Stability Analysis of Control Systems With First-Order Reset Elements, a Frequency-Domain Approach

Reset control systems (RCSs) can achieve performance beyond that of conventional linear time-invariant (LTI) controllers, while also allowing analysis directly in the frequency domain using measured frequency response functions (FRFs). Despite this potential, existing frequency-domain stability approaches are typically restricted to specific RCS architectures and commonly depend on parametric plant models, which limits their applicability in practice. In this paper, a generalizedHβframework is developed for the most comprehensive class of RCS structures, incorporating pre-, post-, and parallel LTI filters, as well as nonzero after-reset values. Based on this formulation, an FRF-based representation corresponding to theHβtransfer function is derived, and frequency-domain sufficient conditions are established to certify theHβ-based quadratic stability criterion. As a result, the proposed framework enables direct FRF-based assessment of quadratic stability and convergence for the considered class of reset control systems, using the measured plant FRF together with the known controller and filter transfer functions, without requiring an explicit parametric plant model. The effectiveness and practical relevance of the method are demonstrated through an illustrative industrial case study.