Time domain modeling

Equation of motion of a WEC

• Frequency domain:

– Ok if all effects/forces are linear

• Time domain:

– If non linear effects/forces are involved (and significant)

– Examples: PTO, control, drag effects, moorings, hydrostatic

( )

(

)

( )

ex

( ) (

H PTO

) ( )

(

( )

PTO

)

( )

others ω ω ω ω ω ω + = − + − + + M A X&&% F% K K X% B B X&%

(

)

( )

₀t

( ) ( )

ex

t

H rad

t

d

PTO

others

τ

τ τ

∞

+

=

−

−

−

+

+

∫

M

µ

X

&&

F

K X

K

X

&

F

Must be linear

( )

= ∞ 0

(

β

−

τ η

) (

β τ τ

)

∫

Outline:

• Integration in time of the equation of motion

• State space approximation of the memory term in the radiation force

Time integration of the equation

of motion

Equation of motion in time domain

• It is a second order ODE (Ordinary Differential Equation):

• But available time integration schemes are for first order ODE…

How to deal with that?

(

M

+

µ

_∞

)

X

&&

=

F

_ex

−

∫

₀t

K t

(

−

τ

)

X

&

( )

τ τ

d

−

K X

_H

+

F

_drag

+

F

_PTO

+

F

_others

(

t

X

X

)

g

X

&

&

=

,

,

&

( )

t

S

f

S

&

=

,

Fluid/structure interactions

PTO force

Other forces (moorings, wind, Coriolis forces…)

Can depend on X (non linear mechanics)

From second order to first order

• Let define the state vector

• The equation of motion can be rewritten as an ODE of first order

( )

( ) (

)

( )

                    + + + − − + + = =

∫

− ∞ others PTO drag H t ex F F F X K d X t K F M X S t f S t f S 0 1 ( ) , ,

τ

τ

τ

µ

& & &













=

X

X

S

&

Discretisation of continuous problem

• Time discretisation is necessary for solving the problem numerically

• Let

∆

t be a small time step

• Discretisation of time:

• Discretisation of the state vector:

• Discretisation of the memory term by the trapezoidal rule (second order accuracy)

t

n

t

n

=

∆

( )

n n

t

S

S

=

( )

(

0 0

)

1

( )

2 0 1

1

(

)

2

n t n n n j j j

K t

τ

X

τ τ

d

K X

K X

t

K

X

t

O

t

− − =

−

=

+

∆ +

∑

∆ +

∆

∫

&

&

&

&

If K and X are sampled at the same times

Discrete equation of motion

(

)

(

)

(

)

(

0 0

)

1 1 1 , 1 , 2 n n n n n n n n n j j n n ex j n n n n

H drag PTO others

S f t S X F K X K X t K X t f t S M K X F F F

µ

− − − = ∞ =        _{+ + ∆ + ∆}  = _  _ + _{ }      _{− + + +}     

∑

& &

The simplest time integration scheme

• Numerical time derivation

• From which:

( )

(

t

S

)

t

O

( )

t

f

S

S

t

O

t

S

S

S

n n n n n n n

∆

+

∆

+

=

∆

+

∆

+

=

+ +

,

1 1

&

( )

t

O

t

S

S

S

n n n

+

∆

∆

−

=

+1

&

It is called ‘Euler explicit scheme’ Advantage: Simple

Drawbacks: Poor accuracy (first order), unstable (divergence)

Second order time integration scheme

• Two steps scheme:

– Step 1: calculate the velocity k₁at tn

– Step 2: make a prediction of the state at tn+1 _{and calculate the}

velocity k₂ at tn+1

– Advance in time using:

• Second order scheme, same as for discretisation of convolution product

(

k

k

)

t

S

S

n+1

=

n

+

₁

+

₂

∆

2

1

(

n n

)

S

t

f

k

₁

=

,

(

t

S

k

t

)

f

k

₂

=

n+1

,

n

+

₁

∆

Higher orders scheme

• There are many higher order schemes

– Runge Kutta 4 – Adams Moutons – …

• MATLAB has validated functions for time integration (ode45, ode23, …)

Summary and recommendations

• Discrete equation of motion:

• Always use a time integration scheme of order at least 2. • Always check the convergence (are the results the same

if you refine the time step?)

(

)

(

)

(

)

(

)

1 0 0 1 1 , 1 , 2 n n n n n n n n n j j n n ex j n n n n

H drag PTO others

S f t S X F K X K X t K X t f t S M K X F F F

µ

− − − = ∞ =        _{+ + ∆ + ∆}  = _  _ + _{ }      _{− + + +}     

∑

& &

May be used in exercise 4 : FD2TD.m

PURPOSE:

• Calculate radiation coefficients in time domain using frequency domain coefficients by using Ogilvie’s formulas:

INPUTS

• w frequency vector

• A added mass coefficients

• B radiation damping coefficients

• T time vector OUTPUTS • K retardation function • Mu added mass Cf help FD2TD [ ] ( ) [ ]( ) ( ) [ ]( ) ( ) ( ) 0 0 1 sin 2 cos rad rad A K d K t B d µ ω τ ωτ τ ω ω ωτ ω π ∞ ∞ ∞ = _ _+ = _ _

∫

∫

For exercise 4 time domain modelling

• Make your own RK2 solver or use MATLAB ode45 • Use of ode45:

1. Create a function f.m

function dS=f(t,S,parameters)

2. Time integration using ode45

[T,S]= ode45(@(t,S) f(t,S,parameters),[ti tf], [IC], options)

Discrete time

Discrete

states parameters _{end time of}Start and simulation

Initial conditions

State space approximation of the

memory term of the radiation force

Direct calculation of the memory term

• Drawbacks:

– Can be CPU time consuming

– Discretisation time of K and t can be different. K needs to be interpolated then not very convenient

Solution: To replace the convolution product by a

function of additional state variables given by additional state equations state space approximation

( )

(

0 0

)

1

( )

2 0 1

1

(

)

2

n t n n n j j j

K t

τ

X

τ τ

d

K X

K X

t

K

X

t

O

t

− − =

−

=

+

∆ +

∑

∆ +

∆

∫

&

&

&

&

(

) ( )

( )

0

t

K t

−

τ

X

τ τ

d

=

g I

Prony’s method

• Approximation using Prony’s method :

( )

∑

=

≈

N i i i

t

t

K

1

)

exp(

β

α

Complex constants calculated by Prony’s method

Prony’s method

( )

∑

=

≈

N i i i

t

t

K

1

)

exp(

β

α

time (s) F o rc e (N /m .s ) 0 2 4 6 8 -20000 -10000 0 10000 20000 30000 K Prony' s approximation

Prony’s method

• Using :

• Let:

• One can show:

( )

∑

=

≈

N i i i

t

t

K

1

)

exp(

β

α

(

) ( )

0 1 N t i i i i i i

K t

X

d

I

I

I

X

τ

τ τ

β

α

=

−

=

=

+

∑

∫

&

&

&

( )

₀t

exp(

(

) ( )

)

i i i

I t

=

∫

α

β

t

−

τ

X

&

τ τ

d

Prony’s method

• Let define the state vector

• The equation of motion can be rewritten in the form an ODE of first order

( )

( ) (

)

1 1 , , N

ex i H drag PTO others

i t X t M F I K X F F F I I X

µ

β

α

− ∞ = =         = _ + _ − − + + + _      _{= +}   

∑

Y F I F I & & & & i

X

X

I









=













Y

&

Calculation of coefficients with Prony.m

PURPOSE:

• Identification of function K using Prony’s method

INPUTS

• T time vector

• K function to be identified OUTPUTS

• Arrays of alpha and beta coefficients

Frequency Domain Identification (FDI)

• State space approximation directly from frequency domain coefficients(FDI, Perez & Fossen)

• Approximation using a rational fraction:

( )

₀t

(

) ( )

rad

F

t

= −

µ

_∞

&&

z

−

∫

K t

−

τ

z

&

τ τ

d

( )

( )

(

( )

)

K j

ω

=

B

ω

+

j

ω

A

ω µ

−

_∞ Fourier transform

( )

P s

( )

( )

K s

Q s

≈

www.marinecontrol.org and

papers by Perez & Fossen for more details

Frequency Domain Identification (FDI)

• Using:

• One can show:

( )

( )

( )

1 1 0 1 1 0

...

...

r r r r n n n

P s

p s

p s

p

K s

Q s

s

q

s

q

− − − −

+

+ +

≈

=

+

+ +

(

) ( )

[

]

( )

( )

( )

{

( )

1 1 0 0 1 2 0

1

1

0

0

0

0

1

0

0

0

0

0

0

t r r n n

K t

X

d

p

p

p

p I t

q

q

q

I t

I t

X t

τ

τ τ

₋ − −

−

=

−

−

−





 





 





 

=





+

 





 





 

= +

∫

R R B A

&

L

L

&

&

O

14444244443

Frequency domain identification

• Let define the state vector

• The equation of motion can be rewritten in the form an ODE of first order

( )

( ) (

)

1 1 1 1 , , r

ex r i i H drag PTO others

i F t X t I M

µ

F p I K X F F F + − ∞ + − = =         = _ + _ − − + + + _      _{= +}   

∑

Y I F I A I B X & & & &

X

X









=













Y

I

&

Frequency domain identification

• How to compute the coefficients of the FDI?

1 Matlab routine invfreqs:

[P,Q] = invfreqs(K,w,r,n)

2 FDI toolbox from Perez & Fossen (www.marinecontrol.org)

[P,Q]=FDIRadMod(w,A33,Mu33,B33,FDIopt,Dof)

( )

( )

( )

1 1 0 1 1 0 ... ... r r r r n n n n P s _{p s p s p} K s Q s q s q s q − − − − + + + ≈ = + + + Coefficients Discrete Transfer function Discrete frequency vector Guess of orders r < n

1

1

n

q

n

r

=





= +



From the physics, the approximation must follow these constraints

Issues with state-space approximation

• Passivity : the radiation force dissipates energy. For the some frequencies, it may not be the case with the

approximation divergence of numerical model 1 With Prony method:

2 With FDI (necessary condition)

( )

∑

=

≈

N i i i

t

t

K

1

)

exp(

β

α

( )

β

_i

0

ℜ

<

( )

( )

k 0 P j Q j

ω

ω

  ℜ_ _ >  

Summary

• Convolution product can be replaced by a state space model

• Coefficients of the state space model can be derived using:

– Prony’s method in Time domain – Frequency Domain Identification

• Matlab routine invfreqs • Perez & Fossen toolbox

Excitation force

Measurement Directional Unidirectional Regular Wave spectrum Wave elevation Incident wave model ( , , )

(

i t

)

I O t Ae ω η β _{= ℜ} − ( ) ( ) 2 S f _ω f π δ = ( ), i j i t I j j O t A e eϕ ω η _{= ℜ} −    

∑

 ( ), i j i t I jl l j O t A e eϕ ω η _{= ℜ} −    

∑∑

 ( , , ) I l l O t η β

∑

Random phases

Directions need to be identified

2 f ω = π       − − = 4 212/ 2 2 5 ) ( σ γ α f e f T B e f A f S 09 . 0 1 07 . 0 1 1 1 = > = < σ σ T f T f 2 1/3 4 4 1 1 5 5 1 16 4 H A B T T = =

( )

2 j j A = S f ∆f

(

)

2 , lj l j A = S β f ∆ ∆f θ

( )

θ θ S f D f S( , ) = ( ) ( )       ₋ = 2 cos2 0 θ θ θ s D D

Excitation force

Measurement Directional Unidirectional Regular

Wave excitation force Wave elevation Incident wave model ( , , )

(

i t

)

I O t Ae ω η β _{= ℜ} − ( )

(

( )

)

, O i t ex ex F t = ℜ AF% β ω e−ω ( )

(

,

)

j j ex i _O i t j ex j j F t A eϕ β ω e−ω =   ℜ  

∑

F  % ( )

(

,

)

jl j ex i _O i t jl ex l j l j F t A eϕ β ω e−ω =   ℜ  

∑∑

F  % ( , , ) I O l t η β

∑

( ) ( ) ( ) ex F t β τ η β τ τ ∞ = −

∑∫

Directions need to be identified

( ), i j i t I j j O t A e eϕ ω η _{= ℜ} −    

∑

 ( ), i j i t I jl l j O t A e eϕ ω η _{= ℜ} −    

∑∑

 Random phases

( )

2 j j A = S f ∆f

(

)

2 , lj l j A = S β f ∆ ∆f θ

May be used in exercise 4 : FD2TDex.m

PURPOSE:

• Calculate force impulse response function in time domain using frequency domain coefficients according to:

INPUTS

• w frequency vector

• Fex excitation force coefficients

• T time vector

OUTPUTS

• Kex force impulse response function

Cf help FD2TDex

( )

1 ₀

(

(

)

)

, , , O i t ex ex K β t F O β ω eω dω π ∞ =

∫

ℜ %

Calc. of wave force with wave measurement

• Force impulse response function Kex

• Calculation of wave excitation force F_ex

( )

,

1

₀

(

(

, ,

)

)

O i t ex

t

ex

O

e

d

ω

β

β ω

ω

π

∞ ₋

=

∫

ℜ

K

F

%

( )

0

(

) (

)

, , , ex t

β

t

τ η

ex O

β τ τ

d ∞ −∞ =

∫

_ex − F K