Higher-Order Sinusoidal Input Describing Function Analysis and Optimal Tuning of Hybrid Integrator-Gain Systems

The hybrid integrator-gain system (HIGS) is a nonlinear control element with significant potential to enhance controller performance in precise motion control. However, existing HIGS-based controller design methods typically rely on describing functions (DFs), which neglect higher-order harmonics. This paper proposes a novel HIGS-based constant in gain lead in phase (CgLp) element, allowing flexible adjustment of the induced phase lead and high-frequency distortion. Higher-order sinusoidal input describing functions (HOSIDFs) are employed as tools for frequency-domain analysis, providing a method to predict the closed-loop behavior of HIGS-controlled systems by connecting open-loop and closed-loop characteristics. Optimal tuning guidelines for HIGS-based controllers are developed using the proposed tools, accounting for the harmonic effects on closed-loop behaviors. Simulation and experimental results on an H-type precise air-floating motion control platform validate the accuracy of the proposed analysis tools and the effectiveness of the optimal tuning method. Note to Practitioners—This paper was motivated by the challenge of closed-loop behavior evaluation and parameter tuning for nonlinear controllers in precise motion control. This paper proposes a nonlinear phase lead element with constant magnitude and adjustable phase lead, which can be easily integrated into linear controllers. In addition, this paper introduces frequency-domain analysis tools based on higher-order describing functions and an iterative closed-loop performance prediction method, which accounts for higher-order harmonics introduced by nonlinearity. An optimization design method, analogous to the loop shaping method in linear control design, is also presented. Application to the H-type precise air-floating motion control platform demonstrates the effectiveness of the proposed analysis tools and controller design methods. Future research will focus on further suppressing high-frequency distortion in nonlinear controllers.