtained that(14)Kpe˙t+KiDt1−uΦet,τ+KdDtε+1Φet,τ=−kst−ηsignst.
Afterwards, the novel fractional-order sliding mode control (FOSMC) based on the proposed FOPID sliding surface and the ERL can be given as follows:(15)iqt=2J3pnϕfKpKpω˙rt+BJωt+TLJ+KiDt1−uΦet,τ+KdDt1+εΦet,τ+kst+ηsignst.
However, the ERL may produce the chattering phenomena in the system. In order to effectively solve the inherent chattering phenomenon, the following ST reaching law (STRL) is chosen in this paper [28]:(16)s˙t=−η1st1/2signst−η2∫signstdt.where η1 and η2 are the positive coefficients.
According to Lemmas1 and 2 and equations (12) and (16), the following equation can be obtained:(17)Kpe˙t+KiDt1−uΦet,τ+KdDtε+1Φet,τ=−η1st1/2signst−∫η2signstdt.
Afterwards, the proposed STFOSMC can be given as(18)iqt=2J3pnϕfKpKpω˙rt+TLJ+BJωt+KiDt1−uΦet,τ+KdDt1+εΦet,τ+η1st1/2signst+η2∫signstdt.
By combining the STFOSMC and a NLSEF, a compound NLSEF-STFOSMC is proposed.
An improved NLSEF is adopted in this paper. The improved NLSEF expression is established as [47](19)u^t=k^pfnewet,a,δ+k^dfnewe˙t,a,δ,(20)fnewx,α,δ=xαsignxx>δ,δα−1α−1−αδαsinδ1−cosδ−δsinδx+1−αδα1−cosδ−δsinδversinxsignxx≤δ,where 0<α,δ<1.
The functionfnewx,a,δ is a smooth and derivable function. A detailed description of the smooth and derivable function (20) can be found in the literature [47].
Integrating the improved NLSEF and the STFOSMC with respect to their effect, the proposed compound NLSEF-STFOSMC can be defined as(21)u¯t=iqt+u^t.


### 3.2. Design of Disturbance Observer

According to (2), define x1=ωt and x2=−TL/J. Equation (2) can be rewritten as(22)x˙1=x2−BJx1+3pnϕf2Jiqt,x˙2=ϑt.where ϑt can be regarded as a disturbance of x2.The expression of conventional nonlinear extended state observer (ESO) is as follows [35]:(23)et=Z21−x1,Z˙21=Z22−β1falet,δ˜,α˜+b0iqt,Z˙22=−β2falet,δ˜,α˜,where(24)falx,α˜,δ˜=xα˜signx,x>δ˜,xδ˜1−α˜,x≤δ˜,where 0<δ˜<1 and 0<α˜<1; β1>0 and β2>0; b0 is a compensation factor; and Z21 and Z22 are used to estimate the values of x1 and x2, respectively. However, the nonsmooth and nonderivable characteristics of the function fal⋅ can easily cause flutter phenomenon in the response of the system. Φ⋅ is a smooth and derivable function which reduces the flutter phenomenon, and when et is greater than τ, the output of Φ⋅ is a constant and the gain caused by the large error is reduced.The novel nonlinear ESO could then be expressed as follows:(25)et=Z21−ωt,Z˙21=Z22−β1Φet,τ+b0iqt,Z˙22=−β2Φet,τ.The largest disturbance of PMSM is external load disturbance, and other disturbances can be ignored [4]. The unknown disturbance −TL/J=Z22−Δdt, Δdt is the estimation error of the ESO.Remark 2.
Due to the application of the nonlinear saturation function, the high-frequency flutter phenomenon can be weakened significantly, and the error of observer estimation also can be reduced.
The traditional SMC method based on the novel ESO can be given as(26)iq1t=2J3pnϕfKpKpω˙rt−Z22+BJωt+KiDt1−uet+KdDt1+εet+kst+ηsignst.
The proposed FOSMC method based on the novel ESO can be given as(27)iq2t=2J3pnϕfKpKpω˙rt−Z22+BJωt+KiDt1−uΦet,τ+KdDt1+εΦet,τ+kst+ηsignst.
The proposed STFOSMC method based on the novel ESO can be given as(28)iq3t=2J3pnϕfKpKpω˙rt−Z22+BJωt+KiDt1−uΦet,τ+KdDt1+εΦet,τ+η1st1/2signst+η2∫signstdt.
Based on equations (19) and (28), the proposed compound NLSEF-STFOSMC method based on the novel ESO can be given as(29)i¯qt=iq3t+u^t.Remark 3.
Compared with other control strategies, the proposed compound NLSEF-STFOSMC method has the robustness of SMC and the high precision and fast convergence of NLSEF.
A novel PMSM speed regulation system based on the proposed compound NLSEF-STFOSMC (29) with the novel ESO (25) is shown in Figure 3.
The structure diagrams of the proposed FOSMC, the proposed STFOSMC, and the proposed compound NLSEF-STFOSMC are shown in Figures4–6, respectively.Figure 3
The configuration of the novel PMSM speed regulation system based on the proposed control strategy.Figure 4
Structure diagram of the proposed FOSMC.Figure 5
Structure diagram of the proposed STFOSMC.Figure 6
Structure diagram of the proposed compound NLSEF-STFOSMC.


### 3.3. Stability Analysis

The stability of the proposed FOSMC system is discussed.Proof.
Step 1: the Lyapunov function for the proposed FOSMC is defined as(30)V=12st2>0.Taking derivative of (30), we get(31)V˙=st⋅s˙t=stKpe˙t+KiDt1−uΦet,τ+KdDtε+1Φet,τ=stKpω˙rt+BJωt−3pnϕf2Jiq2t+TLJ+KiDt1−uΦet,τ+KdDtε+1Φet,τ=st−kst−ηsignst+KpΔdt=−kst2−signstηst+stKpΔdt=−kst2−ηst+stKpΔdt≤−kst2−ηst+stKpΔdt.Assumption 1.
The value ofΔdt has a boundary layer, which satisfies the following condition:(32)Δdt<Ψ,whereΨ∈R+.Ifη>KpΨ is satisfied, it can be obtained that(33)V˙≤−kst2−ηst+stKpΨ<0.Accordingly, we can easily conclude that the sliding surfacest will be bounded as st≤ℓ. ℓ is a small neighborhood of zero.Step 2: according tost≤ℓ, the following formula can be obtained:(34)st=Kpet+KiDt−uΦet,τ+KdDtεΦet,τ,st≤ℓ.Using (34), it can be obtained that(35)et+KiKpDt−uΦet,τ+KdKp−1KpsDtεΦet,τ−1DtεΦet,τ=0.
WhenKd−sDtεΦet,τ−1>0 holds, (35) will still keep the proposed FOPID sliding surface form as (12). Therefore, According to (33) and (34), the trajectory of the system will continue to converge to the proposed FOPID sliding surface until it reaches DtεΦet,τ≤1/Kdℓ.
Using (34) and (36), we get:(36)et+KiKp−1KpstDt−uΦet,τ−1Dt−uΦet,τ+KdKpDtεΦet,τ=0.
So,Dt−uΦet,τ≤1/Kiℓ also can be obtained.
Sequentially, it can be obtained that(37)et=1Kpst−KiKpDt−uΦet,τ−KdKpDtεΦet,τ≤1Kpst+KiKpDt−uΦet,τ+KdKpDtεΦet,τ≤3ℓKp.
According to (37), the control error of the proposed FOSMC can converge to a small neighborhood of zero, and the stable condition has been achieved.
The stability of the proposed STFOSMC system is discussed.Proof.
Step 1: the Lyapunov function for STFOSMC is defined as(38)V=12st2>0.Taking derivative of (38), it can be obtained that(39)V˙=st⋅s˙t=stKpe˙t+KiDt1−uΦet,τ+KdDtε+1Φet,τ=stKpω˙rt+BJωt−3pnφf2Jiq3t+TLJ+KiDt1−uΦet,τ+KdDtε+1Φet,τ=st−η1st1/2signst−∫η2signstdt+KpΔdt≤−η1stst1/2−st∫η2dt+Kpst∫Δd˙tdt.Assumption 2.
The derivative of the value ofΔdt has an upper limit, which satisfies(40)Δd˙t<Θ,whereΘ is the upper limit of the derivative of the value of Δdt, and it is a constant.According to Assumption2 and if KpΘ<η2 is satisfied, it can be obtained that(41)V˙<−η1stst1/2−st∫η2dt+Kpst∫Θdt=−η1stst1/2−st∫η2dt−Kp∫Θdt<η1stst1/2≤0.Accordingly, we can easily conclude that the sliding surfacest will be bounded as st≤ℓ.Step 2: we use the similar procedure from the step 2 of the proof of the FOSMC. It can be obtained that(42)et=1Kpst−KiKpDt−uΦet,τ−KdKpDtεΦet,τ≤1Kpst+KiKpDt−uΦet,τ+KdKpDtεΦet,τ≤3ℓKp.According to (42), the control error of the proposed STFOSMC can converge to a small neighborhood of zero, and the stable condition has been achieved.The stability of the proposed compound NLSEF-STFOSMC system is discussed.Proof.
Step 1: the Lyapunov function for the compound NLSEF-STFOSMC is defined as(43)V=12st2>0.Taking derivative of (43), it can be obtained that(44)V˙=st⋅s˙t=stKpe˙t+KiDt1−uΦet,τ+KdDtε+1Φet,τ=stKpω˙rt+BJωt−3pnϕf2Ji¯qt+TLJ+KiDt1−uΦet,τ+KdDtε+1Φet,τ=st−η1st1/2signst−∫η2signstdt−Kpψu^t+KpΔdt≤−η1stst1/2−st∫η2dt+stKpψu^t+KpstΔdt=−η1stst1/2−st∫η2dt+Kpψst∫u^˙tdt+Kpst∫Δd˙tdt,whereψ=3pnϕf/2J.Assumption 3.
According to the continuous and derivable function (20), the derivative of the value of fnewx,a,δ has a boundary layer, which satisfies the following condition:(45)f˙newx,a,δ<Λ,whereΛ is the upper limit of the derivative of the value of fnewx,a,δ, and it is a constant.According to Assumption3 and if KpψΛk^p+k^d+KpΘ<η2 is satisfied, one can obtain(46)V˙<−η1stst1/2−st∫η2dt+Kpψst∫k^p+k^dΛdt+Kpst∫Θdt=−η1stst1/2−st∫η2dt−Kpψk^p∫Λdt−Kpψk^d∫Λdt−Kp∫Θdt<−η1stst1/2≤0.Accordingly, the sliding surfacest will be bounded in finite time as st≤ℓ.Step 2: the similar procedure from the step 2 of the proof of the FOSMC can be used.Sequentially, it can be obtained that(47)et=1Kpst−KiKpDt−uΦet,τ−KdKpDtεΦet,τ≤1Kpst+KiKpDt−uΦet,τ+KdKpDtεΦet,τ≤3ℓKp.According to (47), the control error of the proposed compound NLSEF-STFOSMC can converge to a small neighborhood of zero, and the stable condition has been achieved.


## 3.1. Design of Controllers

Definition 2.
(see [21–24]). The conventional FOPID sliding surface was defined as(7)st=Kpet+KiDt−uet+KdDtεet,where et=ωrt−ωt, ωrt is desired speed, and Kp∈R+, Ki∈R+, and Kd∈R+.
According to Lemmas1 and 2 and the exponential reaching law (ERL), it can be obtained that(8)Kpe˙t+KiDt1−uet+KdDtε+1et=−kst−ηsignst,where(9)k∈R+,η∈R+,signst=1,st>0,0,st=0,−1,st<0.
Substituting (2) into (8), the following can be derived:(10)Kpω˙rt+BJωt−3pnφf2Jiqt+TLJ+KiDt1−uet+KdDt1+εet=−kst−ηsignst.
Afterwards, the conventional SMC can be given as follows:(11)iqt=2J3pnφfKpKpω˙rt+TLJ+BJωt+KiDt1−uet+KdDt1+εet+kst+ηsignst.Definition 3.
The proposed FOPID sliding surface is given as(12)st=Kpet+KiDt−uΦet,τ+KdDtεΦet,τ,(13)Φet,τ=τ,et>τ,−1τet2+2et,τ≥et≥0,1τet2+2et,0≥et≥−τ,−τ,et<−τ,where Kp>0,Ki>0,andKd>0 and τ is a positive number.
Figure1 shows the corresponding function curve of the Φ⋅ with τ=1. Φ⋅ has the following characteristics [46]: it can be seen that, when et>τ and et<−τ, the error is limited to the value of ±τ and the function has the characteristic of the saturated error; when τ≥et≥0, the minimum value of the function y=Φet,τ−et is zero, that is, Φet,τ≥et is always satisfied; when 0≥et≥−τ, the maximum value of the function y=Φet,τ−et is zero, that is, Φet,τ≤et is always satisfied and the nonlinear function can amplify the error and make the system have better performance even if there is a small error.
The structure diagram of the sliding surface (12) is shown in Figure 2.Figure 1
Characteristic curves ofΦ⋅ with τ=1.Figure 2
Structure diagram of the proposed FOPID sliding surface.Remark 1.
Compared with conventional FOPID sliding surface, the proposed FOPID sliding mode surface with a nonlinear integral term and a nonlinear differential term makes it to effectively meet the control requirements of high quality.
According to Lemmas1 and 2, equations(2) and (12), and ERL, it can be obtained that(14)Kpe˙t+KiDt1−uΦet,τ+KdDtε+1Φet,τ=−kst−ηsignst.
Afterwards, the novel fractional-order sliding mode control (FOSMC) based on the proposed FOPID sliding surface and the ERL can be given as follows:(15)iqt=2J3pnϕfKpKpω˙rt+BJωt+TLJ+KiDt1−uΦet,τ+KdDt1+εΦet,τ+kst+ηsignst.
However, the ERL may produce the chattering phenomena in the system. In order to effectively solve the inherent chattering phenomenon, the following ST reaching law (STRL) is chosen in this paper [28]:(16)s˙t=−η1st1/2signst−η2∫signstdt.where η1 and η2 are the positive coefficients.
According to Lemmas1 and 2 and equations (12) and (16), the following equation can be obtained:(17)Kpe˙t+KiDt1−uΦet,τ+KdDtε+1Φet,τ=−η1st1/2signst−∫η2signstdt.
Afterwards, the proposed STFOSMC can be given as(18)iqt=2J3pnϕfKpKpω˙rt+TLJ+BJωt+KiDt1−uΦet,τ+KdDt1+εΦet,τ+η1st1/2signst+η2∫signstdt.
By combining the STFOSMC and a NLSEF, a compound NLSEF-STFOSMC is proposed.
An improved NLSEF is adopted in this paper. The improved NLSEF expression is established as [47](19)u^t=k^pfnewet,a,δ+k^dfnewe˙t,a,δ,(20)fnewx,α,δ=xαsignxx>δ,δα−1α−1−αδαsinδ1−cosδ−δsinδx+1−αδα1−cosδ−δsinδversinxsignxx≤δ,where 0<α,δ<1.
The functionfnewx,a,δ is a smooth and derivable function. A detailed description of the smooth and derivable function (20) can be found in the literature [47].
Integrating the improved NLSEF and the STFOSMC with respect to their effect, the proposed compound NLSEF-STFOSMC can be defined as(21)u¯t=iqt+u^t.


## 3.2. Design of Disturbance Observer

According to (2), define x1=ωt and x2=−TL/J. Equation (2) can be rewritten as(22)x˙1=x2−BJx1+3pnϕf2Jiqt,x˙2=ϑt.where ϑt can be regarded as a disturbance of x2.The expression of conventional nonlinear extended state observer (ESO) is as follows [35]:(23)et=Z21−x1,Z˙21=Z22−β1falet,δ˜,α˜+b0iqt,Z˙22=−β2falet,δ˜,α˜,where(24)falx,α˜,δ˜=xα˜signx,x>δ˜,xδ˜1−α˜,x≤δ˜,where 0<δ˜<1 and 0<α˜<1; β1>0 and β2>0; b0 is a compensation factor; and Z21 and Z22 are used to estimate the values of x1 and x2, respectively. However, the nonsmooth and nonderivable characteristics of the function fal⋅ can easily cause flutter phenomenon in the response of the system. Φ⋅ is a smooth and derivable function which reduces the flutter phenomenon, and when et is greater than τ, the output of Φ⋅ is a constant and the gain caused by the large error is reduced.The novel nonlinear ESO could then be expressed as follows:(25)et=Z21−ωt,Z˙21=Z22−β1Φet,τ+b0iqt,Z˙22=−β2Φet,τ.The largest disturbance of PMSM is external load disturbance, and other disturbances can be ignored [4]. The unknown disturbance −TL/J=Z22−Δdt, Δdt is the estimation error of the ESO.Remark 2.
Due to the application of the nonlinear saturation function, the high-frequency flutter phenomenon can be weakened significantly, and the error of observer estimation also can be reduced.
The traditional SMC method based on the novel ESO can be given as(26)iq1t=2J3pnϕfKpKpω˙rt−Z22+BJωt+KiDt1−uet+KdDt1+εet+kst+ηsignst.
The proposed FOSMC method based on the novel ESO can be given as(27)iq2t=2J3pnϕfKpKpω˙rt−Z22+BJωt+KiDt1−uΦet,τ+KdDt1+εΦet,τ+kst+ηsignst.
The proposed STFOSMC method based on the novel ESO can be given as(28)iq3t=2J3pnϕfKpKpω˙rt−Z22+BJωt+KiDt1−uΦet,τ+KdDt1+εΦet,τ+η1st1/2signst+η2∫signstdt.
Based on equations (19) and (28), the proposed compound NLSEF-STFOSMC method based on the novel ESO can be given as(29)i¯qt=iq3t+u^t.Remark 3.
Compared with other control strategies, the proposed compound NLSEF-STFOSMC method has the robustness of SMC and the high precision and fast convergence of NLSEF.
A novel PMSM speed regulation system based on the proposed compound NLSEF-STFOSMC (29) with the novel ESO (25) is shown in Figure 3.
The structure diagrams of the proposed FOSMC, the proposed STFOSMC, and the proposed compound NLSEF-STFOSMC are shown in Figures4–6, respectively.Figure 3
The configuration of the novel PMSM speed regulation system based on the proposed control strategy.Figure 4
Structure diagram of the proposed FOSMC.Figure 5
Structure diagram of the proposed STFOSMC.Figure 6
Structure diagram of the proposed compound NLSEF-STFOSMC.


## 3.3. Stability Analysis

The stability of the proposed FOSMC system is discussed.Proof.
Step 1: the Lyapunov function for the proposed FOSMC is defined as(30)V=12st2>0.Taking derivative of (30), we get(31)V˙=st⋅s˙t=stKpe˙t+KiDt1−uΦet,τ+KdDtε+1Φet,τ=stKpω˙rt+BJωt−3pnϕf2Jiq2t+TLJ+KiDt1−uΦet,τ+KdDtε+1Φet,τ=st−kst−ηsignst+KpΔdt=−kst2−signstηst+stKpΔdt=−kst2−ηst+stKpΔdt≤−kst2−ηst+stKpΔdt.Assumption 1.
The value ofΔdt has a boundary layer, which satisfies the following condition:(32)Δdt<Ψ,whereΨ∈R+.Ifη>KpΨ is satisfied, it can be obtained that(33)V˙≤−kst2−ηst+stKpΨ<0.Accordingly, we can easily conclude that the sliding surfacest will be bounded as st≤ℓ. ℓ is a small neighborhood of zero.Step 2: according tost≤ℓ, the following formula can be obtained:(34)st=Kpet+KiDt−uΦet,τ+KdDtεΦet,τ,st≤ℓ.Using (34), it can be obtained that(35)et+KiKpDt−uΦet,τ+KdKp−1KpsDtεΦet,τ−1DtεΦet,τ=0.
WhenKd−sDtεΦet,τ−1>0 holds, (35) will still keep the proposed FOPID sliding surface form as (12). Therefore, According to (33) and (34), the trajectory of the system will continue to converge to the proposed FOPID sliding surface until it reaches DtεΦet,τ≤1/Kdℓ.
Using (34) and (36), we get:(36)et+KiKp−1KpstDt−uΦet,τ−1Dt−uΦet,τ+KdKpDtεΦet,τ=0.
So,Dt−uΦet,τ≤1/Kiℓ also can be obtained.
Sequentially, it can be obtained that(37)et=1Kpst−KiKpDt−uΦet,τ−KdKpDtεΦet,τ≤1Kpst+KiKpDt−uΦet,τ+KdKpDtεΦet,τ≤3ℓKp.
According to (37), the control error of the proposed FOSMC can converge to a small neighborhood of zero, and the stable condition has been achieved.
The stability of the proposed STFOSMC system is discussed.Proof.
Step 1: the Lyapunov function for STFOSMC is defined as(38)V=12st2>0.Taking derivative of (38), it can be obtained that(39)V˙=st⋅s˙t=stKpe˙t+KiDt1−uΦet,τ+KdDtε+1Φet,τ=stKpω˙rt+BJωt−3pnφf2Jiq3t+TLJ+KiDt1−uΦet,τ+KdDtε+1Φet,τ=st−η1st1/2signst−∫η2signstdt+KpΔdt≤−η1stst1/2−st∫η2dt+Kpst∫Δd˙tdt.Assumption 2.
The derivative of the value ofΔdt has an upper limit, which satisfies(40)Δd˙t<Θ,whereΘ is the upper limit of the derivative of the value of Δdt, and it is a constant.According to Assumption2 and if KpΘ<η2 is satisfied, it can be obtained that(41)V˙<−η1stst1/2−st∫η2dt+Kpst∫Θdt=−η1stst1/2−st∫η2dt−Kp∫Θdt<η1stst1/2≤0.Accordingly, we can easily conclude that the sliding surfacest will be bounded as st≤ℓ.Step 2: we use the similar procedure from the step 2 of the proof of the FOSMC. It can be obtained that(42)et=1Kpst−KiKpDt−uΦet,τ−KdKpDtεΦet,τ≤1Kpst+KiKpDt−uΦet,τ+KdKpDtεΦet,τ≤3ℓKp.According to (42), the control error of the proposed STFOSMC can converge to a small neighborhood of zero, and the stable condition has been achieved.The stability of the proposed compound NLSEF-STFOSMC system is discussed.Proof.
Step 1: the Lyapunov function for the compound NLSEF-STFOSMC is defined as(43)V=12st2>0.Taking derivative of (43), it can be obtained that(44)V˙=st⋅s˙t=stKpe˙t+KiDt1−uΦet,τ+KdDtε+1Φet,τ=stKpω˙rt+BJωt−3pnϕf2Ji¯qt+TLJ+KiDt1−uΦet,τ+KdDtε+1Φet,τ=st−η1st1/2signst−∫η2signstdt−Kpψu^t+KpΔdt≤−η1stst1/2−st∫η2dt+stKpψu^t+KpstΔdt=−η1stst1/2−st∫η2dt+Kpψst∫u^˙tdt+Kpst∫Δd˙tdt,whereψ=3pnϕf/2J.Assumption 3.
According to the continuous and derivable function (20), the derivative of the value of fnewx,a,δ has a boundary layer, which satisfies the following condition:(45)f˙newx,a,δ<Λ,whereΛ is the upper limit of the derivative of the value of fnewx,a,δ, and it is a constant.According to Assumption3 and if KpψΛk^p+k^d+KpΘ<η2 is satisfied, one can obtain(46)V˙<−η1stst1/2−st∫η2dt+Kpψst∫k^p+k^dΛdt+Kpst∫Θdt=−η1stst1/2−st∫η2dt−Kpψk^p∫Λdt−Kpψk^d∫Λdt−Kp∫Θdt<−η1stst1/2≤0.Accordingly, the sliding surfacest will be bounded in finite time as st≤ℓ.Step 2: the similar procedure from the step 2 of the proof of the FOSMC can be used.Sequentially, it can be obtained that(47)et=1Kpst−KiKpDt−uΦet,τ−KdKpDtεΦet,τ≤1Kpst+KiKpDt−uΦet,τ+KdKpDtεΦet,τ≤3ℓKp.According to (47), the control error of the proposed compound NLSEF-STFOSMC can converge to a small neighborhood of zero, and the stable condition has been achieved.


## 4. Comparative Simulations

To verify the effectiveness of the proposed FOSMC, STFOSMC, and NLSEF-STFOSMC, we implement the comparative simulations based on the Lenovo Ideapad 320s, Intel(R) Core i7 7500U, CPU @2.70 GHz, RAM 8 GB, 1T BHDD, NVIDIA GeForce 920MX + IntelGMAHD620, 64 bit operating system, Matlab (R2014a)/Simulink.Tables1–6 show the parameters of the PMSM, traditional SMC, FOSMC, STFOSMC, NLSEF-STFOSMC, novel ESO, and traditional ESO, respectively.Table 1
Parameters of the PMSM.
Stator phase resistance (Rs)Inductance (L)Magnetic chain of permanent magnets (φf)Moment of inertia (J)Viscous damping (B)Pole pairs (pn)Power supply voltage (Vs)0.958Ω5.25mH0.1827Wb0.009kg⋅m20.008N⋅m⋅s/rad4400VTable 2
Parameters of the traditional SMC.
KpKiKdεukη0.10.30.30.010.012020Table 3
Parameters of the FOSMC.
KpKiKdεukητ0.10.30.30.010.0120201Table 4
Parameters of the STFOSMC.
KpKiKdεuνη1η2τ0.10.30.30.010.010.05300201Table 5
Parameters of the compound NLSEF-STFOSMC.
KpKiKdεuνη1η2τδαk^pk^d0.10.30.30.010.010.053002010.10.25101Table 6
Parameters of the novel ESO and the traditional ESO.
Traditional ESONovel ESOβ1β2b0α˜δ˜τβ1β2b020001500001110.250.112000150000111Figure7 demonstrates the time responses of the sliding surfaces under the traditional SMC, the proposed FOSMC, the proposed STFOSMC, and the proposed compound NLSEF-STFOSMC. Figure 7(a) shows that the time response of the novel FOPID sliding surface is closer to s=0 than the time response of the traditional FOPID sliding surface. Furthermore, it is clear that the reaching time under the proposed STFOSMC and the proposed compound NLSEF-STFOSMC is shorter than the reaching time under the FOSMC and the traditional SMC. Figure 7(b) reveals that the time response curve of the sliding surface under the proposed compound NLSEF-STFOSMC is the smoothest of the four sliding mode surfaces.Figure 7
(a) Time response curves of the sliding functions under the traditional SMC, the proposed FOSMC, the proposed STFOSMC, and the proposed compound NLSEF-STFOSMC. (b) Partial enlarged views of the image on the left.
(a)(b)The reference speed is set to 100 rad/s at 0s, and the motor starts with3N⋅m load torque. Figure 8 shows the estimated disturbance results of the novel ESO and the traditional ESO. Due to the application of the nonlinear saturation function, the high-frequency flutter phenomenon is weakened significantly and the error of observer estimation is also reduced.Figure 8
(a)Z22 curves under the novel ESO and the traditional ESO. (b) Partial enlarged views of the image on the left.
(a)(b)Figures9 and 10 show the speed response curves and iqt current curves, respectively. Table 7 shows the speed adjustment time with load start-up of the motor under the four control strategies. From Figure 9(a) and Table 7, the results show that the traditional SMC has longer adjustment time than those of the proposed STFOSMC, the proposed FOSMC, and the proposed compound NLSEF-STFOSMC. The proposed compound NLSEF-STFOSMC has shorter adjustment time than the other control strategies. Figure 9(a) shows that the proposed compound NLSEF-STFOSMC exhibits better dynamic performance than the other control strategies. From Figure 9(b), it is obvious that the proposed three strategies have better tracking performances and that the tracking errors in steady state are smaller than those of the traditional SMC. Figure 10 shows that the chattering phenomenon of the traditional SMC, the FOSMC, and the STFOSMC is significantly larger than that of the compound NLSEF-STFOSMC. Noticeably, the comparison results obviously show that the proposed compound NLSEF-STFOSMC has excellent dynamic and static performance.Figure 9
(a) Speed response curves under the SMCs withωr=100. (b) Partial enlarged views of the image on the left.
(a)(b)Figure 10
iq curves under the SMCs.Table 7
The comparative results of the start-up.
Control strategyReference speed (rad/s)Adjustment time (s)The traditional SMC1000.186The proposed FOSMC1000.124The proposed STFOSMC1000.021The proposed compound NLSEF-STFOSMC1000.019The simulation results compare the robustness of the proposed STFOSMC, the traditional SMC, the proposed FOSMC, and the proposed compound NLSEF-STFOSMC under the external disturbance. In the first simulation, when the external load becomes 0N⋅m from 6 N⋅m at 1 s, the corresponding curves of the speed of the different SMC strategies are shown in Figure 11. In the second simulation, when the external load becomes 6 N⋅m from 0 N⋅m at 1 s, the corresponding curves of the speed of the different SMC strategies are shown in Figure 12. In order to better compare the robustness of the four control strategies, speed perturbation and speed recovery time are compared in Table 8. It can be seen from Figures 11–12 and Table 8 that the four SMC strategies have experienced different levels of impact caused by the external load disturbances. The simulation results obvious demonstrate that the proposed compound NLSEF-STFOSMC exhibits the shortest speed recovery time and the smallest speed perturbation. Therefore, the simulation results demonstrate that the proposed compound NLSEF-STFOSMC has stronger robustness than other control strategies.Figure 11
(a) Speed response curves when load suddenly increases at 1s. (b) Partial enlarged views.
(a)(b)Figure 12
(a) Speed response curves when load suddenly decreases at 1s. (b) Partial enlarged views.
(a)(b)Table 8
The comparative results of the sudden load change.
Control strategyThe external load becomes 0N⋅m from 6 N⋅mThe external load becomes 6N⋅m from 0 N⋅mSpeed perturbation (%)Speed recovery time (s)Speed perturbation (%)Speed recovery time (s)The traditional SMC0.710.0380.780.071The proposed FOSMC0.420.0360.520.070The proposed STFOSMC0.180.0080.080.013The proposed compound NLSEF-STFOSMC0.160.0050.070.011


## 5. Conclusions and Future Work

This study successfully presents a novel compound NLSEF-STFOSMC for the speed control of PMSM servo drive based on a novel ESO. Firstly, a novel FOPID switching manifold with a nonlinear integral term and a nonlinear differential term is designed. An STFOSMC involving the novel FOPID switching manifold and an ST reaching law is successfully suggested. Secondly, a novel nonlinear ESO is designed, which can accurately estimate the external disturbance without the high-frequency flutter phenomenon. Thirdly, a compound NLSEF-STFOSMC is proposed combining the advantages of the modified SMC and the NLSEF. Lyapunov approach and Fractional calculus are used to show the stability for the system. Finally, the comparison results show that the proposed hybrid control strategy has excellent dynamic and static performance and strong robustness.In the future, the proposed compound NLSEF-STFOSMC strategy will be further verified by the cSPACE semiphysical control system based on TI TMS320F28335 DSP and Matlab/Simulink. Figure13 shows the PMSM verification platform.Figure 13
The PMSM verification platform.



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*Source: 1843598-2020-06-24.xml*