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

and Applied Analysis 3 Fractional calculus is a generalization of integer-order integration and differentiation to non-integer-order ones. Let symbol aDt denote the fractional-order fundamental operator, defined as follows 20, 21 : D aDt ⎧ ⎪ ⎪⎨ ⎪ ⎪⎩ d dtλ R λ > 0, 1 R λ 0, ∫ t a dτ −λ R λ < 0, 2.1 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 twomost 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 aD λ t f t lim h→ 0 h−λ t−a /h ∑ j 0 −1 j ( λ j ) f ( t − jh, 2.2 where · means the integer part. The RL definition is given as aD λ t f t 1 Γ n − λ d dtn ∫ t a f τ t − τ λ−n 1 dτ, 2.3 where n − 1 < λ < n, and Γ · is the Gamma function. Having zero initial conditions, the Laplace transformation of the RL definition for a fractional-order λ is given by L { aD λ t f t } sF s , 2.4 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 4 Abstract and Applied Analysis sinusoidal. Under the above assumptions, themathematics model of a PMSM can be described in the rotor rotating reference frame as follows 2 : ud Rid −ωeLqiq Ld did dt , uq Riq ωeLdid ωeψf Lq diq dt . 3.1 In the above equations, ud and uq are voltages in the dand q-axes, id and iq are currents in the dand q-axes, Ld and Lq are inductances in the dand 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: Te P [ ψf iq ( Ld − Lq ) idiq ] , 3.2 where Te is the electromagnetic torque, and P is the pole number of the rotor. For surface PMSM, we have Ld Lq; thus, the electromagnetic torque equation is rewritten as follows: Te Pψf iq. 3.3 The associated mechanical equation is as follows: Te − TL J dωm dt Bωm, 3.4 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 ωe Pωm. 3.5 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 position θm strictly follow its desired signal θref. Let x1 θref − θm, x2 ẋ1 θ̇ref − θ̇m, 4.1 Abstract and Applied Analysis 5 where x1 and x2 are the state error variables of the PMSM system, θ̇m ωm, θ̈m ω̇m. 4.2and Applied Analysis 5 where x1 and x2 are the state error variables of the PMSM system, θ̇m ωm, θ̈m ω̇m. 4.2 From 4.1 and 4.2 , it is obvious that ẋ1 x2 θ̇ref − θ̇m, ẋ2 θ̈ref − θ̈m θ̈ref − ω̇m. 4.3 Substituting 3.3 and 3.4 into 4.3 , we have ẋ2 θ̈ref − 1 J [ Pψf iq − TL − Bωm ] . 4.4 Then the state-space equation of the PMSM control system can be written as follows: [ ẋ1 ẋ2 ] [ 0 1 0 0 ][ x1 x2 ] [ 0 E ] U [ 0 F ] , 4.5 where E −f J , F θ̈ref TL Bωm J , U iq. 4.6 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 surface S is designed as follows 4 : S cx1 x2, 4.7 where c is set as a positive constant, and the derivative of 4.7 is as follows: Ṡ cẋ1 ẋ2. 4.8 Substituting 4.3 and 4.4 into 4.8 , we have Ṡ cx2 θ̈ref − 1 J [ Pψf iq − TL − Bωm ] . 4.9 6 Abstract and Applied Analysis When TL 0, and forcing Ṡ 0, then the control output is obtained as follows: Ueq iq J Pψf ( cx2 θ̈ref 1 J Bωm ) . 4.10 Here,Ueq is the equivalent control, which keeps the state variables on the sliding surface. When the system has immeasurable disturbances with upper limit TL-max, then the final control output can be given as U iq Ueq k sgn S J Pψf ( cx2 θ̈ref 1 J Bωm ) k sgn S , 4.11 where k is a positive switch gain, and sgn · denotes the sign function defined as sgn S ⎧ ⎪ ⎨ ⎪ ⎩ 1 S > 0, 0 S 0, −1 S < 0. 4.12 4.3. Stability Analysis The Lyapunov function is defined as V 1 2 S2. 4.13 According to the Lyapunov stability theorem, the sliding surface reaching condition is SṠ < 0. Taking the derivative of 4.13 and substituting 4.11 into 4.9 , we have V̇ SṠ S [ TL-max J − Pψf J k sgn S ] . 4.14 From 4.14 , it is obvious that when k > TL-max Pψf , 4.15 then SṠ < 0, and the system is globally and asymptotically stable; S and Ṡwill 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. Abstract and Applied Analysis 7 5.1. Design of Fractional-Order Sliding Surface First, the fractional-order sliding surface is designed as follows: S kpx1 kdDx1 kpx1 kdDμ−1x2, 5.1 where kp and kd are set as positive constants, the functionD 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 Ṡ kpẋ1 kdD x1 kpx2 kdDμ−1ẋ2. 5.2 Substituting 4.4 into 5.2 , we have Ṡ kpx2 kdDμ−1ẋ2 kpx2 kdDμ−1 { θ̈ref − 1 J [ Pψf iq − TL − Bωm ] } , 5.3 when TL 0, and forcing Ṡ 0, then the control output can be obtained as follows: Dμ−1 { θ̈ref − 1 J [ Pψf iq − Bωm ] } −p kd x2. 5.4 Taking the 1 − μ th-order derivative on both sides of 5.4 will result in θ̈ref − 1 J [ Pψf iq − Bωm ] D1−μ ( −p kd x2 ) . 5.5 From 5.5 , the equivalent control can be obtained as Ueq iq J Pψf ( kp kd D1−μx2 θ̈ref 1 J Bωm ) . 5.6 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 U iq Ueq k sgn S J Pψf ( kp kd D1−μx2 θ̈ref 1 J Bωm ) k sgn S , 5.7 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.and Applied Analysis 7 5.1. Design of Fractional-Order Sliding Surface First, the fractional-order sliding surface is designed as follows: S kpx1 kdDx1 kpx1 kdDμ−1x2, 5.1 where kp and kd are set as positive constants, the functionD 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 Ṡ kpẋ1 kdD x1 kpx2 kdDμ−1ẋ2. 5.2 Substituting 4.4 into 5.2 , we have Ṡ kpx2 kdDμ−1ẋ2 kpx2 kdDμ−1 { θ̈ref − 1 J [ Pψf iq − TL − Bωm ] } , 5.3 when TL 0, and forcing Ṡ 0, then the control output can be obtained as follows: Dμ−1 { θ̈ref − 1 J [ Pψf iq − Bωm ] } −p kd x2. 5.4 Taking the 1 − μ th-order derivative on both sides of 5.4 will result in θ̈ref − 1 J [ Pψf iq − Bωm ] D1−μ ( −p kd x2 ) . 5.5 From 5.5 , the equivalent control can be obtained as Ueq iq J Pψf ( kp kd D1−μx2 θ̈ref 1 J Bωm ) . 5.6 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 U iq Ueq k sgn S J Pψf ( kp kd D1−μx2 θ̈ref 1 J Bωm ) k sgn S , 5.7 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. 8 Abstract and Applied Analysis


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 Fractional calculus is a generalization of integer-order integration and differentiation to non-integer-order ones.Let symbol a D λ t denote the fractional-order fundamental operator, defined as follows 20, 21 : 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 where • means the integer part.
The RL definition is given as where n − 1 < λ < n, and Γ • is the Gamma function.
Having zero initial conditions, the Laplace transformation of the RL definition for a fractional-order λ is given by 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.

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 : u q Ri q ω e L d i d ω e ψ f L q di q dt .

3.1
In the above equations, u d and u q are voltages in the d-and q-axes, i d and i q are currents in the d-and q-axes, L d and L q are inductances in the d-and q-axes, R is the stator resistance, ω e is the electrical angular velocity, and ψ f is the flux linkage of the permanent magnet.
The corresponding electromagnetic torque is as follows: where T e is the electromagnetic torque, and P is the pole number of the rotor.
For surface PMSM, we have L d L q ; thus, the electromagnetic torque equation is rewritten as follows: T e Pψ f i q . 3.3 The associated mechanical equation is as follows: T e − T L J dω m dt Bω m , 3.4 where J is the motor moment inertia constant, T L is the external load torque, B is the viscous friction coefficient, and ω m is the rotor angular speed, and it satisfies ω e Pω m .

3.5
In this paper, the i d 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.

State Equations of PMSM System
The object of the designed controller is to make the position θ m strictly follow its desired signal θ ref . Let Then the state-space equation of the PMSM control system can be written as follows: where

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 surface S is designed as follows 4 : where c is set as a positive constant, and the derivative of 4.
Abstract and Applied Analysis When T L 0, and forcing Ṡ 0, then the control output is obtained as follows: Here, U eq is the equivalent control, which keeps the state variables on the sliding surface.
When the system has immeasurable disturbances with upper limit T L-max , then the final control output can be given as where k is a positive switch gain, and sgn • denotes the sign function defined as

Stability Analysis
The Lyapunov function is defined as

4.13
According to the Lyapunov stability theorem, the sliding surface reaching condition is S Ṡ < 0.
Taking the derivative of 4.13 and substituting 4.11 into 4.9 , we have

4.14
From 4.14 , it is obvious that when then S Ṡ < 0, and the system is globally and asymptotically stable; S and Ṡ will approach zero in a finite time duration.

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

Design of Fractional-Order Sliding Surface
First, the fractional-order sliding surface is designed as follows: where k p and k d are set as positive constants, the function D μ is defined as 2.1 , and 0 < μ < 1.From 5.1 , it can be seen that the fractional-order differentiation of x 1 is used to construct the sliding surface.Meanwhile, as −1 < μ − 1 < 0, the operator D μ−1 x 2 in 5.1 , which means the μ − 1 th-order integration of x 2 , can be seen as a low-pass filter and can reduce the amplitude of high-frequency fluctuations of x 2 .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 .

Design of FOSMC
Taking the time derivative on both sides of 5.1 yields Substituting 4.4 into 5.2 , we have when T L 0, and forcing Ṡ 0, then the control output can be obtained as follows:

5.4
Taking the 1 − μ th-order derivative on both sides of 5.4 will result in From 5.5 , the equivalent control can be obtained as Similar to 4.11 , when the system has load disturbances with upper limit T L-max , then the control output of FOSMC method can be given as where μ is called as the order of FOSMC method.If we set k p c, k d 1, and let A Pψ f /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.

Stability Analysis of FOSMC with Sign Function
When the sign function is used in the control output, then substituting 5.7 into 5.3 , we have From 5.8 , we can get the following.
a When S < 0, then sgn S −1, and we have 5.9 So the μ − 1 th-order fractional integration of δ 1 is higher than zero, that is, which implies that the derivative of the Lyapunov function V S Ṡ < 0.
b When S > 0, then sgn S 1, and we have From 5.11 , it is clear that when then the μ − 1 th-order fractional integration of δ 2 is lower than zero, that is, Ṡ < 0, which means that V S Ṡ < 0. From 5.8 to 5.13 , it is obvious that when k > T L-max Pψ f , 5.14 then the system is globally stable; S and Ṡ 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 D μ−1 • , the variation amplitude of Ṡ in 5.8 is smaller than that of Ṡ 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.

Stability Analysis of FOSMC with Saturation Function
From 5.7 , it can be seen that the sign 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: where ε > 0 denotes the thickness of the boundary layer.
When the saturation function is used, the control output can be rewritten as then, similar to 5.8 , substituting 5.16 into 5.3 , we have 5.17 From 5.17 , the following is clear.
a When S < 0, then sat S < 0, So the μ − 1 th-order fractional integration of δ 3 is higher than zero, that is, which means that the derivative of the Lyapunov function V S Ṡ < 0.
b When S > ε, then sat S 1, and we have Similar with 5.11 -5.13 , when From 5.22 , it can be seen that when Here, it is assumed that a load disturbance with magnitude T L-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 Ṡ < 0 is that the value of k is much higher than T L-max /P ψ f , 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 period 0 < S ε, we have Ṡ 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 As S S * > ε, then from 5.24 , it can be seen that the value of k satisfies condition 5.21 , so Ṡ < 0 and V S Ṡ < 0, that is, the system is globally stable again.In this moment, because S > 0 and Ṡ < 0, then S will decrease with negative velocity.
iii When S 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 Ṡ > 0, then S starts to increase.
iv When S increases until S > ε, then similar to ii , we have S > 0 and Ṡ < 0, then S will decrease with negative velocity.
v After several oscillations and adjustments between stages iii and iv , the sliding surface function S will finally maintain on the point of S ε, and the system is in a stable state with Ṡ 0.
When the system is in the stable state described by v , then from 4.7 or 5.1 , it can be seen that x 2 0 or D μ x 1 0, and the stable position error x 1 can be estimated as follows: Generally, when the load disturbance is T L T L < T L-max , then similar to the above analysis, the stable position error x 1 can be estimated as follows: With the maximum permissible position error x 1 of the PMSM system, 5.26 or 5.28 will be the constraint in designing the parameter ε and c or k p .
Remark 5.1.In the above analysis of parts b and c , the integration effect of the operator D μ−1 • is ignored temporarily.If the integration effect is considered, then the fractional-order μ will decide the phase delay and variation magnitude of Ṡ.When μ is too small, especially when μ 0, then the operator D μ−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 Ṡ, and the stable condition V S Ṡ < 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 D μ−1 • does not have integration action, and Ṡ 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 Ṡ will satisfy the stable condition V S Ṡ < 0, and meanwhile, the appropriate variation magnitude of Ṡ will make the sliding surface S change with small fluctuation, so a better control performance can be obtained.

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

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 term D 1−μ x 2 denotes the 1 − μ th-order differentiation of x 2 , 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 the sgn 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 x 2 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.

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 Here, the operator D μ−1 • means the fractional-order integration since 0 < μ < 1.The following is assumed: i the value of k 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 S Ṡ < 0 or in a stable state i.e., S 0 or constant, and Ṡ 0 .
Then the following three cases will be discussed.a When the system is in a reaching state and S > 0, Ṡ < 0, then If an instant load disturbance T instant which is greater than T L-max is applied on the system, then from 6.1 or 6.2 , it can be seen that in this moment δ 1 > 0 or δ 3 > 0, but because of the integration effect by the fractional-order integration operator D μ−1 • , the integration value that is, D μ−1 δ 1 or D μ−1 δ 3 , 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 It is clear that when an instant load disturbance T instant T instant > T L-max is applied on the system, then Ṡ < 0 immediately, and the system is also unstable at once.b When the system is in a reaching state and S < 0, Ṡ > 0, then Similar to the above analysis, when an instant negative load disturbance i.e., an opposite direction load disturbance T instant which is smaller than −T L-max is applied on the system, then from 6.1 or 6.2 , it can be seen that in this moment δ 1 > 0 or δ 3 > 0, but because of the integration effect by the operator D μ−1 • , the integration value, that is, D μ−1 δ 1 or D μ−1 δ 3 , 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 disturbance T instant T instant < −T L-max is exerted on the system, then according to 6.5 and 6.6 , it can be seen that Ṡ will be smaller than zero i.e., Ṡ < 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 Ṡ 0, then Ṡ k d D μ−1 δ 1 0 ⇒ δ 1 0 6.9If an instant positive or negative load disturbance T instant 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 δ 1 or δ 3 , but because of the integration effect by the fractional-order integration operator D μ−1 • , the integration value, that is, D μ−1 δ 1 or D μ−1 δ 3 , 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 disturbance T instant is greater than T L-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.

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 D μ • by the IRID method 37 , that is, dfod irid fod u, T s , N .In the simulation, the sampling frequency of FOSMC controller is 2 KHz; thus, in the IRID method, T s 0.0005 sec, and the approximation order is N 5.

System Block and Configuration
The block diagram of the PMSM drive system using FOSMC method is shown in Figure 2, 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.
As shown in Figure 2, 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, ω ref 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 Figure 2, the following parameters of SMC and FOSMC are set to be the same, that is,

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 is θ ref π rad, which is a step input with soft-start mode, and the order of the proposed FOSMC method is μ 0.6.
Figure 3 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 the saturation function is replaced by the sign function in the two methods.From Figure 3, 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 D μ−1 x 2 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 Figure 4, 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.

Simulation of Stability Condition
In this simulation, the stability condition will be tested.The position reference is step input θ ref 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 T L-max 2.568 Nm.In each of the following cases, the conventional SMC method and the proposed FOSMC method are executed.
Figures 5-8 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.
In Figures 5 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 T L-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 x 1 ≈ ε/c 0.01, which are consistent with the analysis of Section 5.4 and 5.26 .From Figure 5, one can see that when the external load is exerted on the system at t 0.5 s, 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 Figures 7 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 T L-max .Moreover, an important result can be obtained from Figures 7 and 8, that is, when the load disturbance is greater than T L-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.Figures 9-14 are the time curves of sliding surface function S, position responses, and position error, respectively, in which the sign function is adopted, and different load disturbance is applied at t 0.5 s.
In Figures 9 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 T L-max .Because of the use of sign function, 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 Figures 11 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 T L-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 Figures 13 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 T L-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.

Simulation of Dynamic Position Response with Step Input Signal
In this simulation, the position reference is θ ref π rad, the order of FOSMC is μ 0.6, and the saturation function is adopted.A step disturbance load of 3.1 Nm is applied at t 0.5 s and withdrawn at t 0.6 s, another step disturbance load of −3.1 Nm is applied at t 1.0 s and withdrawn at t 1.1 s.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.    Figure 16 a shows the control output i q between 0 s, 1.5 s , 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 i q between 0 s, 0.02 s , during which the PMSM motor just started; Figure 16 c shows the control output i q between 0.195 s, 0.22 s in this time period, the position output reaches the desired reference value; Figure 16 d shows control output i q between 0.60 s, 0.66 s , 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 i q 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.

Simulation of Dynamic Position Response with Sinusoidal Input Signal
In this simulation, the position reference is a sinusoidal trajectory with θ ref t π sin 10t rad, the order of FOSMC is μ 0.6, and the saturation function is used.A step disturbance load

Controller Performance with Different Fractional Orders
We let the motor angle to track a sinusoidal trajectory θ ref t π sin 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.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
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.

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-order μ.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.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 Figures 21,22,23,24,25,26,27,28,29,30,31, 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.

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-order μ 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.

Figure 2 : 2 a 2 b 6 Figure 3 :
Figure 2: Block diagram of the PMSM position control system.

2 b 6 Figure 4 :Figure 5 :
Figure 4: Phase traces of the conventional SMC method and the proposed FOSMC method with sign function.

Figure 6 : 20 Figure 7 :
Figure 6: Position responses and error with saturation function and load disturbance of 2.5 Nm at t 0.5 s μ 0.6 .

Figure 8 : 5 Figure 9 :
Figure 8: Position responses and error with saturation function and load disturbance of 2.6 Nm at t 0.5 s μ 0.6 .

Figure 10 : 1 Figure 11 :
Figure 10: Position responses and error with sign function and load disturbance of 2.3 Nm at t 0.5 s μ 0.6 .

Figure 12 : 10 Figure 13 :
Figure 12: Position responses and error with sign function and load disturbance of 2.5 Nm at t 0.5 s μ 0.6 .

Figure 14 :
Figure 14: Position responses with sign function and load disturbance of 2.6 Nm at t 0.5 s μ 0.6 .

Figure 15 :
Figure 15: Position responses and velocity responses with load disturbances around t 0.5 s and 1.0 s.

Figure 17 :Figure 18 :
Figure 17: Position responses and error to sinusoidal input signal with load disturbances at t 0.3 s and 0.75 s.

Figure 21 :
Figure 21: Position error and error square with sine reference input 2.3 Nm pulse disturbance, with saturation function .

Figure 22 :
Figure 22: Position error and error square with sine reference input 2.3 Nm pulse disturbance, with sign function .

Figure 23 :
Figure 23: Position error and error square with sine reference input 3.1 Nm pulse disturbance, with saturation function .

Figure 24 :
Figure 24: Position error and error square with sine reference input 3.1 Nm pulse disturbance, with sign function .

Figure 31 :
Figure 31: Position error and error square with trapezoid reference input 3.1 Nm pulse disturbance, with saturation function .

Figure 32 :
Figure 32: Position error and error square with trapezoid reference input 3.1 Nm pulse disturbance, with sign function .

Table 2 :
Controller Performance.Nm is applied at t 0.75 s and vanished at t 0.8 s.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.
Table 2shows the controller performance of the conventional SMC method and the proposed FOSMC method with different fractional-order