# Disturbance Observer-Based Complementary Fractional-Order Sliding Mode Control for PMSM Drive System

Authors: Yong-Hong Lan; Li-Tao Zheng; Zhao-Hong Wang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, a disturbance observer-based complementary fractional-order sliding mode control (CFOSMC) scheme is proposed for the permanent magnet synchronous motor (PMSM) drive system. First, to reconstruct the load disturbance and parameter variations, a nonlinear disturbance observer is designed. Next, a disturbance observer-based fractional-order sliding mode with a saturation function control law is designed to reduce the chattering problem in the existing fractional-order sliding mode control (FOSMC) method. Furthermore, to reduce the thickness of the boundary layer, a CFOSMC scheme is designed. By using the fractional-order Lyapunov stability theorem, the existence condition and the chattering problem are analyzed. Compared with the existing FOSMC, the obtained CFOSMC law does not contain any high-order derivatives of tracking error, which is easier to implement. Finally, the numerical simulations and experimental results are provided to show the superiority of the proposed method. To improve the performance of the permanent magnet synchronous motor (PMSM) drive system in terms of tracking rapidity, accuracy, and robustness, a complementary fractional-order sliding mode control (CFOSMC) scheme with disturbance observer is proposed in this paper.


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## Body

## 1. Introduction

Permanent magnet synchronous motor (PMSM) has many applications in industries due to its excellent features such as superpower density, high torque to current ratio, fast response, and low noise [1, 2]. However, the PMSM is a typical multivariable coupled high nonlinear system, and its performance is sensitive to external load disturbances and parameter uncertainties. Over the last decades, various design methods have been developed, such as robust control [3], predictive control [4], adaptive control [5], and neural network control [6].Recently, to improve the control performances in PMSM diver systems, much attention has been given to the disturbance observer-based (DOB) control method. In [7], a DOB state feedback controller was designed for the PMSM system. On the basis of this paper, a sensorless control method for PMSM drive was developed in [8]. In [9], a generalized predictive current control method combined with sliding mode disturbance compensation was proposed to satisfy the requirement of fast response and strong robustness. By combining the adaptive sliding mode control with sliding mode disturbance observer, a hybrid control strategy was proposed in [10]. In [11], a new estimation method of sensor faults and unknown disturbance in current measurement circuits for the PMSM drive system was presented. However, the above-proposed DOB control schemes only focus on the load torque disturbance but lack the estimation of parameter uncertainties.To counteract the disturbances and uncertainties, sliding mode control (SMC) is a powerful nonlinear control technique, which has been widely used for speed and position control of the PMSM system [12]. To estimate the immeasurable mechanical parameters of PMSM, an intelligent second-order SMC using a wavelet fuzzy neural network estimator was proposed in [13]. Considering the time-varying characteristic and the high-bandwidth property of the uncertainties and disturbances in a PMSM drive system, a disturbance observer-based SMC scheme was introduced in [14]. To estimate the online stator resistance, a new sliding mode observer for sensorless SMC was presented in [15]. To further improve the performance of the SMC, a robust SMC scheme based on a rapid nonlinear tracking differentiator was proposed in [16]. The second-order SMC design problem was discussed in [17], in which a novel saturation function was used. Based on the conventional SMC, adding a generalized error sliding surface, the complementary sliding mode control (CSMC) was proposed for the fault-tolerant control of a six-phase PMSM drive system in [18]. The complementary sliding mode control method based on Elman neural network was proposed in [19], which can not only reduce the system state to the sliding surface time but also guarantee the system tracking accuracy. Considering that the control system in a PMSM has uncertainties and disturbances, in [20], an adaptive switching gain was proposed. Comparisons of experimental results show that the proposed method has a faster adjustment process than SMC. In [21], a disturbance observer-based CSMC design method was proposed for the PMSM control system of the mine traction electric locomotive.As we all know, one obvious disadvantage of the SMC method is the chattering phenomenon caused by discontinuous control law and frequent switching action near the sliding surface. Besides, the upper bound of lumped disturbances is not easy to be determined, which could worsen the chattering phenomenon of SMC strategy. Recently, to overcome these drawbacks, some fractional-order sliding mode control (FOSMC) schemes have been widely adopted to weaken the chattering phenomenon and deal with external disturbances. An FOSMC scheme based on parameters auto-tuning for the velocity control of PMSM was proposed in [22]. A robust FOSMC was proposed for the position control of PMSM in [23]. By selecting a proper fractional-order and designing a fractional-order sliding surface, the proposed FOSMC is distinctly more excellent than that of the conventional SMC. To investigate the position regulation problem of PMSM subject to parameter uncertainties and external disturbances, an FOSMC was proposed and the finite-time stability of the closed-loop system was obtained in [24]. Based on the fractional stability theory, a sliding mode control scheme for synchronization of fractional PMSM was developed in [25]. A novel FOSMC for a class of integer-order systems with mismatched disturbances was proposed in [26]. In [27], a disturbance observer-based composite supertwisting sliding mode control was designed for the PMSM speed regulation problem. To improve the convergence rate over the existing sliding mode control method for the trajectory tracking control, an adaptive fast nonsingular integral terminal SMC method was proposed in [28, 29].Note that the above SMC-based control methods for the PMSM driver system can deal with mismatched disturbance and have better control performance with faster response speed, lower overshoot, and less chattering effect than the traditional control strategy. However, the high-order derivative of the tracking error (or the reference signal) was usually used in the sliding model controller [19, 20, 22]. From a practical point of view, it is difficult to implement. On the other hand, there is still room for improvement when it comes to the chattering phenomenon in classic FOSMC. This observation motivates our current study. In this paper, we mainly investigate a modified FOSMC scheme with a load disturbance observer for the speed control of PMSM. The main contributions are as follows: (1) to estimate the load disturbance and parameter uncertainties, a nonlinear disturbance observer is constructed and the asymptotic stability condition for observation error is also obtained; (2) to reduce the chattering phenomenon in the existing FOSMC method, a disturbance observer-based fractional-order sliding mode with saturation function control law is designed; and (3) to reduce the thickness of boundary layer, a CFOSMC law with disturbance observer is proposed. The tracking performance and robustness of the proposed method are also analyzed and compared with the conventional FOSMC scheme.The rest of this paper is organized as follows: in Section2, the mathematic model of PMSM and problem formulation are presented. The nonlinear disturbance observer design and stability analysis are derived in Section 3. In Section 4, the conventional FOSMC for the PMSM drive system is improved and the disturbance observer-based CFOSMC method is derived. The effectiveness of the proposed algorithm is illustrated in Section 5 through numerical simulations and experiment examples. Finally, some conclusions are drawn in Section 6.


## 2. Mathematical Model of PMSM and Problem Formulation

The mathematics model of a PMSM can be described in the rotor rotating reference frame as follows [10, 22, 23]:(1)uq∗=Rsiq∗+λ˙q+ωfλd,ud∗=Rsid∗+λ˙dq+ωfλq,λq=Lqiq∗,λd=Ldid∗+LmIdf,ωf=npωr∗,where ud∗,uq∗ are the d,q-axis stator voltages; id∗,iq∗ are the d,q-axis stator currents; λd,λq are the d,q-axis stator flux linkages; and Ld,Lq are the d,q-axis stator inductances. While ωf and ωr∗ are the inverter frequency and rotor speed, respectively, Lm is the d-axis mutual inductance; Idf is the equivalent d-axis magnetizing current; np is the number of pole pairs; and Rs is the stator resistance.The electric torque is stated as(2)Te=3npLmIdfiq∗+Ld−Lqiq∗id∗2.Motor dynamics is presented as(3)Te=Jω˙r+Bmωr+Tl,where Tl is the load torque, Bm is the viscous friction coefficient, and J is the moment of inertia.By using the field-oriented mechanism withid=0 [10, 22], we can simplify the electric torque as(4)Te=kp∗iq∗=3npLmIdf2iq∗.Substituting (3) into (4), one can obtain the state equation of servo drive:(5)ω˙r=−aωr+biq−c,a=BmJ,b=kp∗J,c=TlJ.Considering the uncertainties and time-invariant parameters, we can rewrite (5) as(6)ω˙r=−a+Δaωr+b+Δbiq−c+Δc,where Δa,Δb,Δc are the time-invariant parameters.The tracking erroret, in terms of the desired reference speed ωr∗t and the measured actual output speed ωrt, is defined as(7)et=ωr∗t−ωrt.The time derivative ofet is(8)e˙t=−aet−biqt+ϕt+d,ϕt=aωr∗t+c+ω˙r∗t,d=Δaωrt−Δbiqt+Δc,where d is time-variant uncertainties such as mechanical parameter variations, friction force, load disturbances, speed distortion, and current harmonics. It is assumed that(9)d≤Λ>0,limt⟶∞d˙=d∗,where d∗ is a constant.In this paper, the control objective is to find a suitable FOSMC inputiq∗t such that the output speed ωrt can track the desired reference speed ωr∗t asymptotically in the presence of any arbitrary initial conditions and uncertainties.


## 3. Disturbance Observer Design

To estimate time-variant lumped uncertainties and load torque disturbance, we use the following nonlinear disturbance observer [30]:(10)p˙t=−ld^+laet+biqt−ϕt,d^=pt+let,where d^ is the estimation of d and l>0 is the observer gain.Estimation error of the disturbance observer is defined as(11)d˜=d^−d,and error dynamics of disturbance observer is governed by(12)d˜˙=d^˙−d˙.It can be derived from (10)–(12) that(13)d˜˙=−ld˜−d˙.Sincel>0 and limt⟶∞d˙=d∗, we have limt⟶∞d˜=−d∗/l.Now, we give the following result.Lemma 1.
For nonlinear system (8), suppose that the disturbance observer is formulated as (10); then, the disturbance estimation error d˜ is bounded.Remark 1.
It follows from Lemma1 that the observer gain l can change the bound of the disturbance estimation error. Clearly, it should be selected as large enough.


## 4. Disturbance Observer-Based CFOSMC Scheme Design

Considering that the disturbance observer (10) can estimate the load torque disturbance accurately, we will replace d with d^ in the following synthesis.


### 4.1. Traditional FOSMC

Fractional calculus is a generalization of integer-order integration and differentiation to noninteger-order ones. LetDα denote the fractional-order derivative, which is defined as in Definition 1.Definition 1 (see [31]).
The Caputo derivative is defined by(14)Dαft=1Γn−α∫t0tt−τ−α+n−1fnτdτ,where n is the first integer which is not less than α, that is, α∈n−1,n. Γ⋅ is the well-known Gamma function, which is defined by Γz=∫0∞e−ztz−1dt.Definition 2 (see [31]).
The definition of the fractional integral is described by(15)D−αft=1Γα∫t0tt−τα−1fτdτ,α>0.
To proceed with the discussion, the following lemmas will be used.Lemma 2 (see [32]).
Autonomous system is as follows:(16)Dαxt=Axt,with xt0=x0 and 0<α<1, is asymptotically stable if and only if argspecA>απ/2, where spec(A) is the spectrum (set of all eigenvalues) of A. Also, the state vector xt decays towards 0 and meets the condition xt<Mt−α,t>0,M>0.Lemma 3 (see [33]).
Letxt∈ℝn be a vector of a differentiable function. Then, for any time instant t≥t0, the following relationship holds:(17)12DαxTtPxt≤xTtPDαxt,∀α∈0,1,where P∈ℝn×n is a positive definite matrix.Lemma 4 (see [34]).
Letxt=0 be an equilibrium point for the nonautonomous fractional-order system Dαxt=fxt,t, where fxt,t is locally Lipschitz in xt. Assume that there exists a Lyapunov candidate Vxt,t satisfying(18)α1xt2a≤Vxt,t≤α2xt2ab,DβVxt,t≤−α3xt2ab,where α1,α2,α3,a, and b are positive constants and β∈0,1. Then, the equilibrium point is Mittag–Leffler stable.Remark 2.
Mittag–Leffler stability implies asymptotic stability [34].
Clearly, the fractional-order operator has more degrees of freedom than that with integer order. It is likely that a better performance can be obtained with the proper choice of order [22, 23].
As presented in [22, 23], a traditional fractional-order sliding surface S can be chosen as(19)S=λet+Dαet,where λ>0 and 0<α<1.
Taking the time derivative on both sides of (19) and using d^ to estimate d, we have(20)S˙=λe˙t+Dα+1et=λ−aet−biqt+ϕt+d^+Dα+1et.
The equivalent control law can be selected as(21)iq=ueq=1λb−aλet+λϕt+λd^+Dα+1et.
As usual, we adopt the following approach law [22]:(22)S˙=εS+ρsgnS,where ε>0,ρ>0, and sgn⋅ denotes the sign function, which is defined as(23)sgnS=1,S≥0,0,S=0,−1,S<0.
From (21) and (22), the final controller can be designed as(24)iq=ueq+uv=ueq+εS+ρsgnS=1λb−aet+ϕt+d^+Dα+1e+εS+ρsgnS.
When the sliding mode occurs, system (19) can be represented as(25)Dαet=−λet.
Thus, by Lemma2, system (25) is asymptotic stable.Remark 3.
As pointed out in Lemma2, the state et of the fractional-order system (25) decays towards 0 like t−α. But, in the case of the integer-order system, it decays towards 0 like e−t. It means that the energy transfer is slower with fractional-order sliding surface than that with integer-order one [22, 23]. Therefore, the fractional-order sliding surface is smoother compared with the integer-order one. As a result, the chattering can be better attenuated with a fractional-order controller.Remark 4.
It is worth pointing out that ifd=0, then the fractional-order sliding mode controller (24) is reduced to(26)iq=ueq+uv=ueq+εS+ρsgnS=1λb−aet+ϕt+Dα+1e+εS+ρsgnS,which is the same as the one proposed in [22].
It can be seen that thesgn⋅ function is involved in (26). As pointed out in [23], the chattering phenomenon will be caused. In the following, a saturation function sat⋅ is adopted to further reduce the chattering problem, which is described as follows:(27)satS=1,S≥Φ,SΦ,−Φ<S<Φ,−1,S≤−Φ,where Φ>0 denotes the thickness of the boundary layer. Clearly, when the saturation function is used, the final controller (24) can be modified as(28)iq=1λb−aet+ϕt+d^+Dα+1et+εS+ρsatS.
The Lyapunov function is defined as(29)Vt=12S2.
Calculating the derivation of (29) and invoking (20) and (28), it yields(30)V˙t=SS˙=Sλe˙t+Dα+1et=Sλ−aet−biqt+ϕt+d^+Dα+1e=−εS2−SsatSΦ≤0.
According to the Lyapunov stability theorem, the reaching condition of the sliding mode controller is satisfied, which indicates that the system will converge to the switching manifold asymptotically.Remark 5.
Note that the high-order derivativeDα+1e is used in fractional-order sliding model controller (24) and (28). From a practical point of view, it is difficult to implement.Remark 6.
From (19) and (27), it can be seen that the sliding mode surface S will finally maintain on the steady state of S≤Φ. That is, the closed-loop system is in a steady state with et≤Φ/λ. Though the large Φ can reduce the chattering mostly, it also can increase the ultimate bound of the tracking error et. To trade off the chattering and control performance, conservative λ and Φ are always selected by trial and error.


### 4.2. Disturbance Observer-Based CFOSMC

As stated in Remark6, there exists a boundary layer Φ of S in a steady state. In this state, the boundary layer is also the tracking error boundary layer for a given λ. In this section, to further reduce the thickness of the boundary layer, a complementary fractional-order sliding mode controller is designed.Borrowed from [18–20], the complementary fractional-order sliding mode surface is defined as(31)S¯=Sg+Sc,where(32)Sg=Dαet+2λet+λ2D−αet,λ>0,(33)Sc=Dαet−λ2D−αet,λ>0.Taking theα order time derivation on both sides of (32) and (33), respectively, we have(34)DαSg=D2αet+2λDαet+λ2et,(35)DαSc=D2αet−λ2et.From (31) to (35), we can obtain(36)S¯=Sg+Sc=2Dαet+λet,(37)DαSg−DαSc=2λDαet+λet=λS¯.The Lyapunov function candidate is chosen as(38)Vct=12Sg2+Sc2.Taking theα order derivative of (38) and using Lemma 3 as well as (32) to (37), we have(39)DαVct≤SgDαSg+ScDαSc≤SgDαSg+ScDαSg−λSc=Sg+ScDαSg−λSc=S¯D2αet+2λDαet+λ2et−λSc=S¯D2α−1e˙t+2λDαet+λ2et−λSc=S¯D2α−1−aet−biqt+ϕt+d^t+2λDαet+λ2et−λSc.Therefore, if the complementary fractional-order sliding mode controller is designed as(40)iq=ieq+iv,where(41)ieq=1b−aet+ϕt+d^t+2λD1−αet+λ2D1−2αet−λD1−2αSc,(42)iv=ρbD1−2αsatS¯,ρ>0,then(43)DαVct≤−ρS¯satS¯≤0,which means that S¯ will approach zero in a finite-time duration and the system is globally stable.Remark 7.
Note that the CFOSMC scheme (41) can be rewritten as(44)ieq=1b−aet+ϕt+d^t+2λD1−αet+λ2D1−2αet−λD1−αet−λ2D1−3αet.
Since0<α<1, the highest order derivative of et in controller (44) is less than 1, which is easier to implement in practice than FOSMC (24) and (28).Remark 8.
When1/3<α<1, the fractional-order differentiation of et is contained in controller (44). It can be viewed as a low-pass filter and reduce the amplitude of high-frequency fluctuations of et [23]. When α=1/2, especially, controller (44) is a fractional-order PI1/2D1/2 controller. As we all know, fractional-order PID is a trade-off between higher precision (provided by a higher order of integrator) and stability (provided by a higher order of differentiator). Hence, CFOSMC (41) is more smoother than the conventional FOSMC (24) and (28). On the other hand, there are several parameters in the fractional-order sliding mode controller (42). As we all know, one of the difficulties in SMC-based controller design is the determination of the controller parameters. In this paper, we obtained them by trial and error. In fact, even with the effort and knowledge of the control designer, it can not guarantee that the set of parameters is optimal.Remark 9.
When the system is in a steady state, the sliding surface functionS¯ will finally maintain on the state of S¯≤Φ. From (27) and (36), we can see that the closed-loop system is in a steady state with et≤Φ/2λ. Compared with disturbance observer-based FOSMC with saturation function, CFOSMC (41) can eliminate the chattering problem further.


## 4.1. Traditional FOSMC

Fractional calculus is a generalization of integer-order integration and differentiation to noninteger-order ones. LetDα denote the fractional-order derivative, which is defined as in Definition 1.Definition 1 (see [31]).
The Caputo derivative is defined by(14)Dαft=1Γn−α∫t0tt−τ−α+n−1fnτdτ,where n is the first integer which is not less than α, that is, α∈n−1,n. Γ⋅ is the well-known Gamma function, which is defined by Γz=∫0∞e−ztz−1dt.Definition 2 (see [31]).
The definition of the fractional integral is described by(15)D−αft=1Γα∫t0tt−τα−1fτdτ,α>0.
To proceed with the discussion, the following lemmas will be used.Lemma 2 (see [32]).
Autonomous system is as follows:(16)Dαxt=Axt,with xt0=x0 and 0<α<1, is asymptotically stable if and only if argspecA>απ/2, where spec(A) is the spectrum (set of all eigenvalues) of A. Also, the state vector xt decays towards 0 and meets the condition xt<Mt−α,t>0,M>0.Lemma 3 (see [33]).
Letxt∈ℝn be a vector of a differentiable function. Then, for any time instant t≥t0, the following relationship holds:(17)12DαxTtPxt≤xTtPDαxt,∀α∈0,1,where P∈ℝn×n is a positive definite matrix.Lemma 4 (see [34]).
Letxt=0 be an equilibrium point for the nonautonomous fractional-order system Dαxt=fxt,t, where fxt,t is locally Lipschitz in xt. Assume that there exists a Lyapunov candidate Vxt,t satisfying(18)α1xt2a≤Vxt,t≤α2xt2ab,DβVxt,t≤−α3xt2ab,where α1,α2,α3,a, and b are positive constants and β∈0,1. Then, the equilibrium point is Mittag–Leffler stable.Remark 2.
Mittag–Leffler stability implies asymptotic stability [34].
Clearly, the fractional-order operator has more degrees of freedom than that with integer order. It is likely that a better performance can be obtained with the proper choice of order [22, 23].
As presented in [22, 23], a traditional fractional-order sliding surface S can be chosen as(19)S=λet+Dαet,where λ>0 and 0<α<1.
Taking the time derivative on both sides of (19) and using d^ to estimate d, we have(20)S˙=λe˙t+Dα+1et=λ−aet−biqt+ϕt+d^+Dα+1et.
The equivalent control law can be selected as(21)iq=ueq=1λb−aλet+λϕt+λd^+Dα+1et.
As usual, we adopt the following approach law [22]:(22)S˙=εS+ρsgnS,where ε>0,ρ>0, and sgn⋅ denotes the sign function, which is defined as(23)sgnS=1,S≥0,0,S=0,−1,S<0.
From (21) and (22), the final controller can be designed as(24)iq=ueq+uv=ueq+εS+ρsgnS=1λb−aet+ϕt+d^+Dα+1e+εS+ρsgnS.
When the sliding mode occurs, system (19) can be represented as(25)Dαet=−λet.
Thus, by Lemma2, system (25) is asymptotic stable.Remark 3.
As pointed out in Lemma2, the state et of the fractional-order system (25) decays towards 0 like t−α. But, in the case of the integer-order system, it decays towards 0 like e−t. It means that the energy transfer is slower with fractional-order sliding surface than that with integer-order one [22, 23]. Therefore, the fractional-order sliding surface is smoother compared with the integer-order one. As a result, the chattering can be better attenuated with a fractional-order controller.Remark 4.
It is worth pointing out that ifd=0, then the fractional-order sliding mode controller (24) is reduced to(26)iq=ueq+uv=ueq+εS+ρsgnS=1λb−aet+ϕt+Dα+1e+εS+ρsgnS,which is the same as the one proposed in [22].
It can be seen that thesgn⋅ function is involved in (26). As pointed out in [23], the chattering phenomenon will be caused. In the following, a saturation function sat⋅ is adopted to further reduce the chattering problem, which is described as follows:(27)satS=1,S≥Φ,SΦ,−Φ<S<Φ,−1,S≤−Φ,where Φ>0 denotes the thickness of the boundary layer. Clearly, when the saturation function is used, the final controller (24) can be modified as(28)iq=1λb−aet+ϕt+d^+Dα+1et+εS+ρsatS.
The Lyapunov function is defined as(29)Vt=12S2.
Calculating the derivation of (29) and invoking (20) and (28), it yields(30)V˙t=SS˙=Sλe˙t+Dα+1et=Sλ−aet−biqt+ϕt+d^+Dα+1e=−εS2−SsatSΦ≤0.
According to the Lyapunov stability theorem, the reaching condition of the sliding mode controller is satisfied, which indicates that the system will converge to the switching manifold asymptotically.Remark 5.
Note that the high-order derivativeDα+1e is used in fractional-order sliding model controller (24) and (28). From a practical point of view, it is difficult to implement.Remark 6.
From (19) and (27), it can be seen that the sliding mode surface S will finally maintain on the steady state of S≤Φ. That is, the closed-loop system is in a steady state with et≤Φ/λ. Though the large Φ can reduce the chattering mostly, it also can increase the ultimate bound of the tracking error et. To trade off the chattering and control performance, conservative λ and Φ are always selected by trial and error.


## 4.2. Disturbance Observer-Based CFOSMC

As stated in Remark6, there exists a boundary layer Φ of S in a steady state. In this state, the boundary layer is also the tracking error boundary layer for a given λ. In this section, to further reduce the thickness of the boundary layer, a complementary fractional-order sliding mode controller is designed.Borrowed from [18–20], the complementary fractional-order sliding mode surface is defined as(31)S¯=Sg+Sc,where(32)Sg=Dαet+2λet+λ2D−αet,λ>0,(33)Sc=Dαet−λ2D−αet,λ>0.Taking theα order time derivation on both sides of (32) and (33), respectively, we have(34)DαSg=D2αet+2λDαet+λ2et,(35)DαSc=D2αet−λ2et.From (31) to (35), we can obtain(36)S¯=Sg+Sc=2Dαet+λet,(37)DαSg−DαSc=2λDαet+λet=λS¯.The Lyapunov function candidate is chosen as(38)Vct=12Sg2+Sc2.Taking theα order derivative of (38) and using Lemma 3 as well as (32) to (37), we have(39)DαVct≤SgDαSg+ScDαSc≤SgDαSg+ScDαSg−λSc=Sg+ScDαSg−λSc=S¯D2αet+2λDαet+λ2et−λSc=S¯D2α−1e˙t+2λDαet+λ2et−λSc=S¯D2α−1−aet−biqt+ϕt+d^t+2λDαet+λ2et−λSc.Therefore, if the complementary fractional-order sliding mode controller is designed as(40)iq=ieq+iv,where(41)ieq=1b−aet+ϕt+d^t+2λD1−αet+λ2D1−2αet−λD1−2αSc,(42)iv=ρbD1−2αsatS¯,ρ>0,then(43)DαVct≤−ρS¯satS¯≤0,which means that S¯ will approach zero in a finite-time duration and the system is globally stable.Remark 7.
Note that the CFOSMC scheme (41) can be rewritten as(44)ieq=1b−aet+ϕt+d^t+2λD1−αet+λ2D1−2αet−λD1−αet−λ2D1−3αet.
Since0<α<1, the highest order derivative of et in controller (44) is less than 1, which is easier to implement in practice than FOSMC (24) and (28).Remark 8.
When1/3<α<1, the fractional-order differentiation of et is contained in controller (44). It can be viewed as a low-pass filter and reduce the amplitude of high-frequency fluctuations of et [23]. When α=1/2, especially, controller (44) is a fractional-order PI1/2D1/2 controller. As we all know, fractional-order PID is a trade-off between higher precision (provided by a higher order of integrator) and stability (provided by a higher order of differentiator). Hence, CFOSMC (41) is more smoother than the conventional FOSMC (24) and (28). On the other hand, there are several parameters in the fractional-order sliding mode controller (42). As we all know, one of the difficulties in SMC-based controller design is the determination of the controller parameters. In this paper, we obtained them by trial and error. In fact, even with the effort and knowledge of the control designer, it can not guarantee that the set of parameters is optimal.Remark 9.
When the system is in a steady state, the sliding surface functionS¯ will finally maintain on the state of S¯≤Φ. From (27) and (36), we can see that the closed-loop system is in a steady state with et≤Φ/2λ. Compared with disturbance observer-based FOSMC with saturation function, CFOSMC (41) can eliminate the chattering problem further.


## 5. Simulations and Experiments

In this section, the numerical example and experimental results are presented to demonstrate the validity of the proposed method.


### 5.1. Numerical Simulation Results

The configuration of the overall simulation control system is shown in Figure1, in which DO-FOSMC means the disturbance based FOSMC with saturation function (controller (28)), and DO-CFOSMC is the proposed disturbance observer-based CFOSMC (controller (40)). The performances of the proposed DO-CFOSMC are analyzed and compared with that of DO-FOSMC.Figure 1
Block diagram of the PMSM position control system.The parameters of the PMSM system are set as follows: rated powerP=10KW, pole pairs is p=3, friction factor is B=0.001N⋅m⋅s/rad, stator inductance is L=0.0153H, rotor moment of inertia is J=0.0021kg⋅m2, permanent magnet flux is ϕf=0.82wb, and stator resistance is R=0.56Ω. The parameters of DO-CFOSMC and DO-FOSMC are selected as l=5,Φ=0.01,λ=8000,ρ=3000. From the simulation results, we found that the best selection range for the fractional-order of the proposed FOSMC method is α∈0.500.60. In the following, we set α=0.50.The initial rotation speed of the motor is given asrf1=1200 r/min. When t=0.6s, the rotation speed is set to rf2=1000 r/min. The initial load torque of the motor is 0N⋅m while t=0.8s, and the load torque is 10N⋅m, which can be viewed as an external disturbance.The numerical simulation results are shown in Figures2–4. Figure 2(a) indicates the actual speed responses of the proposed DO-CFOSMC and conventional DO-FOSMC in the presence of the above load disturbance. Figure 2(b) shows the rotation speed response when the PMSM motor just started. In the time t=0.6s, the reference rotation speed has a sudden change. The rotation speed response is depicted in Figure 2(c). Figure 2(d) shows the rotation speed response at t=0.8s, in this time, the external disturbance load is added suddenly. From Figure 2, it can be seen that the output can both track precisely the reference input signal. But, under the DO-FOSMC method, the speed response is faster and more stable, whereas the chatter amplitude occurring in conventional DO-FOSMC is bigger than that in DO-CFOSMC.Figure 2
Rotation speed responses.
(a)(b)(c)(d)Figure 3
Three-phase stator current.Figure 4
Load torque responses.With respect to the DO-FOSMC method, the three-phase current of the stator is shown in Figure3. The current frequency is inversely proportional to the rotation speed, while the current amplitude is proportional to the rotation torque and changes rapidly as the load torque varies. The variation of electromagnetic torque is shown in Figure 4 as the speed and load torque change. It can be indicated that the motor has a faster and small overshoot torque response with the DO-CFOSMC than that in DO-FOSMC.In order to evaluate the robustness of the proposed DO-CFOSMC scheme under parameter variations, numerical simulations are also carried out. The tested PMSM electric parameters are the mutual inductanceLm=0.0158, viscous friction coefficient fm=0.006 and stator resistance Rm=0.75. The parameters of the controllers (DO-CFOSMC and DO-FOSMC) are set the same as before.Figure5 shows a comparison of the speed responses between the proposed DO-CFOSMC and conventional DO-FOSMC methods under parameter variations. It is observed that both controllers can make the output speed track the desired reference speed accurately. But the DO-CFOSMC can realize the closed-loop system more robust to parameter variations and guarantee a better tracking performance.Figure 5
Rotation speed responses under parameter variations.According to the above numerical simulations, the effectiveness of the proposed DO-CFOSMC scheme with respect to the rejection of external disturbance and parameter variations has been verified. The results show that the proposed DO-CFOSMC scheme gives better tracking performances than that of the conventional DO-FOSMC.


### 5.2. Experimental Results

To further investigate the effectiveness of the proposed control scheme, some real-time experiments are carried out. The PMSM speed control platform is illustrated in Figure6. The main chip of the inverter adopts the TMS320F28335 digital signal processor (DSP).Figure 6
PMSM speed control platform.The initial rotation speed of the motor is 0 r/min, and the rotation speed is 800 r/min at4s. The load torque of the motor is 0Nm. Figures 7(a) and 7(b) show the dynamic responses of the speed and torque, respectively. From Figure 7(a), it can be seen that the DO-CFOSMC method has a smaller overshoot and a shorter settling time than that of the conventional DO-FOSMC method. Figure 7(b) shows that when the PMSM motor just started, the maximum load torque under the DO-FOSMC method is 9Nm, whereas the DO-CFOSMC method reduces it to 7Nm. Moreover, the adjustment time required for the load torque to return to its original value decreases obviously with the DO-CFOSMC method.Figure 7
Experimental results of the startup procedure.
(a)(b)When a load disturbance torqueTl=14Nm is added and removed suddenly at 0.8s and 1.2s, respectively, the dynamic responses of the speed and torque are shown in Figures 8 and 9, respectively. From these figures, we can see that, under the DO-CFOSMC method, the fluctuation maximum values of the speed and torque are smaller, while the recovering time against load disturbance is shorter than that of the conventional DO-FOSMC method.Figure 8
Experimental results of adding sudden load torque.
(a)(b)Figure 9
Experimental results of removing sudden load torque.
(a)(b)The above experimental results show that the DO-CFOSMC method achieves a smaller fluctuation overshoot with respect to the rotation speed response. In addition, the DO-CFOSMC method guarantees that the adjustment time required for the load torque to return to its original value is smaller than that with the DO-FOSMC method. Therefore, the proposed DO-CFOSMC scheme exhibits a satisfactory tracking rapidity, accuracy, and robustness.


## 5.1. Numerical Simulation Results

The configuration of the overall simulation control system is shown in Figure1, in which DO-FOSMC means the disturbance based FOSMC with saturation function (controller (28)), and DO-CFOSMC is the proposed disturbance observer-based CFOSMC (controller (40)). The performances of the proposed DO-CFOSMC are analyzed and compared with that of DO-FOSMC.Figure 1
Block diagram of the PMSM position control system.The parameters of the PMSM system are set as follows: rated powerP=10KW, pole pairs is p=3, friction factor is B=0.001N⋅m⋅s/rad, stator inductance is L=0.0153H, rotor moment of inertia is J=0.0021kg⋅m2, permanent magnet flux is ϕf=0.82wb, and stator resistance is R=0.56Ω. The parameters of DO-CFOSMC and DO-FOSMC are selected as l=5,Φ=0.01,λ=8000,ρ=3000. From the simulation results, we found that the best selection range for the fractional-order of the proposed FOSMC method is α∈0.500.60. In the following, we set α=0.50.The initial rotation speed of the motor is given asrf1=1200 r/min. When t=0.6s, the rotation speed is set to rf2=1000 r/min. The initial load torque of the motor is 0N⋅m while t=0.8s, and the load torque is 10N⋅m, which can be viewed as an external disturbance.The numerical simulation results are shown in Figures2–4. Figure 2(a) indicates the actual speed responses of the proposed DO-CFOSMC and conventional DO-FOSMC in the presence of the above load disturbance. Figure 2(b) shows the rotation speed response when the PMSM motor just started. In the time t=0.6s, the reference rotation speed has a sudden change. The rotation speed response is depicted in Figure 2(c). Figure 2(d) shows the rotation speed response at t=0.8s, in this time, the external disturbance load is added suddenly. From Figure 2, it can be seen that the output can both track precisely the reference input signal. But, under the DO-FOSMC method, the speed response is faster and more stable, whereas the chatter amplitude occurring in conventional DO-FOSMC is bigger than that in DO-CFOSMC.Figure 2
Rotation speed responses.
(a)(b)(c)(d)Figure 3
Three-phase stator current.Figure 4
Load torque responses.With respect to the DO-FOSMC method, the three-phase current of the stator is shown in Figure3. The current frequency is inversely proportional to the rotation speed, while the current amplitude is proportional to the rotation torque and changes rapidly as the load torque varies. The variation of electromagnetic torque is shown in Figure 4 as the speed and load torque change. It can be indicated that the motor has a faster and small overshoot torque response with the DO-CFOSMC than that in DO-FOSMC.In order to evaluate the robustness of the proposed DO-CFOSMC scheme under parameter variations, numerical simulations are also carried out. The tested PMSM electric parameters are the mutual inductanceLm=0.0158, viscous friction coefficient fm=0.006 and stator resistance Rm=0.75. The parameters of the controllers (DO-CFOSMC and DO-FOSMC) are set the same as before.Figure5 shows a comparison of the speed responses between the proposed DO-CFOSMC and conventional DO-FOSMC methods under parameter variations. It is observed that both controllers can make the output speed track the desired reference speed accurately. But the DO-CFOSMC can realize the closed-loop system more robust to parameter variations and guarantee a better tracking performance.Figure 5
Rotation speed responses under parameter variations.According to the above numerical simulations, the effectiveness of the proposed DO-CFOSMC scheme with respect to the rejection of external disturbance and parameter variations has been verified. The results show that the proposed DO-CFOSMC scheme gives better tracking performances than that of the conventional DO-FOSMC.


## 5.2. Experimental Results

To further investigate the effectiveness of the proposed control scheme, some real-time experiments are carried out. The PMSM speed control platform is illustrated in Figure6. The main chip of the inverter adopts the TMS320F28335 digital signal processor (DSP).Figure 6
PMSM speed control platform.The initial rotation speed of the motor is 0 r/min, and the rotation speed is 800 r/min at4s. The load torque of the motor is 0Nm. Figures 7(a) and 7(b) show the dynamic responses of the speed and torque, respectively. From Figure 7(a), it can be seen that the DO-CFOSMC method has a smaller overshoot and a shorter settling time than that of the conventional DO-FOSMC method. Figure 7(b) shows that when the PMSM motor just started, the maximum load torque under the DO-FOSMC method is 9Nm, whereas the DO-CFOSMC method reduces it to 7Nm. Moreover, the adjustment time required for the load torque to return to its original value decreases obviously with the DO-CFOSMC method.Figure 7
Experimental results of the startup procedure.
(a)(b)When a load disturbance torqueTl=14Nm is added and removed suddenly at 0.8s and 1.2s, respectively, the dynamic responses of the speed and torque are shown in Figures 8 and 9, respectively. From these figures, we can see that, under the DO-CFOSMC method, the fluctuation maximum values of the speed and torque are smaller, while the recovering time against load disturbance is shorter than that of the conventional DO-FOSMC method.Figure 8
Experimental results of adding sudden load torque.
(a)(b)Figure 9
Experimental results of removing sudden load torque.
(a)(b)The above experimental results show that the DO-CFOSMC method achieves a smaller fluctuation overshoot with respect to the rotation speed response. In addition, the DO-CFOSMC method guarantees that the adjustment time required for the load torque to return to its original value is smaller than that with the DO-FOSMC method. Therefore, the proposed DO-CFOSMC scheme exhibits a satisfactory tracking rapidity, accuracy, and robustness.


## 6. Conclusion

In this paper, in order to improve the antidisturbance capability of the PMSM drive system, a disturbance observer-based CFOSMC method has been presented, in which a nonlinear observer was employed to estimate the model uncertainties and load disturbance, while the CFOSMC scheme was utilized to improve the performance of the PMSM in terms of the tracking rapidity, accuracy, and robustness. Both numerical simulations and experimental results have shown the effectiveness of the proposed method. Our future work includes adaptive nonsingular integral terminal CFOSMC with disturbance observer for PMSM. Moreover, the fuzzy logic inference scheme to tune the gains of switching control law will also be included.



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*Source: 8343940-2020-12-03.xml*