MATHEMATICAL MODEL OF THE PMSM & ADRC - VS Sewing Machines

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PMSM is an important category of the electric machines, in which the rotor magnetization is created by permanent magnets attached to the rotor. Many mathematical models have been proposed for different applications, such as the abc-model and the two axis dq-model. Due to the simplicity of the two axis dq-model, it becomes the most widely used model in PMSM engineering controller design. 

The dq-model offers significant convenience for control system design by transforming stationary symmetrical AC variables to DC ones in a rotating reference frame. Based on the dq reference frame theory, the mathematical model of the PMSM can be expressed as the following equations: 1) Circuit equation ( ) d s d r q q d d u R i L i dt di L = − + ω (1) ( ) q s q r d d r f q q u R i L i dt di L = − − ω − ω Ψ (2) 2) Electromagnetic torque equation: [ ( ) ] e p f q d q d q T = n Ψ i − L − L i i (3) 3) Motion equation e L d r T T T dt d J = − − ω (4) 

Where di , qi represent the current of the d-axis and q-axis; d u , uq represent the voltage of the d-axis and q-axis; Ld , Lq represent the inductance of the d-axis and q-axis; Rs is the stator resistance; ωr is the rotor speed; Ψf

is the magnitude of the permanent magnet flux linkage; p n is the number of pole pairs; J is the inertia of the rotor;Te is the electromagnetic torque,TL is the load torque of the motor and Td is the uncertain torque disturbance caused by the external and internal disturbance ;  

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An ADRC consists of three components, a tracking differentiator (TD), an extended state observer (ESO) and a nonlinear state error feedback (NLSEF)[9], as shown in Fig. 1. Fig.1 The structure of ADRC. The function of the n-order TD is to arrange the ideal transient process. It tracks the input V(t) without overshoot and provides the generalized derivatives of the input signal, 1,1 z ,… ,n z1 . The function of (n+1)-order ESO is to observe the state variables 2,1 z , … ,n z2 and estimate the total disturbances 2,n+1 z of the plant. ESO can compensate the entire uncertain external and internal disturbance in real time. The function of NLSEF is to level off the output of the controlled plant and expand the stability region of the whole closed-loop system. 

The control output of NLSEF can be mathematically described by ( ) ( , , ) ( , , ) u0 t = k1 fal ε 1 a δ +…+ k n fal ε n a δ (5) where i ,i ,i z z = 1 − 2 ε (i =1,…n), a ,δ , and i k (i =1,…n) are adjustable parameters. The nonlinear function fal is defined by ⎪⎩ ⎪ ⎨ ⎧ ≤ > = − ε δ ε δ ε ε ε δ ε δ i a i i i a i fal i a 1 / sgn( ) ( , , ) (6) where sgn(x) is a sign function. Thus the actual control for the plant can be expressed as b z u t u t 2,n 1 0 ( ) ( ) + = − (7) According to the equation 4, the rotor speedωr can be described as: T T J T J r e L d = ( − )/ − / • ω (8) Td is considered as the total unknown disturbance of the servo system, which might be estimated and compensated by the ESO. According to the above theory, the ADRC for rotator speed is designed as below. Design the one-order TD as ( , , ) 11 0 0 δ 0 z = −k fal ε a • (9) wher ref = z − n 11 ε , 0 a , δ 0 , 0 k are adjustable parameters. 

Then the ESO can be constructed as ⎪⎩ ⎪ ⎨ ⎧ = − = − + • • ( , , ) ( , , ) ( ) 22 22 1 1 1 21 22 21 1 1 1 ε δ ε δ z k fal a z z k fal a bu t (10) where r ε 1 = z21 −ω , 1 a , δ 1 , 21 k , b, 22 k are adjustable parameters. In this paper, a conventional PI controller is used to replace the NLSEF of the ADRC in the Fig.1, which can enhance the calculating speed of the algorithm and maintain the disturbance rejection advantage of the ADRC. Where ⎪⎩ ⎪ ⎨ ⎧ = − = + ∫ u u z b u k k dt p i / 0 22 0 2 2 ε ε (11) where 2 11 21 ε = z − z , p k and i k are the adjustable parameters of the PI controller. The structural expression of the system control law shows control doesn't attach to the internal parameters of the system, but to the output and the reference input of the system.
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