# Robust Position Control of PMSM Using Fractional-Order Sliding Mode Controller

Authors: Jiacai Huang; Hongsheng Li; YangQuan Chen; Qinghong Xu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A new robust fractional-order sliding mode controller (FOSMC) is proposed for the position control of a permanent magnet synchronous motor (PMSM). The sliding mode controller (SMC), which is insensitive to uncertainties and load disturbances, is studied widely in the application of PMSM drive. In the existing SMC method, the sliding surface is usually designed based on the integer-order integration or differentiation of the state variables, while in this proposed robust FOSMC algorithm, the sliding surface is designed based on the fractional-order calculus of the state variables. In fact, the conventional SMC method can be seen as a special case of the proposed FOSMC method. The performance and robustness of the proposed method are analyzed and tested for nonlinear load torque disturbances, and simulation results show that the proposed algorithm is more robust and effective than the conventional SMC method.


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## Body

## 1. Introduction

Permanent magnet synchronous motor (PMSM) has many applications in industries due to its superior features such as compact structure, high efficiency, high torque to inertia ratio, and high power density [1]. To get fast four-quadrant operation, good acceleration, and smooth starting, the field-oriented control or vector control is used in the design of PMSM drives [1–4]. However, the PMSM is a typical high nonlinear, multivariable coupled system, and its performance is sensitive to external load disturbances, parameter changes in plant, and unmodeled and nonlinear dynamics. To achieve good dynamic response, some robust control strategies such as nonlinear control [5, 6], adaptive control [7, 8], H∞ control [9–11], and sliding mode control (SMC) [12–18] have been developed.The SMC is a powerful nonlinear control technique and has been widely used for speed and position control of PMSM system, because it provides a fast dynamic response and is insensitive to external load disturbances and parameter variations. In [14], a fuzzy sliding mode controller was proposed for the speed and position control of PMSM. In [15], a hybrid controller (HC) which consists of a parallel connected sliding mode controller and a neurofuzzy controller was proposed for the speed control of PMSM. In [16], a robust wavelet-neural-network sliding mode controller was proposed which can achieve favourable decoupling control and high-precision speed tracking performance of PMSM. The design of the SMC mainly contains two steps: the first step is to select the sliding surface, which is usually the linear manifold of the state variables and can guarantee the asymptotic stability; the second step is to determine the control output, which drives the system state to the designed sliding surface and constrains the state to the surface subsequently. Usually, the design of sliding surface for a PMSM is limited to integer order, which means that the sliding surface is constructed by the integer-order integration or differentiation of the state variables.Fractional calculus has a 300-year-old history, and for a long time, it was considered as a pure theoretical subject with nearly no applications. In recent decades, not only the theory of fractional-order calculus is developed greatly, but also the application of fractional controller attracts increasing attention due to the higher degree of freedom provided [19–36]. In [33] the fractional-order adaptation law for integer-order sliding mode control is studied and applied in the 2DOF robot. In [34], the synchronization of chaotic and uncertain Duffing-Holmes system has been done using the sliding mode control strategy and fractional order mathematics. In [35], a robust fractional-order proportion-plus-differential (FOPD) controller for the control of PMSM was proposed. In control practice, it is useful to consider the fractional-order controller design for an integer-order plant. This is due to the fact that the plant model may have already been obtained as an integer-order model in classical sense. In most cases, our objective of using fractional calculus is to apply the fractional-order control (FOC) to enhance the system control performance.This paper applies the fractional calculus into the sliding surface design and proposes a robust fractional-order sliding mode controller (FOSMC) for the position control of a PMSM. The rest of this paper is organized as follows. In Section2, the fractional-order calculus operation is introduced. In Section 3, the mathematical model of PMSM is given. In Section 4, the conventional integer-order SMC for PMSM is reviewed. In Section 5, the FOSMC method for position control of PMSM is derived. In Section 6, the robustness of the proposed FOSMC method is analyzed. In Section 7, the effectiveness of the proposed algorithm is illustrated through numerical examples and compared with the conventional integer-order SMC. In Section 8, the guidance for parameters selection and design is given. Finally, conclusions are presented in Section 9.


## 2. Fractional Calculus

Fractional calculus has been known since the development of the integer-order calculus, but for a long time, it has been considered as a sole mathematical problem. In recent decades, fractional calculus has become an interesting topic among system analyses and control fields.Fractional calculus is a generalization of integer-order integration and differentiation to non-integer-order ones. Let symbolDatλ denote the fractional-order fundamental operator, defined as follows [20, 21]:
(2.1)Dλ≜Datλ={dλdtλR(λ)>0,1R(λ)=0,∫at(dτ)-λR(λ)<0,
where a and t are the limits of the operation, λ is the order of the operation, and generally λ∈R and λ can be a complex number.The two most used definitions for the general fractional differentiation and integration are the Grunwald-Letnikov (GL) definition [22] and the Riemann-Liouville (RL) definition [23]. The GL is given by
(2.2)Datλf(t)=limh→0h-λ∑j=0[(t-a)/h](-1)j(λj)f(t-jh),
where [·] means the integer part.The RL definition is given as(2.3)Datλf(t)=1Γ(n-λ)dndtn∫atf(τ)(t-τ)λ-n+1dτ,
where n-1<λ<n, and Γ(·) is the Gammafunction.Having zero initial conditions, the Laplace transformation of the RL definition for a fractional-orderλ is given by
(2.4)L{Datλf(t)}=sλF(s),
where F(s) is the Laplace transformation of f(t).Distinctly, 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.


## 3. Mathematical Model of PMSM

The PMSM is composed of a stator and a rotor; the rotor is made by a permanent magnet, and the stator has 3-phase windings which are distributed sinusoidally. To get the model of the PMSM, some assumptions are made:(a) the eddy current and hysteresis losses are ignored; (b) magnetic saturation is neglected; (c) no damp winding is on the rotor; (d) the induced EMF is sinusoidal. Under the above assumptions, the mathematics model of a PMSM can be described in the rotor rotating reference frame as follows [2]:
(3.1)ud=Rid-ωeLqiq+Lddiddt,uq=Riq+ωeLdid+ωeψf+Lqdiqdt.
In the above equations, ud and uq are voltages in the d- and q-axes, id and iq are currents in the d- and q-axes, Ld and Lq are inductances in the d- and q-axes, R is the stator resistance, ωe is the electrical angular velocity, and ψf is the flux linkage of the permanent magnet.The corresponding electromagnetic torque is as follows:(3.2)Te=P[ψfiq+(Ld-Lq)idiq],
where Te is the electromagnetic torque, and P is the pole number of the rotor.For surface PMSM, we haveLd=Lq; thus, the electromagnetic torque equation is rewritten as follows:
(3.3)Te=Pψfiq.
The associated mechanical equation is as follows:
(3.4)Te-TL=Jdωmdt+Bωm,
where J is the motor moment inertia constant, TL is the external load torque, B is the viscous friction coefficient, and ωm is the rotor angular speed, and it satisfies
(3.5)ωe=Pωm.
In this paper, the id=0 decoupled control method is applied, which means that there is no demagnetization effect, and the electromagnetic torque and the armature current are the linear relationship.


## 4. Review of Conventional SMC

### 4.1. State Equations of PMSM System

The object of the designed controller is to make the positionθm strictly follow its desired signal θref.  Let
(4.1)x1=θref-θm,x2=x˙1=θ˙ref-θ˙m,
where x1 and x2 are the state error variables of the PMSM system,
(4.2)θ˙m=ωm,θ¨m=ω˙m.
From (4.1) and (4.2), it is obvious that
(4.3)x˙1=x2=θ˙ref-θ˙m,x˙2=θ¨ref-θ¨m=θ¨ref-ω˙m.
Substituting (3.3) and (3.4) into (4.3), we have
(4.4)x˙2=θ¨ref-1J[Pψfiq-TL-Bωm].
Then the state-space equation of the PMSM control system can be written as follows:
(4.5)[x˙1x˙2]=[0100][x1x2]+[0E]U+[0F],
where
(4.6)E=-PψfJ,F=θ¨ref+TL+BωmJ,U=iq.


### 4.2. The Conventional Integer-Order SMC

The design of the SMC usually consists of two steps. Firstly, the sliding surface is designed such that the system motion on the sliding mode can satisfy the design specifications; secondly, a control law is designed to drive the system state to the designed sliding surface and constrains the state to the surface subsequently.The conventional integer-order sliding surfaceS is designed as follows [4]:
(4.7)S=cx1+x2,
where c is set as a positive constant, and the derivative of (4.7) is as follows:
(4.8)S˙=cx˙1+x˙2.
Substituting (4.3) and (4.4) into (4.8), we have
(4.9)S˙=cx2+θ¨ref-1J[Pψfiq-TL-Bωm].
When TL=0, and forcing S˙=0, then the control output is obtained as follows:
(4.10)Ueq=iq=JPψf(cx2+θ¨ref+1JBωm).
Here, Ueq is the equivalent control, which keeps the state variables on the sliding surface.When the system has immeasurable disturbances with upper limitTL-max, then the final control output can be given as
(4.11)U=iq=Ueq+ksgn(S)=JPψf(cx2+θ¨ref+1JBωm)+ksgn(S),
where k is a positive switch gain, and sgn(·) denotes the sign function defined as
(4.12)sgn(S)={1S>0,0S=0,-1S<0.


### 4.3. Stability Analysis

TheLyapunov function is defined as
(4.13)V=12S2.
According to the Lyapunovstability theorem, the sliding surface reaching condition is SS˙<0. Taking the derivative of (4.13) and substituting (4.11) into (4.9), we have
(4.14)V˙=SS˙=S[TL-maxJ-PψfJksgn(S)].
From (4.14), it is obvious that when
(4.15)k>TL-maxPψf,
then SS˙<0, and the system is globally and asymptotically stable; S and S˙ will approach zero in a finite time duration.


## 4.1. State Equations of PMSM System

The object of the designed controller is to make the positionθm strictly follow its desired signal θref.  Let
(4.1)x1=θref-θm,x2=x˙1=θ˙ref-θ˙m,
where x1 and x2 are the state error variables of the PMSM system,
(4.2)θ˙m=ωm,θ¨m=ω˙m.
From (4.1) and (4.2), it is obvious that
(4.3)x˙1=x2=θ˙ref-θ˙m,x˙2=θ¨ref-θ¨m=θ¨ref-ω˙m.
Substituting (3.3) and (3.4) into (4.3), we have
(4.4)x˙2=θ¨ref-1J[Pψfiq-TL-Bωm].
Then the state-space equation of the PMSM control system can be written as follows:
(4.5)[x˙1x˙2]=[0100][x1x2]+[0E]U+[0F],
where
(4.6)E=-PψfJ,F=θ¨ref+TL+BωmJ,U=iq.


## 4.2. The Conventional Integer-Order SMC

The design of the SMC usually consists of two steps. Firstly, the sliding surface is designed such that the system motion on the sliding mode can satisfy the design specifications; secondly, a control law is designed to drive the system state to the designed sliding surface and constrains the state to the surface subsequently.The conventional integer-order sliding surfaceS is designed as follows [4]:
(4.7)S=cx1+x2,
where c is set as a positive constant, and the derivative of (4.7) is as follows:
(4.8)S˙=cx˙1+x˙2.
Substituting (4.3) and (4.4) into (4.8), we have
(4.9)S˙=cx2+θ¨ref-1J[Pψfiq-TL-Bωm].
When TL=0, and forcing S˙=0, then the control output is obtained as follows:
(4.10)Ueq=iq=JPψf(cx2+θ¨ref+1JBωm).
Here, Ueq is the equivalent control, which keeps the state variables on the sliding surface.When the system has immeasurable disturbances with upper limitTL-max, then the final control output can be given as
(4.11)U=iq=Ueq+ksgn(S)=JPψf(cx2+θ¨ref+1JBωm)+ksgn(S),
where k is a positive switch gain, and sgn(·) denotes the sign function defined as
(4.12)sgn(S)={1S>0,0S=0,-1S<0.


## 4.3. Stability Analysis

TheLyapunov function is defined as
(4.13)V=12S2.
According to the Lyapunovstability theorem, the sliding surface reaching condition is SS˙<0. Taking the derivative of (4.13) and substituting (4.11) into (4.9), we have
(4.14)V˙=SS˙=S[TL-maxJ-PψfJksgn(S)].
From (4.14), it is obvious that when
(4.15)k>TL-maxPψf,
then SS˙<0, and the system is globally and asymptotically stable; S and S˙ will approach zero in a finite time duration.


## 5. Proposed Fractional-Order SMC (FOSMC)

In this section, the fractional-order sliding mode controller (FOSMC) for the position control of PMSM will be proposed.


### 5.1. Design of Fractional-Order Sliding Surface

First, the fractional-order sliding surface is designed as follows:(5.1)S=kpx1+kdDμx1=kpx1+kdDμ-1x2,
where kp and kd are set as positive constants, the function  Dμ is defined as (2.1), and 0<μ<1. From (5.1), it can be seen that the fractional-order differentiation of x1 is used to construct the sliding surface. Meanwhile, as -1<μ-1<0, the operator Dμ-1x2 in (5.1), which means the (μ-1)th-order integration of x2, can be seen as a low-pass filter and  can reduce the amplitude of high-frequency fluctuations of x2. In this sense, the fractional-order sliding surface defined by (5.1) is more smooth compared with the conventional sliding surface shown as (4.7).


### 5.2. Design of FOSMC

Taking the time derivative on both sides of (5.1) yields
(5.2)S˙=kpx˙1+kdDμ+1x1=kpx2+kdDμ-1x˙2.
Substituting (4.4) into (5.2), we have
(5.3)S˙=kpx2+kdDμ-1x˙2=kpx2+kdDμ-1{θ¨ref-1J[Pψfiq-TL-Bωm]},
when TL=0, and forcing S˙=0, then the control output can be obtained as follows:
(5.4)Dμ-1{θ¨ref-1J[Pψfiq-Bωm]}=-kpkdx2.
Taking the (1-μ)th-order derivative on both sides of (5.4) will result in
(5.5)θ¨ref-1J[Pψfiq-Bωm]=D1-μ(-kpkdx2).
From (5.5), the equivalent control can be obtained as
(5.6)Ueq=iq=JPψf(kpkdD1-μx2+θ¨ref+1JBωm).
Similar to (4.11), when the system has load disturbances with upper limit TL-max, then the control output of FOSMC method can be given as
(5.7)U=iq=Ueq+ksgn(S)=JPψf(kpkdD1-μx2+θ¨ref+1JBωm)+ksgn(S),
where μ is called as the order of FOSMC method. If we set kp=c,  kd=1, and let  A=Pψf/J, then the block diagram of the proposed FOSMC method can be shown in Figure 1.Figure 1
Block diagram of the proposed FOSMC method.


### 5.3. Stability Analysis of FOSMC with Sign Function

When thesign function is used in the control output, then substituting (5.7) into (5.3), we have
(5.8)S˙=kdDμ-1(TL-maxJ-PψfJksgn(S)).
From (5.8), we can get the following.(a)
WhenS<0, then sgn(S)=-1, and we have
(5.9)δ1≜(TL-maxJ-PψfJksgn(S))=(TL-maxJ+PψfJk)>0.
So the (μ-1)th-order fractional integration of δ1 is higher than zero, that is,
(5.10)S˙=kdDμ-1(TL-maxJ-PψfJksgn(S))>0,
which implies that the derivative of the  Lyapunov function V˙=SS˙<0.(b)
WhenS>0, then sgn(S)=1, and we have
(5.11)S˙=kdDμ-1(TL-maxJ-PψfJk).From (5.11), It is clear that when
(5.12)δ2≜TL-maxJ-PψfJk<0,
that is,
(5.13)k>TL-maxPψf,
then the (μ-1)th-order fractional integration of δ2 is lower than zero, that is, S˙<0, which means that V˙=SS˙<0.From (5.8) to (5.13), it is obvious that when
(5.14)k>TL-maxPψf,
then the system is globally stable; S and S˙ will approach zero in a finite time duration.Moreover, from (5.8), it can be seen that because of the integration effect by the operator Dμ-1(·), the variation amplitude of S˙  in (5.8) is smaller than that of S˙  in (4.14), which means that when the sign function is used, the sliding surface of the proposed FOSMC method has smaller chattering amplitude than the sliding surface of the conventional SMC method.


### 5.4. Stability Analysis of FOSMC with Saturation Function

From (5.7), it can be seen that thesign function is involved in the output, so the chattering phenomenon will be caused. In this paper, a saturation function is adopted to reduce the chattering problem, described as follows:
(5.15)sat(S)={1S>ε,Sε-ε≤S≤ε,-1S<-ε,
where ε>0 denotes the thickness of the boundary layer.When thesaturation function is used, the control output can be rewritten as
(5.16)U=iq=Ueq+ksat(S)=JPψf(kpkdD1-μx2+θ¨ref+1JBωm)+ksat(S),
then, similar to (5.8), substituting (5.16) into (5.3), we have
(5.17)S˙=kdDμ-1(TL-maxJ-PψfJksat(S)).
From (5.17), the following is clear. (a)
WhenS<0, then sat(S)<0,
(5.18)δ3≜(TL-maxJ-PψfJksat(S))>0.So the (μ-1)th-order fractional integration of δ3 is higher than zero, that is,
(5.19)S˙=kdDμ-1(TL-maxJ-PψfJksat(S))>0,
which means that the derivative of the Lyapunov function V˙=SS˙<0.(b)
WhenS>ε, then sat(S)=1, and we have
(5.20)S˙=kdDμ-1(TL-maxJ-PψfJk).
Similar with (5.11)–(5.13), when
(5.21)k>TL-maxPψf,
then S˙<0, which means that V˙=SS˙<0.(c) When0<S≤ε, then sat(S)=S/ε, and we have
(5.22)S˙=kdDμ-1(TL-maxJ-PψfJkSε).
From (5.22), it can be seen that when
(5.23)δ4≜(TL-maxJ-PψfJkSε)<0,
that is,
(5.24)k>TL-maxPψfεS≥TL-maxPψf,
then S˙<0. Here, it is assumed that a load disturbance with magnitude TL-max is exerted on the system. From (5.24), it can be seen that when the value of S is very small, then ε/S≫1, so the condition for S˙<0 is that the value of  k is much higher than TL-max/Pψf, but in fact the parameter k will not be given a so high value. Here, it is assumed that k is assigned a minimum value which meets condition (5.21). Then the sliding surface S will undergo the following stages.(i)
In the period0<S≪ε, we have S˙≫0, so the system is unstable, meanwhile S will rapidly arrive at the peak  value S* (where S*>ε ) in a finite time with large initial positive velocity.(ii)
AsS=S*>ε, then from (5.24), it can be seen that the value of k satisfies condition (5.21), so S˙<0 and V˙=SS˙<0, that is, the system is globally stable again. In this moment, because S>0 and S˙<0, then S will decrease with negative velocity.(iii)
WhenS decreases until S<ε, then from (5.24), it can be seen that the value of k does not satisfy condition (5.21) any longer, which means that S˙>0, then S starts to increase.(iv)
WhenS increases until S>ε, then similar to (ii), we have S>0 and S˙<0, then S will decrease with negative velocity.(v)
After several oscillations and adjustments between stages (iii) and (iv), the sliding surface functionS will finally maintain on the point of S=ε, and the system is in a stable state with S˙=0.When the system is in the stable state described by (v), then from (4.7) or (5.1), it can be seen that x2=0 or Dμx1=0, and the stable position error  x1 can be estimated as follows:
(5.25)S=cx1=ε⇒x1=εc
or
(5.26)S=kpx1=ε⟹x1=εkp.
Generally, when the load disturbance is TL (TL<TL-max), then similar to the above analysis, the stable position error x1 can be estimated as follows:
(5.27)S=cx1=εTLkPψf⟹x1=εTLckPψf
or
(5.28)S=kpx1=εTLkPψf⟹x1=εTLkpkPψf.
With the maximum permissible position error x1 of the PMSM system, (5.26) or (5.28) will be the constraint in designing the parameter ε and c or kp.Remark 5.1.
In the above analysis of parts (b) and (c), the integration effect of the operatorDμ-1(·) is ignored temporarily. If the integration effect is considered, then the fractional-order μ will decide the phase delay and variation magnitude of S˙. When μ is too small, especially when μ=0, then the operator Dμ-1(·) becomes a first-order integer integrator, and the long time integration effect will lead to the largest phase delay and smallest variation magnitude of S˙, and the stable condition V˙=SS˙<0 may not be satisfied promptly, and so the system will become unstable. When μ is too large, especially when μ=1, then the operator Dμ-1(·) does not have integration action, and S˙ has zero-phase delay and the largest variation magnitude, which are the same as the convention SMC method. When μ is selected as a proper value in the range (0,1), then the suitable phase delay of S˙ will satisfy the stable condition V˙=SS˙<0, and meanwhile, the appropriate variation magnitude of S˙ will make the sliding surface S change with small fluctuation, so a better  control performance can be obtained.


## 5.1. Design of Fractional-Order Sliding Surface

First, the fractional-order sliding surface is designed as follows:(5.1)S=kpx1+kdDμx1=kpx1+kdDμ-1x2,
where kp and kd are set as positive constants, the function  Dμ is defined as (2.1), and 0<μ<1. From (5.1), it can be seen that the fractional-order differentiation of x1 is used to construct the sliding surface. Meanwhile, as -1<μ-1<0, the operator Dμ-1x2 in (5.1), which means the (μ-1)th-order integration of x2, can be seen as a low-pass filter and  can reduce the amplitude of high-frequency fluctuations of x2. In this sense, the fractional-order sliding surface defined by (5.1) is more smooth compared with the conventional sliding surface shown as (4.7).


## 5.2. Design of FOSMC

Taking the time derivative on both sides of (5.1) yields
(5.2)S˙=kpx˙1+kdDμ+1x1=kpx2+kdDμ-1x˙2.
Substituting (4.4) into (5.2), we have
(5.3)S˙=kpx2+kdDμ-1x˙2=kpx2+kdDμ-1{θ¨ref-1J[Pψfiq-TL-Bωm]},
when TL=0, and forcing S˙=0, then the control output can be obtained as follows:
(5.4)Dμ-1{θ¨ref-1J[Pψfiq-Bωm]}=-kpkdx2.
Taking the (1-μ)th-order derivative on both sides of (5.4) will result in
(5.5)θ¨ref-1J[Pψfiq-Bωm]=D1-μ(-kpkdx2).
From (5.5), the equivalent control can be obtained as
(5.6)Ueq=iq=JPψf(kpkdD1-μx2+θ¨ref+1JBωm).
Similar to (4.11), when the system has load disturbances with upper limit TL-max, then the control output of FOSMC method can be given as
(5.7)U=iq=Ueq+ksgn(S)=JPψf(kpkdD1-μx2+θ¨ref+1JBωm)+ksgn(S),
where μ is called as the order of FOSMC method. If we set kp=c,  kd=1, and let  A=Pψf/J, then the block diagram of the proposed FOSMC method can be shown in Figure 1.Figure 1
Block diagram of the proposed FOSMC method.


## 5.3. Stability Analysis of FOSMC with Sign Function

When thesign function is used in the control output, then substituting (5.7) into (5.3), we have
(5.8)S˙=kdDμ-1(TL-maxJ-PψfJksgn(S)).
From (5.8), we can get the following.(a)
WhenS<0, then sgn(S)=-1, and we have
(5.9)δ1≜(TL-maxJ-PψfJksgn(S))=(TL-maxJ+PψfJk)>0.
So the (μ-1)th-order fractional integration of δ1 is higher than zero, that is,
(5.10)S˙=kdDμ-1(TL-maxJ-PψfJksgn(S))>0,
which implies that the derivative of the  Lyapunov function V˙=SS˙<0.(b)
WhenS>0, then sgn(S)=1, and we have
(5.11)S˙=kdDμ-1(TL-maxJ-PψfJk).From (5.11), It is clear that when
(5.12)δ2≜TL-maxJ-PψfJk<0,
that is,
(5.13)k>TL-maxPψf,
then the (μ-1)th-order fractional integration of δ2 is lower than zero, that is, S˙<0, which means that V˙=SS˙<0.From (5.8) to (5.13), it is obvious that when
(5.14)k>TL-maxPψf,
then the system is globally stable; S and S˙ will approach zero in a finite time duration.Moreover, from (5.8), it can be seen that because of the integration effect by the operator Dμ-1(·), the variation amplitude of S˙  in (5.8) is smaller than that of S˙  in (4.14), which means that when the sign function is used, the sliding surface of the proposed FOSMC method has smaller chattering amplitude than the sliding surface of the conventional SMC method.


## 5.4. Stability Analysis of FOSMC with Saturation Function

From (5.7), it can be seen that thesign function is involved in the output, so the chattering phenomenon will be caused. In this paper, a saturation function is adopted to reduce the chattering problem, described as follows:
(5.15)sat(S)={1S>ε,Sε-ε≤S≤ε,-1S<-ε,
where ε>0 denotes the thickness of the boundary layer.When thesaturation function is used, the control output can be rewritten as
(5.16)U=iq=Ueq+ksat(S)=JPψf(kpkdD1-μx2+θ¨ref+1JBωm)+ksat(S),
then, similar to (5.8), substituting (5.16) into (5.3), we have
(5.17)S˙=kdDμ-1(TL-maxJ-PψfJksat(S)).
From (5.17), the following is clear. (a)
WhenS<0, then sat(S)<0,
(5.18)δ3≜(TL-maxJ-PψfJksat(S))>0.So the (μ-1)th-order fractional integration of δ3 is higher than zero, that is,
(5.19)S˙=kdDμ-1(TL-maxJ-PψfJksat(S))>0,
which means that the derivative of the Lyapunov function V˙=SS˙<0.(b)
WhenS>ε, then sat(S)=1, and we have
(5.20)S˙=kdDμ-1(TL-maxJ-PψfJk).
Similar with (5.11)–(5.13), when
(5.21)k>TL-maxPψf,
then S˙<0, which means that V˙=SS˙<0.(c) When0<S≤ε, then sat(S)=S/ε, and we have
(5.22)S˙=kdDμ-1(TL-maxJ-PψfJkSε).
From (5.22), it can be seen that when
(5.23)δ4≜(TL-maxJ-PψfJkSε)<0,
that is,
(5.24)k>TL-maxPψfεS≥TL-maxPψf,
then S˙<0. Here, it is assumed that a load disturbance with magnitude TL-max is exerted on the system. From (5.24), it can be seen that when the value of S is very small, then ε/S≫1, so the condition for S˙<0 is that the value of  k is much higher than TL-max/Pψf, but in fact the parameter k will not be given a so high value. Here, it is assumed that k is assigned a minimum value which meets condition (5.21). Then the sliding surface S will undergo the following stages.(i)
In the period0<S≪ε, we have S˙≫0, so the system is unstable, meanwhile S will rapidly arrive at the peak  value S* (where S*>ε ) in a finite time with large initial positive velocity.(ii)
AsS=S*>ε, then from (5.24), it can be seen that the value of k satisfies condition (5.21), so S˙<0 and V˙=SS˙<0, that is, the system is globally stable again. In this moment, because S>0 and S˙<0, then S will decrease with negative velocity.(iii)
WhenS decreases until S<ε, then from (5.24), it can be seen that the value of k does not satisfy condition (5.21) any longer, which means that S˙>0, then S starts to increase.(iv)
WhenS increases until S>ε, then similar to (ii), we have S>0 and S˙<0, then S will decrease with negative velocity.(v)
After several oscillations and adjustments between stages (iii) and (iv), the sliding surface functionS will finally maintain on the point of S=ε, and the system is in a stable state with S˙=0.When the system is in the stable state described by (v), then from (4.7) or (5.1), it can be seen that x2=0 or Dμx1=0, and the stable position error  x1 can be estimated as follows:
(5.25)S=cx1=ε⇒x1=εc
or
(5.26)S=kpx1=ε⟹x1=εkp.
Generally, when the load disturbance is TL (TL<TL-max), then similar to the above analysis, the stable position error x1 can be estimated as follows:
(5.27)S=cx1=εTLkPψf⟹x1=εTLckPψf
or
(5.28)S=kpx1=εTLkPψf⟹x1=εTLkpkPψf.
With the maximum permissible position error x1 of the PMSM system, (5.26) or (5.28) will be the constraint in designing the parameter ε and c or kp.Remark 5.1.
In the above analysis of parts (b) and (c), the integration effect of the operatorDμ-1(·) is ignored temporarily. If the integration effect is considered, then the fractional-order μ will decide the phase delay and variation magnitude of S˙. When μ is too small, especially when μ=0, then the operator Dμ-1(·) becomes a first-order integer integrator, and the long time integration effect will lead to the largest phase delay and smallest variation magnitude of S˙, and the stable condition V˙=SS˙<0 may not be satisfied promptly, and so the system will become unstable. When μ is too large, especially when μ=1, then the operator Dμ-1(·) does not have integration action, and S˙ has zero-phase delay and the largest variation magnitude, which are the same as the convention SMC method. When μ is selected as a proper value in the range (0,1), then the suitable phase delay of S˙ will satisfy the stable condition V˙=SS˙<0, and meanwhile, the appropriate variation magnitude of S˙ will make the sliding surface S change with small fluctuation, so a better  control performance can be obtained.


## 6. Robustness and Effectiveness Analysis of FOSMC

The robustness and effectiveness of the proposed FOSMC method will be analyzed in the following two aspects.


### 6.1. Analysis of the Control Output

From the control output of the FOSMC method shown as (5.7) or (5.16), it can be seen that two important terms are included.(a)
The termD1-μx2 denotes the (1-μ)th-order differentiation of x2, so the fractional dimension accelerating change rate of position error is contained in the output, which means that the output of the FOSMC method is more sensitive to the change rate of position error and can provide a prompt output.(b)
The other term is thesgn(S) in (5.7) or the sat(S) in (5.16), the former is a high-frequency switching signal, and the latter is a relative smooth switch signal. According to the sliding surface S defined by (5.1), it is clear that an (μ-1)th-order integrator for x2 is contained, that is, the proposed sliding surface S is more smooth than the conventional sliding surface. In other words, by using the FOSMC method, the chattering of  sgn(S) in (5.7) is eliminated to some degree, and the term sat(S) in (5.16) is more smooth.


### 6.2. Analysis of Stable Condition

With (5.8) and (5.17), it can be seen that when substituting the control output into the derivative of fractional-order sliding surface S, we have
(6.1)S˙=kdDμ-1(TL-maxJ-PψfJksgn(S))=kdDμ-1δ1
or
(6.2)S˙=kdDμ-1(TL-maxJ-PψfJksat(S))=kdDμ-1δ3.
Here, the operator Dμ-1(·) means the fractional-order integration since 0<μ<1.The following is assumed:(i) the value ofk is set as a constant which is satisfied with condition  (5.14) or (5.21);(ii) the system is in a reaching state (i.e.,V˙=SS˙<0) or in  a stable state (i.e., S=0 or constant, and S˙=0).Then the following three cases will be discussed.(a) When the system is in a reaching state andS>0, S˙<0, then
(6.3)S˙=kdDμ-1δ1<0⇒δ1<0
or
(6.4)S˙=kdDμ-1δ3<0⇒δ3<0.
If an instant load disturbance Tinstant which is greater than TL-max is applied on the system, then from (6.1) or (6.2), it can be seen that  in this moment  δ1>0  or  δ3>0, but because of the integration effect by the fractional-order integration operator Dμ-1(·), the integration value that is, Dμ-1(δ1) or Dμ-1(δ3), will not be greater than zero instantaneous, in other words the system will remain stable for an extra short time.While for the conventional SMC method, from (4.14), it can be seen that the derivative of sliding surface S is
(6.5)S˙=[TL-maxJ-PψfJksgn(S)]
or
(6.6)S˙=[TL-maxJ-PψfJksat(S)].
It is clear that when an instant load disturbance Tinstant (Tinstant>TL-max) is applied on the system, then S˙<0 immediately, and the system is also unstable at once.(b) When the system is in a reaching state andS<0,  S˙>0, then
(6.7)S˙=kdDμ-1δ1>0⇒δ1>0
or
(6.8)S˙=kdDμ-1δ3>0⇒δ3>0.
Similar to the above analysis, when an instant negative load disturbance (i.e., an opposite direction load disturbance) Tinstant which is smaller than (-TL-max) is applied on the system, then from (6.1) or (6.2), it can be seen that in this moment δ1>0 or δ3>0, but because of the integration effect by the operator Dμ-1(·), the integration value, that is, Dμ-1(δ1) or Dμ-1(δ3), will not be smaller than zero instantaneously in other words, the system will continue to be stable for an extra short time.While for the conventional SMC method, it is clear that when an instant negative load disturbanceTinstant (Tinstant<-TL-max) is exerted on the system, then according to (6.5) and (6.6), it can be seen that S˙ will be smaller than zero (i.e., S˙<0) immediately, and thus, the system is also unstable at once.(c) When the system is in a stable state, that is,S=0 or constant, and S˙=0, then
(6.9)S˙=kdDμ-1δ1=0⇒δ1=0
or
(6.10)S˙=kdDμ-1δ3=0⇒δ3=0.
If an instant positive or negative load disturbance Tinstant is applied on the system, then from (6.1) or (6.2), it is obvious that in this moment there is a step change for δ1  or  δ3, but because of the integration effect by the fractional-order  integration operator Dμ-1(·), the integration value, that is, Dμ-1(δ1) or Dμ-1(δ3), will not change greatly in a short time, which means that the sliding surface S will change with smaller fluctuation comparing with the conventional SMC method, so a better  control performance is obtained.In addition, when the load disturbanceTinstant is greater than TL-max, then the same conclusions as those made from the above analysis of (a) and (b) can be obtained.From the above analysis, it is obvious that the proposed FOSMC method is more robust than the conventional SMC method.


## 6.1. Analysis of the Control Output

From the control output of the FOSMC method shown as (5.7) or (5.16), it can be seen that two important terms are included.(a)
The termD1-μx2 denotes the (1-μ)th-order differentiation of x2, so the fractional dimension accelerating change rate of position error is contained in the output, which means that the output of the FOSMC method is more sensitive to the change rate of position error and can provide a prompt output.(b)
The other term is thesgn(S) in (5.7) or the sat(S) in (5.16), the former is a high-frequency switching signal, and the latter is a relative smooth switch signal. According to the sliding surface S defined by (5.1), it is clear that an (μ-1)th-order integrator for x2 is contained, that is, the proposed sliding surface S is more smooth than the conventional sliding surface. In other words, by using the FOSMC method, the chattering of  sgn(S) in (5.7) is eliminated to some degree, and the term sat(S) in (5.16) is more smooth.


## 6.2. Analysis of Stable Condition

With (5.8) and (5.17), it can be seen that when substituting the control output into the derivative of fractional-order sliding surface S, we have
(6.1)S˙=kdDμ-1(TL-maxJ-PψfJksgn(S))=kdDμ-1δ1
or
(6.2)S˙=kdDμ-1(TL-maxJ-PψfJksat(S))=kdDμ-1δ3.
Here, the operator Dμ-1(·) means the fractional-order integration since 0<μ<1.The following is assumed:(i) the value ofk is set as a constant which is satisfied with condition  (5.14) or (5.21);(ii) the system is in a reaching state (i.e.,V˙=SS˙<0) or in  a stable state (i.e., S=0 or constant, and S˙=0).Then the following three cases will be discussed.(a) When the system is in a reaching state andS>0, S˙<0, then
(6.3)S˙=kdDμ-1δ1<0⇒δ1<0
or
(6.4)S˙=kdDμ-1δ3<0⇒δ3<0.
If an instant load disturbance Tinstant which is greater than TL-max is applied on the system, then from (6.1) or (6.2), it can be seen that  in this moment  δ1>0  or  δ3>0, but because of the integration effect by the fractional-order integration operator Dμ-1(·), the integration value that is, Dμ-1(δ1) or Dμ-1(δ3), will not be greater than zero instantaneous, in other words the system will remain stable for an extra short time.While for the conventional SMC method, from (4.14), it can be seen that the derivative of sliding surface S is
(6.5)S˙=[TL-maxJ-PψfJksgn(S)]
or
(6.6)S˙=[TL-maxJ-PψfJksat(S)].
It is clear that when an instant load disturbance Tinstant (Tinstant>TL-max) is applied on the system, then S˙<0 immediately, and the system is also unstable at once.(b) When the system is in a reaching state andS<0,  S˙>0, then
(6.7)S˙=kdDμ-1δ1>0⇒δ1>0
or
(6.8)S˙=kdDμ-1δ3>0⇒δ3>0.
Similar to the above analysis, when an instant negative load disturbance (i.e., an opposite direction load disturbance) Tinstant which is smaller than (-TL-max) is applied on the system, then from (6.1) or (6.2), it can be seen that in this moment δ1>0 or δ3>0, but because of the integration effect by the operator Dμ-1(·), the integration value, that is, Dμ-1(δ1) or Dμ-1(δ3), will not be smaller than zero instantaneously in other words, the system will continue to be stable for an extra short time.While for the conventional SMC method, it is clear that when an instant negative load disturbanceTinstant (Tinstant<-TL-max) is exerted on the system, then according to (6.5) and (6.6), it can be seen that S˙ will be smaller than zero (i.e., S˙<0) immediately, and thus, the system is also unstable at once.(c) When the system is in a stable state, that is,S=0 or constant, and S˙=0, then
(6.9)S˙=kdDμ-1δ1=0⇒δ1=0
or
(6.10)S˙=kdDμ-1δ3=0⇒δ3=0.
If an instant positive or negative load disturbance Tinstant is applied on the system, then from (6.1) or (6.2), it is obvious that in this moment there is a step change for δ1  or  δ3, but because of the integration effect by the fractional-order  integration operator Dμ-1(·), the integration value, that is, Dμ-1(δ1) or Dμ-1(δ3), will not change greatly in a short time, which means that the sliding surface S will change with smaller fluctuation comparing with the conventional SMC method, so a better  control performance is obtained.In addition, when the load disturbanceTinstant is greater than TL-max, then the same conclusions as those made from the above analysis of (a) and (b) can be obtained.From the above analysis, it is obvious that the proposed FOSMC method is more robust than the conventional SMC method.


## 7. Numerical Computation Examples and Simulation

### 7.1. Approximation of Fractional-Order Operator

The Matlab/Simulink is used to simulate the FOSMC control system. In the simulation, a discrete-time finite-dimensional (z) transfer function is computed to approximate the continuous-time fractional-order operator  Dμ(·) by the IRID method [37], that is, dfod=irid_fod(u,Ts,N). In the simulation, the sampling frequency of FOSMC controller is 2 KHz; thus, in the IRID method, Ts=0.0005sec, and the approximation order is N=5.


### 7.2. System Block and Configuration

The block diagram of the PMSM drive system using FOSMC method is shown in Figure2, in which the block “SMC” means the conventional integer-order SMC method, and the block “FOSMC” is the proposed method, which is shown in Figure 1. The performance of the proposed FOSMC is compared with that of the conventional SMC. The rotor of the PMSM is the permanent magnet, and the flux linkage is constant. The specifications of the PMSM are shown in Table 1.Table 1
PMSM specifications.
Features
Values
Rated Voltage
300 V
Maximum Speed (ωm)
2400 rpm
Number of Poles (P)
4
Phase Resistance (Rs)
2.46 Ω
Winding Inductance (Ls)
4.233 mh
Motor Inertia (J)
1.02 × 10−3 Kg·m2
Friction Coefficient (B)
1.0 × 10−4 N·m·s·rad−1
Rotor Flux Linkagesψf
0.214 Wb
Torque (Te)
5.25 NmFigure 2
Block diagram of the PMSM position control system.As shown in Figure2, the drive system has an outer loop of position controller based on FOSMC method and an inner loop including two current controllers, that is, the q-axis and d-axis stator current regulators, both of which are based on PI control algorithm with sampling frequency of 10 KHz, and the d-axis stator current command is set to zero. In the block, ωref is the reference rotor speed in mechanical revolutions per minute, ω is the rotor speed in mechanical revolutions per minute measured by encoder, and the space vector PWM was used for the PWM generation.For comparison, we first determine the optimal parameters of the conventional SMC method, and then the corresponding parameters of the new proposed FOSMC method are set similarely, that is, in Figure2, the following parameters of SMC and FOSMC are set to be the same, that is,
(7.1)k=3,kp=c=100,kd=1,ε=1.


### 7.3. Simulation and Comparison

#### 7.3.1. Simulation of Phase Trace

In this simulation, the phase traces by the conventional SMC method and the proposed FOSMC method are simulated and compared. The given position reference isθref=π  rad, which is a step input with soft-start mode, and the order of the proposed FOSMC method  is μ=0.6.Figure3 shows the simulation results of the phase traces by the conventional SMC method and the proposed FOSMC method with saturation function. Figure 4 is similar to Figure 3, and the only difference is that thesaturationfunction is replaced by the sign function in the two methods.Phase traces of the conventional SMC method and the proposed FOSMC method withsaturation function.
(a)
Phase trace of the conventional SMC method
(b)
Phase trace of the proposed FOSMC method (μ=0.6)Phase traces of the conventional SMC method and the proposed FOSMC method withsign function.
(a)
Phase trace of conventional SMC method
(b)
Phase trace of the proposed FOSMC method (μ=0.6)From Figure3, it can be seen that the phase traces of both methods can reach the sliding surface (S=0) and arrive at the origin finally, but because of the fractional-order integration effect (i.e., the term Dμ-1x2 in S), the phase trace of the proposed FOSMC method is more smooth than that of the conventional SMC method; this also means that the proposed FOSMC has smaller speed vibration, which is consistent with the analysis of Section 5.1.From Figure4, it is obvious that the phase trace of the proposed FOSMC method is more focused on the origin than that of the conventional SMC method, which means that the proposed FOSMC has smaller speed error.


#### 7.3.2. Simulation of Stability Condition

In this simulation, the stability condition will be tested. The position reference is step inputθref=π  rad, the order of FOSMC is μ=0.6, and other parameters are set as (7.1). From (5.13), (5.21), and Table 1, it can be calculated that the maximum load disturbance is TL-max=2.568 Nm. In each of the following cases, the conventional SMC method and the proposed FOSMC method are executed.Figures5–8 are the time curves of sliding surface function S, position responses, and position error, respectively. The saturation function is adopted, and different load disturbance is applied at time t = 0.5 s.Figure 5
Sliding surfaces withsaturation function and load disturbance of 2.5 Nm at t = 0.5 s  (μ=0.6).Position responses and error withsaturation function and load disturbance of 2.5 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorFigure 7
Sliding surfaces withsaturation function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).Position responses and error withsaturation function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorIn Figures5 and 6, the load disturbance is 2.5 Nm, and we can see that the system controlled by SMC or FOSMC is stable, because the load disturbance is less than TL-max. Meanwhile, from Figures 5 and 6(b), it can be seen that the stable value of sliding surface function is S≈ε=1, and the stable position error is x1≈ε/c=0.01, which are consistent with the analysis of Section 5.4 and (5.26).From Figure5, one can see that when the external load is exerted on the system at t=0.5s, the variation amplitude of the sliding surface by the FOSMC method is smaller than that of the conventional SMC method, and consequently, the position error by the FOSMC method is smaller than that by the conventional SMC method, just as shown by Figure 6(b).  The above two simulation results meet the analysis of Section 6.2(c).In Figures7 and 8, the load disturbance is 2.6 Nm, and it is obvious that the system controlled by SMC or FOSMC method is unstable, just because the load disturbance is greater than TL-max. Moreover, an important result can be obtained from Figures 7 and 8, that is, when the load disturbance is greater than TL-max, although the system is unstable any longer, the position error by the proposed FOSMC method is smaller than that by the conventional SMC method, which is keeping with the analysis of Section 6.2.Figures9–14 are the time curves of sliding surface function S, position responses, and position error, respectively, in which the signfunction is adopted, and different load disturbance is applied at t = 0.5 s.Figure 9
Sliding surfaces withsign function and load disturbance of 2.3 Nm at t = 0.5 s (μ=0.6).Position responses and error withsign function and load disturbance of 2.3 Nm at t=0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorFigure 11
Sliding surfaces withsign function and load disturbance of 2.5 Nm at t = 0.5 s (μ=0.6).Position responses and error withsign function and load disturbance of 2.5 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorFigure 13
Sliding surfaces withsign function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).Position responses withsign function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorIn Figures9 and 10, the load disturbance is 2.3 Nm, and we can see that the system is stable under the load disturbance, because the load disturbance is less than TL-max. Because of the use of signfunction, the chattering phenomenon exists in the sliding surface S, just as shown in Figure 9. Meanwhile from Figures 9 and 10, two important results can be seen, that is, (a) the chattering amplitude of the sliding surface S by the FOSMC method is smaller than that by the conventional SMC method; (b) the position error by the proposed FOSMC method is also distinctly smaller than that by the conventional SMC method. The above two results meet the analysis of Sections 5.3 and 6.1.In Figures11 and 12, the load disturbance is 2.5 Nm, and it can be seen that the system is critically stable after the load disturbance is applied, just because the load disturbance is close to TL-max. And we also can see that the chattering amplitude of the sliding surface S and the position error, by the FOSMC method, are also distinctly smaller than those by the conventional SMC method, which meet the analysis of Sections 5.3 and 6.1.In Figures13 and 14, the load disturbance is 2.6 Nm, and it is clear that the system driven by FOSMC or SMC method is unstable after the time 0.5 s, just because the load disturbance is greater than TL-max. Meanwhile, although the system is unstable any longer, the position error by the proposed FOSMC method is smaller than that by the conventional SMC method, which is keeping with the analysis of Section 6.2.All of the above simulation results show the correctness of the stability condition shown by (5.13) or (5.21); meanwhile, the robustness analyses of Sections 5.3, 5.4, and 6 are also verified.


#### 7.3.3. Simulation of Dynamic Position Response with Step Input Signal

In this simulation, the position reference isθref=π  rad, the order of FOSMC is μ=0.6, and the saturation function is adopted. A step disturbance load of 3.1Nm is applied at t=0.5s  and withdrawn at t=0.6s, another step disturbance load of -3.1Nm is applied at t=1.0s and withdrawn at t=1.1s. Figures 15(a) and 15(b) show the dynamic position and velocity responses, respectively, of the conventional SMC method and the proposed FOSMC method in the presence of the above disturbances load. Obviously, the position error by the proposed FOSMC method is significantly smaller than that by the conventional SMC method; in other words, the FOSMC method is of more robustness than the conventional SMC method, which is in agreement with the analysis of Section 6.Position responses and velocity responses with load disturbances aroundt = 0.5 s and 1.0 s.
(a)
Position responses
(b)
Velocity responsesFigure16(a) shows the control output iq between [0s,1.5s], and the particular time period b, c, and d in Figure 16(a) is zoomed in as shown by Figures 16(b), 16(c), and 16(d), respectively. Figure 16(b) shows the control output iq between [0s,0.02s], during which the PMSM motor just started; Figure 16(c) shows the control output iq between [0.195s,0.22s] in this time period, the position output reaches the desired reference value; Figure 16(d) shows control output iq between [0.60s,0.66s], in which the external disturbance load is withdrawn. From Figures 16(b), 16(c), and 16(d), it can be seen that the control output iq of the proposed FOSMC method is more smooth than that of the conventional SMC method; in other words, the system chattering is eliminated to some degree by the FOSMC method.Results of the control outputs.
(a)
Control outputs with load disturbances aroundt = 0.5 s and 1.0 s
(b)
Control outputs during the time [0 s, 0.02 s]
(c)
Control outputs during the time [0.195 s, 0.22 s]
(d)
Control outputs during the time [0.60 s, 0.66 s]


#### 7.3.4. Simulation of Dynamic Position Response with Sinusoidal Input Signal

In this simulation, the position reference is a sinusoidal trajectory withθref(t)=πsin(10t)rad, the order of FOSMC is μ=0.6, and the saturation function is used. A step disturbance load of 3.1Nm is applied at t=0.3s and vanished at t=0.35s, and another step disturbance load of -3.1Nm is applied at t=0.75s and vanished at t=0.8s. Figures 17(a) and 17(b) show the position responses and position error, respectively. From the results, it is clear that the dynamic tracking error of the proposed FOSMC method is smaller than that of the conventional SMC method.Position responses and error to sinusoidal input signal with load disturbances att = 0.3 s and 0.75 s.
(a)
Position responses
(b)
Position error


#### 7.3.5. Controller Performance with Different Fractional Orders

We let the motor angle to track a sinusoidal trajectoryθref(t)=πsin(10t)rad, and a pulse load disturbance with 3.1 Nm amplitude, 50% pulse width, and 100 ms period is applied to the PMSM the total running time is 5 seconds. Table 2 shows the controller performance of the conventional SMC method and the proposed FOSMC method with different fractional-order μ, and the saturation function is adopted in both methods. In Table 2, the error Δe and error square |Δe|2 are defined as follows:
(7.2)Δe=∫|θref(t)-θm(t)|dt,|Δe|2=∫|θref(t)-θm(t)|2dt.
From Table 2, it can be seen that when the fractional order of the proposed FOSMC method is set to small value, the performance is poor, but once the order is got value between μ∈[0.45,0.95], then the position error and error square of the proposed FOSMC method are significantly smaller than those of conventional SMC method, especially when μ∈[0.5,0.6]. This also means that the control performance of the FOSMC method can be improved by selecting a proper fractional-order μ and designing a corresponding fractional-order sliding surface.Table 2
Controller Performance.
Controller Type
Δ
e
|
Δ
e
|
2
SMC
0.8766
0.2899
FOSMC
μ=0.35
575.6
90180
μ=0.4
351.9
34470
μ=0.45
0.4186
0.0669
μ=0.5
0.2695
0.0413
μ=0.55
0.3014
0.0409
μ=0.6
0.328
0.0463
μ=0.65
0.363
0.0549
μ=0.7
0.4057
0.0675
μ=0.75
0.4575
0.085
μ=0.8
0.517
0.1078
μ=0.85
0.5895
0.1384
μ=0.9
0.6738
0.1781
μ=0.95
0.7716
0.2292
μ=0.99
0.8492
0.2745


#### 7.3.6. Controller Performance with Different Fractional-Order and Different Reference Input

In this simulation, we check the effectiveness of the proposed FOSMC method to another position reference input and find the general regularity between the control performance and the different fractional-orderμ. Three position reference inputs, that is, sine wave, triangle wave, and trapezoid wave, are considered and, respectively, shown in Figures 18, 19, and 20. A pulse load disturbance with 50% pulse width, 100 ms period, and alternative amplitude of 3.1 Nm and 2.3 Nm is applied to the PMSM. The total running time is 5 seconds.Figure 18
Sine reference input.Figure 19
Triangle reference input.Figure 20
Trapezoid reference input.In the simulation, for each position reference input, the position error and error square of the proposed FOSMC method and the convention SMC method are regarded as the control performance. The amplitude of the pulse load disturbance is set as 2.3 Nm and 3.1 Nm, respectively, which means that the system is stable under the load disturbance of 2.3 Nm and unstable under the load disturbance of 3.1 Nm.Moreover, under the two kinds of load amplitude, the saturation function and sign function are considered, respectively.For comparison convenience, in Figures21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, and 32, the red dot line represents the error or error square obtained by the conventional SMC method, and it has no relationship with the fractional-order μ, while the green solid line is the error and error square got by the proposed FOSMC method with different fractional-order μ.Position error and error square withsine reference input (2.3 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withsine reference input (2.3 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withsine reference input (3.1 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withsine reference input (3.1 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input (2.3 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input (2.3 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input  (3.1 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input (3.1 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input  (2.3 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input (2.3 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input (3.1 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input (3.1 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squareFrom Figures21–32, it is clear that when μ∈(0,μ*), then the error or error square of the FOSMC method is bigger than that of the conventional SMC method, but when μ∈[μ*,1), the error or error square of the FOSMC method is significantly smaller  than that of the conventional SMC method. Moreover, from Figures 21–32, one can see that the value of μ* is around 0.5; this also means that when the fractional-order μ∈[0.5,1), then the proposed FOSMC method outperforms the conventional SMC method.In fact, from the simulation results shown from Figures21–32, it can be seen that the best selection range for μ is [0.5,0.6].


## 7.1. Approximation of Fractional-Order Operator

The Matlab/Simulink is used to simulate the FOSMC control system. In the simulation, a discrete-time finite-dimensional (z) transfer function is computed to approximate the continuous-time fractional-order operator  Dμ(·) by the IRID method [37], that is, dfod=irid_fod(u,Ts,N). In the simulation, the sampling frequency of FOSMC controller is 2 KHz; thus, in the IRID method, Ts=0.0005sec, and the approximation order is N=5.


## 7.2. System Block and Configuration

The block diagram of the PMSM drive system using FOSMC method is shown in Figure2, in which the block “SMC” means the conventional integer-order SMC method, and the block “FOSMC” is the proposed method, which is shown in Figure 1. The performance of the proposed FOSMC is compared with that of the conventional SMC. The rotor of the PMSM is the permanent magnet, and the flux linkage is constant. The specifications of the PMSM are shown in Table 1.Table 1
PMSM specifications.
Features
Values
Rated Voltage
300 V
Maximum Speed (ωm)
2400 rpm
Number of Poles (P)
4
Phase Resistance (Rs)
2.46 Ω
Winding Inductance (Ls)
4.233 mh
Motor Inertia (J)
1.02 × 10−3 Kg·m2
Friction Coefficient (B)
1.0 × 10−4 N·m·s·rad−1
Rotor Flux Linkagesψf
0.214 Wb
Torque (Te)
5.25 NmFigure 2
Block diagram of the PMSM position control system.As shown in Figure2, the drive system has an outer loop of position controller based on FOSMC method and an inner loop including two current controllers, that is, the q-axis and d-axis stator current regulators, both of which are based on PI control algorithm with sampling frequency of 10 KHz, and the d-axis stator current command is set to zero. In the block, ωref is the reference rotor speed in mechanical revolutions per minute, ω is the rotor speed in mechanical revolutions per minute measured by encoder, and the space vector PWM was used for the PWM generation.For comparison, we first determine the optimal parameters of the conventional SMC method, and then the corresponding parameters of the new proposed FOSMC method are set similarely, that is, in Figure2, the following parameters of SMC and FOSMC are set to be the same, that is,
(7.1)k=3,kp=c=100,kd=1,ε=1.


## 7.3. Simulation and Comparison

### 7.3.1. Simulation of Phase Trace

In this simulation, the phase traces by the conventional SMC method and the proposed FOSMC method are simulated and compared. The given position reference isθref=π  rad, which is a step input with soft-start mode, and the order of the proposed FOSMC method  is μ=0.6.Figure3 shows the simulation results of the phase traces by the conventional SMC method and the proposed FOSMC method with saturation function. Figure 4 is similar to Figure 3, and the only difference is that thesaturationfunction is replaced by the sign function in the two methods.Phase traces of the conventional SMC method and the proposed FOSMC method withsaturation function.
(a)
Phase trace of the conventional SMC method
(b)
Phase trace of the proposed FOSMC method (μ=0.6)Phase traces of the conventional SMC method and the proposed FOSMC method withsign function.
(a)
Phase trace of conventional SMC method
(b)
Phase trace of the proposed FOSMC method (μ=0.6)From Figure3, it can be seen that the phase traces of both methods can reach the sliding surface (S=0) and arrive at the origin finally, but because of the fractional-order integration effect (i.e., the term Dμ-1x2 in S), the phase trace of the proposed FOSMC method is more smooth than that of the conventional SMC method; this also means that the proposed FOSMC has smaller speed vibration, which is consistent with the analysis of Section 5.1.From Figure4, it is obvious that the phase trace of the proposed FOSMC method is more focused on the origin than that of the conventional SMC method, which means that the proposed FOSMC has smaller speed error.


### 7.3.2. Simulation of Stability Condition

In this simulation, the stability condition will be tested. The position reference is step inputθref=π  rad, the order of FOSMC is μ=0.6, and other parameters are set as (7.1). From (5.13), (5.21), and Table 1, it can be calculated that the maximum load disturbance is TL-max=2.568 Nm. In each of the following cases, the conventional SMC method and the proposed FOSMC method are executed.Figures5–8 are the time curves of sliding surface function S, position responses, and position error, respectively. The saturation function is adopted, and different load disturbance is applied at time t = 0.5 s.Figure 5
Sliding surfaces withsaturation function and load disturbance of 2.5 Nm at t = 0.5 s  (μ=0.6).Position responses and error withsaturation function and load disturbance of 2.5 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorFigure 7
Sliding surfaces withsaturation function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).Position responses and error withsaturation function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorIn Figures5 and 6, the load disturbance is 2.5 Nm, and we can see that the system controlled by SMC or FOSMC is stable, because the load disturbance is less than TL-max. Meanwhile, from Figures 5 and 6(b), it can be seen that the stable value of sliding surface function is S≈ε=1, and the stable position error is x1≈ε/c=0.01, which are consistent with the analysis of Section 5.4 and (5.26).From Figure5, one can see that when the external load is exerted on the system at t=0.5s, the variation amplitude of the sliding surface by the FOSMC method is smaller than that of the conventional SMC method, and consequently, the position error by the FOSMC method is smaller than that by the conventional SMC method, just as shown by Figure 6(b).  The above two simulation results meet the analysis of Section 6.2(c).In Figures7 and 8, the load disturbance is 2.6 Nm, and it is obvious that the system controlled by SMC or FOSMC method is unstable, just because the load disturbance is greater than TL-max. Moreover, an important result can be obtained from Figures 7 and 8, that is, when the load disturbance is greater than TL-max, although the system is unstable any longer, the position error by the proposed FOSMC method is smaller than that by the conventional SMC method, which is keeping with the analysis of Section 6.2.Figures9–14 are the time curves of sliding surface function S, position responses, and position error, respectively, in which the signfunction is adopted, and different load disturbance is applied at t = 0.5 s.Figure 9
Sliding surfaces withsign function and load disturbance of 2.3 Nm at t = 0.5 s (μ=0.6).Position responses and error withsign function and load disturbance of 2.3 Nm at t=0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorFigure 11
Sliding surfaces withsign function and load disturbance of 2.5 Nm at t = 0.5 s (μ=0.6).Position responses and error withsign function and load disturbance of 2.5 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorFigure 13
Sliding surfaces withsign function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).Position responses withsign function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorIn Figures9 and 10, the load disturbance is 2.3 Nm, and we can see that the system is stable under the load disturbance, because the load disturbance is less than TL-max. Because of the use of signfunction, the chattering phenomenon exists in the sliding surface S, just as shown in Figure 9. Meanwhile from Figures 9 and 10, two important results can be seen, that is, (a) the chattering amplitude of the sliding surface S by the FOSMC method is smaller than that by the conventional SMC method; (b) the position error by the proposed FOSMC method is also distinctly smaller than that by the conventional SMC method. The above two results meet the analysis of Sections 5.3 and 6.1.In Figures11 and 12, the load disturbance is 2.5 Nm, and it can be seen that the system is critically stable after the load disturbance is applied, just because the load disturbance is close to TL-max. And we also can see that the chattering amplitude of the sliding surface S and the position error, by the FOSMC method, are also distinctly smaller than those by the conventional SMC method, which meet the analysis of Sections 5.3 and 6.1.In Figures13 and 14, the load disturbance is 2.6 Nm, and it is clear that the system driven by FOSMC or SMC method is unstable after the time 0.5 s, just because the load disturbance is greater than TL-max. Meanwhile, although the system is unstable any longer, the position error by the proposed FOSMC method is smaller than that by the conventional SMC method, which is keeping with the analysis of Section 6.2.All of the above simulation results show the correctness of the stability condition shown by (5.13) or (5.21); meanwhile, the robustness analyses of Sections 5.3, 5.4, and 6 are also verified.


### 7.3.3. Simulation of Dynamic Position Response with Step Input Signal

In this simulation, the position reference isθref=π  rad, the order of FOSMC is μ=0.6, and the saturation function is adopted. A step disturbance load of 3.1Nm is applied at t=0.5s  and withdrawn at t=0.6s, another step disturbance load of -3.1Nm is applied at t=1.0s and withdrawn at t=1.1s. Figures 15(a) and 15(b) show the dynamic position and velocity responses, respectively, of the conventional SMC method and the proposed FOSMC method in the presence of the above disturbances load. Obviously, the position error by the proposed FOSMC method is significantly smaller than that by the conventional SMC method; in other words, the FOSMC method is of more robustness than the conventional SMC method, which is in agreement with the analysis of Section 6.Position responses and velocity responses with load disturbances aroundt = 0.5 s and 1.0 s.
(a)
Position responses
(b)
Velocity responsesFigure16(a) shows the control output iq between [0s,1.5s], and the particular time period b, c, and d in Figure 16(a) is zoomed in as shown by Figures 16(b), 16(c), and 16(d), respectively. Figure 16(b) shows the control output iq between [0s,0.02s], during which the PMSM motor just started; Figure 16(c) shows the control output iq between [0.195s,0.22s] in this time period, the position output reaches the desired reference value; Figure 16(d) shows control output iq between [0.60s,0.66s], in which the external disturbance load is withdrawn. From Figures 16(b), 16(c), and 16(d), it can be seen that the control output iq of the proposed FOSMC method is more smooth than that of the conventional SMC method; in other words, the system chattering is eliminated to some degree by the FOSMC method.Results of the control outputs.
(a)
Control outputs with load disturbances aroundt = 0.5 s and 1.0 s
(b)
Control outputs during the time [0 s, 0.02 s]
(c)
Control outputs during the time [0.195 s, 0.22 s]
(d)
Control outputs during the time [0.60 s, 0.66 s]


### 7.3.4. Simulation of Dynamic Position Response with Sinusoidal Input Signal

In this simulation, the position reference is a sinusoidal trajectory withθref(t)=πsin(10t)rad, the order of FOSMC is μ=0.6, and the saturation function is used. A step disturbance load of 3.1Nm is applied at t=0.3s and vanished at t=0.35s, and another step disturbance load of -3.1Nm is applied at t=0.75s and vanished at t=0.8s. Figures 17(a) and 17(b) show the position responses and position error, respectively. From the results, it is clear that the dynamic tracking error of the proposed FOSMC method is smaller than that of the conventional SMC method.Position responses and error to sinusoidal input signal with load disturbances att = 0.3 s and 0.75 s.
(a)
Position responses
(b)
Position error


### 7.3.5. Controller Performance with Different Fractional Orders

We let the motor angle to track a sinusoidal trajectoryθref(t)=πsin(10t)rad, and a pulse load disturbance with 3.1 Nm amplitude, 50% pulse width, and 100 ms period is applied to the PMSM the total running time is 5 seconds. Table 2 shows the controller performance of the conventional SMC method and the proposed FOSMC method with different fractional-order μ, and the saturation function is adopted in both methods. In Table 2, the error Δe and error square |Δe|2 are defined as follows:
(7.2)Δe=∫|θref(t)-θm(t)|dt,|Δe|2=∫|θref(t)-θm(t)|2dt.
From Table 2, it can be seen that when the fractional order of the proposed FOSMC method is set to small value, the performance is poor, but once the order is got value between μ∈[0.45,0.95], then the position error and error square of the proposed FOSMC method are significantly smaller than those of conventional SMC method, especially when μ∈[0.5,0.6]. This also means that the control performance of the FOSMC method can be improved by selecting a proper fractional-order μ and designing a corresponding fractional-order sliding surface.Table 2
Controller Performance.
Controller Type
Δ
e
|
Δ
e
|
2
SMC
0.8766
0.2899
FOSMC
μ=0.35
575.6
90180
μ=0.4
351.9
34470
μ=0.45
0.4186
0.0669
μ=0.5
0.2695
0.0413
μ=0.55
0.3014
0.0409
μ=0.6
0.328
0.0463
μ=0.65
0.363
0.0549
μ=0.7
0.4057
0.0675
μ=0.75
0.4575
0.085
μ=0.8
0.517
0.1078
μ=0.85
0.5895
0.1384
μ=0.9
0.6738
0.1781
μ=0.95
0.7716
0.2292
μ=0.99
0.8492
0.2745


### 7.3.6. Controller Performance with Different Fractional-Order and Different Reference Input

In this simulation, we check the effectiveness of the proposed FOSMC method to another position reference input and find the general regularity between the control performance and the different fractional-orderμ. Three position reference inputs, that is, sine wave, triangle wave, and trapezoid wave, are considered and, respectively, shown in Figures 18, 19, and 20. A pulse load disturbance with 50% pulse width, 100 ms period, and alternative amplitude of 3.1 Nm and 2.3 Nm is applied to the PMSM. The total running time is 5 seconds.Figure 18
Sine reference input.Figure 19
Triangle reference input.Figure 20
Trapezoid reference input.In the simulation, for each position reference input, the position error and error square of the proposed FOSMC method and the convention SMC method are regarded as the control performance. The amplitude of the pulse load disturbance is set as 2.3 Nm and 3.1 Nm, respectively, which means that the system is stable under the load disturbance of 2.3 Nm and unstable under the load disturbance of 3.1 Nm.Moreover, under the two kinds of load amplitude, the saturation function and sign function are considered, respectively.For comparison convenience, in Figures21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, and 32, the red dot line represents the error or error square obtained by the conventional SMC method, and it has no relationship with the fractional-order μ, while the green solid line is the error and error square got by the proposed FOSMC method with different fractional-order μ.Position error and error square withsine reference input (2.3 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withsine reference input (2.3 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withsine reference input (3.1 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withsine reference input (3.1 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input (2.3 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input (2.3 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input  (3.1 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input (3.1 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input  (2.3 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input (2.3 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input (3.1 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input (3.1 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squareFrom Figures21–32, it is clear that when μ∈(0,μ*), then the error or error square of the FOSMC method is bigger than that of the conventional SMC method, but when μ∈[μ*,1), the error or error square of the FOSMC method is significantly smaller  than that of the conventional SMC method. Moreover, from Figures 21–32, one can see that the value of μ* is around 0.5; this also means that when the fractional-order μ∈[0.5,1), then the proposed FOSMC method outperforms the conventional SMC method.In fact, from the simulation results shown from Figures21–32, it can be seen that the best selection range for μ is [0.5,0.6].


## 7.3.1. Simulation of Phase Trace

In this simulation, the phase traces by the conventional SMC method and the proposed FOSMC method are simulated and compared. The given position reference isθref=π  rad, which is a step input with soft-start mode, and the order of the proposed FOSMC method  is μ=0.6.Figure3 shows the simulation results of the phase traces by the conventional SMC method and the proposed FOSMC method with saturation function. Figure 4 is similar to Figure 3, and the only difference is that thesaturationfunction is replaced by the sign function in the two methods.Phase traces of the conventional SMC method and the proposed FOSMC method withsaturation function.
(a)
Phase trace of the conventional SMC method
(b)
Phase trace of the proposed FOSMC method (μ=0.6)Phase traces of the conventional SMC method and the proposed FOSMC method withsign function.
(a)
Phase trace of conventional SMC method
(b)
Phase trace of the proposed FOSMC method (μ=0.6)From Figure3, it can be seen that the phase traces of both methods can reach the sliding surface (S=0) and arrive at the origin finally, but because of the fractional-order integration effect (i.e., the term Dμ-1x2 in S), the phase trace of the proposed FOSMC method is more smooth than that of the conventional SMC method; this also means that the proposed FOSMC has smaller speed vibration, which is consistent with the analysis of Section 5.1.From Figure4, it is obvious that the phase trace of the proposed FOSMC method is more focused on the origin than that of the conventional SMC method, which means that the proposed FOSMC has smaller speed error.


## 7.3.2. Simulation of Stability Condition

In this simulation, the stability condition will be tested. The position reference is step inputθref=π  rad, the order of FOSMC is μ=0.6, and other parameters are set as (7.1). From (5.13), (5.21), and Table 1, it can be calculated that the maximum load disturbance is TL-max=2.568 Nm. In each of the following cases, the conventional SMC method and the proposed FOSMC method are executed.Figures5–8 are the time curves of sliding surface function S, position responses, and position error, respectively. The saturation function is adopted, and different load disturbance is applied at time t = 0.5 s.Figure 5
Sliding surfaces withsaturation function and load disturbance of 2.5 Nm at t = 0.5 s  (μ=0.6).Position responses and error withsaturation function and load disturbance of 2.5 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorFigure 7
Sliding surfaces withsaturation function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).Position responses and error withsaturation function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorIn Figures5 and 6, the load disturbance is 2.5 Nm, and we can see that the system controlled by SMC or FOSMC is stable, because the load disturbance is less than TL-max. Meanwhile, from Figures 5 and 6(b), it can be seen that the stable value of sliding surface function is S≈ε=1, and the stable position error is x1≈ε/c=0.01, which are consistent with the analysis of Section 5.4 and (5.26).From Figure5, one can see that when the external load is exerted on the system at t=0.5s, the variation amplitude of the sliding surface by the FOSMC method is smaller than that of the conventional SMC method, and consequently, the position error by the FOSMC method is smaller than that by the conventional SMC method, just as shown by Figure 6(b).  The above two simulation results meet the analysis of Section 6.2(c).In Figures7 and 8, the load disturbance is 2.6 Nm, and it is obvious that the system controlled by SMC or FOSMC method is unstable, just because the load disturbance is greater than TL-max. Moreover, an important result can be obtained from Figures 7 and 8, that is, when the load disturbance is greater than TL-max, although the system is unstable any longer, the position error by the proposed FOSMC method is smaller than that by the conventional SMC method, which is keeping with the analysis of Section 6.2.Figures9–14 are the time curves of sliding surface function S, position responses, and position error, respectively, in which the signfunction is adopted, and different load disturbance is applied at t = 0.5 s.Figure 9
Sliding surfaces withsign function and load disturbance of 2.3 Nm at t = 0.5 s (μ=0.6).Position responses and error withsign function and load disturbance of 2.3 Nm at t=0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorFigure 11
Sliding surfaces withsign function and load disturbance of 2.5 Nm at t = 0.5 s (μ=0.6).Position responses and error withsign function and load disturbance of 2.5 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorFigure 13
Sliding surfaces withsign function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).Position responses withsign function and load disturbance of 2.6 Nm at t = 0.5 s (μ=0.6).
(a)
Position responses
(b)
Position errorIn Figures9 and 10, the load disturbance is 2.3 Nm, and we can see that the system is stable under the load disturbance, because the load disturbance is less than TL-max. Because of the use of signfunction, the chattering phenomenon exists in the sliding surface S, just as shown in Figure 9. Meanwhile from Figures 9 and 10, two important results can be seen, that is, (a) the chattering amplitude of the sliding surface S by the FOSMC method is smaller than that by the conventional SMC method; (b) the position error by the proposed FOSMC method is also distinctly smaller than that by the conventional SMC method. The above two results meet the analysis of Sections 5.3 and 6.1.In Figures11 and 12, the load disturbance is 2.5 Nm, and it can be seen that the system is critically stable after the load disturbance is applied, just because the load disturbance is close to TL-max. And we also can see that the chattering amplitude of the sliding surface S and the position error, by the FOSMC method, are also distinctly smaller than those by the conventional SMC method, which meet the analysis of Sections 5.3 and 6.1.In Figures13 and 14, the load disturbance is 2.6 Nm, and it is clear that the system driven by FOSMC or SMC method is unstable after the time 0.5 s, just because the load disturbance is greater than TL-max. Meanwhile, although the system is unstable any longer, the position error by the proposed FOSMC method is smaller than that by the conventional SMC method, which is keeping with the analysis of Section 6.2.All of the above simulation results show the correctness of the stability condition shown by (5.13) or (5.21); meanwhile, the robustness analyses of Sections 5.3, 5.4, and 6 are also verified.


## 7.3.3. Simulation of Dynamic Position Response with Step Input Signal

In this simulation, the position reference isθref=π  rad, the order of FOSMC is μ=0.6, and the saturation function is adopted. A step disturbance load of 3.1Nm is applied at t=0.5s  and withdrawn at t=0.6s, another step disturbance load of -3.1Nm is applied at t=1.0s and withdrawn at t=1.1s. Figures 15(a) and 15(b) show the dynamic position and velocity responses, respectively, of the conventional SMC method and the proposed FOSMC method in the presence of the above disturbances load. Obviously, the position error by the proposed FOSMC method is significantly smaller than that by the conventional SMC method; in other words, the FOSMC method is of more robustness than the conventional SMC method, which is in agreement with the analysis of Section 6.Position responses and velocity responses with load disturbances aroundt = 0.5 s and 1.0 s.
(a)
Position responses
(b)
Velocity responsesFigure16(a) shows the control output iq between [0s,1.5s], and the particular time period b, c, and d in Figure 16(a) is zoomed in as shown by Figures 16(b), 16(c), and 16(d), respectively. Figure 16(b) shows the control output iq between [0s,0.02s], during which the PMSM motor just started; Figure 16(c) shows the control output iq between [0.195s,0.22s] in this time period, the position output reaches the desired reference value; Figure 16(d) shows control output iq between [0.60s,0.66s], in which the external disturbance load is withdrawn. From Figures 16(b), 16(c), and 16(d), it can be seen that the control output iq of the proposed FOSMC method is more smooth than that of the conventional SMC method; in other words, the system chattering is eliminated to some degree by the FOSMC method.Results of the control outputs.
(a)
Control outputs with load disturbances aroundt = 0.5 s and 1.0 s
(b)
Control outputs during the time [0 s, 0.02 s]
(c)
Control outputs during the time [0.195 s, 0.22 s]
(d)
Control outputs during the time [0.60 s, 0.66 s]


## 7.3.4. Simulation of Dynamic Position Response with Sinusoidal Input Signal

In this simulation, the position reference is a sinusoidal trajectory withθref(t)=πsin(10t)rad, the order of FOSMC is μ=0.6, and the saturation function is used. A step disturbance load of 3.1Nm is applied at t=0.3s and vanished at t=0.35s, and another step disturbance load of -3.1Nm is applied at t=0.75s and vanished at t=0.8s. Figures 17(a) and 17(b) show the position responses and position error, respectively. From the results, it is clear that the dynamic tracking error of the proposed FOSMC method is smaller than that of the conventional SMC method.Position responses and error to sinusoidal input signal with load disturbances att = 0.3 s and 0.75 s.
(a)
Position responses
(b)
Position error


## 7.3.5. Controller Performance with Different Fractional Orders

We let the motor angle to track a sinusoidal trajectoryθref(t)=πsin(10t)rad, and a pulse load disturbance with 3.1 Nm amplitude, 50% pulse width, and 100 ms period is applied to the PMSM the total running time is 5 seconds. Table 2 shows the controller performance of the conventional SMC method and the proposed FOSMC method with different fractional-order μ, and the saturation function is adopted in both methods. In Table 2, the error Δe and error square |Δe|2 are defined as follows:
(7.2)Δe=∫|θref(t)-θm(t)|dt,|Δe|2=∫|θref(t)-θm(t)|2dt.
From Table 2, it can be seen that when the fractional order of the proposed FOSMC method is set to small value, the performance is poor, but once the order is got value between μ∈[0.45,0.95], then the position error and error square of the proposed FOSMC method are significantly smaller than those of conventional SMC method, especially when μ∈[0.5,0.6]. This also means that the control performance of the FOSMC method can be improved by selecting a proper fractional-order μ and designing a corresponding fractional-order sliding surface.Table 2
Controller Performance.
Controller Type
Δ
e
|
Δ
e
|
2
SMC
0.8766
0.2899
FOSMC
μ=0.35
575.6
90180
μ=0.4
351.9
34470
μ=0.45
0.4186
0.0669
μ=0.5
0.2695
0.0413
μ=0.55
0.3014
0.0409
μ=0.6
0.328
0.0463
μ=0.65
0.363
0.0549
μ=0.7
0.4057
0.0675
μ=0.75
0.4575
0.085
μ=0.8
0.517
0.1078
μ=0.85
0.5895
0.1384
μ=0.9
0.6738
0.1781
μ=0.95
0.7716
0.2292
μ=0.99
0.8492
0.2745


## 7.3.6. Controller Performance with Different Fractional-Order and Different Reference Input

In this simulation, we check the effectiveness of the proposed FOSMC method to another position reference input and find the general regularity between the control performance and the different fractional-orderμ. Three position reference inputs, that is, sine wave, triangle wave, and trapezoid wave, are considered and, respectively, shown in Figures 18, 19, and 20. A pulse load disturbance with 50% pulse width, 100 ms period, and alternative amplitude of 3.1 Nm and 2.3 Nm is applied to the PMSM. The total running time is 5 seconds.Figure 18
Sine reference input.Figure 19
Triangle reference input.Figure 20
Trapezoid reference input.In the simulation, for each position reference input, the position error and error square of the proposed FOSMC method and the convention SMC method are regarded as the control performance. The amplitude of the pulse load disturbance is set as 2.3 Nm and 3.1 Nm, respectively, which means that the system is stable under the load disturbance of 2.3 Nm and unstable under the load disturbance of 3.1 Nm.Moreover, under the two kinds of load amplitude, the saturation function and sign function are considered, respectively.For comparison convenience, in Figures21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, and 32, the red dot line represents the error or error square obtained by the conventional SMC method, and it has no relationship with the fractional-order μ, while the green solid line is the error and error square got by the proposed FOSMC method with different fractional-order μ.Position error and error square withsine reference input (2.3 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withsine reference input (2.3 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withsine reference input (3.1 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withsine reference input (3.1 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input (2.3 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input (2.3 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input  (3.1 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtriangle reference input (3.1 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input  (2.3 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input (2.3 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input (3.1 Nm pulse disturbance, with saturation function).
(a)
Position error
(b)
Position error squarePosition error and error square withtrapezoid reference input (3.1 Nm pulse disturbance, with sign function).
(a)
Position error
(b)
Position error squareFrom Figures21–32, it is clear that when μ∈(0,μ*), then the error or error square of the FOSMC method is bigger than that of the conventional SMC method, but when μ∈[μ*,1), the error or error square of the FOSMC method is significantly smaller  than that of the conventional SMC method. Moreover, from Figures 21–32, one can see that the value of μ* is around 0.5; this also means that when the fractional-order μ∈[0.5,1), then the proposed FOSMC method outperforms the conventional SMC method.In fact, from the simulation results shown from Figures21–32, it can be seen that the best selection range for μ is [0.5,0.6].


## 8. Guidance for Parameters Selection and Design of FOSMC

In the proposed FOSMC method, there are five parameters, that is,kp,kd,k,ε,andμ, which need to be designed. From the above analyses and numerical simulation results, one can select and design the five parameters through the following procedures:(i)
select a value in the range[0.5,0.6] or [0.5, 1) for the fractional-order μ;(ii)
estimate the maximum load disturbanceTL-max;(iii)
according to theTL-max and (5.21), compute the value range of parameter k and then select a suitable value for k;(iv)
becausekd is the coefficient of fractional-order differentiation of position error x1, if the value of kd is big, it will be too sensitive to the variation of position error x1 and cause oscillation. In general, kd can be set as a suitable small value, for example, in this paper kd=1;(v)
with the maximum permissible position errorx1 of the PMSM system and the parameter k designed by the procedure (iii), then parameters ε and kp can be designed and selected according to (5.28).


## 9. Conclusions

A new and systematic design of the fractional-order sliding mode controller (FOSMC) for PMSM position control system is presented. By selecting a proper fractional-orderμ and designing a fractional-order sliding surface, the control performance such as control precision  and system robustness of the proposed FOSMC method is distinctly more excellent than that of the conventional SMC method, because an extra fractional order, the real parameters μ, is involved. The robustness of the proposed FOSMC method is analyzed in detail, and the guidance for parameters selection and design is given. The numerical simulation results demonstrate the effectiveness and robustness of the proposed FOSMC method.



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*Source: 512703-2012-11-22.xml*