modelling and identification of radiation-force models for ships … perez.pdf · modelling and...

Modelling and Identification of Radiation-force Models for Ships and Marine Structures

A/Prof Tristan Perez Leader Autonomous Systems & Robotics

School of Engineering, the University of Newcastle AUSTRALIA

Collaborators: Thor Fossen, Reza Taghipour, Kari Unneland, Togeir Moan, Olav Egeland

My work at CeSOS

2004-2007 •  Ship roll stabilisation

•  System identification of manoeuvring models

•  System identification of seakeeping models (rigid and flexible structures)

•  Dynamic positioning with constrained control

•  Control allocation for ride control of high-speed vessels

•  Adaptive wave filtering

•  Parametric roll (modeling and observer design)

2007-2012

•  Risk assessment of autonomous systems (aerospace)

•  Fault-tolerant systems

•  Energy-based control vehicle dynamics

CeSOS Conference 2013 | 27-29 March

Motivation


Hydrodynamic

CodeIden/fica/on CumminsEqua/on

Experiments&CFDModelwithViscous

Correc/on

Hull geometry

and loading condition

Non-parametric

models: frequency response functions

Parametric fluid

memory model


MSS FDI toolbox

Non-parametric seakeeping models

System identification of parametric fluid-memory models


Non-parametric seakeeping Models

RB Dynamics & Hydrodynamic forces


Two-day Tutorial, Lecce, Italy, July 2009 22

Potential Theory for Ships in Waves

The fluid forces are due to variations in pressure on the surface

of the hull

It is normally assumed that the forces (pressure) can be made of

different components

Excitation loads Radiation loads Total loads

⌧ = ⌧rad

+ ⌧res

+ ⌧exc

⇠ = J(⇠)⌫

MRB⌫ +CRB(⌫)⌫ = ⌧

Cummins’s Equation (1961)

Approximation RB dynamics:


MRB ⇠ = ⌧

Radiation forces:

⌧ rad = �A1 ⇠ �Z t

0K(t� t0)⇠(t0) dt0

(M+A1)⇠ +

Zt

0K(t� t0)⇠(t0) dt0 +G⇠ = ⌧

exc

Cummins’s Equation:

⌧ = ⌧rad

+ ⌧res

+ ⌧exc

Zero forward speed and small angles

⇠ = J(⇠)⌫

MRB⌫ +CRB(⌫)⌫ = ⌧

Frequency-domain Model

Radiation forces:

Frequency-domain non-parametric model:

Abuse of notation:


⌧ rad(j!) = !2A(!)⇠(j!)� j!B(!)⇠(j!)

[M+A(!)]⇠ +B(!)⇠ +G⇠ = ⌧exc

(�!2[M+A(!)] + j!B(!) +G)⇠(j!) = ⌧exc

(j!)

Summary non-parametric models

•  Time domain:

•  Frequency domain:


(M+A1)⇠ +

Zt

0K(t� t0)⇠(t0) dt0 +G⇠ = ⌧

exc

(�!2[M+A(!)] + j!B(!) +G)⇠(j!) = ⌧exc

(j!)

Ogilvie’s Relations (1964)

•  Frequency-dependent added mass and potential damping:

•  Retardation functions


A(!) = A1 � 1

!

Z 1

0K(t) sin(!t) dt,

B(!) =

Z 1

0K(t) cos(!t)

K(t) =2

⇡

Z 1

0B(!) cos(!t) d!

K(j!) =

Z 1

0K(t)e�j!t d! = B(!) + j![A(!)�A1]


System Identification of parametric fluid-memory models

Replacing the convolution

Due to Markovian properties of the state-space model, significant gains in simulation speed can be obtained.

The parametric model is also convenient for analysis of stability and for control and estimation.


(M+A1)⇠ +

Zt

0K(t� t0)⇠(t0) dt0 +G⇠ = ⌧

exc

µ =

Z t

0K(t� t0)⇠(t0) dt0 ⌘ x = A x+ B ⇠

µ = C x+ D ⇠,

Replacing the convolution

•  Time domain identification:

•  Frequency-domain identification:


B(!) K(t)

K(s)

A BC D

�Data Mapping Indentification

B(!),

A(!),

[A1]

Mapping

K(j!)

Indentification Mapping A BC D

�Data

Properties – background information


The following properties derive from the hydrodynamics, and have implications on the parametric models:

Consistency and rationality requires us to use all this information in the identification problem.

coe�cients of the transfer functions are obtained, a state space realisation can be obtained using any of the

standard canonical forms—see, for example, Kailath (1980).

4. Constraints on Parametric Approximations

An important aspect of any identification problem is the amount of a priori information available about the

dynamic system under study and how this information is used in the identification method to set constraints on the

model structure and parameters. Table 1 shows on the left-hand side column five properties of the fluid memory

models. These properties were derived by Perez and Fossen (2008b) using hydrodynamic properties of the radiation

potential. The right-hand side column of Table 1 summarises implications the propertieshave on the parametric

model approximations (12).

Table 1: Properties of Retardation Functions and Implications on Parametric Approximations.

Property Implication on Parametric Models

1) lim!!0 K(j!) = 0 There are zeros at s = 0.

2) lim!!1 K(j!) = 0 Strictly proper.

3) limt!0+ K(t) 6= 0 Relative degree 1.

4) limt!1 K(t) = 0 Input-output stable.

5) The mapping ⇠ 7! µ is Passive K(j!) is positive real.

The low-frequency asymptotic value (property 1 in Table 1), establishes that the transfer functions Kik

(s) in

(12) have a zero at s = 0. This can be used to set the constraint b0 = 0 in (12):

Kik

(s) =P

ik

(s)

Qik

(s)=

pr

sr + pr�1s

r�1 + ... + p1s

sn + qn�1sn�1 + ... + q0

. (13)

The high-frequency asymptotic value (property 2 in Table 1) establishes that the trasfer function models are

strictly proper, that is deg(Qik

) > deg(Pik

) since this result in the denominator growing faster than the numerator

as the frequency increases, and hence, the frequency response tends asymptotically to zero. The initial-time value

(property 3 in Table 1), however, imposes a further constrain on the relative degrgee. This property establishes

that the relative degree of Kik

(s) is exactly 1. Indeed, for the entries Kik

(j!) that are not uniformly zero due to

symmetry of the hull, the initial-value theorem of the Laplace transform establishes that

limt!0+

Kik

(t) =2

⇡

Z 1

0B

ik

(!) d! 6= 0,

= lims!1

sKik

(s) = lims!1

sP

ik

(s)

Qik

(s)=

sr+1pr

sn

.

(14)

This will hold if n = r+1. Hence, the relative degree of the approximations must be 1, that is deg(Qik

)-deg(Pik

)=1.

The final-time value (property 4 in Table 1) establishes that the impulse response tends to zero as time goes to

infinity. This is a necessary and su�cient condition of bounded-input bounded-output stability of the models.

With regards to the property 5 in Table 1, the negative feedback interconnection of passive systems is passive; and

thus, stable under observability conditions (Khalil, 2000)—further details about passivity of fluid-memory models

5

System Identification

System Identification = Model structure selection + Parameter estimation

Time-domain identification:

•  LS-fitting of the retardation function (Yu & Falness 1998)

•  Realisation theory (Kristiansen & Egeland 2003)

Frequency-domain identification:

•  LS-fitting of the retardation frequency response (Jeffreys 1984), (Damaren 2000)

•  LS fitting of added mass and damping (Soding 1982), (Xia et. al 1998), (Sutulo & Guedes-Soares 2006).


Distortion of the retardation function


K(t) =2

⇡

Z 1

0B(!) cos(!t) d!

LS-fitting retardation function

§  Complicated Optimisation problem.

§  The non-unique model structure.

§  Order of and initial parameters not easy to infer.

§  Makes no use of the prior knowledge.


Realisation Theory


Discrete-time approximation:

Steps: 1)  Form a Hankel matrix with the impulse response samples. 2)  Do a singular value decomposition (SVD). 3)  Obtain the order from the number of non-zero singular values. 4)  Obtain the model matrices via factoriastion. 5)  Convert the model to continuous time.

)

Realisation Theory

•  Relatively easy to implement. •  Does not require initial parameter estimates. •  Allows order detection.

•  Starts from a distorted impulse response. •  Requires conversion to continuous time (further distortion). •  Makes no use of prior information. •  In general poor model quality.


Realisation theory – order


The distortion of the impulse response may affect the order selection. The containership example suggests K55(s) order 2 to 5

Example containership DOF3,3


Reconstruction of damping and added mass from the parametric approximation:

Model not passive B33(w)<0!

Frequency-domain identification


Then we can estimate the parameters via LS using the frequency response computed using the data generated by hydrodynamic code:

The i,k entry of K(s) can be approximated by a rational transfer function:

This problem is non-linear in the parameters, but it can be linearised.

K(j!) = B(!) + j![A(!)�A1]

Prior knowledge and constraints

From the physics of the problem we know that the transfer functions have

1.  Relative degree 1

2.  Hik(s)=0 for s=0

3.  Stable

4.  Passive

5.  Minimum order approximation is 2

Some of these properties can be enforced in the structure of the model and its parameters without complicating the optimisation.


Example containership


Realization Theory FD-Id with constraints

By imposing constraints on the model structure and parameters, we obtain a model that satisfy all the properties of the retardation functions and have a better quality.

Example semisub


Data from marinecontrol.org

Surge – order 5


Heave – order 8


Roll-sway – order 8


Couplings are not necessarily passive B(w)<0

What are we doing estimation wise?

Can we relate our point estimates to characteristic of the posterior for the parameter vector?

Bayes-Laplace Theorem:


p(✓|DI) =p(D|✓, I)p(✓|I)

p(D|I)

Measurement model:

Assumption 1 - Gaussian uncertainty:

Assumption 2 – i.i.d. uncertainties Assumption 3 – diffuse prior for the parameters



K(j!) = U(!) + jV (!)

nl ⇠ N (0,⌃)

Dl = f(!l, ✓) + nl

,U(!l)V (!l)

�= f(!l, ✓)


Data:

The likelihood function (measurement model):


D = {D1, D2, . . . , DN}Dl = f(!l, ✓) + nl

U(!l)V (!l)

�= f(!l, ✓)

p(D|✓, I) /Y

l

exp

1

2

(Dl � f(!l, ✓))T⌃

�1(Dl � f(!l, ✓))

�

=exp

"1

2

X

l

(Dl � f(!l, ✓))T⌃

�1(Dl � f(!l, ✓))

#


Our estimates based on NLS correspond to the maximum a posteriori (MAP) estimates

under the following assumptions:

Assumption 1 - Gaussian uncertainty:

Assumption 2 – i.i.d. uncertainties

Assumption 3 – diffuse prior for the parameters


✓NLS ⌘ ✓MAP = argmin✓

X

l

(Dl � f(!l, ✓))T⌃�1(Dl � f(!l, ✓))

Extension to the 2D data

•  Some Hydrodynamic codes use strip theory to compute hydrodynamic non-parametric models.

•  The infinite frequency added mass is not available.

•  This is a simple extension to the frequency-domain method.


Example FPSO


DOF 33 joint estimation


DOF 35 and 53


Estimated added mass

Infinite Frequency Added mass coefficient estimates:


MSS FDI toolbox


FDIRadMod.mω,A(ω),B(ω),A∞

FDIopt

Krad(s),

[A∞]

EditAB.m Ident_retarda>on_FD.m

Fit_siso_fresp.m

Ident_retarda>on_FDna.m

Demo_FDIRadMod_WA.m

Demo_FDIRadMod_NA.m

Conclusion

•  Parametric fluid-memory models are used in simulators and also in analysis and design of wave–energy converters.

•  Several methods have been proposed to identify parametric models from data computed by hydrodynamic codes.

•  The identification based on the frequency response of the the retardation function have advantages over other methods:

–  No need to compute high or low frequency data –  No need to guess initial parameters

–  Automatic model order selection

–  Incorporates prior knowledge

–  Simple parameter estimation problem.


References

Perez, T. and T.I Fossen (2011),”Practical Aspects of Frequency-Domain Identification of Marine Structures from Hydrodynamic Data”, Ocean Engineering, Volume 38, Issues 2-3, February 2011, Pages 426-435.

Perez, T. and T.I. Fossen (2008) “A Matlab Toolbox for Parametric Identification of Radiation-Force Models of Ships and Offshore Structures.” Modelling Identification and Control, Vol. 30, No. 1, pp. 1-15. DOI:10.4173/mic.2009.1.118.

Perez, T. and T.I. Fossen (2008) “Joint Identification of Infinite-frequency Added Mass and Fluid-Memory Models of Marine Structures.” Modelling Identification and Control, Vol. 29, No. 3, pp. 93-102. DOI: 10.4173/mic.2008.3.2.


Thank you


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