NUMERICAL SOLUTION METHOD OF NOETHER SYMMETRY FOR PIEZOELECTRIC MICRO/NANO LEVEL HIGH-PRECISION POSITIONING SYSTEM

The hysteresis nonlinear effect can induce significant positioning errors in piezoelectric micro/nano level positioning system. To mitigate the impact of hysteresis nonlinearity, it is essential to perform dynamic modeling and investigate the nonlinear behavior of the system, and build a controller to compensate for hysteresis to enhance positioning accuracy. Due to the presence of non-conservative quantities such as hysteresis and dissipation, it is very difficult for traditional methods to find the Noether symmetry of the system, which leads to the inability to construct symmetry solutions for the system, thereby affecting the optimization design of the system. To address this issue, this paper proposes a Noether symmetry numerical solution method for piezoelectric micro/nano level positioning system. The study focuses on the P721.CDQ Z-axis flexible objective scanner manufactured by Physik Instrumente in Germany. Based on its structural characteristics, the Lagrange function of the system is obtained by studying its energy, and the dynamic equations of the system are established. By introducing the infinitesimal group transformations of time, charge and displacement coordinates, the generalized Noether theorem for piezoelectric micro/nano level positioning system is given. The concept of Noether symmetry for special cases is proposed, where non-conservative variables such as input, hysteresis, and external forces are treated as step sequences within each sampling time interval. The generalized Noether symmetry generators of the system are found, and the corresponding conservation quantities are obtained, leading to the construction of symmetry solutions for these special cases. On the basis of the symmetry solutions of the system, the calculation flow of Noether symmetry numerical solution method is designed by using the loop control technology, and several calculation programs are developed in MATLAB environment, which can be used to solve the hysteresis dynamic equation of piezoelectric micro/nano level positioning system. Finally, experimental verifications are carried out, comparing the output displacement responses under the same input voltage using the Noether symmetry numerical solution method, the Runge-Kutta fourth-fifth order method, and experimental measurements. Large-scale computational simulations are also performed. The results show that the Noether symmetry numerical solution method not only achieves high precision but also significantly improves the calculation speed of output response, which is more advantageous in scenarios involving large-scale computations.