Application Research Of Lie Group To The Dynamic Analysis Of Piezo-actuated Positioning System

In recent years,with the quick development of nano science and technology,the positioning system actuated by piezoceramic gradually becomes the key component realizing micromanipulation in many precision manufacturing and precision measuring equipments,and it is also an important research direction of intelligent structure.Hysteresis nonlinearity is one of the important factors which reduce positioning accuracy of the piezoelectric positioning system and restrict its applications in in the field of ultraprecision positioning.Due to the existence of non-conservative force such as hysteresis and dissipation,the numerical solutions are generally adopted to solving the piezo-actuated positioning system.However,the calculating speed of popular numerical solving methods for hysteresis nonlinear dynamic equations of piezo-actuated positioning system is not ideal in the occasions with large amount of calculations and real-time control.This dissertation takes piezo-actuated positioning system as the research object,builds appropriate dynamic model that can describe the hysteresis of piezo-actuated positioning system,introduces the Lie group method into the analysis of the positioning system,establishs the general theory of piezo-actuated positioning system,presents a new Noether symmetric numerical solving method,which can meet the needs of the engineering applications,of piezo-actuated positioning system by studying symmetries and the corresponding conserved quantities of piezo-actuated positioning system in special case,and then applies the obtained results to simulation calculation of the output displacement of piezo-actuated positioning system,parameter identification of the system model and design of controller for compensating hysteresis nonlinear effect of the system.The main research contens and achievements are listed as follows.A modified electromechanical coupling model for piezo-actuated positioning system is presented,and dynamic equations of the system are built.The classical model is derived from constitutive relations of piezocearmic actuators,and a modified electromechanical coupling model is presented combining the characteristics of the other components of piezo-actuated positioning system.By studying the energy of the system,the total electromechanical coupling energy is given for the first time,from which the Lagrange function of the system can be obtained,and then the dynamic equtions are established consequently.The Prandtl-Ishlinskii(P-I)model and Duhem model are employed to describe the hysteresis nonlinearitity of the system,repectively.The Noether symmetry and Lie symmetry are studied,and the symmetries of the system in special case are solved.The piezo-actuated positioning system of axial movement is selected as the research object,and the infinitesimal transformations of displacement,electric charge and time are introduced.According to the invariant of the Hamilton action for the piezo-actuated positioning system,the generalized Noether identity,the generalized Killing equtaions and the generalized Noether theorem of the piezo-actuated positioning system are presented.For the convenience of solving the Noether symmetries of the piezo-actuated positioning system,the concept of Noether symmetries in special case is proposed for the first time,the generators of Noether symmetries of the system and the gauge function are derived,and then the corresponding conserved quantities of the system are obtained in this case.This practice solves the problem that it is more difficult to seek the solutions of the generalized Noether identity and the generalized Killing equtaions.According to the invariant of the motion differential equations for the piezo-actuated positioning system,the determining equations as well as structural equation of Lie symmetries and Lie theorem are presented,and the generators of Lie symmetries of the piezo-actuated positioning system are derived in special case.A new Noether symmetric numerical solving method of the hysteresis nonlinear dynamic equations for piezo-actuated positioning system is presented.Utilizing obtained conserved quantities in special case,the symmetric solutions of the system in each sampling interval are derived.The hysteresis nonlineartity is described by the Prandtl-Ishlinskii model and Duhem model,a new Noether symmetric numerical solving method of nonlinear dynamic equations with hysteresis for the piezoelectric positioning system is provided by the designed fast algorithm.Some comparisons of the new Noether symmetric numerical solving method,the several numerical methods and experimental measurement are conducted,and the results verify the effectiveness of the new Noether symmetric numerical solving method.More importantly,the further experiment shows that compared with common numerical algorithm the new Noether symmetric numerical solving method not noly has higher accuracy but also can greatly improve the computing speed,and it has more advantages in the occasions with large amount of calculations and real-time control.An improved particle swarm optimization algorithm is proposed for parameter identification of hysteresis nonlinear dynamic model for piezo-actuated positioning system.On the basis of the classical PSO algorithm,through three improvements of the introduction of the adaptive inertia weights and acceleration coefficients,adding the adaptive mutation module and the introduction of the new Noether symmetric numerical solving method in the calculation of particle objective function,the improved particle swarm optimization algorithm is developed in MATLAB,which is applied to parameter identification experiments of hysteresis nonlinear dynamic model for the piezo-actuated positioning system.It can bee seen from the identified results that the improved PSO has faster convergence speed and smaller running time.With identified parameters,the error between the simulation data and experimental data is very small.The results indicated the effectiveness of the improved PSO algorithm.Based on the research results of applying Lie group method to piezo-actuated positioning system,a controller is designed to suppress the hysteresis effect.Taking the Duhem hysteresis nonlinear dynamic model as a simulator of the system,where the output displacement of the system is calculated by the new Noether symmetric numerical solving method.A z-axis piezo-actuated objective positioning system is selected as the research object,using the control method based on the hysteresis inverse model,a controller including feedforward compensation and feedback compensation is designed with the Duhem hysteresis nonlinear dynamic model,where the feedforward compensation is implemented by the inverse Duhem hysteresis dynamic model and the feedback compensation use the proportional-integral-derivative(PID)control,and then the precision positioning control of piezo-actuated objective positioning system can be realized.Based on the improved PSO algorithm,the parameter tuning method of PID controller is proposed,and the optimal parameters of PID controller are obtained.Finally,the simulation experiments of the designed controller are conducted.