Nonlinear analysis of hybrid phase-controlled systems in z-domain with convex LMI searches

Abstract

Phase-controlled systems such as phase-locked loops (PLLs) have been used in numerous applications ranging from data communications to speed motor control. The hybrid case where only the phase detector is digital while others are analog has advantages over the classical PLLs in the sense that it provides a wider locking range and is more suitable when the input and output signals come in digital waveforms. Although such systems are inherently nonlinear due to the phase detector's characteristics, the nonlinearity is often bypassed in order to ease the analysis and design methods. This, however, will give erroneous results when the phase difference between input and output falls into the nonlinear range. Another source of inaccuracies in modeling PLLs is the continuous-time approximation, which is only useful if the operating frequencies of interest are much less than the incoming data transition rate. In this paper, we present a nonlinear analysis of a hybrid PLL in the z-domain where the stability is established via the discrete-time Lur'e-Postnikov Lyapunov function, and the performance is evaluated via the induced $\ell_2$ norm objective. The results are formulated in the form of linear matrix inequality searches, which are computationally tractable. We also extend the result for analysis of a PLL-based frequency synthesizer and provide several numerical examples to illustrate the effectiveness of the results compared to the existing ones.