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wlyh/ ]U iru }hur dqg E/ P dqg V uhvshfwlyho| phdq elj/ phglxp dqg vpdoo1 Wkh uxoh edvh ri wkh SG0olnh VHFV xvhg lq wklv sdshu lv vkrzq lq Wdeoh 4> wkh fruuhvsrqglqj phpehuvkls ixqfwlrqv iru wkh huuru/ lwv ghulydwlyh dqg iru wkh frqwuroohu rxwsxw duh vkrzq lq Ilj1 6 dqg Ilj1 71 Reylrxvo|/ wkh uxoh edvh dqg wkh phpehuvkls ixqfwlrqv surylgh frpsohphqwdu| lqirupdwlrq= iru lqvwdqfh/ iurp Wdeoh 4 rqh pd| frqfoxgh wkdw li wkh huuru lv cQhjdwlyh Phglxp* dqg lwv ghulydwlyh lv c]hur*/ wkhq wkh frqwuroohu rxwsxw kdv wkh surshuw| ri ehlqj cSrvlwlyh Vpdoo*1 Krzhyhu/ wklv vwdwhphqw lv vwloo txdolwdwlyh dqg pxvw eh frpsohwhg zlwk wkh lqglfdwlrq ri wkh phpehuvkls ixqfwlrqv wkdw vshfli| wkh ghjuhh wr zklfk d sduwlfxodu txdqwlwdwlyh fulvs ydoxh ri wkh huuru ehorqjv wr wkh sk|vlfdo ydoxh ri huuru qhjdwlyh phglxp1 Wkh vdph frqfhsw dssolhv wr wkh phpehuvkls ixqfwlrq ri wkh wkh huuru ghulydwlyh ru ri wkh frqwuroohu rxwsxw1 D ihz frpphqwv wr Ilj1 6 dqg Ilj1 7 duh zruwk qrwlqj1 Zh qrwh uvw wkdw wkh lqsxw phpehuvkls ixqfwlrqv duh Jdxvvldq dqg wkh rxwsxw phpehuvkls ixqfwlrqv duh wul0 dqjxodu/ d vkdsh zklfk lv yhu| frpprq lq ix}}| frqwuro1 Lq frqwudvw zlwk wuldqjxodu phpehuvkls ixqfwlrqv/ lq wkh fdvh ri Jdxvvldq phpehuvkls ixqfwlrqv doo wkh +vhyhq,
:
;
]U
SV
SP
SE
SP
SV
]U
]U
SP
SV
]U
]U
QV
SV
]U
]U
QV
QP
SP
]U
]U
]U
QP
QE
SP
SV
]U
]U
QV
QP
QE
SP
SV
]U
]U
QV
QP
QE
QE
SE
]U
]U
QV
QP
QE
QE
QE
QE
SE
SE
SE
QP QV
SE
SE
SE
SP
]U
SE
SV
NB 1
NM
NS
ZR
PS
0
0.2
PM
PB
0.8
Degree of membership
e q e QE QP QV
0.6
0.4
0.2
Wdeoh 4= Uxoh edvh iru wkh ix}}| frqwuroohu1
0 NB 1
NM
NS
ZR
PS
PM
PB
-1
-0.6
-0.4
-0.2
δ
0.4
0.6
0.8
1
e
Iljxuh 7= Phpehuvkls ixqfwlrq ri wkh frqwuroohu rxwsxw1
0.8
Degree of membership
-0.8
0.6
lqsxwv wr lwv rxwsxw ghshqglqj rq wkh ydoxh ri wkh fhqwudo lqsxw/ zklfk lv fdoohg 0.4
wkh frqwuro lqsxw1 Iru lqvwdqfh/ irfxvlqj wkh dwwhqwlrq rq wkh xsshu vzlwfk ri wkh
0.2
lv juhdwhu wkdq d {hg ydoxh %1 Zkhq
h dv orqj dv mhm mhm lv ohvv wkdq %> wkh vzlwfk dup dvvxphv wkh orzhu srvlwlrq dqg wkh rxwsxw vljqdo ehfrphv htxdo wr h nR2 1 Wkh lqwurgxfwlrq vfdolqj eorfn/ wkh rxwsxw vljqdo wr wkh vzlwfk lv htxdo wr wkh huuru
ri wkh vzlwfk eorfnv lv lpsruwdqw wr lpsuryh wkh frqwuroohu shuirupdqfh qhdu wkh
0 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
e, de/dt
Iljxuh 6= Phpehuvkls ixqfwlrq ri huuru h dqg ri ghulydwlyh ri huuru h= b
vwhdg|0vwdwh/ l1h1 zkhq wkh ydoxhv ri
h
dqg hb ehfrph vx!flhqwo| vpdoo1 Iru wklv
uhdvrq wkh ydoxhv ri nR2 dqg n_2 pd| eh wxqhg wr rswlplvh wkh v|vwhp shuirupdqfh1 Wkh lqwurgxfwlrq ri wkh qrqolqhdu vdwxudwlrq eorfnv suhyhqwv wkh ix}}| frqwuroohu iurp ehlqj gulyhq e| vljqdov h{fhhglqj wkh qrupdolvhg udqjhv1 Wklv lv xvhixo pdlqo|
ix}}| vhwv duh ryhuodsslqj/ zklfk phdqv wkdw doo ix}}| uxohv duh uhg vlpxowdqhrxvo|1 Wklv lq wxuq ohdgv wr d vruw ri vprrwklqj h hfw rq wkh rxwsxw vljqdo1 Qrwh dovr wkdw wkh glylvlrq ri ix}}| vhwv lq Ilj1 6 dqg Ilj1 7 lv pdgh lq vxfk d zd| wkdw lq wkh fhq0 wudo uhjlrq +durxqg wkh }hur, ix}}| vhwv duh qduurzhu wkdq lq wkh shulskhudo uhjlrqv1
gxulqj wkh lqlwldo wudqvlhqw uhvsrqvh ri wkh v|vwhp/ zkhq wkh huuru pd| dvvxph kljk ydoxhv1 Ilqdoo|/ qrwh wkdw wkh jdlq n uhsuhvhqwv wkh ghqrupdolvdwlrq eorfn1 Lw pd| eh fkrvhq e| wkh ghvljqhu e| uhfdoolqj wkdw lwv ydoxh gh qhv wkh pd{lpxp doorzdeoh h{fxuvlrq ri wkh frqwuro vljqdo x1
Wklv hqdeohv d odujh lqsxw frqwuro wr eh hpsor|hg zkhq wkh huuru ru lwv ghulydwlyh lv idu iurp }hur dqg surgxfhv d vruw ri ghvdwxudwlrq qhdu wkh fruuhfw +}hur huuru, srvlwlrq1
9
D slwfk udwh frqwuro v|vwhp
Wkh duudqjhphqw ri wkh VHFV lv vkrzq lq Ilj1 81 Wkh vfdolqj eorfn lv frqvwlwxwhg
Dv d uvw h{dpsoh vkrzlqj wkh sursrvhg phwkrgrorj|/ zh frqvlghu wkh ghvljq ri d
e| irxu sdudphwhuv nR > nR2 > n_ dqg n_2 zklfk duh wkh vfdolqj idfwruv/ wzr vzlwfk eorfnv dqg wzr vdwxudwlrq eorfnv1 Wkh vdwxudwlrq eorfnv lpsrvh xsshu dqg orzhu
vlpsoh slwfk udwh frqwuro v|vwhp1 Wkh uhihuhqfh dlufudiw prgho lv frqvwlwxwhg e| wkh
erxqgv wr wkh rxwjrlqj vljqdov/ zkloh wkh vzlwfk eorfn sursdjdwhv rqh ri wkh wzr
lv wkh Odsodfh yduldeoh1 Wr vlpsoli| wkh glvfxvvlrq zh dvvxph wkdw wkh vkruw shulrg
wudqvihu ixqfwlrq t +v,@e +v, zkhuh
t lv wkh slwfk udwh/ e lv wkh hohydwru dqjoh dqg v
g|qdplfv rqo| duh wdnhq lqwr dffrxqw dqg wkdw wkh rshq0orrs wudqvihu ixqfwlrq ri
wkh vzlwfk dup dvvxphv wkh orzhu srvlwlrq dqg wkh rxwsxw vljqdo ehfrphv htxdo wr h nR2 1 Wkh lqwurgxfwlrq vfdolqj eorfn/ wkh rxwsxw vljqdo wr wkh vzlwfk lv htxdo wr wkh huuru
ri wkh vzlwfk eorfnv lv lpsruwdqw wr lpsuryh wkh frqwuroohu shuirupdqfh qhdu wkh
0 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
e, de/dt
Iljxuh 6= Phpehuvkls ixqfwlrq ri huuru h dqg ri ghulydwlyh ri huuru h= b
vwhdg|0vwdwh/ l1h1 zkhq wkh ydoxhv ri
h
dqg hb ehfrph vx!flhqwo| vpdoo1 Iru wklv
uhdvrq wkh ydoxhv ri nR2 dqg n_2 pd| eh wxqhg wr rswlplvh wkh v|vwhp shuirupdqfh1 Wkh lqwurgxfwlrq ri wkh qrqolqhdu vdwxudwlrq eorfnv suhyhqwv wkh ix}}| frqwuroohu iurp ehlqj gulyhq e| vljqdov h{fhhglqj wkh qrupdolvhg udqjhv1 Wklv lv xvhixo pdlqo|
ix}}| vhwv duh ryhuodsslqj/ zklfk phdqv wkdw doo ix}}| uxohv duh uhg vlpxowdqhrxvo|1 Wklv lq wxuq ohdgv wr d vruw ri vprrwklqj h hfw rq wkh rxwsxw vljqdo1 Qrwh dovr wkdw wkh glylvlrq ri ix}}| vhwv lq Ilj1 6 dqg Ilj1 7 lv pdgh lq vxfk d zd| wkdw lq wkh fhq0 wudo uhjlrq +durxqg wkh }hur, ix}}| vhwv duh qduurzhu wkdq lq wkh shulskhudo uhjlrqv1
gxulqj wkh lqlwldo wudqvlhqw uhvsrqvh ri wkh v|vwhp/ zkhq wkh huuru pd| dvvxph kljk ydoxhv1 Ilqdoo|/ qrwh wkdw wkh jdlq n uhsuhvhqwv wkh ghqrupdolvdwlrq eorfn1 Lw pd| eh fkrvhq e| wkh ghvljqhu e| uhfdoolqj wkdw lwv ydoxh gh qhv wkh pd{lpxp doorzdeoh h{fxuvlrq ri wkh frqwuro vljqdo x1
Wklv hqdeohv d odujh lqsxw frqwuro wr eh hpsor|hg zkhq wkh huuru ru lwv ghulydwlyh lv idu iurp }hur dqg surgxfhv d vruw ri ghvdwxudwlrq qhdu wkh fruuhfw +}hur huuru, srvlwlrq1
9
D slwfk udwh frqwuro v|vwhp
Wkh duudqjhphqw ri wkh VHFV lv vkrzq lq Ilj1 81 Wkh vfdolqj eorfn lv frqvwlwxwhg
Dv d uvw h{dpsoh vkrzlqj wkh sursrvhg phwkrgrorj|/ zh frqvlghu wkh ghvljq ri d
e| irxu sdudphwhuv nR > nR2 > n_ dqg n_2 zklfk duh wkh vfdolqj idfwruv/ wzr vzlwfk eorfnv dqg wzr vdwxudwlrq eorfnv1 Wkh vdwxudwlrq eorfnv lpsrvh xsshu dqg orzhu
vlpsoh slwfk udwh frqwuro v|vwhp1 Wkh uhihuhqfh dlufudiw prgho lv frqvwlwxwhg e| wkh
erxqgv wr wkh rxwjrlqj vljqdov/ zkloh wkh vzlwfk eorfn sursdjdwhv rqh ri wkh wzr
lv wkh Odsodfh yduldeoh1 Wr vlpsoli| wkh glvfxvvlrq zh dvvxph wkdw wkh vkruw shulrg
wudqvihu ixqfwlrq t +v,@e +v, zkhuh
t lv wkh slwfk udwh/ e lv wkh hohydwru dqjoh dqg v
g|qdplfv rqo| duh wdnhq lqwr dffrxqw dqg wkdw wkh rshq0orrs wudqvihu ixqfwlrq ri
Ww @ 4v> @ 3=: dqg $ @ 5udg v3 1 Qrwh wkdw dqg $ uhvshfwlyho| duh wkh vkruw shulrg gdpslqj dqg iuhtxhqf|1 Dq lqwhjudo frqwuroohu kdv ehhq ghvljqhg wr jxdudqwhh }hur vwhdg|0vwdwh huuru ri wkh uhihuhqfh prgho zkhq gulyhq e| d vwhs uhihuhqfh vljqdo1 Wkh jdlq ri wkh frqwuroohu kdv ehhq fkrvhq wr surylgh d skdvh pdujlq ri 63 wr wkh v|vwhp1 Vxssrvh qrz wkdw wkh uhdo sodqw gl huv iurp wkh uhihuhqfh prgho= 4, wkh vkruw shulrg gdpslqj lv vpdoohu + @ 3=8, dqg 5, wkhuh duh xqprghohg kljk iuhtxhqf| g|qdplfv zklfk lv uhsuhvhqwhg e| wkh wudqvihu ixqfwlrq 43@+v . 43,1 Dv suhylrxvo| vwdwhg/ wkh vdph frqwuroohu xvhg iru wkh uhihuhqfh prgho lv dssolhg wr wkh uhdo sodqw1 Lw lv lqwhuhvwlqj wr qrwh wkdw/ zlwk wkh deryh frqwuroohu/ wkh lqqhu orrs ri wkh uhdo sodqw wxuqv rxw wr eh xqvwdeoh1 Wkh huuru vljqdo/ jhqhudwhg dv wkh gl huhqfh ehwzhhq wkh rxwsxw ri wkh uhdo sodqw dqg wkh rxwsxw ri wkh uhihuhqfh prgho lv xvhg wr gulyh wkh VHFV zklfk/ lq wxuq/ surylghv dq dgglwlrqdo lqsxw vljqdo
x wr wkh uhdo sodqw dv vkrzq lq Ilj1 91
Wkh vwuxfwxuh ri wkh VHFV kdv ehhq suhylrxvo| ghvfulehg1 Wkh irxu lqsxw vfdolqj idfwruv nR > nR2 > n_ dqg n_2 kdyh ehhq rewdlqhg e| plqlplvlqj/ e| phdqv ri jhqhwlf U| dojrulwkpv/ wkh remhfwlyh ixqfwlrq M @ f r6 h2 gw zkhuh wr6 lv wkh vlpxodwlrq wlph1
44
6
time (s)
Iljxuh := Wlph uhvsrqvhv ri wkh frqwuroohg v|vwhp zlwk d uhihuhqfh xqlw vwhs lqsxw Wkh uhvxowv duh vkrzq lq Ilj1 := wkh ehkdylrxu ri wkh frqwuroohu lv h{fhoohqw1 Qrwh wkdw wkh slwfk udwh ri wkh uhdo v|vwhp lv doprvw lghqwlfdo wr wkdw ri wkh prgho/ wkh huuru h ehlqj vpdoohu wkdq 3=84332 ghj @v= Lw lv lqwhuhvwlqj wr vkrz krz wkh frqwuroohu dfwv zkhq wkh sodqw prgho lv gl huhqw iurp lwv qrplqdo ydoxh1 Wr ghprqvwudwh wklv/ zh kdyh vxssrvhg wkdw wkh vkruw shulrg gdpslqj lv rqo| 3=4 + yh wlphv vpdoohu wkdq wkh qrplqdo ydoxh,1 Wkh uhvxowv duh vxppdulvhg lq Ilj1 ;1 D qrwdeoh ihdwxuh lv wkdw wkh huuru uhpdlqv doprvw lghqwlfdo wr wkh rwkhu fdvh= wklv lv pdgh srvvleoh iurp d khdylhu hpsor|phqw ri wkh ix}}| frqwuroohg vljqdo x1
:
D qrupdo dffhohudwlrq FDV
Zh qrz wxuq wkh dwwhqwlrq wr d pxfk pruh lqyroyhg h{dpsoh/ qdpho| wkh ghvljq ri d qrupdo dffhohudwlrq Frqwuro Dxjphqwdwlrq V|vwhp +FDV,1 Wkh jhqhudo duudqjhphqw lv vkrzq lq Ilj1 < zkhuh wkh vxshuvfulsw +p, ghqrwhv yduldeohv zklfk ehorqj wr wkh uhihuhqfh prgho1 Dv xvxdo/ wkh lqqhu orrs iru wkh uhihuhqfh prgho lv ghvljqhg uvw1 Lw lv edvhg rq d olqhdu frqwuroohu= wkh qrupdo dffhohudwlrq
q5
lv phdvxuhg dqg lv
sdvvhg wkurxjk dq lqwhjudo frqwuroohu> d slwfk gdpshu lv xvhg wr lqfuhdvh +e| phdqv ri wkh slwfk udwh t , wkh vkruw shulrg gdpslqj ri wkh v|vwhp dqg d wklug vljqdo lv
45
G GW
N S
saturation switch
Fuzzy controller
N
pitch rate ( deg/s)
N S
H
X
NG
H
1.4
2
1.2
1
1 0.8 0.6 0.4
0
-3
-1 -2 -3
0.2
NG
x 10
0
e (deg/s)
H
-4 0
2
4
6
8
-5
10
0
2
4
time (s)
6
8
10
8
10
time (s)
Scaling
U + -
N V
0.4
0.2
0.3
0.1
c
X
0.3
u (deg)
u (deg)
Iljxuh 8= Duudqjhphqw ri wkh VHFV1
0.5
X +
T V δ V
F
+
T
0.2 0.1
0 -0.1
H
0
0
2
4
6
8
-0.2
10
0
2
4
time (s)
Iljxuh 9= Uhdo sodqw= lqqhu0orrs frqwuro v|vwhp zlwk wkh lqsxw x ri wkh VHFV wkh uhihuhqfh prgho lv jlyhq e|
PB +v . 4@Ww , t+v, @ e +v, v2 . 5$v . $2 zlwk PB @ 43udg3 v32 > Ww @ 4v> @ 3=: dqg $ @ 5udg v3 1 Qrwh wkdw dqg $ uhvshfwlyho| duh wkh vkruw shulrg gdpslqj dqg iuhtxhqf|1 Dq lqwhjudo frqwuroohu kdv ehhq ghvljqhg wr jxdudqwhh }hur vwhdg|0vwdwh huuru ri wkh uhihuhqfh prgho zkhq gulyhq e| d vwhs uhihuhqfh vljqdo1 Wkh jdlq ri wkh frqwuroohu kdv ehhq fkrvhq wr surylgh d skdvh pdujlq ri 63 wr wkh v|vwhp1 Vxssrvh qrz wkdw wkh uhdo sodqw gl huv iurp wkh uhihuhqfh prgho= 4, wkh vkruw shulrg gdpslqj lv vpdoohu + @ 3=8, dqg 5, wkhuh duh xqprghohg kljk iuhtxhqf| g|qdplfv zklfk lv uhsuhvhqwhg e| wkh wudqvihu ixqfwlrq 43@+v . 43,1 Dv suhylrxvo| vwdwhg/ wkh vdph frqwuroohu xvhg iru wkh uhihuhqfh prgho lv dssolhg wr wkh uhdo sodqw1 Lw lv lqwhuhvwlqj wr qrwh wkdw/ zlwk wkh deryh frqwuroohu/ wkh lqqhu orrs ri wkh uhdo sodqw wxuqv rxw wr eh xqvwdeoh1 Wkh huuru vljqdo/ jhqhudwhg dv wkh gl huhqfh ehwzhhq wkh rxwsxw ri wkh uhdo sodqw dqg wkh rxwsxw ri wkh uhihuhqfh prgho lv xvhg wr gulyh wkh VHFV zklfk/ lq wxuq/ surylghv dq dgglwlrqdo lqsxw vljqdo
x wr wkh uhdo sodqw dv vkrzq lq Ilj1 91
Wkh vwuxfwxuh ri wkh VHFV kdv ehhq suhylrxvo| ghvfulehg1 Wkh irxu lqsxw vfdolqj idfwruv nR > nR2 > n_ dqg n_2 kdyh ehhq rewdlqhg e| plqlplvlqj/ e| phdqv ri jhqhwlf U| dojrulwkpv/ wkh remhfwlyh ixqfwlrq M @ f r6 h2 gw zkhuh wr6 lv wkh vlpxodwlrq wlph1
44
6
time (s)
Iljxuh := Wlph uhvsrqvhv ri wkh frqwuroohg v|vwhp zlwk d uhihuhqfh xqlw vwhs lqsxw Wkh uhvxowv duh vkrzq lq Ilj1 := wkh ehkdylrxu ri wkh frqwuroohu lv h{fhoohqw1 Qrwh wkdw wkh slwfk udwh ri wkh uhdo v|vwhp lv doprvw lghqwlfdo wr wkdw ri wkh prgho/ wkh huuru h ehlqj vpdoohu wkdq 3=84332 ghj @v= Lw lv lqwhuhvwlqj wr vkrz krz wkh frqwuroohu dfwv zkhq wkh sodqw prgho lv gl huhqw iurp lwv qrplqdo ydoxh1 Wr ghprqvwudwh wklv/ zh kdyh vxssrvhg wkdw wkh vkruw shulrg gdpslqj lv rqo| 3=4 + yh wlphv vpdoohu wkdq wkh qrplqdo ydoxh,1 Wkh uhvxowv duh vxppdulvhg lq Ilj1 ;1 D qrwdeoh ihdwxuh lv wkdw wkh huuru uhpdlqv doprvw lghqwlfdo wr wkh rwkhu fdvh= wklv lv pdgh srvvleoh iurp d khdylhu hpsor|phqw ri wkh ix}}| frqwuroohg vljqdo x1
:
D qrupdo dffhohudwlrq FDV
Zh qrz wxuq wkh dwwhqwlrq wr d pxfk pruh lqyroyhg h{dpsoh/ qdpho| wkh ghvljq ri d qrupdo dffhohudwlrq Frqwuro Dxjphqwdwlrq V|vwhp +FDV,1 Wkh jhqhudo duudqjhphqw lv vkrzq lq Ilj1 < zkhuh wkh vxshuvfulsw +p, ghqrwhv yduldeohv zklfk ehorqj wr wkh uhihuhqfh prgho1 Dv xvxdo/ wkh lqqhu orrs iru wkh uhihuhqfh prgho lv ghvljqhg uvw1 Lw lv edvhg rq d olqhdu frqwuroohu= wkh qrupdo dffhohudwlrq
q5
lv phdvxuhg dqg lv
sdvvhg wkurxjk dq lqwhjudo frqwuroohu> d slwfk gdpshu lv xvhg wr lqfuhdvh +e| phdqv ri wkh slwfk udwh t , wkh vkruw shulrg gdpslqj ri wkh v|vwhp dqg d wklug vljqdo lv
45
kα r + −
kI s
10 s + 10
− +
uc Actuator
−
Aircraft model
α (m) nz(m) q(m)
kq
1.4
5
x 10
-3
Iljxuh nU dqg n^ kdyh ehhq fkrvhq zlwk vwdqgdug whfkqltxhv1 Wkh lqqhu orrs kdv wkh iroorzlqj fkdudfwhulvwlfv zlwk uhvshfw wr d xqlw vwhs uhihuhqfh
0
-0.2 0.1
prgho wdnhq iurp Vwhyhqv^46`1 Wkh uhihuhqfh +wulp, frqglwlrq lv fkdudfwhulvhg e| d
+uhihuuhg wr d vwhdg| vwdwh huuru ohvv wkdq 5(,1 Zlwk vxfk d prgho dw kdqg/ wkh 0
2
4
6
8
10
time (s)
Iljxuh ;= Wlph uhvsrqvhv ri wkh frqwuroohg v|vwhp zlwk d uhihuhqfh xqlw vwhs lqsxw1 Wkh qrp0 lqdo ydoxh ri wkh vkruw shulrg gdpslqj kdv ehhq uhgxfhg wr
@ 3=4=
zkroh frqwuro v|vwhp lv duudqjhg dv vkrzq lq Ilj1 431 Wkh qrqolqhdu sodqw uhsuhvhqwv d ixoo qrqolqhdu I49 prgho1 Wkh vdph lqqhu0orrs frqwuro v|vwhp lv dssolhg wr wkh uhdo +qrqolqhdu, sodqw1 Lq wklv fdvh/ wkh dfwxdwru ri wkh hohydwru lv prghoohg zlwk d qrqolqhdu prgho frqvwlwxwhg e| d wudqvihu ixqfwlrq htxdo wr J +v, +dv iru wkh uhihuhqfh prgho, soxv d udwh olplwhu zklfk olplwv wkh ghulydwlyh ri wkh vljqdo sdvvlqj wkurxjk lw1 Lw frqvwudlqv wkh hohydwru yhorflw| lq vxfk d zd| wkdw be ? 93 ghj @v1 Qrwh wkdw wkuhh huuru vljqdov duh qrz jhqhudwhg/ htxdo wr wkh gl huhqfhv ehwzhhq wkh uhdo dqg prgho ydoxhg vljqdov jlyhq e| wkh dqjoh ri dwwdfn/ wkh slwfk udwh dqg
6
wkh ordg idfwru1 Lq sduwlfxodu/ zh gh qh h @ q5 q5 1 Lw vkrxog eh qrwhg wkdw/ lq {
E
wklv fdvh/ wkh zkroh VHFV lv frpsrvhg ri wkuhh gl huhqw ix}}| frqwuroohuv/ hdfk rqh ehlqj lq wkh irup ghvfulehg lq ¢81 Wkh vfdodu rxwsxwv ri wkh wkuhh ix}}| frqwuroohuv duh wkhq vxpphg dqg gh qh wkh ix}}| frqwuro vljqdo x zklfk lv dgghg dv dqrwkhu frqwuro lqsxw iru wkh uhdo sodqw1
46
47
kα r + −
kI s
10 s + 10
− +
uc Actuator
−
Aircraft model
α (m) nz(m) q(m)
kq
1.4
5
x 10
-3
Iljxuh nU dqg n^ kdyh ehhq fkrvhq zlwk vwdqgdug whfkqltxhv1 Wkh lqqhu orrs kdv wkh iroorzlqj fkdudfwhulvwlfv zlwk uhvshfw wr d xqlw vwhs uhihuhqfh
0
-0.2 0.1
prgho wdnhq iurp Vwhyhqv^46`1 Wkh uhihuhqfh +wulp, frqglwlrq lv fkdudfwhulvhg e| d
+uhihuuhg wr d vwhdg| vwdwh huuru ohvv wkdq 5(,1 Zlwk vxfk d prgho dw kdqg/ wkh 0
2
4
6
8
10
time (s)
Iljxuh ;= Wlph uhvsrqvhv ri wkh frqwuroohg v|vwhp zlwk d uhihuhqfh xqlw vwhs lqsxw1 Wkh qrp0 lqdo ydoxh ri wkh vkruw shulrg gdpslqj kdv ehhq uhgxfhg wr
@ 3=4=
zkroh frqwuro v|vwhp lv duudqjhg dv vkrzq lq Ilj1 431 Wkh qrqolqhdu sodqw uhsuhvhqwv d ixoo qrqolqhdu I49 prgho1 Wkh vdph lqqhu0orrs frqwuro v|vwhp lv dssolhg wr wkh uhdo +qrqolqhdu, sodqw1 Lq wklv fdvh/ wkh dfwxdwru ri wkh hohydwru lv prghoohg zlwk d qrqolqhdu prgho frqvwlwxwhg e| d wudqvihu ixqfwlrq htxdo wr J +v, +dv iru wkh uhihuhqfh prgho, soxv d udwh olplwhu zklfk olplwv wkh ghulydwlyh ri wkh vljqdo sdvvlqj wkurxjk lw1 Lw frqvwudlqv wkh hohydwru yhorflw| lq vxfk d zd| wkdw be ? 93 ghj @v1 Qrwh wkdw wkuhh huuru vljqdov duh qrz jhqhudwhg/ htxdo wr wkh gl huhqfhv ehwzhhq wkh uhdo dqg prgho ydoxhg vljqdov jlyhq e| wkh dqjoh ri dwwdfn/ wkh slwfk udwh dqg
6
wkh ordg idfwru1 Lq sduwlfxodu/ zh gh qh h @ q5 q5 1 Lw vkrxog eh qrwhg wkdw/ lq {
E
wklv fdvh/ wkh zkroh VHFV lv frpsrvhg ri wkuhh gl huhqw ix}}| frqwuroohuv/ hdfk rqh ehlqj lq wkh irup ghvfulehg lq ¢81 Wkh vfdodu rxwsxwv ri wkh wkuhh ix}}| frqwuroohuv duh wkhq vxpphg dqg gh qh wkh ix}}| frqwuro vljqdo x zklfk lv dgghg dv dqrwkhu frqwuro lqsxw iru wkh uhdo sodqw1
46
47
q
nz(m)
(m )
−
u
Fuzzy controller
+
α Real plant
e
nz
−
1
+
+
0.8
u z
+
Reference model
Fuzzy controller
+
load factor, n
r
1.2
−
α (m)
Fuzzy controller
q +
0.6
0.4
0.2
0
SECS -0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time (s)
Iljxuh 43= Duudqjhphqw ri wkh zkroh qrupdo dffhohudwlrq FDV zlwk ix}}| frqwuroohu1
Iljxuh 44= Frpsdulvrq ehwzhhq wkh wlph uhvsrqvhv ri wkh qrqolqhdu prgho zlwk uhvshfw wr d Gxh wr wkh suhvhqfh ri wkuhh ix}}| frqwuroohuv/ lq wklv fdvh wkh wxqlqj surfhvv ri wkh VHFV frpsulvhv wzhoyh lqsxw vfdolqj idfwruv/ irxu iru hdfk frqwuroohu1 Dv lq wkh suhylrxv h{dpsoh/ wkhlu ydoxhv kdyh ehhq rewdlqhg e| plqlplvlqj/ e| phdqv ri jh0 qhwlf dojrulwkpv/ wkh remhfwlyh ixqfwlrq M @
U
xqlw vwhs uhihuhqfh lqsxw1 Wkh vrolg olqh fruuhvsrqgv wr wkh lghdo uhvsrqvh/ wkh rwkhu wzr fdvhv uhsuhvhqw wkh v|vwhp zlwk wkh ix}}| frqwuroohu +olqh zlwk vwduv, dqg zlwkrxw lw +olqh zlwk flufohv,1
|r6 h2 gw/ zkhuh w lv wkh vlpxodwlrq r6
f
wlph1 Wkh uhvxowv duh vkrzq lq Ilj1 44 dqg lq Ilj1 451 Wkh ehwwhu ehkdylrxu ri wkh
0.15 0.1
ri wkh v|vwhp lq dq r 0qrplqdo vlwxdwlrq zh frqvlghu wkh vdph uhihuhqfh ljkw frq0 glwlrq iru wkh dlufudiw/ exw zlwk d gl huhqw srvlwlrq ri wkh fhqwuh ri judylw| +f1j1,1
e
ix}}| frqwuroohg v|vwhp lv hylghqw1 Krzhyhu/ lq rughu wr hpskdvlvh wkh ehkdylrxu
0.05 0 -0.05
Zh zloo dvvxph wkdw wkh f1j1 srvlwlrq lq wklv qhz frq jxudwlrq lv pryhg diw ri d
3
txdqwlw| htxdo wr 3=38 f/ zkhuh f lv wkh phdq dhurg|qdplf fkrug ohqjwk1 Lq sdvvlqj
2
lv qrw xvhg/ lv vkrzq lq Ilj1 46 dqg Ilj1 471 Wkhvh jxuhv fohduo| ghprqvwudwh wkh
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
time ( s) 2
u (deg)
vxshulrulw| ri wkh ix}}| frqwuro v|vwhp ryhu d fodvvlfdo rqh1
0 -2 -4
D jhqhudo phwkrgrorj| kdv ehhq gh qhg wr ghvljq frpsoh{ dlufudiw frqwuro v|vwhpv1
1.5
c
0 -1
Frqfoxvlrqv
1
1
ehkdylrxu ri wkh frqwuroohg v|vwhp/ dv frpsduhg wr wkh fdvh zkhuh wkh ix}}| frqwuro
;
0.5
time ( s)
u (deg)
zh revhuyh wkdw/ zlwk wkh qhz f1j1 srvlwlrq/ wkh dlufudiw lv rshq0orrs xqvwdeoh1 Wkh
0
0
0.5
1
1.5
2
2.5
time ( s)
Wkh sursrvhg dssurdfk frpelqhv d fodvvlfdo +lqqhu orrs, ghvljq edvhg rq d olqhdu
Iljxuh 45= Frpsdulvrq ehwzhhq wkh wlph uhvsrqvhv ri wkh qrqolqhdu prgho zlwk uhvshfw wr
frqwuro v|vwhp sklorvrsk|/ dqg dq rxwhu orrs zklfk lv vxshuylvhg e| d ix}}| orjlf
d xqlw vwhs uhihuhqfh lqsxw1 Wkh wzr fdvhv uhsuhvhqw wkh v|vwhp zlwk wkh ix}}|
qrqolqhdu frqwuroohu1 Vhyhudo uhdvrqv prwlydwh wkh fkrvhq dufklwhfwxuh= 4, wkh lqqhu
frqwuroohu +wklq olqh, dqg zlwkrxw lw +wklfn olqh,1
orrs lv hdvlo| rswlplvhg e| iroorzlqj wkh zhoo0hvwdeolvkhg jxlgholqhv iru wkh ghvljq ri
48
49
q
nz(m)
(m )
−
u
Fuzzy controller
+
α Real plant
e
nz
−
1
+
+
0.8
u z
+
Reference model
Fuzzy controller
+
load factor, n
r
1.2
−
α (m)
Fuzzy controller
q +
0.6
0.4
0.2
0
SECS -0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time (s)
Iljxuh 43= Duudqjhphqw ri wkh zkroh qrupdo dffhohudwlrq FDV zlwk ix}}| frqwuroohu1
Iljxuh 44= Frpsdulvrq ehwzhhq wkh wlph uhvsrqvhv ri wkh qrqolqhdu prgho zlwk uhvshfw wr d Gxh wr wkh suhvhqfh ri wkuhh ix}}| frqwuroohuv/ lq wklv fdvh wkh wxqlqj surfhvv ri wkh VHFV frpsulvhv wzhoyh lqsxw vfdolqj idfwruv/ irxu iru hdfk frqwuroohu1 Dv lq wkh suhylrxv h{dpsoh/ wkhlu ydoxhv kdyh ehhq rewdlqhg e| plqlplvlqj/ e| phdqv ri jh0 qhwlf dojrulwkpv/ wkh remhfwlyh ixqfwlrq M @
U
xqlw vwhs uhihuhqfh lqsxw1 Wkh vrolg olqh fruuhvsrqgv wr wkh lghdo uhvsrqvh/ wkh rwkhu wzr fdvhv uhsuhvhqw wkh v|vwhp zlwk wkh ix}}| frqwuroohu +olqh zlwk vwduv, dqg zlwkrxw lw +olqh zlwk flufohv,1
|r6 h2 gw/ zkhuh w lv wkh vlpxodwlrq r6
f
wlph1 Wkh uhvxowv duh vkrzq lq Ilj1 44 dqg lq Ilj1 451 Wkh ehwwhu ehkdylrxu ri wkh
0.15 0.1
ri wkh v|vwhp lq dq r 0qrplqdo vlwxdwlrq zh frqvlghu wkh vdph uhihuhqfh ljkw frq0 glwlrq iru wkh dlufudiw/ exw zlwk d gl huhqw srvlwlrq ri wkh fhqwuh ri judylw| +f1j1,1
e
ix}}| frqwuroohg v|vwhp lv hylghqw1 Krzhyhu/ lq rughu wr hpskdvlvh wkh ehkdylrxu
0.05 0 -0.05
Zh zloo dvvxph wkdw wkh f1j1 srvlwlrq lq wklv qhz frq jxudwlrq lv pryhg diw ri d
3
txdqwlw| htxdo wr 3=38 f/ zkhuh f lv wkh phdq dhurg|qdplf fkrug ohqjwk1 Lq sdvvlqj
2
lv qrw xvhg/ lv vkrzq lq Ilj1 46 dqg Ilj1 471 Wkhvh jxuhv fohduo| ghprqvwudwh wkh
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
time ( s) 2
u (deg)
vxshulrulw| ri wkh ix}}| frqwuro v|vwhp ryhu d fodvvlfdo rqh1
0 -2 -4
D jhqhudo phwkrgrorj| kdv ehhq gh qhg wr ghvljq frpsoh{ dlufudiw frqwuro v|vwhpv1
1.5
c
0 -1
Frqfoxvlrqv
1
1
ehkdylrxu ri wkh frqwuroohg v|vwhp/ dv frpsduhg wr wkh fdvh zkhuh wkh ix}}| frqwuro
;
0.5
time ( s)
u (deg)
zh revhuyh wkdw/ zlwk wkh qhz f1j1 srvlwlrq/ wkh dlufudiw lv rshq0orrs xqvwdeoh1 Wkh
0
0
0.5
1
1.5
2
2.5
time ( s)
Wkh sursrvhg dssurdfk frpelqhv d fodvvlfdo +lqqhu orrs, ghvljq edvhg rq d olqhdu
Iljxuh 45= Frpsdulvrq ehwzhhq wkh wlph uhvsrqvhv ri wkh qrqolqhdu prgho zlwk uhvshfw wr
frqwuro v|vwhp sklorvrsk|/ dqg dq rxwhu orrs zklfk lv vxshuylvhg e| d ix}}| orjlf
d xqlw vwhs uhihuhqfh lqsxw1 Wkh wzr fdvhv uhsuhvhqw wkh v|vwhp zlwk wkh ix}}|
qrqolqhdu frqwuroohu1 Vhyhudo uhdvrqv prwlydwh wkh fkrvhq dufklwhfwxuh= 4, wkh lqqhu
frqwuroohu +wklq olqh, dqg zlwkrxw lw +wklfn olqh,1
orrs lv hdvlo| rswlplvhg e| iroorzlqj wkh zhoo0hvwdeolvkhg jxlgholqhv iru wkh ghvljq ri
48
49
fodvvlfdo dlufudiw ljkw frqwuro v|vwhpv> 5, wkh rxwhu orrs juhdwo| lpsuryhv wkh zkroh
1.4
v|vwhp shuirupdqfh/ lqfuhdvhv lwv urexvwqhvv dqg h{whqgv wkh rshudwlyh ehkdylrxu> 1.2
6, wkh zkroh surfhvv lv qhdu dxwrpdwlf/ gxh wr wkh {hg vwuxfwxuh ri wkh ix}}| frq0 wuroohu1 Edvlfdoo|/ wkh frqwuroohu ghvljq lv wudqvodwhg lqwr dq rswlplvdwlrq surfhgxuh
1
zkhuh d vhw ri vfdolqj idfwruv qhhg wr eh vxlwdeo| wxqhg wr gh qh wkh ehkdylrxu ri 0.8
load factor, n
z
wkh ix}}| frqwuroohu1 Wklv odwwhu sureohp lv/ lq wxuq/ hdvlo| vroyhg e| phdqv ri d jhqhwlf dojrulwkp dssurdfk1 Wkh zkroh phwkrgrorj| lv vlpsoh wr kdqgoh dqg pd|
0.6
eh h hfwlyho| xvhg wr jlyh txlfn dqg h!flhqw uhvsrqvhv wr wkh ghvljqhu1 Wzr gh0 0.4
wdlohg fdvh vwxglhv kdyh ehhq suhvhqwhg wr ydolgdwh wkh sursrvhg dssurdfk1 Ixuwkhu vlpxodwlrqv kdyh ehhq frqgxfwhg e| wkh dxwkru wr lqyhvwljdwh wkh frqwuroohg v|vwhp
0.2
ehkdylrxu lq suhvhqfh ri glvwxuedqfhv +olnh zlqg jxvwv,/ vhqvru qrlvh dqg uhihuhqfh 0
-0.2
lqsxwv wkdw gl hu iurp wkrvh hpsor|hg lq wkh ghvfulehg h{dpsohv1 Wkh rewdlqhg 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
uhvxowv/ dowkrxjk qrw suhvhqwhg khuh iru vsdfh olplwdwlrqv/ frq up wkh h hfwlyhqhvv ri wkh ghvljq phwkrgrorj|1 D qdo uhpdun kdv wr gr zlwk wkh ljkw fhuwl fdwlrq
time (s)
Iljxuh 46= Frpsdulvrq ehwzhhq wkh wlph uhvsrqvhv ri wkh qrqolqhdu prgho zlwk r 0qrplqdo frqglwlrqv1 Wkh vrolg olqh fruuhvsrqgv wr wkh lghdo uhvsrqvh/ wkh rwkhu wzr fdvhv uhsuhvhqw wkh v|vwhp zlwk wkh ix}}| frqwuroohu +olqh zlwk vwduv, dqg zlwkrxw lw +olqh zlwk flufohv,1
dvshfwv ri wkh suhvhqwhg phwkrgrorj|/ zklfk duh uhodwhg wr wkh vwdelolw| ri wkh zkroh ix}}| frqwuro v|vwhp1 Wklv nlqg ri sureohp lv qrw vlpsoh wr kdqgoh/ vlqfh ix}}| frqwuro v|vwhpv duh hvvhqwldoo| qrqolqhdu v|vwhpv1 Dfwxdoo|/ vrph forvhg0irup whfkqltxhv kdyh ehhq sursrvhg wr lqyhvwljdwh wkh vwdelolw| ri wkh ryhudoo vfkhph^
