Adaptive control of servo system based on LuGre model

This paper established a mechanical model of feed system based on LuGre model. In order to solve the influence of nonlinear factors on the system running stability, a nonlinear single observer is designed to estimate the parameter z in the LuGre model and an adaptive friction compensation controller is designed. Simulink simulation results show that the control method can effectively suppress the adverse effects of friction and external disturbances. The simulation show that the adaptive parameter kz is between 0.11-0.13, and the value of gamma1 is between 1.9-2.1. Position tracking error reaches level 10-3 and is stabilized near 0 values within 0.3 seconds, the compensation method has better tracking accuracy and robustness.


Introduction
With the progress of science and technology, more and more machining accuracy requirements are put forward for NC machine tools [1][2]. However, due to the nonlinear friction disturbance, the ball screw feed system in the traditional control methods cannot meet the requirements of high precision control [3].
The key of friction compensation is to establish accurate friction model, and calculate the friction force of the system according to the speed, position etc. The friction model can be divided into static and dynamic [4]. At present, the LuGre model is the most widely used, the model accurately describes the dynamic and static characteristics of frictional, and has a good compensation effect. Kamalzadeh [5] proposed for the adaptive sliding mode controller for axial vibration characteristics of ball screw feed system model; Choi J[6] establish dynamic friction model according to LuGre model, show the friction hysteresis characteristic of the system, and apply it to the system sliding mode controller; Zhou Jinzhu [7] has designed a nonlinear observer according to the LuGre model, and the integral backstepping adaptive control algorithm is used.
This paper set up mechanics model of the NC system based on LuGre model, an adaptive friction compensation method is designed, analyses the stability of the adaptive control method and asymptotic convergence, and the validity of the compensation method is verified by simulation.

Feed system dynamics modeling
The simplified physical model of the feeding system of NC machine tools is shown in Figure 1： Where J m -equivalent rotational inertia of feed system; m  -Motor shaft-screw rotation angle; K-motor torque constant; u-control quantity; D-equivalent damping coefficient of feed system; F d -screw load force; R-lead screw radius. Then: LuGre dynamic model can describe all kinds of dynamic and static characteristics of friction accurately, such as creeping, limit cycle shock, pre sliding deformation, friction memory, variable static friction and static Stribeck curve. The friction compensation based on the LuGre friction model can represent the friction torque of the system as: The LuGre friction model is divided into two parts, one part is related to z, to reflect the changes of z and the relevant parts of the mane deformation, The other part is related to the viscous friction factor, and  is used to reflect the change of viscous friction factor. Therefore, the friction torque M f in the LuGre model is modified as follows: Then: Where 0 = m k  , and k 0 is the conversion factor between  and m  .

Design of friction compensation controller
Because of the poor robustness of the ordinary PID control, the high-precision tracking requirements cannot be achieved, the Backstepping method has unique advantages in dealing with nonlinear control problems. Therefore, for the above dynamic equations, a robust adaptive friction compensation controller is designed by using the Backstepping design method. In order to observe the z values in the LuGre friction model, a nonlinear observer is used in the simplified manner, and the equation is as follows: Set z as the error of the state observer: Where k 4 -is a positive real number, J and D -measurements of parameter J and D. Define the Lyapunov function further: