Z[IAO Xi-ren,et al:Probability modeling for robustness of muhivariate LQG designing based on ship lateral motion 9 19 9 Table 1
Hydrodynamic parameters of a ship
f~ =
D-~A~X +D-IB1LR +D-~CIWf AAX +Bu +CWz,
a,, hy(Iro(lynamicparameters b~i hydrodynamic i)arameters
(3)
az~
3. 6825e +005
bz2
a_,~
-2. 983 5e +005
b24
6.6819(,+003 -5. 2408e +003
a:~
-2.792 le +006
b~6
- 3. 3675e +006
A =D-IAI ,B =D-IB~ ,C =D-ICI ,u = L R , w e choose
6/42 a,,~
-2. 983 5e +005 8. 2524e +005
b~,_ b~
- 5. 2408e +003 1.1192e +004
swaying displacement y, roll angle ~ and yawing angle
6/4~
3.3642e +006
b46
2. 8100e +006
ae,2
2.621 7 e + 006
b62
6/~
3. 4447e +006
b~
3. 454 8 e + 006 -2. 721 3e +006
6/~
5. 894 7e + 008
b~
2. 3002e + 006
where
as measurement variables, then we can get the measuring equation
r = ttx + v,
Notes: the speed of 18 kn and the course of 45 ~ under Rank 5
H=
where,
state of sea.
(4)
i, 0 0 0 0 i] 0
0
1
0
0
-0
0
0
0
1
.
Usually we call the hydrodynamic parameters giv-
V is 3 dimensions measurement noise, which is often
en in Table 1 nominal value. For navigation conditions
regarded as white noise and whose covariance is Q,, =
change and testing calculation is not completely pre-
diag[0.81
cise, the hydrodynamic parameters will fluctuate a-
of Rank 1.
9 •
16 •
chosen by sensor
round nominal value. Formula ( 1 ) is differential equation of the ship
3
LQG DESIGNING
lateral motion model.
Based on separation principle, the design fi)r LQG
Suppose: X "r = I X l Ey
:;+
x2
,/,
x3
x4
x5
LF:
=
x6]
=
F(,], For-
C1WI,
(2)
1
0
0
0
+ a2~_ 0
m
0
0
0
a~_.I
0
a2~
0
0
1
0
0
0
0
a L,
0
I~ + a44
0
a46
0
0
0
0
1
0
0
a6~"
0
a6~
0
[66 + a66
BI : [ o
Aj =
K a l m a n state o p t i m u m e s t i m a t i o n with extended state variable
Random waves disturbing force and motion can be
where,
D--
estimation and calculating linear quadratic form laws. 3.1
mula ( 1 ) equals the following matrix equations: DJ( = A I X + B I L R +
controller consists of two parts: Kalman state optimum
recognized stationary process, ust, ally we call it color noise. Ira order to obtain optimum estimation, we can fit sea wave spectrum by forming filter" of wave disturbing force in rational form. Then we have white noise as input of the new extended system and use Kalman state optimum estimation to estinaate it.
07
1
0
0
0
0 0
0 0
1 0
00ll, 0
n 0
nZ~ 0
We have the following transfer function of forming fiher of wave disturbing force and momen( 4 : C(s)
bts
s
nXr],
+ als +
(5) a 2
Now we take sea condition 5, ship speed 18kn
-0
1
0
0
0
0
0
- b22
0
- b24
0
- b26
0
0
0
1
0
0
0
-b~2
-c,~
-b~
0
-b46
Filters
0
0
0
0
0
1
Gz(s)
_0
-b62
0
-b64
0
-b6o
and course heading 45 ~ for example and give the parameters of forming filter which is given in Table 2. Table 2
G,(s)
G~(s)
The parameters of forming filter
bj 6.29e04 9.05e04 1.18e06
a~ 0.214 0.230 0.232
6/2 0.125 0.113 0.110
with Eq. ( 2 ) .we can have the continuous state equaNotes: the speed of 18 kn and the course of 45 ~ under Rank 5 tion
state of sea.
Z[IAO Xi-ren,et al:Probability modeling for robustness of muhivariate LQG designing based on ship lateral motion 9 19 9 Table 1
Hydrodynamic parameters of a ship
f~ =
D-~A~X +D-IB1LR +D-~CIWf AAX +Bu +CWz,
a,, hy(Iro(lynamicparameters b~i hydrodynamic i)arameters
(3)
az~
3. 6825e +005
bz2
a_,~
-2. 983 5e +005
b24
6.6819(,+003 -5. 2408e +003
a:~
-2.792 le +006
b~6
- 3. 3675e +006
A =D-IAI ,B =D-IB~ ,C =D-ICI ,u = L R , w e choose
6/42 a,,~
-2. 983 5e +005 8. 2524e +005
b~,_ b~
- 5. 2408e +003 1.1192e +004
swaying displacement y, roll angle ~ and yawing angle
6/4~
3.3642e +006
b46
2. 8100e +006
ae,2
2.621 7 e + 006
b62
6/~
3. 4447e +006
b~
3. 454 8 e + 006 -2. 721 3e +006
6/~
5. 894 7e + 008
b~
2. 3002e + 006
where
as measurement variables, then we can get the measuring equation
r = ttx + v,
Notes: the speed of 18 kn and the course of 45 ~ under Rank 5
H=
where,
state of sea.
(4)
i, 0 0 0 0 i] 0
0
1
0
0
-0
0
0
0
1
.
Usually we call the hydrodynamic parameters giv-
V is 3 dimensions measurement noise, which is often
en in Table 1 nominal value. For navigation conditions
regarded as white noise and whose covariance is Q,, =
change and testing calculation is not completely pre-
diag[0.81
cise, the hydrodynamic parameters will fluctuate a-
of Rank 1.
9 •
16 •
chosen by sensor
round nominal value. Formula ( 1 ) is differential equation of the ship
3
LQG DESIGNING
lateral motion model.
Based on separation principle, the design fi)r LQG
Suppose: X "r = I X l Ey
:;+
x2
,/,
x3
x4
x5
LF:
=
x6]
=
F(,], For-
C1WI,
(2)
1
0
0
0
+ a2~_ 0
m
0
0
0
a~_.I
0
a2~
0
0
1
0
0
0
0
a L,
0
I~ + a44
0
a46
0
0
0
0
1
0
0
a6~"
0
a6~
0
[66 + a66
BI : [ o
Aj =
K a l m a n state o p t i m u m e s t i m a t i o n with extended state variable
Random waves disturbing force and motion can be
where,
D--
estimation and calculating linear quadratic form laws. 3.1
mula ( 1 ) equals the following matrix equations: DJ( = A I X + B I L R +
controller consists of two parts: Kalman state optimum
recognized stationary process, ust, ally we call it color noise. Ira order to obtain optimum estimation, we can fit sea wave spectrum by forming filter" of wave disturbing force in rational form. Then we have white noise as input of the new extended system and use Kalman state optimum estimation to estinaate it.
07
1
0
0
0
0 0
0 0
1 0
00ll, 0
n 0
nZ~ 0
We have the following transfer function of forming fiher of wave disturbing force and momen( 4 : C(s)
bts
s
nXr],
+ als +
(5) a 2
Now we take sea condition 5, ship speed 18kn
-0
1
0
0
0
0
0
- b22
0
- b24
0
- b26
0
0
0
1
0
0
0
-b~2
-c,~
-b~
0
-b46
Filters
0
0
0
0
0
1
Gz(s)
_0
-b62
0
-b64
0
-b6o
and course heading 45 ~ for example and give the parameters of forming filter which is given in Table 2. Table 2
G,(s)
G~(s)
The parameters of forming filter
bj 6.29e04 9.05e04 1.18e06
a~ 0.214 0.230 0.232
6/2 0.125 0.113 0.110
with Eq. ( 2 ) .we can have the continuous state equaNotes: the speed of 18 kn and the course of 45 ~ under Rank 5 tion
state of sea.
9 20 9
Journal of Marine Science and Application,Vol. 4, No. 4, December 2005
In Table 2, G: (s) is swaying force molding filter, G4 ( s )
is swaying moment of inertia molding filter,
G6 (s) is yawing moment of inertia naolding filter.
/r
is
Kalman
stable
state.
3.2
Linear quadratic form laws Criterion rvl of performance fi~r controlling is N
m.~ = fitim, ' 4- Ci + lot,
F~
i = 2,4,6.
(6)
J = El ~. [ x l " ( k ) Q , X ( k )
= Hirni,
uT(k
[00.1254 1
[
] C~: 0.21371 ' ~
[o o.,,31 ]
0
[o]
[00.1089
9.05e004 ' [
] C6= 0. 23241'
0
1]
H4'0
(11)
1
Q, = 100
'
QI = 1c6 x diag[5e2 3e3 5e9 2cl 4e7 8cl. ~'
H6[0
1
oooooo3 (12)
-'
combining them anti we have
vf
- 1) ] }.
]
1. 182e006 '
m = AfM
-- 1 ) Q 2 u ( k
N = 1 000, the whole controlling prtw.css
]
H~IO 6.294e004 ' -
C4 =
1 0. 22961'
1
+
k=l
Where
A6=
of
X( k + 1/k ) = ~ X ( k ) + -flu (k) is one step prediction
space :
A4 =
gain
value 76
Convert the above fln'ee transfer functions to state
A,: "
filtering
According Io Bellman optimization p r m c l p n n n - , we will have the optimum controlling laws.
+
Of W,
u(k) =-L-(k)X(k).
(7)
hyMn=Wf
(13)
L ( k ) is derived from the equations below
in the equation
{,k : [Q, + F r S ( k ~ 1)b'!-'FrS(k + 1 ) ~
M T= i'n2
m~ m6i,
S(k) = q~rS(k + l ) i ~ + F L ( k )
W ~ : i,,,~_
,,~ ~ , r~ r~,~,
S ( N ) = Ql,]f
Fjr- : iF2 Ar = diag~A2
N
-
1,N - 2 , - " , 1 , 0 .
When the hydrodynamic parameters fluctuate, the
Take white noise W as input of the new extended system, consider wave disturbing W f as intermediate
lateral motion model of ship will change, X expresses as ~;. The model for LQG designing is as follows
quantity and regard forming filter as a part of the new
~(k + l) = ,t,x(~) + F%(k) + ~W(k),
system, then the new extended system /-~l is as fol-
(15)
lowed :
Y(k+l)
.~ = A X + Bu + CW Y
(14)
STATISTICS M O D E L I N G M E T H O D
Q I,HI!A2 H~ H~!.
C4
Cr : d i a g [ C 2
4
A 4 A 6(,
=
+Q,-
=HX(k+I)
+V(k+l),
(16)
where q~, F, H are obtained from ~1~,F, H when their ,
(8)
HX + V
hydrodynamic parameters fluctuate at random on the proportional distributing with a range of + 10%, +
where,
~=
[A 0
2 0 % , _+30%. The concrete descriptions are as fol-
CHI],~ = [ B ] ,
lows.
AI J
1 ) When all the hydrodynamic parametcrs azj,bij
e: [o),,_
(i,j = 2 . 4 , 6 )
dora after the proportional distributing once, we obtain
discreted equation ( 8 ) 13:
one sample, we will sample 100 samples tbr LQ(, de-
[X(k + 1) = @X(k) + Fu(k) + i:w(k) tY(k+l)
HX(k+I)
Suppose Qff = Q,,. = diagl 1
fluctuate around nominal value at ran-
(9)
+V(k+l) 1
low.
1 ] , then
System in o p e . loop ~ ( k
X(k + 1) = X( k + l/k ) + K,,,[ Y( k + 1) H X ( k + 1//,-) 3.
signing and explain the meaning of No. k sample he-
(10)
+ I ) : 4' X ( k )
( k ) , whose component is Xop~~ closed loop X(k + 1) : ~ X ( k )
+ ]~V
; System !8; in
+FU(k) +FW(k),
ZIIAO Xi-ren,et a/:Pmbability modeling for rolmstness of nmhivariatc LQG designing based on ship lateral motion 9 21 Table 3
whose the component is Nch.se ( i, k) ; i = 1 , 2 , " - , N ; k = 1,2,--.,100;N
= 1 000
Results for roll angle
subinterval/%
rti
np,
( n i -np,. ) =/npi
( - zr ,22.5020)
0
0.0004
0. (X104
(22.5020,29. 13281
0
0.0257
0.0257
(29. 1328,35.76351
0
0.6345
0.6345
(35.7635,42.3943)
5
6. 1928
0.2297
(42.3943,49.0250)
28
24. 1554
0.6119
3 ) Calculate standard deviation of the model in
(49.0250,55.6558)
41
37.9823
0.2398
open cycle and closed loop, relative controlling effect
(55.6558,62.2865)
17
24. 1554
2. 1196
of closed loop against open cycle
(62.2865,68.9173)
8
6. 1928
0.5274
(68.9173,75.548(/)
1
0.6345
0.2105
( 75. 5480, ~ )
0
0. 0261
0. 0261
2) Calculate the mean of the model in open loop and closed loop A
x: .......... ( i , k )
- N - 1 100i___~t~s
( i, k) '
N --
1
;oh. .....
(i,k)
=
- N - 100,_.~mox ' ' l ' ' ' ' ( i ' k ) "
l
'~
=
(17)
=
1
cr,>~(k) = ,XS-N
Notes: + 10% I)erlu,'bation of hydrodynamic t)a,'ameter
113(32 [Xdo~,(i,k) -.L.,.,~.,,,,(i,k)22 ,
cG = [ (o',,,,,,_o',.a,,,)/O-o,~..- • 100%.
(18)
5
41 Hypothesis 1to: relative controlling effec! osubmits to normal distrilmtion
SIMULATION RESULTS In this artMe we do all sinmlation nnder Matlab6.
1. When a given ship navigation at the speed of 18 k n / {,r_ml)2
f ( ,,,~ ,~r ,, ,o" ) : - ~ e
h and the course of 45 degree under Rank 5 state of
=4 ,
sea, and the hydrodynamic parameters of the ship fluc-
I00
1 m I - 1002o'e(k),
tuate at random on the proportional distributing with a
k=l
1 ,,,~
range of - + 1 0 % ,
100
_ lo0Y_, o-;(k), k=l
_+20%,
_ + 3 0 % , t h e modeling re-
suits of all amounts are given in table 4.
o',, = ~ - m ~ .
(19)
5) P e a r s o n - x 2 check method. We will introduce the Pearson method by taking the example that a given
m r is the
mean value of the relative controlling effect, m~, is the mean value without perturbation of hydrodynamic parameter and tr:, is standard deviation. Table 4
Simulation Results
ship navigationg at the speed of 18 kn and the course of 45 ~ under Rank 5 state of sea anti its hydrodynamic
[)aJ lall~-'te'r~
m0
+ 10% mi
o-~
+ 20% m~ o'~,
+ 3(1% mt tre
parameters of the ship fluctuate at random on the proportional distributing with a range of + 10%. For relative controlling effect of roll angle q~, we know its normal distribution density function is f~. ( In.1 ,
O'p ~ X )
f ( 5 2 . 3 4 0 , 4 . 6 . 689 ;x) after the Hypothesis lIo,
:
and
the level of significance c~ = 0. 05. In order to verify
y
61.92 59.65 15.33 60.80 8.75 63.91
8.90
y
64.39 67.20 tl.46 66.82 9.13 62.59 9.65
4~
57.18 56.34 6.69 60.83 8.08 55.02 1(I.29
qb
66.11 65.48 9.92 68.36 10.14 67.64 13.69
~'J
71.16 68.12 15.61 72.11
7.56 73.18 7.50
75.88 80.00 6.63 79. 19 5.78 76.79 6.41
whether f, ( m I , (rp ; x ) can be accepted, we divide the precision interval into 10 subintervals and compute the
Notes: perturbation _+10% , +20% , +30% of hydrodynamic parameters
100 samples. The result is given in table 3. S
Because of ~ ( n, - npi )"-~rip = 4. 6256 < x 2 ( 9 , i: 1
0.05)
= 16. 92,
we accept H 0,f~ ( m l , c G ; x )
f(52. 3404,6. 6896;x).
=
6
APPLICATION Tile algorithm we describe above can hel l) us to
calculate the reliability of the LQC, designing. Taking sea condition 5, ship speed 18 kn and course heading
9 22 9
Journal of Marine Science and Application. Vol. 4, No. 4, l)eeember 2005
45 ~ for cxantple, from Table 4 we know the relative
"3iSUN Zheng-qi. Theolw of eomtmte corm'o[ and appLieationg
corttrolling effect of roll angle 6 submits to normal dis-
i Mj. Belling: Tsinghua University Press, 2000(in Chi-
tribution f , ( m l , o-p ; x ) ,f, (
I T / l , O'p ; X
) = f ( 52. 340 4 ,
nese ) . [4 1PENG Xiuyan. Engineering rando,n process [ M . llarbin:
6. 6896 ;X) When the reliability value is L, we cart calculate
llarbin Engineering University F'ress, 2000 ( in Chinese). 5 ! NIKOUIIAH R. WILI,SKY A S, BERNARD C I,. Kalman
the estimation precision b after lhe formula below: (m I ,o'p;x)dx =
Wlten L = 0 . 9 5 , 41.35%,
~ e ~ d x ~/2'rr
= L.
fihering and ficcati equations for dcsel'iplor systems [ J J. IEEE Trans Automatic Control, 1992. 37(9) :106 - 108. [6~ VALAPPIL J, GI:ORGAKIS C. Systematic estimation of
(20)
state noise statistics for extended kalman filters[ J]. AiChE
from Eq. ( 2 0 ) we can have b =
Journal/American Institute of Chemical Engince,s, 2000,
It sitows that when a given ship navigates at
the speed of 18 kn and the course of 45 ~ under Rank 5
46(2) :62 -65. L7 IGAUTIEt/ M, P()IGNET P. Extended Kalman fihering and
state of sea, and its hydrodynantic parameters fluctuate
weighted least squares dynamic identification of robot ~J ].
at random on the proportional distributing wilh a range
Control Engineering Practice, 2001 , 9( 12 ) : 1361 - 1372.
of -+ 10% , under probability of 95% the relative controlling effect of roll angle 4b is better than 4 1 . 3 5 % , and the mean
of the
relative controllirtg effect is
'- 8 ! GRIMBI,E M J. LQG Feed forward/feedback stochastic optimal control and marine applicalionl J ~. Transactions of the Institute of Measurenmnt and Conlrol, 1999, 21 ( 1 ) :30 31.
53.34%. 7
ZHAO Xiren is a professor of Control
Theory and Control Engineering in llarbin
CONCLUSIONS
Engineering Uniw'rsity. He received his 1 ) Bring up the method to model fi)r relative con-
B. S in the departrrlent of automaiio,1 con-
trolling effect of LQG designing based on ship lateral
trol of Tsinghua University in 1964 and re-
motion when the hydrodynamic parameters of ship fluc-
ceived P. E in the department of auto,nation control of llarbin Engineering Univel,'sity in 1980. IIe has
tuates. When the hydrodynamic parameters of ship
long pursuing the researches of random control theory, non-line
fluctuatcs, and we can give the reliability of LQG de-
system modeling arm prediction. He has gained many science
2)
signing under some probability. 3)
When the hydrodynamic parameters of ship
and technology, awards, published 2 books and more than 70 articles. He was honored to one of the leaders of this field.
fluctuates around nominal value at random on the pro-
PENG Xiuyan is a professor of Control
portional distributing , the relative controlling effect of
Theory and Control Engineering in llarbin
LQG designing submit to normal distribution and the
Engineering University. She received her
mean of relative controlling effect has no remarkable
B. S in the depa,'tment of automation con-
changes comparing to that without perturbation of hy-
trnl of Harbin Engineering University in
drodynamic parameter.
1985 and received P. E in the departmevt of automation control ot' llal'bin Engineering UJfiversity in 1988.
REFERENCES
She has long pursuing the searches of random cont,'ol theory,
[ l ] GAO llanqiu. Report of modeli,,g and control for ship lateral
non-line system modeling and prediction. She has gai,,ed many
motion r A-. Technical tteporl, Report NO. 32- Rj. Wuxi:
science and technoh~gy awards, published 1 books and more
Ship Research (2enter of China, 2004, 7 - 11 ( in Chinese).
than 30 articles. She was honorcd to one of the leaders ~>f this
:2~ KEI,LER J Y, I)AROUACII M. Optimal two-stage Kahnan fiher in the presence of random, bias [ J ]. Automatica, 1997, 33 (9) : 1745 - 1748.
field.
