Диссертация (1145368), страница 4
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Ðèñ. 1.4).Ñêðûòûå êîëåáàíèÿ â êîíòðïðèìåðàõ ê ãèïîòåçàì Àéçåðìàíà èÊàëìàíà 1949 ãîäó Ì.À. Àéçåðìàí ñôîðìóëèðîâàë ïðîáëåìó [5], êîòîðàÿ ñðàçóïðèâëåêëà âíèìàíèå ìíîãèõ èçâåñòíûõ ó÷åíûõ. Ãèïîòåçà Àéçåðìàíà ìîæåòáûòü ñôîðìóëèðîâàíà ñëåäóþùèì îáðàçîì.Ðàññìîòðèì ñèñòåìó ñ îäíîé21D[E[ \\///[/[Ðèñóíîê 1.4: Âèçóàëèçàöèÿ ÷åòûðåõ ïðåäåëüíûõ öèêëîâ (çåëåíûé öâåò óñòîé÷èâûå öèêëû, êðàñíûé íåóñòîé÷èâûå) â äâóìåðíîé êâàäðàòè÷íîéïîëèíîìèàëüíîé ñèñòåìå22ẋ = −(a1 x + b1 xy + c1 y + α1 x + β1 y), ẏ = −(a2 x2 + b2 xy + c2 y 2 + α2 x + β2 y), ñêîýôôèöèåíòàìè a1 = b1 = β1 = −1, c1 = α1 = 0, b2 = −2.2, èc2 = −0.7, a2 = 10, α2 = 72.7778, β2 = −0.0015.
Òðè âëîæåííûõ ïðåäåëüíûõöèêëà (L1,2,3 ; L2 - ñêðûòûé àòòðàêòîð) âîêðóã óñòîé÷èâîãî íóëåâîãî ñîñòîÿíèÿðàâíîâåñèÿ (çåëåíàÿ òî÷êà) ñïðàâà îò òðàíñâåðñàëüíîé ïðÿìîé x = −1, è îäèíïðåäåëüíûé öèêë (L4 ñàìîâîçáóæäàþùèéñÿ àòòðàêòîð) âîêðóãíåóñòîé÷èâîãî ñîñòîÿíèÿ ðàâíîâåñèÿ (êðàñíàÿ òî÷êà) ñëåâà îò ïðÿìîé x = −1.ñêàëÿðíîé íåëèíåéíîñòüþdx= Px + qψ(r∗ x),dtãäå∗P ïîñòîÿííàÿ ìàòðèöàn × n, q, r îïåðàöèÿ òðàíñïîíèðîâàíèÿ,÷òîψ(0) = 0.ψ(σ)x ∈ Rn ,(1.4) ïîñòîÿííûå âåêòîðà ðàçìåðíîñòèn,íåïðåðûâíàÿ ñêàëÿðíàÿ ôóíêöèÿ, òàêàÿÏóñòü ëþáàÿ ëèíåéíàÿ ñèñòåìà (1.4) ñψ(σ) = μσ,μ ∈ (μ1 , μ2 )(1.5)ÿâëÿåòñÿ àñèìïòîòè÷åñêèé óñòîé÷èâîé (ò.å. íóëåâîé ñîñòîÿíèå ðàâíîâåñèÿÿâëÿåòñÿ ãëîáàëüíûì àòòðàêòîðîì).
Áóäåò ëè íóëåâîå ðåøåíèå ñèñòåìû(1.4) ñ ïðîèçâîëüíîé íåëèíåéíîñòüþ ψ(σ), èç ñåêòîðàμ1 <ψ(σ)< μ2 ,σ∀σ = 0,22ãëîáàëüíûì àòòðàêòîðîì? 1957 ãîäó Ð. Êàëìàí âûäâèíóë ñëåäóþùóþ ãèïîòåçó [145]: If f (e) inFig. 1 [see Fig. 1.5] is replaced by constantsofK corresponding to all possible valuesf (e), and it is found that the closed-loop system is stable for all such K , then itis intuitively clear that the system must be monostable; i.e., all transient solutionswill converge to a unique, stable critical point.
Çàìåòèì, ÷òî âûïîëíåíèå óñëîâèére6f(e)fG(s)cÐèñóíîê 1.5: Nonlinear control system. G(s) is a linear transfer function, f (e) is asingle-valued, continuous, and dierentiable function [145]ãèïîòåçû Êàëìàíà âëå÷åò çà ñîáîé âûïîëíåíèå óñëîâèé ãèïîòåçû Àéçåðìàíà. 1952 ãîäó ãèïîòåçà Àéçåðìàíà áûëà ïîëíîñòüþ èññëåäîâàíà äëÿ n = 2È.Ã. Ìàëêèíûì [16], Í.Ï. Åðóãèíûì [10], è Í.Í. Êðàñîâñêèì [14]. Äëÿ ñëó÷àÿn = 2 ãèïîòåçà Àéçåðìàíà âñåãäà âåðíà êðîìå ñëó÷àÿ, êîãäà ìàòðèöà (P+μ1 qr∗ )èìååò äâîéíîå íóëåâîå ñîáñòâåííîå ÷èñëî è∞ψ(σ) − μ1 σ dσ = +∞0èëè−∞ ψ(σ) − μ1 σ dσ = −∞.0Í.Í.
Êðàñîâêèé ïîêàçàë [14], ÷òî â ýòîì ñëó÷àå ñèñòåìà (1.4) ìîæåòèìåòü ðåøåíèå, óõîäÿùåå íà áåñêîíå÷íîñòü. Ýòî áûë ïåðâûé êîíòðïðèìåðê ãèïîòåçå Àéçåðìàíà. 1958 ãîäó Â.À. Ïëèññ [23] ðàçðàáîòàë ìåòîäïîñòðîåíèÿ òðåõìåðíûõ íåëèíåéíûõ ñèñòåì, óäîâëåòâîðÿþùèõ óñëîâèÿìãèïîòåçû Àéçåðìàíà è èìåþùèõ ïåðèîäè÷åñêîãî ðåøåíèÿ. Çàòåì ýòîò ìåòîäáûë îáîáùåí íà ñèñòåìû ïðîèçâîëüíîãî ïîðÿäêà [11, 234].ïðèìåðû íå óäîâëåòâîðÿëè óñëîâèÿì ãèïîòåçû Êàëìàíà.Îäíàêî âñå ýòè231.520301512010100.55f (e)00ex20−10−5−20−10−0.5−1−15−30−30−20−10x101020−2030010203040t 5060708090100−1.5−2−1.5−1−0.5e00.511.52Ðèñóíîê 1.6: Êîíòðïðèìåð ê ãèïîòåçàì Àéçåðìàíà è Êàëìàíà. Íåëèíåéíàÿñèñòåìà ẋ1 = −x2 − 10f (e), ẋ2 = x1 − 10.1 f (e), ẋ3 = x4 , ẋ4 =−x3 − x4 + f (e), e = x1 − 10.1 x3 − 0.1 x4 , f (e) = tanh(e) èìååò åäèíñòâåííîåñîñòîÿíèå ðàâíîâåñèÿ (ðèñóíîê ñëåâà, çåëåíàÿ òî÷êà), êîòîðîå ÿâëÿåòñÿëîêàëüíî óñòîé÷èâûì; íåëèíåéíîñòü (ðèñóíîê ñïðàâà, ñèíÿÿ êðèâàÿ) è ååïðîèçâîäíàÿ ïðèíàäëåæàò ñåêòîðó ëèíåéíîé óñòîé÷èâîñòè (K ∈ (0, 9.9),ïðàâûé ðèñóíîê, êðàñíûå ëèíèè); ñèñòåìà èìååò óñòîé÷èâîå ïåðèîäè÷åñêîåðåøåíèå, êîòîðîå ÿâëÿåòñÿ ñêðûòûì êîëåáàíèåì (ëåâûé ðèñóíîê, çåëåíàÿêðèâàÿ, ïðîåêöèÿ íà ïëîñêîñòü (x1 , x2 )).Ïîçäíåå áûëî ïîêàçàíî, ÷òî èç ÷àñòîòíîãî êðèòåðèÿ ßêóáîâè÷à ñëåäóåòñïðàâåäëèâîñòü ãèïîòåçû Êàëìàíà ïðè n = 2 è n = 3 [53, 206].Â1966 ãîäó Ð.
Ôèòö (Fitts) [108] ÷èñëåííî ïîñòðîèë ñåðèþ êîíòðïðèìåðîâ êãèïîòåçå Êàëìàíà, ðàññìàòðèâàÿ ñèñòåìó ÷åòâåðòîãî ïîðÿäêà ñ êóáè÷åñêîéíåëèíåéíîñòüþ, îäíàêî ïîçäíåå íåêîòîðûå èç åãî ýêñïåðèìåíòîâ íå óäàëîñüïîâòîðèòü [53, 176]. äàëüíåéøåì ðàçëè÷íûå ïîäõîäû ê ïîñòðîåíèþêîíòðïðèìåðîâ ê ãèïîòåçå Êàëìàíà íà îñíîâå ñèñòåì ñ êóñî÷íî-ëèíåéíûìèíåëèíåéíîñòÿìè (sign, sat) áûëè ïðåäëîæåíû â ðàáîòàõ [53, 56].Íèæåïðèâåäåí êîíòðïðèìåð ê ãèïîòåçå Êàëìàíà ñ ãëàäêîé íåëèíåéíîñòüþ tanh(·),ïðåäëîæåííûé àâòîðîì, èç [175, 176, 179].Ñîâðåìåííîìó ñîñòîÿíèþ èññëåäîâàíèé ïî ãèïîòåçàì Àéçåðìàíà è Êàëìàíàè òåîðèè àáñîëþòíîé óñòîé÷èâîñòè ïîñâÿùåíû, íàïðèìåð, ñëåäóþùèå ðàáîòû[118,215,224,257,261]. Ñì.
òàêæå [44,175,176,179182].  ðàáîòå [125] ðàññìîòðåíäèñêðåòíûé àíàëîã ãèïîòåçû Êàëìàíà è ïîñòðîåí êîíòðïðèìåð äëÿ n = 2.Îáùåíèå ãèïîòåçû Êàëìàíà íà âåêòîðíûå íåëèíåéíîñòè íàçûâàåòñÿãèïîòåçîé Ìàðêóñà-ßìàáå (Markus-Yamabe conjecture) [226]:ẋ = f (x),f : R n → Rn ,f ∈ C 1,Ïóñòü ñèñòåìàf (0) = 0(1.6)24⎛⎞df (x) ⎠èìååò åäèíñòâåííîå ñîñòîÿíèå ðàâíîâåñèÿ è ìàòðèöà ßêîáè ⎝èìååòdxâñå ñîáñòâåííûå ÷èñëà ñ îòðèöàòåëüíûìè âåùåñòâåííûìè ÷àñòÿìè äëÿ âñåõx ∈ Rn . Âåðíî ëè, ÷òî ñîñòîÿíèå ðàâíîâåñèÿ (1.6) ÿâëÿåòñÿ ãëîáàëüíûìàòòðàêòîðîì.Ýòà ïðîáëåìà èìååò ïîëîæèòåëüíîå ðåøåíèå äëÿ n = 2 [107, 117, 121] èîòðèöàòåëüíîå, â îáùåì ñëó÷àå, äëÿ n ≥ 3. Ðàññìîòðèì ñîîòâåòñòâóþùèéïðèìåð ïîëèíîìèàëüíîé ñèñòåìû, ðàññìîòðåííîé â [249],ẋ1 = −x1 + x3 (x1 + x2 x3 )2 ,ẋ2 = −x2 − (x1 + x2 x3 )2 ,ẋ3 = −x3 .ãäå ìàòðèöà ßêîáè èìååò òðè ñîáñòâåííûõ ÷èñëà ðàâíûõ −1, íî ñèñòåìà èìååòíåîãðàíè÷åííîå ðåøåíèåx1 (t), x2 (t), x3 (t) = 18et , −12e2t , e−t .Îòìåòèì, ÷òî ñ ïðèêëàäíîé òî÷êè çðåíèÿ ýòîò êîíòðïðèìåð è êîíòðïðèìåðÊðàñîâñêîãî ÿâëÿþòñÿ íå åñòåñòâåííûìè, òàê Ð.
Êàëìàí â ñâîåé ðàáîòå ïèøåò: Undoubtedly the most important instability of interest in control systems are limitcycles; the circumstances which lead to creation or destruction of limit cycles are tobe studied in considerable detail. Solutions tending to innity are also fairly readilyavoidable ....1.2.3 Ñêðûòûå àòòðàêòîðû â ïðèêëàäíûõ ìîäåëÿõÑêðûòûå àòòðàêòîðû â ýëåêòðîìåõàíè÷åñêèõ ñèñòåìàõ áåçñîñòîÿíèÿ ðàâíîâåñèÿÑêðûòûìè àòòðàêòîðàìè ÿâëÿþòñÿ àòòðàêòîðû â ñèñòåìàõ áåç ñîñòîÿíèÿðàâíîâåñèÿ, îïèñûâàþùèõ ðàçëè÷íûå ìåõàíè÷åñêèå è ýëåêòðîìåõàíè÷åñêèåñèñòåìû ñ âðàùåíèå è ýëåêòðè÷åñêèå öåïè ñ öèëèíäðè÷åñêèì ôàçîâûìïðîñòðàíñòâîì.Îäèí èç ïåðâûõ òàêèõ ïðèìåðîâ áûë îïèñàí ÀðíîëüäîìÇîììåðôåëüäîì (Arnold Sommerfeld) â 1902 ãîäó [282].
Îí èçó÷àë çàïóñê ðîòîðà25íà óïðóãîì îñíîâàíèè è îïèñàë ýôôåêò çàõâàòà ÷àñòîòû âðàùåíèÿ ðîòîðà ýôôåêò Çîììåðôåëüäà (Sommerfeld eect): Thisexperiment corresponds roughlyto the case in which a factory owner has a machine set on a poor foundation running at 30 horsepower. He achieves an eective level of just 1/3, however, becauseonly 10 horsepower are doing useful work, while 20 horsepower are transferred tothe foundational masonry [96].Äëÿ äåìîíñòðàöèè ýôôåêòà Çîììåðôåëüäà ðàññìîòðèì ñëåäóþùóþýëåêòðîìåõàíè÷åñêóþ ìîäåëü: ìîòîð ïîñòîÿííîãî òîêà (DC motor) ïðèâîäèòâ äâèæåíèå ýêñöåíòðèê, ñâÿçàííûé ñ ïëàòôîðìîé, êîòîðàÿ â ñâîþ î÷åðåäüñâÿçàíà ïðóæèíîé ñî ñòåíîé è ìîæåò ñîâåðøàòü òîëüêî ãîðèçîíòàëüíûåäâèæåíèÿ.Ýòà ñèñòåìà îïèñûâàåòñÿ â ðàáîòàõ [105, 109] ñëåäóþùèìèóðàâíåíèÿìè:(M + m)ẍ + k1 ẋ + ml(θ̈ cos θ − θ̇2 sin θ) + kx = 0,J θ̈ + kθ θ̇ + mlẍ cos θ = u,(1.7)ãäå M ìàññà ïëàòôîðìû, m ìàññà ýêñöåíòðèêà, l ýêñöåíòðèñèòåò, θ óãîë ïîâîðîòà ðîòîðà, x ëèíåéíîå ãîðèçîíòàëüíîå ñìåùåíèå ïëàòôîðìûìàññû M îò ñîñòîÿíèÿ ðàâíîâåñèÿ, u êðóòÿùèé ìîìåíò ìîòîðà, k æåñòêîñòü ïðóæèíû, k1 è kθ êîýôôèöèåíòû äåìïôèðîâàíèÿ, J ìîìåíòèíåðöèè ðîòîðà â îòñóòñòâèå äèñáàëàíñà.Íà Ðèñ.
1.7 äëÿ ñèñòåìû áåç ñîñòîÿíèé ðàâíîâåñèÿ (1.7) ñ ïàðàìåòðàìèJ = 0.014, M = 10.5, m0 = 1.5, l = 0.04, kθ = 0.005, k = 5300, k1 = 5,u = 0.48ïðåäñòàâëåíàëîêàëèçàöèÿñîñóùåñòâóþùèõïðîñòðàíñòâå (x, ẋ, θ̇) (ñì. [152]).ñêðûòûõàòòðàêòîðîââÎòìåòèì, ÷òî çäåñü ôèçè÷åñêèé çàïóñêñèñòåìû (ẋ = x = θ = θ̇ = 0) ïðèâîäèò ê ëîêàëèçàöèè îäíîãî èçñêðûòûõ àòòðàêòîðîâ (ñîîòâåòñòâóåò ýôôåêòó Çîììåðôåëüäà), â òî âðåìÿ êàêëîêàëèçàöèÿ äðóãîãî ñêðûòîãî àòòðàêòîðà (ñîîòâåòñòâóåò íîðìàëüíîì ðåæèìóðàáîòû) òðåáóåò ñïåöèàëüíîãî ïîäáîðà íà÷àëüíûõ äàííûõ â ìîäåëèðîâàíèè(íàïðèìåð, ẋ = x = θ = 0, θ̇ = 40).Äðóãèì øèðîêî èçâåñòíûì ïðèìåðîì õàîòè÷åñêîé ñèñòåìû áåç ñîñòîÿíèéðàâíîâåñèÿ ÿâëÿåòñÿ NoseHoover îñöèëÿòîð [139, 238, 284].
Òàêæå ñêðûòûå26Normal operation8060Normal operation80ș..x. = x = 0ș= 40Sommerfeld effect.60Sommerfeld effectș404000ïx0ïï..x00.x = x = ș= 0xïïï0.xÐèñóíîê 1.7: Ñîñóùåñòâîâàíèå ñêðûòûõ àòòðàêòîðîâ â ýëåêòðîìåõàíè÷åñêîéìîäåëè áåç ñîñòîÿíèé ðàâíîâåñèÿ. Ñèíÿÿ êðèâàÿ ñîîòâåòñòâóåò ôèçè÷åñêîìóçàïóñêó ñèñòåìû èç íóëåâûõ íà÷àëüíûõ äàííûõ è ýôôåêòó Çîììåðôåëüäà,êðàñíàÿ êðèâàÿ ñîîòâåòñòâóåò æåëàåìîìó ðàáî÷åìó ðåæèìó ñèñòåìû).õàîòè÷åñêèå êîëåáàíèÿ áûëè íàéäåíû â ìîäåëè ýíåðãîñèñòåìû áåç ñîñòîÿíèéðàâíîâåñèÿ [293].Ñêðûòûå êîëåáàíèÿ ñ ìîäåëÿõ áóðîâûõ óñòàíîâîêÈññëåäîâàòåëè èç Òåõíè÷åñêîãî óíèâåðñèòåòà Ýéíäõîâåíà â ðàáîòàõ [45, 82]èçó÷àëè ïîâåäåíèå ýêñïåðèìåíòàëüíîé áóðîâîé óñòàíîâêè è ñðàâíèâàëè åãî ñïîâåäåíèåì ìàòåìàòè÷åñêîé ìîäåëèJu θ̈u + kθ (θu − θl ) + b(θ̇u − θ̇l ) + Tf u (θ̇u ) − km v = 0,Jl θ̈l − kθ (θu − θl ) − b(θ̇u − θ̇l ) + Tf l (θ̇l ) = 0,(1.8)êîòîðàÿ îïèñûâàåò âðàùåíèå ìîòîðîì ïîñòîÿííîãî òîêà äâóõ äèñêîâ, ñâÿçàííûõãèáêèì ñòåðæíåì, íà êîòîðûå äåéñòâóþò ñèëû òðåíèÿ.Çäåñü θu è θl îòêëîíåíèÿ óãëîâ âåðõíåãî è íèæíåãî äèñêà; Ju è Jl ïîñòîÿííûå ìîìåíòû èíåðöèè; b êîýôôèöèåíò äåìïôèðîâàíèÿ; kθ êðóòèëüíàÿ æåñòêîñòü ïðóæèíû; km ïîñòîÿííàÿ ìîòîðà; v ïîñòîÿííîåíàïðÿæåíèå; Tf u (θ̇u ) è Tf l (θ̇l ) ìîìåíòû òðåíèÿ, äåéñòâóþùèå íà íèæíèé èâåðõíèé äèñêè.Ñèëà òðåíèÿ Tf l (θ̇l ) õàðàêòåðèçóåò òðåíèå íèæíåãî äèñêàñ ïîðîäîé, Tf u (θ̇u ) − km v îòðàæàåò âçàèìîäåéñòâèå ìîòîðà è ñòåðæíÿ.Âëèÿíèå ìîòîðà çäåñü îïèñûâàåòñÿ ïîñòîÿííûì íàïðÿæåíèåì è ñèëîé òðåíèÿ,27θu-θl ≈ 0,..θu≈ 6.1, θl ≈6.115Stable equilibrium10.θl5Stable limit cycle0−58θu-θl = 0,..θu= 0, θl =06.θu420−50510θu-θlÐèñóíîê 1.8: Ñêðûòûå êîëåáàíèÿ â ìîäåëè áóðîâîé óñòàíîâêè: â ïðîñòðàíñòâå(θu − θu , θ̇u , θ̇l ) óñòîé÷èâîå ñîñòîÿíèå ðàâíîâåñèÿ ñîñóùåñòâóåò ñ óñòîé÷èâûìïåðèîäè÷åñêèì êîëåáàíèåì.28äåéñòâóþùåé íà âåðõíèé äèñê. [45, 82]) äëÿ Tf u (θ̇u ) è Tf l (θ̇l ) áûëèýêñïåðèìåíòàëüíî íàéäåíû ñëåäóþùèå âûðàæåíèÿ⎧⎪⎨Tf l (ω + θ̇l ) ∈ ⎪⎩Tcl (ω + θ̇l )sign(ω + θ̇l ), ω + θ̇l = 0[−T0 , T0 ] ,ω + θ̇l = 0,(1.9)ãäåTcl (ω + θ̇l ) =T0Tsl (Tpl−|+ (Tsl − Tpl )eω+θ̇l δslωsl |+ bl |ω + θ̇l |).(1.10)Çäåñü T0 , Tsl , Tpl , ωsl , δsl è bl ïîëîæèòåëüíûå êîýôôèöèåíòû.
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