Time‐discretizations of differentiators: Design of implicit algorithms and comparative analysis

Abstract

A complete review of the known differentiators and their time‐discretizations have been addressed in this study. To resolve the drawbacks of the explicit (forward Euler) discretization, which is commonly utilized in sliding‐mode‐based differentiators, implicit time discretization methods are proposed to handle the set‐valued functions. The proposed schemes are supported by some analytical results to show their crucial properties, for example, finite‐time convergence, exactness, invariant sliding‐surface, chattering elimination, insensitivity to the gains during the sliding‐phase, and the well‐posedness. The causal implementation of the proposed implicit schemes has been addressed clearly and supported by flowcharts. Semi‐implicit schemes are also presented to provide a compromise between the performance and the ease of implementation. Finally, an exhaustive comparison is made using numerical experiments among 25 different state‐of‐the‐art differentiators to evaluate the behaviors of the differentiators facing the noise and initial error. Realistic conditions are considered in the simulations to provide useful information based on practical conditions. General conclusions are that implicit discretizations can supersede explicit and semi‐implicit ones.

Adaptive Integral Sliding Mode Based Course Keeping Control of Unmanned Surface Vehicle

“…The dynamics generated by an homogeneous controller can be seen as a lineal dynamic system with an adaptive gain that grows to ∞ as |x(t)| → 0, generating the well know singularity at the origin which is undesired for real applications. Nevertheless, as commented in [45], the practical implementation of homogeneous dynamics system designed in the continuous time domain prevents the use of explicit Euler discretization scheme to achieve a mere copy of the continuous time approach due to its simplicity. This type of discretization is considered inappropriate, especially when set-valued functions has to be considering, causing numerical chattering and sensitivity to the gains.…”

Toward Embedded System Resources Relaxation Based on the Event-Triggered Feedback Control Approach

“…The selection of the SMC controller gains is related directly to the sampling time [34,35]. Different discrete implementations of SMC are proposed in [34][35][36][37][38], where implicit techniques overperformed explicit integration schemes. Both techniques introduced a fixed sampling period.…”

Differentiator for Noisy Sampled Signals With Best Worst-Case Accuracy

“…E STIMATING in real-time the derivatives of a signal affected by noise is a fundamental problem in control theory and continues to be an active area of research, see, e.g., the special issue [1], the comparative analysis [2], and the references therein. Differentiators are often used, for instance, for state estimation [3], Proportional-Derivative controllers, fault detection [4], [5], and unknown input observers [6].…”

“…Several techniques for that purpose have therefore been proposed, cf. [2], [15], [16]. In any case, the inherent performance limitations of continuous-time differentiators cannot be surpassed in the discrete domain via discretization.…”