1231833

Modelado, control y navegacion para el vuelo autonomo
de dirigibles
Leonardo Enrique Solaque Guzman
To cite this version:
Leonardo Enrique Solaque Guzman. Modelado, control y navegacion para el vuelo autonomo de
dirigibles. Automatic. INSA de Toulouse, 2007. Español. �tel-00136325�
HAL Id: tel-00136325
https://tel.archives-ouvertes.fr/tel-00136325
Submitted on 13 Mar 2007
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INSA de Toulouse
No. d’ordre 866
Universidad
de los Andes
École Doctorale Systèmes
Modelado, control y navegación para el
vuelo autónomo de dirigibles
TESIS
presentada y sustentada publicamente el 30 de Enero 2007
para la obtención del
Doctorado de la Universidad de los Andes y
l’Institut National des Sciences Appliquées de Toulouse
(especialidad sistemas automáticos)
por
Leonardo Enrique Solaque Guzmán
Composición del jurado
Revisores :
Anibal Ollero
José Raul Azinheira
Profesor de la Universidad de Sevilla-España
Profesor del I.S.T. de Lisboa-Portugal
Examinadores :
Simon Lacroix
Mauricio Duque Escobar
Director de investigación, LAAS-CNRS
Profesor asociado Universidad de los Andes
Alain Gauthier
Michel Devy
Profesor titular Universidad de los Andes
Director de investigación, LAAS-CNRS
Laboratoire d’Analyse et d’Architecture des Systèmes
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# $ '- !5 dIC
dt
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+ Ω ∧ Pd|Rd = Fext
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+ Ω ∧ IC + V
dPd|Rd
dt
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~˙ C|Rd + Ω
~ ∧ mV
~C − Ω
~ ∧ mS(rc )Ω
~ = P F~ext
mV
P
~˙ C|Rd + mS(rc )V
~˙ C|Rd + Ω
~ ∧ IC Ω
~ +Ω
~ ∧ mS(rc )V
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~ ∧ mS(rc )Ω
~ = M
~ ext
IC Ω
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mV
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IC Ω
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p
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2
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K2 = −0.061 − 0.4132δe + 0.6899δe2 − 1.276δe3
K3 = 0.1069 − 0.0087δe + 0.0932δe2
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K1
2
p
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K1 = −0.0553 + 0.0129δg + 0.0488δg2
K2 = −0.061 + 0.4132δg + 0.6899δg2 + 1.276δg3
K3 = 0.1069 + 0.0087δg + 0.0932δg2
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1
ρSref Lref
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2
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p
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β sin2 α cos2 β + sin2 β + 0.173 cos α
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100
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Time [s]
Angular speed in R
0
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100
200
Time [s]
Linear position in Ro
0
−500
100
200
Time [s]
Angular position in R
0
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100
200
Time [s]
Angular position in R
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200
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200
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Angular position in R
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100
200
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Linear position in Ro
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φ [rad]
0
z [m]
0
−0.2
d
ψ [rad/s]
0
100
200
Time [s]
Angular speed in R
0.5
500
y [m]
x [m]
0
−0.5
100
200
Time [s]
Linear position in Ro
200
−200
0
d
0
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100
200
Time [s]
Angular speed in R
0.5
θ [rad/s]
φ [rad/s]
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u [m/s]
v [m/s]
0
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Linear speed in Rd
0.2
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Ø
Linear speed in Rd
10
−10
Ø
0
100
Time [s]
0
−5
200
0
100
Time [s]
200
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Linear speed in R
o
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y
x
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V [m/s]
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5
0
−10
100
200
Time [s]
Linear position in R
0
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0
−400
100
200
Time [s]
Angular position in Ro
0
100
Time [s]
!
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0.5
0
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200
0
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100
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0
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200
0
Linear speed in Rd
0
w [m/s]
v [m/s]
4.6
0.08
0.06
98
100
102
Time [s]
Angular speed in R
d
−0.1
98
100
102
Time [s]
Angular speed in R
0
−0.01
−0.02
98
100
102
Time [s]
Linear position in Ro
−0.115
−0.12
−0.125
98
170
160
98
−49
−270
−280
98
100
102
Time [s]
Angular position in R
100
102
Time [s]
Angular position in R
o
−49.5
98
100
102
Time [s]
Angular position in R
o
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0.08
98
!
100
Time [s]
Ø o‘
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102
−2
ψ [rad]
0.05
θ [rad]
0.12
100
102
Time [s]
Linear position in Ro
z [m]
−260
y [m]
180
100
102
Time [s]
Linear position in Ro
100
102
Time [s]
Angular speed in R
d
ψ [rad/s]
0.005
−0.05
d
θ [rad/s]
0.01
200
Linear speed in Rd
0.1
0
98
100
Time [s]
" $ &-8:9!/4+,8<;>=?143$,/>5 8<1 /G$ 143,5 $ @E>( " '*)
4.8
4.4
98
100
200
Time [s]
Angular position in Ro
5
Linear speed in Rd
u [m/s]
−60
100
200
Time [s]
Angular position in Ro
ψ [rad]
θ [rad]
φ [rad]
0
−40
1
0
−0.5
z [m]
0
y [m]
x [m]
100
−200
100
200
Time [s]
Linear position in R
o
0
0
0
o
200
0.5
φ [rad/s]
−5
200
−100
x [m]
0
100
200
Time [s]
Linear position in R
o
φ [rad]
o
10
0
−10
ٝ
Linear speed in R
o
10
0
−0.05
98
100
Time [s]
102
−2.5
−3
98
100
Time [s]
+'37(0+'/C9 8<3,=&E2/ 56/237$ &-8:9!/4+,8<;>=?143 3 $ &/41 EG$ @/>(0/($ " '*)
102
f›2+ 3+,#+, $1-9b+ ( b (
Linear speed in R
Linear speed in R
o
V [m/s]
−2
−4
−6
92
100
102
Time [s]
Linear position in R
−49.4
−270
−275
98
100
102
Time [s]
Angular position in Ro
Øo
8
Ÿ
−0.1
98
102
−3
98
102
Linear speed in Rd
θ [rad/s]
0.1
0
w [m/s]
0
0
−5
100
200
Time [s]
Angular speed in Rd
0
0
0
−500
100
200
Time [s]
Angular position in Ro
0
0
100
Time [s]
!
200
Ø o™
Ÿ
0
−1
−50
0
100
200
Time [s]
Angular position in Ro
5
ψ [rad]
0
100
200
Time [s]
Linear position in Ro
−100
100
200
Time [s]
Angular position in Ro
1
θ [rad]
0.2
0
0
z [m]
0
100
200
Time [s]
Angular speed in Rd
0
−0.2
100
200
Time [s]
Linear position in Ro
0
0.2
500
y [m]
200
0
0.5
−0.5
100
200
Time [s]
Linear position in Ro
" '*)
Linear speed in Rd
0
−0.2
100
200
Time [s]
Angular speed in Rd
102
5
ψ [rad/s]
0
0
100
Time [s]
$ +'3,(0+'/>9 8<37= & E/!56/3 $ &)8:9D/ +78<;C= 1 356/>$$,/>5 8<1 /G$ @/>(0/ $ v [m/s]
u [m/s]
φ [rad/s]
100
Time [s]
−2.5
0.2
0
100
102
Time [s]
Angular position in Ro
−2
0
Linear speed in Rd
x [m]
−49.6
98
100
102
Time [s]
Angular position in Ro
ψ [rad]
φ [rad]
θ [rad]
100
Time [s]
10
φ [rad]
−49.5
0.1
0.1
!
z [m]
−265
160
100
102
Time [s]
Linear position in R
o
170
y [m]
x [m]
o
−49.3
0
98
−0.2
96 98 100 102
Time [s]
Linear position in R
−260
0.2
−200
−0.2
98
94
180
150
98
−0.1
0
z
Vy [m/s]
Vx [m/s]
−4
0.2
o
−10
o
0
−3
−5
98
Linear speed in R
o
−2
0
100
Time [s]
200
"%$'&)8:9!/4+,8<;>=?143 3 $ &/414E>$143,5 $ 0
−5
0
100
Time [s]
200
@E>( ' )
ّ
! "#%$'&%()$*,+.-0/12+ 3+,#+, $1%
Linear speed in R
Linear speed in R
o
0
y
x
Vz [m/s]
V [m/s]
V [m/s]
5
0
−10
100
200
Time [s]
Linear position in R
0
0
−20
0
−400
100
200
Time [s]
Angular position in Ro
0
100
Time [s]
!
Ø o†”
Ÿ#"
0.5
0
−0.5
200
0
0
−5
200
d
−0.05
−0.1
98
100
102
Time [s]
Angular speed in R
d
0.01
0
−0.01
98
100
102
Time [s]
Linear position in Ro
−0.1
−0.12
−0.14
98
160
−270
−280
98
100
102
Time [s]
Angular position in R
o
−50
98
100
102
Time [s]
Angular position in R
Ø o†Œ
Ÿ
102
−2
ψ [rad]
θ [rad]
!
100
Time [s]
0
−0.05
98
100
102
Time [s]
Angular position in R
o
0.05
0.1
−48
o
0.12
100
102
Time [s]
Linear position in Ro
−46
z [m]
−260
y [m]
180
100
102
Time [s]
Linear position in Ro
100
102
Time [s]
Angular speed in R
d
ψ [rad/s]
θ [rad/s]
0.005
0
w [m/s]
v [m/s]
d
0.01
200
Linear speed in R
0.08
0.06
98
100
102
Time [s]
Angular speed in R
0.08
98
100
Time [s]
d
4.6
140
98
0
@E>( '*)
0.1
0
98
100
200
Time [s]
Angular position in Ro
Linear speed in R
4.8
u [m/s]
100
Time [s]
$ &)8:9D/ +78<;C= 1 3$,/>5 8<14/G$D143,5 $ d
4.4
98
0
5
Linear speed in R
φ [rad/s]
−60
100
200
Time [s]
Angular position in Ro
ψ [rad]
θ [rad]
φ [rad]
0
−40
1
0
−0.5
z [m]
100
−200
100
200
Time [s]
Linear position in R
o
0
0
0
o
0.5
x [m]
−5
100
200
Time [s]
Linear position in R
200
−100
φ [rad]
0
200
y [m]
x [m]
o
&8
o
10
0
−10
Ù
Linear speed in R
o
10
100
Time [s]
102
−2.5
−3
98
100
Time [s]
$ +'3,(0+'/>9 8<37= & E/!56/3 $ &)8:9D/ +78<;C= 1 3 37$ & / 14EG$ @/>(F/($ ' )
102
C% (%Xb ()[ 2 $;#%$'%($1b.
Linear speed in R
Linear speed in R
o
−5
98
0.2
Vz [m/s]
Vy [m/s]
Vx [m/s]
−4
−2
−3
−4
98
100
102
Time [s]
Linear position in R
o
−265
−49.3
z [m]
170
−270
−275
98
100
102
Time [s]
Angular position in Ro
−49.4
−49.5
98
100
102
Time [s]
Angular position in Ro
0.2
θ [rad]
0.1
0
−0.1
98
100
Time [s]
0
−0.2
98
102
100
102
Time [s]
Angular position in Ro
−2
ψ [rad]
0.2
100
102
Time [s]
Linear position in R
o
−49.2
150
98
φ [rad]
−0.2
98
100
102
Time [s]
Linear position in R
−260
y [m]
x [m]
o
0
180
160
o
−1
−3
Linear speed in R
o
−2
|
100
Time [s]
−2.5
−3
98
102
100
Time [s]
102
Ø oL˜ Ÿ%$ +F3,(0+'/>9 8<3,=&E / 5 /237$'&)8:9!/4+,8<;>=?14356/G$ $,/>5 8<1 /[email protected]/>(F/"$ ' )
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zk = Υk (xk , ξk ) + ηk
#"
$"
™
! "#%$'&%()$*,+.-0/12+ 3+,#+, $1%
Gaussian distribution
Gaussian distribution
−1
0
1
f(x)
0.5
0
−2
0.5
0
−0.1
1
*
0.5
*
0
−0.5
0.5
0
−1
0
1
f(x)
1
0.5
0
−1
2
*
0
0.5
0
−20
1
0
1
0.5
0
−0.5
0.1
error, x=[φ − φ*]
poq˜c˜ Ÿ#"
"
Gaussian distribution
0
x
0
−0.2
0.5
x
0
−2
0.2
y
Gaussian distribution
1
0
−1
2
*
Gaussian distribution
f(x)
0
0
−20
1
*
f(x)
0.5
0
0.2
*
error, x=[φ − φ ]
poq˜ Ÿ#"
"
"
0
10
*
1
0.5
0
−0.5
−10
error, x=[z − z ]
Gaussian distribution
1
0
−0.2
0.5
error, x=[y − y ]
Gaussian distribution
1
4
z
1
0.5
error, x=[x − x ]
Gaussian distribution
2
z
Gaussian distribution
0.5
0
error, x=[V − V* ]
y
1
f(x)
f(x)
0
0.5
error, x=[V − V* ]
1
f(x)
Gaussian distribution
0.5
error, x=[V − V* ]
0
0.1
1
f(x)
f(x)
f(x)
0.5
0
−1
0
error, x=[ψ − ψ*]
1
−0.5
0
−0.1
0.5
( (0E>(037$143D37$'&)8:9!/4+,8<;>=?143 /C( 8</ ,563 $D143 3 $ &/414E 97&E1 E " '*)
Gaussian distribution
0
−1
0.5
error, x=[θ − θ*]
1
!
0
f(x)
!
10
error, x=[z − z ]
Gaussian distribution
f(x)
f(x)
0.5
0
*
error, x=[y − y ]
Gaussian distribution
1
0
−0.1
−10
*
error, x=[x − x ]
Gaussian distribution
1
0.5
error, x=[r − r ]
Gaussian distribution
1
0.5
0
*
error, x=[q − q ]
Gaussian distribution
f(x)
f(x)
0
0.5
*
error, x=[p − p ]
Gaussian distribution
1
4
1
0
−0.5
0.5
2
*
f(x)
f(x)
f(x)
0.5
0
0
error, x=[w − w ]
Gaussian distribution
1
0
−0.5
f(x)
0
−2
0.1
error, x=[v − v ]
Gaussian distribution
1
c”
0
0.5
*
error, x=[u − u ]
Gaussian distribution
Gaussian distribution
1
f(x)
f(x)
1
0
0.5
*
error, x=[θ − θ ]
0.5
0
−0.2
0
0.2
*
error, x=[ψ − ψ ]
( (0E>(037$143D37$'&)8:9!/4+,8<;>= 1 3$7/C5 8<14/>$ 143,5 $ 8 $ & 3,9!/ 97&E1 E " ' )
C% (%Xb ()[ 2 $;#%$'%($1b.
Gaussian distribution
Gaussian distribution
0
−1
0
1
0.5
0
−0.1
2
*
0
−0.5
0.1
0
−0.1
0
f(x)
1
0.5
0
−0.1
0.1
0.5
error, x=[w − w ]
Gaussian distribution
1
0.5
0
*
error, x=[v − v ]
Gaussian distribution
f(x)
f(x)
0
0.5
*
error, x=[u − u ]
Gaussian distribution
1
1
f(x)
0.5
Gaussian distribution
1
f(x)
f(x)
1
|
*
0
0.5
0
−0.1
0.1
*
error, x=[p − p ]
Gaussian distribution
0
0.1
*
error, x=[q − q ]
Gaussian distribution
error, x=[r − r ]
Gaussian distribution
−10
0
0.9
0.8
0.7
10
f(x)
f(x)
−5
*
0.5
0
−4
5
0
1
f(x)
f(x)
−0.02
0.5
0
−0.1
0.02
error, x=[φ − φ*]
Gaussian distribution
0.5
0
−0.5
4
error, x=[V − V* ]
x
x
0.9
0.9
0.8
0.7
0
0.5
error, x=[V − V* ]
y
z
Gaussian distribution
1
f(x)
f(x)
0
−0.5
0.5
y
Gaussian distribution
z
Gaussian distribution
1
0.8
0.5
0.7
−10
0
error, x=[x − x*]
10
−5
Gaussian distribution
0
error, x=[y − y*]
0
−10
5
Gaussian distribution
0.5
0
−0.2
0
0.2
*
error, x=[φ − φ ]
poq˜] Ÿ
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20
1
0.5
0
−0.2
0
10
error, x=[z − z*]
Gaussian distribution
1
f(x)
1
f(x)
0
0.5
error, x=[V − V* ]
1
!
1
f(x)
f(x)
f(x)
2
0.2
Gaussian distribution
1
0.5
0
error, x=[ψ − ψ*]
Gaussian distribution
1
0
0
−0.2
0.1
3 $ &)8:9D/ +78<;C= 1 3 />( 8</ 75637$ 1 3 37$ & / 14E %9 ,&E1 E '*)
0
−2
0.5
error, x=[θ − θ*]
poq˜ Ÿ#"%( (FE>(037$143
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0
f(x)
!
2
error, x=[z − z ]
Gaussian distribution
1
0.5
0
*
error, x=[y − y ]
Gaussian distribution
1
0
−0.04
−2
*
error, x=[x − x ]
Gaussian distribution
f(x)
0
f(x)
f(x)
1
0.9
0.8
0.7
0
0.2
*
error, x=[θ − θ ]
0.5
0
−0.5
0
0.5
error, x=[ψ − ψ*]
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k k−
EW€NPIKJRYUN%IKHMT'JRYHcTINPT fUKJCDLHcW0FZH IKJDLHcWlIKJCDLHcW0WJV#WHcT'J0W mDeNPW0iPN%T'QeN%VKwN%W0IKJCW'UKW€FJIKQLIN%W
WHcVmR]N%DLRUKDN%IN%WlN~SNPT'O*QLTIKJ.DeNaT'JW'SKUKJW_ObNIKQV PFZQLR][email protected][IKJ.DeNPWObN%tKDN%WRN%T*N%RO*JYT W_O*QLR]N%W
JV2O*T'Ju2N%IKN%WMSHcT)JY
D N%t#TQRN%V2O*JPo JYD 1. §zdIKH%VKIKJzW'JzSKTJYWJYVfO*NDeNQVKW_O*TU#FZJYVfO*N%R/Q cV IKJD
IKQT'QucQLtKDJPdxWJ[O*QLJVKJ fUKJZSN%T*N^JDEOQSHmI#J M¨ JYFatN%T'R]N%IKH#dhDNMi%NPTQeNPRQ %VJYVvDNmFJI#QIN9IKJDeN
dsJYVnDNFJIKQLIN~IKJCDN%W"icJYDH RQIKN%IKJW0JW
^SKN%T*NDeN JV2O*TN%DhQVKJYTRQN%Dd
SsHcW'QRYQ cVmJW
±2m
±0.5m/s
DHcW %V#ucUKDH%WO*QJYVKJV^iPN%TQN%R/Q cVMIKJ
o
$
±0.05rad
Ĭ . 61 !0)*
Va /T ¡pQsO*HcFZN%FHcWJDxFZH IKJDLH~TJYIKUKRQLIKH[SN%TN[DNCicJYDH RQIKN%IJYRUN%RYQcVX˜co 2‚")RHcV#WQI#JT*NPFZHcW fUKJ
JDicJYRYOHcTZI#JmJYW'O*N%IKHSN%T*N!DNzQIKJYVfO/Q RN%R/Q cV¹JW
dEJD"FHpIKJYDHzI#J EUKDLJT~SNPT*NXJD
χ = [x, κ, τ ]T
WQW_O*JYF)N~WQLFZS#DQ/KR]N%IKHCSKUKJI#J„W'JTlJW'RTQqO*HRHcFH9ž
. 1.
.$(!01
3
"
x(k + 1) = x(k) +
τ (k + 1) = τ (k)
κ(k + 1) = κ(k)
Tm
(κ(k)u(k)
τ (k)
¡pJ.SKU#JIKJ>FZH%W'O*TN%T fUKJ>IKJ poq˜c˜.WJ.Hct#OQJV#J\DeN~F)N O*TQLw
"


(1 − Tτm )
0
F =
0
Tm u
τ
1
0
F
Tm
(−κu
τ2
0
1
− x(k))
 poq˜c˜ ‚
#"
SKT'JW'JV2ObN%IKNZN[RHcV2OQVfUN%RYQcVXž
+ x)


 poq˜ c‚
#"
"
C% (%Xb ()[ 2 $;#%$'%($1b.
|
ŠNu%UKT*N poL˜,‘0SKT'JWJYV2ObN€DH%WjT'JWU#DLObNPIKHcWhIKJ;DN`QFSKDLJFJV2ObN%RYQcV\I#J;DeNPWjJYRUN%RYQHcV#JWƒNPVfOJT'QHcT'JW
JYV“nNPO*DN%t ·¡pQFaU#DQV#‰o ŠNMRYUKT'iPN9WUKSJT'QHcTgQw fUKQLJT'INMFaUKJYW'O*TNMDeN)JYW'OQFZN%RQ/cVzIKJYDƒJW_ObN%IKH9I#J~DeN
i%JDH RQLIN%I›dsRHcFSN%TN%IN9RH%VfOT*N9DeN)i%JDLHpRYQIN%IXIKJ[WN%DQLINMIKJDEW'QW_O*JFZN !DNMIKJD kT'JV2O*JnW'UKSsJYTQLHcT
IKJYTJYRb}N2‚>FCUKJW_O*T*N3JDRHcFSHcT'O*N%FZQLJV2O*H^IKJZDeN9iPN%TQN%VKw]N^IKJYDBJYW'ObNPIKHnJYW'O*QLF)NPIKHKo ŠN%WCRYUKT'iPN%W„IKJ
dlRYHcV JYD
DNSN%T'OJXQ+V kJYTQH%TMQw U#QJT'IN¹SKTJYWJYVfO*N%V DNRHcV2icJYTucJYVKRQN¦IKJDH%W9RH J RYQJV2OJW
κ τ
RYHcFZSHcT_ObN%FQJYVfOHaIKJ.WU#WliPN%T'QeN%V#w]N%W"JYV kT'JV2O*JPo
&`VN%V %DQLWQWgIKJ~DeN)iPN%T'QeN%RYQ cVzI#JDLHcW\SHcDHcWgJVXJDESKDN%VKH9RYHcFZS#DJ H9FCUKJW_O*T*N fUKJ[JDEW'QW_O*JFZN
SKT'HcSKUKJYW'OHSJT'F)NPVKJRYJ€JW_ObN%tKDLJ€IKUKTN%V2O*J\D
N N%WJ\I#Jgi UKJDLH~RTU#RJT'HKdKSKT'JW'JV2ObN%V#IKHZU#VNCiPN%TQN%R/Q cV
SNPT*N%F YOTQRNZIKJ~W'UKW\RH J KRQJYV2O*JW fUKJ~WJaI#JtJaO*JV#JT\SKT'JWJYV2O*JW>JV!JYDƒIKQLWJº#H9IKJaDH%W€TJYucUKDeN IKH%TJWYo W_O*J„TJW'UKDLO*N%IKHSKUKJYIKJCW'JT`N%SKTJYRQN%IKHJVnDN ucUKTN poq˜ 8 dIKH%VKIKJ.ObN%Fat#Q VmWJ>FaUKJYW'OT*NZJYD
JYTTH%T"IKJ.JW'OQFZN%R/Q cV›o
"
!
"
Estimated of the speed state, x in blue and x* in red
14
Variance
2
12
1.5
1
σSpeed
Speed
10
8
6
0.5
4
0
2
0
0
50
100
Time [s]
150
−0.5
200
0
50
100
Time [s]
150
200
Variance of σκ in blue and στ in red
κ in blue and τ in red
2
8
1.5
σκ and στ
κ and τ
6
4
1
0.5
2
0
0
0
50
100
Time [s]
150
200
−0.5
0
50
100
Time [s]
150
200
!#"$% $&&"(')"($*%+-,.$%+'/0 "("21%+'!35476!$047'!*"($0% 8'/"$%+'947$"!:; "'
<>[email protected]?BADCE=GFHCJIKILM?ONQPSRTPVUHAXWYADZ[Z[KE\^]_R`K7aGb[Idc;KEZfehg
z/δe i
j kmlon9p(qfrsloktsn(u.vwn`xmyzq t{n`xqwt{n9xv|n`p(}mv~p`kSfr f€f p`qfr
tsnƒV}sxSn`u„mv~u/v#n`[email protected]/kSlo/n`y‚vzv#kStsn`r[.k †mp‡vTun`l‰ˆ
z(k + 1) =
θ(k + 1) =
q(k + 1) =
kz (k + 1) =
k1 (k + 1) =
k2 (k + 1) =
k3 (k + 1) =
χ1 = [z, θ, q, kz , k1 , k2 , k3 ]T 
z(k) + Tm kz (k)θ(k)
θ(k) + Tm q(k)
q(k)(1 + Tm k1(k)) + Tm (k2 (k)θ(k) + k3 (k)δe (k))
kz (k)
k1 (k)
k2 (k)
k3 (k)
tsn d‹Ž n`x~n`p.qfu-tsn‰n`l!v~t{q
χ1
l/n‰.kn(rsnwx‘v#y‚vT.uok’
F
n(xmy‚qDtsn`xSq
Š S‡‹[Œ
„su/n(l/n(rX.v~tmvzn`r S`“ 
“s
%$'&%($*,+.-0/2+3+#+, $
Pole mouvement
Imaginary axe
0.02
0.01
0
−0.01
−0.02
−1.8
−1.6
−1.4
−1.2
−1
Real axe
−0.8
−0.6
−0.4
−0.2
Gaussian distribution
1
0.8
f(x)
0.6
0.4
0.2
0
−1.5
!
poq˜ 8 Ÿ
"
−1
−0.5
0
error, x=[u − u*]
0.5
1
1.5
E 8:9 8<3,=&E 1 3 @E>5 EG$ 3,=375 @ 56/C= E+FEC9 @563"!,E$#23,( (0EC(143 3 $ &)8:9D/ +78<;C= @/>(0/256/ 3,56E +78<1 /41

1 Tm kz
0
Tm θ 0
0
0
0
1
Tm
0
0
0
0 


0 Tm k2 (1 + Tm k1) 0 Tm q Tm θ Tm δe 


0
0
1
0
0
0 
F =

0

0
0
0
0
1
0
0


0
0
0
0
0
1
0 
0
0
0
0
0
0
1

 poq˜]f‚
#"
`DpT'J]N%DLQw]NPThDN`QLFZSKDLJFJV2ObNPRQ%V>I#JJW_ObN%WƒJYRUNPRQ%VKJWƒJYVaDH%WGN%DLucHcT'QLO*[email protected]*N%T'THcDLDeN%I#HcWjW'Ju&%#V
DeN\O*JH%T eN.IKJ .N%DFZN%V[N%V2O*JYW;FJVKRYQHcVKN%IN#d2W'J`HctpO*QJYVKJVZDHcWT'JWU#DLObNPIKHcWrSKTJYWJV2O*N%IKHcWBJVMDN KucUKT*N
poL˜ ™ IKHcVKIKJ\DN%WRYUKT'iPN%W"SKT'JW'JV2ObN%V^DNaRHcV2icJYTucJYVKRQNaObN%V2O*H~I#J\DHcWBJW'O*N%IKHcWlIKJYDxW'QW'OJFZNaRHcFHaIKJ
DHcWSN%T %FJYO*T'HcWIKJW'RHcV#HpRYQIKHcWIKJ0JW_O*JPo#¡ QW'J`JYW'OUKIKQeN.UKVZSsH RH.FZN%WJV)I#JYObNPDDJ"JYW'OHcWBT'JW'UKDLO*N%IKHcW
WJgOQJV#J fUKJ`JDWQW_O*JYF)N.SKDeNPVfOJ]N%I#HCSJTFZN%VKJYRJ`JW'O*N%tKDLJgNCDLH„DN%Tu%H>I#JgDeN N%W'JgIKJ0ipU#JDH„RT'UKRJYTHKd
N TFZN%RYQ cV fUKJmJW[WH%SsHcT_ObN%IKNXSHcT~DN ucUKTN oL˜,”!IKH%VKIKJ^WJ^FaUKJYW'OT*NXJD"FH i QLFZQLJV2O*HzI#JmDLHcW
SsHcDLHcW 9DNCi%NPTQeNPRQ %[email protected]%T"IKJ.DHcW"JYW'ObNPIKHcW0JW'OQFZN%IKHcWYo
"
"
. 1.
! $&$($(!0)*
3
' ( '
$($(!)*+
!0,( !$&
ψ/δg cH V#WQI#JT*NPVKIKH^DHcW\FQW'FZH%W€SKTQLVKRQLSKQH%WgIKJ[DH%W\IKHcW LOJFW€SKTJYRJYIKJV2O*JYWdhDeNMJYRUN%RYQcV ˜co ™
RHcFH~i%NPTQeNPtKDJYWIKJ.JW_ObN%IKH
T [email protected]#JDHCIKJU#[email protected]*J~ž
"
χ1 = [ψ, r, k1 , k2 ]
ψ(k + 1) = ψ(k) + Tm r(k)
r(k + 1) = r(k)(1 + Tm k2 (k)) + Tm k1 (k)δg (k)

IKJ.IKH%VKIKJ.DeNaFZNPO*T'Qw
"
F
SN%T*N[DeNaS#TJI#QRRYQcV^JYW>ž
 poq˜ ‘c‚
#"
( Xb () ( $ %$ %($b.
z
σ
0
−1
50
0
50
100
150
100
150
100
150
100
150
Time [s]
Variance
2
0
1
θ
0.2
0
−0.2
−1
50
100
150
Time [s]
Estimation: q state, x in blue & x* in red
0
50
Time [s]
Variance
2
1
σ
q
0.2
0
−0.2
−0.4
0
−1
100
150
Time [s]
kz in blue, k1 in red, k2 in black & k3 in magenta
k2
10
σ ,σ ,σ
0
100
4
2
k1
50
50
Time [s]
Variances
0
kz
−10
0
0
k3
50
&σ
0
κ&τ
1
100
150
Time [s]
Estimation: Pitch state, x in blue & x* in red
150
0
50
Time [s]
!
Variance
0
0
|
2
σ
θ [deg]
50
0
q [deg]
Estimation: Altitude state, z in blue & z* in red
100
g
z & δ *100 in black
poL˜ ™ Ÿ#"
Time [s]
$ &-8:9!/4+,8<;>=?143,537$ & / 14E 14356/2/C5 &)8:&-*1 !# 56EG$%@/>(0BC9D3,&-(FEG$ 143,5 9!E1 375 E (03'1C*+78<1 E
"
Imaginary axe
Pole mouvement
5
0
−5
−1.8
−1.6
−1.4
−1.2
−1
−0.8
Real axe
Gaussian distribution
−0.6
−0.4
−0.2
0
f(x)
1
0.5
0
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.1
0.15
0.2
0.25
error, x=[z − z*]
Gaussian distribution
f(x)
1
0.5
0
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
error, x=[θ − θ*]
Gaussian distribution
f(x)
1
0.5
0
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
*
error, x=[q − q ]
!
poq˜,” Ÿ
"
E 8:9 8<37= & E 1 3 @E>56E>$ 37= 375 @ 56/C= E2+'EC9 @563 !,E 2# 3,( (0E>(1 3 37$ &-8:9!/4+,8<;>[email protected]/>(0/256/2/C5 &)8:&-*1

%$'&%($*,+.-0/2+3+#+, $

1
Tm
 0 1 − Tm
τ (k)
F =
 0
0
0
0
0
Tm δe (k)
τ (k)
1
0

0
m
− τ T2 (k)
(κ(k)δe (k) − θ(k)) 


0
1
 poq˜ 8 ‚
#"
`DE}N%RJYT„DN9QFSKDJYFZJYVfO*N%RQ/cV!IKJDeN%W\JYRUNPRQH%VKJWCN%V2O*JT'QHcT'[email protected][O*QJYVKJVvDeN%W KucUKT*NPW poq˜,Œ
po IKH%VKIKJ>WJ>SKTJYWJV2O*N)DeN~RH%Vfi%JT'ucJVKRYQeNIKJ„DLHcWlJW_ObN%IKH%WgN[UKV^i%NPDHcT ^JYDhRYHcFSsHcT_ObN%FQJYV2O*HaJYV
[email protected]%VKH[RHcFSKDLJ [email protected]_O*JYF)N~QI#JV2O*Q R]N%I#HT'JW'SsJYRYO*QqiPN%FZJYV2O*J%o
"
"
"
*
Estimation: ψ state, ψ in blue and ψ in red
Variance
ψ & δg in black
0.6
2
0.4
σψ
1
0.2
0
0
−0.2
0
−1
50
100
150
Time [s]
*
Estimation: dψ/dt state, r in blue and r in red
0
50
σr
r [deg]
1
0
0
−0.02
0
50
100
Time [s]
k1 in blue and k2 in red
−1
150
−0.5
−1
0
50
100
0
50
100
Time [s]
Variances: σk1 in blue and σk2 in red
1
0
−1
150
0
50
Time [s]
poL˜]Œ Ÿ#"
150
2
σk1 and σk2
k1 and k2
0
©"ª0ª‡©
150
2
0.02
!
100
Time [s]
Variance
100
150
Time [s]
$ &-8:9!/4+,8<;>=?143,537$ & / 14E143 56/21C8:(03'+'+78<;C= !# 56E>$%@ /C(0B>9!37&)(0E>$1 375 9DE 143,56E2(03'1>*+,8<14E
"
²rµ77g²´ _¶¸ g­@µ#!`­@µ7 ® ,³¯€µ7 €´x² g
­ ­@µ7"´%7g² ` ²ƒ±®`­@µ
°g¸ #­@²rµ
0VNnicJYwmJYW'OUKIKQeNPIKHcWaDLHcWaR]NPWHcW~JYV W'QFaU#DeN%RYQcV›dŠW'J9IKJtJV N%SKDLQRN%T„JYW'OHcWN%DucH%TQLOFZH%W„NzDLHcW
INPO*H%WT'J]N%DLJW~IKJ9U#V i UKJYDHKo ŠN%W KucUKT*NPW o p˜ po !S#TJW'JV2ObNPV¦DLHcW~TJYWUKDqObN%IKH%W~Hct#O*JYVKQI#HcW[N%D
TJ]NPDQwN%T;DNCQI#JV2O*Q R]N%RYQ cV)IKJD›FH IKJDLHCIKJ€RYHcV2O*T'HcD›SN%TNaDeN„i%JDLHpRYQIN%IMN%JT'HpIKQLV %FQRN#d2RHcV^IN O*HcW
N%I fUKQT'QIKH%W\IKJ[UKV!ipU#JDH^T'J]N%D o EVvJYDDeNPW€WJN%SKT'JRYQeN9JDƒRYHcFSsHcT_ObN%FQJYV2O*HMIKJYDESsH%DH9IKJYDEWQW_O*JYF)N
SKDeN%V2OJ]N%IKH [email protected]*H[[email protected]'O*N%IKHZN[DLH~DeN%T'ucH~IKJYD›ipU#JDH#o
&
©"ª0ª
g­@¸B±
€´s²r´ ·¶r¸ g
­@µ
"
"
!g­@µ7
"
"
"
´ ³¯€µ·­
hHpRN%W3t#QtKDLQHcucTN eNPW)}N%t#DeN%VIKJDeN¹JW_O*QFZN%RYQcV IKJSN%T %FZJO*T'HcWmN%JT'HpIKQLV %FQRYHcW^N SKN%T'OQT
IKJ9INPOHcWaIKJ)i UKJYDH!T'J]N%D djucJYVKJTN%DFJV2OJMWJMT'JFQLOJV¹NzDNnTJN%DQLw]N%RYQcVIKJ9FHpIKJYDHcW„NzJW'R]N%DN
WUKSKU#JW'OHcW fUKJCSJT'FZQqO*JYV3}NPRJT`DeN[QIKJYV2O*/Q KR]N%RYQ cVmVfUKF YTQLR]NNZSN%T_O*QLT0I#JCINPOHcWlO*HcFZN%IKHcW0JVnJD
O %KVKJYDŠIKJ\ipQLJV2O*H3icJT„‹ $ pd #.y`Œ%” e‚o
c
( b () (
b.
Pole mouvement
Imaginary axe
1
0.5
0
−0.5
−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
Real axe
Gaussian distribution
−0.3
−0.2
−0.1
0
f(x)
1
0.5
0
−8
−6
−4
−2
0
2
4
6
*
error, x=[ψ − ψ ]
Gaussian distribution
−4
x 10
f(x)
1
0.5
0
−8
−6
−4
−2
0
2
4
6
*
−3
error, x=[r − r ]
!
po Ÿ
"
"
x 10
E 8:9 8<3,=&E 1 3 @E>5 EG$ 3,[email protected] 56/C= E +FEC9 @563"!,E$#23,( (0EC(143 3 $ &)8:9D/ +78<;C= @/>(0/256/1>8:(03'+'+78<;C=
Measured State (blue) & Estimated State (red .)
Speed, u=Vx
15
10
5
0
0
50
100
150
200
250
300
Tiem [s]
Coefficient [k]
350
400
450
500
0
50
100
150
200
250
300
Iterations
Coefficient [τ]
350
400
450
500
0
50
100
150
200
250
300
Iterations
350
400
450
500
Static gain [k]
1
0.5
0
time constant [τ]
15
10
5
0
! po p˜ Ÿ " E>5 *+78<;C=143,5 37$'&/41 E 1 375 9!E1 375 E 143 3,56E +78<1 /41 1>*.(0/C=&3D*.= *3,56E (F3F/C5 u #$'* $!+'E3 +,8<37= & 37$
& /C= /C= +,8</37$ & BC&-8<+'/ κ #2+'E>=$'&/>= & 3 143&-8<3,9 @ E τ
"
"
GNPT'O*QLJVKI#HnIKJMDN%W.IK/Q kJYTJYVfOJWa}#QS%O*JYWQW SKDeNPVfOJ]N%FQJYV2O*HcW.TJN%DQLw]N%I#HcW„JYV{O*TN%tN H%WCIKJZWQW OJFZN%W\RHcFH^JDIKQLTQu%QtKDLJ)‹ #.ŒcŒpd hUKR Kd jU#R 8 d ”#˜cdh“nN- 8 dh“nGd # %t›d# cN  [email protected]
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0.1
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0.05
0
−0.05
−0.1
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−0.25
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−0.15
Real axis
−0.1
−0.05
0
4
6
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1
0.8
f(x)
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0.4
0.2
0
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v
w
p
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0.0177
0.0034
0.0156
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0.0057
0.0439
0.0670
0.0217
0.0626
0.0179
0.0153
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0.0006
0.0003
0.0001
0.0001
0.0001
0.0015
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0.0009
0.0001
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Cm3
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0.0190
0.0058
0.0287
0.0017
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0.0020
0.0639
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0.0651
0.0209
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( B %!
a11
a22
a33
a44
a55
a66
a15 = a51
a24 = a42
a26 = a62
a35 = a53
a46
m13
m33
xm11
xm22
xm13
x2 m11
x2 m22
CT 1
CT 2
CT 3
CL1
CL2
CL3
CN 1
CN 2
CN 3
Cl1
Cl2
Cl3
Cm1
Cm2
Cm3
Cn1
Cn2
Cn3
1.5276
21.0933
20.4220
16.3905
382.1290
388.0972
0
0
−69.6939
67.7045
0
1.2801
−49.7019
25.6919
23.6878
−4.5582
−173.4906
−166.3538
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
1.2633
20.8660
20.2018
15.9151
380.0030
379.9988
0.1309
−0.0958
−59.9030
70.0684
0.0801
2.0621
−48.0249
23.0748
20.0129
−9.1165
−150.0044
−149.9994
−1.8974
0.1071
−2.8752
50.8756
−15.9874
−1.2234
−0.0417
0.5487
1.0258
0.0205
−0.1919
0.0170
−0.0405
0.7975
0.2930
1.0833
−0.8574
0.0450
1.9660
21.1896
21.4239
16.8548
380.1463
384.0155
0.0001
0.1628
−69.8978
70.3361
0.0101
1.5235
−48.5013
24.5183
21.0075
−5.4699
−170.8227
−158.8524
−0.6579
0.1069
−0.6877
15.4789
−11.5582
−7.2243
−0.0664
0.5789
0.389
0.0304
−0.1271
0.0266
−0.0415
0.9982
0.2173
0.6207
−0.7589
−0.0010
% a11
a22
a33
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a26 = a62
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11.2428
90.4922
70.5841
42.3387
383.7979
419.9314
6.9103
1.2382
−195.3407
59.4323
−28.5030
33.7772
−93.7707
76.4093
54.7163
75.3240
−201.9972
−224.8353
−2.9074
−0.2250
−0.7970
15.0799
−7.6177
−3.2706
−2.1196
−13.9818
0.6837
5.1576
2.9208
1.0168
−0.0725
−1.8937
1.1017
−0.1082
−0.5101
0.0115
ii 0.1580
0.0925
0.0922
0.0901
0.0837
0.0872
0.1309
0.1240
0.1269
0.1053
0.1053
0.0621
0.0982
0.0905
0.0192
0.0962
0.0335
0.0896
0.1290
0.0446
0.0767
0.0744
0.0644
0.0249
0.0676
0.0949
0.0508
0.0538
0.1509
0.0582
0.1442
0.0814
0.0762
0.0942
0.0415
0.0227
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40
60
80
100
Time [s]
HQPGMRJK(SN K(H H1MS K J(KS GHFNQH N G KS+HŠMRJ>I>H K KJ-S&NSM TV
Pole mouvement
0.04
0.03
0.02
Imaginary axe
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
−0.06
−2.2
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8:9<;>=-4
ÆÐ
−2
−1.8
−1.6
−1.4
Real axe
−1.2
−1
−0.8
−0.6
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z(s)
k k
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k
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2
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1
p
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Speed Control, v
Altitude Control
r|red
v [m/s] & α [deg]
10
0
s
z in red & z in blue, [m]
r
−10
r
−20
0
50
100
0
Signal Control of Altitute
50
150
Signal Control of Speed
T [N]
0
e
0
50
100
20
z
0
−20
0
50
100
Time [s]
150
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40
20
0
150
w in blue & V in red, [m/s]
δ [deg]
100
60
Time [s]
Angular Speed [Ωθ]
q [deg/s]
black
Time [s]
50
−40
&α
−10
Time [s]
−50
s|blue
0
−20
150
&v
10
0
50
100
Time [s]
Speed on the axe Z in R & R
d
150
0
2
0
−2
0
50
100
150
Time [s]
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dψ
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δ (s) = K ψ +
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s + s (τ − κK ) + v (κK )s − v κK
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i
g
r
p
3
d
2
p
a
d
2
i
0
50
100
Time [s]
150
10
5
0
−5
200
0
Signal Control of Heading
δg [deg]
T [N]
0
50
100
150
Time [s]
Angular Speed [Ωψ]
200
20
0
50
100
150
Time [s]
Speed on the axe Y in R
200
50
200
d
0.1
v [m/s]
r [deg/s]
150
40
0
200
5
0
0
50
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100
Time [s]
60
0
−5
50
Signal Control of Speed
20
−20
n
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50
0
d
Heading Control
100
−50
i
2
n
n
v [m/s] & β [deg]
ψr in red & ψs in blue, [deg]
p
a
100
Time [s]
150
HQPGMRJK(SN K(H:K NOH
200
0
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[email protected]
0
100
Time [s]
150
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<,5=?>@/,.2PBB.H02Df4<^BZ/V*1Zy Y0=zI0={<;/fD>@1/fD,|BS6 fD=0f468f4r;B.90, *x1;2P9)1;*H)2B.r;B.6 fD1;=?>@,'901;2PA0,./f4r,5=?>@1WHCBZ/@B
24B}/1;<I0HX1;/@BZ<;f %.=nBW<^BZ9CBWHCB.*,89012BW*a1 'CB.2G9)1k<,5=?>@/,52%K
P='1*>~B> 2<;=0fD<^B1;*gf468H€,./>~BZ=s>1e>@1=01;/gI0= 6 ,l901;2D,,fD901^B9012,5/901= 9)1;26 ,l90124,0vZ<;,5=0,l<;1/72B
)UI0=)<;f %.= 901>@/B.d* )U1/1=0<;f4Bk,k1;2,5/901;= 901‚2D,5*7H€,.24f4=),56 fD,5*pU=sI06 1/@B.9),5\
/ *901;=0,.6 fD=CB.90,5/@tƒv?68,:9)1;24,.*
fD=01;*M>~B.-02D1;*H0I01901;=n*1;/3<;,.=0*fD901;/B.90,5*€K [„y.ICB.24681;=?>1k*19)1;-X1<,5=?>~B./3<;,5=FI0=b6 ,l901;2D,901H€1/>@I)/ -CBZ<;f4,.=01;*3<;,568,'12 2D2B.6 B.90,ST†S‡ ˆ‰†Šp { ,5=?>/,52D241;9  I:>@, „‹31y5/1**fji51WBZ=09 [„=?>@1y5/19 ` ,uilf4=0y
 i51/@B.=0y.1utƒv
vh9),5=0901 C(z−1) y51;=01/@B.2D6 1=?>@1W1;*WJ.Œv ]WpUr ~Jut‚I0={H€,.24f4=),56 fD, blI)1
n(t) = C(z−1)e(t)
D(z−1)
1 )H)2 .4<fD>@B.6 1=?>@1V>f41;=)1I0=bfD=?>@1;y./@B.90,./ ∆ = 1 − z−1 * e(t) 1*‚/I0f49),W-02B.=)<;,<;,5=F681;90f4B}<;1;/,0K
,ŒB )UI)=0<;&f %5=,5
- 01>fDi.,Žp[‘ tHCBZ/@B\1;2[(RWT>f41=01|<;,568, 3C=CB.2Df49CB.912*1;y.I0f468f41=s>,{901|2B
/(1 )U1;/1;=0<fB
B *{2B|6 fD=0f468f4r;B.<;&f %5=n9)1 2D,5*1;d* )UI)1;/r,5*9)1'<;,.=s>/,52d‡K ,ŒBF1 )H)/1;*f %5=’y51;=)1;/@BZ2gH0B./@Bb“eK•”
"
"
!
!
–
n Speed Control
10
5
0
100
200
Time [s]
300
0
−10
−20
Signal Control of Speed
δe [deg]
0
0
100
200
Time [s]
8:9<;>=D4:K Š?
0
0
−20
100
200
300
Time [s]
Angular Speed [Ω ]
0
−20
H C GM <JK(SC KH
100
200
300
Time [s]
Angular Speed [Ω ]
ψ
5
0
−40
0
100
200
Time [s]
0
−5
300
0
100
200
Time [s]
SNOS M Q SQJ MnH1JCFH KHdGHFMRS"1N G HFNOS €
NHPGMRJ(KSN T-VW
Airship position during the flight
12
Position on axis −Z [m], Up
10
8
6
4
cruise flight
2
start
0
−2
−4
recovery of the system
−6
0
−100
400
300
−200
200
−300
100
−400
−500
−200
Position on axis −Y [m], West
–
8:9<;>=-4)K~?
300
20
θ
300
100
200
Time [s]
Signal Control of Heading
40
20
q [deg/s]
, [m/s]
red
&w
0
blue
v
0
−50
100
200
300
Time [s]
Speed on the axe Y & Z in Rd
−2
−200
r [deg /s]
T [N]
20
0
300
0
50
40
2
100
200
Time [s]
200
Signal Control of Altitute
60
0
0
Heading Control
400
δg [deg]
0
Altitude Control
10
ψr in red & ψs in blue, [deg]
z d ‘ x)
zr in red & zs in blue, [m]
ur in red & us in blue, [m/s]
s€<O 0
−100
Position on axis X [m], North
H1C GM QJ(KSC K(HsMRJ+HS M GK(HsMJC ƒSC S H1C ON HQP GMJKS NUTVOW
300
:
Q
1*Š’
J(N1 , N2 , Nu ) = E
N2
X
2
δ(j)[y(t + j|t) − w(t + j)] +
Nu
X
λ(j)[∆u(t + j − 1)]2
p4)K ]?t
j=1
j=N1
09 ,5=)901 E 1*24BS1;*H€1/@B.=0r;Bb90124BF[‘ KjvG2BH0/f4681;/BS*I06 BZ>@,./fBS1*1;2P>@1/68f4=0,|B.*,:<fB.9),|B.2
*1;y5I0fD6 fD1;=?>@,W9)1I0=CBW/1)U1;/1=0<;f4B *'2B}*1;y5I0=)9CB BW24B}6 fD=0f468f4r;B.<;f&%5= 9)124,5*1*d)UI01/r;,.*‚90124B}*1('CB.2
901W<,5=?>@/,52%KG& B./@BI0=z241<>,5/f4=?>@1/1*@B.90,|1=’H0/, )UI0=09)f4r^BZ/[*,5-)/11;*M>@1<;,5=?>/,52dvH0I01901 <,5=0*I02D>@B./
_ { ] – c‚,q1;2‚24fD-0/,’901 ,ŒB.=09CBZI R_ ,‡, ` ] c%_K ,QBo[‘ H0/1;*1;=?>~BqI0=CB’*,52DI0<;&f %5= B.=CBZ2 .D>f4<;BzHX,5/'12
</fD>1;/f4,<;ICBZ90/ `Z>f4<, <ICB.=090,'1;2Œ6 ,l90124,1;*‚=0,'2Df4=01;B.2 *F=0, >f41=01/1;*M>@/fD<;<f4,5=01*;v901k24,8<;,5=?>/@B./f4,
24Bb*,.24I0<f %5=m*1S901-€1SA0B.<;1/}901I0=CBb6'B.=)1;/@BbfD>@1/@BZ>fDiZBmpU<;,5=o2BxB *lI09CBx901'<;,568H0I)>@B.90,5/@tƒK X‚=CB
H0I)-024fD<^B.<f %5=q901F2BzB.H02Df4<;B.<;&f %5=q901F2B )UI0=0<f %5= ,5- 01ƒ>@fDi.,’<,5= /1;*M>@/fD<;<f4,5=01* 1* H)/1;*1;=?>~BZ9CBoHX,5/
{ B.6 B.<~A0, _ { B.6'] 4 cdK  H024fD<^B.<f4,5=)1;*gB24BfD=090I0*M>@/fB<;,5= /1;*I02D>@B.90,5*g*@B >@f4F* )B.<>,5/fD,5*P*,.='/1;B.24fDr^B.9),5*
HX,5/ { 24B./YZ1_ { 24B cdK { ,.=0<;2DzI *51=090,0vC24Bf4901;B}- `.*f4<^B}9)1;2(RWT 1;*<^BZ24<;I)2B./ I0=CBW*1<;I01=0<;f4B9)12B
*a1 'CB.-2 )UI)>I0/@B9)1V<;,.=s>/,52dvs1;='i .Bk901V<,5=0*1;y5I0fD/e24Bk68f4=0fD6 fDr^B.<f %5=901V24B[‘ *,.-0/1‚I0=SA0,5/f4r,5=?>@1
901H0/1;90fD<;<f %5=K
fG*1>f41=01I0=F6 ,l901;2D, [ ” <,568,npU6 ,l90124, T†S‡ ˆ‰†|v
t ’
p 4)KDJ?t
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9012*fD*>1;6 B)v ∆ 1*I0=Ff4=?>@1y5/@B.9),5/;v e(t) 1*3I)=F/I0f49),W-02B.=)<;,W<;,.=|iZB./fB.=)r^B}<;1/, *  v * { *,5=
HX,524fD=0,568f4,5*vCI0=CB241*x901<,5=?>@/,.2>@fDH€,b(SRTH0I01901*1;/H024B.=?>@1;B.9CB)K ,QB|3Cy.I0/@B 4)K ]S/1H0/1*1;=?>@B
I0=o*f4*M>@16'BF1;=q2BZr;,F<;1;//@B.9),x<;,5=qI)=m<;,5=?>/,52 $ & { 12e<;I0B.2 *f4/Mi51'<,568,nH0I0=?>,\901'/(1 )U1;/1;=0<fB
HCBZ/@BW24,5*9)1;*@BZ//,.2424,.* bsI01*1kH0/1*1;=?>@B.=xBW<;,.=s>f4=sICB.<f%5=hK
A(z −1 )y(t) = z −d B(z −1 )u(t − 1) + C(z −1 )
e(t)
∆
!#"%$&(')+*,-/.0$1"32,4567&87#9"3$;:<=>:)?",2%:3$;:<[email protected]
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l~_bcfVd(bgmheco_bcJcod~ce‚ogmrAic hog%fVpVzmd~fV_bcƒIr„_{fVgml~_bcfVg zm_aq‚oho_bl#† ‡
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4)KDc
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p 4)KjJ5J t
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I0=0,0K
,GI01y5,0vB.H02Df4<;B.=090,W<ICB.2bsI0fD1;/B9)1k2D,5*B.2Dy5,5/fj>@68,5*eH)/1;*1;=?>~BZ90,5*V1=o_ {`ba ZcŒ,b_ { ] – cdvC<;,5=
v
v
v024B2D1(*'901<;,5=?>@/,52lbsI01*1k1;=0<I01;=?>/@B 1c* ’
N = 1 N = N = 10 λ = 1.2
0.0198z −1
H(z ) =
1 − 0.9802z −1
−1
1
2
u
T (k) = T (k − 1) − 4.7404va (k) + 4.0845va (k − 1) + 0.0151va|ref (k + 1)
+0.0288va|ref (k + 2) + · · · + 0.1084va|ref (k + 10)
%
(*
^3
~4 5
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p4)KjJƒ65t
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*,5= H0/1*1;=?>@B.90,5* 1;='24B Cy5I0/B )KjJ )v?9),5=09013*13,5-0*1;/iZBkI0= *,5-0/1 „H0B.*,6'B.*7y5/B.=0901Zv?<;,568HCB./B.90,
<;,5= 12V/1*I02j>~B.9),m901;2V& K P*>1x<;,.6 HX,5/>@B.68f41;=?>,’1*>~BqBZ*,l<;fBZ90,mBo24B’*f4=?>@,5=)f4r^BZ<;f%.= /1;B.24f r^B.9CB8HCB./B 1;*M>@1/1;y.I02B.9),5/;v0>/@BZ>@B.=090,89012D241ƒi.BZ/I0=nHX,:<,'6 B.*/ .H)f490, B.2Œ*fD*>@16'B8B 12Gi.BZ24,5/901
/1()U1/1=0<;f4B)K
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Speed Control
6
s
v in red & v in blue, [m/s]
8
4
r
2
0
0
20
40
60
80
100
Time [s]
120
140
160
180
200
120
140
160
180
200
Signal Control
50
T [N]
40
30
20
10
0
0
20
40
60
80
100
Time [s]
8:9<;>=€4)KjJ5Š? 
SNOSMzKHHFMRS K(JK Œ
NOHQPGMRJK(SN€T
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P=F1;2G90fD*1a'),89)1;2G<;,.=s>/,52G9)1BZ2D>@fj>@I09HX,5/ $ & { Cv *1kB.*I)6 f&%W1;2G68,:90124,B’
"
H(z −1 ) =
b
p4)KjJ4?t
,5=)*f49)1;/@BZ=090,24,.*V*fDy5I0fD1;=?>@1*[i.BZ24,5/1;*VH0B./@BS24B.*‚iZB./fB.-02D1;*‚901W2B UI)=0<;f 5=x,5-01ƒ>@fji5, N = 1 v
v
v)2B}241 *901<;,.=s>/,52G<;B.24<I02BZ9CB}*1H0/1;*1=?>~B 1=b42 B}1;<ICB.<;f&%5=‰4)KDJ0K 1
= N = 10 λ = 6.2
{
N2
0.000603 + 0.002424z −1 + 0.000609z −2
1 − 3.021z −1 + 3.041z −2 − 1.02z −3
)
&%
u
:
Q
δe (k) = 0.9484δe (k − 1) − 0.0516δe (k − 2) − 24.2363z(k) + 65.6447z(k − 1) − · · ·
+18.3751z(k − 3) + 0.00004zref (k + 1) + · · · + 0.0384zref (k + 10)
4 , 3
|4
&%
*
"
&3
)
/.
&% nb
%
)
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*
10m
w
q
0
(' !*
%
Speed Control, vr|red & vs|blue & αblack
Altitude Control
v [m/s] & α [deg]
0
−5
s
z in red & z in blue, [m]
p )KDJ st
ŒB Cy5I)/@B )KjJ5JH0/1*1;=?>@B|2D,5*3/1;*I02D>@B.90,5*‚1;=\*f46}I024B.<;f 5=|24,5y./@B.90,.*‚HX,5/V1*>@B'241 b9)1}<;,5=?>/,52dK
P*>@B.*<I0/iZB.*6'BZ=0f C1*>@B.=’1271;//,5/k901'1*>~BZ90,n1*>~BZ-02418901;2g/1y5I024B.90,5/1;=oI0=oH0I)=s>,x90f U1;/1=?>@1
B.2€H0I0=?>@,890,5=)901*1/1;B.24fDr;,}24B* D=?>@1;*f4*kpUH0I0=?>@,8901,5HX1;/B.<;f 5=tƒv sI01HCB./@B}12<^BZ*,901*fD6}I02BZ<;f .=
UI01901
v€f4y5I0B.24681;=?>@1*1WH0/1;*1;=?>~B.=z-CB BZ*Vi5124,l<;f490B.901;*V901 n901 vh<;I)6 H02Df41=090,S<,5={24,5*
,5- 1>@fji5,5*90190fD*1 0, 6'BZ=s>1;=0fD1;=09),812G*fD*>@16'BW1=b24B}/1;y.f 5=|24fD=01^BZ2%K
−10
r
−15
0
50
100
150
10
0
−10
−20
0
50
Time [s]
Signal Control of Altitute
T [N]
e
0
50
100
Time [s]
Angular Speed [Ωθ]
q [deg/s]
40
20
0
0
50
100
Time [s]
150
40
20
0
150
w in blue & Vz in red, [m/s]
δ [deg]
60
0
−20
150
Signal Control of Speed
50
−50
100
Time [s]
0
50
100
Time [s]
Speed on the axe Z in Rd & R0
150
2
0
−2
−4
0
50
100
150
Time [s]
!"#%$&('$)*+-,/.0
13254/6)78259;:=<>:@?A78<CBDBD?FE/4HGI9FJK4=2ML=27N?FOD254/6)JK9
PIQNRSQUTWVYX[Z%\]T^[_`X[TWVYa_ Bb ¡SR]_cVdX[TeX[ZRfTWaaWZgcbhTWbiTWVYj[VQcb[_Uk[_ RfZl_ bB¡QcV0\]TeaW_ b \]ZX TWR]g`TVYmn_CX[TWV%_
X[Z%\]aR],T ¡]_e\]Zocp[ZTTb ¡]T ¸
H(z −1 ) =
0.001839z −1 + 0.001722z −2
1 − 1.822z −1 + 0.8219z −2
st_ \eucQNV_ RfTW\vaW_ b[\fZX[TRSQcX _ \nj[QcRSQiV(QhwFp[b[aWZ%g bx_cyCzfT,¡]Z%u _iwFp[TWRf_ b
Ks=Q|VT+}UX[T~aW_ bB¡]R]_ V€p[T‚\]T~Tb[aW_ bB¡]R]gƒT\j¸
λ = 0.6
q % º r
N1 = 1 { N2 = Nu = 10 {
„
n s€<O z d ‘ x)
δg (k) = 0.9717δg (k − 1) − 0.0282δg (k − 2) − 38.5727ψ(k) + 64.3467ψ(k − 1) − · · ·
+0.0029ψref (k + 1) + 0.0108ψref (k + 2) + · · · + 0.1609ψref (k + 10)
4 p )KjJ ?t
,G,5*/1;*I)2D>~BZ90,5*,5-)>1;=0fD90,5*B.2GBZH024fD<^B./ 2B}241*S901<,5=?>@/,.2GHCB./@BW24B}90f4/1;<;<f%5=*,5-0/1k12G6 ,l90124,
1;=\*fD6}I024B.<;f&%5=b*,5=xH0/1;*1;=?>~B.9),5*1;=x24B53Cy.I0/@B|4:KDJƒ6lK P*M>@,/1*I02j>~B.9),5*V6}I)1;*>/@B.=nI0=CBS1 :fDy51;=0<fB
6'B *5,./B|24,.*BZ<>@I0B.90,5/1;*;vG*fD>ICB.<;f&%5={901;*1^B.90BbHCB./BF2D,5y5/B./I0=CB|1;*M>~B.-0fD24fDr^B.<f%5=x9)1;27*fD*>@16'B1=
I0='>@fD1;68H€,6 B.*7<;,5/M>@,B.2*,52Df4<fD>~BZ90,HCB./B1;2X&\[1]vbsI01;/f41;=)90,68,5*>/@B./sbsI0131;2X1 :f4y.f4/7B.2*fD*>1;6 B)v
1;*>@B.-0fD24f4r;B.<;fD,5=01*36 B.*/`ZH0f49CBZ*VB82B90f4=`Z6 fD<^B}9012 *f4*>1;6 B)v*1k/1C110B.=n1;=\B.<<;fD,5=01;*3901<;,5=?>@/,52
6'B.* )UI)1;/>1;*K
Speed Control, vr|red & vs|blue & βblack
Heading Control
100
v [m/s] & β [deg]
ψr in red & ψs in blue, [deg]
"
50
0
−50
0
50
100
Time [s]
150
10
5
0
−5
200
0
Signal Control of Heading
δg [deg]
T [N]
0
50
100
150
Time [s]
Angular Speed [Ωψ]
200
40
20
0
200
0
50
100
150
Time [s]
Speed on the axe Y in Rd
200
0
50
200
0.2
v [m/s]
10
r [deg/s]
150
60
0
0
−10
100
Time [s]
Signal Control of Speed
50
−50
50
0
50
;K
»
100
Time [s]
150
200
0
−0.2
100
Time [s]
150
MK +;"# " 3 * ) " f!&/('+$)*+-,/.0
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Q ¶Ko p RSQe [
„T»
3D
:
5
0
100
200
Time [s]
300
−5
−10
−15
Signal Control of Speed
δe [deg]
−200
0
−50
0
0
−50
100
200
300
Time [s]
Angular Speed [Ω ]
q [deg/s]
0
100
200
Time [s]
8:9<;>=-4)KjJ4~?
0
H C GM <JK(SC KH
100
200
300
Time [s]
Angular Speed [Ω ]
10
0
−20
300
300
ψ
20
−2
100
200
Time [s]
0
θ
2
0
0
Signal Control of Heading
50
r [deg /s]
T [N]
20
100
200
300
Time [s]
Speed on the axe Y & Z in Rd
0
100
200
Time [s]
5
0
−5
300
0
100
200
Time [s]
S&NOS M < S<JMnH1JCFH KH/GH1MRS FN GHFNOS-NOHQP GMRJ(KS N -T
300
Airship position during the flight
12
Position on axis −Z [m], Up
vblue & wred, [m/s]
300
0
50
40
−4
100
200
Time [s]
200
Signal Control of Altitute
60
0
0
Heading Control
400
δg [deg]
0
Altitude Control
0
ψr in red & ψs in blue, [deg]
zr in red & zs in blue, [m]
Speed Control
10
r
s
u in red & u in blue, [m/s]
Q
10
8
6
cruise flight
4
recovery of the system
2
0
100
0
start
−100
−200
−300
−200
−400
−300
Position on axis −Y [m], West
8:9<;>=-4)KDJ?
−100
0
100
200
400
300
Position on axis X [m], North
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Op 4)KjJ 5t
ẋ = f (x, u)
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901<,:,5/901;=0B.9CB K P=n1;2Q<^BZ*,S9012 9)f4/fDy5f4-)241.v 1*VI0=n<^B.68H€, 90~1 )UI)1;/r;B.v* *F1*V68,:90f 3C<^BZ90,H€,5/
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= */,.>@,./901<;,524B)K
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"
#
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#
p )KjJ ?t
4 ∂f 0 0
∂f 0 0
f (x, u) = f (x0 , u0 ) +
(x , u )(x − x0 ) +
(x , u )(u − u0 ) + ς(x − x0 , u − u0 )
∂x
∂u
2
*
δx = x − x0 δu = u − u0
&% 4 #
B’
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2BW1;<ICB.<;f 5= )KjJ }H0I01;9)1*1;//1*<;/fD>@B8<,56 ,
Op 4)KjJu]?t
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δ ẋ =
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∂f 0 0
(x , u )δx +
(x , u )δu
∂x
∂u
v
€
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δu
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/1()U1;/1;=0<fB δv *1k>f41;=)!1 ’
δ ẋ = F (x0 , u0 )δx + G(x0 , u0 )δu
% 4 6 c)
(*
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Op 4)K6:J t
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δu = −K(x0 , u0 )δx + δv
"
"
Op 4)K6Z65t
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8 9<;>=D4)KjJ
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m
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2
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Op 4)K6 ?t
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1//,./1;=F1;*>@B.90,81;*M>~B.-02D1=lI)24,0v0*1>@f41=01 ’
Z
δFm = −K(u0 , Fm 0)δu + (δuref − δu)dt
(*
€4 6 a'
*
s2 + 2ξwo s + wo2
p 4)K6 5t
‹1^B.2Df4r;B.=090,2BWfD=s>1;y5/B.<;f&%5=|90124BW241(*|9)1<,5=?>@/,.2 4:K76 :vC*1>@f41=01k2BW1 :H0/1;*f&%5=xHCBZ/@B1;2Œ<;,56 H€1=0*@B.9),5/901ki.1;2D,:<f49CB.9*f4y5I)f41;=?>!1 ’
−2ξwo m−ρSref CT t uo
m
cosµ
mwo2
ka (uo ) = cosµ
k(uo ) =
Op 4:K76 5t
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1;=1;2€<;,5=?>/,52€& v0HX,5/e2D,>~BZ=s>,0vl*1[H0/1;*1;=?>~BW1;2 UI0=0<f4,5=0B.6 fD1;=?>@,901;2€<;,5=?>/,52€B.=?>@1[H)/I01-CB.*E901
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blI)1'1;27/1;y5I024B.90,5//1^B.<<;f4,.=CBbB|2BZ*kHX1;/M>@I0/-CB.<f4,5=01*9)1;27/I0fD90,0vG68,5*>/@B.=)90,b*1 CBZ241;*9018<;,5=?>@/,52
blI)1/1^B.<<;f4,.=CB.=v0*f4=F*@B.<;B./9012BWr,5=CBW9011;*M>~B.-0fD24fD9CB.9|B.2*f4*M>@1;6 B)K
Z
−2ξwo mu(t) 1 ρSref CT t u2 (t) mwo2
Fm (t) =
(uref (t) − u(t))dt
−
+
cosµ
2
cosµ
cosµ
b
d.
\[1]
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‡,
&%
3
54 a'
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x
/ c€ ‘Cd
Speed Control
6
5
4
s
v in red & v in blue, [m/s]
7
3
r
2
1
0
0
20
40
60
80
100
Time [s]
120
140
160
180
200
120
140
160
180
200
Signal Control
50
T [N]
40
30
20
10
0
0
20
40
60
80
100
Time [s]
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ż = −u sin θ
θ̇ = q
q̇ = k2 q + k1 θ + k3 δe
!’
p4)K76Z]?t
δe0 = 0
θ0 = 0
q0 = 0
p4)K4?t
δ ż = −u cos θ0 δθ
δ ψ̇ = δq
δ q̇ = k2 δq + k1 δθ + k3 δδe
B’
p4)K4:J t
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"
fG24B}241(*S901<;,5=?>@/,52*11!)H)/1;*B <,56 ,
δδe = −K(z0 , θ0 , q0 , δe0 )δx +
Z
(δzref − δz)dt
pO4)KR465t
09 ,5=)901 δx 1*2B}iZB./f4B.<;f&%5=901;2i.1;<ƒ>@,5/3901k1;*>@B.90,0v K = [k , k , k ]T 12Œi.1;<ƒ>@,5/901ky?B.=CBZ=0<;f4B.*
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(s2 + 2ζwns + wn2 )(s + 1/τ1 )(s + 1/τ2 )
q2
%
ZY
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p4)K4 4?t
n s€<O z d ‘ x)
δe =
2
wn
−u cos θk3 τ1 τ2
R
(zref − z)dt −
τ2 +τ1 +2ζwn
q+
τ 1 τ 2 k3
2
2ζwn +τ1 +τ2
z
τ1 τ2 k3 u cos θ
1 τ2 (wn +k1 )
− 1+2ζwn (τ1 ττ21 )+τ
θ+
τ 2 k3
p4)KR4 st
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{
Speed Control, vr|red & vs|blue & αblack
Altitude Control
0
v [m/s] & α [deg]
zr in red & zs in blue, [m]
#
−5
−10
−15
0
50
100
150
10
0
−10
−20
0
50
Time [s]
Signal Control of Altitute
T [N]
0
50
100
Time [s]
Angular Speed [Ωθ]
q [deg/s]
40
20
0
0
50
100
Time [s]
150
40
20
0
150
w in blue & Vz in red, [m/s]
δe [deg]
60
0
−20
150
Signal Control of Speed
50
−50
100
Time [s]
0
50
100
Time [s]
Speed on the axe Z in Rd & R0
150
2
0
−2
−4
0
50
100
150
Time [s]
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jKQcRSQ TWV X[TW{ \]QcR]Rf_ VV%_ X[TnV(Q`VT+} X TƒaW_cbT¡]R]_ V TWV mv_X[TV_ R]TX[p[aWZ%X[_ j[QcRSQ V(Q X ZR]TaWaZg b TWbMTVj[VQcb[_
q r R]TwF_cR]m p[VQcb[X[_|Tb VQ|TWaWp[QcaWZ%g b`\]Zocp[ZTTb ¡]T
x, y {
{
„
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ψ̇ = r
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q * º r
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x
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δg 0 = 0
r0 = 0
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"
p4)K4 5t
δ ψ̇ = r
δ ψ̇ = k1 δψ + k2 δg
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wn2
δg =
k1 τ
Z
!
(ψref
v [m/s] & β [deg]
s
50
0
r
50
100
Time [s]
150
10
5
0
−5
200
0
Signal Control of Heading
T [N]
δ [deg]
g
0
50
100
150
Time [s]
Angular Speed [Ωψ]
150
200
40
20
0
200
0
50
100
150
Time [s]
Speed on the axe Y in Rd
200
0
50
200
0.1
v [m/s]
5
r [deg/s]
100
Time [s]
60
0
0
−5
50
Signal Control of Speed
50
−50
Speed Control, vr|red & vs|blue & βblack
Heading Control
0
'901<;,5=?>/,52G*,5=bH)/1;*1;=?>~BZ90,5*31;=F2B Cy5I)/@B54)KDJ )K
100
−50
p4)K4 ?t
wn2 τ + 2ξwn
2ξwn τ + 1
r−
ψ
− ψ)dt −
k1 τ
k1 τ
(*
@3
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ψ in red & ψ in blue, [deg]
v)*1V>@fD1;=01^bsI01V2B1 )H)/1;*f%5=9)1;2h<,5=?>@/,52B.9),5/
2
o
s3 + s2 (2ξw0 + τ1 ) + s(wo 2 + 2ξw
) + wτo
τ
0
50
5
100
Time [s]
150
200
0
−0.1
100
Time [s]
"# " c c ) " f!&/('+$)*+
150
.
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j[RfTW\fTWBb ¡QvTWb V(‚
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3D
„
n Speed Control
10
5
0
100
200
Time [s]
300
−5
−10
−15
Signal Control of Speed
δe [deg]
0
−50
0
0
100
200
300
Time [s]
Angular Speed [Ω ]
−2
0
100
200
Time [s]
8:9<;>=-4)KDJ^]@?
300
20
0
−20
0
100
200
300
Time [s]
Angular Speed [Ω ]
ψ
0
0
100
200
Time [s]
300
5
0
−5
0
100
200
Time [s]
H1C GM QJ(KSCsK(H SNOSM < S<JMzH1JCFH_K(H /GHFMRS"FN G H1NS NOHQPGMRJK(SN&T
Airship position during the flight
recovery of the system
Position on axis −Z [m], Up
12
10
8
6
4
2
0
cruise flight
100
0
start
−100
−200
−300
−400
−500
−200
Position on axis −Y [m], West
8 9<;>=D4)K6Š?
300
10
20
−20
100
200
Time [s]
Signal Control of Heading
40
40
0
0
θ
2
q [deg/s]
, [m/s]
red
&w
blue
−200
r [deg /s]
T [N]
20
100
200
300
Time [s]
Speed on the axe Y & Z in Rd
v
300
0
50
40
−4
100
200
Time [s]
200
Signal Control of Altitute
60
0
0
Heading Control
400
δg [deg]
0
Altitude Control
0
ψr in red & ψs in blue, [deg]
z d ‘ x)
zr in red & zs in blue, [m]
ur in red & us in blue, [m/s]
s€<O −100
0
100
200
400
300
Position on axis X [m], North
H C GM <JK(SCsKH:MRJvHSM G!KH MRJC SC SZH C D
NHPGMRJ(KSN
&T
300
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‡
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90f )U1;/1;=?>@1*S1>@B.HCB.* ’ I0=CBofD=0*>/I0681;=?>@B.<;f&5% =Ž9012V90fD/f4y.f4-02D1.v+*1;241<;<f5% = 901n*f4*M>@16'Bo,.H€1/@BZ>fDi5, *
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1=1;2<,56 H)I)>~B.9),5/e16}-CB./<^B.90,)KqŒ, B.*90,5*eH0/fD6 1/@B.* HCB./>1;**1901;*<;/fD-€1=|<;,5=I0=|/1;2B >@fDi.,}y5/B.90,
901bH0/, U) I0=090fD9CB.9 1= 24,5* B.=01!:,5*  * /1;*H€1<>@fjiZB.6 1=?>@1.K "P2‚H0I)=s>,q901n24B’fD6 H02D1;681;=?>~BZ<;f.% =
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& [1]v $ & { ,k=0,k24fD=01^B.2
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"
Reference (red) & Outpu (blue) system
10
5
0
200
400
600
800
1000
Time [s]
Signal Control
1200
1400
1600
200
400
600
800
1000
Time [s]
Type of Control
1200
1400
1600
200
400
600
1200
1400
1600
80
60
40
20
ELC=1,GPC=2,PID=4
Thrust propeller [N]
Aerodynamic speed [m/s]
"
0
5
4
3
2
1
0
"!$#&%('*)+#-,/.1032&04,5#7698:.<;1.
800
Time [s]
1000
09%=09,?>@8>@'A09B;)+0C;&,D.-8E)@8GF-8:H9,I0KJ/)+LM%-N OQPQ)@8EB04)3)[email protected]>@ST,U';<.1#
V n z d ‘ x)
reference (red) & Outpu (blue) system
15
10
5
0
300
400
500
600
700
800
Time [s]
Signal Control
900
1000
1100
1200
1300
300
400
500
600
700
800
Time [s]
Type of Control
900
1000
1100
1200
1300
300
400
500
600
700
900
1000
1100
1200
1300
80
60
40
20
0
ELC=1,GPC=2,PID=4
Thrust propeller [%]
Aerodynamic speed [m/s]
s€<O 5
4
3
2
1
0
800
Time [s]
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Reference (red) & Outpu (blue) system
−18.5
−19
z [m]
−19.5
−20
−20.5
−21
−21.5
160
180
200
220
240
260
Time [s]
280
300
320
280
300
320
Signal Control with GPC
40
δe [deg]
20
0
−20
−40
160
180
200
220
240
260
Time [s]
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3
Reference (red) and Output (blue −−) system
Speed [m/s]
8
6
4
2
Forecepropeller [%]
0
250
300
350
400
450
Time [s]
Signal Control
500
550
600
250
300
350
400
450
500
Time [s]
FO=1,GPC=2,PID2=3,PID=4
550
600
250
300
350
550
600
100
50
0
Type of Control
5
4
3
2
1
400
450
Time [s]
500
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2
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Reference (red) & Outpu (blue−−) system
400
ψ [dec]
300
200
100
0
1000
1050
1100
1150
1100
1150
Time [s]
Signal Control (PID)
40
δg [%]
30
20
10
0
1000
1050
Time [s]
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Reference (red) & Outpu (blue−−) system
−360
ψ [dec]
−380
−400
−420
−440
−460
1360
1370
1380
1390
1400
1410
Time [s]
1420
1430
1440
1450
1420
1430
1440
1450
Signal Control (GPC)
30
20
g
δ [dec]
10
0
−10
−20
−30
−40
1360
1370
1380
1390
1400
1410
Time [s]
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Y position [m], East
Displacement in the horizontal plane (2D), Dubins in red & SmoothingM1 in blue & SmoothingM2 in black
200
150
100
goal
start
50
0
0
20
40
60
80
100
120
X position [m], North
ψ evolution
140
160
180
200
ψ [deg]
80
60
40
20
dψ/dt [deg/sec]
0
direction of movement
0
5
10
15
20
25
30
Time [s]
dψ/dt evolution
40
45
50
Smoothing (model 1)
Smoothing (model 2)
Dubins
10
0
−10
0
5
10
15
20
25
30
Time [s]
!#"$&%')(+*,-.0/,12- %"3#"54"$6-.#7
8
35
35
40
45
ζi = [0, 0, 0]
50
55
ζf = [250, −250, 0]
6G4M,;wRy/*#x+,;4%(+w/;5
º
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<"€$2#xζ+f w!=
[250, −50, −90]
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T¹
I¹ к
T¹
YÀ
Y position [m], East
Displacement in the horizontal plane (2D), Dubins in red & SmoothingM1 in blue & SmoothingM2 in black
50
0
start
goal
−50
0
50
100
150
X position [m], North
ψ evolution
200
250
50
ψ [deg]
0
−50
−100
0
5
10
15
20
25
Time [s]
dψ/dt evolution
30
35
40
45
dψ/dt [deg/sec]
10
Smoothing (Model 1)
Smoothing (Model 1)
Dubins
5
0
−5
−10
direction of movement
0
5
10
15
20
25
Time [s]
30
23 d ~ !"$# %&!'(*)+!,5&!7*-,K7w/,.B"%&Q<(+*-,:,7*-x+;w,;wx'&!#(+*-,
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35
40
45
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50
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ζi = [0, 0, 0]
ζf = [−100, 500, 180]
ζ1 = [250, −50, −90] ζ2 = [200, −300, 180] ζ3 = [−100, −250, −270] ζ4 =
/.[x\w,7&Q
[−100, −100, 90] ζ5 = [−50, 200, 0] ζ6 = [250, 250, −270] ζ7 = [200, 500, −180]
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ψ evolution of Dubins plan.
Positions in 2D of Dubins plan.
0
500
ψ [deg]
change of goal
y [m]
goal
0
start
−200
−400
−500
−200
−100
0
100
x [m]
200
300
0
200
300
Time [s]
ψ evolution of 1st Model plan.
Positions in 2D of 1st Model plan.
goal
0
change of goal
−500
−200
−100
0
100
x [m]
200
300
0
ψ [deg]
y [m]
change of goal
0
start
187+
400
−400
200
300
Time [s]
ψ evolution of 2sd Model plan.
0
goal
−500
−200
100
−200
Positions in 2D of 2sd Model plan.
500
400
0
start
ψ [deg]
y [m]
500
100
−100
0
−200
−400
100
x [m]
200
300
0
!#"$&%' ( *, - 0/, - %"3#"54"$6 -. 7
100
ζi = [0, 0, 0]
200
Time [s]
300
ζf = [−100, 500, 180]
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ṡ = 1−dc(s)
cos ψe
d˙ = va sin ψe
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7+
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20
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60
80
Time [s]
in red and ψsys in blue
200
100
0
100
0
20
40
60
80
100
Time [s]
2D position, Posref en red and Possys in blue
0
100
200
300
X position [m]
ψerror [deg]
400
500
0
20
40
60
Time [s]
80
100
300
Y position [m]
100
50
0
0
20
40
60
Time [s]
derror [m]
80
200
100
0
100
20
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5
0
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40
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80
100
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Xref in red and Xsys in blue
Yref in red and Ysys in blue
[m]
300
position
400
200
0
Y
Xposition [m]
600
0
20
100
0
0
20
40
60
Time [s]
derror [m]
80
40
60
80
100
Time [s]
2D position, Posref en red and Possys in blue
0
100
200
300
X position [m]
ψerror [deg]
400
500
0
20
40
60
Time [s]
80
100
100
10
Error [deg]
Error [m]
20
200
0
100
5
0
−5
0
300
50
−50
100
0
100
Y position [m]
ψdirection [deg]
ψref
40
60
80
Time [s]
in red and ψsys in blue
200
0
20
40
60
Time [s]
80
0
−10
−20
100
U % !"/#6'&)(*% ,"+&.(& 02&)A,(„3!"+&‡"Q#%7 "/2& ,2(7
X
ref
in red and X
sys
in blue
Y
ref
[m]
position
400
200
0
0
20
0
0
20
40
60
Time [s]
derror [m]
80
100
0
20
40
60
80
100
Time [s]
2D position, Posref en red and Possys in blue
0
100
200
300
X position [m]
ψerror [deg]
400
500
0
20
40
60
Time [s]
80
100
200
100
0
100
5
Error [deg]
2
Error [m]
in blue
300
50
0
−2
sys
200
0
100
Y position [m]
ψdirection [deg]
ψref
40
60
80
Time [s]
in red and ψsys in blue
100
−50
in red and Y
300
Y
Xposition [m]
600
0
20
40
60
Time [s]
80
100
0
−5
−10
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red
and Sys
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blue(without wind) & green (with wind)
and Sys
red
blue(without wind) & green (with wind)
300
[m]
400
position
Xposition [m]
200
100
Y
0
100
200
Time [s]
300
400
Refred and Sysblue(without wind) & green (with wind)
−200
−400
0
100
200
Time [s]
300
400
derror [m], Sysblue(without wind) & green (with wind)
200
300
400
Time [s]
2D position, Refred and Sysblue(without wind) & green (with wind)
Goal
400
0
20
50
−40
3 Ò
0
100
200
Time [s]
300
0
−50
−100
400
) !" % # !" wind (1.5m/s)
−200
100
−20
without wind
200
40
0
100
with wind
−400
−100
0
100
200
300
Start
X position [m]
ψerror [deg], Sysblue(without wind) & green (with wind)
Error [deg]
Error [m]
0
Y position [m]
ψdirection [deg]
0
0
−200
−100
0
200
0
100
200
Time [s]
-5# !, / 300
400
!, + Airship position within 3D, tracking task
Position on axis −Z [m], Up
without wind
40
with wind
20
Start
0
−20
Wind (1.5m/s)
400
300
200
100
300
Goal
0
−100
−200
−300
−400
−500
Position on axis −Y [m], West
−600
Ò
3 (
I~
200
100
0
−100
Position on axis X [m], North
−200
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blue(without wind) & green (with wind)
I¹
Y
X
0
−200
100
200
Time [s]
300
400
0
−200
0
200
300
400
Time [s]
2D position, Refred and Sysblue(without wind) & green (with wind)
−200
−400
0
100
200
Time [s]
300
0
Wind (1.5m)/s
Start
−500
−200
400
ψ
20
20
−20
3 (9(
§©¨p§
100
200
Time [s]
300
[dec], Sys
blue(without wind) & green (with wind)
−20
0
5 ) !, % & # !, & 300
0
−40
400
with wind
0
100
200
X position [m]
−100
error
40
Error [deg]
Error [m]
derror [m], Sysblue(without wind) & green (with wind)
0
without wind
Goal
40
0
100
500
Y position [m]
0
[deg]
200
−400
0
Refred and Sysblue(without wind) & green (with wind)
−40
blue(without wind) & green (with wind)
400
position
position
200
direction
T¹ šº
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red
[m]
[m]
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200
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300
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(+w",;4, &!w$9>#;],-j
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Airship position within 3D, tracking task
Position on axis −Z [m], Up
without wind
40
with wind
20
Wind (1.5m/s)
0
−20
Start
400
200
Goal
0
400
300
−200
200
100
−400
0
Position on axis −Y [m], West
−100
−600
23 (Ž~
wº
Position on axis X [m], North
−200
! !' (*) + !, ! # - # !, !, !
+x(+w, +"P<$24%w/xI&!*My/*-x #w/"*3y/4%(>#(Wy/*-x+, &Q#xI&;w-,7wK&!4%w/x+w
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0.01482332769204x−30.66539713900427
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6000
35
5000
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4000
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0.01004850893208x −8.05131775009724
3000
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2500
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4000
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400
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The control signal
1750
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PWM [ms]
1650
1600
1550
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