The Gardner Problem on the Lock-In Range of Second-Order Type 2 Phase-Locked Loops

• 成果类型: Article
• 署名作者: Kuznetsov, Nikolay V.; Lobachev, Mikhail Y.; Yuldashev, Marat V.; Yuldashev, Renat V.; Tavazoei, Mohammad Saleh
• 署名单位: Saint Petersburg State University; Russian Academy of Sciences; Sharif University of Technology
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2023.3277896
• 发表日期: 2023
• 页码: 7436-7450
• 关键词: Cycle slipping; Gardner problem; lock-in range; Lyapunov methods; nonlinear control systems; phase-locked loops (PLLs); global stability analysis

摘要

Phase-locked loops (PLLs) are nonlinear automatic control circuits widely used in telecommunications, computer architecture, gyroscopes, and other applications. One of the key problems of nonlinear analysis of PLL systems has been stated by Floyd M. Gardner as being to define exactly any unique lock-in frequency. The lock-in range concept describes the ability of PLLs to reacquire a locked state without cycle slipping and its calculation requires nonlinear analysis. This work analyzes a second-order type 2 PLL with a sinusoidal phase detector characteristic. Using the qualitative theory of dynamical systems and classical methods of control theory, we provide stability analysis and suggest analytical lower and upper estimates of the lock-in range based on the exact lock-in range formula for a second-order type 2 PLL with a triangular phase detector characteristic. Applying phase plane analysis, an asymptotic formula for the lock-in range, which refines the existing formula is obtained. The analytical formulas are compared with computer simulation and engineering estimates of the lock-in range. The comparison shows that engineering estimates can lead to cycle slipping in the corresponding PLL model and cannot provide a reliable solution for the Gardner problem, whereas the lower estimate presented in this article guarantees frequency reacquisition without cycle slipping for all parameters, which provides a solution to the Gardner problem.