Êàðà÷óí Â.Â.,
Ìåëüíèê Â.Ì., Ëîçîâèê Ò.Ì., Ñàâåð÷åíêî Â.Ã., Êðèâåöü Î.Î.
Íàö³îíàëüíèé
òåõí³÷íèé óí³âåðñèòåò Óêðà¿íè «Êϲ»
²ÄÅÍÒÈÔ²ÊÀÖ²ß ÄÈÍÀ̲×ÍÎÃÎ ÎÁ’ªÊÒÓ
Ïîñòàâèìî íà
ìåò³ ³äåíòèô³êàö³þ äèíàì³÷íîãî îá’ºêòó çà ðÿäîì òî÷îê éîãî
àìïë³òóäíî-ôàçîâî¿ õàðàêòåðèñòèêè.
Ìåòîä
íàéìåíøèõ êâàäðàò³â ç éîãî çäàòí³ñòþ çãëàäæóâàííÿ ìîæëèâèõ ïîõèáîê ó âèõ³äíèõ
äàíèõ, ùî âèêîðèñòîâóþòüñÿ äëÿ ³äåíòèô³êàö³¿ äèíàì³÷íîãî îá’ºêòà, à òàêîæ ç
éîãî íåçàëåæí³ñòþ â³ä íåïîâíî¿ â³äïîâ³äíîñò³ àïðîêñèìóþ÷î¿ ñòðóêòóðè ãëèáèííèì
âëàñòèâîñòÿì äîñë³äæóâàíîãî îᢺêòó, ìîæå áóòè
çàñòîñîâàíèì äëÿ ³äåíòèô³êàö³¿ çà ðÿäîì òî÷îê àìïë³òóäíî-ôàçîâî¿ õàðàêòåðèñòèêè
â³äïîâ³äíîãî êàíàëó.
Îòæå,
ïðèïóñòèìî, ùî º â íàÿâíîñò³ ðÿä òî÷îê àìïë³òóäíî-ôàçîâî¿ õàðàêòåðèñòèêè 
, 
. Òóò 
, ³íôîðìàö³ÿ
çîñåðåäæåíà â òðüîõ ìàñèâàõ, â³äïîâ³äíî 
. Àïðîêñèìóþ÷ó ïåðåäàòî÷íó
ôóíêö³þ øóêàºìî ó âèãëÿä³
. (1)
Çàì³íþþ÷è â (1) 
íà 
òà ïîçíà÷àþ÷è 
, îäåðæóºìî
.
Çà êðèòåð³é ÿêîñò³ àïðîêñèìàö³¿ ïðèéìàºìî íàñòóïíèé –
(2)
Óìîâè ì³í³ì³çàö³¿ Å:
(3)
ϳäñòàâëÿþ÷è
â (3) âèðàç (2), îäåðæóºìî ñèñòåìó ç (2n+1) ë³í³éíèõ àëãåáðà¿÷íèõ ð³âíÿíü
â³äíîñíî øóêàíèõ êîåô³ö³ºíò³â ïåðåäàòî÷íî¿ ôóíêö³¿ (1).
Ðîçãëÿíåìî
àëãîðèòì ôîðìóâàííÿ ðîçøèðåíî¿ ìàòðèö³ ö³º¿ ñèñòåìè äëÿ ðÿäó ïîñë³äîâíèõ
çíà÷åíü n.
Íåõàé 
(4)
. (5)
Óìîâè
îïòèì³çàö³¿ Å ìàþòü âèãëÿä:
(6)
Ðîçøèðåíó
ìàòðèöþ äëÿ ñèñòåìè (6) íàâåäåìî ó âèãëÿä³:
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4 |
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ϳä
ñòîâï÷èêàìè ðîçøèðåíî¿ ìàòðèö³ îçíà÷åí³ çíà÷åííÿ íåâ³äîìèõ, êîåô³ö³ºíòàìè ïðè
ÿêèõ ñëóãóþòü åëåìåíòè â³äïîâ³äíèõ ñòîâï÷èê³â.
Íåõàé òåïåð: 
. (7)
(8)
Óìîâè
ì³í³ì³çàö³¿ Å ì³ñòÿòü:
Ðîçøèðåíó
ìàòðèöþ äàíî¿ ñèñòåìè çàïèøåìî òàê –
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Äëÿ 
ìàºìî:
; (9)
(10)
Çà
áðàêîì ì³ñöÿ îïóñêàºìî ôîðìóëè äëÿ ïîõ³äíèõ â³ä Å, íàâîäèìî â³äðàçó ñõåìó çàïîâíåííÿ ðîçøèðåíî¿ ìàòðèö³, ïðè÷îìó ðîç³á’ºìî
¿¿ íà äâ³ ÷àñòèíè: ïåðøà – 4 ñòîâï÷èêè, äðóãà – ðåøòà.
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- |
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m+1 |
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Ïðîöåñ
íàðîùóâàííÿ n ìîæíà ïðîäîâæóâàòè. Íà ïðàêòèö³, î÷åâèäíî, äîö³ëüíî îáìåæèòèñü
ìàêñèìàëüíèìè çíà÷åííÿìè n ïîðÿäêà 4 ÷è 5. Êîðèñòóâà÷ ëåãêî ìîæå ñàì ñòâîðèòè
â³äïîâ³äíå ïðîãðàìíå çàáåçïå÷åííÿ.
Ðîçâ¢ÿçàííÿ ñèñòåìè ð³âíÿíü, ÿê³é
â³äïîâ³äຠðîçøèðåíà ìàòðèöÿ, ñôîðìîâàíà îïèñàíèì ñïîñîáîì, ðåàë³çóºòüñÿ
âèêëèêîì ï³äïðîãðàìè SystUr â òàê³é ôîðì³
SystUr (2* n+1, U,A),
äå U –ðîçøèðåíà
ìàòðèöÿ òèïó Matr
(type Matr=array[1..2*Nmax+1,1..2*Nmax+2]
of real),
A-ìàñèâ
òèïó Coef.
(type
Coef=array[-1..30] of real).
Ðîçâ¢ÿçîê ó âèãëÿä³ ìàñèâó À ðîçíîñèòüñÿ íà äâà ìàñèâè: À òà Â
(îáèäâà òèïó Coef) òàêèì ÷èíîì
A[-1]=n; A[0]=1;
B[-1]=n;
For s:=0 to n do B[s]:=A[n+1+s].
Íà öüîìó
âèçíà÷åííÿ êîåô³ö³ºíò³â ïîë³íîì³â B(p)
òà À(ð) ÷èñåëüíèêà òà çíàìåííèêà
â³äïîâ³äíî¿ ïåðåäàòî÷íî¿ ôóíêö³¿ çàâåðøóºòüñÿ.
Îïèñàíèé
àëãîðèòì ðåàë³çóºòüñÿ ï³äïðîãðàìîþ WpCMNK
Procedure WpCMNK(W,R,J:coef;n:integer; var B,A:coef);
var z,s,m:integer; v:real;
WP:array [2..2*n] of coef;
E,RJ,R2,J2,R2J2:coef;
U:Matr;
function C(E,F:coef):real;
var s:integer; D:real;
begin
D:=0;
for
s:=0 to m do D:=D+E[s]*F[s]; C:=D
end;
function Zn(s:integer):integer;
begin if odd(s) then
Zn:=1 else Zn:=-1 end;
begin
m:=round(W[-1]);
for
s:=0 to m do
begin
E[s]:=1; R2[s]:=sqr(R[s]);
WP[2,s]:=sqr(W[s]);
J2[s]:=sqr(J[s]); RJ[s]:=R[s]+J[s];
R2J2:=R2[s]+J2[s]
end;
for z:=3 to 2*n do
for
s:=0 to m do WP[z,s]:=WP[z-1,s]*W[s];
V:=C(R2J2,WP[2]); U[1,1]:=V;
case
N of
1: begin
U[2,2]:=m+1; V:=C(J,W); U[1,2]:=V;
U[2,1]:=V; V:=C(R,WP[2]);U[1,3]:=-V;
U[3,1]:=-V; U[2,4]:=C(R,E); U[3,4]:=C(J,W)
for s:=1 to 3 do U[s,5-s]:=0;
U[3,3]:=C(W,W)
end;
2: begin
U[2,6]:=V; U[1,2]:=0;
U[2,1]:=0;
V:=C(J,W); U[1,3]:=V;
U[3,1]:=V;
U[2,2]:=C(R2J2, WP[4]);
V:=C(R,WP[2]);
U[1,4]:=-V; U[2,3]:=V;
U[3,2]:=V;
U[4,1]:=-V; V:=C(J,WP[3]);
for s:=1 to 5 do U[s,6-s]:=-V*Zn(s);
U[3,3]:=m+1; V:=C(R,WP[4]);
U[1,6]:=0;
U[2,5]:=-V; U[3,4]:=0; U[4,3]:=0;
U[5,1]:=-V; V:=C(WP[2],E);
for s:=3 to 5 do U[s,8-s]:=-V*Zn[s];
V:=C(R,E); U[3,6]:=V;
U[4,5]:=0;
U[5,4]:=0; U[5,5]:=C(WP[4],E)
end;
3:begin
U[2,8]:=V; U[1,2]:=0; V:=(R2J2,
WP[4]);
for s:=1 to 3 do U[s,4-s]:=-V*Zn[s];
V:=C(J,W); U[1,4]:=V;
U[2,3]:=0;
U[3,2]:=0; U[4,1]:=V; U[5,8]:=V;
V:=C(R,WP[2]);
for s:=1 to 5 do U[s,6-s]:=-V*Zn(s);
U[6,8]:=-V;
U[3,3]:=C(R2J2,WP[6]);
V:=C(J,WP[3]); U[7,8]:=-V;
for s:=1 to 3 do
begin
U[s,7-s]:=-V*Zn(s);
U[7-s,s]:=-V*Zn(s)
end;
V:=C(R,WP[4]);
for s:=1 to 7 do U[s,8-s]:=V*Zn(s);
U[4,4]:=m+1;
V:=C(J,WP[5]);U[1,8]:=0;
U[2,7]:=-V; U[3,6]:=V; U[4,5]:=0;
U[5,4]:=0; U[6,3]:=V; U[7,2]:=-V;
V:=C(R,WP[6]); U[3,7]:=V;
U[7,3]:=-V;
V:=C(WP[2],E);
for s:=4 to 6 do U[s,10-s]:=V*Zn(s);
for s:=3 to 7 do U[s,11-s]:=0;
U[4,8]:=-C(R,E); V:=C(WP[4],E);
for s:=5 to 7 do U[s,12-s]:=-V*Zn[s];
U[7,6]:=0; U[6,7]:=0;
U[7,8]:=-C(WP[6],E)
end;
4:begin
U[1,2]:=0; U[2,1]:=0;
V:=C(R2J2,WP[4]);
for s:=1 to 3 do U[s,4-s]:=-V*Zn(s);
for s:=1 to 4 do U[s,5-s]:=0;
V:=C(J,W); U[1,5]:=V;
U[5,1]:=V;
U[6,10]:=V; V:=C(R2J2,WP[6]);
for s:=2 to 4 do U[s,6-s]:=V*Zn(s);
V:=C(R,W); U[1,6]:=-V; U[2,5]:=V;
U[3,4]:=0; U[4,3]:=0;
U[5,2]:=V;
U[6,1]:=-V; U[7,10]:=V;
V:=C(J,WP[3]);
for s:=1 to 7 do U[s,8-s]:=-V*Zn[s];
U[4,4]:=C(R2J2,WP[8]);
V:=C(R,WP[4]);
for s:=1 to 3 do
begin U[s,9-s]:=V*Zn(s);
U[9-s,s]:=V*Zn(s) end;
U[4,5]:=V; U[5,4]:=-V;
U[9,10]:=-V;
V:=C(J,WP[5]); U[6,5]:=0;
for s:=1 to 9 do U[s,10-s]:=V*Zn(s);
U[5,5]:=m+1; U[1,10]:=0;
U[5,6]:=0;
V:=C(R,WP[6]); U[2,10]:=C(RJ,WP[2]);
for s:=2 to 4 do
begin
U[s,11-s]:=-V*Zn(s);
U[11-s,s]:=-V*Zn(s) end;
V:=C(J,WP[7]); U[3,9]:=-V;
U[4,8]:=V;
U[8,4]:=V; U[9,3]:=-V;
V:=C(WP[2],E);
for s:=5 to 7 do U[s,12-s]:=-V*Zn(s);
for s:=3 to 8 do U[s,13-s]:=0;
V:=C(R,WP[8]); U[4,9]:=-V;
U[9,4]:=-V;
U[4,10]:=C(RJ,WP[4]); V:=C(WP[4],E);
for s:=5 to 9 do U[s,14-s]:=V*Zn(s);
for s:=6 to 9 do U[s,15-s]:=0;
U[5,10]:=C(R,E); V:=C(WP[6],E);
for s:=7 to 9 do U[s,16-s]:=-V*Zn(s);
U[8,9]:=0; U[9,8]:=0;
U[8,10]:=C(J,WP[3]);
U[9,9]:=C(WP[8],E)
end
end;
SystUr (2*n+1,U,A); A[-1]:=n; A[0]:=1; B[-1]:=n;
for
s:=0 to n do B[s]:=A[n+1+s]
end;