William F. Egan (1936 – December 16, 2012 [1]) was well-known expert and author in the area of PLLs. The first and second editions of his book Frequency Synthesis by Phase Lock [2] [3] as well as his book Phase-Lock Basics [4] [5] are references among electrical engineers 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. [2]: 176 [3]: 245 [4]: 192 [5]: 161 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. [6]: 96 and [7]: 6 for type II CP-PLL. The hold-in and pull-in ranges of type II APLL for a given parameters may be either (theoretically) infinite or empty, [8] 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 [9]). Although it is known that for the second-order type II APLL the conjecture is valid, [10] [5]: 146 the work by Kuznetsov et al. [9] shows that the Egan conjecture may be not valid in some cases.
A similar statement for the second-order APLL with lead-lag filter arises in Kapranov's conjecture on the pull-in range and Viterbi's problem on the APLL ranges coincidence. [11] [12] 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). [13] [14] For control systems, a similar conjecture was formulated by R. Kalman in 1957 (see Kalman's conjecture).
References
- ^ https://www.legacy.com/us/obituaries/mercurynews/name/william-egan-obituary?id=18926339
- ^ a b Egan, William F. (1981). Frequency synthesis by phase lock (1st ed.). New York: John Wiley & Sons. Bibcode: 1981wi...book.....E.
- ^ a b Egan, William F. (2000). Frequency Synthesis by Phase Lock (2nd ed.). New York: John Wiley & Sons.
- ^ a b Egan, William F. (1998). Phase-Lock Basics (1st ed.). New York: John Wiley & Sons.
- ^ a b c Egan, William F. (2007). Phase-Lock Basics (2nd ed.). New York: John Wiley & Sons.
- ^ Aguirre, S.; Brown, D.H.; Hurd, W.J. (1986). "Phase Lock Acquisition for Sampled Data PLL's Using the Sweep Technique" (PDF). TDA Progress Report. 86 (4): 95–102.
- ^ Fahim, Amr M. (2005). Clock Generators for SOC Processors: Circuits and Architecture. Boston-Dordrecht-London: Kluwer Academic Publishers.
- ^ 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. S2CID 12292968.
- ^ a b Kuznetsov, N.V.; Lobachev, M.Y.; Yuldashev, M.V.; Yuldashev, R.V. (2021). "The Egan problem on the pull-in range of type 2 PLLs" (PDF). IEEE Transactions on Circuits and Systems II: Express Briefs. 68 (4): 1467–1471. doi: 10.1109/TCSII.2020.3038075.
- ^ Viterbi, A. (1966). Principles of coherent communications. New York: McGraw-Hill.
- ^ Kuznetsov, N.V.; Lobachev, M.Y.; Mokaev, T.N. (2023). "Hidden Boundary of Global Stability in a Counterexample to the Kapranov Conjecture on the Pull-In Range". Doklady Mathematics. 108: 300–308. doi: 10.1134/S1064562423700898.
- ^ Kuznetsov N.V. (2020). "Theory of hidden oscillations and stability of control systems" (PDF). Journal of Computer and Systems Sciences International. 59 (5): 647–668. doi: 10.1134/S1064230720050093. S2CID 225304463.
- ^ Leonov G.A.; Kuznetsov N.V. (2013). "Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits". International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 23 (1): 1330002–219. Bibcode: 2013IJBC...2330002L. doi: 10.1142/S0218127413300024.
- ^ Kuznetsov, N.V.; Leonov, G.A.; Yuldashev, M.V.; Yuldashev, R.V. (2017). "Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE" (PDF). Communications in Nonlinear Science and Numerical Simulation. 51: 39–49. Bibcode: 2017CNSNS..51...39K. doi: 10.1016/j.cnsns.2017.03.010.
William F. Egan (1936 – December 16, 2012 [1]) was well-known expert and author in the area of PLLs. The first and second editions of his book Frequency Synthesis by Phase Lock [2] [3] as well as his book Phase-Lock Basics [4] [5] are references among electrical engineers 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. [2]: 176 [3]: 245 [4]: 192 [5]: 161 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. [6]: 96 and [7]: 6 for type II CP-PLL. The hold-in and pull-in ranges of type II APLL for a given parameters may be either (theoretically) infinite or empty, [8] 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 [9]). Although it is known that for the second-order type II APLL the conjecture is valid, [10] [5]: 146 the work by Kuznetsov et al. [9] shows that the Egan conjecture may be not valid in some cases.
A similar statement for the second-order APLL with lead-lag filter arises in Kapranov's conjecture on the pull-in range and Viterbi's problem on the APLL ranges coincidence. [11] [12] 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). [13] [14] For control systems, a similar conjecture was formulated by R. Kalman in 1957 (see Kalman's conjecture).
References
- ^ https://www.legacy.com/us/obituaries/mercurynews/name/william-egan-obituary?id=18926339
- ^ a b Egan, William F. (1981). Frequency synthesis by phase lock (1st ed.). New York: John Wiley & Sons. Bibcode: 1981wi...book.....E.
- ^ a b Egan, William F. (2000). Frequency Synthesis by Phase Lock (2nd ed.). New York: John Wiley & Sons.
- ^ a b Egan, William F. (1998). Phase-Lock Basics (1st ed.). New York: John Wiley & Sons.
- ^ a b c Egan, William F. (2007). Phase-Lock Basics (2nd ed.). New York: John Wiley & Sons.
- ^ Aguirre, S.; Brown, D.H.; Hurd, W.J. (1986). "Phase Lock Acquisition for Sampled Data PLL's Using the Sweep Technique" (PDF). TDA Progress Report. 86 (4): 95–102.
- ^ Fahim, Amr M. (2005). Clock Generators for SOC Processors: Circuits and Architecture. Boston-Dordrecht-London: Kluwer Academic Publishers.
- ^ 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. S2CID 12292968.
- ^ a b Kuznetsov, N.V.; Lobachev, M.Y.; Yuldashev, M.V.; Yuldashev, R.V. (2021). "The Egan problem on the pull-in range of type 2 PLLs" (PDF). IEEE Transactions on Circuits and Systems II: Express Briefs. 68 (4): 1467–1471. doi: 10.1109/TCSII.2020.3038075.
- ^ Viterbi, A. (1966). Principles of coherent communications. New York: McGraw-Hill.
- ^ Kuznetsov, N.V.; Lobachev, M.Y.; Mokaev, T.N. (2023). "Hidden Boundary of Global Stability in a Counterexample to the Kapranov Conjecture on the Pull-In Range". Doklady Mathematics. 108: 300–308. doi: 10.1134/S1064562423700898.
- ^ Kuznetsov N.V. (2020). "Theory of hidden oscillations and stability of control systems" (PDF). Journal of Computer and Systems Sciences International. 59 (5): 647–668. doi: 10.1134/S1064230720050093. S2CID 225304463.
- ^ Leonov G.A.; Kuznetsov N.V. (2013). "Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits". International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 23 (1): 1330002–219. Bibcode: 2013IJBC...2330002L. doi: 10.1142/S0218127413300024.
- ^ Kuznetsov, N.V.; Leonov, G.A.; Yuldashev, M.V.; Yuldashev, R.V. (2017). "Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE" (PDF). Communications in Nonlinear Science and Numerical Simulation. 51: 39–49. Bibcode: 2017CNSNS..51...39K. doi: 10.1016/j.cnsns.2017.03.010.