William F. Egan (1936 – 2018) was well-known expert and author in the area of

PLL A phase-locked loop or phase lock loop (PLL) is a control system that generates an output Signal (electrical engineering), signal whose phase (waves), phase is related to the phase of an input signal. There are several different types; the simp ...

s. The first and second editions of his book ''Frequency Synthesis by Phase Lock'' as well as his book ''Phase-Lock Basics'' are references among

electrical engineers Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

specializing in areas involving PLLs.

Egan's conjecture on the pull-in range of type II APLL

In 1981, describing the high-order PLL, William Egan conjectured that type II APLL has theoretically infinite the hold-in and pull-in ranges. From a mathematical point of view, that means that the loss of global stability in type II APLL is caused by the birth of self-excited oscillations and not hidden oscillations (i.e., the boundary of global stability and the pull-in range in the space of parameters is trivial). The conjecture can be found in various later publications, see e.g. and for type II

CP-PLL Charge-pump phase-locked loop (CP-PLL) is a modification of phase-locked loop A phase-locked loop or phase lock loop (PLL) is a control system that generates an output signal whose phase is related to the phase of an input signal. There are ...

. The hold-in and pull-in ranges of type II APLL for a given parameters may be either (theoretically) infinite or empty, thus, since the pull-in range is a subrange of the hold-in range, the question is whether the infinite hold-in range implies infinite pull-in range (the Egan problem). Although it is known that for the second-order type II APLL the conjecture is valid, the work by Kuznetsov et al. shows that the Egan conjecture may be not valid in some cases. A similar statement for the second-order APLL with lead-lag filter is known as Kapranov's conjecture on the pull-in range of type I APLL. In general, his conjecture is not valid and the global stability and the pull-in range for the type I APLL with lead-lag filters may be limited by the birth of hidden oscillations (hidden boundary of the global stability and the pull-in range). For control systems, a similar conjecture was formulated by R. Kalman in 1957 (see

Kalman's conjecture Kalman's conjecture or Kalman problem is a disproved conjecture on absolute stability of nonlinear control system with one scalar nonlinearity, which belongs to the sector of linear stability. Kalman's conjecture is a strengthening of Aizerman' ...

References

{{DEFAULTSORT:Egan, William F. 1936 births American electrical engineers Hidden oscillation