直线电机数学模型

A simple and often adequate approach is to regard friction force as a static nonlinear function of the velocity, i.e., Ff ( y ) , which is given by Ff ( y ) By Ffn ( y ) (2)


The linear motor considered here is a current-controlled threephase epoxy core motor driving a linear positioning stage supported by recirculating bearings. The mathematical model of the system can be described by the following equations:


Advanced Motion Control: From Classical PID to Nonlinear Adapti来自百度文库e Robust Control （2010） 同上
Mq u Bq Ffn (q) (4)


where Fd Fr represents the lumped disturbance. Let qr (t ) be the reference motion trajectory, which is assumed to be known, bounded with bounded derivatives up to the second order . The control objective is to synthesize a control input u such that the output q(t) tracks qr (t ) as closely as possible in spite of various model uncertainties.




This phenomenon is still captured by (1), since the bounded variation of position-dependent friction can be lumped into the external disturbance Fd . Thus, considering (2), we can rewrite (1) as x1 x2
f cogging ( y P) f cogging ( y )


Ripple force is developed due to the periodic variation of the force constant; when the position of the translator (or load) change, the winding self-inductance varies, which in turn results in the variation of the force constant. Thus, when current flow through the coils, a positiondependent periodic force is resulted.
Cogging force and ripple force





Cogging force is a magnetic force developed due to the attraction between the permanent magnets and the iron cores of the translator. It depends only on the relative position of the motor coils w.s.t the magnets, and is independent of the motor current. Thus, this force can be described as a position-dependent function f cogging ( y ) . If the permanent magnets of the same linear motor are all identical and are equally spaced at a pitch of P, then f cogging ( y ) is a periodic function with a period of P, i.e.

where B is an equivalent viscous friction coefficient of the system, Ffn is the nonlinear friction term which can be modeled as y ys Ffn ( y ) f c ( f s f c )e sgn( x) (3)
Dynamic models of the linear motor
2013.11
Adaptive robust precision motion control of linear motors with negligible electrical dynamics: theory and experiments(2001)
Output feedback adaptive robust precision motion control of linear motors(2001)




The linear is a current-controlled three-phase epoxy core motor driving a linear positioning stage supported by recirculating bearings. To fulfill the high performance requirements, the model is obtained to include most nonlinear reflects like friction and force ripple. In the derivation of the model , the current dynamics is neglected in comparison to the mechanical dynamics due to the much faster electric response. The mathematical model of the system can described by the following equation:
Mq u F (q, q) F (q, q) Ff Fr Fd (1)
Mq u F (q, q) F (q, q) Ff Fr Fd

(1)

where q(t ), q(t ), q(t ) represents the position, velocity and acceleration of the inertia load, respectively, M is the normalized mass of the inertia load plus the coil assembly, u is the input voltage to the motor, F is the normalized lumped effect of uncertain nonlinearities such as friction F f , ripple force Fr and external disturbance Fd (e.g. cutting force in machining ) While there have been many friction models proposed, a simple and often adequate approach is to regarded the friction force as a static nonlinear function of the velocity, i.e., Ff (q) which is given by
My u F F F f Fr Fd
(1)



where y represents the position of the inertia load, M is the mass of the inertia load plus the coil assembly, u is the input voltage to the motor, F is the normalized lumped effect of uncertain nonlinearities such as the friction F f , ripple forces Fr , and external disturbance Fd (e.g., cutting force in machining).
Ff (q) Bq Ffn (q) (2)

Where B is the equivalent viscous coefficient of the system, Ffn is the nonlinear function term that can be modeled as
Ffn (q) [ f c ( f s f c )e
Mx2 u Bx2 Ffn


y x1 (4) where x [ x1 , x2 ]T represents the state vector of the position and velocity , y is the position output , and Fd Fr represents the lumped disturbance. Let yr (t ) be the reference motion trajectory. The objective is to synthesize a control input u such that the output y tracks yr (t ) as closely as possible in spite of various model uncertainties.

q qs

]sgn(q)
(3)

where f s is the level of stiction, f c is the level of Coulomb friction, and qs and empirical parameters used to describe the Stribeck effect. Thus, considering (2), one can rewrite (1) as
where f s is the level of static friction, f c is the minimum level of Coulomb friction, and ys and are empirical parameters used to describe the Stribeck effect. In practice, due to the inaccuracy of the positioning stage and ball bearings, the friction force may also depend on position y.