CONTOUR PLOT OF TRACEtPCK K+Nl t2(Kgt 11 AS FUNCTIOt- Cl= CZltK)11 HORIZ [2CK)J2 VERT EXAMPLE TO SHSW GROWTH OF TXACEtr(KKNgt3 SUff AGE WITH TIME T(KNgt ITS SHAPE APPROACHES THAT OF CP(KK)311 SURFACE rSVPTOTICALLY FOR LARGE N
555 44 44 44 444 3555 5355S 5555 5555 535 444 44 444 444 4444 44-44 44-14-14 444 bull144 bull444-144 3 444-J4 3 444-14 3 44444 4444
333 333 44 333 333 44 333 3333 44 333 3333 pound4 333 3333 44 333 333 At 33 3333 4 333 333 4-333 22 333 333 222222 333 333 222222222 333 33 222222J2222 333 (33 222222222222 33 13 2232 22222222252 333
22222 333 44 550 EGtmejGGGSS 222 33 44 555 C5e6tweampe6u66eGfl0^6eS666666666 2222 33 444 55tgt3 666666o6666S6GG6666l3S
222 33 44 5ti055amp 222 33 44 555S5iij555S555555SS555555555 222 33 444 55355555555555
222 33 444444^44 444444444 V2Z 3333 2222 33333333233333333333333333333333^3333333
1 1 1 1 1 1 1 1 1 1 1 111 111 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
f i i t u r n i i 2222222222222222222222222222 11111111 222222222 222222 111111 22222222 33333333333333333333 22222 111 22222 33333 3333 2222 3333 444444444444444 3333 222322 3333 44444 4444 Clt33 22222^2 33333 4444 4444 333 2222222 3333 4444 4444 333 22222pound2 3333 --4444 44444 3333 222222 33333 444444444444444 333 2222 22 3333 3333 2222 bull222222 3J333333333333333333 2222 2222222222 22222 2222^222^2222222 2222222222 2222i2ii22222222222222222222e22222pound222 22222222pound22-i2222222 2222222222222 +33333 222H2222222222222222222222222222222pound 111111111111 333333 222222222222222 222222222222222222222222222 444444 3333 2222222 33333333333333Ct3333 44444 3333 33333 333333333333333333333333 35 444 3333 3333 444444444^4-la 5555 444 333 33333 4444444444444-1444
111111111111111111111111111 1111 111111 111111111 1 i i m i u m i i n t i n i i a
^ =^ i f (91 (9) l^llgl lt8) IIg3f|gl (7) (7gt lSiil tS) pound6) i83I--8 (5) t5gt i3^igi (4) (4) l8sSgi f3I (3) lf^gl C21 (2 li5SIgl ( 1 ) (1) P | (0) _l18537E 02_
CONTOUR PLOT OF T R A C E C P f K K + N ) lt Z ( K ) ) 3 AS FUNCTION OF t Z t K U l HORIZ pound Z ( K ) ] 2 VERT EXAMPLE TO SHOW GROWTH OF T R A C E [ P ( K K + N ) 3 SURFACE WITH TIME T I K + N ) I T S SHAPE APPROACHES THAT OF C P ( K K gt ] 1 1 SURFACE ASYMPTOTICALLY FOR LARGE N
5S3 44 333333 555 444 333333
5555 44 33333 S5SSS 44 3333
_ S555S 444 3333 +555 44 333
44 3333 444 3333
4444 333 444444 333
CZ(K)12
09
3333333 333333 3333333
333333 33333
33333 44 55 65 777 3333 44 55 66 777 0888G888BS
3333 44 55 66 777 660688886888 3333 444 55 6S 77777
3333 44 55 G66 777777777
4 4 4 55 6 77 889 pound39999999 0 5 6 77 8C8 993399999
4 4 5 66 77 860EI 9999999999 4 4 55 66 77 eSEIS 9999999999 4 4 55 66 77 009688 999999999999S999
44444 U33 222222222 333 44 55 4444 333 22222222222222 333 444 55 444 333 2222222222222222 333 44 51 44 33 222222 22222222 333 44
333 2222 22222 33 444 333 2222 2222 33 44
333 222 2222 333 222 1111111 222
3333 222 11111111111 222 333
$656 777777777777 66666 7777777777777777777
lta 6563566 777777777 555 66666GS66666
555 666666666656666666666 G66666666666666-555E5
14 55SS5o335 444 5553S5555amp5S55SS5555
_ _ _ _ _ 444 amp55555lgt535555555555555 33333 222 1111111111111 222 333 4444444
333333 222 111111111111111 222 333 444444444444444444444444444444444+ 33 2222 1 1 1 1 11 1111111 222 33333
2222 111111 11111 2222 3333333333333333333333333333333333 222222 1111 11111 pound22222222 22222222222
11111 1111111 1111111111 1111 H i l l 111 1111111111111111111 1111111111111111-11111111 111111111111111 1111111 11111111111111111111111 1111111111t 111111111111111111111111 I -bull 111 11111111 2222222222 111111111
222222 22222 11111111111111111111111111111 2222222 3333333333333333333 22222 11111111111111111111111111111111
22222 3333 4444444144 333 22222 3333 4444 4444 333 2222223222222222222222222222
33333 444 555555555 444 333 222i2222222222222222222222222222222 +3333333 444 555555b555555 44 333 22Ppound2222222poundpound222222222222222222222-3333333 444 5555Si5o555355 44 333 22^ZV32222222222222222222222222222 33333333 444 55S55L555 444 333 2222 213222222222222222222222222
3333 4444 4444 333 2222ZT22Z 22222222 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 ^ 4 3 3 3 2 2 2 2 2 1 1 1 1 1 1 1 1 1
+ 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 _
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 P 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 Q 2 2 2 2 2 2 2 2 2 2 2 2 2 pound 2 2 2 2 2 2 t 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2pound222222
3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 ^ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 3 3 2 J 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 33C-333333333
4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 AAA 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 -
T ( K + N ) laquo 1 4 0 0 0 E - 0 1 T ( K ) = 9 0 0 0 0 E - 0 2 N = 5 STEPS AFTER F I R S T MEASUREMENT
SYMB
( 0 )
LEVEL RANGE
3 6 1 1 7 E - 0 2
( 9 ) ( 9 )
3 5 5 5 5 E - 0 2 3 4 9 9 2 E - 0 2
( 8 1 ( 8 )
3 4 4 2 S E - 0 2 3 3 0 5 6 E - 0 2
( 7 ) (7)
3 3 3 0 4 E - 0 2 3 2 7 4 1 E - 0 2
( 6 ) ( 6 )
3 J 17BE-02 3 I 6 1 6 E - 0 2
( 5 ) (5gt
3 1 0 5 3 E - 0 2 3 0 4 9 0 E - 0 2
( 4 ) lt4)
2 9 9 2 7 E - 0 2 2 9 3 3 5 E - 0 2
( 3 ) ( 3 )
2 8 6 0 2 E - 0 2 2 8 2 3 9 E - 0 2
( 2 ) ( 2 )
2 7 6 7 C E - 0 2 2 7 1 1 4 E - 0 2
( 1 ) ( 1 )
2 6 - 5 1 E - 0 2 2 5 9 0 8 E - 0 2
(copygt 2 5 4 2 5 E - 0 2
ESTIMATION ERROR CRITERION CONSTRAINT =
7 3 0 0 0 E - 0 2
Figure 615B Contour plot of Tr measurement amp 5 (0] a t t in tbdquo = 014 five time steps after first LKt5
CCM-OUR PLOT OF T R A C E t P ( K K N K 2 ( K ) I AS FUNCTION OP t Z ( K ) 7 1 HORIZ EZ fKJJS VERT EXAMPLE TO SHOW GROWTH OF TRACECP(KKN)3 SURFACE WITH TIME T ( K + H ) I TS SHAPE APPROACHES THAT OF [ P ( K K ) 3 U SURFACE ASYMPTOTICALLY FOR LARGE N
4 4 4 46 AC A
r5 66 - 7 7 7
GG 7 7 7 PSb 77
6G6 5 66 55 666
0 bull 555 144 333333333333 55f 44 333333333333
555 44 03333333333333 _ 55555 444 33333353333333 55555 44 333^333033333333 bull555 444 333333333333333333
4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 XH M 4 4 4 3 3 3 3 3 3 3 3 3 3 3 4 4 5
4 4 4 4 3 3 3 3 3 3 3 3 3 3 4 4 4 4 1 4 4 4 3 3 3 3 3 3 3 3 4 4 4
1 + 4 4 4 4 3 3 3 3 3 3 3 4 4 4 4 ^ 4 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 44 5 5 5
3 3 3 3 222222222222P gt 33 4 4 5 5 5 333 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 4 4 5 5 5 5
3 3 3 3 2 2 2 2 2 2 2 2 2 3 3 4 4 5 5 3 3 3 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 g
3 3 3 3 2 2 2 2 2 2 333 4 4 4 4 3 3 3 3 2 2 2 1 I t 11111 2 2 2 33 4 4 4 4
3 3 3 3 3 3 3 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 444
3 3 3 3 3 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 4 bull 3 3 3 2 2 2 U 1 1 M 1 1 1 U 1 U 1 1 2 2 2 3 3 3 3
2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 2 2 2 2 2 2 2 1111 11111 2 2 2 2 2
1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 raquo I 1 1 1 ) 1 1 1 1 1 1 bull 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 111111 1 gt
2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2
2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 2 2 2 2 2 2 3333 4444 53535 444 333 2222
3333333 444 5555555 5555555 444 333 33333 444 555 555 444 333 33333 444 5555 5555 444 333 333333 44 55555 55555 444 333 33333333 444 555555555 444 333 222
3333 444444 44444 330 221222 222 33333 T^33 22222 222222222 3333333333333 22222
2222222222222222 2222222 22222222222222222222
2222222222222 222222222222
333333 222222222222 222222222222222222 33333 2222222222222222222
4444444 333 22222222222 33333333333 4444 3333 333333
4444 3333 3333
JSiJ 3Sfl e raquo 3 8
9 9 9 9 9 9 9 9 9 9 9 9 9 S 9 S 9 9
9 9 9 9 9 9 9 9 9 9 9 9 3 9 9 9 9 9 9 9
iSBraquolaquo 9 9 9 9 9 9 9 S 9 9 9 9 S 9 9 858cea3e 999999999-
7777 7777777
7777777777 iGi i 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 ei5666 7777777777777
S6666666666 6666G66666666666
S35 SGS6S066666666B )5i S55555
HJ5555555S5U555555 5555555555^555555555
14 55555 1444444444444444444444
4 4 4 4 4 4 4 4 4 4 4 J 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 I222222222222222222222222222222
r i u i u i u i u i i i u n u n i i i i i i
1 i n 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 bull m i n i
2 2 2 2 2 2 2 2 2 2 2 gt 2 2 2 2 2 2 2 2 2 2 2 2 2 lt 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 pound
2 2 2 2 2 2
u u i n 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1
m i n i m i m i n i m i 1 1 1 1 1 1 1 m 11 1111 111 1111111111
1 1 1 1 1 1 1 1 1 1 m i m m 1 1 m 2J22222
222222222222222222222 i33333333 3303
33333333333333333333332 3 333333333333333333
T(KraquoN)= ISOOOE01 TIK) = 90000E-02 N = 10 STEPS ftFTE F IRST MEASUREMENT
CONTOUR LEVELS ANO SYMBOLS
SYMS LEVEL RANGE
t O ) 4 2 3 1 9 1 1 - 0 2
( 9 ) ( 9 )
4 1 7 9 7 E - 0 2 4 1 2 7 4 E - 0 2
3 ) t e gt
4 0 7 5 1 E - 0 2 4 0 2 2 0 E - 0 2
(7gt ( 7 )
3 9 7 0 5 E - 0 2 3 9 l a 2 E - 0 2
( 6 ) (Ggt
3 6 amp 3 9 C - 0 2 3 amp 1 3 C E - 0 2
( 5 ) ( 5 )
3 7 t e l 3 E - 0 2 3 7 0 9 1 E - 0 2
( 4 ) ( 4 )
3 6 5 G R E - 0 2 3 6 0 4 5 E - 0 2
C3gt ( 3 )
3 5 5 2 2 E - G 2 3 4 S amp 9 pound - 0 2
( 2 ) 3 4 4 7 6 C - 0 2 3 3 S b 3 E - C 2
(1 ) ( 1 )
3 3 4 C O H - 0 2 3 2 9 U 6 E - 0 2
(0) 3 2 3 0 5 E - Q 2
EST) MAT 1 Oi l EKROR CRITERION CONSTRAINT =
7 5 O 0 C F - 0 2
1 - 2 5 0 Q E - 0 1 1
Figure 615C Contour plot of Tr measurement
bullK+10AK (h) at time t K+10 019 ten time steps af ter f i r s t
cz(Kgtia 03
CONTOUR PLOT OF T R A C E t P t K K N ) t Z ( K gt ) 3 AS FUNCTION OF t Z ( K ) ] T HOR1Z t Z ( K H 2 VERT EXAMPLE TO SHOW GROWTH OF TRACEEPCKKraquoNgt1 SURFACE WITH TIME T ( K N ) ITS SHAPE APPROACHES THAT OF [ P lt K K ) ] 1 1 SURFACE SVYPTOTICALLY FOR LARGE N
555 44 33323333 555 4 333023333 555 444 333333(333
5b55 44 3333tngt33333 5S55S 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 55L5 444 333333333333333
444 33333333333333333 444 33333333333333333333
444 55 6 444 55 444 55 444 S 5
77 BE 6 77 OEGfl
7 7 pound9118 777 ease
4404 33333 444444 3333 44444 3333 444 3333 222
33333333 444 5 333333 444 3333 444 333 444
55 66 777 44 55 66 777 444 55 666 7777 666 77777
999999999 999S90999 9S9SS39999 99999999999 99999999999999 99999999
333 2222P222222222 333 22222222222222222 3333 222222 22222222 3333 22222 2222 3333 222 222
680e88666038B68 6S6 7777777 BC3QBQSBBB gtamp 66GC 7777777777 555 6i6fiS 77777777777777 777 bull 555 6056666 77777777777
3333 222 333333 222 11111111111 33333 222 11111111111111 33 2222 111111111111111111 2222 11111 111111 222222 1111 11111
444 5555 666666366666 I3 444 555S 66GS66666S6666666 33 444 5amp05S5 6666666666666 333 444 t5Sy555555S5 333 444 555555555555555555 55555555S55555555 222 333 4444 222 333 444444444 222 3333 44 14444444444444444444441 mdash2 333333 44444444 222 333313333333333333333333333333 222222 111111 221-22222222222222222222222222222 111111111111111111111111111111 1111111111111
llll1111111111 111111111 1111 111111)1111 22222222222 11111111 22222 22222 11111111 222222222 3333 3333 22222 2 3333 444444 444444 333S 222222221 3333 144 555555535 444 3333 ZZpoundZ 333233 444 5555 5555 444 333 3333 444 555 555 444 3333 333 444 5556 555 444 3333 3333 44 5555 5555 444 3333 333333 444 5555555555555 444 3333 Zt 33333 4444 4444 333 2222222 33333 444444 3333 22222 22222222 3333333333333333 22222 111 22rgt2pound222222222 222222 11111111 2^2 2e2Sgtpound22222222222222222 1111111111 2gt2212222Ve^^-^2^222 1111111 222222poundZi2222
3333333 22222222222222222222222222222222222222 33333 222222222222222222 4441444 0333 22222222 3333333333333 444 3333 33333
111111111111111111111111111111
444 3333 33333
111111111111111111111111111111 11111111111II 111111 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 - 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2 2 2 = 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
T t K N ) = 2 4 0 0 0 E - 0 1 TCKl = 9 0 0 0 0 E - 0 2 N = 15 STEPS AFTER F IRST MEASUREMENT
CONTOUR LEVELS AND SYMBOL5 SYM0 LEVEL RAN3E tOgt 46551E-02 (9gt (9
4 4 9039E-7D27E--02 02
4 4 701-1E-eao2pound-02 -02
lt7 (7raquo
4 4 59fSE-5477E--02 -02 lt6J (6gt
4 4 49GEE 44S2E--02 -02
(5J 4 4 39C0E-34pound7E--02 -02
(4j (4J 4 4 291
rJE-2-103 E-
bull02 -02 I3J (3)
4 4 1 830E-I37SE- 02 -02 (2gt 12)
4 4 06C5E- 03L3E--02 -02 J (1)
3 3 93-IIE-3323E--02 -02 lt0 36310E-02
EST 1 HAT I ON ERRPR CRITERION CONSTRAINT = 7taOOOE-02
Figure 615D Contour plot of T r EK+^^K) a t time t K +_ = 024 fifteen time steps after first measurement L J
CONTOUR PLOT OF TRACpound[PCKKNgtCZ(KgtgtJ AS FUNCTION pff C Z lt K ) ] 1 HORIZ t Z ( K gt 1 2 VERT EXAMPLE TO SHOW GROWTH OF T R A C E t P ( K K + N H SURFACE WITH TIME T lt K N ) I T S SHAPE APPROACHES THAJ OF C P f K K l l U SURFACE AgtV1PT0TICALLY FOR LAROE N
TJME= 9 0 0 0 0 E - 0 2 FIRCT MEASUREMENT ELEMENT 1 1)
555 444 444 55 6G 55 444 33 444 53 66
555 44 0333 4444 55 66 555 444 3333333 444 55 66
553555 AAA 3333333333 4444 55 6pound 5555 444 3 3 33 33 i 133333 444 553 S 444 333333333333333 444 ~
6D3 8 0 3e
3 3 F 9 7 7 7 3poundJt
939909039 9999S9999
990030099 39J999999
7 7 7 1 3 8 8 0 8 6 9 9 9 9 D 9 9 S 9 9 9 9 9 9 66 777 eaiaaena 99999999-
_ 666 7 7 7 7 8 6 6 e 8 8 - 8 8 8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 55 6 6 77777 8e38688C8O880OO(38
4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 5 3 6 6 6 7 7 7 7 7 7 7 7 8 8 0 6 8 8 8 8 8 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 4 4 4 5 5 656G 7 7 7 7 7 7 7 7 7 7 7 4 4 4 4 4 3 3 3 3 3 3 3 3 4 4 4 5 5 5 6G3E-6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 - -4 4 4 3 3 3 3 3 3 3 144 sect55 5pound-SG6666 7 7 7 7 7 7 7 7 7
333 2 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 = 5 5 666665G5GGG6 3 3 3 2 R a R a raquo K 2 2 2 S 3 3 3 4 4 4 505 CGtJ6ampo6-6GGGCrGCGfiC6
3333 r y 2 2 2 2 r i 2 2 L 2 2 2 2 2 33 4 4 4 SS55 -gtb 66Gl5CCftgtG0tgt5 3 3 3 3 2gtZ2 2 2 2 2 Z 33 4-14 5E- 3 j ^ S S r i S W S 3 3 3 3 2r-22 2 2 2 2 333 4 4 4 4 55555503555511555555
3 3 3 3 3 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 0 5 5 5 5 5 amp 5 5 5 5 5 5 5 5 3 3 3 3 3 3 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 333 4 4 4 4 4lt 4-14444
3 3 3 3 3 2 2 2 1 1 1 1 1 1 1 1 1 I I I 11 2 2 2 2 333 1 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 + 3 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 4 4 4 4 4
2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 V3Z 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 11111 1 1 1 1 1 1 2 2 2 2 2 2 2 ^
1111T1 111111 2 2 2 2 2 ^ 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1111
11 1 11 111 1 1 1 -
11111111111111111111111111 1111111111 2222222222222 11111111
2222^ 22222 111117 1 22222222 33i^3 3333 2222
333 4-S44 44444 333 pound2222222 333333 444 55555555555 444 333 1
33333 444 5555 S555 444 3333 33 44 55 3 6G666 D55 44 393333
444 505 6665066 555 44 33333 333 444 555 555 444 3333 333333 444 55555555555555S 444 333 3333333 4444 4444 333 2222221
33333 4444444444 3333 222222 2222722 3333333333^3333333 22222 2222222222222 222222 111 1111 22P222i2-22l22P22222222222222 U11 U 1111 2ir2ai22-222i22irr2222 1111 11
22222r2-2Ki2 22 3333333 22pound2J22222Z22222222222222222Jai
3333 22222222222222222 4344444 3333 2222222 333333333 33C-
4444 3333 33333 444 33333 33333
11111111111111111111111111111 22222222222222222222222222222
33333333 3333333333333333333 333333333333033333
33353 22222
bull22222222222222222222222222222 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 22222222
2 2 2 2 2 2 2 2 2 1 3 3 3 3 3 3 3 3 3 ^ 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 J
( 0 )
LEVEL RAKCE
1 6 0 S 3 E - 0 2
13) ( 9 )
1 6 3 4 S E - 0 2 1 5 0 4 0 E - O 2
1 5 3 3 4 E - C 2 1 4 C 2 E - 0 2
it ( 7 )
1 4 r 2 U - 0 2 1 3 t 1- t 02
( S ) ( 6 )
i sacaoos 1 2 8 0 2 E - 0 2
(5gt ( 5 )
1 2 2 9 5 f - 0 2 1 1 7 6 9 c - 0 2
( 4 ) ( 4 )
1 l pound P E - 0 2 1 0 7 7 C E - 0 2
( 3 J 133
1 O27OE-02 9 7 o 3 ^ pound - 0 3
(2) ( 2 )
9 2 5 t t l E - 0 3 8 7 j O ^ E - 0 3
(1 ) (1 )
BZnopound-03 7 7 3 7 5 E - 0 3
tOgt 7 2 3 1 2 E - 0 3
ESTMATUN ERtiR CRITERION C L l t T R U I H =
7 t r n o e - 0 2
SOURCE NPUr COVAKlANCE I W 1 - 1 2 5 0 f E - 0 1 1
Figure 616 Contour plot degl [amph at f i r s t measurement t ime t bdquo = 009 compare with asymptotic
response of Tr [ppound + N (z K )1 surface at t K + l g = 024 in Figure 615D
188
at the next sample at time t K + N when (645) is next satisfied From Conclusion X the minimax problem in (647) separates into finding zt
such that
[ E^4i = IK L - ^ that z which
^n-lr-1 $ 5$ pound
and independently findino that z which leads to
4 T max c(z) c(z)
(648)
(649)
for N large Various properties of the solution of th is problem are
demonstrated by example in what fol lows
631 Asymptotic Responses of Output Estimation Error - to demonshy
strate the asymptotic separation of the minimax problem in (647) into
the independent problems of vector minimization in (648) and scalar
maximization 1n (649) the problem of Section 61 was solved but as a
monitoring problem of the second kind with
~005 p 002
000001 (650) 000001
^ 000001 _
and with thi bound on maximum variance in the output estimate
Pdeg = ~0
lim 01 (651)
For this case a plot of the evolution of o^+(j(S((z) t n e gtin1max probshylem statement In (647) as a function of time t K + N 1s shown in Figure 617
The asymptotic separation of the minimax problem is demonstrated in Figures 618 and 619 The former 1s a plot of a^[z0z) as a function of the position 1n the medium z for values of time t R = 0 T 2T 9T
1OOOOE-01
6BO0OE-O2
S2000E-02
OeOOOE-02
4C000E-DZ
X X
X X
X
X
X X
XX
gt XX
X
X X
X XX
X X
X X
X X
X X
X X
X X
X X
X XX
X X
X X
X X
X
X X
X
X X
X
X X
X X
X
X
X X
X
X X
X X
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Figure 617 Time response of aLwU((laquoz)gt t h e P e l f deg r m a n c e criterion for the optimal monitoring probshylem with bound on error in the output estimate for a = 010 samples occur at t = 011 047 and 085
EXAMPLE TO SHOW EVOLUTION OF VARIANCE IN OUTFUT ESTIMATE WITH TIME POSITION OF MAXIMUM VARIANCE APPROACHES STEADY-$ I At IT VALUE FOR LARGE TIME
80000E-02
74000E-02
96 7777 6 7 709 9 76666 e 876 6 7 9 976 555 6 78 6 55 56 9
1 0 0
865 4444 56 S 87 44 4 9 7654 4 5676
8654 33 9 754 33 33 4 567
3 SZZ100 965 3333g2H00 754
6BOOOE-02 4444pound2110 8343 5 55-JJgt3322 1002533222 777ii -514332293222 S^SS tiS314i65 0111
g- 03779S7 0 S99 (
62O00E-O2 1 2 3458
1 6 1 2 34579
36 1 2 4 79
1 35 8 2 6
i 1 34576 O I 2 6 J
C 1 23457) 6 9
12345 B 1 234 i7f)
1 gt579 0 123 13539
00 12 J4M5S9 41 OC 1 3-Ti67 9 9567
00 i345 6 6300 OOOOOO 0 00
SYMB TIME TK+N (0) 0E00 CI) 5000CE-C3 (2) 0001^-03 CD GOOCOE-03 (4) 80000C-03 t5 ) IOuOCE-02 (6) 12000E-02 ( 7 lJidOOC-02 0000 (6) 1600CC-02 000 I1 (9) 1 OOOOE-02 00 111
00111222222 0122 30333 P0112233344444 011223S4445SS-3 01 23341553 C306 012g34J50tt b67777 01254553077770360
12334Lpound67736999 12345My88393 12345677S99 12345^769 123b67699 1245S7S9 12456099 1245789 1246709 1246G99 134689 135799 13579 14R89 i99 2589 04799 2599
4000E-01 PtSl ION Z
Figure 618 Plot of performance criterion oilaquo[z) as a function of position z in the medium for K + N- -- - 2 _ _ _ J times t K+N 00 002 004 018 note how position z
changes with time of o + N(z ) = max a K + N U)
130O0E-O1
1 32O0E-O1
1 1 4 0 0 E - 0 1 ODDOnOOOCOO
raquoe00Dpound-02
oooooouooo
60D00E-02
Figure 619 Plot of asymptotic shape of performance c r i te r ion deg K + N ( z ) as a function of position z in the medium as N-raquo compare posit ion z =
totic position of maximum in Figure 618
the medium as N+degdeg compare posit ion z = 03 for Urn r j x apound + M (z) in th is curve with asymp-im n x N-~gt z
192
where T = (t K + - tbdquo) = 0002 zbdquo was taken as the initial guess at the best measurement locations z Q = [015015] The latter plot is a plot of
lti(z)T a c(z) (652) SS
2 the steady-state term in the asymptotic response of crJ + N fo r N large
Thus comparison of the asymptotic approach in time of the curves in
Figure 618 to the steady-state curve in Figure 619 shows that
N
c ( z ) T V n 1 M n 1 d(z) - c ( z ) T a c(z) (653) imdash S~S~ n=l
As a special case it shows that
max o+fzz)mdashgt max c(z) q c(z) SS
(654)
at the position of maximum variance z Note here that as expected the position of maximum variance is directly over the source position
(655)
632 The Effect of a priori Statistics mdash To demonstrate the efshyfect of the uncertainty in the initial state estimate x = m upon the optimal monitoring design problem consider variations in the a priori
statistics given in the initial state estimate error covariance matrix Pg = M- For this example fix the time interval of interest at 0 lt t lt 20 and set o | i m 5 02
(656A)
Compare the f i r s t case for which
000001 o E o s 8 o
0 0 00001
193
with the case where
E g - H o
oi 000001
o
o
000001
(656B)
The first choice results in the evolution of obdquo+bdquo(ztz) shown in Figure 620 resulting in one measurement at t = 126 The corresponding con-tour plot of [ E K ( K ) ] ] I as a function of [ z j and [jd for that meashysurement is shown in Figure 621
The plot of o^+f(zJz) for the second choice of M as in (656B) 2 is shown in Figure 622 where owing to the higher initial value of aQ
two sample times result at t = 046 and t = 160 The corresponding conshytour plots for those measurements are shown in Figure 623
Study of Figures 621 and 623 show that the locations of optimal measurement positions are not effected by the a priori statistics given in MQ provided that the time to the firsc sample is sufficiently long for the infrequent sampling approximations to apply
For the first case the time to the first sample is t = 126 for the second case the first sample occurred at t K = 046 Thus the only
effect that the choice of Mbdquo has upon the optimal monitoring design probshylem is the detirnrination of the time of the first sample
Thus the results of Conclusion V are substantiated here within the context of a monitoring problem with bound jn output estimation error
To illustrate the transient effects at play in the general monitorshying problem effects that exist before the infrequent sampling requireshyments of (518) and (520) are met consider the same problem as in the
20000E-01
16000E-01
taoooE-oi
raquo XX XX X XX X XX XX X
X Xt XX X XX X XX XX X
X X XX XX X XX X XX
I XX
X sx
XX X XX
X XX XX X XX X XX X 1 XX 1 X I X I XX I X I X I X I X I X
XX X X X X X X
X 1 X IX
X
X
1 600E+CO
2 2 0 Figure 620 Time response of ai+ufivtZ J f o r degi- = 0- 2 with initial covariance matrix P Q H H Q given in (656A) one sample occurs at t = 126
CONTOUR PLOT OF CP(KK) tZ(K)) J11 AS A FUNCTION CF CZCOU HORIZ AND EZtKgt32 VERT
bull4444 33 22222222222222222 4444 333 222222222222222222 4444 33 222222222222222 444 33 22222222222222222J 333 22222 2222222222222 333 2222
fZCKHZ
03
3333 __ 3333 22
33333 222 3333 2222 333 222 333 222 33 222 3 222
222222222E222 222222222222 2222222222222
2222222222222 222222222222
222 222
222 1 2222 11
22222 t11 1111
11111 bull1111111
22222222222 2222 31
1111 2222 31 11 111111 222 11(1111111111111 222
111111111111111111111 222 1111111111111111111111 22
1111111111 22 1 1111 I 22
11111 1111
1111
33 AA RK 7 aesss 999939 0 33 AA UK 7 7 eaaeo 99999 333 AA KH 7 a ieaa 333 A fifi 77 888908 999999
33 Ai HH 333 A 55 6t 7777 CAB 188 99999999
33 44 bullgt B 77777 888883 9953 333 AA Vgt 6(i 77777 0888883
3 3 44 Hfgt lies 777777 8880088885 3 3 3 AA 55 8SS6 777777 889Pd3S8
66666 7777777 44 555 6G6666 77777777 444 E5gt3 6666666 7777T777777
I 44 5SS5 66D6666 7777777 I 44 i5555 666G665 13 444 5555S5 666G66G66 J3 444 55055555 6665666666
1111 1111111111 22 33 AApoundA 5555555555 66666 22 333 J4I44 555555555 222 333 44AA4A4AAA 55555555553
222 3333 4444444444444 222 3323gt33 444444444444
111 222 33333333333333333 11U1 222E222 3333333333
11111 222222222222222222222 11111 1111111111 2222222-
H I 11 i i i i m i i i i i i i n i i i i n i u i u n i i i m i n 11111 m m 111111111111111111111
11111 222222222222 1 1 m m m i m - m m 1111111 222 33333333 222 11111 11111111111111111111 11111111 22 33 444 33 22 111111 11111111111111111111111111 1111 2E2 33 44 444 33 222 1 11 11 11 1 1 11 1 11 1 1111 1111 222 33 44 555 555 4 33 222 1111)11 2222 3 4 55 66666666 55 44 3 222 22222222222222 222222 33 4 5 G6 666 55 ltJ 33 222 222bull22222222222222222222 bull22222 33 44 55 66 777 66 35 44 33 22222 2222222222222222pound22222222-22222 33 44 53 66 777 6 5S 4 33 2222 2222222222222222222222222 22222 33 4 5 66 666 55 44 3 222 2222222222222222 222222 33 A 55 6666666 35 44 33 222 1 2222 33 44 655 555 44 33 22 11111111111111111 1111111111 222 33 444 44 33 22 1111111 1111111111111111111111111 222 333 333 222 11111 1111 ^
bull11 O 111111111111111111111 1111111 111111 111111111111111111111 1 22222222 22222 222222 22222222222222222222222222222 2222 222222222222 222 2222222
11111 bull2222 1 11 11
2222 1111 333 2222 11
3333 224 333 222 333 222
222222 222 111 m m
i m i m i i 111111 1111111111 111111 m i m 111 m m i i 11111 n
m m i i m n
CONTOUR LEVELS AND SYMBOLS
SYMB LEVEL RAIiGE
~76) iTs^ ie -o i 19) (9)
2 2
4972E 4402E-
02 02
( 8 ) 2 2
303i 3263E
02 02
C7) pound7)
2 2
2S94I-2124b
02 02
(61 (6)
2 2
155ipound 0985g
02 02
(5) (5)
2 1
011 5pound 98-562
02 02
t4 ) (4)
1 1
927ampE 87071J
02 02
(3) (3)
1 1
6137E 75S8E
02 02
(2) (2)
1 1
6996E S428E
02 02
(1 ) n 1
1 1
5059E 52QUE
02 02
(copy) 1 J720E 0 2
EST1 HATION ERROR CRITERION CONSTRAINT =
SOOCC^-Ol
12500E-O13
F i g u r e 6 2 1 C o n t o u r p l o t o ^ F K ^ K ^ l n w 1 t h i n i t i a l cdegvariance matrix E Q = - 0 9 i v e n i n ^ 6 5 6 A f o r
the sample at t j 126
20000E-01
95000E-02
6 OOOOE-OS
SS000E-02
Figure 622 Time response of C J | + N ( Z Z ) for ltm = 02 with i n i t i a l covariance matrix P 0 i MQ given in (656B) two samples occur at t K = 046 and 160
CONTOUR PLOT OF t P ( K K ) ( Z ( K ) ) 311 AS A FUNCTION OF CZCfOJ I HORIZ AND r Z ( K gt 1 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPUT ESTIMATE WITH T IME P O S I T I O N OF MAXIMUM VARIANCE APPROACHES STEADY-SiTATE VALUE FOR LARGE T I M E
CZ(Kgt]2 05
4444 33 222222222222222222 4444 333 222222222222222222 444 33 222222222222222222 444 33 222222222222222222 333 22222 2222222222222 333 2 22 333 2222 3333 2222 33333 222 3333 222 333 222 222
222 t 222 11
2222222222222 2222222222222 2222222222222
22 22 22 22 111
2222222222222 pound22222222222 2222222222 22222
23 44 55 6G 77 33 44 5 66 777 333 AC 5 66 777 333 4 55 66 777 33 44 55 C3 777 333 4 55 56 7777 33 44 5 e3 77777 333 4 55 i36 77777
999999 99999 93999 999999 99999999 99989999 9999 8888866
0
2222 222 222
111 222 222 222 2222 111 22222 111 111 1 11111 1111111
11111 11111111111 11111111111111 1111111111111111
1111111111 111 1 I I
11111 1111
111
55 666 77777 4 53 6666 777777 68688688 4 tgt55 66666 7777777 44 3E5 666666 77777777 444 5J55 6666666 77777777777
44 S55S 66665C6 777777 44 5555 6666666 Aamp1 555555 66666666
2 a 3 J14 555555555 6666666666 2 33 4144 555555555 66666 22 333 44444 555555555 222 333 4444444444 55555555555
222 3331 4444444444444 222 3133333 444444444444
1111 222 333333333333333333 11111 22 2^22 3333333333
111111 322222222222222222222 222222 11111
111111111111111 -11111 111111 111111111111111111111
11U1 222222222222 11111 1111111)11111111 1111111 222 33333333 222 11111 11111111111111111111
bull11111111 22 33 4444 33 22 111111 11111111111111111111111111 111 222 33 44 44 33 222 11111111111111111111 11111
222 33 44 555 555 4 33 222 11111111 22222 33 4 55 66666666 55 4 33 222 22222222222222
222222 33 4 5 66 66 5 4 33 2222 2L 1222222222222222222222 22222 33 44 55 66 7777 66 55 44 33 22222 2222222222222222222222222 2222 33 44 5 66 7777 66 SS 44 33 2222 2222222222222222222222222 22222 33 4 5 66 666 55 4 3 222 2222222222222222 222222 33 4 55 66666666 55 44 3 222
2222 33 44 555 555 44 33 22 111111111111111111 111 11 111 111 222 333 44 444 33 22 1111111 1111111111111111111111111
2222 333 333 222 1111 11111 2222 3333333 222 1111 11 22222222222222 22222 22222 3333 2222 333 222 333 222 333 222
111 11 0 11111111111111111111 1111111 111111 111111111111111111111 1 2222222222 22222 222222 22222222222222222222222222222 2222 22222222222 2222 222222
SYMB
t b i
LEVEL RANGE
z 5 5 1 9 pound - b 2 _
( 9 ) ( 9 )
2 2
4952E 4384E
0 2 C2
I B ) ( 8 )
2 2
3816E 3248E
0 2 0 2
( 7 ) ( 7 )
2 2
2G60E 2112E
0 2 0 2
( 6 ) ( 6 )
2 2
1544E 0977E
0 2 0 2
( 5 ) lt5gt
2 1
0409E 984 I E
0 2 0 2
( 4 ) ( 4 )
1 1
9273E 8705E
0 2 0 2
( 3 ) ( 3 1
1 1
8137E 7570E
0 2 0 2
( 2 ) t 2 )
1 1
7002E E 4 3 4 E
0 2 0 2
( 1 ) ( 1 )
1 1
5 8 6 6 E 5298E
- 0 2 - 0 2
( 0 ) 1 4 7 3 0 E - Q 2
ESTIMATION ERROR CRITERION CONSTRAINT =
2 0 0 0 Q E - O 1
1Z300E-011
Figure 623A Contour plot of Ppound( K )1 with in i t ia l covariance matrix f 0 = MQ given in (656B) and ulim = 02 for the first sample at tbdquo = 046
CONTOUR PLOT OF t P I K K lt ^ C K J gt 1 1 AS A FUNCTION O t 2 ( K ) J 1 HOBI2 AND t Z ( K gt ) 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPU ESTIMATE WITH T I M E POSIT ION OF KAXtrUlK VARIANCE APPROACHES STEADY-31A7E VALUE FOR LARGE T I M E
09
333 44 14 33 44J 33 333 333 2222 3333 2222 3333 222 33333 222 3333 2222 06 +333 222 333 222 222 I 222 11
07
CZCK132 O S
222 222 222 2222 22222 1i till lllli bull1111111
22222222222232222 22222222222222222 222222222222222222 222222222pound2222Z222 22222 2222^2^222222 2L2 222 222222 222222222222 2222222222222 2322222222222 222222222222 22222222222 22222 1 2222 1111 1 111111 Mil 111111111111111 11111111111)111 11111111 111
3 3 4 4 7 7 7 3 3 3 4 4 S 3 3 4 59 66 7 7
3 53 6 7 aar 4 4 ss i6 3 4-1 tgt
333 44 S3 tgttgt6
777
333
222 222
44 SS 66 77 USB66 993939 0- 3 G 6 99999 8388 59939 eeeoas gposgfl 00866 99999599 ltft aeoeoo 9999399s-77777 888068 3999 77777 8638880 665 777777 6608800888 4 OS 6SE6 777777 86000680 4 55= 66666 7777777 44 ESi 66SCC6 77777777 444 5i3 60EG666 77777777777 44 iSC5 6GGGGG6 7777777 44 35355 G61606G 1 44t 555553 GoGG66G66 22 33 114 355553C3 G6GG66G6G6 I 22 33 44 4 5355330553 CS666 II 22 333 4444 535555553 III 222 333 1444444444 33353515533 111 222 3333 4444444444444 till 222 3313J33 444444444444 1111 222 33333333333333333 11111 222122 3333333333 11)11 222222222222222222222 11111 Hill It II 222222 111 M1111111111111 11 II111111 I 1111 III11111I1 11111 111111 111111111111 11M111 11111 222222222222 11111 1 It 1M111111111 111111 222 33333333 222 11111 bull 1 111 11 1111 11111M 11111111 22 33 44 33 22 111111 11 111 11 I 111 11111 1111 I till 22 33 44 444 33 22 11111 11 bull 111111111 11 II 222 33 44 553 335 4 33 222 1 1111 III 2222 3 4 55 C666666G 53 44 3 222 222222raquo22222 222222 33 4 5 G6 666 53 4 33 222 22222222222^222272222222 22222 33 44 55 C 777 5 05 44 33 2222 2S222Wr2S2222222222 22222 33 44 5E 66 777 6 53 4 33 2222 22L-22rT22E22222 222222 22222 33 4 5 66 6SG 55 44 3 222 2227222222222222 2222222 33 4 55 G66GC66 53 44 33 222 11 2222 33 44 555 533 44 33 22 111 111 1 11 I I 111 111 11 I11 1 11 111
1111 111 HI
222 33 444 222 333 -111111 222 333333 222 11P1H 22222222222222
33 22 22
11111
2222 111 t i n 11 t i n 1 3traquo3 222 1111111111111
3333 222 11111111111 333 222 1111111111
2 111 m m
m n i u l i i i i i n m i n i i m i 11111
m m 11111 mi 11 1 11111 m 1 m 111111 1 22222gt222
222222 2222 222
m m 111 m 1111111111111
111111 m i n i m i m m i i m t 22222 222222222Z3222222222232222222 2222222222222 2222222
SYI-3 LEVEL RANGE (0) 25540E-02
l 2 2
4970E-02 440IE-02
2 3B31E-02 32G1E-02
i l l 2 2
2GXE-02 21225-02
1 2 1352E-02 0963E-02
11 2 1
O4I3E-02 9843E-Q2
i I I
9274E-02 8704E-02
II 1 8I3-JE-02 -756-S-02
si 1 1 6S93E-02
GJ25 -i-02
1 I
3333pound-02 GZilLC-02
lt0gt
g trade -12uorE-oil
Figure 623B Contour plot of [ P pound ( Z K j L with i n i t i a l covariance matrix PQ = HQ given in (656B) and
degl lim = 02 for the second sample at t R = 160
199
2 last case above with HQ defined in (656B) but with a = 016 instead
This results in the curve for o K + N(zJJz) shown in Figure 624 for the
shorter time interval 0 lt t lt 10 Two sample times result at t bdquo = 011
and t K + r ) = 086 Corresponding plots for [pound(lt)] and [ P pound + [ | ( Z K + H ) ]
are given in Figure 625 Notice how in this case that the optimal meashy
surement positions it and z bdquo + N at the two samples are different The o
reason for this is that here the estimation error l i m i t o is so low
that the infrequent sampling approximations do not apply at the f i r s t
sample t ime This is inferred by the response of degV+N^K Z^ i 9 U r e
624 where i t is seen that zhe steady-state slope [ f tJ i i = 000125 for
this problem has not been reached yet at the f i r s t sample whereas i t has
at the second thus the steady-state simpl i f icat ions 1o not apply at the
f i r s t sample For th is reason in practical applications step (3) of the
algorithmic procedure given in (572) is important where at each sample
i t is necessary to check whether or not steady-state conditions have
been adequately approached for the infrequent sampling approximations to
apply
833 Problems with a Fixed Number of Samplers aid Constant Error
Bound - Consider a problem withm = 2 samplers to be used in every 2
measurement with a time-invariant error bound o = 0075
The i n i t i a l covariance matrix
000001 O 1
eS = y 0 (657)
O 000001 Conclusion V and XI are substantiated in the context of this problem with bound on output error
laquo vV
X X K
- w XX XX XX XX XX XX XX XX
X
XX XX XX XX XX XX XX XX XX
xx m
X XX XX
gt X X X X
X X XX X X X X X
S5QQ0E-Opound
X X
X
X
X X X x
X
Figure 624 Time response of a K + fzpoundz) for a = 015 with initial covariance matrix P Q = M Q
given in (656B) Two samples occur at t = 011 and 086 compare with Figure 622 for case with a = 02
CONTOUR PLOT CF CP(KKMZltKgt1311 AS A FUNCT0^ t r [ZC EXAMPLE TO SHOW EVOLUTION OF VARIANCE ID C - J _ P C rSrl POSITION CF MAXIMUM VARIANCE APPROACHES S T C ^ V bullpound ATE i
Ji HOTIZ AND tZ(Kgt]2 VERT E WITH TI ME LUE FOR LAHGE TIME
tZ(K)12 05
aa 33 44 4 -_ -
4444 33 222 4 4 4 33 2222 4 4 333 222
33 222i 33 222
3333 222 33333 222 33333 222 33333 222
2222222222222222222 2222222222222r222 22222 2 t2222^^22
22222 2 2222 2f P 22 2222^22P2 22j2^^2r^22
22 ^lt7ih
3333 3333 3333 3333 3333 33 33 3333 333
22 22
2 2
i n n i m n n i n 11111111111111
2222 222 222
77 7 A C e R B 9C99 0 77 c-rrc-rs 90909 77 SCT638 S3^99 7 77 0^036 099999 777 CC3C36 92999999 7777 G363G3 99999999-
7777 eee 9999 i j 7777 e^cr pound33 (--bull 77777 iJZWrampec V G 7777 7 6^000833
j GMJ-5 7777777 o U -CG 777777 gtbullgt Ev -ro 777777777777 bulljT -5 CCSG^GS 7 7 7 7 7 7 7 7
11111111111 11111 j
1111 m i
22 33 AA
2 2 2 2 2 2 2 2 111
1111 bull i m m 1111
1511 2 2 33
i l l
111111111111 11111 11111
1111 222222222222 1111 111111 222 3333333 222 1111
11111111 22 33 4444 33 222 11111 i m 222 33 44 44 33 22 11111
222 33 44 55 555 4 33 22 11 22222 33 4 5 666666666 55 A 3 222
22222 33 44 55 G6 66 5 4 3 2Z2 2222 33 4 5 6 7777777 CO 55 4-1 33 2222 2222 33 4 5 66 7777777 56 55 44 33 2222 22222 33 4 4 5 5 6 6 7 66 5 4 3 222 222222 33 44 55 G66S C666 55 4 3 222
22222 33 4 555 55J 4 33 22 11 2222 33 444 44 33 pound2 11111
222 33 44444 T3 222 111 11
4l4 4fCltits44-44 53355-ltt44-144444
J333333333 r333533 33333333333
2222^^^^22^22222222 11111 1 I I 2222
m m m m i 11111 m 11 m m m i i m u m
m m m t m u u u u-u m i m 1111 m m i m m m m 11 n m M TVZ
222ytgt gtr 222222 2 2 2 - f v SW2V2vbullgt222222
22 - ^ ^ 2 ^ 2 2 2 2 2 2 2 2 2 ^ V 2 2 2 2 2
11111 bull m i m i m u m m m M U U 1 1 1 1 1
i raquo i 11 I I 111 m i 2 2 2
2 2 2 2 2 2 2 2 2 2
333333 222 333 222
4444 33 222 44444 33 222
2 2 2 2 2 2 111 m m
11 M l 111 1
111 M i l l 1 1 1 11 t m i l i u m m u i 11 U U 1 1 U 1
1111 u
22222 2222
2222 33 222 333
1111111111111111 1111111111111111-1111111
2222222222222 22222222222222 222222222
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3133333333 3333333333
CONTOUR LEVELS AND SYMBOLS
SYMB LEVEL RANGE
( 0 ) 2 C ^2E-02
( 9 1 113^151 ca t I-I13II--S1 pound71 pound71 iiS51ESf ( 6 ) (6) flIIlsecti ( 5 ) ( 5 1 UI|g| ( 4 ) pound41 i laquoSIS ( 3 ) ( 3 ) ^IIsectI ( 2 ) ( 2 ) sectvSgSI pound1 ) pound1 1 ssiis (0) 1 4302-02
ESTIMATION ERROR Jraquo TERION C0NampTR i - r =
C W 1 =
pound - 0 1 )
Figure 625A Contour plot of te)]u wi th initial covariance matrix P = H given in (656B) and cC HO15 for the first sample at t K = 011 case with a s 02
Lim
Compare with Figure 623A for
CONTOLR PLOT OF tPCKK) CZIK)) 311 AS A FUNCTION 3F [ Z ( m W3R1Z AND tZ tK) )2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE IN OUTPUT ESTIMATE UTH TIME POSITION OF MAXIMUM VARIANCE APPROACHES STEADY-STATE VALUE fOR LARGE TIME
tZ(K)12 os
44444 333 22222222 44444 333 2222222 44444 333 22222-222 4444 33 2222222 4-14 333 2222222 A 33 22PZZZ
333 22ZS-K^^2222 333 22222^ 22222 333 2222222-222 22222
333333 222222222222222222222 33333 22222 33333 2222 3333 2222 3333 222 333 222 bull333 222 333 22 333 222
222
9333 At 3333 A 3333 A 333 gt 3333 333 bull 3333 333 333 33
22 222
2222 11 pound2222 1111
22222222222222 2222222222222 3
22222222222 22222 2222
111 11 222 11111111111 222
111111111 222 1111111111111111 1 11111111111
1111111 1111111
11111
99299 0 909999
S3 GG TIT B06BB 939999 55 66 77 85BG03 993299
A 5 65 77 03BBB 99999999 4 55 66 7777 66G86 99D999999 AA 5 6 3 7777 BBB30G 999999 AA 55 6 56 77777 QBOB600
44 55 56U 77777 eeBSBBBO AA 555 6GS6 777777 8008806008
44 S5S 0666 777777 66800 44 55 i 666C6 7777777
i3 44 5-iS 666666 777777777 33 44 550 6GG6666 7777777777 33 444 raquo5Si5 6G666SS6 77777 333 44 S)iS35 GGGGGGG6 33 444 3555535 6GG6660GG0 333 444 5555555555 66G666666
222 33 44 14 5555555S5S5 22 33 14144 5555555555 22 333 4444444444444 5535555 222 333 1 4444lt1444444 2222 I3lt13333333333 4144444
33333333333333
111111111111 1111111111111111111
1111 111111 1111 2222222222222 11111
11111 222 33333 2222 11 11111 222 333 333 222
1111111 222 33 44444444 333 111111 22 33 444 444 33 ez 1111 222 33 44 5555 44 33 Zi 11 22 33 44 55555555 44 33 2 11 22 33 44 55SS5 444 33 f 1111 222 33 444 444 33 22i 111111 222 33 3444 4444 233 222 11111111 22 333 44 333 222
111111 222 3333333333 222 11 11111 2222 22L1
11111 22222 1111111 111111
11111 11111111111 111111111
11111 222222 11111
2222 1111 222 11111
33 222 11111
11111 2222 11111 222222222222222 1111111 222222222222222222
i i11111 i 11111 n i i i I 11 i m i n i i i i i i
n i n m i i i i i i i i i n i n i i i i i i 111U1111111U1111111111111
m i l i i n u n i i i n i i 1111 i i m i m i i-1111111 I 111 i i n i n i 11111 11 1 111
1 1 2222222222
1111
1111 11111 11111 111111
222222 222W222222222 1 2222P222 HiP2222222222 2 2222222i^22222222222
1 111 222 1111 1 I I 1 1 1 1 1
1 1 1 1 1 1 1 lt i m i l m 1111 in 1111 m i ii 1111 ii i i lt i i i i i i i i i i i i i i i i i
m i i i i i i i
n m i n i m i i 222
222 - 2222222222222222222222222222Z22 222222 222222222222222222
22222 2222222222222
SYK3 LEVEL KAKEJE
(01 25171E-02
l 2 2
d570E-02 397CE-02
2 2
33G3E-02 27tiOE-02
2 2
21amp8E-02 15G7E-02
i 2 2
OQti7E-02 OatiiSE-QZ
i 1 0765E-02 9163E-0Z
1 05G4E-C2 79G4E-02
1 1
73G H-02 O7r3E-02
sect 1 1
eir2pound-02 55G1E-02
1 1
49G1E02 43G0E-0Z
tQl 137G0E-02
ESTIMATION EMWJ3 CPlTpoundRtCN CCNS^MNT laquo
I SOJSt-Ot
HIAfCL IWJ
Figure 625B Contour plot of | EK(^K) w i t h i n i t i a l covariance matrix p[j = HQ given in (656B) and a =015 for the second sample at t K = 086 Compare with Figure 623B for
case with a 7 - ~ 02
203
Supoose the problem starts at time tbdquo As discussed in Section 63 and according to Conclusion XI the position z of maximum variance in the estimate of the pollutant concentration at all measurement times is independent of time and is thus calculated at the beginning of the problem With this value z relationships among the various optimal measurement position vectors z at Ihe measurement times are to be conshysidered
Assume that the time the first measurement is required is at timj t iy is found to maximize Ktt) the time the next measurement is reshyquired Then at t K + N gt it+bdquo is found to maximize the next time interval to a measurement etc A typical plot of a (zz) over values of tbdquo is shown in Figure 626 For each measurement time t bdquo + N gt zJ +bdquo is to be
found to minimize [ P S ( Z K + N ) ] so that to corroborate the optimizations K+N over K + N contour plots are made at every measurement time for [ P K + N
(z K + N)] as a function of [ji+N] horizontally and [Zj+NJ vertically Plots for the four resulting measurement times in this problem at t = 027 048 069 and 090 are shown in Figure 6-7 Notice that the contours at all samples are the same leading to the eame optimal design for z] + N at all measurement times t K +^ thus Conclusion VI is demonshystrated
Comparing the first two measurement time intervals in Figure 626 that is (t K - t Q) = 027 compared with ( t K + N - t K) = 021 shows that for N large the only effect that the choice of U Q has upon the optimal measurement design at the first sample at time t is in determining the time of the required measurement t K it has no effect upon the optimal locations zt which demonstrates again Conclusion V
RUN N3 1 EAMfgtLE 7 0 - T I C W IPOLUTION OF VARIANCE I N O U T f U I ESTIMATE WITH T I M r S I G ( t ) POSIT ION OF r A X I M W I VARIANCE Prf iOACHES STFAIV -STATE VALUE FOR LArtCE T I M
60000E-02
4B0DEE-02
1-6000E-02
x x raquo X
X X
X X
X
X X
X X
X X
I X
I X I X I X I X I X I X
X
X X
X X
X
x
x x x X X X x x x
x x
I X I X
I X
X X
X
X
X X
X
X
X X
X
X
X X
X
X
1 X
i x
IX
X X X laquo(
X
l - f y r s ^ - ^
Figure 626 Time response of o K + t Yz z ) fcr obdquo - 0075 fojr samples occur at t f deg-7 048 069 and 090
205
deg gK Slt1
1 ss rjti on OO OO s
Vr gK Slt1
1 is 5 1 T 3
ore 2-5--
co iZ ^ pound3 Sm mdash SS raquo N
T 3
ore 2-5--
o tfgt W laquo WWttWW r-r- bullft w laquo NWWWW r-- ID n v ^ n WWVWftl r-f^ o m raquoltT f t WWWWCd S lO V o WWWWW
o rt V WWfV-W N T iT o ftiwwcvw N r u w N M V N N
bulla L i V laquo ltj laquobull IV o V o n wywcvcv
t o o i n lt o n WflWftWfti bull bull M O O m T WltoeJW
O t f rt V WWftftftiftJ O O w T o r a OlttKiV-jAiAW p laquo T WWMMftAlMW - N L I V WftiXFMAiiVOi
- N 1 bulli l V OCT L i ft
pound o irw 7 o ft ltt -v
t ID o ttvfitirv i m laquo w bullcjftCnWW
^ tvft fNPJVWPi o Ift W o W f - gt bull laquo ( raquo gt laquo OHO ifl bull o laquo c (M^Cft(M lOul n ^r Vi Nfftl O O - iv iww
(D^-gt bull c- laquo wwv luWNUi 10OO - 1 n n wwcv vwni
ww o o bull
mdash mdash mdash CJW
mdash mdash - mdash mdash Wftl
- ^ N N N N r v
www bull inmdash
bull (Oioininraquo-))0in
H 5 S 5 2
ftjft www Mftt
WiMCU
mdash ^ - w
c^v fJSJCl mdash - mdash -
iiiisis mdashmdash WW bull O mdashmdash (M J bull bull bull bull o
CONTOUR PLtff OF EP(KK)(Z(K))311 AS A FUNCTION F rZ(K)J1 H0R1Z AND CZ(Kgt32 VERT EXAMPLE TO SH3W EVOLUTION OF VARIANCE IN OUTPU1 ESTIMATE WITH TIME POiilTION Of MAXIMUM VARIANCE APPHOACHES STEADY- J T M T E VALUE FOR LARQE TIME
1
CZ0012 05
444 444
4444 4344 4VV
4144 444
4444 bull4444
44444 44-14-14
444-144 444444 44 14 4 4414
444444 5 444444 5 444444 5 4lt14lti44 SI 444M 44444 4444 44-144 4444 4 114
777 777 777
66
114 333 3333 3333 3333 3333 3333333 3333 22JU22 2222 2222222 22221-2
3333 333C333 333S$33333 I33333J373333 $33313raquo33S33333 4-144 555 666 3$3333amp3i33333333 444 55 666 33$^33J33J33333333333 444 55fgt 66J 3333 33333 333333 44 55 6E-6 333$3 3J33C333 444 55 GS 05S33 444 505 33333 44 b-S 222222 3333 444 555
0080 ueeo H388 SC30
OC038P occecoo
9990339 99093999 9S9S999 99303399 1S999 _ 9999S99raquo 999959999b 88663098 S99999 77 388833083 7777 8063000068 777777 808EJ8C88380860 7777777 0S03C3SQyC8B 777777777 6838008 777777777 Jifi 77777777777 ltJ 0C6 7777777777777777 fgtiit36SC 77777777777 66b5Eil3S^GC6 222222222222 333 44ltJ 555 22222-gt^I22amp2pound22 3333 bull 222pound222222222222 33 222222222 333 2222222 3333 -4^4ltM1414444
222222 33333 4441444444444444444444 222222 333333333
50305555355555 555555505555555555
222222 pound22222-2 2222
2222 2222222 ^ 2 2 2 2
1111111 1111 111 1111 111 II i n i n m i n i m m m i i n m m i m m i m i m i m i 1111
i n i m n
m m n n m i i
11111 m i l
2222
22
111
n
m i n i I n 11iin11 I I ii i m i n i m i IinI1111 n i n m 111111111 in-1
111 Ull 22222222-111111 22222222poundK22222 t 22222222222
11111 2222 2222 2222222222222 11111 2222222
11111 22222 33333333333 J3333 oo ^22222 ii i n ^ ^ i m i i H2222 333333c
SYMamp LEVEL RANGE
tO 21520E-6pound
(6t C6gt lISISi (5[ (5f l3ililgl (4) 14) 15SfI8i
(2J 1026oE-02
ita
I250UE-01J
F i g u r e 6 2 7 B C o n t o u r p l o t o f fe)jn for the second sample at tv = 048 K
CONTCLrt PLOT OF I P f K K ) ( Z ( K ) ) J 1 1 A3 A FUNCTION O f [ Z t K l l l HOR12 AND t Z ( K gt 3 2 VERT EXAMPLE Tr- SiTOW EVO^UTIDN OF VARIANCE I N OUTPUT r S l M A T E WITH T IME POSIT ION Oi MAXIMUM VARIANCE APPROACHES STEADY-G ATI VALUE FOR LARG T IME
444444 55 66 777
41 V pound4 tgt5 SS6 77 SS 66 777
44 444 oL-5 06 77
4 4 4 4 4 4
4 4 4 4
4 4 4 44-11 33333
444 1 3 3 3 3 3 3 3 A4-aA 3 3 3 3 3 3 J 3 3 3 [4ltii 3333Cgt J3073033 44 4 3 3 3 5 J i 3 J 3 3 3 3 3 3 3 IJ44 33 3V333o3-raquo3333333 M4 3 3 1 3 3 i 3 3 ^ J j J 3 a 3 3 3 3 a 3 144 3 3 J 3 3 3 3 3 3 3 3 3 3 3 3 44 5 S 3 6(gt 14 3 2 3 3 3 3 3 3 3 3 3 444 5S 5C I 3 3 3 3 3 3 2 3 3 3 AAA C 5
3 3 S 3 3 3 3 3 3 4 4 035 3 3 3 3 E222222 3333 -144 j-5
3 3 3 3 ZZZsrlte22 C3C3 4 4 4 5555 bull 3 3 3 3 322 2 22SV2222 3 3 3 3 4 4 4 5 3 3 3 3 3 3 3 23222gt2222-2raquoPgtpound22 3i3 4 4 4 4 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 1 2 3133 4-1444-i
2 2 2 R 2 r t 2 2 2 3 3 3 3 4- 1
2222-222 2 2 2 2 ^ 2 3 3 3 3 3 bull 2 2 2 2 2 2 222 3 3 3
1 1 1 1 1 1 1 1 1 1 2J222
-1-114 J 44-1
4 4 4
m i n i i i i i i i n i m i i i i i i i i t i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 bull i i i n m m 11 I n i i
m i i n n i m i m l i m n
u i m 11 i i t 1 1 I 11 II 1111 1111111
bull111111 111111111
11111 m i l
2222 1111 +pound2222 1111
111
22222 22222222 2222222
m u m 111111111111111 11 111 1111111111111111111111
11111111111111111111111111111 m m i u n i i i i i
m m i i m n u m n m
l i m n m m
11 11 1 22-gt22 11111 2222222 11111 22222 11111 22222
1 C8 9 3 9 9 9 9 9 0+ B t3 9 9 5 9 3 9 9 9 U CBS 93S--99
EU3S 3SJSJ3U39 E-r-so 9 r j099S99
CC30a 33S-SSE9 CfiSBOO 9999 pound999999
383S3S8 9 9 9 9 2 9 9 3 9 9 77 8S33C308 9 9 9 9 9 9 bull717 GC^raaraquoSB 7T777 amp088 iS9QeS
777 777 e8oSSr 30808388 777777 6 3 a 0 8 3 8 3 3 8 0 a
7 7 7 7 7 7 7 7 7 8 8 8 8 6 0 8 7 7 7 7 7 7 7 7 7
Ht 7777777777
CCfiSS 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 i l Egt6 -amp3S 7 7 7 7 7 7 7 7 7 7
S-j^tiGfcG666SG
0 j55 6C6eSCi66e666 _x^CJ50tgtSS555553
S5Cgt5055C55DS5oS5S5 -4M44444444A
4444444444--1444444444 3333333
33333333033333333333133333 3333oJ33333 2r^222 2- i^^22222222
22 pound 3ft laquoraquoamp 2 22222P2S2 222 22i^lamp r PP-2-2222^22222e2
2222 vr^- amp2222222 2 r ^ g 2 L - - ^ 2 2 pound 2 2 2 2 2 2 2 2 2
2^2 r 22 gt22222HS222222S22 P22^252i-pound-HSpoundHS-222i 12K 22c
2222^222gt2222222P22 22222222 2 222 222^22
pound22222222222 m i 1 bull m i n 11111111111111
i i i 1 1 1 1 1 1 i i i 1111111 11H11111 i l l 111
22222222-2^222222222222222^22222222222 bullbulliiiiL22ZZgt2Z-ZZZt
SYMB (01 mm (91 i OC03E-02 0152E-Q2 (8) (B)
9450E-02 B748E-02
(7) (7)
C04SZ-02 7344E-02
(6) (6)
GC43E-02 5341C-02
15) (5)
5235E-02 4 33E-ca
14) (4)
3S36E-02 3134E 02
(3) 13)
2432E-02 1730E-02
(2) C21
103pound-C2 Oj27t--02
(1) (1 J i 6252E-03 9234pound-03 (copyJ 8 - 2 2 1 7 E - 0 3
ESTIMATION ERROR C-RIrEKl f lN CONSTRAINT =
7 0 0 0 P S - 0 3
KlANCE [WJe
1 2 S 0 0 E - 0 1 1
Figure 627C Contour plot of [bullft M i l for the th i rd sample at t K = 069
CONTOUR degLOT OF tPCKK)(2(K))I1 AS A FUNCTIOM CF [ Z C K U I HORI2 AND (Z(K)13 VERT EXANPLF TO SiampU EVOLUTION OF VARIANCE N OUTPUT ESllMATE WITH TIME POSITION OF MAXIMUM VARIANCE APPROACHES STEADY-STATE VALUE FOR LARGE TIKE
3b55 5Sgt3 S5S6 555
444 4444 444 AAH 444
aaaa aaa
4444 44 3-4
lt4444 44- 114 444-44 44441
444444 444444 444 4 11 444414 4444-1 44444 1444-1 -14414 4444
53 G6 777 55 66 777 55 66 777 55 (JPS 77 GSS SS 77 55 GG 7 55 S6 7
I o
4 t44 Sco SG$
535
IZ(K)J2
05
33333 3333333
333333J333 333333330
33-raquor-ltgt3^ii333 V J 33ogt-i333ampJ^33333 444
3 3 3 3 S 3 3 S S 3 3 3 3 3 3 3 3 3 3 J 3 4 4 4 3 3 3 1 3 3 3 3 3 3 3 3 3 3 3 4 4 4
I 3 3 3 3 3 3 3 3 3 3 3 4 4 4 3 3 3 3 3 3 3 3 3 4 4 4
3 3 3 3 3 3 3 3 3 44 3 3 3 3 2 2 2 2 2 2 2 2 3 3 3 3 4lt
3 3 3 3 222222gt22 3 2 y a 4 4 4 3333 27-1- 2222Z 3333
3333333 S2Sk4gtgtZSfgtamp2lrfS32 033
3 5 0
4 4 4
3 3 3 3 22222 2 2 2 2
2222222 + 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 U U 1 1 1 1 1 + 1 1 1 1 1 1 1 1 1
l l t t l l l 11 1 1 1 1 1 1 1 ) 1 1 1 1 1 1 1 1 1 1 1 - 1 1 1 U I U M 1 i i i m n 1111111 n u n 11111 bull i i i n 11111 m m
m m i m m
Z-2222P2 3333 414 2222222 3333
222222 33333 222222 3
pound222322 22222r
222
222222 2222J222 2222222
1 1 1 1 1 1 1 1 M M M M M M I
1 1 1 1 1 1 1 1 1 1 111111 111 111111 1 1 M l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 M I M t n i 11 M l
m m m 11111 m 0 1 1 m
m m 1 1 1 1 1 1 1 1 1
u r n 11111
2 2 2 2 11111 +22222 Mill
1111M1M 1111111 1 M 1 M Mill ZZM 11111 222raquo222 11111 22222 Hill 22222
9S39399 0393339 UvV9 9S0S999 8bamp3 30S0S3999 B08CSS S99SS3S999 Oer668 9999999999999 6800836 939S3S9939 77 8SC8PC03 999999
777 08SS bull-iOPOS 7777 uoaac^osae 777777 5031^GOBpound3338
7 7 7 7 777 8S08S3 l 38J 08 7 7 7 7 7 7 7 6080668
bullrraquo 7 7 7 7 7 7 7 7 7 7 G6 7 7 7 7 7 7 7 7 7 7 t-se66 77777777777777
coorgts6eeu 7777777777
iJ amplaquo053 660CC666C666 i J5S5055oj5C55
14 5535555S^0li055555 --444444444444
4 4 4 4 4 4 4 4 4 4 ^ 4 4 4 4 4 4 4 4 4 4 r )33333339
33333333333333333332233333 33pound-3333333 gt22222222 22P22 gt2222222
222222gtpound2222222222 2P^222 igt222222222222 22-222poundgt ^22^22^2^22222 2gt=-r^^c-^i7iVgt^y2^2
2poundf 2222 pound laquo 2t222-poundT2222 222222pound^2 222222
222L22222222P2^22222 1 22222-222222 1 1 1 M 1 1 1 1 1 1 1 1 1 1 1 111 M l 11 1111111
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111 M l 111 111 111 1 I M M
22222222-gt22r222gt2222222 12222222222222
^2^22222^2222222
CONTOUR LEVELS AND SYMBOLS
SYC1 LEVEL BADGE
( 0 ) 2 1 5 6 2 E - 0 2
( 9 ) ( S I Isectlil81 ( 8 ) ( 8 ) i l^Ig| 17gt ( 7 ) SMIgI ltS) (6gt lWSUi ( 5 ) ( 5 ) iI5SIsectI ( 4 ) 1 4 ) V^f-Si ( 3 ) (3gt f^gl C2gt 12 ) JSISi ( M ( 1 ) lIii8i lt0gt 8 2 3 3 E - 0 3
E-STIMAIICI-I E R O J CUTEFUQN CONSTPAlMT =
7 i i C 3 C E - 0 2
12oOCE-013
Figure 627D Contour plot of [laquo)] for the fourth sample at tbdquo = 090
20
634 The effect of Level of Estimation Error Bound upon the Opti-niaJ_jhpoundrtoring Problem - In the examples of the previous two sections a comparison is now made of the effect of the level of the estimation error limit upon Jie outcomes of the optimal monitoring problems of design and management In both cases start with H given in (656A) or (657) In the first example in Section 632 o r 02 whereas in that of Section 633 j v 007b
In the first case o+(zjtz] is shown in Figure 620 in the secshyond in Figure 626 Notice immediately that there is a diieat effect upon the bullbullbullbull bullt- problem a lower estimation error limit leads to higher sampling frequency as would be expected
However a more interesting point comes in the effect of the value of o v upon the optincl design problem the optimal placement of moni-
tors Comparison of the contour plots of [P^(zbdquo)l for sample times 2 2
tbdquo in Figure 621 for a r 02 with those in Figure 627 for a = 0075 shows that the optimal design problems are vastly different leadshying to entirely different positions zt for the global minima in the two problems
Notice also that the shape of the contour in Figure 621 is differshyent from those in 627 the predominant difference being the cmaller height of the rise around the source location z = 03 This can be exshyplained as fallows la the case of the flrst samples far the problem with a = 0075tbdquo = 027 whereas for o = 020 tbdquo = 126 Thus
urn J K ivn K
the stochastic source has longer to act upon the system with te higher error bound The effect of this can be seen by considering ihe form of the predicted covariance matrix P^ in (624) and (628) For the asympshytotic case of infrequent sampling from Section 532
210
Pdeg Mbdquo Ktg]
(628)
o o n s~s
(Jo] + K C ^n)
L ss
(658)
Thus as K grows the first element of fdeg get larger relative to the other steady-state terms in Pdeg as seen on the right-hand side of (658) This results in different values for the inverse [ pound ( 2 K ) P S C ( J K ) T + V] in the equation for the corrected covanance matrix in (626) Thus with T = (t K + 1 - t R) = 001 oZ
tim = 02 leads to K = 126 for the probshylem in Section 633 whereas that in 632 with cr = 0075 leads to K = 27 this results in the different contours in Figures 621 and 627 Thus the optimal design of the measurement locations is seen to be a function of the level of the error bound which substantiates Conclusion IV
635 Examples of Various Levels of Bound upon Output Error -The same problem as in the last examples was solved but with a range of error bound levels as follows o ^ H 005 0075 01 0125 015 02 and 05 Resultant contours of [Ppound(Z)]bdquo at the first sample time tbdquo for each case are shown in Figure 628
As the time interval grows before a sample is made the uncertainty in the estimate of the state in the area near the source z w s 03 beshycomes large relative to that elsewhere in the medium These plots further
CONTOUR PLOT OF t P ( K K ) ( Z ( K gt ) 3 I 1 AS A FUNCTION C CZ(K )31 HORIZAND t Z ( K ) J 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPUT E-STlMATE WITH T l W E POSIT ION OF KAXir iUM VARIANCE APPUOACHES STfeADY-STATE VALUE FOR LARGE T I M E
CZ(K)32 05
555 555 553 555 555 S55 555
444444 444444 44444 44444 44444 4444 41444 4444 4444 4444 4444 4444 444
4444444 4444444 4444144 4444144 AAA 144 44 1-144 444144
55 G6 77 083
4444 444 444 444 444 444 44
44444 444444 44444 raquo5 et 4444 555 I 44444 55 I 4444 55 I 4444 55 lt 33 444 55 3333333 4444 55 33333333333 444 555 33333333333333 444 33J333333333333 44 33333 3333333 444
999999999 992919339 53 66 i i JBB 53993399 55 66 77 CSS 099S9S939 55 66 77 608 999329999 55 66 77 copySi 9029099993 555 66 77 CU-iS 09 Oji 309999 555 f-6 77 BCe 9S23DS99S9999 55 66 77 Or60 999990999999999 55 56 777 FEd9 99993999999999993999 535 GB 77 C0U98 9S9P9999992999993-777 8U930O 99999999999999 77 03311388 939999939 6 777 S0ii008338
6 7 7 7 7 s a o a a a a e e a s 6 7 7 7 7 Q880aBCelt23688e tiG 7 7 7 888dC0e0LC388Ca8C338888
_ 6 6 6 7 7 7 7 7 8 8 8 8 3 8 0 3 8 0 8 8 3 8 8 8 9 5 5 66S 7 7 V 7 7 7 7 7
665 777777777777 4444 3333 33333 144 555 6666 777777777777777777777777
4444 3333 3333 444 553 6C6C866 7777777 444444 333 2222 3333 44 5555 6666666565606066666
3333 222222222222 3333 444 S55t3S 566S66666 33333 22222222222222222 3333 4444 55D55555555S555555j55555555
3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2
pound 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 i m i m t m u
bull 1 1 1 1 1 1 1 I M 1 1 1 1 K 1 1 1 1 1 1 1 1 1 U 1 1 11111
i i i
n u n 1 1 U 1 1 I 1 1 1
m i l l 2 2 2 2 2 11111 2 2 2 2 2 2 2 2 1 1 1 1 1 1
2 2 2 2 2 1 1 1 1 1 1 1 1
22222222 333 4444 222222 33333 4444444444444444444444444444
22222 333353333 333333333333333333333333333333
222 333333 22222222222
2S 25 722222222222222222222 2^2222222^22222222222222222222
22222lt222222222227222222 Z22222222222232222222
22222222222222222-11 1111 111111 1111111 11111111 111111
1111111 22222222222222222222 111111 22222222222222222222222222222222222 11111 2 2 222-2 pound2 111111 22222 333333333333333333333333d 1111111 2222 33333333333333333333333333 11111111 222 3333333
1111111 2 2 2 2 2 2 u i m u n 1 2 2 2 2 2 2 1 1 1 1 U 1 1 1 1 1111 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1111111111 n n u m i i i 1111111111
1111111111111111 i m i n t t i i i i i i
l i m n l i m i t 11111111
SYM3 LEVEL RANGE (6) 13141E-02 ( 9 ) ( S
1 2 6 8 7 E - 0 2 1 2 2 3 4 E - Q Z
( 0 ) ( 6 )
1 1 7 6 1 E - 0 2 1 1 3 2 8 E - 0 2
(7gt (7gt
1 0 8 7 4 E - 0 2 1 0 4 2 1 E - 0 2
( 6 ) ( 6 )
9 9 3 7 0 E - 0 3 9 5 1 4 5 E - 0 3
( 5 ) ( 5 )
9 O 6 1 2 E - 0 3 8 6 0 7 9 E - 0 3
( 4 ) ( 4 )
8 1 5 4 6 E - 0 3 7 7 0 1 3 E - D 3
(33 lt3gt
7 2 4 3 0 E - 0 3 6 7 3 4 7 E - 0 3
( 2 1 ( 2 )
6 3 4 1 5 E - 0 3 5 6 0 9 2 E - 0 3
( 1 ) ( 1 1
5 4 3 4 9 E - 0 3 4 9 8 1 6 E - 0 3
(Q) 4 5 2 3 3 E - 0 3
ESTIMATION ERROR CRITERION CONSTRAINT =
5 0 0 0 0 E - 0 2
12500E-01J
Figure 628A Contour plot of B ^ ( z K ) l 1 1 at f i r s t sample tirr t K = on for o ^ = 005
CONTOUR PLO T OF [P(KKIZ(K))JM AS A FUNCTION O r Z(K)11 H3RIZ AND tZ(K)J2 VERT EXAILE TQ SIOW EVOLUTION or VARIANCE I N OUTPUT E I M A T E WITH T I M E POSITION OF MAX MUH VARIANCE APTtOACKES SrCADY-SrAE VALUE FOR LAHQE TIME
C Z lt K gt J 2
0 5
4 4 4 1 4 4 4 5 5 5 6G 4 4 4 4 1 - 1 4 gtSgt 6 6 4 4 4 4 1 4 1 SOS G5 7 7 7 4 4 I - 4 - 4 0 5 eC 7 7
4 1 4 4 4 4 5 5 GC 7 7 7 444 - = i14 5 5 5 5G 7 7 7
4 4 4 4 - 1 4 5 5 5 GS 7 7 7 4 4 4 4 4 5 5 6 S 7 7
3 3 3 3 4 3 144 5 5 5 0 5 6 77 3 3 3 3 3 3 3 4 - 1 4 4 4 5 5 6 C 7
3 3 3 2 3 3 3 3 3 3 1444 5 5 5 tgteuro6 3 3 raquo 3 3 - j ^ - 3 i 3 3 4 4 4 4 S 5 6 3 6
333333-gt gtraquo3 -gt3333 4 4 4 4 5 5 5 C S G I - ^ v 3 3 3 o 3 3 5 - j 3 3 3 3 3 3 3 3 4 4 4 5 S 5 6GGS 4 4 3 1 r--ijgt333 3 5 3 3 3 0 3 3 1 3 4 1 4 5 J 3 6 -4 4 4 3 3 3 S 3 3 3 i 3 3 r ^ 3 3 4 1 4 -i CC 4 4 4 4 33T-2 3 3 i J 3 3 454 j ^ 5 f 4 4 3 ^ 3 3 3 ^ J 3 4 4 4 5 3 5
3 3 3 3 3 3 3 4 4 4 5 5 1 5 3 3 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 555E-
3 3 3 3 2 2 2 r - i ^ 2 2 2 3 3 3 3 4 4 4 5 S 3 3 3 3 3 3 22 laquo - - yraquo jraquo2 3 3 3 4 4 4 4
_ _ - - - r ^ amp ^ 2 ^ i 2 2 3 3 1 3 4 4 4 4 2 2 2 2 2 C 2 r 3 3 3 3 4 4
2222ltgt2 3 3 3 3 3 2 2 laquo 2 2 2 2 3 3 3
P 2 2 2 gt
5555 444 5555 444 555 4-i4 5-5 444 i 55 444
44-14 4444 4444 4444 44414 44444
4444- 14 444444
33333 222222
22222 222 -2
1111111 1111111 1UI1 11
2 - 2 pound bull
11111 11111111111 11 n m i n i i i n n i n m m n i
1111 n m 111111 m m m 1111 111111m 111111
m i
1111 11111111
111111 11111 ftfraquofgti- bull
1 1 1 1 1 WWZZZ
JErJSe pound 1 9 3 9 9 9 9 0lt
S L B 3 9 9 0 i T 9 - 9 f - a 3 D O - bull s - s s
bull i 3 3 3 O 3-3999 eccose ss-v9S3999
8 t S S C 8 9 9 0 9 9 0 9 9 9 9 9 9 9 9 8tt81B8 99S999999S9
V e t J f i380 t i 9 9 9 9 9 9 9 7 7 c s s o e r G O y77 e o s u c c - i i s n o
7 7 7 7 7 fcampceooaaeoeoe 7 7 7 7 7 7 7 a p 3 3 C 8 e e e e e 3 9
7 7 7 7 7 7 7 7 o c e o B e o s i 777f77777 Jo 77771(1777 3 3 77tn7777 pound 0 6 5 3 6 7 7 7 7 7 7 7 7 7 7
iGeampampG6CgtGS6 3poundGC66SC(GpoundGQ
i 5 i 3 6 G amp a amp 6 6 G 6 6 6 6 C G 5 5 J 3 5 5 5 5 5 W S 5 5
3 5 5 5 5 5 1 J S C - 5 5 5 5 5 G 5 5 5 5 5 1 1 - 1 1 4 4 4 4 4
44444 44 44444444-T444444444 J333 4444
3-3Cn3S333J3L--J33333 3 3 3 J J 3 3 3 3 3 J 3 3 3 3 0 3
pound 2 - 2 2 2 r i - 1 H i i 2 2 2 2
2-raquo i- raquogtr---2igt2 j j - r gt V ^ - l 2322
222 - bullgtbullbull2 raquo2222222raquoa 2 2 gt V 2 ^ gt i gt - S P 2 2 2 2 2 2 2 2
- 2 r ^ - gt 2 K 2 2 2 2 2 2 2 2 2 2 ^ 2 - ^ - - V 2 ^ 2 2 2 2 2 2
2fc i 2^22^ -2lt i 222
m m bull m m 1111 m 1 1 m 11111 i i i i m - i i
222222222 bull bull bull 2 i r - ^ 2 2 r ^ 2 R 2 2 2 2 2 2 2 2 2 2 2 - 2 ^ r ^ - ^ ^ 2 2 2 2 2 2 2
SYM3
( 0 ) mm ( 9 ) ( 9 - iJiiI8i ( 8 ) ( 6 ) Wiiiii lt 7 t ( 7 ) J5JiSi ( C ) ( 6 ) I8Sf8 ( 5 ) ( 5 ) 3i5i|g| ( 4 ) 4 gt lHIgI ( 3 ) ( 3 ) lfJ|8i ( 2 ) ( 2 ) HSSiSi ( I ) ( 1 gt I2iJIsect lt0gt 7 0 S W E - O J
ESTt l - ATITN tlrila C1C TCR10N C C K - r r A f T =
7 S 0 J C E - 0 S
SOURCE 1VPUT CUVAFUANCE [ W l raquo t 1 2 6 0 0 E - 0 1 J
Figure 628B Contour plot of fell at f i r s t sample time t 027 for o- i 0075
CONTOUR PLOT OF [ P f K K X Z C K ) ) 111 AS A FUNCTION CF I Z O O J 1 H 0 R I 2 AND t Z ( K 1 1 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPUT ESTIMATE WITH T I M E P O S I T I O N OF MAKIKUH VARIANCE APPROACHES S T E A D Y - E T A T E VALUE FOR LARGE T I M E
t Z lt K gt 3 2
0 0
44 444 AAA
4114 44444 A 4 4 L I 41 44 44 4 4 4 4 444
33333333333 33333333333 35 S 3 3 3 3 laquo33
SiJ^JyS gtlt33 32 i i - - 3^ - gt33 33-gt3- bull -
05 66 77
33 444 444
444 333 313 r i 33 laquo - i n333 3 2 ^ 3 3 J i3i bullbullraquo33333
3 3 3 3 3 3 3 3 gt t j r 3 3 3 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 444 3 3 3 3 3 3 3 3 3 3 3 3 3 3 AAA
3 3 3 3 3 3 3 3 3 3 3 3 Ad 3 1 3 3 3 3 3 3 4 4 4
3 3 3 3 2 2 2 2 2 3 3 3 3 44 2 V 2 2 P 2 3 3 3 3
63 66
i 66G
8000 0 3336
7 60G0B 7 Q6CS0 77 seaoe 777 flSJSi
777
9999939 9999999
9 9 9 9 9 3 9 93939999
S9Q09999 osaa 9993099999S9 80COB0B 999999999ltgt
533 3333
3i33 333333 33333 ZtZZ 333 gtZZ
2^22 222P2222 22222 1
1 M I 11111 11111
111111111111 11H1M 111
111 - 1111
11111 111111
ifpFte gt222 -gt22222 a 2pound-2P2
22222222 2S2ii2^
2 22
7777 O00C36 66 7777 0050008
gt 06 77777 6G03Ceea iS 656 777777 0030830888088-i5 6tGti 7777777 060088308 53 CM a 77777777 BB 555 6006 777777777
^1 533 ( J6GS6 77777777777 144 Su fJ3 60695096 777777777777
44 5355 6660CC-66S6 777777 3 44 G3C3 6CS56GG=S06 3 ^ 4 4 4 ^ s s - s s e e i i c c s e e s G c s 3laquo3 444 o35S355SS 66Gpounde66666
333 441 Sb5335rgtS55j5 333 444441 igtS5Sgt5SS55S55535
1 1
2 2 2 2 3 3 3 lti 1 4 4 4 4 4 4 4 4 4 4 4 4 4 111 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 111111 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 M 1 I 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 2 2 2 2 2 - 2 2 2 2 2 2 2 11 2 pound 2 2 2 2 2 2 2 2 2 2 2 gt2222
2222r-V j2222222222
22222222222 222i-rt 2 222r-222222 2222222222222222 2222--i2 22poundPamp22
1 1 111111
111 euro 3333
222222 222222 22222 22222 22222
22222 222222
1 1 1 1 1 1 1 1 11111
1111 2 2 2 2 2 bull 2 2 2 2 2 1111
222222- V222JV222J-P22222 22^22 -- ^^22222laquo22
22--V-J W J2gt2gtJ 22
222f Pr - gt 225r^laquo2J 2222 2222raquo fi 2r-2^igt22222
11111111111 22222 1111 222222222222 11111111111111111111 Kill 11 II 11111 111 11111 1 i 111111111 111
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 m i l 111 m i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
g) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1111111 2 2 2 2 2 2 2
11111 2Pgt 2222 2222 =V 22222222222 222 11111 22l- bull 22Vv22222
11111 222222 11111 222222 333
SYtu
( 0 )
LEVEL RANGE
2 4 0 S J E - b 2
9 ) 9 )
2 2
33Z 1E-02 J 6 2 C E - 0 2
( 6 ) ( 3 )
2 1 9 2 7 E - 0 2 1 2 2 r i E - 0 2
( 7 ) ( 7 1
2 05271E-02 0 C 2 3 pound - r ) 2
t o ( 6 i
1 1
9 i r 2 r - o e - S ^ l E - 0 2
( 5 ) ( 5 )
1 1
7 7 2 G E - 0 2 7 0 1 3 E - 0 2
( 4 ) ( 4 ) 6 3 1 7 E - 0 2
S61 (3pound -02
f 3 ) ( 3 ) 1
4 9 1 5 E - 0 2 4 2 1 4 E - 0 2
(2gt lt2J 1 331 3- -02
2 8 1 I E - 0 2
( 1 ) ( 1 )
1 1
21 I O E - 0 2 1 4 0 9 E - 0 2
(0) 1 0 7 0 8 E - 0 2
s^fc 1 2 Q 0 0 E - 0 1 ]
Figure 628C Contour plot of fe)]n at f i r s t sample time t bdquo = 046 for a = 010
CONTOUR PLOT OF tPltXK)(Z(K))J1 1 AS A FUNCTION 01 tZ(K)I HORIZ AND [Z(K)J2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE IN OUTPUT KSUMATE WITH TIME POSITION OF MAXIMUH VARIANCE APPROACHES S1EA0Y-SAVE VALUE FOR LARGE TIME
333 444 4444 4444 333 44444 333 44444 333 +4444 333 444 333 333 222 333 2222222 3333 222222222 3333
33C3
CZ(K)]2 OS
333TJj3 333333 33333 33333 33333 3333 3333 333
33333333 44 6 68 77 33333333 44 S3 66 77 3333333 44 55 65 7 3333333 44 55 66 7 33333333 444 S ^6 3333333 44 55 J6 3333333 44 55 666 33333 44 55 666 33333 444 55 G6i 3333 4-1 55 6-222K2222222 333 44 55 i 222222222222222 333 44 2^2222222222222222222 22222 2222222222222 333 44 222 22222222222 2222 222 222 111 222 111 1111111
222 m n i i i i i i 222 1 HI 11 11 111 1 22 11ll 1111ll 111
33 33 333
44 444 55
11111 1111
11
2222222 2222 2222 33 444 222 333 4444 222 333 444 222 33C 4 222 333 2222 3333 2222
mil limiii ii i i 1111 2222222 1111 22222 22222 Mill 222 3 2222 22
22
222222 2 1111 11111111 11111111 11111111 11111111 11111111 1111111
68BG8 999999 eSCfiS 093999 86838 999999 bull7 8SC83 9399999 77 eoooee 99999999 777 7777 77777 i 77777 S 777777 S58ECSBQBC30 bull SM5 7777777 60830860+ 6ilaquoC6 7777777 66666 77777777 66666GE 77 77777777777 i 6SG6C666 777777777 iSf 6pound 6666566 7-i5amp05 666666666 50555595 6666666666666
555Q5555C35 6666666 I 5 5 U 5 ^ 5 5 5 5 5 14lt144 5555553555555-
444444444444444 13 4laquoi444444444444 333333333333333333 3333333333333 22222222222222222
22222222222222222 1111 11111111111 11 imiimt
222 33333333333 222 333 333 2222 iiii 33 4 333 2222 333 44444 333 222 33 4444 333 2222 333 3333 222 11111 J 33333 333333 222 11111111 222 3333 2222 1111111111
+11111 1111m mi 22222 111 222222 111 2222 111
11111
copy
22222 222222 1111 m m m m m i m
urn
m m m i 2222 m i 222222 1111 2222222
I 2222222222222222222 222222222222222222222 222222222222222222222 22222222222222222222 II 1 11111111 111111111111111111111111111)1 1 m u m m i n i m u m
i i n m m m i m m m m m i l i m u m i m m m 2222222
2i22222222222222222222222222222 22222222222222222e22
22222222222222
TIME laquo 66D00E-O1 FIRST MEASUREMENT
CONTOUR LEVELS AND SYMBOLS
SYM3 LEVEL RANGE (01 2 4793E 02 (9gt 2 pound9gt 2 4158E 3523E 02 02 (0gt 2 (8) 2 2363E 2252E 02 02 (7) 2 (7) 2 1617E 0982E 02 02 (6) 2 (6) 1 0347E 9712E 02 02 (51 1 (5) 1 9077E 9441E 02 02 (4) (4) 1 7806E 7171E 02 02 (3) 1 (3) I
6536E 5901E 02 02 (2) 1 (2) 1 52S5E 463DE 02 02 (1) 1 (1) 1 39S5E 3350E -02 02 (0) 1 2725E 02
ESTIMATION EPROR CRITERION CONSTRAINT = 1-2500E-01 SOURCE COVARi INPUT AHCE Wl-
12500E-01J
Figure 628D Contour plot of feMi at f i r s t sample time t K = 066 for o ^_ = 0125 2
CONTOUR PLOT OF E P ( K K H Z lt K gt ) 3 1 1 AS A FUNCTION 0 Z ( K ) 1 1 KORIZ AND C Z ( K ) 3 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPUT i S T I M A T E WITH T I M E P O S I T I O N OF MAXIMUM VARIANCE APPROACHES STEADY-S A f t VALUE FOR LARGE T I M E
bull 4 4 4 4 4 3 3 3 2 2 2 2 2 2 2 2 46640 3 3 3 2 2 2 2 2 2 2 44444 3 3 3 V2Z2Z9ZZ 4444 33 22222c J 22 4 4 4 3 3 3 2 2 2 2 2 2 2 2 bullA 3 3 2 gt 2 2 2 V 2 2 2
3 3 3 2 2 2 2 ^ ^ 2 2 2 2 3 3 3 2 2 ^ ^ f - 2 2 2
3 3 3 22^V22^ 2 2 2 2 2 3 3 3 3 ^ 3 2 2 2 2 2 i 2 ^ f r 2 2 2 2 2 2 2
[ 2 1 K gt 3 2
05
3333 4 3333 4 3333 4
333 3333
333 3333
3^3
11 65308
2 2 2 33333 2222i 33333 3353 3333 333 333 E33 33 222
222
3 3 3 3 3 3 3 3
44
2 2 2
2 2 I t 2 2 2 111
2 2 2 2 1 11 2 2 2 2 2 1111
1111 l i n t
2 2 2 2 2 2 2 2 2 2 2 2 2 pound 2 2 2 2 2 2 2 2 2 2 2 2 2 3 a
2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 3
2222 I 11111 2 2 2 2 11111111111 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2
1 1 1 1 1 1 1 1 1 1 1 2 2 11111111
3 9 9 9 9 9 9 3 9 9 9
9 S 9 9 9 9 9 9 9 9 9 9
t i s s u e 9 9 9 9 9 9 9 9 7 7 7 7 09888 993S99399-
6 7777 esesao 999999 5 77777 8300886
flfl -Jigt 66 77777 88030688 44 5 5 5 ltSlt~C 777777 8008808888
44 SS5 liCSS 777777 86665+ 44 S55 66S66 7777777 44 5SE 6G66G6 77777777
44 556 666S666 7777777777 13 444 5t 5raquo 66666666 77777 3 44 SJ55 66666666 33 444 pound5555555 6666066666 333 444 55505S5555 666666666
33 444fl 53555555555 33 44-AV 555555S555
333 4144444444444 5555335
1111111 m i l l m i 111 n I mi mm in
It T1111 222 3333 44444444444 111111 2222 3(333333333333 4444444
111111 2222 333333333333333 111111 222J 2222222222222
1111111 2222222222222222222-111111111111 1111111111-11111
1111111111111111111 11111i111111111111111111 1111 111111 1111111111111111111111111111
1111 2222222222222 111111 111111111111111111111111111 11111 222 33333 222 11111111 1T11111111111111111111111111111
11111 222 333 333 222 11111111111111111111 222 33 22 33 bull 222 3 44 22 39 44 22 33 44
33 44444444 333
444 444 ~ S555 44
553SS3 444 555055 444
444 11 22
33 4444 33
222 4444 333
333 2 3333333333 222
222 u m m uui 222 11111 222 222 222 222 1111
33 2222222222 2222222Zamp22amp222222222 -2222222222222222222222 222222222222222222Z22
222 11111111111111 111111
1111 2222 2222 11111 11111 22222 11111 copy
11111111 1111111 11111 11111111111 11H11111 niituut nnniniv mu mmiimi i m mimiim urn m 222222 11111 111111 222
2222 1111 11ll 1 2222222222222222222222222222222222222 222 1111 11111 222222 222222222222222222
33 222 IHtl 11111 22222 2222222222222
TIME 6 6 0 0 O E - O f 1RST MEASUREMENT
CONTOU LEVELS AND SYMBOLS
SYHB LEVEL RANGE
lt0) 2 5 1 6 G E 0 2
( 9 ) ( 9 1
2 4 5 6 5 E 2 3 9 6 4 E
0 2 0 2
( 8 ) ( 6 )
2 3 3 6 2 E 2 2 7 6 1 E
0 2 0 2
( 7 1 lt71
2 2 1 6 0 E 2 1 5 5 S E
0 2 0 2
( 6 ) (6gt
2 0 0 5 7 E 2 0 3 5 6 E
0 2 0 2
lt5) ( 5 )
I 9 7 5 5 E 1 9 I 5 4 E
0 2 0 2
( 4 ) 14 )
1 0 5 5 3 E 1 7 9 5 1 E
0 2 0 2
( 3 ) ( 3 )
1 7 3 5 0 E 1 6 7 4 9 E
0 2 0 2
12) ( 2 )
t 6 1 4 G E 1 5 0 4 7 E
0 2 0 2
1 ) ( 1 )
1 4 9 4 5 E 1 4 3 4 4 E
0 2 0 2
l O ) 1 3 7 4 3 E - 0 2 ESTIMATION ERROR CRITERION CONSTRAINT =
1 5 0 D 0 E - 0 1
SUUSCE INPUT COVTMANCe pound 1 2300E
MEASUREMENT ERR03 COVAR
I 0 5 0 I - 0
W]=
on tv)laquo - 0 1 D233
Figure 628E Contour plot of [ P ^ K J I a t f i r s t Spoundp1e time t K = 086 for a l i m = 015
CONTOUR PLOT OF I P ( K K ) ( Z ( K ) ) 1 1 1 AS A FUNCTION ( F t Z I K I I I HCRIZ AND t Z ( K ) 1 2 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE I N OUTPUT ESTIMATE WITH T I M E P O S I T I O N OF MAXIMUM VARIANCE APPROACHES STEADY- ITAVE VALUE FOR LARGE T I M E
CZltK)J2 0 3
^IPllI 33 44 55 6G 7 J
3330 2222 3333 222 33353 222 3333 2222 333 pound22 222 pound22 pound22 333
pound2 22 22 pound2
22-gt222 2222Z_222Z 22222 T-K222 2222- 0272ZZ 33
2d2i7gt2922 33 22lt2gt-222 3 22222 1 2222 1111 222 11111111111 222 111111111111111 222 1111111111111111 22 1111111111 22 111111
333 44 5 lt 333 4 55 I 33 44 55 333 44 55
0CCSO 0S8GO 83808
333 44 55
7 i 777 bull 777 til bulllt 7777 pound C 77777 t Gi 77777 rgt66 777777
999999 03099 9D399 999399 99939999 99999999 686830 9999 8608069 8088366368
111 222 222 222 II 2222 111 22222 111 1111 11111 1111111 11111
111 111111111111111 11111 11 11111 222222222222 1111111 222 33333333 222 11111111 22 33 444 33 1111 222 33 44 444 3 222 33 44 555 555 4 2222 3 4 5 66666C66
6665 777777 44 55 66G66 7777777 3 44 55 GSG666 77777777 3 444 5-5 66GCCCC 77777777777 33 -14 5555 6605666 777777 33 44 gt5535 666G66G 33 444 555555 606660666 33 AAe 5tgt5lgt5555 666G666G66 z 33 44I4 5553355S55 6GG66 22 333 4-144 555555555
bull 33 506 55 4 33 222 777 66 55 A- 33 22222 777 6 55 4 33 2222 i 66 665 55 44 3 222 55 6666G6G 55 44 33 222 2222 33 44 555 555 44 33 22 1111 222 33 444 44 33 22 1111111 222 333 333 222 11111 11111 222 333333 222 1111
11111 222 333 -1-544444444 55555555555 Ill 222 3333 4444444444444 1111 222 33C3323 444444444444 1111 222 33333333333333333 11111 2222 pound2 3333333333 11111 222222222pound22222222222 11U11IMI 2222222 1111-111111111111111111111 11 1111111 111 1111 111111 11111 11111
111111 111 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 2 2 2 1 1 1 1 1 1 1 1 3 2 2 2
111 H I 1 111 1111111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1111
33 44 53 66 33 44 55 66 33 4 ~
223J222222222 22222 ^2poundf22^2 2222222
22XgtM2V-gtpoundlt2V2Z_WW2PZZZ 22222 e222222gt22222
22222L-2222222222
1 1 1 1 1 1 11111
2 2 2 2 11 2 2 2 pound
333 2 2 2 2 3 3 3 3 222
3 3 3 2 2 2 3 3 3 2 2 2
22H22222222222 11 1 uiiinninniniii mini iniii iiiinmniiinimi 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ^ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ^ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
SYK3 LEVEL RANGE
oi 2 5 5 4 1 E 0 2
sect I 4 3 7 2 E 4402E
0 2 0 2
sect I 383 3S 32C5F
0 2 0 2
n I 2 0 9 E 21 EZ
0 2 0 2
ni I 1554E 0ampC6E
0 2 0 2
fl 0 7 1 5 E 9S45E
0 2 0 2
n 927SE 67D7E
0 2 - 0 2
sect 8 I 3 7 E 7560E
0 2 - 0 2
I 6 9 M E 6 4 2 J E
- 0 2 - 0 2
i 55C5E 5a5C
- 0 2 - 0 2
a -L 4 7 2 0 E 0 2
ESTIMATION ERHCrt CR f E i d O N CONSTRAINT =
2 C 0 0 0 E - 0 1
1 2500E-01]
Figure 628F Contour plot of P ^ i O m a t f i r S t s a m p l e t i m e tK = 1 2 G f o r deglw = deg 2 0
217
C O i O O bull O O i O O ss OO i
i mdash tfgt i W mdash 1 mm gt turn CUM I bull n n 55 flH
^ w J I
H U J U O
Si mdashbull- ltgtjltvwlaquotvw
O l o r -
E D gt o o O C O O f -
KM (-^-gt -gt - 3 V J mdash w n n laquo j - mdash mdash o o bull O D H W o o o n W - - o bull Z 10 - ltl O O O O WftJ wv 3 K - - lti o o ft l L - ^ 0 - W O laquo ^ 1 1 laquo W M fu
HI - W gt T 1 gt O N bull t U T n -v i i i o bull=bull w w
o o - w T I m i l i i c raquo ltgt l i v - w n igt t i v W C J bullVft -lt lt o - o v i n I O O O O ifgt n w i i y bull
laquo mdash W m t o I D T O laquo w - e n mdash W O f ( N - M v i 3 laquo J t ^ - laquo o - w n v m o huraquo n laquo ^ (
bull-gtlt - N 0 ( 0 0 (OTTO ft-lt bull - laquo (0 h - U J i f l W gt
w _ O O N raquo t u r n o r n ftikM w bull o o ftlt - 2 laquo o ^ E a N lt 0 sect W lt n sect rt N T ^ lt WCgtVtfgt 0) O N V O - o - ftt-gtv P - M i laquo i i laquo r ^ mdash o N laquo I O O O ^ V L I T C K I gt I - ( w o v O X - N O ^ V c
o gt p P - n ogt O N I - gt T c x -i
- - - - - - O R - n v o laquo o o r - T o n
- D E - - - - - - O w f t v a o s o c o t a T I laquo - D E - - - - - - O laquo W O N ) lt O O O - laquo r o N O i 1 o
o U 1 X
- laquo r o N O i 1 o
o OO
l u - w B i o N N ifgt o o o o -- - W O ^ r i O o m i T O
O u W O 10 Q U O igt T O O J O O [ j bdquo _ _ mdash _ _ _ _ _ - - M V i f t O 3 ( i o o D-t- - w w w w w _ _ _ _ _ _ bdquo _ - - - W 0 gt T - W u l l O L I T O
z ( C W O n i z ( C W O +_laquolaquoOKV f t JgJlaquo l ~ _ W w o Slt n T5 SS lt- n i 3 _ 1 ~ ftftjftjlt) _ ft O - 3 1 T V [J laquo 0 C H mdash _ j o W T S J - C o o o 1 laquoSp ^ojci^S^^Jv^^^NN^ bullbull w ^ v i - j ^ 2 5 ^ laquo laquo - gt laquo laquo W W ft I j - W N W l ^ C f l J W O T o o o o L1U1 bull o x o 0 - ~ 0 W M M ( laquo gt N A i M mdash - M W O O O O O t O f i -O a J t t laquo f ^ O U N T W W W - - - w w o o o o o in1) bull
0 0 ( 0 W W W W W bdquo _ _ (u Pgt n n o n laquo laquo raquo bull
218
substantiate the existence of a functional relationship between the optishymal measurements zt and the level of the output error bound o
636 The Effect of Time-Varying Error Bound upon the Optimal Meashysurement Design - Consider here an example where the output estimation error limit cC is allowed to vary in time For this problem let
lim 01 (659)
at the first sample time and then
Aim - degL + deg- 0 2 5 (660) for each sample thereafter
The resultant plot of o^ + N(jtz) over time for the interval 0 lt t S 2 is shown in Figure 629 where the initial covariance P^ E M n is as before in (657)
Notice how the curve asymptotically approaches the slope [Q]- =
00025 just before each sample in accordance with the infrequent samshypling approximations
v
At each samplecontour plots of lEDU^)] a r e 9 e n e r iraquoted and preshysented in Figure 630 for sample ti mes t| - 046 104 180 As can be seen from these plots the contours change with the error level as shown in the previous sections in fact they directly compare with those of the previous section Thus the converse of Conclusion VI may be stated as
Conclusion VIB The optimal measurements found at one measurement time may not in general be optimal for other measurement times if the bound on estimation error varies with time (CVIB)
Further verifications of the effects of the a priori statistics and level of estimation error bound upon the optimal design problem can be
1 2 0 0 0 E - 0 1
6 0 0 Q O E - 0 2
1 X
X
x x X
XX x
X X X
X X
x x X
XX x
X X
X X
X X
X
x X
X X X
X X
X X
X
x X
X
X
XX X
X X
X X
X X
X X
X X
X
X X
X X
X X
X X
X
X
X
X X
X
X
X
X
X
X
x x
X
X
X
c
X X X
Figure 629 Time response of ^+n(K z) f o r t lt n e v a r y i n S estimation error l imit o z ^( t) = 010 0125 and 0150 at sample times t K = 046 104 and 180 respectively
CONTOUR PLOT OF t F ( K K ) ( 2 ( K gt ) i 11 AS A FUNCTION = I Z I K H I HOfIZ AND I Z i K ) ] 2 VERT EXAMPLE TO S1ICW EVOLUTION OF VARIANCE I N OUTPUT r l 11 MATE WITH T IME POSITION CF MAXIMUM VARIANCE APPROACHES S I EADY- -T TE VALUE FOR LAKOE T I M E
C6
tZltKgt12
444 444 4444 444
44 33333333 444 444 3333333lt33 444 3333333J333 444 33C-^rS3J3333 444 33333S3333333 4444 3333333i333333 444444 3333333333333333 444444 333333333333333333 44444 33333333333333333333 4444 33333 3333333333333 444 33333 333333333333 3333 3333333333
55 6G 77 bulljV 66 77 eoaee 9900J 0 3
93 li9
3333 3333 3333 333333 33333 333 2222 2222 22222222 22222 1 1111 1111111111 m m i m i l 1U1111 m
i n m i
i n n m m
111111 m m 111111
i i i m i n n
i n n i m
i n
2222 333 lt 4444444444-1414
33333333 i 33333 I 22222 3333 2^^-^^2-222 3333 2222222222222222 333 2222J2222222222222 333 22222222 2222222222222 333 2222 2222
22222222222 22222222^22222222 2222 222222 222 222322 222 33 22222 222 3333 22222 2222 22222 2222 22222 222222 222222 1111111111 222222222222 111111111111111111
^222iV-2v_iV bullbull VJlaquo
222 2 L 22 2 2 r-^ gt L2 22I-22 22222
11111 11111111111U11111111
1111111111111
11111 11111111
u r n 22 11 11 22222 1111 22222 1111
11111111111 111111111111111
11111
n u n i m i n i i i i i i i i i i n m i i n bull m 11111 n i i i m i m i - i i i i i n m i i i m 1111151111111111111 ill 1 1 2222222 222222222222222222222222222 22222J-=2 2222222222222222r 222222 222222 333
t o t z i-o-
( 9 ) ( 9 )
2 KiSi ( 0 )
2 bulltJi-ll ( 7 1 (7)
2 1 degri-pound
ltegt 1 -vmii lt5gt ( 5 )
1 1 STSIgl
( 4 gt ( 4 )
1 -mii-n ( 3 ) ( 3 ) bullm-E ( 2 ) ( 2 )
i i if8f
C 1 ) ( 1 )
i i bullVW-ll
) O70 pound e ii ON
- 0
lwAa v i i E U T [W] =
C 5 C 0 C 3 E bull 3 1 1
ESamp sr EV3-
I -5g =pound
Figure 630A Contour plot of Figure 628C [4i a t f i r s t s a m p l e t i m e t bull deg 4 6 f 0 r deglin 0 1 0 compare w i t h
CONTOUR PLOT OF I P ( K K M Z ( K ) ) ] U AS A FUKCUOH poundlPLE TO SHDW evOLUT ON OF VUiJAhCE IN OHIrJ COS TIOM OF MAXIMUM VARIANCE APliCACULi STL-HY
pound2(KH2 03
d4At 33 4444 333 44444 333
444 44 333 44-1dfl 333 4J44 333 3^3
3333lt33 4 3333333 4-0333333 4 3333333 J 3333333 333333 333333 333J3
3333
bull ^ 3 9lti9nlaquo
33333 33333 33333 33333 3333 3333
32 2p||p-gtill p 044 55
2222 222 222
2222222222 333 222222 33 444 22222 333 44 33 444 1 J-2 333 44 2^2 333 laquo 222 333 2232 333 2222
11111 222 222 111111111111 22 33 22 1111111111111111 2 222 1111111 111 11 1 1 ] 1 111 222 i n u n u u u i u n n
222222 111 11 111111111 222 11111 1111111 111111 1 1 1111 11111111 1 I U U 1 U 1111 111111111111 11111111111111111111111111111111 inn i m n 11 n
1111 2222222 111 111 Tll 22222 22222 1 1 1 1111 222 3 1 2222 111 11 222 333333333333 222 111111 22 333 333 2J-22 m m m u 22 33 44pound 333 2222 bull11111111 22 333 444-144 333 222 11111111 22 333 44444 333 2EKpound 1 lllllll 222 333 333 222 lilt 1 1111 22 33333 33333 222 UUUi 1111 222 33333 222 111111111 111 22222 222222 11111 1111 22 11111 111111111111111111 11111111 1 1 1
bull4444444I4444444 C _ r 4^44444444444
m 1 r i i m 111 m
illllll
111 111 22222 111
I 1 M 111 ill 11 1 1 1 111 111 1111 1111 1 111 111111111111 m m m
2222222 bullit bull-222222^SfTl - 2222222222222 bullZ 222222222222 2 ^222 22222222222222
Figure 630B Contour plot of [ l $ (z K ) with Figure 628D
at second sample time t ^ = 104 for ^lln
CONTOUR PLOT OF tP(KK)(Z(Kgt)311 AS A FUNCTION V IZ(K)JI IflRIZ AND tZltKgt12 VERT EXAMPLE TO SHOW EVOLUTION OF VARIANCE IN OUTPUT EI-M HATE WITH TIHE POSITION OF MAXIMUM VARIANCE APPROACHES STEAD-li TATE VALUE FOR LARGE TIHE
i 444d4 333 22222222 44444 333 22222222 44444 333 22222222 4444 33 22222222 444 333 2222222222 I a 33 22222222222 333 222222222222 333 22222222222222 333 2222222222222222 333333 222222222222222222222 bull33333 22222 33333 2222 3333 2222 3333 222 333 222 bull333 222 333 22 333 222 1 222 1
39399 999939 999939 999399
CZ(K))2 06
3333 44 5 66 77T 6BI 3333 44 0 66 777 861 3333 44 55 66 777 81 333 4 55 66 777 I 3333 44 9 66 7 77 333 44 55 66 7777 3333 44 5 60 7777 333 44 55 665 77777 333 44 53 CiSe 77777 33 44 35^ St 66 777777 333 44 555 6666 777777 2222222222222 33 44 555 66666 7777777 22222222222 33 44 555 666666 777777777 222222 333 44 535- 6666666 7777777777 2222 33 444 55S-5 66666666 77777 111111 2222 33 44 515555 66666666 111111111111 222 33 444 5555555 111111111111111 222 333 444 5555555555 1111111111111111 222 33 4444 555555555SS 1 11111111111 22 33 444lt44 5555555555 11111111 22 333 444444444444 5555555 1111111 222 3333 44444444444 11111 2222 33G33333333333 4444444 111111 2222 333333333333333 11111 22221222222222222 1 11111 2222222222222222222+ 111111111111 1111111111111111 1111111111111111111 111111111111111111111111 1111 bdquobdquobdquobdquobdquo A 111111 1111111111111111111111111111 1111 2222222222222 111111 111111111111111111111111111 11111 222 33333 222 11111111 111111111111111111111111111111+ 11111 222 333 333 222 11111111111111111111 1111 222 33 44444444 333 222 1111111111111 111 22 33 444 444 33 222 11111 2222222222 1 222 3 44 5555 44 33 222 222322222222222222222 22 33 44 55555555 444 33 222 2222222222222222222222+ 22 33 44 055555 444 33 222 222222222222222222222 222 33 44 444 33 222 11111 222
M 33 4444 4444 333 222 U l l l l i m u U 33 44 333 222 1111111111111111111111111111111111111111 222 3333333333 222 11111 111111111111111111111 2222 2222 1111
222 111 2222 111 22222 1111 1111 11111 +111111
111111 11111111 2 bull 111111 1111 11111 22222 11111 1111111 1111111 11111 11111111111 +111111111 1111111111 11111111111+ 11111 111111111111111111111111111111111111 11111 111111 222 2 2 1 1 1 1 11111 2222222 2222222222222222r 222222222222 n n n 1111 11111 222222 222222222222222222
CONTOUR LEVELS AND SYMBOLS SYMBLEVEL RANGE (0) 25168E-02 (9) (9) 24567E-02 239G6E-02 (6) (6) 23365E-02 22764E-02 17) (7) 22164E-02 21563E-02 (6) (6) 20962E-02 20361E-02 (5) (5)
19760E-02 19159E-02 C4gt (4) 18558E-02 1795SE-02 (3) (3) 17357E-02 16756E-02 (2) (2) 16155E-02 15554E-02 (1) 14953E-02 14353E-02 (reg) 1375EE-02
ESTIMATION ERROR CRITERION CONSTRAINT =
15000E-01
5Q000E-0J1
^ 2 2 11111 111111 22222 2222222222222
Figure 630C Contour plot of [ p ^ z ^ at third sample time t K - 180 for o ^ = 0150 compare with Figure 628E
223
obtained by comparison of the contours in Figure 630 with those for the cases with a^ = 01 0125 and 05 in Figure 628 in the previshyous section
637 The Effect of Time-Varying Disturbance and Measurement Statistics upon the Optimal Monitoring Design and Management Problems Consider a problem with
_2 Ums0-
0125
005
(661A)
(661B)
0025 (661C)
and with PQ = M given in (657) Consider two cases F i r s t f i x the
measurement s ta t i s t i cs V to the values given above in (661C) but l e t
the disturbance s ta t i s t i cs vary For this case for the time interval
0 lt t lt 2 sample times occur at t K = 046 and 122 The time-varying
disturbance s ta t is t i cs between samples start ing with W in (661B) is
then given by
j W 0 lt t lt 046 W(t) = lt 05 W 046 lt t lt 122
025W 122 S t lt 20 (662)
The resultant plot of cC + N(zpoundz) as a function of time t K + N is shown in Figure 631 wrere the effects of variable W(t) in (662) are readily seen As W(t) decreases so does the rate at which the uncertainty in the estishymate of the maximum variance in the output grow Thus times between samples change greatly changing the nature of the management problem
i
Though the plots of [PudSt)] are omitted for brev i ty for reasons slnri-K K 11
la r to those in the example of Section 534 the contours change from
sample to sample affect ing nonconstant solutions to the design problem
10COOE-O1 L t 1 bull bull XX i gt t X I X [ X I X I X
X XX X XX XX X XX
laquo t X I X 1 X I X I X I X
X x x x
XX X X XX X X
XX xxx xxx xxx xxx xxx xxx xxx xxx I X I X I X i x I X
X X X X
X X
XX X
X X X
1 X
1 X
IX
X
X
x
X
X 1600E00
Figure 631 Time response of ^ + M ( Z | ( raquo Z ) for time-varying disturbance statistics W(t) given in (662)
225
Thus Conclusion VIC The solutions for the optimal
monitoring design and management problems may not in general be the same for all measurement times if the disturbance noise statistics are allowed to vary with time (CVIC)
Second fix the disturbance noise statistics W to the value given in (661B) but now let the measurement error statistics vary from sample to sample In this case the sample times occur at t = 046 080 112
138 162 180 and 194 over the interval 0 lt t lt 2 Starting with V given in (661C) for the first sample let the measurement statistics be given by
V(t) = lt
[ - t = 046
15 y t = 080
(1-5) 2 V t = 112
( i 5 ) 3 y t = 138
( i 5 ) 4 y t = 162
( i 5 ) 5 y t = 180
( i - 5 ) 6 y t = 194
(663)
The plot of c^+N(zjjIz) for V(t) is shown in Figure 632 Note that V(t) specified in (663) may be interpreted as taking consecutively worse and worse measurements from sample to sample Thus as the quality of the measurements decreases the uncertainties in the estimate of the maxishymum variance in the output increase leading to higher initial conditions for the branches of at after each measurement and resulting in shorter and shorter times between measurements This completes the countershyexamples for Conclusion VI which are summarized in
Conclusion VIP The solutions for the optimal deshysign and management problems may not in general be the same for all measurement times if the measurement error statistics at each sample are allowed to vary (CVID)