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