Диссертация (1145368), страница 16
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Äëÿ ôèëüòðîâ áîëåå âûñîêîãî ïîðÿäêà, ìíîæåñòâîΩhold-in ìîæåò áûòü áîëåå ñëîæíûì. Ïîýòîìó ñ èíæåíåðíîé òî÷êè çðåíèÿñëåäóåò òðåáîâàòü, ÷òîáû ωΔfree = 0 ïðèíàäëåæàëî ìíîæåñòâó óäåðæàíèÿ èîïðåäåëÿòü ïîëîñó çàõâàòà êàê íàèáîëüøèé èíòåðâàë [0, ωh) èç ìíîæåñòâàóäåðæàíèÿ:[0, ωh ) ⊂ Ωhold-in ,òàêîé ÷òî íåêîòîðîå ñîñòîÿíèå ðàâíîâåñèÿ íåïðåðûâíî èçìåíÿåòñÿ ïðèèçìåíåíèè ωΔfree âíóòðè èíòåðâàëà9.
Çäåñü ωh íàçûâàåòñÿ ÷àñòîòîéóäåðæàíèÿ ( hold-in frequency [110, ñòð.38]).Çàìå÷àíèå 4.  îáùåì ñëó÷àå, êîãäà íåò ñèììåòðèè ïî îòíîøåíèþ ê ωΔfree(ñì. (3.24)) ìíîæåñòâî óäåðæàíèÿ ìîæåò íå áûòü ñèììåòðè÷íûì è âÎïðåäåëåíèè 8 íåîáõîäèìî ðàññìàòðèâàòü ωΔfree ∈ Ωhold-in.200ïg(t)Ƨ¨(t)ïï0ï02040t6080100ï02040t6080100freeÐèñóíîê 3.6: ωΔ= 3: óñòîé÷èâîå ñîñòîÿíèå ðàâíîâåñèÿ ñóùåñòâóåò.3.2.2 Ãëîáàëüíàÿ óñòîé÷èâîñòü è ïîëîñà çàõâàòàÏðåäïîëîæèì, ÷òî ïèòàíèå ñõåìû áûëî èçíà÷àëüíî îòêëþ÷åíî è çàòåìâêëþ÷åíî â ìîìåíò âðåìåíè t = 0. Ïóñòü íà÷àëüíàÿ ðàçíîñòü ÷àñòîò äîñòàòî÷íîâåëèêà, òîãäà ñõåìà ìîæåò âîéòè â ñèíõðîííûé ðåæèì ðàáîòû (ðàáî÷èé ðåæèì)íå "ñðàçó", à ÷àñòîòà Ïà áóäåò ïîñòåïåííî ïîäñòðàèâàòüñÿ ïîä ÷àñòîòó âõîäíîãî9Âîáùåì ñëó÷àå (êîãäà óñòîé÷èâûå ñîñòîÿíèÿ ðàâíîâåñèÿ ñîñóùåñòâóþò è ìîãóò ïîÿâëÿòüñÿ è èñ÷åçàòü),óñòîé÷èâîå ñîñòîÿíèå ðàâíîâåñèÿ ìîæåò ðàññìàòðèâàòüñÿ êàê ìíîãîçíà÷íàÿ ôóíêöèÿ ïåðåìåííîéòðåáóåòñÿ ñóùåñòâîâàíèå íåïðåðûâíîé âåòâè äëÿ|ωΔ | ∈ [0, ωh ).freefreeωΔ.Òîãäà121400030002000g(t)Ƨ¨(t)ï100000ï0x 1002040t6080100ï0ï702040t6080100freeÐèñóíîê 3.7: ωΔ= 35: íåò óñòîé÷èâûõ ñîñòîÿíèé ðàâíîâåñèÿ (ñì.
òàêæåÐèñ. 3.4).0.51.560.51.55950.51.559g(t)Ƨ¨(t)1.55850.51.5580.51.557502040t60801000.499902040t6080100freeÐèñóíîê 3.8: ωΔ= 39.997: óñòîé÷èâîå ñîñòîÿíèå ðàâíîâåñèÿ ñóùåñòâóåò.ñèãíàëà è ñõåìà áóäåò ïîñòåïåííî âòÿãèâàòüñÿ â ñèíõðîíèçì (ýòîò ïðîöåññíàçûâàåòñÿ acquisition process ).óñòîé÷èâîñòüþ (transient stability ).Òàêîé ýôôåêò íàçûâàåòñÿ äèíàìè÷åñêîéÏîíÿòèå ïîëîñû çàõâàòà (pull-in range)èñïîëüçóåòñÿ äëÿ îáîçíà÷åíèÿ òàêèõ îòêëîíåíèé ÷àñòîò, äëÿ êîòîðûõ òàêîåâòÿãèâàíèå â ñèíõðîíèçì âîçìîæíî (ñì., íàïðèìåð, [110, ñòð.40], [57, ñòð.61]).×òîáû ñòðîãî îïðåäåëèòü ïîëîñó çàõâàòà ïðèâåäåì îïðåäåëåíèå èç òåîðèèóñòîé÷èâîñòè.Îïðåäåëåíèå 9. Åñëèfreeêàæäàÿ òðàåêòîðèÿ ñèñòåìûäëÿ íåêîòîðîãî ωΔ(3.10) ñòðåìèòñÿ ê íåêîòîðîìó ñîñòîÿíèþ ðàâíîâåñèÿ, òî ñèñòåìà ñ òàêèìfreeçíà÷åíèåì ωΔíàçûâàåòñÿ ãëîáàëüíî àñèìïòîòè÷åñêè óñòîé÷èâîé (ñì.Ðèñ.
3.9).Òåïåðü ðàññìîòðèì âîçìîæíîå ñòðîãî îïðåäåëåíèå ìíîæåñòâà çàõâàòà.122x0ïï010e620freeÐèñóíîê 3.9: Ôàçîâûé ïîðòðåò äëÿ ωΔèç ïîëîñû çàõâàòà: êàæäàÿòðàåêòîðèÿ ïðèòÿãèâàåòñÿ ê íåêîòîðîìó ñîñòîÿíèþ ðàâíîâåñèÿ.Îïðåäåëåíèå 10. Ìíîæåñòâî âñåõ îòêëîíåíèé ÷àñòîò |ωΔ | òàêèõ, ÷òîìàòåìàòè÷åñêàÿ ìîäåëü â ïðîñòðàíñòâå ôàç ñèãíàëîâ ÿâëÿåòñÿ ãëîáàëüíîàñèìïòîòè÷åñêè óñòîé÷èâîé, íàçûâàåòñÿ ìíîæåñòâîì çàõâàòà Ω .Çàìå÷àíèå 5.
 îáùåì ñëó÷àå, êîãäà íåò ñèììåòðèè ïî îòíîøåíèþ ê ωΔ ,â Îïðåäåëåíèè 10 íåîáõîäèìî ðàññìàòðèâàòü ìíîæåñòâî ωΔ ∈ Ω .Çàìå÷àíèå 6. Ìíîæåñòâî çàõâàòà ÿâëÿåòñÿ ïîäìíîæåñòâîì ìíîæåñòâàóäåðæàíèÿ: Ω ⊂ Ω , è òàêæå íå îáÿçàòåëüíî ÿâëÿåòñÿ èíòåðâàëîì.Ñ èíæåíåðíîé òî÷êè çðåíèÿ, ðàçóìíî òðåáîâàòü, ÷òîáû ωΔ = 0ïðèíàäëåæàëî ìíîæåñòâó çàõâàòà è îïðåäåëÿòü ïîëîñó çàõâàòà, êàêìàêñèìàëüíûé èíòåðâàë [0, ωp) èç ìíîæåñòâà çàõâàòà:freepull-infreefreepull-inpull-inhold-infree[0, ωp ) ⊂ Ωpull-in .Çäåñü ωp íàçûâàåòñÿ ÷àñòîòîé çàõâàòà pull-in frequency ( [110, ñòð.40]).123Çàìå÷àíèå 7.
Åñëè âîçìîæíûå ñîñòîÿíèÿ ôèëüòðà îãðàíè÷åíû:x ∈ Xreal (e.g. Xreal = {x : cmin < |x| < cmax }),â ñèëó ôèçè÷åñêîé ðåàëèçàöèè ñõåìû (íàïðèìåð, êîíäåíñàòîðû èìåþòîãðàíè÷åííóþ ìàêñèìàëüíóþ åìêîñòü, ÷àñòîòà Ïà îãðàíè÷åíà), òî âîïðåäåëåíèè ìíîæåñòâà çàõâàòà ñëåäóåò òðåáîâàòü, ÷òîáû òîëüêî ðåøåíèÿñ íà÷àëüíûìè äàííûìè èç x(0) ∈ Xreal ñòðåìèëèñü ê ñòàöèîíàðíîìóìíîæåñòâó. Òðàåêòîðèè ñ íà÷àëüíûìè äàííûìè âíå îáëàñòè, îïðåäåëÿåìîéx(0) ∈ Xreal (çäåñü θΔ (0) ìîæåò ïðèíèìàòü ëþáûå çíà÷åíèÿ), íå îáÿçàòåëüíîäîëæíû ïðèòÿãèâàòüñÿ ê ñòàöèîíàðíîìó ìíîæåñòâó.Äëÿ ñõåìû áåç ôèëüòðà (ò.å.
H(s) = const) ìíîæåñòâî çàõâàòà ñîâïàäàåòñ ìíîæåñòâîì óäåðæàíèÿ. Ìíîæåñòâî çàõâàòà äëÿ ÔÀÏ ñ ôèëüòðîì ïåðâîãîïîðÿäêà ìîæåò ýôôåêòèâíî îöåíèâàòüñÿ ïðè ïîìîùè àíàëèçà ïîâåäåíèÿòðàåêòîðèé íà ôàçîâîé ïëîñêîñòè [9, 290], îäíàêî â îáùåì ñëó÷àå ýòî òðóäíàÿçàäà÷à [225, 285, 294].Äëÿ ìîäåëåé ñ ïàññèâíûì ïðîïîðöèîíàëüíî-èíòåãðèðóþùèì ôèëüòðîì, âðàáîòå [225, ñòð.123] îòìå÷àåòñÿ, ÷òî thedetermination of the width of the capture range together with the interpretation of the capture eect in the second ordertype-I loops have always been an attractive theoretical problem.
This problem hasnot yet been provided with a satisfactory solution.  ðàáîòàõ [35, 119, 241] äëÿìîäåëè ÔÀÏ ñ ôèëüòðîì H(s) =1+sτ21+s(τ1 +τ2 )ïîêàçàíî, ÷òî îáëàñòü ïðèòÿæåíèÿñòàöèîíàðíîãî ìíîæåñòâà ìîæåò áûòü îãðàíè÷åíà, íàïðèìåð, ïîëóóñòîé÷èâûìöèêëîì è äàíû ñîîòâåòñòâóþùèå áèôóðêàöèîííûå äèàãðàììû, ïîëó÷åííûå ïðèïîìîùè àíàëèòèêî-÷èñëåííûõ ìåòîäîâ.Çàìåòèì, ÷òî ïðîñòîå ìîäåëèðîâàíèå ìîæåò ïðèâîäèòü â ýòîì ñëó÷àå êíåâåðíûì âûâîäàì è äîëæíî ïðèìåíÿòüñÿ îñòîðîæíî. Íèæå áóäåò ðàññìîòðåíïðèìåð [211], ãäå SIMetrics SPICE ìîäåëü äâóõôàçíîé ÔÀÏ ñ ïðîïîðöèîíàëüíîèíòåãðèðóþùèì ôèëüòðîì ïðè îäíèõ è òåõ æå ïàðàìåòðàõ äåìîíñòðèðóåòäâà êà÷åñòâåííî ðàçëè÷íûõ ïîâåäåíèÿ äëÿ øàãà äèñêðåòèçàöèè auto è 1m.Àíàëîãè÷íàÿ ïðîáëåìà íàáëþäàåòñÿ ïðè ìîäåëèðîâàíèè â MatLab Simulink[212, 235, 274].êîëåáàíèÿìè.Ýòè ïðèìåðû ñâÿçàíû ñ ìóëüòèóñòîé÷èâîñòüþ è ñêðûòûìè124Çàìåòèì, ÷òî íåëèíåéíûé àíàëèç ÔÀÏ òðåáóåò ðàçðàáîòêè è ïðèìåíåíèÿñïåöèàëüíûõìåòîäîâðàçðàáîòàííûåâàäàïòèðîâàíûäëÿèññëåäîâàíèÿðàìêàõóñòîé÷èâîñòè,êëàññè÷åñêîéèññëåäîâàíèÿòåîðèèñèñòåìñòàêêàêóïðàâëåíèÿ,öèëèíäðè÷åñêèììåòîäû,÷àñòîíåôàçîâûìïðîñòðàíñòâîì.
Íàïðèìåð, â êëàññè÷åñêîì êðèòåðèè ãëîáàëüíîé óñòîé÷èâîñòèÁàðáàøèíà-ÊðàñîâñêîãîËàÑàëëÿ(Barbashin-Krasovskii-LaSalleprinciple)ôóíêöèÿ Ëÿïóíîâà äîëæíà íåîãðàíè÷åííî âîçðàñòàòü (ò.å. V (x, θΔ ) → +∞ïðè ||(x, θΔ )|| → +∞).  òî âðåìÿ êàê äëÿ àíàëèçà ìîäåëåé ÔÀÏ îáû÷íîèñïîëüçóþòñÿ ôóíêöèè Ëÿïóíîâà ïåðèîäè÷åñêèå ïî θ (íàïðèìåð, V (x, θΔ )â Çàìå÷àíèè 10 ÿâëÿåòñÿ îãðàíè÷åííîé äëÿ ëþáîãî ||(0, θΔ )|| → +∞), èîáñóæäåíèÿ ýòîãî ôàêòà ÷àñòî îïóñêàåòñÿ (ñì., íàïðèìåð, ïàòåíò [42] èðàáîòû [39, 41]).Íåîáõîäèìûåíåëèíåéíûõìîäèôèêàöèèñèñòåìñêëàññè÷åñêèõöèëèíäðè÷åñêèìêðèòåðèåâôàçîâûìóñòîé÷èâîñòèïðîñòðàíñòâîìáûëèðàçðàáîòàíû âî âòîðîé ïîëîâèíå 20 âåêà, ñì., íàïðèìåð, [204, 206, 207, 295].Çàìåòèì, ÷òî äëÿ ïðèìåíåíèÿ ðàçëè÷íûõ êðèòåðèåâ óñòîé÷èâîñòè ÷àñòî óäîáíîçàïèñûâàòü ìîäåëü ÔÀÏ (3.10) â ôîðìå Ëóðüå:⎛⎞˙⎜ x̄ ⎟⎠θ̇Δ⎝ãäå⎛⎞⎛⎞⎛⎞A 0⎟ ⎜ x̄ ⎟ ⎜ b ⎟⎝⎠⎝ ⎠ + ⎝⎠ ϕ̄(θΔ ),=⎜−Lc∗ 0 θΔ−Lh(3.42)x̄ = x − xeq = x + A−1 bϕ(θeq ), ϕ̄(θΔ ) = ϕ(θΔ ) − ϕ(θeq ),freeϕ(θeq ) = ωΔL−1 (c∗ A−1 b − h)−1 .3.2.3 Ïðîñêàëüçûâàíèå öèêëîâ è ïîëîñà çàõâàòà áåçïðîñêàëüçûâàíèÿÂâåäåì ïîíÿòèå ïðîñêàëüçûâàíèÿ öèêëîâ (cycle slipping) â ôàçîâîìïðîñòðàíñòâåÎïðåäåëåíèå 11.
Åñëèlim sup |θΔ (0) − θΔ (t)| > 2π,t→+∞(3.43)125òî èìååò ìåñòî ïðîñêàëüçûâàíèå öèêëà (ñì., ïóíêòèðíóþ òðàåêòîðèþ íàÐèñ. 3.9).Èíîãäà âìåñòî ïðåäåëüíîãî çíà÷åíèÿ çäåñü ðàññìàòðèâàþò ìàêñèìóìðàçíîñòè ( [285, ñòð.131]).Îïðåäåëåíèå 11' Åñëèsup |θΔ (0) − θΔ (t)| > 2π,(3.44)t>0òî èìååò ìåñòî ïðîñêàëüçûâàíèå öèêëà.Çàìåòèì, ÷òî â îáùåì ñëó÷àå Îïðåäåëåíèå 11' íå îáÿçàòåëüíî âëå÷åò çàñîáîé íåâûïîëíåíèå (ïîñëå ïåðåõîäíîãî ïðîöåññà) óñëîâèÿ (3.43).Èíîãäà ðàññìàòðèâàþò êîëè÷åñòâî ïðîñêàëüçûâàíèé.Îïðåäåëåíèå 12. Åñëè2kπ < lim sup |θΔ (0) − θΔ (t)| < 2(k + 1)π,(3.45)t→∞òî èìååò ìåñòî ïðîñêàëüçûâàíèå k öèêëîâ.×èñëåííûé àíàëèç ïðîñêàëüçûâàíèÿ öèêëîâ â êëàññè÷åñêîé ÔÀÏ âûïîëíåíâ [49].
Àíàëèòè÷åñêèå ìåòîäû äëÿ îöåíêè ÷èñëà öèêëîâ ïðîñêàëüçûâàíèÿ âçàâèñèìîñòè îò ïàðàìåòðîâ ìîäåëè ïðåäñòàâëåíû, íàïðèìåð, â [104, 207].÷àñòîòû çàõâàòà áåç ïðîñêàëüçûâàíèÿ (lock-in frequency) èïîëîñû çàõâàòà áåç ïðîñêàëüçûâàíèÿ (lock-in range) áûëà ïðåäëîæåíà äëÿÊîíöåïöèÿîïèñàíèÿ îòêëîíåíèé ÷àñòîò, òàêèõ ÷òî ÔÀÏ âòÿãèâàåòñÿ â ñèíõðîíèçì áåçïðîñêàëüçûâàíèÿ (çà îäèí öèêë ÏÃ).  [110, ñòð.40] Ãàðäíåðîì äàíî ñëåäóþùååIf, for some reason, the frequency dierence between input and VCOis less than the loop bandwidth, the loop will lock up almost instantaneously withoutslipping cycles. The maximum frequency dierence for which this fast acquisition ispossible is called the lock-in frequency.îïðåäåëåíèå: free=Îäíàêî â îáùåì ñëó÷àå, äàæå äëÿ íóëåâîãî îòêëîíåíèÿ ÷àñòîò (ωΔ0) è äîñòàòî÷íî áîëüøèõ ñîñòîÿíèé ôèëüòðà (x(0)), ìîæåò èìååò ìåñòîïðîñêàëüçûâàíèåöèêëîâ(ñì.,íàïðèìåð,ïóíêòèðíóþòðàåêòîðèþíàÐèñ.
3.10, ñëåâà). Ïîýòîìó íåîáõîäèìî ðàññìàòðèâàòü âñå ïåðåìåííûå ñîñòîÿíèÿäëÿ èññëåäîâàíèè ïðîñêàëüçûâàíèÿ öèêëîâ, è, ñëåäîâàòåëüíî ïðåäëîæåííàÿ126Ãàðäíåðîì êîíöåïöèÿ òåðÿåò ñìûñë äëÿ óïðîùåííîé êëàññè÷åñêîé ìîäåëè(3.27), òàê êàê îíà íå âêëþ÷àåò â ñåáÿ íà÷àëüíîå ñîñòîÿíèå ôèëüòðà.Ïðèâåäåííîåâûøåîïðåäåëåíèåáûëîçàòåìïðèâåäåíîâðàçëè÷íûõìîíîãðàôèÿõ.0.060.060.060.040.040.040.020.020.02xx0x00−0.02−0.02−0.02−0.04−0.04−0.04−0.06−50ș¨510−0.06−50ș¨510−0.06−50ș¨510Ðèñóíîê 3.10: Ôàçîâûé ïîðòðåò êëàññè÷åñêîé ÔÀÏ äëÿ ïàðàìåòðîâ:1+sτ2H(s) = 1+s(τ, τ1 = 4.48 · 10−2 , τ2 = 1.85 · 10−2 , L = 250, ϕ(θΔ ) = 12 sin(θΔ ) è1 +τ2 )ðàçëè÷íûõ îòêëîíåíèé ÷àñòîò. ×åðíûé öâåò èñïîëüçóåòñÿ äëÿfree .
Êðàñíûé öâåò èñïîëüçóåòñÿ äëÿ ñîîòâåòñòâóþùåãîïîëîæèòåëüíîãî ωΔ= |ω|free . Îáëàñòè ïðèòÿæåíèÿ áåç ïðîñêàëüçûâàíèÿîòðèöàòåëüíîãî ωΔ= −|ω|freeçàøòðèõîâàíû (ñâåðõó ÷åðíàÿ ãîðèçîíòàëüíàÿ øòðèõîâêà äëÿ ωΔ> 0, ñíèçófreefreeêðàñíàÿ âåðòèêàëüíàÿ øòðèõîâêà äëÿ ωΔ < 0). Ñëåâà: ωΔ = 0;  ñåðåäèíå:freefreeωΔ= ±65; Ñïðàâà: ωΔ= ±68..freeÄëÿ ëþáîãî çíà÷åíèÿ ωΔôàçîâûé ïîðòðåò ìîäåëè (3.10) èìååòîáëàñòè, ãäå òðàåêòîðèè íå ïðîñêàëüçûâàþò öèêëû (îáëàñòü ïðèòÿæåíèÿ áåçïðîñêàëüçûâàíèÿ). Îáëàñòü ïðèòÿæåíèÿ áåç ïðîñêàëüçûâàíèÿ (lock-in domain)ÿâëÿåòñÿ îáúåäèíåíèåì ëîêàëüíûõ îáëàñòåé ïðèòÿæåíèÿ áåç ïðîñêàëüçûâàíèÿ(local lock-in domains), êàæäàÿ èç êîòîðûõ ñîîòâåòñòâóåò îäíîìó èç ñîñòîÿíèéðàâíîâåñèÿ è èìååò ñâîþ ôîðìó (ñì., íàïðèìåð, çàøòðèõîâàííûå îáëàñòèíà Ðèñ.
3.10, ñëåâà, îïðåäåëÿåìûå ñîîòâåòñòâóþùèìè ñåïàðàòðèñàìè). Ôîðìàfree.  ìîíîãðàôèè [294, ñòð.50]) îáëàñòüîáëàñòåé èçìåíÿåòñÿ ïðè èçìåíåíèè ωΔïðèòÿæåíèÿ áåç ïðîñêàëüçûâàíèÿ íàçûâàåòñÿ frequency lock. Íåêîòîðûå àâòîðû(íàïðèìåð, [285, ñòð.132] ) èñïîëüçóþò êîíöåïöèþ lock-in range äëÿ îáîçíà÷åíèÿ.lock-in domainÓ÷èòûâàÿ âûøåèçëîæåííîå, âî âòîðîì èçäàíèè ñâîåé ìîíîãðàôèè ÃàðäíåðíàïèñàëThere is no natural way to dene exactly any unique lock-in frequency[111, ñòð.70], [112, ñòð.188].127Íèæå ïîêàçàíî êàê îáîéòè ýòè òðóäíîñòè è îïðåäåëèòü ïîëîñó çàõâàòà áåçïðîñêàëüçûâàíèÿ.freefreeÐàññìîòðèì ωΔè ââåäåì îáîçíà÷åíèå Dlock-in (ωΔ) äëÿ ñîîòâåòñòâóþùåéîáëàñòè çàõâàòà áåç ïðîñêàëüçûâàíèÿ.free|ëþáîãî |ωΔ∈Ωhold-in ,Òàêàÿ îáëàñòü ñóùåñòâóåò äëÿïîòîìó ÷òî,êàê ìèíèìóì,âñå ñîñòîÿíèÿfree∈ Ω ðàññìîòðèìÄëÿ ìíîæåñòâà ωΔðàâíîâåñèÿ âõîäÿò â ýòó îáëàñòü.ïåðåñå÷åíèå ñîîòâåòñòâóþùèõ îáëàñòåé çàõâàòà áåç ïðîñêàëüçûâàíèÿ (ñì.,free íà= ±|ω|íàïðèìåð, ïåðåñå÷åíèå ëîêàëüíûõ îáëàñòåé äëÿ ðàçëè÷íûõ ωΔÐèñ.
3.10 îáëàñòè çàøòðèõîâàííûå îäíîâðåìåííî âåðòèêàëüíûìè êðàñíûìèè ãîðèçîíòàëüíûìè ÷åðíûìè ëèíèÿìè):0Dlock-in (Ω) =freeωΔ∈ΩfreeDlock-in (ωΔ).Îïðåäåëåíèå 13. Ïîëîñà çàõâàòà áåç ïðîñêàëüçûâàíèÿ ýòî òàêîéíàèáîëüøèé èíòåðâàë [0, ωl ), ÷òî äëÿ |ωΔ | ∈ [0, ωl ) ìàòåìàòè÷åñêàÿ ìîäåëüÔÀÏ â ïðîñòðàíñòâå ôàç ñèãíàëîâ ÿâëÿåòñÿ ãëîáàëüíîé àñèìïòîòè÷åñêèóñòîé÷èâîé (ò.å. [0, ωl ) ⊂ [0, ωp)) è ñëåäóþùàÿ îáëàñòüfreeDlock-in (−ωl , ωl ) =0free|<ωl|ωΔfreeDlock-in (ωΔ).ñîäåðæèò âñå ñîîòâåòñòâóþùèå ñîñòîÿíèÿ ðàâíîâåñèÿ:freefreexeq (ωΔ), θeq (ωΔ) ∈ Dlock-in (−ωl , ωl ) .Òàêóþ îáëàñòü Dlock-in = Dlock-in (−ωl , ωl ) áóäåì íàçûâàòü ðàâíîìåðíîéîáëàñòüþ çàõâàòà áåç ïðîñêàëüçûâàíèÿ (uniform lock-in domain) (ðàâíîìåðíàÿïî îòíîøåíèÿ ê èíòåðâàëó (−ωl , ωl )), ωl áóäåì íàçûâàòü ÷àñòîòîé çàõâàòà áåçïðîñêàëüçûâàíèÿ (lock-in frequency) (ñì. [110, ñòð.40]).Íà Dlock-in ìîãóò áûòü íàëîæåíû ðàçëè÷íûå äîïîëíèòåëüíûå óñëîâèÿ,íàïðèìåð, ÷òî îíà äîëæíà ñîäåðæàòü ïðÿìóþ, îïðåäåëÿåìóþ x(ñì., íàïðèìåð, [154, ñòð.258]) èëè ïîëîñó, îïðåäåëÿåìóþ |x|<≡0cmax .Åñëè âìåñòî ãëîáàëüíîé óñòîé÷èâîñòè â îïðåäåëåíèè ìíîæåñòâà çàõâàòàðàññìàòðèâàëàñü óñòîé÷èâîñòü â îáëàñòè, îïðåäåëÿåìîé Xreal , òî áóäåì1281òðåáîâàòü, ÷òîáû ïåðåñå÷åíèå Dlock-in Xreal ñîäåðæàëî âñå ñîîòâåòñòâóþùèåñîñòîÿíèÿ ðàâíîâåñèÿ.Çàìå÷àíèå 8.
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