Contraction-Based Nonlinear Controller for a Laser Beam Stabilization System Using a Variable Gain

In this letter, we propose a contraction-based variable gain nonlinear control scheme for the laser-beam stabilizing (LBS) servo-system, which guarantees that the closed-loop system is convergent. With the variable gain acting on the velocity error, the well known waterbed effect of the low-frequency/bandwidth trade-off can be overcome. Moreover, the contraction-based framework allows us to extend the linear control performance metrics for analyzing the closed-loop nonlinear system behavior. The closed-loop system's performance is evaluated in numerical simulations under input disturbances and/or white noise measurements and its efficacy is compared to that using PID and LQG controllers.

I. Introduction

Laser beam steering systems have been used in modern engineering technologies, in which high precision and robustness are required. For instance, in laser-based manufacturing processes and printing, surgical robotics, optical communications, advanced scientific instruments in physics and astronomy, optical storage drive, bar code scanning, among others [1], [2], [3]. Control techniques for beam steering are key in the aforementioned opto-mechatronics applications. The LBS problem, roughly speaking, refers to dynamically control the beam’s direction in order to stabilize the beam’s image at a target point [4]. The main difficulties for solving the LBS problem arises from the narrow beam divergence angle and vibration of the pointing system. In order to obtain precision pointing of the laser beam and high-bandwidth rejection of jitters produced by the platform vibrations, one uses active mirrors in the beam stabilizer. Then, by sampling a small percentage of the beam, the active mirrors can stabilize the beam’s motion by using feedback control from position sensing detectors [1], [5]. The necessity of high accuracy in the pointing of the laser beams poses a real challenge for the successful operation due to low-frequency/bandwidth trade-off. In order to deal with these problems, many control approaches have been designed and evaluated for such systems, e.g., adaptive control [4], [6], frequency weighting method [7], PID-based controllers [5], fractional order PID control [8], approaches [2], [3], integral resonant control [9]; to name a few. From linear control literature, the waterbed effect has been known and recognized by control practitioners where design trade-off must be made in increasing the closed-loop bandwidth and the low-frequency disturbance rejection properties at the cost of deteriorating the sensitivity to high-frequency measurement noise. On the other hand, nonlinear control schemes like nonlinear PID and sliding mode controllers [10] can take into account the low-frequency/bandwidth trade-off. Similar to the linear control counterpart, the performance in terms of noise measure attenuation is increased without unnecessarily deteriorating the time response of the closed-loop system. Nevertheless, the aforementioned trade-off in the nonlinear setting is less intuitive and the design procedure is not straightforward.