2018/10/26 by Maben Rabi, Rabi, Maben
Computer Science · Engineering · Mathematics · #Control and Stability of Dynamical Systems #FOS: Electrical engineering #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Numerical methods for differential equations #Optimization and Control (math.OC) #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1810.11371
openalex publication_date 2018/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a relay feedback system (RFS) having an ideal relay element and a\nlinear, time-invariant, second order plant. We model the relay element using an\nideal on-off switch. And we model the second order plant with a transfer\nfunction that: (i) is Hurwitz stable, (ii) is proper, (iii) has a positive real\nzero, and (iv) has a positive DC gain.\n We analyze this RFS using a state space description, with closed form\nexpressions for the state trajectory from one switching time to the next. We\nprove that the state transformation from one switching time to the next: (a)\nhas a Schur stable linearization, (b) is a contraction mapping, and (c) maps\npoints of large magnitudes to points with lesser magnitudes. Then using the\nBanach contraction mapping theorem, we prove that every trajectory of this RFS\nconverges asymptotically to an unique limit cycle. This limit cycle is\nsymmetric, and is unimodal as it has exactly two relay switches per period.\nThis result helps understand the behaviour of the relay autotuning method, when\napplied to second order plants with no time delay.\n We also treat cases where the plant either has no finite zero, or has exactly\none zero and that is negative.\n