Modeling And Control Of Piezoelectric Ceramics With Non-linear Hysteresis

Many researchers favor smart material-based actuators for their excellent characteristics,such as fast response and precise positioning.Due to the inverse piezoelectric effect of piezoelectric ceramics,it is possible to obtain small displacements with high precision,so this paper uses this property to design precision positioning equipment based on piezoelectric ceramics,which can be used in high-end machine tools,medical services,and aerospace.However,piezoelectric ceramics also have many nonlinear characteristics,such as hysteresis nonlinearity and creep nonlinearity,and this paper investigates the modeling and control of a piezoelectric ceramic-driven positioning system with nonlinear hysteresis.For the modeling of the piezoelectric positioning system,this paper proposes a discrete Hammerstein structural model that connects the static and dynamic characteristics of the piezoelectric positioning system in series,uses the discrete PI(Prandtl-Ishlinskii,PI)model to describe the static hysteresis nonlinear phenomenon,uses the discrete second-order transfer function to represent the dynamic characteristics,and uses the output of the PI model.The output of the PI model is used as the input of the dynamic transfer function to model the entire piezoelectric positioning system.Since the discrete system facilitates the implementation of complex control methods without reducing the computational speed,and the transmission signal of the discrete system has higher accuracy,better control results can be obtained.Compared with the traditional Hammerstein structure,the weight coefficients of the PI model are identified using nonlinear least squares,and the linear part is obtained by most miniature squares identification,which facilitates the modularization of complex systems and simplifies the computational process while satisfying the fitting accuracy.Since the accuracy of piezoelectric positioning systems generally requires a micronanometer level,changing the system model’s parameters caused by vibration outside the system is fatal to the model-dependent control method.This paper proposes a robust tracking control method combining feedforward compensation and adaptive feedback to solve this problem.A feedforward pseudo-inverse compensator with nonlinear hysteresis characteristics is designed,and adaptive feedback control is applied for the inverse compensation error and dynamic characteristics.Here,the model’s coefficients are adaptively estimated online in realtime based on the model change characteristics,i.e.,the model’s parameters are adjusted.Then the feedback control is used to track the reference displacement.The advantage is that the controller can adapt to the change of the model parameters in real-time to keep the system in the optimal state,and the stability of the proposed control system is demonstrated theoretically.The results show that the proposed adaptive control is better than the PID control.On the other hand,for precision positioning systems with specific constraint requirements,this paper proposes a prescribed performance sliding mode control method with feedforward compensation,which has the advantage that the performance of the controller can be predetermined and the prescribed performance index can be satisfied by a practical design of the controller.Specifically,the idea of prescribed performance control is introduced into the discrete-time sliding mode control.The new prescribed performance function and dynamic convergence function are used to describe the convergence region of the error.Then the conversion error is treated as the original error to design the discrete-time sliding mode controller.The goal of the control is to converge the original error to the predetermined convergence region.At the same time,it is proved theoretically that once the error enters the predetermined convergence region,it will stay within the region.It is also theoretically proved that once the error enters the preset convergence region,it will remain in the region,and the quasi-sliding mode band of the sliding mode control is obtained.Finally,the experiments compare the conventional sliding mode control;the prescribed performance sliding mode control converges faster,the error accuracy is higher,and the sliding mode control jitter phenomenon is effectively improved.