Approximate differentiator with varying bandwidth. (c2019)

Abstract

In the 1960’s, innovative research in the former Soviet Union laid grounds for the evolution of the Sliding Mode Control (SMC). Known for its robustness and insensitivity to parameter variations, SMC immediately became an efficient tool for the control of highly non-linear systems, such as rigid robot manipulators. Despite its significant advantages, SMC was criticized for an inherent drawback manifested by the chattering phenomenon. The latter may cause vibrations that could eventually compromise the safety of the manipulator as well as degrade its tracking performance. In its simplest form, SMC requires measurements of joint positions and velocities as the terminal sliding variable vector has a non-linear term of both joint positions and velocities. Since most industrial manipulators are not equipped with velocity sensors, typical SMC applications resort to different methodologies to approximate joint velocities from measurements of joint positions. The Approximate Differentiator, also referred to as the Dirty Derivatives Filter (DF), is a first order filter that estimates the joint velocity error commonly used in feedback control. In this thesis, we exploit key differences between the continuous-time model of the DF and its discrete-time model. We show that the discrete-time filter shares the characteristics of an exponentially weighted moving average; in particular, the filter smooths the derivative of its input. We integrate the discrete-time DF with a conventional SMC and show the stability of the closed-loop system. We numerically and experimentally demonstrate how the filter estimation performance follows a convex trend in function of the filter bandwidth. We further demonstrate how the bandwidth at which the filter achieves “optimal” performance varies with the frequency of the filter input. Inspired by the latter, we propose an Approximate Differentiator with Varying Bandwidth (ADVB) where the filter bandwidth varies based on the magnitude of the position tracking error. We illustrate the superiority of the proposed ADVB over the “optimal” DF numerically and experimentally on a four-degree-of-freedom (DOF) robot manipulator. We also demonstrate that DF outperforms a High-Gain Observer for the closed-loop control system under consideration.