Fractional-order single state reset element

Abstract

This paper proposes a fractional-order reset element whose architecture allows for the suppression of nonlinear effects for a range of frequencies. Suppressing the nonlinear effects of a reset element for the desired frequency range while maintaining it for the rest is beneficial, especially when it is used in the framework of a “Constant in gain, Lead in phase” (CgLp) filter. CgLp is a newly introduced nonlinear filter, bound to circumvent the well-known linear control limitation—the waterbed effect. The ideal behaviour of such a filter in the frequency domain is unity gain while providing a phase lead for a broad range of frequencies. However, CgLp’s ideal behaviour is based on the describing function, which is a first-order approximation that neglects the effects of the higher-order harmonics in the output of the filter. Although CgLp is fundamentally a nonlinear filter, its nonlinearity is not required for all frequencies. Thus, it is shown in this paper that using the proposed reset element architecture, CgLp gets closer to its ideal behaviour for a range of frequencies, and its performance will be improved accordingly.

Stacking Integrators Without Sacrificing the Overshoot in Reset Control Systems

“…It is worth mentioning that the CgLp element has a constant gain and since according to Remark 2, CR architecture will not change the gain and phase behaviour of CgLp, the overall control loop will have a DF gain and phase matching to that of PI n D. Thus the control design in open-loop can be done in frequency domain. According to the guidelines presented in [10], [14], [19] for tuning of CgLp elements, the following values can be proposed for CgLp parameters:…”

“…Reset control has also recently been used to approximate the complex-order filters [9], [10]. Advantage of using reset control over linear control has been shown in many studies especially in precision motion control [8], [11]- [15], [15]- [19]. However, these studies are mostly focused on solving one problem.…”

“…Describing Function (DF) analysis of this element shows that it can provide broadband phase lead while maintaining a constant gain. Such an element is used in the literature to replace some part of the differentiation action in PID controllers as it will help improve the precision of the system according to loop-shaping concept [10], [13], [14], [19]. In [13], [19], it is suggested that DF analysis for reset control systems can be inaccurate as it neglects the higher-order harmonics created in response of reset control systems.…”

“…Such an element is used in the literature to replace some part of the differentiation action in PID controllers as it will help improve the precision of the system according to loop-shaping concept [10], [13], [14], [19]. In [13], [19], it is suggested that DF analysis for reset control systems can be inaccurate as it neglects the higher-order harmonics created in response of reset control systems. It is concluded that higher-order harmonics can adversely affect the steady-state precision of the system.…”

Optimal reset law design based on guaranteed cost control method for Lipschitz nonlinear systems

“…This controller was generalized to first order reset element (FORE) in Reference 2. In the last three decades, various stability results for reset control systems are reported in the literature 3‐10 . For instance, stability conditions, 11 reset times‐dependent stability, 12 delay‐independent and delay‐dependent stability, 13,14 and passivity‐based approach 15,16 .…”