Time domain modeling

Size: px

Start display at page:

Download "Time domain modeling"

Barnard Chase
8 years ago
Views:

1 Time domain modeling

2 Equationof motion of a WEC Frequency domain: Ok if all effects/forces are linear M+ A ω X && % ω = F% ω K + K X% ω B ω + B X% & ω ( ) H PTO PTO + others Time domain: Must be linear If non linear effects/forces are involved (and significant) Examples: PTO, control, drag effects, moorings, hydrostatic F t && & M + µ X = F t K X K t τ X τ dτ + F H rad PTO + others t = K β, t τ η O, β, τ dτ

effects/forces are involved (and significant) Examples: PTO, control, drag effects, moorings,

3 Outline: Integration in time of the equation of motion State space approximation of the memory term in the radiation force Excitation force

4 Time integration of the equation of motion

5 Equation of motion in time domain Can depend on X (non linear mechanics) t && & PTO force Other forces (moorings, wind, Coriolis forces ) M + µ X = F K t τ X τ dτ K X + F + F + F H drag PTO others Fluid/structure interactions It is a second order ODE (Ordinary Differential Equation): X & = g t, X, X& But available time integration schemes are for first order ODE S & = f t, S How to deal with that?

others Fluid/structure interactions It is a second order ODE (Ordinary Differential Equation): X &

6 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 F F F X K d X t K F M X S t f S t f S ) (,, τ τ τ µ & & & = X X S &

of first order + + + + + = = others PTO drag H t F F F X

7 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: t n = n t n S = S ( n t ) Discretisation of the memory term by the trapezoidal rule (second order accuracy) t n n n n j j K( t τ ) X& τ dτ = K X& + K X& t + K X& t + O t 2 ( ) ( 2 ) j= If K and X are sampled at the same times

n t n S = S ( n t ) Discretisation of the memory term by the trapezoidal rule (second order accuracy) t

8 Discrete equation of motion S& = f t S ( n n, S ) (, ) n n n f t X& n n ( n n ) n j j = F + K X& + K X& t + K X t & ( M µ ) 2 + j= n n n n KH X + Fdrag + FPTO + F others n

9 The simplest time integration scheme Numerical time derivation n+ S S S& n = t From which: S S n+ = S = S n + S n+ n & n + f n + O t + O ( t) ( t) ( n n t, S ) t + O( t) Never used in practice It is called Euler plicit scheme Advantage: Simple Drawbacks: Poor accuracy (first order), unstable (divergence)

t, S ) t + O( t) Never used in practice It is called Euler plicit scheme

10 Second order time integration scheme Two steps scheme: Step : calculate the velocity k at t n Step 2: make a prediction of the state at t n+ and calculate the velocity k 2 at t n+ Advance in time using: = f ( t n S n ) ( n+ n = f t S + k t) k, k S 2, n+ ( k + k ) t Second order scheme, same as for discretisation of convolution product May be used in practice = S n + 2 2

Advance in time using: = f ( t n S n ) ( n+ n = f t S + k t) k, k S 2, n+ ( k + k ) t Second

11 Higher orders scheme There are many higher order schemes Runge Kutta 4 Adams Moutons MATLAB has validated functions for time integration (ode45, ode23, )

12 Summary and recommendations Discrete equation of motion: n n n S& = f t, S n X& n ( n n ) n ( n n, ) n j j F K X K X t K X t f t S = + & + & + & ( M ) 2 + µ j= n n n n KH X Fdrag FPTO F others 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?)

X Fdrag FPTO F + + + others Always use a time integration scheme of order at least 2.

13 May beused in ercise 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 µ A ω Krad τ ωτ dτ ω [ ] = + [ ] sin 2 Krad t B ω ωτ dω π [ ] = cos OUTPUTS K retardation function Mu added mass Cf help FD2TD

by using Ogilvie s formulas: INPUTS w frequency vector A added mass coefficients B radiation

14 For ercise 4 time domain modelling Make your own RK2 solver or use MATLAB ode45 Use of ode45:. 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 Start and end time of simulation Initial conditions

Time integration using ode45 [T,S]= ode45(@(t,s) f(t,s,parameters),[ti tf], [IC],

15 State space approximation of the memory term of the radiation force

16 Direct calculation of the memory term t n K( t τ ) X& τ dτ = K X& + K X& t + K X& t + O t 2 ( n n ) n j j ( 2 ) j= 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 t K t τ X & τ d τ = g I I& = h( I, X& ) New states variables

K needs to be interpolated then not very convenient Solution: To replace the convolution product by a function of

17 Prony s method Approximation using Prony s method : K N ( t) i= α i p(β t) i Compl constants calculated by Prony s method

18 Prony s method Force (N/m.s) K Prony' s approximation time (s) K N ( t) i= α i p(β t) i

19 Prony s method Using : Let: K N ( t) i= α i p(β t) t I ( t) = α p( β ( t τ )) X& ( τ ) dτ i i i i One can show: t K t τ X& τ dτ = I I& = β I + α X& i i i i N i= i

20 Prony s method Let define the state vector Y X = X& I i The equation of motion can be rewritten in the form an ODE of first order ( t, ) Y& = F I F X& N t = M + F I K X + F + F + F I& i = βiii + αi X& (, I) ( µ ) i= i H drag PTO others

21 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 Cf help Prony

22 Frequency Domain Identification (FDI) State space approximation directly from frequency domain coefficients(fdi, Perez & Fossen) F rad t = µ && z K t τ z& τ dτ Approximation using a rational fraction: K s P s Q s t Fourier transform K ( jω ) = B( ω) + jω A( ω ) µ The unknowns are the compl coefficients of polynomials P and Q and papers by Perez & Fossen for more details

23 Frequency Domain Identification (FDI) Using: K s One can show: t r P s prs + p s p = n Q s s + q s + + q r r n n... ( ) & = [ L ] K t τ X τ dτ p p p p I t qn qn 2 L q I& ( t) = I ( t) + X& t O { ( n = r + ) A R r r B R

24 Frequency domain identification Let define the state vector Y X = X& I The equation of motion can be rewritten in the form an ODE of first order Y& F = (, I) F t X& r+ t I = M + F p I K X + F + F + F I& = ARI + BRX& (, ) ( µ ) r+ i i H drag PTO others i=

25 Frequency domain identification How to compute the coefficients of the FDI? K s r P s prs + p s p = n Q s q s + q s + + q r r n n... Matlab routine invfreqs: [P,Q] = invfreqs(k,w,r,n) Coefficients 2 FDI toolbox from Perez & Fossen ( From the physics, the approximation must follow these constraints n Discrete Transfer function q n = n = r + [P,Q]=FDIRadMod(w,A33,Mu33,B33,FDIopt,Dof) Cf help FDIRadMod & Perez, Fossen (29) Guess of orders r < n Discrete frequency vector

26 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 With Prony method: K N ( t) i= α i p(β t) i R ( β ) i < 2 With FDI (necessary condition) ( ωk ) ( ω ) P j R > Q j k

27 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

28 Excitation force

29 Excitation force Incident wave model Wave elevation Wave spectrum Regular Unidirectional Directional Measurement η η I I i t (, β, ) ηi O t Ae ω j O, t A e e = R S ( f ) = δ ( f ) iϕ iωt = R j j Aj = 2S ( f j ) f iϕ j iωt O, t = R Ajle e l j Random phases ( β ) A = 2 S, f f θ lj l j Directions need to be identified ηi ( O, βl, t) l S( f ω = 2π f A ) = α f 5 e B f 5 H 5 A = = 6 T 4 2 /3 B 4 4 T 4 e γ ω 2π f 2 T 2 / 2 2 σ S ( f, θ ) = S( f ) D θ D θ θ 2 = cos 2 s θ D f < T f > T σ =.7 σ =.9 Picture from D-J Doong, B-C Lee, C.C. Kao, (2)

30 Excitation force Incident wave model Wave elevation Wave citation force Regular i t (, β, ) ηi O t Ae ω ( i t ) O = R = R F % ( β, ω) F t A e ω Unidirectional Directional Measurement η η I I iϕ iωt = R j j j O, t A e e Aj = 2S ( f j ) f iϕ j iωt O, t = R Ajle e l j Random phases ( β ) A = 2 S, f f θ lj l j Directions need to be identified ηi ( O, βl, t) l F R F t R F t l ( t) ϕ j A e F % j = ( β, ω ) i O iω jt j j e ϕ jl A e F % l = = j ( β, ω ) i O iω jt jl l j e (, ) (,, ) K β t τ η O β τ dτ O l I l

31 May beused in ercise 4 : FD2TD.m PURPOSE: Calculate force impulse response function in time domain using frequency domain coefficients according to: INPUTS w frequency vector F citation force coefficients T time vector OUTPUTS K force impulse response function O ( iωt K, (,, ) ) β t = R F% O β ω e dω π Cf help FD2TD

32 Calc. of wave force with wave measurement Force impulse response function K O (, ) ( (,, ) iωt K ) β t = R F % O β ω e dω π Calculation of wave citation force F F = K (, ) (,, ) t β t τ η O β τ dτ See ample in FDI.m

Controller Design in Frequency Domain

ECSE 4440 Control System Engineering Fall 2001 Project 3 Controller Design in Frequency Domain TA 1. Abstract 2. Introduction 3. Controller design in Frequency domain 4. Experiment 5. Colclusion 1. Abstract