Design of a novel discrete dual-switching tracking differentiator

Abstract

Signal filtering and differential acquisition are classic yet challenging issues in control engineering. The discrete-time optimal control (DTOC) based on classic tracking differentiator (TD) can effectively extract differentiation signals and filter signals, while eliminating the chattering problem that arises during the discretization of the continuous solution. However, under external disturbance, the convergence mode may change, leading to overshoot and noise amplification. In this paper, a dual-switching strategy is proposed, which can alternate between the base double-integral system and its dual system according to the quadrant of the system’s state. And a novel linearized control law is also introduced, deriving a novel dual-switch tracking differentiator. Further analysis of system convergence and time optimality is provided. Simulation results show that the application of this dual-switching strategy notably reduces overshoot in both tracking and differential signals while enhancing noise filtering performance. Moreover, experiments conducted on a permanent magnet synchronous motor (PMSM) platform, where the proposed TD acts as a filter in the speed feedback loop, demonstrate that the standard deviation between the reference speed and the target speed (at a constant speed of 378 r/min) decreased from 5.63 r/min to 4.93 r/min, compared to the moving average algorithm.

1. Introduction

Signal filtering and differentiation have long been the key challenges in control engineering, particularly since signal differentiation is crucial in many control algorithms. Extensive research has been conducted on differentiator design, including linear time tracking differentiator [1], based on the continuous linear system, high-gain observer-based differentiators [2], which requires high gain values to ensure differentiation accuracy, and sliding-mode based differentiators, such as high-order super-twisting based differentiator [3] and a hybrid robust finite-time differentiator [4] among others [5−8].

Initially developed by HAN [9], the discrete tracking differentiator was proposed using the isochronic region (IR) method, denoted as Fhan-TD [10], which solved the high-frequency chattering problem that occurs in the direct digitization of continuous differentiator solution. This tracking differentiator has several advantages: it requires weaker constraints on input signals [11], avoiding setpoint jumps [12]. It also plays an important role in active disturbance rejection control (ADRC) [9] and has been applied in various scenarios, including speed and position detection for maglev trains [13, 14], speed filtering for electric motors [15], and speed estimation on optical encoders [16], etc. [17, 18]. Moreover, Ref. [10] and relevant research demonstrate that the discrete-time optimal control (DTOC) is not a bang-bang control [19], and the essence of the control law is a sliding-mode control with boundary levels [10].

However, overshoots persist in signal tracking and differentiation when using the discrete tracking differentiator Fhan-TD [9], which reduces the anti-interference capability in the differential signal. This issue occurs in at least three scenarios: 1) when it approaches the steady state; 2) when the system step size is too large; and 3) when the input signal is heavily polluted by noise. Regarding the first case, SUN et al [20] found that the actual state trajectory does not always align with the discrete points on the minimum time state trajectory, causing the trajectory to cross the coordinate axes before reaching the steady state. XIE et al [21, 22] also pointed out that the trajectory is suboptimal, with larger step sizes leading to bigger overshoots and static tracking errors. Both SUN et al and XIE et al modified the control synthesis function based on two factors: the positional relationship between the state and the optimal trajectory points, and a time criterion comparing the sampling step to the time required for the state to return to the origin. Many researchers also focus on improving the control synthesis function, including ZHANG’s linearized control strategy proposed by ZHANG et al [23], a novel control algorithm based on the hyperbolic tangent function [24], and a control law based on the position relationship of the state and corresponding characteristic points on the boundary curves [25]. These methods are computationally simpler and generally easier to apply, with some already implemented in industrial applications [26, 27]. Notably,