“Constant in Gain Lead in Phase” Element– Application in Precision Motion Control

Abstract

This work presents a novel 'Constant in gain Lead in phase' (CgLp) element using nonlinear reset technique. PID is the industrial workhorse even to this day in high-tech precision positioning applications. However, Bode's gain phase relationship and waterbed effect fundamentally limit performance of PID and other linear controllers. This paper presents CgLp as a controlled nonlinear element which can be introduced within the framework of PID allowing for wide applicability and overcoming linear control limitations. Design of CgLp with generalized first order reset element (GFORE) and generalized second order reset element (GSORE) (introduced in this work) is presented using describing function analysis. A more detailed analysis of reset elements in frequency domain compared to existing literature is first carried out for this purpose. Finally, CgLp is integrated with PID and tested on one of the DOFs of a planar precision positioning stage. Performance improvement is shown in terms of tracking, steady-state precision and bandwidth.

Complex order control for improved loop-shaping in precision positioning

“…Recently, 'Constant-gain Lead-phase' (CgLp) filter was presented to obtain constant unity gain with phase lead in the required frequency range, thereby enabling the use of reset for broadband phase compensation. CgLp integrated into PID-framework results in better open-loop shape; and significant improvements in tracking and steady-state precision are reported in [13], [14]. CgLp is designed by using a reset first (or second)-order lag filter R in series with a corresponding linear first (or second)-order lead filter L where the two filters are designed to have the same cut-off frequency as given below.…”

“…where the arrow indicates the resetting action. Although both filters are to be designed to have same cut-off frequency, it can be seen that a different term ω rα is used for filter R. This is to account for the shift in the corner frequency of reset elements as noted in detail in [13]. An LPF at ω f is primarily used to make filter L proper.…”

“…Describing function analysis shows similar magnitude behaviour, but with reduced phase lag of 38.1 • compared to its linear counterpart. This has been extended in literature to include first-order reset element (FORE) [10], second-order reset element (SORE) [11] and even fractional-order (intended for all non-integer real numbers) reset elements (FrORE) [12] including generalization [13] to take advantage of this phase behaviour. However, most works in literature limit the potential of reset by utilizing it mainly as part of the integrator and/or using the reduced phase lag property.…”

“…However, most works in literature limit the potential of reset by utilizing it mainly as part of the integrator and/or using the reduced phase lag property. Recently, a novel reset filter termed 'Constant-gain Lead-phase' (CgLp) was designed combining FORE or SORE with its corresponding linear lead filter [13]- [15]. The describing function of CgLp yields a constant unity gain with a corresponding phase lead in the designed frequency range of interest enabling significant improvement in open-loop shape.…”

Reset Control for Vibration Disturbance Rejection

“…The cut-off frequency of R is not exactly the same as L, but instead also takes in a correction factor to accommodate for the small change in gain behaviour seen with the introduction of reset. This is explained in greater detail in Saikumar et al (2019a). This phase lead which can be achieved through CgLp can be used to compensate for the phase margin difference seen between PID and PI 2 D, where CgLp is introduced in series with PI 2 D to form CgLp-PI 2 D. Further, we should note again that CgLp is designed to provide phase lead without altering the gain characteristics of the open-loop.…”