Sea trials of high speed crafts data processing and model identification

IntevnationalConfevence on Fast Sea Tfansportation FAST'2005, June 2005, St.Petersburg, Russia

SEA T R I A L S OF H I G H SPEED C R A F T S DATA PROCESSING AND

M O D E L I D E N T I F I C A T I O N

Victor M.Ambrosovsky, Elena B.Ambrosovskaya St. Petersburg State Electrical Engineering University

prof.Popova str.,, 5,197376, St. Petersburg, Russia NAVIS CO. Obvodny emb.,14, 192019, St.-Petersburg, Russia

A B S T R A C T

The paper is devoted to the metliodic of sea trials data processing and mathematical model of ACV identifcation. The proposed methodic include: non-linear Kalman filtering of sensor data M'ithfaidt detection; smoothing of data using frequency-domain filtering; linear ACV model identification. For identification as topical maneuvers like turning as special maneuver - random binary rudder command ("telegraph" signal) are used. The approach is illustrated by the results of trial data processing of cargo ACV.

I N T R O D U C T I O N

In the problems of high speed craft control systems development, it is very important to use proper mathematical models of ship motion. Unfortunately, it is very hard to build such model using towing tank data, because it requires very large number of different experiments, and, besides that, these models may include large errors due to scale effect. That's why usage of sea trial data for obtaining high speed craft model is one of the most precision methods. However, in literature, practically, there isn't any data about high ACV identification. The aim of our paper is filling up of this gap.

In this paper, the approach to the sea trial data processing is illustrated by cargo ACV sea trials results. The main particulars of cargo ACV are: weight about 100 t, aii- cushion aspect ratio L/B= 2.0, ftill speed ahead Fn= 1.9, two Aero Variable Pitch Propellers (AVPP) and two air Rudders behind AVPP.

The aim of the trials was to estimate seakeeping and maneuverability of ACV. The measurement complex includes: gyrosemicompass, yaw, roll and pitch rate sensors, vertical reference sensor, 2-component Doppler radio log (speed and drift angle) and feedback units for AVPP and air rudders.

The following tests were done: tuming 30 degrees port and starboard, turning 30 degrees port and starboard with AVPP difference 6 degrees, 2 crash stops, 2 stoppings and special maneuver for identification.

:y

" • 13 ; 1 :; • • • / 14 i 6

*~^: y^

ii i f i i ! Ml — ^ 5 - - —

Fig. 1: ACV motion osciilogranm. The fragment of record 9

The fragment of record is shown in fig. 1. This record contains following signals: (1) - ship speed V = V u ^ + V ^ (4) - drift angle y5, (5) - a r rudda-s command, (6) - a r rudder port, (7) - air rudder starboa-d, (8) - AVPP port, (9) - AVPP sta-|x3ad, (10) - roll angle ^3, (11) - pitch angle 6», (13) - yaw angle ^ , (14) yaw rate r , (15) roll rate p ^ (16) -pitch rate q .

Data processing includes the foi I owing stages: 1. Kal man fil tering with fault detection based

on kinematical modës pairs "heading/yaw rate", "roll/roll rate", "pitch/pitch rate". 2. Data smoothing (based on frequency-domain

3. Calculation of stiip track (estimation of ship coordi nates usi ng speed, head! ng and dri ft values).

4. Raramdric identif ication of ship motion model.

1. K A L M A N F I L T E R I N G W I T H FAULT DETECTIONS

Let's consider par of signals "angle' angular rate" (heading/yaw hate, roll/roll rate, pitch/pitch rate). Simple ship motion model for these signds can be written as

>^(t) = <'(t);

x,(t) = X3(t);

)^(t) = x,(t);

where: x,(t) angle, x^it) angular rate; X3(t) -angular acceleration, <^(t) - disturbance (white noise).

I n di sa^e-ti me form equati ons can be ravri tten

as.-)C3(t + 1) = X3(t) + A.C(t);

X2(t + 1) = X2(t) + A.X3(t);

x,(t + 1) = x,(t) + A - X 2 ( t ) ; where A - sampling period.

State vector is X ( t ) = ( x 3 ( t) , X 2( t ) , X , ( t ) ) ' A matrix is

f

1

C matri X is:

A =

A

0 6"

1 0

A'/2

A 1

Measurement equati ons with faults ae:

y,(t) = >^(t) + c^(t)v,(t) + (c^(t)-1);7,(t);

y2(t) = X 2 ( t ) - f - q 2 ( t H ( t ) + (q2(t)-1);72(t);

cov(;7i(t))»cov(Vi(t)),

where

Vj (t) - low noise of i-th sensor (normal mode);

7 i ( t ) - high noise of i-th sensor (fault mode);

q (t) - flag of i-th sensor fault (q^ (t) = 1 when no fault, q ( t ) = 0 whan fault).

C =

0 1 0

0 0 1

Covai ance matri x of di sturbance i s

^cov(w) 0 0^

cov^-) 0 0

0 0 0

0 0 0

= A'

0

0

0 0

0 0

where w - i s di sturbance inti me-conti nuous system. The above model provi des to si mul ate f aul ts usi ng flag switching from q (t) = 1 to q (t) = 0. The essence of method of integrated filtering with alternated covariance is foi I owing. Output of both sensorsaeentered to Kalman filter input. When fault occurred, the vari ance of filtering error grows Filter detects this fact, and fault isd^ected. For fault localization, we use alternating R j matrix: when no faults

cov(v^(t)) 0

, 0 cov(v,(t))

when ecjiH fault of 1st sensor (angle sensor) is detected

'cov(;7,(t))

0

^ 0 cov(v2(t))^

when fault of 2nd sensor (angular rate sensor) is detected

^cov(v,(t)) 0 '

, 0 cov(;;,(t))J'

when fault of both sensors is detected

^cov(/7,(t)) 0 ^

. 0 cov(;7,(t))^ •

When R2 alternating as above, gan of filter provides the decreasing of influence of d a n c e d sensor measurements

Thus, f i I teri ng al gori thm i s f ol I owi ng: 1. CalculatePmatrix ^ f o l l o w s

P(t +

1)

= AP(t)A•^ +

R,-- AP(t)C^ [

R

^ + CP(t)C^

y

CP(t) •

initial condition is

P(0)

=

2,

O

0 ^

O « 2 O

O O

a^j

where large >0.

2. Matrix R2 is calculated using ^ v e rule. 3. Gain of Kalman filter is calculated:

K ( t ) = P(t)C^[F^ + C P ( t ) C ^ [ V

4. Esti mati on of state vector at ti me ^ 4 .

m = X(t - 1 ) + K ( t ) ( y ( t - 1 ) - CX(t - 1 ) )

where

X(0) = (0, y,(0), y 2( 0 ) ) \

y ( t ) = (ca,(t), M/(t))',

5. Predi cti on of state vector at ti me t + 1

x ( t + 1|t) = A x ( t ) ;

6. Opti mal esti mati on at ti me t + 1

= C x ( t ) ,

^X2

2

(tr

X i ( t )

where XjCt) estimation of angular rate, x.,(t) -estimation of anie

There is no fault, when

M easurements are i n range; Esti mati ons are i n range;

Absolutevalueof filter error issmaller then predefined limit;

A bsol ute val ue of 2"^ di ff erence of measurements i s smal I er then predefined limit.

For angle sensor, limits are

Range for measurements and estimations a^,^,a^ Limit for filter error = k^ör^g^A Limit for 2"" difference = k , ( : ^ _ A ^

z max For angular rate sensor, limits are:

Range for measurements and estimations = k^a^

Limit for filter error =k4t:^^3^A Limit for 2""^ difference =kö'^g^A^

Coefficients k,,...,k5 are determined from simulation (in range 1..2).

Whenfaultof one of sensor is detected, ki,...,k5 a-e increased in 2..4 times.

2. A P P L I C A T I O N OF K A L M A N F I L T E R I N G W I T H FAULT DETECTION TO SEA T R I A L S DATA PROCESSI NG

Resul ts of f il teri ng are shown i n f i g.2-4. Sampling period A =0.172a

Fig.2. Filtration of ACV roll and roll rate (record 9) with faults of roll angle sensor and roll rate sensor.

• •• f i il" •

-• - . |.; i.V,, . ^ .

*' ' ' s

••••-"•rj * •.• - v - - - T ' — ! l y . . .. — a — ' —ti—'•

• ——4

Fig.3. Filtration of ACV roll and roll rate (record 9, fragment) with faults of roll angle sensor a i d roll rate sensor. 1- sensor output, 2-filtered

Fig.4. Filtration of ACV yaw and yaw rate (record 9)

From f i gure 4 i t i s cl ear how to el i mi nate probI em of gyrosemicompass limitations using angle rate sensor

and compass measurement integration. Gyrosanicompass is measuring heading angle in range where parameter i / / ^ depends on concr^e type of compass. After achievement of value i / ^ , output of semi-compass s^stoO, This fact can fc)e considered as fault.

3. DATA S M O O T H I N G (FREQUENCY-D O M A I N F I L T E R I N G )

Sea trial data usually includes high-frequency components, induced by actual disturbances aid measurement noises. These high-frequency components contain no information and cai be filtered. Spectral analysis methods allow to cdculste the filtered agnal without its del a/ relative to measured signal (because we process sll measurements together, "off-liné').

The essence of é gori thm i s the foi I owi ng.

If w e h a / e measured signd x ( t ) , t e [ 0 , t ^ ] with constant sample A . Then we may use the foi I owing procedure:

1. Weassumethat our signal isperiodical with period t ^ . Under this assumption, we calculate discrete Fourier transformation X ( f )

( f e [ - 1 / 2 A , 1 / 2 A ] ) using FFT dgorithm [ 2] . ( X ( f ) is compi ex-valued function). 2. We choose windowing function in

frequency-domain W ( f ) (seefig.5). Thiswindowing function must satisfy following requirements

- W( f ) must be z ^ o at I f |> F^^,, where we suppose that frequency baid of signal is less than and all measurements with frequency

I f l> are noises

- Besides that, W ( f ) must be sufficiently smooth (for example, it can't be a rectangular window). This condition provides absence of extra (paasitic) oscillated processes after rdurni ng i nto ti me-domai n.

3. T h e n w e c d c u l a t e Y( f ) = X ( f ) W ( f ) (see fig.6).

4. We calculate inverse FFT and obtan y ( t ) = filtered x ( t ) .

The main problem on this way is that, if x(t) isnot periodical but transient (for ©(ample, ship speed -fig.4) then after filtering y ( t ) will be periodicd a i d itsvaluesin points t = 0 and t = will beequd

y(o) = y ( U ) = ( x ( o ) + x ( u ) ) / 2 .

We offer the modification of m^hodic described above

Let's add signal x ( t ) symmdrically before filtering

[x(t),t > 0

^x(-t),t < 0

Thissignd isdefined on intervd [ L a x . t m a x ] -For X ( t ) we use the above given procedure of frequency-doman filtering. As a result, we'll ha/e filtered agnd y ( t ) on interval

[-t^,t^].

y ( t ) = y ( t ) , t e [ 0 , 1 , 3 , ]

We have used f requency-domai n f i I teri ng d gori thm to process all signals (kinematic paanders), measured during cargo ACV t r i d s

Tirtie-domain v-indov.'W(tJ

Fig.5. Window function for frequency-domain filtering

f.Hl

|Y(i)l=l>;(DW(iii

li

Fi g.6.111 ustration for f requency-doma n f i I teri ng

The result of frequency-domain filtering of speed (acceleration maneuver), (NattolI'swindowing function = 0.1Hz) isshown in figure?.

Dnft angle,'

Fig.7. Results of frequency-domain filtering

Using filtered ship speed V = a / U ^ + V ^ , drift angle [5 and filtered heading ^

val ues, we can restore shi p tra;k (coordi nates Xg, Yg) by integration of following relations

Xg = ucos^-i-vsin^

Vg = - u s i n i / / +

vcos^

where tg^^ = V / U .

Ship track (turning maneuver) is shown in fig.8.

ACV Track

Fig.8. Ship track

2. I D E N T I F I C A T I O N

I n thi s secti on the probI em of i denti fi cati on of model of ACV yaw is considered. The structure of model is usually given from ship theory; but its coeffi dents it i s better to cal cul ate by usi ng paanetri c

i denti fi cati on procedure [ 1 ], [4]. The probI em of i denti fi cati on of shi p steeri ng dy nam! cs h ^ been considered by K.Astrom and C.KdIstrom [3].

ACV modë can be written in form [5]:

mu + mvr=X3,,;

m\/-mur=Y3,,;

where m isshipmass, J , ^ J y aeshipinertia moments, is rol I damping cod^ficient,

X^^^,^

^ s u m , ^sjm are sum forces and moments At small drift and roll angles, shipyaw lineaized modë can be rewritten:

^ + p„/? + p.^r + p,,<p = b,,S, + h,,S, + f, + g, (t);

r -f P 2 , / ? + p^^r + P23^ = b^,^^, +

b^^S, + f, +

9 2 ( t ) ;

^ + P31/? + P32r + P33^ + P340 = b,,S, + h,,S, + f3 + 93(1); where confidents p^ and b,j depends on ship

speed, is rudder angle, i s A V P P a i g l e di ff erence, f.,, f 2 , f3 are f uncti ons of speed a i d direction of encounted wind, g.,, 9 2 , g 3 a-e random ti me-f uncti ons

(dependson intensity, frequency and direction of wave).

Let's consider, for example, linear ACV yaw model without roll angle (we neglect the rol I influenceon yaw and dri ft) - i t i s acceptabi e i n course st^Di I i zi ng mode In this case, model has 2nd order:

= a , , ^ + ^ 2 ' - + A + M 2 + fi + g i ( t ) ;

r = a,,/3 + a^r + b^,S, + b^S, + f , + g,{t).

Ld's note that when we transfer this model into discrete form (rewrite this equations relative /? a i d

r measurements in discrete time), model (without wind and wave disturbance) will have ARX structure

a(q)y(t) = b(q)u(t-nk) + e{t)

but model (with wind and wave) will ha/e ARMAX structure

a(q)y(t) = b(q)u(t-nk) + c(q)e(t)

where q is dday, y(t) measurement, u(t) -control,

e(t)

- discrete noise,

a(q),b(q),c(q)

-polynomialswith constant coeffidents

We are considering active identification. In this case we can make special maneuvers for identification purposes. Input signals must satisfy the following

conditions: property of "persistent excitation" (i.e excite ttie ship at all frequencies in its frequency band). Besides that, input signal must be "realiz^le": not very long; its spectrum can't includes high frequency (because rudder actuator has low-frequency dynamics); its amplitude can't be too smd I (to a/oid nonlinear effects).

During the trials, we use the maneuver when input signal is "telegraph" signal)- fig.10.

It is characterized by two paameters amplitude a and minimal period T = k^-A . At low frequencies (npH si n ;!fr w ;dT ) spectrum of "telegraph" agnal is close to white noise spectrum - fig.9.

S(f)

f,

Fig.9. Spectrum of telegraph signal ^ x(t)

f. r«

Fig.10. Tëegraph signal

This maneuver satisfies the above given conditions Rudder angle as "tdegraph" signal realization (experiment 9) is shown in fig.10.

Fig.11. Rudder angle^"tëegraph" agnal realization (eKperiment 9)

For identification of ARX or A R M A X discrëe models coefficients we use mëhod of least squa-es (LS), method of general least squaes (GLS) a i d mëhod of instrumental variables (IV) [4].

This investigation shows that when disturbance actions (wind, wave) are essential, we cai use only IV and GLS mëhods, because LS estimations a e bi ased i n thi s case. There i sn't the essenti ë di ff erence between IV and GLS estimations for ACV identification. At low disturbance levë, the results of usi ng L S mëhod a e acceptabi a

The results of ACV yaw model identification a-e the following.

1. Frequency responses of modës identified by di ff erent expai ments are shown i n f i g. 12.

As it is clear from investigations, the maneuvers like turning provide good low-frequency part idaitlfication (identification of gan), but middle-frequency part can be better identified using special input signals("tdegraph" signal).

ai" • ••- ii. . • . - 1

osr.::C:Sv. -.1

»* m' yff tjtitt!}

:ï>;s.

\

^

Fig.12. Frequency responses of identified modës

2. Identified modë coefficientsae

^-0.0043 0.0309 ^

A =

0.0135 -0.0895

B =

^0.0040 0.0013^

0.0240 -0.0065

V

These results correspond to the "average" transfer function (hatched band in fig.12).

3. The results of simulation (identified modë) a i d triës results, experiment no9, are shown in fig. 13. Some high-frequency errors can be induced by wave disturbance and measurement noise.

f r r

-Fig.13. Identified modë (1)and triës results (2) experiment 9

CONCLUSIONS _REFERENCES

I n thi s paper the methodi c of sea tri al data process! ng are proposed. It includes: Kalman filtering with fault detection of sensor data based on kinematic models; smoothing of data using frequency-domain filtering; model identification. The approach is illustrated by the trials results of cargo ACV. In palicula", data processing of sea trials is implonanted, and paameters of simple linear yaw model of ACV a e obtained.

The designed methodic and software (Matlab 5) can be used for processing of sea trial data for different kinds of ships

This software was used in NAVIS co. (St.-Petersburg) for data processing of sea trids a i d control systems development for high speed crafts

1. K.J.Astrom, B.Wittenmak. Computer controlled systems Theory and design. Prentice-Hdl, Inc. Englewood Cliffs, 1984.

2. J.S.Bendat, A.G.Rersol. Random Data Andysis and M e ^ r e m e n t Procedures John W i l ^ , 1986. 3. C.G.Kallstrom, K.J.Astrom. Experi ences of System I daiti f i cati on A ppl i ed to Shi p Steeri ng.

In: Automatica, Vol.17, N o i , 1981.

4. L.Ljung. System Identification: Theory for the User. Prentice-Hdl, Inc. 1987

5. ShipTheory Handbook. Edby Y.I.Voitkounski. In 3 Volumes Vol.3 Manoeuvrability of convention^ Ships Hydrodynamics of Gliders, Hydrofoilsaid Hovercraft.- Leningrad: "Sudostroenié', 1985. (in Russian)