Hysteresis nonlinearity is found in many control system applications such as piezo-actuated nanopositioners. The positioner is represented as a linear system preceded by hysteresis. This hysteresis nonlinearity is usually modeled by operators in order to simulate their effects in the closed-loop system or to use their inverse to compensate for their effects. In order to reduce the hysteresis effect, an approximate inverse operator is used as a feedforward compensator. The first part of our work considers driving an upper bound on the inversion error using the hysteresis model. This bound is a function of the input references, which is much less conservative than constant bounds. It is used in designing the closed-loop control systems. The second part is to design feedback controller to achieve the desired performance. Three different methods are used throughout this work and a comparison between them is also provided. First, we use the conventional proportional Integral (PI) control method, which is extensively used in commercial applications. However, in our method we add a feedforward component which improves the performance appreciably. Second, a sliding-mode-control (SMC) scheme is used because it is one of the very powerful nonlinear robust control methods. Other schemes like high gain feedback and Lyapunov redesign have close results to SMC and hence it is not included in this work. The third control is H∞ control. It is a robust linear control method, which deals with uncertainty in the system in an optimal control structure. Unlike the PI controller, the H∞ controller uses the features of the linear plant in the design which allows to accomplish more than the simple PI controller. Mainly, it can shape the closed-loop transfer function of the system to achieve the design objectives. Including the operators in the closed-loop system, makes it hard to obtain explicit solutions of the dynamics using conventional methods. We exploit two features of piezoelectric actuators to provide a complete solution of the tracking error. First, the hysteresis is approximated by a piece-wise linear operator. Second, the linear plant has a large bandwidth which allows using singular perturbation techniques to put the system in a time-scale structure. We show that the slope of a hysteresis loop segment plays an important role in determining the error size. Our analysis also shows how error is affected by increasing the frequency of the reference input. We verify that the accumulation of the error, which is propagating from segment to another is bounded and derive its limit. We provide a comparison between simulation and the analytic expressions of the tracking error at different frequencies. Experimental results are also presented to show the effectiveness of our controllers compared with other techniques.
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