Special class of positive definite functions for formulating adaptive micro/nano manipulator control

This paper presents a special class of positive definite functions for the formulation of adaptive control strategies, specifically in the research of an effective control algorithm for piezoelectric actuation systems in micro/nano manipulation. To deal with the control problems of unknown system parameters, nonlinear hysteresis effects, and disturbances in the piezoelectric actuation systems, an adaptive control methodology is proposed. Using the saturation function derived from a positive definite function to formulate the control methodology, the closed-loop system stability can be guaranteed. Furthermore, the control methodology is proposed to track a desired motion trajectory in position, velocity, and acceleration. In this paper, a special positive definite function is introduced and a control formulation as well as a stability analysis is detailed. Implementation of the control methodology is practical and requires only a knowledge of the initial estimate of the system parameters. In this study, control experiments conducted using the proposed control approach on a piezoelectric actuation system has demonstrated a promising ability in tracking a specified motion trajectory. With this control capability in the presence of unknown system parameters, hysteresis, and external disturbance, the adaptive methodology is very attractive in the field of micro/nano manipulation in which high performance control applications could be realised.