Polynomials-Based Terminal Control of Affine Systems

Abstract

One of the approaches to solving terminal control problems for affine dynamical systems is based on the use of polynomials of degree 2n − 1, where n is the order of the system in question. In this paper, we investigate the terminal control problem for which the final state of the system coincides with the origin in the phase space. We seek a set of initial states such that the solution of the terminal control problem can be constructed by using a polynomial of degree 2n − 2.Note that solution of the terminal control problem in question can be used to solve the problem of stabilizing the zero equilibrium in a finite time.For the second-order systems we prove the necessary and sufficient conditions for existence of the polynomial of the second degree which determines the solution of the terminal problem. The solutions of the terminal control problem based on the polynomials of second and third degree are given. As an example, the terminal control problem is considered for the simple pendulum.We also discuss solution of the terminal problem for affine systems of the third order, based on the use of the fourth and fifth degree polynomials. The necessary and sufficient conditions for existence of the fourth-degree polynomial such that its phase graph connects an arbitrary initial state of the system and the origin are obtained.For systems of arbitrary order n we obtain the necessary and sufficient conditions for existence of a solution of the terminal problem using the polynomial of degree 2n − 2. We also give the solution of the problem by means of the polynomial of degree 2n − 1.Further research can be focused on extending the results obtained in this note to terminal control problems where the desired final state of the system is not necessarily the origin.One of the potential application areas for the obtained theoretical results is automatic control of technical plants like unmanned aerial vehicles and mobile robots.

Full text

Íàóêà è Îáðàçîâàíèå. ÌÃÒÓ èì. Í.Ý. Áàóìàíà.
Ýëåêòðîí. æóðí. 2015. ¹ 2. Ñ. 101{114.
DOI: 10.7463/0215.0758826
Ïðåäñòàâëåíà â ðåäàêöèþ: 05.03.2015
c
ÌÃÒÓ èì. Í.Ý. Áàóìàíà
ÓÄÊ 519.71
Ðåøåíèå òåðìèíàëüíîé çàäà÷è óïðàâëåíèÿ
äëÿ àôôèííîé ñèñòåìû ïðè ïîìîùè ìíîãî÷ëåíîâ
Ãîëóáåâ À. Å.1,*, Êðèùåíêî À. Ï.1*
1ÌÃÒÓ èì. Í.Ý. Áàóìàíà, Ìîñêâà, Ðîññèÿ
 ñòàòüå èññëåäîâàíà òåðìèíàëüíàÿ çàäà÷à óïðàâëåíèÿ äëÿ àôôèííûõ äèíàìè÷åñêèõ ñèñòåì.
Ðàññìîòðåí ñëó÷àé, êîãäà êîíå÷íîå ñîñòîÿíèå ñèñòåìû ñîâïàäàåò ñ íà÷àëîì êîîðäèíàò â ôàçîâîì
ïðîñòðàíñòâå ñèñòåìû. Ðåøåíèå òåðìèíàëüíîé çàäà÷è ïîñòðîåíî íà îñíîâå ìíîãî÷ëåíîâ ñòåïåíè
2n−2, ãäå n| ïîðÿäîê ñèñòåìû. Ïîëó÷åíû íåîáõîäèìûå è äîñòàòî÷íûå óñëîâèÿ ñóùåñòâî-
âàíèÿ ìíîãî÷ëåíà ñòåïåíè 2n−2, ôàçîâûé ãðàôèê êîòîðîãî ñîåäèíÿåò ïðîèçâîëüíîå íà÷àëüíîå
ñîñòîÿíèå ñèñòåìû è íà÷àëî êîîðäèíàò â ôàçîâîì ïðîñòðàíñòâå. Ïðèâåäåíî òàêæå ðåøåíèå òåð-
ìèíàëüíîé çàäà÷è ïðè ïîìîùè ìíîãî÷ëåíà ñòåïåíè 2n−1. Ðàññìîòðåí ïðèìåð ðåøåíèÿ çàäà÷è
óïðàâëåíèÿ äëÿ ìàòåìàòè÷åñêîãî ìàÿòíèêà. Âîçìîæíîé îáëàñòüþ ïðèìåíåíèÿ ïîëó÷åííûõ â
ðàáîòå òåîðåòè÷åñêèõ ðåçóëüòàòîâ ÿâëÿåòñÿ ðåøåíèå çàäà÷ àâòîìàòè÷åñêîãî óïðàâëåíèÿ òåõíè÷å-
ñêèìè ñèñòåìàìè, íàïðèìåð, áåñïèëîòíûìè ëåòàòåëüíûìè àïïàðàòàìè è ìîáèëüíûìè ðîáîòàìè.
Êëþ÷åâûå ñëîâà: óïðàâëåíèå, àôôèííàÿ ñèñòåìà, òåðìèíàëüíàÿ çàäà÷à, ìíîãî÷ëåíû
Ââåäåíèå è ïîñòàíîâêà çàäà÷è
Ðàññìàòðèâàåòñÿ òåðìèíàëüíàÿ çàäà÷à óïðàâëåíèÿ äëÿ àôôèííîé äèíàìè÷åñêîé ñèñòåìû,
çàïèñàííîé â âèäå
y(n)+f(y, ˙y,...,y(n−1)) = g(y, ˙y,...,y(n−1))u, (1)
ãäå y=y, ˙y, . . . , y(n−1)∈Rn| âåêòîð ñîñòîÿíèÿ ñèñòåìû; u∈R| óïðàâëåíèå; f(y)è
g(y)| ãëàäêèå ôóíêöèè ñâîèõ àðãóìåíòîâ; g(y)6= 0 ïðè âñåõ y∈Rn.
Íàïîìíèì, ÷òî ïîä òåðìèíàëüíîé çàäà÷åé ïîäðàçóìåâàþò íàõîæäåíèå óïðàâëåíèÿ, ïå-
ðåâîäÿùåãî äèíàìè÷åñêóþ ñèñòåìó çà íåêîòîðûé îòðåçîê âðåìåíè èç çàäàííîãî íà÷àëüíîãî
ñîñòîÿíèÿ â çàäàííîå êîíå÷íîå ñîñòîÿíèå [1]. Îäèí èç ñïîñîáîâ ðåøåíèÿ òåðìèíàëüíûõ
çàäà÷ äëÿ ñèñòåì âèäà (1) îñíîâàí íà èñïîëüçîâàíèè ìíîãî÷ëåíîâ ñòåïåíè 2n−1[1, 2].
Ðåøåíèå ðàçëè÷íûõ òåðìèíàëüíûõ çàäà÷ ñ èñïîëüçîâàíèåì ìíîãî÷ëåíîâ ðàññìàòðèâàëîñü,
íàïðèìåð, â ðàáîòàõ [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].
Íàóêà è Îáðàçîâàíèå. ÌÃÒÓ èì. Í.Ý. Áàóìàíà 101

 íàñòîÿùåé ñòàòüå èññëåäóåòñÿ òåðìèíàëüíàÿ çàäà÷à, äëÿ êîòîðîé êîíå÷íîå ñîñòîÿíèå
ñèñòåìû (1) ñîâïàäàåò ñ íà÷àëîì êîîðäèíàò y= 0. Èùåòñÿ ìíîæåñòâî íà÷àëüíûõ ñîñòî-
ÿíèé òàêèõ, ÷òî ðåøåíèå òåðìèíàëüíîé çàäà÷è ìîæíî ïîñòðîèòü ïðè ïîìîùè ìíîãî÷ëåíà
ñòåïåíè 2n−2.
Îòìåòèì, ÷òî ðåøåíèå ðàññìàòðèâàåìîé òåðìèíàëüíîé çàäà÷è óïðàâëåíèÿ ìîæåò áûòü
èñïîëüçîâàíî äëÿ ðåøåíèÿ çàäà÷è ñòàáèëèçàöèè íóëåâîãî ïîëîæåíèÿ ðàâíîâåñèÿ çà êîíå÷íîå
âðåìÿ [15, 16, 17, 18].
 ðàçä. 1 äëÿ ñèñòåì âòîðîãî ïîðÿäêà äîêàçûâàþòñÿ íåîáõîäèìûå è äîñòàòî÷íûå óñëîâèÿ
ñóùåñòâîâàíèÿ ìíîãî÷ëåíà âòîðîé ñòåïåíè, êîòîðûé îïðåäåëÿåò ðåøåíèå òåðìèíàëüíîé çà-
äà÷è. Ïðèâîäèòñÿ ðåøåíèå òåðìèíàëüíîé çàäà÷è óïðàâëåíèÿ ñ èñïîëüçîâàíèåì ìíîãî÷ëåíîâ
âòîðîé è òðåòüåé ñòåïåíè. Ðàññìîòðåí ïðèìåð ðåøåíèÿ çàäà÷è óïðàâëåíèÿ äëÿ ìàòåìàòè-
÷åñêîãî ìàÿòíèêà.  ðàçä. 2 íàéäåíî ðåøåíèå òåðìèíàëüíîé çàäà÷è äëÿ àôôèííûõ ñèñòåì
òðåòüåãî ïîðÿäêà, îñíîâàííîå íà èñïîëüçîâàíèè ìíîãî÷ëåíîâ ÷åòâåðòîé è ïÿòîé ñòåïåíè.
Äëÿ ñèñòåì ïðîèçâîëüíîãî ïîðÿäêà nâ ðàçä. 3 ïîëó÷åíû íåîáõîäèìûå è äîñòàòî÷íûå óñëî-
âèÿ ñóùåñòâîâàíèÿ ðåøåíèÿ òåðìèíàëüíîé çàäà÷è c èñïîëüçîâàíèåì ìíîãî÷ëåíà ñòåïåíè
2n−2. Ïðèâåäåíî òàêæå ðåøåíèå çàäà÷è ïðè ïîìîùè ìíîãî÷ëåíà ñòåïåíè 2n−1.
1. Ñèñòåìû âòîðîãî ïîðÿäêà
Ïðè n= 2 ñèñòåìà (1) çàïèøåòñÿ ñëåäóþùèì îáðàçîì:
¨y+f(y) = g(y)u, (2)
ãäå y= (y, ˙y).  êà÷åñòâå íà÷àëüíîãî ñîñòîÿíèÿ ñèñòåìû ðàññìîòðèì ïðîèçâîëüíóþ òî÷êó
y= (y0,˙y0)ôàçîâîé ïëîñêîñòè, îòëè÷íóþ îò íà÷àëà êîîðäèíàò y= (0,0). Òîãäà èñêîìàÿ
ïðîãðàììíàÿ òðàåêòîðèÿ y(t),u(t)ñèñòåìû (2) äîëæíà óäîâëåòâîðÿòü ãðàíè÷íûì óñëîâèÿì
y(0) = y0,˙y(0) = ˙y0èy(T)=0,˙y(T)=0, ãäå T > 0íåêîòîðîå êîíå÷íîå çíà÷åíèå
íåçàâèñèìîé ïåðåìåííîé t.
Íàïîìíèì, ÷òî ôàçîâûì ãðàôèêîì [1] ôóíêöèè ϕ(t)∈Cn[0, T ]â ôàçîâîì ïðîñòðàíñòâå
ñèñòåìû (1) íàçûâàþò êðèâóþ, çàäàííóþ ïàðàìåòðè÷åñêè ïðè ïîìîùè óðàâíåíèé y(i)=
=ϕ(i)(t),i= 0, n−1,t∈[0, T ].
Íàéäåì ìíîãî÷ëåí p(t), ôàçîâûé ãðàôèê p(t) = (p(t),˙p(t)),t∈[0, T ], êîòîðîãî ñîåäè-
íÿåò òî÷êè y= (y0,˙y0)èy= (0,0) â ôàçîâîì ïðîñòðàíñòâå ñèñòåìû (2).
Ðàññìîòðèì ñíà÷àëà ñëó÷àé, êîãäà Tçàäàíî. Ñîãëàñíî [1, 2] ñóùåñòâóåò òàêîé åäèíñòâåí-
íûé ìíîãî÷ëåí ñòåïåíè 3, èìåþùèé âèä
p(t) = y0+ ˙y0t+c1t2+c2t3,(3)
÷òî âûïîëíåíû óñëîâèÿ p(T) = 0,˙p(T) = 0. Îòìåòèì, ÷òî ïðè ëþáûõ çíà÷åíèÿõ ïîñòîÿííûõ
c1èc2ñïðàâåäëèâû ðàâåíñòâà p(0) = y0è˙p(0) = ˙y0.
Íàóêà è Îáðàçîâàíèå. ÌÃÒÓ èì. Í.Ý. Áàóìàíà 102

Êîýôôèöèåíòû c1,c2íàõîäÿòñÿ èç ñèñòåìû ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé


T2T3
2T3T2



c1
c2

=

−y0−˙y0T
−˙y0

,(4)
ðåøåíèå êîòîðîé ñóùåñòâóåò è åäèíñòâåííî â ñèëó íåâûðîæäåííîñòè ìàòðèöû ýòîé ñèñòåìû.
Òàêèì îáðàçîì, ôàçîâûé ãðàôèê p(t)=(p(t),˙p(t)),t∈[0, T ], ìíîãî÷ëåíà (3) ïðè óêà-
çàííûõ çíà÷åíèÿõ êîýôôèöèåíòîâ c1,c2ñîåäèíÿåò òî÷êè y= (y0,˙y0)èy= (0,0) â ôàçîâîì
ïðîñòðàíñòâå ñèñòåìû (2), ïðè÷åì p(T) = 0,˙p(T) = 0.
 ñëó÷àå, åñëè çíà÷åíèå Tíå çàäàíî, èìååò ìåñòî ñëåäóþùèé ðåçóëüòàò.
Òåîðåìà 1. Äëÿ ñóùåñòâîâàíèÿ ìíîãî÷ëåíà p(t)ñòåïåíè 2, ôàçîâûé ãðàôèê p(t) =
= (p(t),˙p(t)),t∈[0, T ], êîòîðîãî ñîåäèíÿåò òî÷êè y= (y0,˙y0)èy= (0,0) â ôàçîâîì
ïðîñòðàíñòâå ñèñòåìû (2), íåîáõîäèìî è äîñòàòî÷íî âûïîëíåíèÿ óñëîâèÿ
y0˙y0<0.(5)
Ä î ê à ç à ò å ë ü ñ ò â î. Ðàññìîòðèì ìíîãî÷ëåí
p(t) = y0+ ˙y0t+c1t2,(6)
ãäå c1| íåêîòîðàÿ êîíñòàíòà, ïîäëåæàùàÿ îïðåäåëåíèþ. Îòìåòèì, ÷òî ïðè ëþáîì çíà÷åíèè
ïîñòîÿííîé c1ñïðàâåäëèâû ðàâåíñòâà p(0) = y0è˙p(0) = ˙y0.
Äàëåå, óñëîâèå p(T) = 0 âûïîëíåíî òîãäà è òîëüêî òîãäà, êîãäà èìååò ìåñòî ðàâåíñòâî
y0+ ˙y0T+c1T2= 0.(7)
Ãðàíè÷íîå óñëîâèå ˙p(T) = 0 çàïèñûâàåòñÿ â âèäå ñîîòíîøåíèÿ
˙y0+ 2c1T= 0.(8)
Ðåøèâ óðàâíåíèÿ (7) è (8) îòíîñèòåëüíî íåèçâåñòíûõ c1èT, ïîëó÷èì
c1=−˙y0
2T, T =−2y0
˙y0
.
Òàêèì îáðàçîì, äëÿ âûïîëíåíèÿ óñëîâèÿ T > 0íåîáõîäèìî è äîñòàòî÷íî, ÷òîáû áûëî
ñïðàâåäëèâî íåðàâåíñòâî (5).
Ïðè íàéäåííûõ çíà÷åíèÿõ c1èTìíîãî÷ëåí (6) ïðèìåò âèä
p(t) = y0+ ˙y0t+˙y2
0
4y0
t2.(9)
Ôàçîâûé ãðàôèê p(t) = (p(t),˙p(t)),t∈[0, T ], ìíîãî÷ëåíà (9) ñîåäèíÿåò òî÷êè y= (y0,˙y0)
èy= (0,0) íà ôàçîâîé ïëîñêîñòè, ïðè÷åì p(T) = 0,˙p(T) = 0. Òåîðåìà äîêàçàíà.
Íàóêà è Îáðàçîâàíèå. ÌÃÒÓ èì. Í.Ý. Áàóìàíà 103

Ïðîãðàììíîå óïðàâëåíèå, ÿâëÿþùååñÿ ðåøåíèåì ðàññìàòðèâàåìîé òåðìèíàëüíîé çàäà÷è
äëÿ ñèñòåìû (2), çàïèøåòñÿ ñëåäóþùèì îáðàçîì [1, 2]:
u(t) = 1
g(p(t))¨p(t) + f(p(t)),(10)
ãäå p(t)| ñîîòâåòñòâóþùèé ìíîãî÷ëåí, èìåþùèé âèä (3), (4) èëè (9).
Ïðèìåð. Ðàññìîòðèì òåðìèíàëüíóþ çàäà÷ó óïðàâëåíèÿ äëÿ ìàòåìàòè÷åñêîãî ìàÿòíèêà,
óðàâíåíèÿ äâèæåíèÿ êîòîðîãî èìåþò âèä



˙x1=x2,
˙x2=csin x1−dx2+u, (11)
ãäå x= (x1, x2)ò∈R2| âåêòîð ñîñòîÿíèÿ ñèñòåìû; u| óïðàâëÿþùèé ìîìåíò, êîíñòàíòû
cèdïîëîæèòåëüíû.
 êà÷åñòâå æåëàåìîãî êîíå÷íîãî ñîñòîÿíèÿ ñèñòåìû (11) âîçüìåì òî÷êó x= 0, ñîîòâåò-
ñòâóþùóþ âåðõíåìó íåóñòîé÷èâîìó ïîëîæåíèþ ðàâíîâåñèÿ ìàÿòíèêà.
 ïåðåìåííûõ y=x1,˙y=x2ñèñòåìà (11) ïðèìåò âèä (2), ãäå f(y) = −csin y+d˙y,
g(y) = 1.
Ïîñòðîèì ðåøåíèå òåðìèíàëüíîé çàäà÷è ïðè ïîìîùè ìíîãî÷ëåíîâ âòîðîé è òðåòüåé
ñòåïåíè.  êà÷åñòâå íà÷àëüíîãî ñîñòîÿíèÿ ñèñòåìû (2) ðàññìîòðèì ïðîèçâîëüíóþ òî÷êó
y= (y0,˙y0)ôàçîâîé ïëîñêîñòè òàêóþ, ÷òî âûïîëíåíî íåðàâåíñòâî (5). Òîãäà ñîãëàñíî
òåîðåìå 1 ôàçîâûé ãðàôèê p(t)=(p(t),˙p(t)),t∈[0,−2y0/˙y0], ìíîãî÷ëåíà (9) cîåäèíÿåò
òî÷êè y= (y0,˙y0)èy= (0,0) â ôàçîâîì ïðîñòðàíñòâå ñèñòåìû (2).
Ñëåäîâàòåëüíî, ïàðàìåòðè÷åñêè çàäàííàÿ êðèâàÿ x1=p(t),x2= ˙p(t),t∈[0,−2y0/˙y0],
ñîåäèíÿåò â ïðîñòðàíñòâå ñîñòîÿíèé ñèñòåìû (11) òî÷êè x= (y0,˙y0)òèx= 0.
Îòìåòèì, ÷òî êðèâîé, ñîåäèíÿþùåé òî÷êè x= (y0,˙y0)òèx= 0 â ôàçîâîì ïðîñòðàíñòâå
ñèñòåìû (11), ÿâëÿåòñÿ òàêæå ôàçîâûé ãðàôèê p(t) = (p(t),˙p(t))ò,t∈[0, T ], ìíîãî÷ëåíà (3),
ãäå êîýôôèöèåíòû c1,c2óäîâëåòâîðÿþò ñèñòåìå ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé (4),
T > 0| íåêîòîðîå çàäàííîå êîíå÷íîå çíà÷åíèå íåçàâèñèìîé ïåðåìåííîé.
Ðåøåíèå ðàññìàòðèâàåìîé òåðìèíàëüíîé çàäà÷è äëÿ ñèñòåìû (11) èìååò âèä ïðîãðàìì-
íîãî óïðàâëåíèÿ (10), ãäå p(t)| ñîîòâåòñòâóþùèé ìíîãî÷ëåí (9) âòîðîé ñòåïåíè èëè ìíî-
ãî÷ëåí (3) òðåòüåé ñòåïåíè.
Ðåçóëüòàòû ÷èñëåííîãî ìîäåëèðîâàíèÿ ñèñòåìû (11) ñ óïðàâëåíèåì (10) ïðåäñòàâëåíû
íà ðèñ. 1 è 2 ïðè ñëåäóþùèõ çíà÷åíèÿõ ïàðàìåòðîâ è íà÷àëüíûõ çíà÷åíèé: c= 9,81,d= 0,1,
x1(0) = π/6,x2(0) = −0,5.
Çàìåòèì, ÷òî ïðè ïîñòðîåíèè ïðîãðàììíîé òðàåêòîðèè äëÿ ñèñòåìû (2) íà îñíîâå ìíî-
ãî÷ëåíà òðåòüåé ñòåïåíè âîçìîæíû êîëåáàòåëüíûå ïðîöåññû, ÷òî íåæåëàòåëüíî.  ñëó÷àå
èñïîëüçîâàíèÿ ìíîãî÷ëåíà âòîðîé ñòåïåíè ëåãêî ïîêàçàòü, ÷òî äëÿ ðàññìàòðèâàåìîé òåðìè-
íàëüíîé çàäà÷è îáðàùåíèå â íîëü ïðîèçâîäíîé ˙p(t)íà ôàçîâîé ïëîñêîñòè âîçìîæíî òîëüêî
â òî÷êå y= (0,0) ïðè t=−2y0/˙y0.
Íàóêà è Îáðàçîâàíèå. ÌÃÒÓ èì. Í.Ý. Áàóìàíà 104

Ðèñ. 1. Ôàçîâûå òðàåêòîðèè ñèñòåìû, ñîîòâåòñòâóþùèå ìíîãî÷ëåíó âòîðîé
(ñïëîøíàÿ ëèíèÿ) è òðåòüåé (ïóíêòèð) ñòåïåíè (T > −2x1(0)/x2(0))
Ðèñ. 2. Ôàçîâûå òðàåêòîðèè ñèñòåìû, ñîîòâåòñòâóþùèå ìíîãî÷ëåíó âòîðîé
(ñïëîøíàÿ ëèíèÿ) è òðåòüåé (ïóíêòèð) ñòåïåíè (T < −2x1(0)/x2(0))
Íàóêà è Îáðàçîâàíèå. ÌÃÒÓ èì. Í.Ý. Áàóìàíà 105

2. Ñèñòåìû òðåòüåãî ïîðÿäêà
Ðàññìîòðèì ñèñòåìó (1), èìåþùóþ òðåòèé ïîðÿäîê è çàïèñàííóþ â âèäå
y(3) +f(y) = g(y)u, (12)
ãäå y= (y, ˙y, ¨y). Åñëè çíà÷åíèå T > 0ôèêñèðîâàíî, òî ñóùåñòâóåò åäèíñòâåííûé ìíîãî÷ëåí
ïÿòîé ñòåïåíè
p(t) = y0+ ˙y0t+¨y0
2t2+c1t3+c2t4+с3t5,(13)
ôàçîâûé ãðàôèê p(t) = (p(t),˙p(t),¨p(t)),t∈[0, T ], êîòîðîãî ñîåäèíÿåò òî÷êè y= (y0,˙y0,¨y0)
èy= (0,0,0) â ôàçîâîì ïðîñòðàíñòâå ñèñòåìû (12), ïðè÷åì p(T) = 0,˙p(T) = 0,¨p(T) = 0
[1, 2]. Ïîñòîÿííûå c1,c2èc3èùóòñÿ èç ñëåäóþùåé ñèñòåìû ëèíåéíûõ àëãåáðàè÷åñêèõ
óðàâíåíèé:





T3T4T5
3T24T35T4
6T12T220T3










c1
c2
c3





=





−y0−˙y0T−¨y0
2T2
−˙y0−¨y0T
−¨y0





,(14)
ðåøåíèå êîòîðîé ñóùåñòâóåò è åäèíñòâåííî, òàê êàê ìàòðèöà ýòîé ñèñòåìû íåâûðîæäåíà.
 ñëó÷àå, êîãäà çíà÷åíèå Tíå çàäàíî, ââåäåì îáîçíà÷åíèÿ a= 6 ˙y0/¨y0,b= 12y0/¨y0è
ñôîðìóëèðóåì ñëåäóþùóþ òåîðåìó.
Òåîðåìà 2. Äëÿ ñóùåñòâîâàíèÿ ìíîãî÷ëåíà p(t)ñòåïåíè 4, ôàçîâûé ãðàôèê p(t) =
= (p(t),˙p(t),¨p(t)),t∈[0, T ], êîòîðîãî ñîåäèíÿåò òî÷êè y= (y0,˙y0,¨y0)èy= (0,0,0) â
ôàçîâîì ïðîñòðàíñòâå ñèñòåìû (12), íåîáõîäèìî è äîñòàòî÷íî âûïîëíåíèÿ óñëîâèé
b < 0
èëè
b≥0, a < 0, a2−4b≥0
èëè
¨y0= 0, y0˙y0<0.
Ä î ê à ç à ò å ë ü ñ ò â î. Ðàññìîòðèì ìíîãî÷ëåí
p(t) = y0+ ˙y0t+¨y0t2
2+c1t3+c2t4,(15)
ãäå c1,c2| íåêîòîðûå êîíñòàíòû, ïîäëåæàùèå îïðåäåëåíèþ. Çàìåòèì, ÷òî ïðè ëþáûõ
çíà÷åíèÿõ ïîñòîÿííûõ c1èc2âûïîëíÿþòñÿ ðàâåíñòâà p(0) = y0,˙p(0) = ˙y0è¨p(0) = ¨y0.
Ãðàíè÷íûå óñëîâèÿ p(T)=0,˙p(T)=0è¨p(T)=0ýêâèâàëåíòíû ñëåäóþùåé ñèñòåìå
óðàâíåíèé îòíîñèòåëüíî íåèçâåñòíûõ c1,c2èT:













y0+ ˙y0T+¨y0T2
2+c1T3+c2T4= 0,
˙y0+ ¨y0T+ 3c1T2+ 4c2T3= 0,
¨y0+ 6c1T+ 12c2T2= 0.
(16)
Íàóêà è Îáðàçîâàíèå. ÌÃÒÓ èì. Í.Ý. Áàóìàíà 106

Ðàññìîòðåâ ïåðâûå äâà óðàâíåíèÿ ñèñòåìû (16) êàê ñèñòåìó ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâ-
íåíèé îòíîñèòåëüíî ïîñòîÿííûõ c1èc2, íàéäåì
c1=−˙y0
T2−2¨y0
3T, c2=˙y0
2T3+¨y0
4T2.(17)
Ïîäñòàâèâ ñîîòíîøåíèÿ (17) â ïîñëåäíåå óðàâíåíèå ñèñòåìû (16), ïðè ¨y06= 0 îòíîñè-
òåëüíî íåèçâåñòíîãî Tïîëó÷èì êâàäðàòíîå óðàâíåíèå
T2+aT +b= 0.(18)
Äèñêðèìèíàíò êâàäðàòíîãî òðåõ÷ëåíà, ñòîÿùåãî â ëåâîé ÷àñòè ðàâåíñòâà (18), èìååò âèä
D=a2−4b. Êâàäðàòíîå óðàâíåíèå (18) èìååò ïîëîæèòåëüíîå ðåøåíèå T > 0òîãäà è
òîëüêî òîãäà, êîãäà âûïîëíåíû óñëîâèÿ b < 0èëè b≥0,a < 0,D≥0.
 ñëó÷àå ¨y0= 0 îòíîñèòåëüíî íåèçâåñòíîãî Tèìååì óðàâíåíèå
6 ˙y0T+ 12y0= 0.
Ñëåäîâàòåëüíî, äëÿ ïîëîæèòåëüíîñòè T=−2y0/˙y0íåîáõîäèìî è äîñòàòî÷íî âûïîëíåíèÿ
óñëîâèÿ y0˙y0<0.
Äàëåå, â ñëó÷àå ïîëîæèòåëüíîñòè Tôàçîâûé ãðàôèê p(t)=(p(t),˙p(t),¨p(t)),t∈[0, T ],
ìíîãî÷ëåíà (15), (17), (18) ñîåäèíÿåò òî÷êè y= (y0,˙y0,¨y0)èy= (0,0,0) â ôàçîâîì
ïðîñòðàíñòâå ñèñòåìû (12), ïðè÷åì p(T) = 0,˙p(T) = 0,¨p(T) = 0. Òåîðåìà äîêàçàíà.
Ïðîãðàììíîå óïðàâëåíèå, ÿâëÿþùååñÿ ðåøåíèåì ðàññìàòðèâàåìîé òåðìèíàëüíîé çàäà÷è
äëÿ ñèñòåìû (12), ñîãëàñíî [1, 2] çàïèøåòñÿ ñëåäóþùèì îáðàçîì:
u(t) = 1
g(p(t))p(3)(t) + f(p(t)),
ãäå p(t)| ñîîòâåòñòâóþùèé ìíîãî÷ëåí, èìåþùèé âèä (13), (14) èëè (15), (17), (18).
3. Ñèñòåìû ïîðÿäêà n > 3
Ðàññìîòðèì ñèñòåìó (1) ïðîèçâîëüíîãî ïîðÿäêà n. Ïðåäïîëîæèì, ÷òî çíà÷åíèå T > 0
çàäàíî. Òîãäà ñîãëàñíî [1, 2] ñóùåñòâóåò åäèíñòâåííûé ìíîãî÷ëåí ñòåïåíè 2n−1, èìåþùèé
âèä
p(t) =
n−1
X
k=0
y(k)
0
k!tk+
n
X
k=1
cktn−1+k,(19)
ôàçîâûé ãðàôèê p(t) = p(t),˙p(t), . . . , p(n−1)(t),t∈[0, T ], êîòîðîãî ñîåäèíÿåò òî÷êè
y= (y0,˙y0, . . . , y(n−1)
0)èy= (0, . . . , 0) â ôàçîâîì ïðîñòðàíñòâå ñèñòåìû (1), ïðè÷åì
p(T)=0,˙p(T)=0, ..., p(n−1) (T)=0. Ïîñòîÿííûå c1,c2, ..., cníàõîäÿòñÿ èç ñèñòåìû
Íàóêà è Îáðàçîâàíèå. ÌÃÒÓ èì. Í.Ý. Áàóìàíà 107

ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé








TnTn+1 . . . T 2n−1
nT n−1(n+ 1)Tn. . . (2n−1)T2n−2
......................
n!
1! T(n+ 1)!
2! T2. . . (2n−1)!
n!Tn
















c1
c2
.
.
.
cn








=













−
n−1
P
k=0
y(k)
0
k!Tk
−
n−1
P
k=1
y(k)
0
(k−1)!Tk−1
.
.
.
−y(n−1)
0













,(20)
ðåøåíèå êîòîðîé ñóùåñòâóåò è åäèíñòâåííî â ñèëó íåâûðîæäåííîñòè ìàòðèöû ýòîé ñèñòåìû.
Ðàññìîòðèì òåïåðü ñëó÷àé, êîãäà çíà÷åíèå Tíå ôèêñèðîâàíî. Èìååò ìåñòî ñëåäóþùåå
óòâåðæäåíèå.
Òåîðåìà 3. Äëÿ ñóùåñòâîâàíèÿ ìíîãî÷ëåíà p(t)ñòåïåíè 2n−2, ôàçîâûé ãðàôèê p(t) =
=p(t),˙p(t), . . . , p(n−1)(t),t∈[0, T ], êîòîðîãî ñîåäèíÿåò òî÷êè y= (y0,˙y0, . . . , y(n−1)
0)
èy= (0, . . . , 0) â ôàçîâîì ïðîñòðàíñòâå ñèñòåìû (1), äîñòàòî÷íî, ÷òîáû ÷èñëî ïåðåìåí
çíàêîâ â ñèñòåìå íà÷àëüíûõ çíà÷åíèé y0,˙y0,...,y(n−2)
0,y(n−1)
0áûëî íå÷åòíî.
Ä î ê à ç à ò å ë ü ñ ò â î. Ðàññìîòðèì ìíîãî÷ëåí
p(t) =
n−1
X
k=0
y(k)
0
k!tk+
n−1
X
k=1
cktn−1+k,(21)
ãäå c1,c2, ..., cn−1| êîíñòàíòû, ïîäëåæàùèå îïðåäåëåíèþ. Îòìåòèì, ÷òî ïðè ëþáûõ
çíà÷åíèÿõ ïîñòîÿííûõ c1,c2,...,cn−1äëÿ ìíîãî÷ëåíà (21) âûïîëíÿþòñÿ óñëîâèÿ p(0) = y0,
˙p(0) = ˙y0,...,p(n−1)(0) = y(n−1)
0.
Ñîîòíîøåíèÿ p(T) = 0,˙p(T) = 0,...,p(n−1)(T) = 0 çàïèøåì â âèäå ñèñòåìû óðàâíåíèé
îòíîñèòåëüíî íåèçâåñòíûõ c1,c2,...,cn−1èT, êîòîðàÿ èìååò âèä









TnTn+1 . . . T 2n−2
nT n−1(n+ 1)Tn. . . (2n−2)T2n−3
.....................
n!
1! T(n+ 1)!
2! T2. . . (2n−2)!
(n−1)! Tn−1

















c1
c2
.
.
.
cn−1








=













−
n−1
P
k=0
y(k)
0
k!Tk
−
n−1
P
k=1
y(k)
0
(k−1)!Tk−1
.
.
.
−y(n−1)
0













.(22)
Ðàññìîòðèì, íàïðèìåð, ïåðâûå n−1óðàâíåíèé ñèñòåìû (22) êàê ñèñòåìó ëèíåéíûõ
àëãåáðàè÷åñêèõ óðàâíåíèé îòíîñèòåëüíî ïîñòîÿííûõ c1,c2,. . .,cn−1. Ðåøèâ ýòó ñèñòåìó è
ïîäñòàâèâ íàéäåííûå çíà÷åíèÿ c1,c2,. . .,cn−1â ïîñëåäíåå óðàâíåíèå ñèñòåìû (22), ïîëó÷èì
îòíîñèòåëüíî íåèçâåñòíîãî Tñëåäóþùåå óðàâíåíèå:
y(n−1)
0Tn−1+n!
1!(n−2)!y(n−2)
0Tn−2+(n+ 1)!
2!(n−3)!y(n−3)
0Tn−3+. . .
. . . +(2n−3)!
(n−2)!1! ˙y0T+(2n−2)!
(n−1)! y0= 0.(23)
Íàóêà è Îáðàçîâàíèå. ÌÃÒÓ èì. Í.Ý. Áàóìàíà 108

Ñîãëàñíî [19] ÷èñëî ïîëîæèòåëüíûõ êîðíåé ìíîãî÷ëåíà, ñòîÿùåãî â ëåâîé ÷àñòè ðàâåí-
ñòâà (23), ðàâíî ÷èñëó ïåðåìåí çíàêîâ â ñèñòåìå êîýôôèöèåíòîâ y0,˙y0...,y(n−1)
0èëè ìåíüøå
ýòîãî ÷èñëà íà ÷åòíîå ÷èñëî. Òàêèì îáðàçîì, åñëè ÷èñëî ïåðåìåí çíàêîâ â óêàçàííîé ñè-
ñòåìå çíà÷åíèé êîýôôèöèåíòîâ íå÷åòíî, òî ñóùåñòâóåò õîòÿ áû îäíî ïîëîæèòåëüíîå ðåøå-
íèå T > 0óðàâíåíèÿ (23).  ýòîì ñëó÷àå ôàçîâûé ãðàôèê p(t) = p(t),˙p(t), . . . , p(n−1)(t),
t∈[0, T ], ìíîãî÷ëåíà (21), (22) ñîåäèíÿåò òî÷êè y= (y0,˙y0, . . . , y(n−1)
0)èy= (0, . . . , 0) â
ôàçîâîì ïðîñòðàíñòâå ñèñòåìû (1), ïðè÷åì p(T) = 0,˙p(T) = 0,...,p(n−1)(T) = 0. Òåîðåìà
äîêàçàíà.
Ç à ì å ÷ à í è å . Íåîáõîäèìûì è äîñòàòî÷íûì óñëîâèåì ñóùåñòâîâàíèÿ ìíîãî÷ëåíà
ñòåïåíè 2n−2, ôàçîâûé ãðàôèê êîòîðîãî ñîåäèíÿåò òî÷êè y= (y0,˙y0, . . . , y(n−1)
0)è
y= (0, . . . , 0) â ôàçîâîì ïðîñòðàíñòâå ñèñòåìû (1), ÿâëÿåòñÿ íàëè÷èå ïîëîæèòåëüíîãî
êîðíÿ ó ìíîãî÷ëåíà, ñòîÿùåãî â ëåâîé ÷àñòè ðàâåíñòâà (23).
Íàêîíåö, ïðîãðàììíîå óïðàâëåíèå, ÿâëÿþùååñÿ ðåøåíèåì ðàññìàòðèâàåìîé òåðìèíàëü-
íîé çàäà÷è äëÿ ñèñòåìû (1), ñëåäóþùåå: [1, 2]:
u(t) = 1
g(p(t))p(n)(t) + f(p(t)),(24)
ãäå p(t)| ñîîòâåòñòâóþùèé ìíîãî÷ëåí (19), (20) èëè (21), (22).
Çàêëþ÷åíèå
 íàñòîÿùåé ðàáîòå èññëåäîâàíà òåðìèíàëüíàÿ çàäà÷à óïðàâëåíèÿ äëÿ àôôèííûõ ñèñòåì,
èìåþùèõ âèä (1). Ðàññìîòðåí ñëó÷àé, êîãäà êîíå÷íîå ñîñòîÿíèå ñèñòåìû ñîâïàäàåò ñ íà-
÷àëîì êîîðäèíàò. Ðåøåíèå òåðìèíàëüíîé çàäà÷è ïîñòðîåíî íà îñíîâå ìíîãî÷ëåíà ñòåïåíè
2n−2. Ïðèâåäåíî òàêæå èçâåñòíîå ðåøåíèå ïðè ïîìîùè ìíîãî÷ëåíà ñòåïåíè 2n−1.
Îòìåòèì, ÷òî íåîáõîäèìûå è äîñòàòî÷íûå óñëîâèÿ ñóùåñòâîâàíèÿ çàìåíû ïåðåìåííûõ,
ïðåîáðàçóþùåé àôôèííóþ äèíàìè÷åñêóþ ñèñòåìó ê âèäó (1), ìîæíî íàéòè, íàïðèìåð, â
ìîíîãðàôèè [1].
Äàëüíåéøèå èññëåäîâàíèÿ ìîãóò áûòü ñâÿçàíû ñ ðåøåíèåì íà îñíîâå ìíîãî÷ëåíîâ ñòå-
ïåíè 2n−2òåðìèíàëüíûõ çàäà÷ óïðàâëåíèÿ ïðè ïðîèçâîëüíîì êîíå÷íîì ñîñòîÿíèè ñè-
ñòåìû (1), íå ñîâïàäàþùèì ñ íà÷àëîì êîîðäèíàò.
Ðàáîòà âûïîëíåíà ïðè ôèíàíñîâîé ïîääåðæêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ
(ïðîåêòû ¹736 ïðîãðàììû <Îðãàíèçàöèÿ ïðîâåäåíèÿ íàó÷íûõ èññëåäîâàíèé> è ¹1711
ãîñóäàðñòâåííîãî çàäàíèÿ ÐÔ) è Ðîññèéñêîãî ôîíäà ôóíäàìåíòàëüíûõ èññëåäîâàíèé (ïðîåêò
¹14-01-00424).
Ñïèñîê ëèòåðàòóðû
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åêòîðèè áåñïèëîòíîãî ëåòàòåëüíîãî àïïàðàòà â âåðòèêàëüíîé ïëîñêîñòè // Íàóêà è
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15. Perruquetti W., Floquet T., Orlov Y. Finite time stabilization of interconnected second order
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P. 6460{6465. DOI: 10.1109/CDC.2012.6425999
17. Polyakov A. Fixed-time stabilization of linear systems via sliding mode control //
Proc. of the 12th Workshop on Variable Structure Systems. IEEE, 2012. P. 1{6. DOI:
10.1109/VSS.2012.6163469
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systems // IEEE Trans. on Automatic Control. 2012. Vol. 57, no. 8. P. 2106{2110. DOI:
10.1109/TAC.2011.2179869
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Íàóêà è Îáðàçîâàíèå. ÌÃÒÓ èì. Í.Ý. Áàóìàíà 111

Science and Education of the Bauman MSTU,
2015, no. 2, pp. 101{114.
DOI: 10.7463/0215.0758826
Received: 05.03.2015
c
Bauman Moscow State Technical University
Polynomials-Based Terminal Control of Affine Systems
Golubev A. E.1,*, Krishchenko A. P.1*
1Bauman Moscow State Technical University, Russia
Keywords: control, affine system, terminal problem, polynomials
One of the approaches to solving terminal control problems for affine dynamical systems
is based on the use of polynomials of degree 2n−1, where nis the order of the system in
question. In this paper, we investigate the terminal control problem for which the final state
of the system coincides with the origin in the phase space. We seek a set of initial states such
that the solution of the terminal control problem can be constructed by using a polynomial of
degree 2n−2.
Note that solution of the terminal control problem in question can be used to solve the problem
of stabilizing the zero equilibrium in a finite time.
For the second-order systems we prove the necessary and sufficient conditions for existence of
the polynomial of the second degree which determines the solution of the terminal problem. The
solutions of the terminal control problem based on the polynomials of second and third degree are
given. As an example, the terminal control problem is considered for the simple pendulum.
We also discuss solution of the terminal problem for affine systems of the third order, based
on the use of the fourth and fifth degree polynomials. The necessary and sufficient conditions for
existence of the fourth-degree polynomial such that its phase graph connects an arbitrary initial
state of the system and the origin are obtained.
For systems of arbitrary order nwe obtain the necessary and sufficient conditions for existence
of a solution of the terminal problem using the polynomial of degree 2n−2. We also give the
solution of the problem by means of the polynomial of degree 2n−1.
Further research can be focused on extending the results obtained in this note to terminal control
problems where the desired final state of the system is not necessarily the origin.
One of the potential application areas for the obtained theoretical results is automatic control
of technical plants like unmanned aerial vehicles and mobile robots.
Science and Education of the Bauman MSTU 112

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