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Robust stability analysis and improved design of phase‐locked loops with non‐monotonic nonlinearities: LMI‐based approach

Due to nonlinear nature of several phase detectors, linear approximation method often leads to performance degradation in many phase‐locked loops (PLLs), particularly when the phase errors are sufficiently large. A third or higher order PLL, in spite of the ability to track a wider variety of inputs and having higher operating‐frequency range, requires more design attention in order to ensure stable tracking. In this work, with the nonlinearities inserted into the system's model, suitable criteria that take into account the nonlinearities' non‐monotonicity, sector and slope bounds are employed to establish robust stability conditions. The result is applicable to any PLLs without order and type restrictions. For Type‐1 PLLs, the resulting condition can be used to search for the maximum stable loop gain, which is also linked to the lock‐in range of the system. In the later part of this work, the focus is devoted towards designing PLLs with high lock‐in range, which is performed via mixing the proposed method with H∞ synthesis. The searches for the parameters in both PLL analysis and design are expressed in terms of convex linear matrix inequalities, which are computationally tractable. To illustrate the improvement introduced via this approach, several numerical examples and simulations are included with comparisons over conventional methods. Copyright © 2017John Wiley & Sons, Ltd.