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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
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Motivation
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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
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MSS FDI toolbox
Non-parametric seakeeping models
System identification of parametric fluid-memory models
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Non-parametric seakeeping Models
RB Dynamics & Hydrodynamic forces
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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:
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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:
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⌧ 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:
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(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
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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]
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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.
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(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:
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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
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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).
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Distortion of the retardation function
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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.
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Realisation Theory
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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.
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Realisation theory – order
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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
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Reconstruction of damping and added mass from the parametric approximation:
Model not passive B33(w)<0!
Frequency-domain identification
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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.
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Example containership
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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
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Data from marinecontrol.org
Surge – order 5
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Heave – order 8
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Roll-sway – order 8
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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:
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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
What are we doing estimation wise?
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K(j!) = U(!) + jV (!)
nl ⇠ N (0,⌃)
Dl = f(!l, ✓) + nl
,U(!l)V (!l)
�= f(!l, ✓)
What are we doing estimation wise?
Data:
The likelihood function (measurement model):
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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, ✓))
#
What are we doing estimation wise?
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
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✓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.
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Example FPSO
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DOF 33 joint estimation
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DOF 35 and 53
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Estimated added mass
Infinite Frequency Added mass coefficient estimates:
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MSS FDI toolbox
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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.
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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.
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Thank you
CeSOS Conference 2013 | 27-29 March
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