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Nonlinear stochastic optimal control of offshore platforms under wave loading
ARTICLE IN PRESS JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 296 (2006) 734–745 www.elsevier.com/locate/jsvi
Nonlinear stochastic optimal control of offshore platforms under wave loading M. Luoa,b, W.Q. Zhua, a
Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, 310027, People’s Republic of China b College of Information Science and Engineering, East China University of Science and Technology, 130 Meilong Road, Xuhui District, Shanghai, 200237, People’s Republic of China Received 6 April 2004; received in revised form 2 May 2005; accepted 31 January 2006 Available online 2 June 2006
Abstract A nonlinear stochastic optimal (NSO) control strategy for wave-excited jacket-type offshore platforms is proposed. Wave force is determined according to linearized Morison equation. Spectral density functions of water particle velocity and acceleration are approximated by some rational forms, respectively. Wave force vector is then treated as output of a linear filter driven by white noise. A set of partially averaged Itoˆ equations for controlled modal energies are derived by applying stochastic averaging method for quasi-integrable Hamiltonian systems. Optimal control law is then determined by using stochastic dynamical programming principle. To demonstrate the effectiveness and efficiency of the proposed control strategy, performances of uncontrolled, linear quadratic Gaussian (LQG)-controlled and NSO-controlled example platforms under different sea states are evaluated. Numerical results show that the NSO controller has better control effectiveness and efficiency than the LQG controller. r 2006 Elsevier Ltd. All rights reserved.
1. Introduction Active control of civil engineering structures has been investigated for more than twenty years [1]. Many control strategies and control devices have been proposed for active control of civil engineering structures. However, most studies care about structures excited by wind and earthquakes and only a few studies concern wave-excited offshore structures [2–8]. Active control of wave-excited offshore platforms has not been appropriately investigated. As platform moves far and far into deep water, extreme wave in deep water can induce large motion of offshore platform, which can threaten safety and operation of platform. To prevent fatigue damage and to protect operation and crew of platform, more attention should be paid to active control of wave-excited offshore structures. Since dynamic loading such as earthquake, wind and wave are usually modeled as random processes, to apply stochastic optimal control theory to civil engineering structures is natural and reasonable. Recently, Corresponding author. Tel.: +86 571 8795 3102; fax: +86 571 8795 2651.
E-mail address: [email protected] (W.Q. Zhu). 0022-460X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2006.01.071
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based on the stochastic averaging method for quasi-Hamiltonian systems and stochastic dynamical programming principle, a nonlinear stochastic optimal control strategy was proposed by the present second author and his co-workers [9,10]. It has been applied to the control of buildings structures under wind and earthquake excitation, and proved to be more effective and efficient than linear quadratic Gaussian (LQG) control strategy [10,11]. The present paper is devoted to the application of the nonlinear stochastic optimal control strategy to wave-excited offshore structures. Although the basic control strategy is the same, the structures and loadings are different here and in Refs. [10,11]. So, the control effectiveness and efficiency are different. Besides, the control system here is assumed to be completely observable rather than partially observable as in Refs. [10,11]. To apply the nonlinear stochastic optimal strategy to wave-excited offshore structures, it is necessary to establish a set of stochastic differential equations for random wave loading, which is usually characterized by power spectral density function. An filter approaches to wave kinematics was presented [12], extended and used for active control of offshore platform subject to wave loading using H2 controller [13]. Here this filter approach is used in the active control of offshore platform subject to wave loading using the nonlinear control strategy. 2. Model of wave load It is assumed that wave surface elevation Z(t) is a zero-mean, stationary, Gaussian random process, characterized by its spectral density function SZ(o). One of the most commonly used spectral density function of wave surface elevation is JONSWAP spectral density [14] 5H 2s 5 5 4 b S Z ðoÞ ¼ (1) ðo=o0 Þ exp ðo=o0 Þ g , 4 16o0 where Hs is significant wave height; o0 is dominant (peak) wave frequency, g is sharpness magnification factor (2) b ¼ exp ðo o0 Þ2 =2t2 o20 in which t ¼ 0.07 for oro0 and t ¼ 0.09 for oZo0. In order to obtain a rational form for the spectral density function of wave force, the following approximate JONSWAP spectral density is taken [12] S Z ðoÞ
Gðo=o0 Þ4 ,
2
2 2 2 2 2 o=o0 k1 þ c1 o=o0 o=o0 k2 þ c2 o=o0
(3)
where the parameters G,k1,k2,c1,c2, are determined by using a least-squares algorithm. Following the linear wave theory, the spectral density function of horizontal water-particle velocity u˙ at level z is 2 2 2 cosh ðkzÞ SZ ðoÞ, Su_u_ ðo; zÞ ¼ H uZ _ ðo; zÞ S Z ðoÞ ¼ o sinh2 ðkdÞ
(4)
where Hu˙Z(o,z) is the frequency response function for transformation from wave surface elevation to horizontal water-particle velocity at level z; k ¼ o2/g is wave number, g is the gravity acceleration. The spectral density function of horizontal water-particle acceleration u¨ at level z is S u€u€ ðo; zÞ ¼ o2 S u_u_ ðo; zÞ.
(5)
The frequency response function in Eq. (4) can be expanded as [13] 2 4 6 2 1 1 1 2 2 cosh ðkzÞ 2 1 þ 2 ðkzÞ þ 24 ðkzÞ þ 720 ðkzÞ þ H uZ ¼ o , _ ðo; zÞ ¼ o 1 1 1 þ 16 ðkd Þ3 þ 120 sinh2 ðkd Þ ðkd Þ5 þ 5040 ðkd Þ7 þ
(6)
where d is the water depth at structural site. Keeping the first two terms of Fourier expansions of numerator and denominator, respectively, of |Hu˙Z(o,z)|2 in Eq. (6) and substituting k ¼ o2/g into Eq. (6),
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one obtains 2 H uZ _ ðo; zÞ
z2 6 o þ o2 2g2
1þ
d3 6 o : 6g3
Thus, the approximate spectral density function of horizontal water-particle velocity is
3Ggz2 o40 2g2 4 o þ o6 3 2 z d i h i
Su_u_ ðo; zÞ ¼ h 2 2 3 o2 k1 o20 þ c21 o20 o2 o2 k2 o20 þ c22 o20 o2 o6 þ 6g 3 d and the approximate spectral density function of horizontal water-particle acceleration is
3Ggz2 o40 2g2 4 o þ o8 3 2 z d i h i
. S u€u€ ðo; zÞ ¼ h 2 2 3 o2 k1 o20 þ c21 o20 o2 o2 k2 o20 þ c22 o20 o2 o6 þ 6g 3 d Similarly, the approximate cross-spectral density function of horizontal water-particle velocity is
3Ggzm zn o40 2g2 4 6 o þ zm zn o d3 i h i
S u_u_ ðo; zm ; zn Þ ¼ h 2 2 3 o2 k1 o20 þ c21 o20 o2 o2 k2 o20 þ c22 o20 o2 o6 þ 6g d3 and the approximate cross-spectral density function of horizontal water-particle acceleration is
3Ggzm zn o40 2g2 4 8 o þ 3 zm zn o d h i h i
. Su€u€ ðo; zm ; zn Þ ¼ 2 2 3 o2 k1 o20 þ c21 o20 o2 o2 k2 o20 þ c22 o20 o2 o6 þ 6g d3
(7)
(8)
(9)
(10)
(11)
According to the Morison equation, wave force acting on the structure at level z is of the form _ uj _ þ kM u, € F ðz; tÞ ¼ kD uj
(12)
where kD ¼ ð1=2ÞrC D Ap ;
kM ¼ rC M V p
(13)
in which r is density of seawater, CD and CM are drag and inertia coefficients, respectively, and Ap and Vp are projected area and volume of structure at level z, respectively. The linearized Morison equation is of the form [14] pffiffiffiffiffiffiffiffi € F ðz; tÞ ¼ 8=pkD su_ u_ þ kM u, (14) where su˙ ¼ su˙(z) is the standard deviation of horizontal water-particle velocity at level z. Then the crossspectral density of the approximate wave forces acting on the structure at zm and zn is 8 S F ðo; zm ; zn Þ ¼ kM ðzm ÞkM ðzn ÞSu€u€ ðo; zm ; zn Þ þ kD ðzm ÞkD ðzn Þsu_m su_n S u_u_ ðo; zm ; zn Þ, p
(15)
where Su€u€ ðo; zm ; zn Þ and S u_u_ ðo; zm ; zn Þ are the cross-spectral densities of horizontal water-particle velocity and acceleration. Suppose that the offshore structure is simplified as a lumped mass system with N lumped masses. The Ndimensional wave force vector F(t), where FðtÞ ¼ ½F N ðtÞ; F N1 ðtÞ; ; F 1 ðtÞT , acting on N lumped mass with spectral density matrix S(o) obtaining from discreting Eq. (15) can be modeled as the output of the following linear filter with a unit intensity Gaussian white noise process w(t) as an input FðtÞ ¼ Cf y;
y_ ¼ Af y þ Bf wðtÞ,
(16)
ARTICLE IN PRESS M. Luo, W.Q. Zhu / Journal of Sound and Vibration 296 (2006) 734–745
where
737
3 0 2 3 0 6 .. 7 7 6 0 0 1 . 7 607 6 6 7 7 6 .. 6 7 7 6 0 607 . 1 7 6 6 7 6 6 7 .. 7 7 6 ; B Af ¼ 6 0 ¼ 607 ; f 1 . 7 6 7 7 6 607 . 7 6 .. 6 7 6 0 1 0 7 6 7 7 6 405 7 6 0 0 0 1 5 4 1 71 a0 a1 a2 a3 a4 a5 a6 qffiffi h qffiffi
qffiffi
i 8 8 k b þ k b k b þ k b b Cf ¼ 0; 0; 0; p8ru_ kD b0 ; r r ; ; k _ _ D 1 M 0 D 2 M 1 M 2 u u p p 2
0
1
0
N7
a0 ¼
k1 k2 o40
sffiffiffiffiffiffiffi 6g3 ; d3
a1 ¼ ðc1 k2 þ
sffiffiffiffiffiffiffi 6g3 a3 ¼ k1 k2 o40 þ ðc2 þ c1 Þo0 ; d3
c2 k1 Þo30
sffiffiffiffiffiffiffi 6g3 ; d3
a2 ¼ ðk2 þ c1 c2 þ
sffiffiffiffiffiffiffi 6g3 a4 ¼ ðc1 k2 þ c2 k1 Þo30 þ , d3
k1 Þo20
(17)
;
sffiffiffiffiffiffiffi 6g3 , d3
a5 ¼ ðk2 þ c1 c2 þ k1 Þo20 ; a6 ¼ ðc2 þ c1 Þo0 , ru_ ¼ diagðsu_zN ; su_zN1 ; . . . su_z1 Þ, kD ¼ diagðkDZN ; kDZN1 ; . . . ; kDZ1 Þ; kM ¼ diagðkM ZN ; kM ZN1 ; . . . ; kM Z1 Þ, 3 2 2 pffiffiffiffiffiffi 3 2 3 zN zN 1 7 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6 pffiffiffiffiffiffiffiffiffiffi 7 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi6 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 7 6 zN1 7 6 zN1 7 17 2 8 7 7 4 72G g4 o 6 3Ggo40 6 6Gg3 o40 6 6 7 06 7 7 6 ; b ; b b0 ¼ ¼ ¼ , 6 1 2 . .. 7 . 7 6 . 7 7 d3 6 d6 d3 6 6 .. 7 6 . 7 . 5 4 5 5 4 4 pffiffiffiffiffi 1 N1 z1 z1
ð18Þ
where diag denotes diagonal matrix. Note that, since the highest power of o in the denominator of approximate rational spectral density function of linearized wave force is 14, the dimension of matrix Af is 7 7. 3. Equation of controlled system The simplified controlled system equation is of the form € þ CX _ þ KX ¼ FðtÞ þ PU; MX
Xð0Þ ¼ X0 ,
(19)
_ X € are N-dimensional vectors of structural displacements, velocities and accelerations, where X; X; respectively.M,C and K are mass, damping and stiffness matrices of the system, respectively; U ¼ [U1,U2,y,Um]T is m-dimensional vector of control forces produced by m control devices; P is N m control device placement matrix. Introduce model transform (20)
X ¼ UQ,
where Q ¼ ½Q1 ; Q2 ; ; Qn T is the dominant modal displacement vector, and U is a N n dominant modal matrix consist of n dominant modal vectors. The equation for the n dominating modes can be written as € þ 2fXQ _ þ X2 Q ¼ UT FðtÞ þ UT PU; Q
Qð0Þ ¼ Q0 ,
(21)
where 2fX ¼ UT CU;X2 ¼ diagðo2i Þ ¼ UT KU. oi and zi are the frequency and damping coefficient of the ith mode, respectively.
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Eqs. (16) and (21) can be combined and converted into the following Itoˆ stochastic differential equation ~ dZ ¼ ðAZ þ BUÞ þ C dBðtÞ;
Zð0Þ ¼ Z0 , (22) ~ ˙ ,y ,y˙ ] ; BðtÞ is a standard Wiener process, Z0 is a Gaussian random vector representing the where Z ¼ [Q ,Q ~ initial state of the system, which is independent of BðtÞ, 2 3 2 3 0 2 3 0 0 I 0 0 6 UT P 7 6 0 7 2 6 7 6 7 6 7 A ¼ 4 X 2fX Cf 0 5; B ¼ 6 (23) 7; C ¼ 6 7. 4 0 5 4 Bf 5 0 0 Af 0 0 0 T
T
T
T T
4. Stochastic averaging To simplify the equation of controlled system and reduce its dimension, the stochastic averaging method for ˙ in Eq. (22) except the control quasi-integrable Hamiltonian system [15] is applied to the equations for Q and Q force terms. Then the obtained partially averaged Itoˆ stochastic differential equations for the n dominant modal energies are ¯ i ðtÞ; i ¼ 1; 2; :::; n, dH i ¼ mi ðH i Þ þ oðqH i =qQ_ i ÞFTik Pkr U r 4 dt þ si ðH i Þ dB (24) where o 4 denotes averaging operation with respect to time t; 2 H i ¼ ðQ_ i þ o2i Q2i Þ=2;
i ¼ 1; 2; :::; n
(25)
¯ i ðtÞ are standard Wiener processes; denotes the energy of the ith mode; B mi ðH i Þ ¼ 2zi oi H i þ FTil Fki Slk ðoi Þ, s2i ðH i Þ ¼ 2H i FTil Fki S lk ðoi Þ;
ð26Þ
Slk ðoi Þ ¼ S F l F k ðoi Þis the l, kth element of spectral density matrix S(o) evaluated at oi. 5. Optimal control law Eq. (24) implies that Hi are controlled diffusion processes. The objective of stochastic optimal control is to seek the optimal feedback control U* for minimizing the following semi-infinite time-interval performance index Z 1 tf LðHðtÞ; oUðtÞ4Þ dt, (27) JðUÞ ¼ lim tf !1 tf 0 where L(H(t),oU(t)4) is the so-called cost function. Based on the stochastic dynamical programming principle, the following dynamical programming equation can be established [9] ( ) T qV qH T 1 q2 V l ¼ min LðHðtÞ; oUðtÞ4Þ þ mðHÞ þ o rðHÞrT ðHÞ , (28) U PU4 þ tr _ U qH 2 qH2 qQ where 1 tf !1 tf
Z
l ¼ lim
tf
½LðHðtÞ; oU ðtÞ4 dt
(29)
0
is the optimal average cost, V is the value function and U* is the optimal control force vector. The expression for U* can be obtained from minimizing the right-hand side of Eq. (28) with respect to U. Suppose that the cost function L is of the form L ¼ gðHÞ þ oUT RU4,
(30)
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where R is a n n positive-define matrix. Then U* is of the following form: 1 _ T ðqV =qHÞ (31) U ¼ R1 PT UðqH=qQÞ 2 ˙ with respective to t yield the final Substituting Eq. (31) into Eq. (28) and completing averaging Q and Q dynamical programming equation for semi-infinite time-interval control problem " # n X qV 1 qV 2 1 2 q2 V l ¼ gðHÞ þ mi ðH i Þ Dii H i þ si ðH i Þ , (32) qH i 4 qH i 2 qH 2i i¼1 where Dii ¼ ½UT PR1 PT Uii : qV/qH can be obtained by solving the final dynamical programming equation (32) and the optimal control U* can be obtained by substituting the resultant qV/qH into the Eq. (31). 6. Response prediction of controlled structure AveragingðqH i =qQi ÞFTik Pkr U r Þ and then substituting it into equation (24) to replace ðqH i =qQ_ i ÞFTik Pkr U r , one obtains the completely averaged Itoˆ equation for Hi h i ¯ i ðtÞ, dH i ¼ mi ðH i Þ þ mðUÞ ðH Þ dt þ si ðH i Þ dB (33) i i where 1 T qV 1 T mðUÞ . i ðH i Þ ¼ Fik Pkr Rrs Psl Fli H i 2 qH i
(34)
The stationary probability density p(H) can be obtained from solving the reduced Fokker-plank-kolmogorov (FPK) equation associated with Itoˆ Eq. (33). It is of the following form ( Z ) n HX qV pðHÞ ¼ C 0 exp Di þ E i dH i , (35) qH i 0 i¼1 where C 0 is a normalization constant, Di ¼ 4zi oi =FTil Fki S lk ðoi Þ, T T E i ¼ FTik Pkr R1 rs Psj Fji =Fil Fki S lk ðoi Þ.
ð36Þ
The stationary probability density of the modal displacement and velocity of the controlled structure is then [15] _ ¼ ½pðHÞ=TðHÞ pðQ; QÞ 2 , (37) H ¼ðQ_ þo2 Q2 Þ=2 i
i
i
where TðHÞ ¼
n I Y i¼1
n Y dQi 2p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ . oi i¼1 2H i o2i Q2i
The mean square values of the modal displacement and velocity of the controlled structure are Z 1 Z 1 _ dQ dQ _ ¼ 1 Q2i pðQ; QÞ H i pðHÞ dH, E Q2i ¼ o2i 0 1 Z 1 Z 1 h 2i 2 _ dQ dQ _ ¼ E Q_ i ¼ H i pðHÞ dH. Q_ i pðQ; QÞ 1
0
(38)
ð39Þ
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Finally, the mean square displacement and acceleration of each floor are obtained from Eq. (39) by using modal transformation (20) as follows: 2 !2 3 n X 2 E X i ¼ E4 Fij Qj 5, 2
j¼1
n h 2i X E X€ i ¼ E 4 Fij o2j Qj þ 2zi oi Q_ j UT FðtÞ þ UT PU j
j¼1
!2 3 5.
ð40Þ
The response statistics of the uncontrolled structure can be obtained in a similar way by letting control force vanish.
7. Performance criteria The main purpose of vibration control of wave-excited offshore platforms is to reduce displacement to prevent fatigue damage, and to reduce deck acceleration to protect operations and crew of offshore platforms. So, to evaluate a control strategy, the control effectiveness and efficiency of the displacements and deck acceleration are considered. Control effectiveness is defined as [9] K ¼ ðsu sc Þ=su ,
(41)
where s stands for the standard deviation of displacement or deck acceleration; subscripts u and c denote uncontrolled and controlled offshore platforms, respectively. Control efficiency is defined as m ¼ K=sF , (42) 2 ffi Pm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where sF ¼ i¼1 E U i =G~ is the standard deviation of control force, normalized by total weight G~ of offshore platform. It is seen from the definitions that the larger K and m are, the better the control strategy is.
Z7=144.78m d=121.92m Z6=118.87m Z5=99.06
Z4=79.06
Z3=59.44
Z2=39.62
Z
Z1=19.81
Fig. 1. Model of jacket-type offshore platform used for numerical example.
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8. Numerical example A simplified jacket-type platform with 7 lumped masses is taken as an example (see Fig. 1). The height of the platform is 144.78 m. The mass, stiffness and damping matrices are M ¼ diagf 4818 1475 1302 1533 1840 2205 3738 g 103 kg; 3 2 344 289 17 1 26 4 16 6 670 366 4 33 23 10 7 7 6 7 6 7 6 725 368 14 22 9 7 6 7 6 777 410 23 23 7 103 kN=m; K¼6 7 6 6 879 491 56 7 7 6 7 6 sym 1005 595 5 4 1344
8 0.7
7 Control Effectiveness of STD Displacements
Floor Number
6 5 4 3 2
Uncontrolled LQG NSO
1
0.6 0.5 0.4 0.3
0
0.2 0.02
0.04 0.06 0.08 STD Displacements
Control Effieciency of STD Displacement
(a)
LQG NSC
(c)
0.10
0.12
0
1
3 4 5 Floor Number
2
(b) 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16
6
7
8
LQG NSO
0
1
2
3 4 5 Floor Number
6
7
8
Fig. 2. (a) STD displacements of uncontrolled, NSO- and LQG-controlled offshore platforms; (b) control effectiveness and (c) control efficiency of NSO and LQG control strategies. T ¼ 1:5 s, R ¼ 107 I7 , s1 ¼ ½1; 1; 1; 1; 1; 1; 1T for NSO and Rl ¼ 1, Ql ¼ 107 I14 for LQG.
ARTICLE IN PRESS M. Luo, W.Q. Zhu / Journal of Sound and Vibration 296 (2006) 734–745
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2
2522 1244 6 2892 6 6 6 6 6 C¼6 6 6 6 6 sym 4
3 59 30 7 7 7 106 22 7 7 7 161 113 7 103 N s=m: 7 1292 122 7 7 7 4311 1605 5 6769
9 144
263 936
135 188
2808
920
161
3142
1072 3652
18 7
(43)
The natural frequencies of this platform are 0.7432, 1.2497, 2.1678, 2.9563, 3.6921, 4.3649, 4.9773 (Hz), and the modal damping ratios are 5% for all modes. The water depth d is 121.92 m. The vertical coordinates of lumped masses are z ¼ ½z7 ; z6 ; ; z1 T ¼ ½144:78; 118:87; 99:06; 79:06; 59:44; 39:62; 19:81T m;
(44)
the drag coefficient matrix is kD ¼ diagf0; 790; 676; 719; 757; 828; 1557g 103 kg=m;
(45)
8 0.6
7 Control Effectiveness of STD Displacements
Floor Number
6 5 4 3 Uncontrolled LQG NSO
2 1
LQG NSO
0.5
0.4
0.3
0.2
0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 (a) STD Displacements
0.1 0
1
2
3
6
7
(b)
4 5 Floor Number
6
7
8
80 75 70
LQG NSC
Control Efficiency of STD Displacement
65 60 55 50 45 40 35 30 25 20
0
(c)
1
2
3
4 5 Floor Number
8
Fig. 3. (a) STD displacements of uncontrolled, NSO- and LQG-controlled offshore platforms; (b) control effectiveness and (c) control efficiency of NSO and LQG control strategies. T ¼ 2:5 s, R ¼ 107 I7 , s1 ¼ ½1; 1; 1; 1; 1; 1; 1T for NSO and Rl ¼ 1, Ql ¼ 107 I14 for LQG.
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and the inertia coefficient matrix is kM ¼ diagf0; 1744; 1674; 1939; 2563; 3150; 6735g 103 kg:
(46)
A JONSWAP spectrum with significant wave height H s ¼ 10 m, sharpness magnification factor g ¼ 3.0 and periods T ¼ 1.5, 2.5, 3.5 (s) is used to characterize the sea states. The parameters for the approximate JONSWAP spectrum are G ¼ 23.72, c1 ¼ 0.2, c2 ¼ 2.32, k1 ¼ 0.97 and k2 ¼ 0.44. It is difficult to obtain the exactly analytical solution to Eq. (32). However, it is possible to obtain its approximately analytical solution. For example, if gðHÞ ¼ s0 þ
7 X
7 X
H i ðs1i þ s2i H i þ s3i H 2i Þ þ
i¼1
sbij H i H j þ 0ðH i H j H k Þ
(47)
i;j¼1;iaj
then a polynomial solution ^¯ ¼ V ðHÞ
7 X
7 X
H^ i ðp1i þ p2i H^ i Þ þ
i¼1
pbij H^ i H^ j
(48)
i;j¼1;iaj
8 0.6
7 Control Effectiveness of STD Displacement
Floor Number
6 5 4 3 Uncontrolled LQG NSO
2 1
LQG NSO
0.5
0.4
0.3
0.2
0.1
0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 (a) STD Displacements
0
2
1
3
(b)
5 4 Floor Number
6
7
8
80 75 70
LQG NSO
Control Efficiency of STD Displacement
65 60 55 50 45 40 35 30 25 20 0
(c)
1
2
3
4
5
6
7
8
Floor Number
Fig. 4. (a) STD displacements of uncontrolled, NSO- and LQG-controlled offshore platforms; (b) control effectiveness and (c) control efficiency of NSO and LQG control strategies. T ¼ 3:5 s, R ¼ 107 I7 , s1 ¼ ½1; 1; 1; 1; 1; 1; 1T for NSO and Rl ¼ 1, Ql ¼ 107 I14 for LQG.
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Table 1 Standard deviation of deck acceleration and control force Sea states
T ¼ 1.5 T ¼ 2.5 T ¼ 3.5
STD deck acceleration sa (m2/s)
Control effectiveness K (%)
Control efficiency m
Uncontrolled
NSO
LQG
NSO
LQG
NSO
LQG
2.8004 1.1950 0.7285
1.2287 0.5802 0.3442
1.7941 0.7999 0.4839
56.12 51.45 52.75
35.93 33.06 33.58
36.10 67.68 110.14
23.11 43.49 70.11
to Eq. (32) may be obtained. Note that only some coefficients in Eq. (47) can be predetermined. The other coefficients in Eq. (47) and the coefficients in Eq. (48) can be determined by substituting them into dynamical programming Eq. (32). Numerical results by using LQG control strategy are also obtained. Rewriting system Eq. (21) as system state equation _ ¼ Al ZðtÞ þ Bl U þ Cl F, ZðtÞ
(49)
T T
_ ; where ZðtÞ ¼ ½QT ; Q " Al ¼
#
0
I
X2
2fX
;
Bl ¼
0 UT P
;
Cl ¼
0 UT
.
(50)
Let the cost function L(Z, U) be LðZ; UÞ ¼ ZT Ql Z þ UT Rl U,
(51)
where Ql is a semi-definite symmetric matrix and Rl is a positive-definite symmetric matrix. In the case of semiinfinite time-interval control with performance index 1 T!1 T
Z
J l ¼ lim
T
LðZ; UÞ dt
(52)
0
the optimal control force can be determined easily [10]. The active control device is installed on the top floor. Numerical results of STD displacement responses of uncontrolled, LQG- and NSO-controlled offshore platforms for three different sea states are shown in Figs. 2–4. Numerical results of STD deck acceleration responses are listed in Table 1. It can be seen from these numerical results that the control effectiveness and efficiency of the proposed NSO control strategy are greater than those of LQG control strategy.
9. Conclusions In this paper a nonlinear stochastic optimal (NSO) control strategy has been proposed for offshore platform subject to wave loading. The proposed control strategy has several advantages. By using the stochastic averaging, the dimension of control system is reduced to a half of that of the original controlled system, and thus the dimension of dynamical programming equation is also reduced by a half. Numerical results for an example offshore platform subject to three different sea states shows that the proposed control strategy is more effective and efficient than LQG control strategy. Note that this is the primary study on the application of the nonlinear stochastic strategy on wave-excited offshore structures. The more practical study considering all details including measuring and estimating system state will be conducted in future.
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Acknowledgements The work reported in this paper was supported by the National Natural Science Foundation of China under Key Grant no. 10332030 and the special Fund for Doctor Programs in the Institutions of Higher Learning of China under Grant no. 20020225092. References [1] G.W. Housner, L.A. Bergman, T.K. Caughey, A.G. Chassiakos, R.O. Claus, S.F. Masri, R.E. Skelton, T.T. Soong, B.F. Spencer, J.T.P. Yao, Structural control: past, present, and future, ASCE Journal of Engineering Mechanics 123 (1997) 897–971. [2] S. Sirlin, C. Paliou, R.W. Longman, M. Shinozuka, E. Samaras, Active control of floating structures, ASCE Journal of Engineering Mechanics 112 (1986) 947–965. [3] A.M. Reinhorn, G.D. Manolis, C.Y. Wen, Active control of inelastic structures, ASCE Journal of Engineering Mechanics 113 (1987) 315–333. [4] T. Hirayama, N. Ma, Dynamic response of a very large floating structure with active pneumatic control, Proceedings of the Seventh International Offshore and Polar Engineering Conference, Honolulu, vol I, 1997, pp. 269–276. [5] K. Yoshida, H. Suzuki, D. Nam, M. Hineno, S. Ishida, Active control of coupled dynamic response of TLP hull and tendon, Proceedings of the Forth International Offshore and Polar Engineering Conference, Osaka, Japan, 1994, pp. 10–15. [6] M. Abdel-Rohman, Structural control of a steel jacket platform, Structural Engineering and Mechanics 4 (1996) 125–138. [7] J. Suhardjo, A. Kareem, Structural control of offshore platforms, Proceedings of the Seventh International Offshore and Polar Engineering Conference, ISOPE-97, Honolulu, 1997. [8] J.L. Hua, J.H. Sau-Lon, J. Christopher, H2 active vibration control for offshore platform subjected to wave loading, Journal of Sound and Vibration 263 (2003) 709–724. [9] W.Q. Zhu, Z.G. Ying, T.T. Soong, An optimal nonlinear feedback control strategy for randomly excited structural systems, Nonlinear Dynamics 24 (2001) 31–51. [10] W.Q. Zhu, Z.G. Ying, Nonlinear stochastic optimal control of partially observable linear structures, Engineering Structures 24 (2002) 333–342. [11] W.Q. Zhu, M. Luo, Z.G. Ying, Nonlinear stochastic optimal control of tall buildings under wind loading, Engineering Structures 26 (2004) 1561–1572. [12] P.T.D. Spanos, Filter approaches to wave kinematics approximation, Applied Ocean Research 8 (1986) 2–7. [13] J. Suhardjo, A. Kareem, Feedback-feedforward control of offshore platforms under random waves, Earthquake Engineering and Structural Dynamics 30 (2001) 213–235. [14] T. Sarpkaya, M. Isaacson, Mechanics of Wave forces on Offshore Structures, Van Nostrand Reihhold, New York, 1981. [15] W.Q. Zhu, Z.L. Huang, Y.Q. Yang, Stochastic averaging of quasi-integrable Hamiltonian systems, ASME Journal of Applied Mechanics 64 (1997) 975–984.
Nonlinear stochastic optimal control of offshore platforms under wave loading M. Luoa,b, W.Q. Zhua, a
Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, 310027, People’s Republic of China b College of Information Science and Engineering, East China University of Science and Technology, 130 Meilong Road, Xuhui District, Shanghai, 200237, People’s Republic of China Received 6 April 2004; received in revised form 2 May 2005; accepted 31 January 2006 Available online 2 June 2006
Abstract A nonlinear stochastic optimal (NSO) control strategy for wave-excited jacket-type offshore platforms is proposed. Wave force is determined according to linearized Morison equation. Spectral density functions of water particle velocity and acceleration are approximated by some rational forms, respectively. Wave force vector is then treated as output of a linear filter driven by white noise. A set of partially averaged Itoˆ equations for controlled modal energies are derived by applying stochastic averaging method for quasi-integrable Hamiltonian systems. Optimal control law is then determined by using stochastic dynamical programming principle. To demonstrate the effectiveness and efficiency of the proposed control strategy, performances of uncontrolled, linear quadratic Gaussian (LQG)-controlled and NSO-controlled example platforms under different sea states are evaluated. Numerical results show that the NSO controller has better control effectiveness and efficiency than the LQG controller. r 2006 Elsevier Ltd. All rights reserved.
1. Introduction Active control of civil engineering structures has been investigated for more than twenty years [1]. Many control strategies and control devices have been proposed for active control of civil engineering structures. However, most studies care about structures excited by wind and earthquakes and only a few studies concern wave-excited offshore structures [2–8]. Active control of wave-excited offshore platforms has not been appropriately investigated. As platform moves far and far into deep water, extreme wave in deep water can induce large motion of offshore platform, which can threaten safety and operation of platform. To prevent fatigue damage and to protect operation and crew of platform, more attention should be paid to active control of wave-excited offshore structures. Since dynamic loading such as earthquake, wind and wave are usually modeled as random processes, to apply stochastic optimal control theory to civil engineering structures is natural and reasonable. Recently, Corresponding author. Tel.: +86 571 8795 3102; fax: +86 571 8795 2651.
E-mail address: [email protected] (W.Q. Zhu). 0022-460X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2006.01.071
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based on the stochastic averaging method for quasi-Hamiltonian systems and stochastic dynamical programming principle, a nonlinear stochastic optimal control strategy was proposed by the present second author and his co-workers [9,10]. It has been applied to the control of buildings structures under wind and earthquake excitation, and proved to be more effective and efficient than linear quadratic Gaussian (LQG) control strategy [10,11]. The present paper is devoted to the application of the nonlinear stochastic optimal control strategy to wave-excited offshore structures. Although the basic control strategy is the same, the structures and loadings are different here and in Refs. [10,11]. So, the control effectiveness and efficiency are different. Besides, the control system here is assumed to be completely observable rather than partially observable as in Refs. [10,11]. To apply the nonlinear stochastic optimal strategy to wave-excited offshore structures, it is necessary to establish a set of stochastic differential equations for random wave loading, which is usually characterized by power spectral density function. An filter approaches to wave kinematics was presented [12], extended and used for active control of offshore platform subject to wave loading using H2 controller [13]. Here this filter approach is used in the active control of offshore platform subject to wave loading using the nonlinear control strategy. 2. Model of wave load It is assumed that wave surface elevation Z(t) is a zero-mean, stationary, Gaussian random process, characterized by its spectral density function SZ(o). One of the most commonly used spectral density function of wave surface elevation is JONSWAP spectral density [14] 5H 2s 5 5 4 b S Z ðoÞ ¼ (1) ðo=o0 Þ exp ðo=o0 Þ g , 4 16o0 where Hs is significant wave height; o0 is dominant (peak) wave frequency, g is sharpness magnification factor (2) b ¼ exp ðo o0 Þ2 =2t2 o20 in which t ¼ 0.07 for oro0 and t ¼ 0.09 for oZo0. In order to obtain a rational form for the spectral density function of wave force, the following approximate JONSWAP spectral density is taken [12] S Z ðoÞ
Gðo=o0 Þ4 ,
2
2 2 2 2 2 o=o0 k1 þ c1 o=o0 o=o0 k2 þ c2 o=o0
(3)
where the parameters G,k1,k2,c1,c2, are determined by using a least-squares algorithm. Following the linear wave theory, the spectral density function of horizontal water-particle velocity u˙ at level z is 2 2 2 cosh ðkzÞ SZ ðoÞ, Su_u_ ðo; zÞ ¼ H uZ _ ðo; zÞ S Z ðoÞ ¼ o sinh2 ðkdÞ
(4)
where Hu˙Z(o,z) is the frequency response function for transformation from wave surface elevation to horizontal water-particle velocity at level z; k ¼ o2/g is wave number, g is the gravity acceleration. The spectral density function of horizontal water-particle acceleration u¨ at level z is S u€u€ ðo; zÞ ¼ o2 S u_u_ ðo; zÞ.
(5)
The frequency response function in Eq. (4) can be expanded as [13] 2 4 6 2 1 1 1 2 2 cosh ðkzÞ 2 1 þ 2 ðkzÞ þ 24 ðkzÞ þ 720 ðkzÞ þ H uZ ¼ o , _ ðo; zÞ ¼ o 1 1 1 þ 16 ðkd Þ3 þ 120 sinh2 ðkd Þ ðkd Þ5 þ 5040 ðkd Þ7 þ
(6)
where d is the water depth at structural site. Keeping the first two terms of Fourier expansions of numerator and denominator, respectively, of |Hu˙Z(o,z)|2 in Eq. (6) and substituting k ¼ o2/g into Eq. (6),
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one obtains 2 H uZ _ ðo; zÞ
z2 6 o þ o2 2g2
1þ
d3 6 o : 6g3
Thus, the approximate spectral density function of horizontal water-particle velocity is
3Ggz2 o40 2g2 4 o þ o6 3 2 z d i h i
Su_u_ ðo; zÞ ¼ h 2 2 3 o2 k1 o20 þ c21 o20 o2 o2 k2 o20 þ c22 o20 o2 o6 þ 6g 3 d and the approximate spectral density function of horizontal water-particle acceleration is
3Ggz2 o40 2g2 4 o þ o8 3 2 z d i h i
. S u€u€ ðo; zÞ ¼ h 2 2 3 o2 k1 o20 þ c21 o20 o2 o2 k2 o20 þ c22 o20 o2 o6 þ 6g 3 d Similarly, the approximate cross-spectral density function of horizontal water-particle velocity is
3Ggzm zn o40 2g2 4 6 o þ zm zn o d3 i h i
S u_u_ ðo; zm ; zn Þ ¼ h 2 2 3 o2 k1 o20 þ c21 o20 o2 o2 k2 o20 þ c22 o20 o2 o6 þ 6g d3 and the approximate cross-spectral density function of horizontal water-particle acceleration is
3Ggzm zn o40 2g2 4 8 o þ 3 zm zn o d h i h i
. Su€u€ ðo; zm ; zn Þ ¼ 2 2 3 o2 k1 o20 þ c21 o20 o2 o2 k2 o20 þ c22 o20 o2 o6 þ 6g d3
(7)
(8)
(9)
(10)
(11)
According to the Morison equation, wave force acting on the structure at level z is of the form _ uj _ þ kM u, € F ðz; tÞ ¼ kD uj
(12)
where kD ¼ ð1=2ÞrC D Ap ;
kM ¼ rC M V p
(13)
in which r is density of seawater, CD and CM are drag and inertia coefficients, respectively, and Ap and Vp are projected area and volume of structure at level z, respectively. The linearized Morison equation is of the form [14] pffiffiffiffiffiffiffiffi € F ðz; tÞ ¼ 8=pkD su_ u_ þ kM u, (14) where su˙ ¼ su˙(z) is the standard deviation of horizontal water-particle velocity at level z. Then the crossspectral density of the approximate wave forces acting on the structure at zm and zn is 8 S F ðo; zm ; zn Þ ¼ kM ðzm ÞkM ðzn ÞSu€u€ ðo; zm ; zn Þ þ kD ðzm ÞkD ðzn Þsu_m su_n S u_u_ ðo; zm ; zn Þ, p
(15)
where Su€u€ ðo; zm ; zn Þ and S u_u_ ðo; zm ; zn Þ are the cross-spectral densities of horizontal water-particle velocity and acceleration. Suppose that the offshore structure is simplified as a lumped mass system with N lumped masses. The Ndimensional wave force vector F(t), where FðtÞ ¼ ½F N ðtÞ; F N1 ðtÞ; ; F 1 ðtÞT , acting on N lumped mass with spectral density matrix S(o) obtaining from discreting Eq. (15) can be modeled as the output of the following linear filter with a unit intensity Gaussian white noise process w(t) as an input FðtÞ ¼ Cf y;
y_ ¼ Af y þ Bf wðtÞ,
(16)
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where
737
3 0 2 3 0 6 .. 7 7 6 0 0 1 . 7 607 6 6 7 7 6 .. 6 7 7 6 0 607 . 1 7 6 6 7 6 6 7 .. 7 7 6 ; B Af ¼ 6 0 ¼ 607 ; f 1 . 7 6 7 7 6 607 . 7 6 .. 6 7 6 0 1 0 7 6 7 7 6 405 7 6 0 0 0 1 5 4 1 71 a0 a1 a2 a3 a4 a5 a6 qffiffi h qffiffi
qffiffi
i 8 8 k b þ k b k b þ k b b Cf ¼ 0; 0; 0; p8ru_ kD b0 ; r r ; ; k _ _ D 1 M 0 D 2 M 1 M 2 u u p p 2
0
1
0
N7
a0 ¼
k1 k2 o40
sffiffiffiffiffiffiffi 6g3 ; d3
a1 ¼ ðc1 k2 þ
sffiffiffiffiffiffiffi 6g3 a3 ¼ k1 k2 o40 þ ðc2 þ c1 Þo0 ; d3
c2 k1 Þo30
sffiffiffiffiffiffiffi 6g3 ; d3
a2 ¼ ðk2 þ c1 c2 þ
sffiffiffiffiffiffiffi 6g3 a4 ¼ ðc1 k2 þ c2 k1 Þo30 þ , d3
k1 Þo20
(17)
;
sffiffiffiffiffiffiffi 6g3 , d3
a5 ¼ ðk2 þ c1 c2 þ k1 Þo20 ; a6 ¼ ðc2 þ c1 Þo0 , ru_ ¼ diagðsu_zN ; su_zN1 ; . . . su_z1 Þ, kD ¼ diagðkDZN ; kDZN1 ; . . . ; kDZ1 Þ; kM ¼ diagðkM ZN ; kM ZN1 ; . . . ; kM Z1 Þ, 3 2 2 pffiffiffiffiffiffi 3 2 3 zN zN 1 7 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6 pffiffiffiffiffiffiffiffiffiffi 7 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi6 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 7 6 zN1 7 6 zN1 7 17 2 8 7 7 4 72G g4 o 6 3Ggo40 6 6Gg3 o40 6 6 7 06 7 7 6 ; b ; b b0 ¼ ¼ ¼ , 6 1 2 . .. 7 . 7 6 . 7 7 d3 6 d6 d3 6 6 .. 7 6 . 7 . 5 4 5 5 4 4 pffiffiffiffiffi 1 N1 z1 z1
ð18Þ
where diag denotes diagonal matrix. Note that, since the highest power of o in the denominator of approximate rational spectral density function of linearized wave force is 14, the dimension of matrix Af is 7 7. 3. Equation of controlled system The simplified controlled system equation is of the form € þ CX _ þ KX ¼ FðtÞ þ PU; MX
Xð0Þ ¼ X0 ,
(19)
_ X € are N-dimensional vectors of structural displacements, velocities and accelerations, where X; X; respectively.M,C and K are mass, damping and stiffness matrices of the system, respectively; U ¼ [U1,U2,y,Um]T is m-dimensional vector of control forces produced by m control devices; P is N m control device placement matrix. Introduce model transform (20)
X ¼ UQ,
where Q ¼ ½Q1 ; Q2 ; ; Qn T is the dominant modal displacement vector, and U is a N n dominant modal matrix consist of n dominant modal vectors. The equation for the n dominating modes can be written as € þ 2fXQ _ þ X2 Q ¼ UT FðtÞ þ UT PU; Q
Qð0Þ ¼ Q0 ,
(21)
where 2fX ¼ UT CU;X2 ¼ diagðo2i Þ ¼ UT KU. oi and zi are the frequency and damping coefficient of the ith mode, respectively.
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Eqs. (16) and (21) can be combined and converted into the following Itoˆ stochastic differential equation ~ dZ ¼ ðAZ þ BUÞ þ C dBðtÞ;
Zð0Þ ¼ Z0 , (22) ~ ˙ ,y ,y˙ ] ; BðtÞ is a standard Wiener process, Z0 is a Gaussian random vector representing the where Z ¼ [Q ,Q ~ initial state of the system, which is independent of BðtÞ, 2 3 2 3 0 2 3 0 0 I 0 0 6 UT P 7 6 0 7 2 6 7 6 7 6 7 A ¼ 4 X 2fX Cf 0 5; B ¼ 6 (23) 7; C ¼ 6 7. 4 0 5 4 Bf 5 0 0 Af 0 0 0 T
T
T
T T
4. Stochastic averaging To simplify the equation of controlled system and reduce its dimension, the stochastic averaging method for ˙ in Eq. (22) except the control quasi-integrable Hamiltonian system [15] is applied to the equations for Q and Q force terms. Then the obtained partially averaged Itoˆ stochastic differential equations for the n dominant modal energies are ¯ i ðtÞ; i ¼ 1; 2; :::; n, dH i ¼ mi ðH i Þ þ oðqH i =qQ_ i ÞFTik Pkr U r 4 dt þ si ðH i Þ dB (24) where o 4 denotes averaging operation with respect to time t; 2 H i ¼ ðQ_ i þ o2i Q2i Þ=2;
i ¼ 1; 2; :::; n
(25)
¯ i ðtÞ are standard Wiener processes; denotes the energy of the ith mode; B mi ðH i Þ ¼ 2zi oi H i þ FTil Fki Slk ðoi Þ, s2i ðH i Þ ¼ 2H i FTil Fki S lk ðoi Þ;
ð26Þ
Slk ðoi Þ ¼ S F l F k ðoi Þis the l, kth element of spectral density matrix S(o) evaluated at oi. 5. Optimal control law Eq. (24) implies that Hi are controlled diffusion processes. The objective of stochastic optimal control is to seek the optimal feedback control U* for minimizing the following semi-infinite time-interval performance index Z 1 tf LðHðtÞ; oUðtÞ4Þ dt, (27) JðUÞ ¼ lim tf !1 tf 0 where L(H(t),oU(t)4) is the so-called cost function. Based on the stochastic dynamical programming principle, the following dynamical programming equation can be established [9] ( ) T qV qH T 1 q2 V l ¼ min LðHðtÞ; oUðtÞ4Þ þ mðHÞ þ o rðHÞrT ðHÞ , (28) U PU4 þ tr _ U qH 2 qH2 qQ where 1 tf !1 tf
Z
l ¼ lim
tf
½LðHðtÞ; oU ðtÞ4 dt
(29)
0
is the optimal average cost, V is the value function and U* is the optimal control force vector. The expression for U* can be obtained from minimizing the right-hand side of Eq. (28) with respect to U. Suppose that the cost function L is of the form L ¼ gðHÞ þ oUT RU4,
(30)
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where R is a n n positive-define matrix. Then U* is of the following form: 1 _ T ðqV =qHÞ (31) U ¼ R1 PT UðqH=qQÞ 2 ˙ with respective to t yield the final Substituting Eq. (31) into Eq. (28) and completing averaging Q and Q dynamical programming equation for semi-infinite time-interval control problem " # n X qV 1 qV 2 1 2 q2 V l ¼ gðHÞ þ mi ðH i Þ Dii H i þ si ðH i Þ , (32) qH i 4 qH i 2 qH 2i i¼1 where Dii ¼ ½UT PR1 PT Uii : qV/qH can be obtained by solving the final dynamical programming equation (32) and the optimal control U* can be obtained by substituting the resultant qV/qH into the Eq. (31). 6. Response prediction of controlled structure AveragingðqH i =qQi ÞFTik Pkr U r Þ and then substituting it into equation (24) to replace ðqH i =qQ_ i ÞFTik Pkr U r , one obtains the completely averaged Itoˆ equation for Hi h i ¯ i ðtÞ, dH i ¼ mi ðH i Þ þ mðUÞ ðH Þ dt þ si ðH i Þ dB (33) i i where 1 T qV 1 T mðUÞ . i ðH i Þ ¼ Fik Pkr Rrs Psl Fli H i 2 qH i
(34)
The stationary probability density p(H) can be obtained from solving the reduced Fokker-plank-kolmogorov (FPK) equation associated with Itoˆ Eq. (33). It is of the following form ( Z ) n HX qV pðHÞ ¼ C 0 exp Di þ E i dH i , (35) qH i 0 i¼1 where C 0 is a normalization constant, Di ¼ 4zi oi =FTil Fki S lk ðoi Þ, T T E i ¼ FTik Pkr R1 rs Psj Fji =Fil Fki S lk ðoi Þ.
ð36Þ
The stationary probability density of the modal displacement and velocity of the controlled structure is then [15] _ ¼ ½pðHÞ=TðHÞ pðQ; QÞ 2 , (37) H ¼ðQ_ þo2 Q2 Þ=2 i
i
i
where TðHÞ ¼
n I Y i¼1
n Y dQi 2p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ . oi i¼1 2H i o2i Q2i
The mean square values of the modal displacement and velocity of the controlled structure are Z 1 Z 1 _ dQ dQ _ ¼ 1 Q2i pðQ; QÞ H i pðHÞ dH, E Q2i ¼ o2i 0 1 Z 1 Z 1 h 2i 2 _ dQ dQ _ ¼ E Q_ i ¼ H i pðHÞ dH. Q_ i pðQ; QÞ 1
0
(38)
ð39Þ
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Finally, the mean square displacement and acceleration of each floor are obtained from Eq. (39) by using modal transformation (20) as follows: 2 !2 3 n X 2 E X i ¼ E4 Fij Qj 5, 2
j¼1
n h 2i X E X€ i ¼ E 4 Fij o2j Qj þ 2zi oi Q_ j UT FðtÞ þ UT PU j
j¼1
!2 3 5.
ð40Þ
The response statistics of the uncontrolled structure can be obtained in a similar way by letting control force vanish.
7. Performance criteria The main purpose of vibration control of wave-excited offshore platforms is to reduce displacement to prevent fatigue damage, and to reduce deck acceleration to protect operations and crew of offshore platforms. So, to evaluate a control strategy, the control effectiveness and efficiency of the displacements and deck acceleration are considered. Control effectiveness is defined as [9] K ¼ ðsu sc Þ=su ,
(41)
where s stands for the standard deviation of displacement or deck acceleration; subscripts u and c denote uncontrolled and controlled offshore platforms, respectively. Control efficiency is defined as m ¼ K=sF , (42) 2 ffi Pm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where sF ¼ i¼1 E U i =G~ is the standard deviation of control force, normalized by total weight G~ of offshore platform. It is seen from the definitions that the larger K and m are, the better the control strategy is.
Z7=144.78m d=121.92m Z6=118.87m Z5=99.06
Z4=79.06
Z3=59.44
Z2=39.62
Z
Z1=19.81
Fig. 1. Model of jacket-type offshore platform used for numerical example.
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8. Numerical example A simplified jacket-type platform with 7 lumped masses is taken as an example (see Fig. 1). The height of the platform is 144.78 m. The mass, stiffness and damping matrices are M ¼ diagf 4818 1475 1302 1533 1840 2205 3738 g 103 kg; 3 2 344 289 17 1 26 4 16 6 670 366 4 33 23 10 7 7 6 7 6 7 6 725 368 14 22 9 7 6 7 6 777 410 23 23 7 103 kN=m; K¼6 7 6 6 879 491 56 7 7 6 7 6 sym 1005 595 5 4 1344
8 0.7
7 Control Effectiveness of STD Displacements
Floor Number
6 5 4 3 2
Uncontrolled LQG NSO
1
0.6 0.5 0.4 0.3
0
0.2 0.02
0.04 0.06 0.08 STD Displacements
Control Effieciency of STD Displacement
(a)
LQG NSC
(c)
0.10
0.12
0
1
3 4 5 Floor Number
2
(b) 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16
6
7
8
LQG NSO
0
1
2
3 4 5 Floor Number
6
7
8
Fig. 2. (a) STD displacements of uncontrolled, NSO- and LQG-controlled offshore platforms; (b) control effectiveness and (c) control efficiency of NSO and LQG control strategies. T ¼ 1:5 s, R ¼ 107 I7 , s1 ¼ ½1; 1; 1; 1; 1; 1; 1T for NSO and Rl ¼ 1, Ql ¼ 107 I14 for LQG.
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2
2522 1244 6 2892 6 6 6 6 6 C¼6 6 6 6 6 sym 4
3 59 30 7 7 7 106 22 7 7 7 161 113 7 103 N s=m: 7 1292 122 7 7 7 4311 1605 5 6769
9 144
263 936
135 188
2808
920
161
3142
1072 3652
18 7
(43)
The natural frequencies of this platform are 0.7432, 1.2497, 2.1678, 2.9563, 3.6921, 4.3649, 4.9773 (Hz), and the modal damping ratios are 5% for all modes. The water depth d is 121.92 m. The vertical coordinates of lumped masses are z ¼ ½z7 ; z6 ; ; z1 T ¼ ½144:78; 118:87; 99:06; 79:06; 59:44; 39:62; 19:81T m;
(44)
the drag coefficient matrix is kD ¼ diagf0; 790; 676; 719; 757; 828; 1557g 103 kg=m;
(45)
8 0.6
7 Control Effectiveness of STD Displacements
Floor Number
6 5 4 3 Uncontrolled LQG NSO
2 1
LQG NSO
0.5
0.4
0.3
0.2
0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 (a) STD Displacements
0.1 0
1
2
3
6
7
(b)
4 5 Floor Number
6
7
8
80 75 70
LQG NSC
Control Efficiency of STD Displacement
65 60 55 50 45 40 35 30 25 20
0
(c)
1
2
3
4 5 Floor Number
8
Fig. 3. (a) STD displacements of uncontrolled, NSO- and LQG-controlled offshore platforms; (b) control effectiveness and (c) control efficiency of NSO and LQG control strategies. T ¼ 2:5 s, R ¼ 107 I7 , s1 ¼ ½1; 1; 1; 1; 1; 1; 1T for NSO and Rl ¼ 1, Ql ¼ 107 I14 for LQG.
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and the inertia coefficient matrix is kM ¼ diagf0; 1744; 1674; 1939; 2563; 3150; 6735g 103 kg:
(46)
A JONSWAP spectrum with significant wave height H s ¼ 10 m, sharpness magnification factor g ¼ 3.0 and periods T ¼ 1.5, 2.5, 3.5 (s) is used to characterize the sea states. The parameters for the approximate JONSWAP spectrum are G ¼ 23.72, c1 ¼ 0.2, c2 ¼ 2.32, k1 ¼ 0.97 and k2 ¼ 0.44. It is difficult to obtain the exactly analytical solution to Eq. (32). However, it is possible to obtain its approximately analytical solution. For example, if gðHÞ ¼ s0 þ
7 X
7 X
H i ðs1i þ s2i H i þ s3i H 2i Þ þ
i¼1
sbij H i H j þ 0ðH i H j H k Þ
(47)
i;j¼1;iaj
then a polynomial solution ^¯ ¼ V ðHÞ
7 X
7 X
H^ i ðp1i þ p2i H^ i Þ þ
i¼1
pbij H^ i H^ j
(48)
i;j¼1;iaj
8 0.6
7 Control Effectiveness of STD Displacement
Floor Number
6 5 4 3 Uncontrolled LQG NSO
2 1
LQG NSO
0.5
0.4
0.3
0.2
0.1
0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 (a) STD Displacements
0
2
1
3
(b)
5 4 Floor Number
6
7
8
80 75 70
LQG NSO
Control Efficiency of STD Displacement
65 60 55 50 45 40 35 30 25 20 0
(c)
1
2
3
4
5
6
7
8
Floor Number
Fig. 4. (a) STD displacements of uncontrolled, NSO- and LQG-controlled offshore platforms; (b) control effectiveness and (c) control efficiency of NSO and LQG control strategies. T ¼ 3:5 s, R ¼ 107 I7 , s1 ¼ ½1; 1; 1; 1; 1; 1; 1T for NSO and Rl ¼ 1, Ql ¼ 107 I14 for LQG.
ARTICLE IN PRESS M. Luo, W.Q. Zhu / Journal of Sound and Vibration 296 (2006) 734–745
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Table 1 Standard deviation of deck acceleration and control force Sea states
T ¼ 1.5 T ¼ 2.5 T ¼ 3.5
STD deck acceleration sa (m2/s)
Control effectiveness K (%)
Control efficiency m
Uncontrolled
NSO
LQG
NSO
LQG
NSO
LQG
2.8004 1.1950 0.7285
1.2287 0.5802 0.3442
1.7941 0.7999 0.4839
56.12 51.45 52.75
35.93 33.06 33.58
36.10 67.68 110.14
23.11 43.49 70.11
to Eq. (32) may be obtained. Note that only some coefficients in Eq. (47) can be predetermined. The other coefficients in Eq. (47) and the coefficients in Eq. (48) can be determined by substituting them into dynamical programming Eq. (32). Numerical results by using LQG control strategy are also obtained. Rewriting system Eq. (21) as system state equation _ ¼ Al ZðtÞ þ Bl U þ Cl F, ZðtÞ
(49)
T T
_ ; where ZðtÞ ¼ ½QT ; Q " Al ¼
#
0
I
X2
2fX
;
Bl ¼
0 UT P
;
Cl ¼
0 UT
.
(50)
Let the cost function L(Z, U) be LðZ; UÞ ¼ ZT Ql Z þ UT Rl U,
(51)
where Ql is a semi-definite symmetric matrix and Rl is a positive-definite symmetric matrix. In the case of semiinfinite time-interval control with performance index 1 T!1 T
Z
J l ¼ lim
T
LðZ; UÞ dt
(52)
0
the optimal control force can be determined easily [10]. The active control device is installed on the top floor. Numerical results of STD displacement responses of uncontrolled, LQG- and NSO-controlled offshore platforms for three different sea states are shown in Figs. 2–4. Numerical results of STD deck acceleration responses are listed in Table 1. It can be seen from these numerical results that the control effectiveness and efficiency of the proposed NSO control strategy are greater than those of LQG control strategy.
9. Conclusions In this paper a nonlinear stochastic optimal (NSO) control strategy has been proposed for offshore platform subject to wave loading. The proposed control strategy has several advantages. By using the stochastic averaging, the dimension of control system is reduced to a half of that of the original controlled system, and thus the dimension of dynamical programming equation is also reduced by a half. Numerical results for an example offshore platform subject to three different sea states shows that the proposed control strategy is more effective and efficient than LQG control strategy. Note that this is the primary study on the application of the nonlinear stochastic strategy on wave-excited offshore structures. The more practical study considering all details including measuring and estimating system state will be conducted in future.
ARTICLE IN PRESS M. Luo, W.Q. Zhu / Journal of Sound and Vibration 296 (2006) 734–745
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Acknowledgements The work reported in this paper was supported by the National Natural Science Foundation of China under Key Grant no. 10332030 and the special Fund for Doctor Programs in the Institutions of Higher Learning of China under Grant no. 20020225092. References [1] G.W. Housner, L.A. Bergman, T.K. Caughey, A.G. Chassiakos, R.O. Claus, S.F. Masri, R.E. Skelton, T.T. Soong, B.F. Spencer, J.T.P. Yao, Structural control: past, present, and future, ASCE Journal of Engineering Mechanics 123 (1997) 897–971. [2] S. Sirlin, C. Paliou, R.W. Longman, M. Shinozuka, E. Samaras, Active control of floating structures, ASCE Journal of Engineering Mechanics 112 (1986) 947–965. [3] A.M. Reinhorn, G.D. Manolis, C.Y. Wen, Active control of inelastic structures, ASCE Journal of Engineering Mechanics 113 (1987) 315–333. [4] T. Hirayama, N. Ma, Dynamic response of a very large floating structure with active pneumatic control, Proceedings of the Seventh International Offshore and Polar Engineering Conference, Honolulu, vol I, 1997, pp. 269–276. [5] K. Yoshida, H. Suzuki, D. Nam, M. Hineno, S. Ishida, Active control of coupled dynamic response of TLP hull and tendon, Proceedings of the Forth International Offshore and Polar Engineering Conference, Osaka, Japan, 1994, pp. 10–15. [6] M. Abdel-Rohman, Structural control of a steel jacket platform, Structural Engineering and Mechanics 4 (1996) 125–138. [7] J. Suhardjo, A. Kareem, Structural control of offshore platforms, Proceedings of the Seventh International Offshore and Polar Engineering Conference, ISOPE-97, Honolulu, 1997. [8] J.L. Hua, J.H. Sau-Lon, J. Christopher, H2 active vibration control for offshore platform subjected to wave loading, Journal of Sound and Vibration 263 (2003) 709–724. [9] W.Q. Zhu, Z.G. Ying, T.T. Soong, An optimal nonlinear feedback control strategy for randomly excited structural systems, Nonlinear Dynamics 24 (2001) 31–51. [10] W.Q. Zhu, Z.G. Ying, Nonlinear stochastic optimal control of partially observable linear structures, Engineering Structures 24 (2002) 333–342. [11] W.Q. Zhu, M. Luo, Z.G. Ying, Nonlinear stochastic optimal control of tall buildings under wind loading, Engineering Structures 26 (2004) 1561–1572. [12] P.T.D. Spanos, Filter approaches to wave kinematics approximation, Applied Ocean Research 8 (1986) 2–7. [13] J. Suhardjo, A. Kareem, Feedback-feedforward control of offshore platforms under random waves, Earthquake Engineering and Structural Dynamics 30 (2001) 213–235. [14] T. Sarpkaya, M. Isaacson, Mechanics of Wave forces on Offshore Structures, Van Nostrand Reihhold, New York, 1981. [15] W.Q. Zhu, Z.L. Huang, Y.Q. Yang, Stochastic averaging of quasi-integrable Hamiltonian systems, ASME Journal of Applied Mechanics 64 (1997) 975–984.