Modelling and optimal control of nanopositioning piezo stage
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Nanopositioners have a wide variety of applications in many fields, such as micro and nano manufacturing, medical research and study of micro and nano material properties. They are mainly used to induce forces and movements in micrometer and nanometer range. They are used as a part of special equipment's like atomic force microscope and scanning probe microscope which are widely used for the study of microscale or nanoscale material properties. Research efforts on nanopositioners can be broadly classified into two categories, modelling and control. Modelling of nanopositioners involve modelling of both their linear dynamics as well as their nonliner dynamics like creep and hysteresis. Many efforts are being made in understanding and control of these nonlinearities. Optimal control is one of the most widely used approach for a nanopositioner in order to achieve high speed high precision control.
To address the modelling issue of creep nonlinearity, traditionally, approximate linear models or logarithmic models were used. Unlike creep, hysteresis nonlinearity is quite complex to model. Hence many efforts were made to understand and mathematically formulate hysteresis, the most popular of hysteresis models are Bouc-Wen models, Prandtl-Ishlinskii models, Duhem models etc. the problem with these models is that they are hard to linearize or invert for the purpose of control, especially if they are used for the control of wide range of frequency profiles. In the last decade numerous efforts were made in modelling the nonlinear behavior of the nanopositioners using neural networks. Due to the inherent nonlinearities the optimal control of a nanopositioner is difficult. Recently many tools were developed for the nonlinear optimal control using neural network models. Model Predictive control(MPC) is one of the most widely used optimal control techniques. The main advantage of MPC are that it provides essential tools to apply constraint to the control problem. Many techniques were developed in past for linear MPC control and nonlinear MPC control using neural networks. Due to this advantages, in this work we are using MPC for the optimal control of nanopositioners.
Since nanopositioners are involved in high speed operations in sub-micrometer ranges, using purely physics-based models to formulate the dynamics may not result in accurate models. Consequently, purely data driven models or hybrid models (data+physics) are widely utilized. In the methods proposed in this work we make use of purely data driven models. The advantage of using data driven models is that they can be built without the prior knowledge of the internal physics of a system. After a data driven model is built it can latter be analyzed to understand the internal physics of a system.
In this work we present the use of traditional linear methods for the purpose of modelling and control of nonlinear behavior of nanopositioners. A way to model nonlinear behavior using linear methods is by using adaptive linear models whose parameters depend on the operating point i.e., they are time varying linear parameters. In this work we have investigated two approaches in solving the nonlinear control problem of a nanopositioner. In the first method we make use of concept of cascade control where we first optimally control the nanopositioner system using MPC techniques and then we use an additional controller on the MPC+nanopositioner system to minimize the tracking error. To demonstrate the efficacy of the proposed methods, they were developed and used for the trajectory tracking of a nanopositioner. And the result were then compared to the traditional control schemes like linear MPC etc.
The second method proposed is adaptive Model Predictive Control to achieve the optimal tracking control of the nanopositioner. Adaptive MPC is a novel method in which we use adaptive time varying linear model at each operating point to achieve the control goal. Nanopositioners are generally fast responding systems hence the sampling frequency used for their practical operations is high. Consequently, adaptive MPC will lead to optimal control performance. Since models are built for each operating point, adaptive MPC can be used for wide variety of input profiles. To demonstrate its performance, an adaptive MPC controller was developed and implemented for the trajectory tracking of a nanopositioner and its results were then compared with traditional control techniques like PID controller.
The results of both the proposed methods show that they can be used for a wide range of frequency profiles. Unlike many other data-driven techniques where the developed systems will be biased to the profile of the data used to develop the system.