On The Compensation Algorithm Of Nonlinear Systems With Backlash

In recent years, there has been increasing concern about the study of nonlinearity, it has become a hot research and focus. As related to nonlinear science and the humanities and social sciences and many other disciplines and fields, it has a very important scientific value and significance. The main difference of linear and nonlinear lies in whether or not it meets the principle of superposition. Nonlinear characteristics can be expressed in the interaction, dissipation, saturation, multi-value and several other aspects.Backlash is a typical nonlinear, exists in robotics systems, servo control, mechanical actuators, and other fields of electromagnetic, study of control systems with backlash is an important and challenging area.Backlash exists in mechanical servo control systems and transmission systems, it is an unconventional, non-smooth typical nonlinear, its complexity is manifested in the following two aspects:①Multi-mapping: The same input can have different values of the output, or the same output value can be different input;②Memory: The current output of backlash is not only related to he current input value, but also to the historical value.Considering various factors such as machinery installation, lubrication and so on, the existence of gear clearance is inevitable, so the backlash caused by gear clearance also exists objectively. The impact of backlash can be negligible for the systems whose control performance is not strict. But for the control of high precision servo control system, the existence of backlash severely restricted the control precision of the system, affecting the control performance. Therefore, the compensation of backlash is necessary.The object studied in this paper is ball and plate experimental platform BPVS-JLU II which has been independently developed by the laboratory of control theory and intelligent system in Jilin University. The image information of plate with the small ball on it can be collected by the digital camera at the top of the plate. The image coordinates of the ball can be acquired after processing by the image processing algorithms. The position information of the ball can be acquired after coordinate transformation. The information of the deviation with given location is transmitted to the system controllers, and then the stepper motors drive the plate rotate in the two mutually orthogonal directions after computing and reasoning by the controllers. Thus the ball can run to the given position of the plate or get the given movement state.Ball and plate system is a typical multi-variable, non-linear, strong coupling servo control system. Because of its complexity and typicality, an increasing number of domestic and foreign scholars have concern about it. At the same time, many kinds of ball and plate systems have been developed. There are so many kinds of drive mechanism, such as wire transmission, a direct link-driven, gear, etc. Every transmission way has its own advantages and disadvantages. The transmission of ball and plate system developed by the author's laboratory is turbo worm gear speed reducer. This transmission mode with a simple mechanical structure, modeling convenience, transmitting torque, easy to control can achieve good control performance under the conditions of the ball low-speed movement. However, when asked on the ball high-speed high-precision movement, the impact caused by the backlash can not be neglected. Since the existence of gear clearance, when the driven part in backlash, the system is temporarily not controllable, that is clearly negative for the ball and plate. Therefore, in this paper, a backlash model was built based on the analysis of backlash, and fixed inverse model and adaptive inverse model compensation controllers have been developed to compensate the backlash.The analysis of backlash and study of backlash compensation algorithm are based on the theory of inverse and adaptive control. The main research objective are: (1) To build the backlash model of backlash after the analysis of characteristics of backlash, and to take into account the overall ball and plate system and improve the system mathematical model. (2) To design the fixed inverse compensation controller and the adaptive inverse model compensation controller, and to design the adaptive law of parameters online identification, then to prove that the convergence of the adaptive laws theoretically. (3) To apply the compensation control algorithm to simulation and experimental research, and improvement dynamic characteristics of ball and plate system.Building the backlash model and the integrated mathematical model of ball and plate system is very necessary for the analysis of backlash nonlinear characteristics and designing compensation control algorithm. Three common nonlinear mathematical model of gear clearance are: backlash nonlinear model, dead zone model and "vibration - raiding" model. Backlash nonlinear model is built on the premise that the driving and driven parts contact rigidly, the output of the driven part is unchangeable. Dead-zone model is established from the perspective of torque, required that the parameters of ball and plate are more precise. "Vibration - raiding" model is usually used for dynamics and kinematics analysis of multi-gear systems. Comparison of three models, dead zone model requires higher accuracy of the model parameters, but the stepper motors output torque can not be measured or calculated precisely, and "vibration - raiding" model is usually not be applied to the control system analysis, although there are some nonlinear model assumptions in backlash nonlinear model, ball and plate system can meet these assumptions, the model also has the advantages of modeling simple and easy to realize. Therefore, considering complexity and realization of the model, and the demand of the system, we built the backlash nonlinear model of ball and plate system. And the model parameters are measured by experiments and the integrity mathematical model has been built. The describing function of backlash is derived, and stability of the system is analyzed by the method of describing function.Backlash nonlinear compensation control algorithm design is the main body of this paper. The basic principles of inverse model compensation is that, to add a backlash nonlinear inverse model in front of the backlash nonlinear model, to offset the impact of backlash nonlinear on the system, and then the backlash nonlinear model and its inverse model can constitute a bogus linear system, so that we can use various control algorithms of linear systems. The paper design the nonlinear inverse model compensation controller based on the above principle. And it is proved that the designed inverse model could theoretically achieve full compensation. We design the parameters gradient projection adaptive laws to update inverse model parameters because of the time-variability of the actual parameters of the nonlinear system. Then we overcome the shortcomings of model match error. And it shows the convergence of the adaptive laws by adopting Barl(a|ˇ)lat lemma.Finally, we have done a lot of simulation and experimental study in different experimental conditions by applying the fixed inverse model and the adaptive inverse model control compensation algorithm.In summary, this paper do a lot of academic and experimental research on the backlash nonlinear of ball and plate system, but there are still some issues that need further study. The author believes that the results of this study may be a certain extent useful for the accurate compensation of backlash nonlinear of the ball and plate system. To obtain better compensation effect, we need to make more unremitting efforts on both the hardware equipment and control compensation algorithms.