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

Transcription

1 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

2 My work at CeSOS 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) Risk assessment of autonomous systems (aerospace) Fault-tolerant systems Energy-based control vehicle dynamics

3 Motivation Non-parametric models: frequency response functions Parametric fluid memory model Hydrodynamic Code Iden/fica/on Cummins Equa/on Experiments & CFD Model with Viscous Correc/on Hull geometry and loading condition

4 Non-parametric seakeeping models System identification of parametric fluid-memory models MSS FDI toolbox

5 Non-parametric seakeeping Models

6 RB Dynamics & Hydrodynamic forces Excitation loads Radiation loads Total loads = rad + res + exc = J( ) M RB + C RB ( ) =

7 Cummins s Equation (1961) Approximation RB dynamics: = J( ) M RB + C RB ( ) = = rad + res + exc Zero forward speed and small angles M RB = Radiation forces: rad = A 1 Z t 0 K(t t 0 ) (t 0 ) dt 0 Cummins s Equation: Z t (M + A 1 ) + 0 K(t t 0 ) (t 0 ) dt 0 + G = exc

8 Frequency-domain Model Radiation forces: rad (j!) =! 2 A(!) (j!) j!b(!) (j!) Frequency-domain non-parametric model: (! 2 [M + A(!)] + j!b(!)+g) (j!) = exc (j!) Abuse of notation: [M + A(!)] + B(!) + G = exc

9 Summary non-parametric models Time domain: (M + A 1 ) + Z t 0 K(t t 0 ) (t 0 ) dt 0 + G = exc Frequency domain: (! 2 [M + A(!)] + j!b(!)+g) (j!) = exc (j!)

10 Ogilvie s Relations (1964) Frequency-dependent added mass and potential damping: 1 A(!) =A 1! B(!) = Z 1 0 Z 1 0 K(t) cos(!t) K(t) sin(!t) dt, Retardation functions K(t) = 2 Z 1 0 B(!) cos(!t) d! K(j!) = Z 1 0 K(t)e j!t d! = B(!)+j![A(!) A 1 ]

11 System Identification of parametric fluid-memory models

12 Replacing the convolution (M + A 1 ) + Z t 0 K(t t 0 ) (t 0 ) dt 0 + G = exc µ = Z t 0 K(t t 0 ) (t 0 ) dt 0 ẋ = ˆµ = Âx+ ˆB Ĉx+ ˆD, 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.

13 Replacing the convolution Time domain identification: Data Mapping Indentification B(!) K(t) appleâ Ĉ ˆB ˆD Frequency-domain identification: Data B(!), A(!), [A 1 ] Mapping Indentification Mapping appleâ ˆB K(j!) ˆK(s) Ĉ ˆD

14 Properties background information The following properties derive from the hydrodynamics, and have implications on the parametric models: 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) lim t!0 + K(t) 6= 0 Relative degree 1. 4) lim t!1 K(t) =0 Input-output stable. 5) The mapping 7! µ is Passive K(j!) is positive real. Consistency and rationality requires us to use all this information in the identification problem.

15 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).

16 Distortion of the retardation function K(t) = 2 Z 1 0 B(!) cos(!t) d!

17 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.

18 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.

19 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.

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

21 Example containership DOF3,3 Model not passive B33(w)<0! Reconstruction of damping and added mass from the parametric approximation:

22 Frequency-domain identification The i,k entry of K(s) can be approximated by a rational transfer function: Then we can estimate the parameters via LS using the frequency response computed using the data generated by hydrodynamic code: K(j!) =B(!)+j![A(!) A 1 ] This problem is non-linear in the parameters, but it can be linearised.

23 Prior knowledge and constraints From the physics of the problem we know that the transfer functions have 1. Relative degree 1 2. H ik (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.

24 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.

25 Example semisub Data from marinecontrol.org

26 Surge order 5

27 Heave order 8

28 Roll-sway order 8 Couplings are not necessarily passive B(w)<0

29 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)

30 What are we doing estimation wise? K(j!) =U(!)+jV (!), apple U(!l ) V (! l ) = f(! l, ) Measurement model: D l = f(! l, )+n l Assumption 1 - Gaussian uncertainty: Assumption 2 i.i.d. uncertainties Assumption 3 diffuse prior for the parameters n l N (0, )

31 What are we doing estimation wise? Data: D = {D 1,D 2,...,D N } apple U(!l ) V (! l ) = f(! l, ) The likelihood function (measurement model): D l = f(! l, )+n l p(d,i) / Y apple 1 exp 2 (D l f(! l, )) T 1 (D l f(! l, )) l " # 1 X =exp (D l f(! l, )) T 1 (D l f(! l, )) 2 l

32 What are we doing estimation wise? Our estimates based on NLS correspond to the maximum a posteriori (MAP) estimates NLS MAP = arg min under the following assumptions: X (D l f(! l, )) T 1 (D l f(! l, )) l Assumption 1 - Gaussian uncertainty: Assumption 2 i.i.d. uncertainties Assumption 3 diffuse prior for the parameters

33 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.

34 Example FPSO

35 DOF 33 joint estimation

36 DOF 35 and 53

37 Estimated added mass Infinite Frequency Added mass coefficient estimates:

38 MSS FDI toolbox ω,a(ω),b(ω),a FDIopt FDIRadMod.m Krad(s), [A ] EditAB.m Ident_retarda>on_FD.m Ident_retarda>on_FDna.m Demo_FDIRadMod_WA.m Demo_FDIRadMod_NA.m Fit_siso_fresp.m

39 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.

40 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 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 DOI: / mic 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 DOI: /mic

41 Thank you