Автореферат (Аналитико-численные методы исследования скрытых колебаний), страница 5

Gardner (èçâåñòíûé àìåðèêàíñêèé èíæåíåð, IEEE Fellow)25 : If, for some reason, the frequency dierencebetween input and VCO is less than the loop bandwidth, the loop will lock upalmost instantaneously without slipping cycles. The maximum frequency dierencefor which this fast acquisition is possible is called the lock-in frequency .Ýòî ïîíÿòèå ïîëó÷èëî øèðîêîå ðàñïðîñòðàíåíèå è àêòèâíî èñïîëüçóåòñÿ èíæåíåðàìè âíàñòîÿùåå âðåìÿ26 .

Îäíàêî ñ ìàòåìàòè÷åñêîé òî÷êè çðåíèÿ ïîíÿòíî, ÷òî óñòîé÷èâûå ñîñòîÿíèÿ ðàâíîâåñèÿ ìàòåìàòè÷åñêîé ìîäåëè ÔÀÏ â ïðîñòðàíñòâå ôàçñèãíàëîâ (30) èìåþò îãðàíè÷åííóþ îáëàñòü ïðèòÿæåíèÿ áåç ïðîñêàëüçûâàíèÿè äëÿ äîñòàòî÷íî áîëüøèõ ñîñòîÿíèé ôèëüòðà âòÿãèâàíèå â ñèíõðîíèçì áóäåòïðîèñõîäèò ñ ïðîñêàëüçûâàíèåì öèêëîâ.  1979 ãîäó F. Gardner ñôîðìóëèðîâàëïðîáëåìó òî÷íîãî îïðåäåëåíèÿ ïîëîñû çàõâàòà áåç ïðîñêàëüçûâàíèÿ27 : there isno natural way to dene exactly any unique lock-in frequency, âìåñòå ñ òåì óêàçàâíà âàæíîñòü ýòîãî ïîíÿòèÿ: despite its vague reality, lock-in range is a usefulconcept .24 A.Viterbi, Principles of Coherent Communications.

New York: Mc- Graw-Hill, 1966.Øàõãèëüäÿí Â.Â., Ëÿõîâêèí À.À., Ñèñòåìû ôàçîâîé àâòîïîäñòðîéêè ÷àñòîòû. Ñâÿçü, 1972.25 F. Gardner, Phaselock Techniques, John Wiley & Sons, 1966. P. 40.26 R. Best, Phase-Lock Loops: Design, Simulation and Application, 6th ed. McGraw-Hill, 2007.V. Kroupa, Frequency Stability: Introduction and Applications, Wiley-IEEE Press, 2012.27 F. Gardner, Phaselock Techniques, 2nd ed. New York: John Wiley & Sons, 1979. P. 70.F.

Gardner, Phaselock Techniques, 3rd ed. Wiley, 2005. PP. 187-188.230.060.060.060.040.040.040.020.020.02xx0x00−0.02−0.02−0.02−0.04−0.04−0.04−0.06−50θ∆510−0.06−50θ∆510−0.06−50θ∆510e èÐèñ. 5: Ôàçîâûå ïîðòðåòû êëàññè÷åñêîé ÔÀÏ äëÿ ïîëîæèòåëüíîãî ω∆ = |ω|e ïðè ðàçëè÷íûõ |ω|e . Îáëàñòè ïðèòÿæåíèÿ áåç ïðîñêàëüçûâàíèÿîòðèöàòåëüíîãî ω∆ = −|ω|çàøòðèõîâàíû (ñâåðõó ãîðèçîíòàëüíàÿ øòðèõîâêà äëÿ ω∆ > 0, ñíèçó âåðòèêàëüíàÿøòðèõîâêà äëÿ ω∆ < 0)..freefreefreefreefreeôàçîâûé ïîðòðåò ìîäåëè (30) èìååò îáëàñòè, ãäåÄëÿ ëþáîãî çíà÷åíèÿ ω∆ó òðàåêòîðèé íåò ïðîñêàëüçûâàíèÿ öèêëîâ (îáëàñòü ïðèòÿæåíèÿ áåç ïðîñêàëüçûâàíèÿ).

Îáëàñòü ïðèòÿæåíèÿ áåç ïðîñêàëüçûâàíèÿ ÿâëÿåòñÿ îáúåäèíåíèåìëîêàëüíûõ îáëàñòåé ïðèòÿæåíèÿ áåç ïðîñêàëüçûâàíèÿ, êàæäàÿ èç êîòîðûõ ñîîòâåòñòâóåò îäíîìó èç ñîñòîÿíèé ðàâíîâåñèÿ è èìååò ñâîþ ôîðìó (ñì., íàïðèìåð çàøòðèõîâàííûå îáëàñòè íà Ðèñ. 5, ñëåâà, îïðåäåëÿåìûå ñîîòâåòñòâóþùèìèfreefreeñåïàðàòðèñàìè). Ðàññìîòðèì ω∆è ââåäåì îáîçíà÷åíèå Dlock-in (ω∆) äëÿ ñîîòfree∈ Ωâåòñòâóþùåé îáëàñòè çàõâàòà áåç ïðîñêàëüçûâàíèÿ.

Äëÿ ìíîæåñòâà ω∆ðàññìîòðèì ïåðåñå÷åíèå ñîîòâåòñòâóþùèõ îáëàñòåé çàõâàòà áåç ïðîñêàëüçûâàíèÿ:\freeDlock-in (Ω) =Dlock-in (ω∆).free ∈Ωω∆ ðàáîòå ïðåäëîæåíî ñëåäóþùåå îïðåäåëåíèå:Îïðåäåëåíèå 4. Ïîëîñà çàõâàòà áåç ïðîñêàëüçûâàíèÿ ýòî òàêîé íàèáîëü-øèé èíòåðâàë[0, ωl ),÷òî äëÿfree|ω∆| ∈ [0, ωl )ìàòåìàòè÷åñêàÿ ìîäåëü ÔÀÏ âïðîñòðàíñòâå ôàç ñèãíàëîâ ÿâëÿåòñÿ ãëîáàëüíîé àñèìïòîòè÷åñêè óñòîé÷èâîé (ò.å.[0, ωl )ñîäåðæèòñÿ â ïîëîñå çàõâàòà) è ñëåäóþùàÿ îáëàñòü:Dlock-in (−ωl , ωl ) =\free|ω∆|<ωlfreeDlock-in (ω∆)ñîäåðæèò âñå ñîîòâåòñòâóþùèå ñîñòîÿíèÿ ðàâíîâåñèÿ:freefreexeq (ω∆), θeq (ω∆) ∈ Dlock-in (−ωl , ωl ) .24Èç îïðåäåëåíèÿ ñëåäóåò, ÷òî åñëè ñõåìà ÔÀÏ íàõîäèòñÿ â ñèíõðîííîì ðåæèìå (ñîîòâåòñòâóåò àñèìïòîòè÷åñêè óñòîé÷èâîìó ñîñòîÿíèþ ðàâíîâåñèÿ ìàòåìàòè÷åñêîé ìîäåëè â ïðîñòðàíñòâå ôàç ñèãíàëîâ), òî ïîñëå ðåçêîãî èçìåíåíèÿfreeâíóòðè ïîëîñû çàõâàòà áåç ïðîñêàëüçûâàíèÿ [0, ωl ) ñèñòåìà âòÿãèâàåòñÿ âω∆ñèíõðîíèçì áåç ïðîñêàëüçûâàíèÿ, åñëè íå ïðåðûâàòü ïåðåõîäíûé ïðîöåññ.Íà Dlock-in ìîãóò áûòü íàëîæåíû ðàçëè÷íûå äîïîëíèòåëüíûå óñëîâèÿ, íàïðèìåð, ÷òî îíà äîëæíà ñîäåðæàòü ïðÿìóþ, îïðåäåëÿåìóþ x ≡ 0.

Åñëè âìåñòîãëîáàëüíîé óñòîé÷èâîñòè â îïðåäåëåíèè ìíîæåñòâà çàõâàòà ðàññìàòðèâàåòñÿóñòîé÷èâîñòü â îáëàñòè, îïðåäåëÿåìîé Xreal , òî áóäåì òðåáîâàòü, ÷òîáû ïåðåTñå÷åíèå Dlock-in Xreal ñîäåðæàëî âñå ñîîòâåòñòâóþùèå ñîñòîÿíèÿ ðàâíîâåñèÿ. Âfreeîáùåì ñëó÷àå, êîãäà íåò ñèììåòðèè ïî îòíîøåíèþ ê ω∆, â îïðåäåëåíèè íåîáõîäèìî ðàññìàòðèâàòü èíòåðâàë, ñîäåðæàùèé íóëü.Íà îñíîâå ïðåäëîæåííîãî ïîäõîäà áûëè ïîñòðîåíû28 áèôóðêàöèîííûå äèàãðàììû ïîëîñû çàõâàòà áåç ïðîñêàëüçûâàíèÿ äëÿ êëàññè÷åñêèõ ÔÀÏ ñ òèïîâûìè ôèëüòðàìè ïåðâîãî ïîðÿäêà. 50õ ãîäàõ ïðîøëîãî âåêà J. Costas (èíæåíåð General Electric) èçîáðåë ñõåìóÊîñòàñà, êîòîðàÿ ïîçâîëÿåò ïðîâîäèòü äåìîäóëÿöèþ è âîññòàíîâëåíèå íåñóùåéñèãíàëîâ (Ðèñ.

6). Çäåñü âõîäîì ñõåìû ÿâëÿåòñÿ ïðîèçâåäåíèå ñèãíàëà äàííûõвходm(t)sin(θ1(t))=данныенесущаяφ1(t)= 1 m(t)(cos(θΔ(t))-cos(θ1(t)+θ2(t)))2sin(θ2(t))ПГg(t)ФФНЧ1g1(t)φ(t)o90φ2(t)= 1 m(t)(sin(θΔ(t))+sin(θ1(t)+θ2(t)))2ФНЧ2g2(t)Ðèñ. 6: Ñõåìà Êîñòàñà â ïðîñòðàíñòâå ñèãíàëîâ íà óðîâíå ôèçè÷åñêîé ðåàëèçàöèè.m(t) ∈ {+1, −1} è íåcóùåé sin(θ1 (t)); sin(θ2 (t)) ñèãíàë ïîäñòðàèâàåìîãî ãåíåðàòîðà (ÏÃ); θ1,2 (t) ôàçû; θ∆ (t) = θ1 (t) − θ2 (t); ⊗ ïåðåìíîæèòåëü; ÔÍ×1,ÔÍ×2 ôèëüòðû íèçêèõ ÷àñòîò; Ô ëèíåéíûé ôèëüòð; áëîê 90o ñäâèãàåòôàçó ñèãíàëà íà π2 ; g1,2 (t) âûõîäû ôèëüòðîâ íèçêèõ ÷àñòîò; g(t) ñèãíàë íàóïðàâëÿþùåì âõîäå ÏÃ.

 íàñòîÿùåå âðåìÿ ñõåìà Êîñòàñà øèðîêî ïðèìåíÿåòñÿ,28 M.V.Blagov, E.V. Kudryashova, N.V. Kuznetsov, G.A. Leonov, M.V. Yuldashev, R.V. Yuldashev,Computation of lock-in range for classic PLL with lead-lag lter and impulse signals, IFAC-PapersOnLine, 49(14),2016, 42-44.K.D. Aleksandrov, N.V. Kuznetsov, G.A. Leonov, P. Neittaanmaki, M.V. Yuldashev, R.V. Yuldashev,Computation of the lock-in ranges of phase-locked loops with PI lter, IFAC-PapersOnLine, 49(14), 2016, 36-41.25íàïðèìåð, â ñèñòåìàõ ñâÿçè è ñïóòíèêîâîé íàâèãàöèè.

Êëàññè÷åñêàÿ ìàòåìàòè÷åñêàÿ ìîäåëü ñõåìû29 , ïðåäëîæåííàÿ Êîñòàñîì, ñîâïàäàåò ñ ñèñòåìîé (30) äëÿϕ(θ∆ ) = 18 sin(2θ∆ ) (äëÿ ñèãíàëîâ îáùåãî âèäà ôîðìóëû õàðàêòåðèñòèêè ïîëó÷åíû â [20]). íàñòîÿùåé ðàáîòå ïîêàçàíî, ÷òî ñòðîãîå ïðèìåíåíèå ìåòîäà óñðåäíåíèÿïðèâîäèò ê ñëåäóþùåé ìàòåìàòè÷åñêîé ìîäåëè ñõåìû Êîñòàñà1ẋ1 = A1 x1 + b1 cos(θ∆ ),21ẋ2 = A2 x2 + b2 sin(θ∆ ),2∗ẋ = Ax + b(c1 x1 )(c∗2 x2 ),(32)freeθ̇∆ = ω∆− L(c∗ x) − Lh(c∗1 x1 )(c∗2 x2 ). ðàáîòå äîêàçàíà òåîðåìà, ïîêàçûâàþùàÿ ïðèíöèïèàëüíîå îòëè÷èå â ïîâåäåíèè ìàòåìàòè÷åñêîé ìîäåëè ñõåìû Êîñòàñà (32) îò óïðîùåííîé êëàññè÷åñêîéìîäåëè:Òåîðåìà4.Ïîëîñàóäåðæàíèÿìîäåëè(32)cïðîïîðöèîíàëüíî-èíòåãðèðóþùèì ôèëüòðîì (Ô) ïåðâîãî ïîðÿäêà ñ ïåðåäàòî÷íîé ôóíêöè1+sττ1 > τ2 > 0 è íèçêî÷àñòîòíûìè ôèëüòðàìè (ÔÍ×1, ÔÍ×2) ñåé 1+sτ2 ,11ïåðåäàòî÷íîé ôóíêöèåé 1+ s èìååò âèäω3vu2Lu−1 − ω3 τ1L4freet< |ω∆|< ,1−8L −τ1 + τ2 + ω3 τ1 τ28!−1 − ω3 τ1τ1 − τ2 4ïðè ω3 <, || < 1,τ1 τ2L −τ1 + τ2 + ω3 τ1 τ2!L−1−ωττ−τ41231free|ω∆| < , ïðè ω3 <, || > 1,8τ1 τ2L −τ1 + τ2 + ω3 τ1 τ2Lτ1 − τ2free|ω∆| < , ïðè ω3 ≥.8τ1 τ2!!(33)Ïðè ýòîì ïîëîñà óäåðæàíèÿ äëÿ êëàññè÷åñêîé ìàòåìàòè÷åñêîé ìîäåëè ÊîLfreeñòàñà áåç áîêîâûõ ôèëüòðîâ èìååò âèä |ω∆ | < 8 è íå çàâèñèò îò ω3 .Ïðèìåðû êà÷åñòâåííîãî îòëè÷èÿ â ïîâåäåíèè ðàçëè÷íûõ ìîäåëåé ñõåìû Êîñòàñà ïðè ÷èñëåííîì ìîäåëèðîâàíèè ïðåäñòàâëåíû â [25].Îäíîé èç òðóäíîñòåé ÷èñëåííîãî àíàëèçà ìîäåëåé êëàññè÷åñêîé ÔÀÏ è ñõåìÊîñòàñà ÿâëÿåòñÿ âîçìîæíîñòü íàëè÷èÿ ñêðûòûõ êîëåáàíèé, êîòîðûå ìîãóòáûòü íå îáíàðóæåíû ñòàíäàðòíûì ìîäåëèðîâàíèåì [6, 24, 25, 29].

 ðàáîòå [24]29 J.Costas, Synchronous communications, Proceedings of the IRE, 44 (12), 1956, pp. 1713171826íà ïðèìåðå äâóõôàçíîé ÔÀÏ (ìîäåëü êîòîðîé â ïðîñòðàíñòâå ñèãíàëîâ è â ïðîñòðàíñòâå ôàç ñèãíàëîâ îïèñûâàåòñÿ ñèñòåìîé (30)) ïîêàçàíî ÷òî ïðè íàëè÷èèñêðûòûõ êîëåáàíèé ìîäåëèðîâàíèå ÔÀÏ â øèðîêî èñïîëüçóåìîì èíæåíåðíîìïàêåòå SIMetrix (Ðèñ. 7) äëÿ îäíèõ è òåõ æå íà÷àëüíûõ äàííûõ ìîæåò äàâàòüêà÷åñòâåííî ðàçíûå ðåçóëüòàòû ïðè ðàçëè÷íûõ óñëîâèÿõ íà òî÷íîñòü ìîäåëèðîâàíèÿ. Äëÿ ïàðàìåòðîâ ìîäåëèðîâàíèÿ ïî óìîë÷àíèþ â SIMetrix ìîäåëü âòÿsin_InputV(N1)*V(N2)ARB11500mN1 OUTN20 Sine(0 1 1.5915494k 0 0)V1PD_outputfilter_out4.48kR2E3E5+filter_outC210u IC=185mR11.85kvco_frequencycos_inputV(N1)*V(N2)ARB20 Sine(0 1 1.5915494k -157.03518u 0)V2N1 OUTN2vco_sin_output9.8211kV3sin(V(N1))ARB3OUT N1vco_cos_outputintegrator_outintegrator_in50kcos(V(N1))ARB4OUT N110u1-500R1b1E1C15GE2E6Ðèñ.

7: SPICE ðåàëèçàöèÿ äâóõôàçíîé ÔÀÏ â SIMetrixãèâàåòñÿ â ñèíõðîíèçì, îäíàêî äëÿ øàãà äèñêðåòèçàöèè 10−3 ìîäåëèðîâàíèåïîêàçûâàåò íàëè÷èå ñêðûòîãî êîëåáàíèÿ. çàêëþ÷åíèè ïðèâåäåíû îñíîâíûå ðåçóëüòàòû ðàáîòû. ïðèëîæåíèè ïðèâåäåíû ïðèìåðû ðåàëèçàöèè â MATLAB àëãîðèòìîââû÷èñëåíèÿ ñêðûòûõ àòòðàêòîðîâ è ëÿïóíîâñêîé ðàçìåðíîñòè.27Îñíîâíûå ïóáëèêàöèè àâòîðà1. Kuznetsov N. Hidden attractors in fundamental problems and engineeringmodels. A short survey // Lecture Notes in Electrical Engineering.