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Floyd M. Gardner introduced a lock-in range concept for PLLs and posed the problem on its formalization (known as Gardner problem on the lock-in range[1][2]). In the 1st edition of his book he introduced a lock-in frequency concept for the PLL in
the following way:[3]: 40 If, for some reason, the frequency difference between input and VCO is less than the loop bandwidth, the loop will lock up almost instantaneously without slipping cycles. The maximum frequency difference for which this fast acquisition is possible is called the lock-in frequency. Later, in the 2nd and 3rd edition of his book, Gardner noted that despite its vague reality, lock-in range is a useful concept and there is no natural way to define exactly any unique lock-in frequency[4]: 70 [5]: 187–188 .
^Kuznetsov, N.V.; Lobachev, M.V.; Yuldashev, M.V.; Yuldashev, R.V. (2019). "On the Gardner problem for phase-locked loops". Doklady Mathematics. 100 (3): 568–570. doi:10.1134/S1064562419060218.
^Leonov, G. A.; Kuznetsov, N. V.; Yuldashev, M. V.; Yuldashev, R. V. (2015). "Hold-in, pull-in, and lock-in ranges of PLL circuits: rigorous mathematical definitions and limitations of classical theory". IEEE Transactions on Circuits and Systems I: Regular Papers. 62 (10). IEEE: 2454–2464. arXiv:1505.04262. doi:10.1109/TCSI.2015.2476295.
