Adaptive Inverse Control of Offshore Jacket Platform Based on Grey prediction and Rough Neural Network

(Department of Naval Architecture,Dalian University of Technology,Dalian 116085,China)

1 Introduction

The offshore jacket platform usually produces harmful vibration under the action of waves and winds loads.During the past decades,many domestic and overseas researchers have done lots of research on the problem for vibration control of the offshore jacket platform,and have made great achievements[1].Although the traditional passive method can control the harmful vibration in some extent by increasing the stiffness and damping of the platform,we still can not achieve the ideal effect utilizing the traditional passive method because of the complex structure and the variable parameters of the platform.The active control method can effectively reduce the vibration level of the offshore jacket platform in the whole frequency domain.However,the traditional active control method based on accurate mathematical model is not an ideal method because the offshore jacket platform is a non-linear,strong-coupling,multi-variation,uncertainty,time-varying and the control signal transmission delay problem.The intelligent methods[2]which are independent of the accurate mathematical model have the properties of strong stability,robustness and non-linear processing,such as neural network,so the problem of vibration control can be solved effectively.

Grey prediction theory[3]finds out the law of system development from the raw data of the happened behaviors,and then determines the corresponding control decision according to the future developing trend of the system.The grey prediction is a lead control method,which has advantages of simple calculation frame,superior characteristics of adaptability,real-time and high accuracy.Rough set theory[4]can be used to analyze and process large amount of experiential data,discover hidden information and disclose potential rules effectively.The neural network has strong ability of anti-noise and generalization.The rough neural network(RNN)integrates the advantages of the rough set theory and the neural network,such as clear structure and strong error-tolerance capability.

Dynamic stiffness matrix(DSM)method is a powerful tool for solving vibration problems in structural engineering.The DSM method can be devoted to calculate the natural frequency and mode shape by using fewer elements.The method can also decrease the degree-of-freedoms.With the increase of the calculated frequency,the DSM method does not require to further refine elements.When the W-W algorithm[5]is adopted,the calculation speed of the DSM method will be accelerated.

In order to control the vibration of the platform,a method combining the grey prediction,the RNN techniques and the DSM which is used to build an adaptive inverse control model is proposed in this paper.The structure model of offshore jacket platform is built by the DSM method.The inverse model of the platform identified by the RNN combining the grey prediction is used as the adaptive feed-forward predictive inverse controller to carry out the predictive inverse control,and the time delay during the signal transmission is handled by this predictive control.The errors emerged during model-establishing,identifying and feed-forward control processing of the platform system and wind load are treated as the disturbance signals.The influence of the above factors on the control system performance is reduced by the disturbance eliminator which is the inverse model built by RNN.Experiments results show that our method is effective.

2 Offshore jacket platform model based on DSM method

2.1 Dynamic stiffness matrix method

The element stiffness matrix of DSM was derived from the analytic solution of the element dynamic differential equations.The dynamic differential equation[5-6]of the Timoshenko beam may be written as:

where w represents the transverse displacement,E is the Young’s modulus,G is the shear modulus,ρ is the density of the material,A is the cross section area,Asis the cross section shear area,I is the moment of inertia of cross section,and k is the shape factor for correction section.

It is supposed that the transverse displacement w and the rotation angle ψ of the cross section vary with time t in a sinusoidal form when the beam is vibrating freely,that is where W（x） and Ψ （x） represent the amplitude of transverse displacement and rotation angle respectively.Let ξ=x/L,where L is the length of the beam element.Combined with beam coupling equations,Eq.(1)can be written as follows:

When b2r2s2>1,the vibration frequency is so high that it exceed the using range of the Timoshenko beam theory.In this paper,it is assumed that b2r2s2＜1,then the general solutions of the above differential equations are as follows:

According to Fig.1,when bending vibration of the uniform straight Timoshenko beam element occurs,the boundary conditions of displacement and force are respectively given as

The expressions of bending moment and shear force are shown as follows:

Thus,a general expression can be obtained according to Eqs.(6)～(11):

where F=[V1M1V2M2]Tis a force vector,Δ=[W1Ψ1W2ψ2]Tis a displacement vector of the element and Kdis DSM of the Timoshenko beam element.The detailed expressions of all the components of DSM are then given as follows:

By using standard finite element assembly techniques,the global dynamic stiffness matrix D（ω ） which is the function of circular frequency ω can be obtained.The relationship between the global mass matrix of structure and the global dynamic stiffness matrix can be determined by the Leung’s theorem:

where M（ω） is global mass matrix.

2.2 Establishment of calculation model for the offshore jacket platform

In this paper,the Timoshenko beam element based on the DSM method is used to establish the offshore jacket platform model.The offshore jacket platform is reduced to a plane rigid frame structure,the weights of machinery and equipment,etc,distributed on the corresponding rigid frames.The boundary condition is the rigid fastening at the distance of six times pile diameter under the platform.The well-known W-W algorithm is used to compute natural frequencies and mode shape of the simplified jacket platform model.Then the mode superposition method is applied to calculate the vibratory response of the platform.Then the motion equation of the controlled platform with n degrees of freedom can be written as:

where M（ω） and D（ω）are the global mass and stiffness matrix of order（n×n）,C（ω） is the global damping matrix of order（n×n）,when Rayleigh damping is used,C（ω） can be replaced by α′D （ω）+β′M （ω）,where α′and β′are proportional constants.Y （t）,）and）are the n dimensional displacement,velocity and acceleration vector respectively,U（t）is the m dimensional control force vector;F（t）is the r dimensional external disturbance force vector;L1is the location matrix of order（n×m） control force,and L2is the location matrix of order（n×r） external disturbing force.Finally,the Duhamel integration is used directly to calculate the each mode responses of the controlled platform.The superposition of those responses is used to obtain the vibration response of the whole controlled platform system.

3 Adaptive inverse control system based on grey prediction and RNN

3.1 Grey prediction

In this paper,GM(1,1)model[7]is applied to predict the system output at the time k+d.For SISO system,the time series of original input and output is denoted by where n is the number of modeling dimension.

Because original series(16)and(17)are grey series influenced by random disturbances,the first-order accumulated generating operation(AGO)series of original series is Through the original series and the AGO series,the GM(1,1)model can be built:

For previewing the system development rules,it is necessary to pick up the suitable number of modeling dimension and prediction steps in grey prediction control theory,which will improve the performance of the predictive control system,such as accuracy and real-time.The number of modeling dimension（n） of grey prediction is 5,and the prediction step（d） is 6 in this paper.

3.2 Rough neural networks

(1)Basic concept of the rough set

An information system S′can be defined as:S′=（U′,A′,V′,F′）.U′={x1,x2,…,xn}is a set of samples,A′={a1,a2,…,am}is a set of attributes,and V′is the attribute value domain defined as V′=∪a∈A′Va′,where Va′is the value domain of attribute a.F′:U′×A′→V′is an information function,that is,for every x∈U′,a∈A′has F′（x,a）∈Va′.If the set of attributes A′can be divided into the set of condition attributes C′and the set of decision attributes D′,then A′=C′∪D′,and the information system is called as decision system[8].

For sets X′,X′⊆U′,R′is the equivalence relation about domain U′,that is,according to R′,U′can be divided into some disjoint sets of equivalence class U′/R′.[x]R′shows that the equivalence class in which element x is included;rough set can be expressed by two accurate sets,that is,the lower approximation′（X′）and the upper approximation′（X′）of the rough set:

(2)Structure and training of RNN

The rough set theory is used to extract rules from the given data,and establish the model structure of neural network by these rules.The RNN proposed in this paper consists of five layers:

The first layer:Input layer.

Input vector:x=（x1,x2,…,xn）T.

The second layer:Classification layer of the input equivalence class.

The n input components xi（i=1,2,…,n）are discretized into ridifferent values by some discretization method,the output of neuronis

where cij,σijare the center and width value of jth（j=1,2,…,ri）class variable respectively.

The third layer:Rule layer.

The fourth layer:Classification layer of the output equivalence class.

All outputs of the same class of the neurons in the third layer are taken as the inputs for the fourth layer.Let rdto be the discretization number of output variables,then the output of neuronis

The fifth layer:Output layer.

The output of the neural network yris

where ydis the expected output of the network.

3.3 Structure of adaptive inverse control system based on grey prediction and RNN

For Adaptive inverse control method,the first step is to identify the inverse model of the controlled object,and then in order to control the dynamic performance,the identified model as a controller was linked with the controlled object[9].In this paper,RNN is used to identify the inverse model of the offshore jacket platform.The inverse model structure is shown in Fig.2,where the RNNis the platform inverse model identified by RNN,P is the platform system model,us（k） is the force which acts on the top of the platform,ys（k） is the response of the top of the platform under the force us（k）,and（k） is the identification output of RNNThe platform system is regarded as SISO system for convenience of discussion,and the response can be showed in the following equation:

where m＜n,φ（·）is a non-linear function.As Eq.(35)shows,the reversible system of SISO system can be demonstrated as:

When the training sample set{Xs（k）,us（k）}of RNN is built,the inverse model of the platform system can be identified after the learning course of RNN,leads to（k）→us（k） and error es（k）→0.The output value（k） of RNNC＾is the input value us（k） of the platform system P.For the new input data Xh（k）,the output of inverse system RNNof the platform system is（k）.Because of the existence of identification error in RNN,RNNcan be given by:

There are two parts in the adaptive inverse control system:feed-forward control part and the disturbance elimination control part.In the feed-forward control part

herein,the output signal of the controller RNN1 is taken as the input signal of the grey prediction model GM（ 1,1）,then the predictive control force u1can be obtained through the model GM（ 1,1）.This control signal u1is exerted on the top of the platform in order to control the response of vibration at the top of the platform system caused by the wave load fa.

In the disturbance elimination control part

The control force u2produced by disturbance elimination controller RNNis exerted on the top of the platform to control the response nbof the top of the platform system caused by the wind load and all the other forms of disturbance.When sampling at very high speed,the influence of unit delay is very small.

In the adaptive inverse control system of the offshore jacket platform based on grey prediction and RNN,the feed-forward control error,dynamic stiffness matrix model error and RNN identification error can be treated as the additive noises of the platform control system response,and can be expressed by Nb（k）,thus

The control system structure is equivalent to the structure shown in Fig.4.

In Fig.4,the response signal y on the top of the platform can be directly sampled by the sensor,which is also taken as the input signal of the disturbance elimination controller RNNC＾2.This will be helpful to carry out the on-line real-time control.

4 Numerical example

4.1 Calculation model of the offshore jacket platform

To verify the effectiveness of the proposed method in this paper,an example of offshore jacket platform is demonstrated.The water depth of the sea where the platform works is 80m.The total height of the platform is 140m.There are two equipment layers and three layers of the living area blocks on the top deck with the total height of 20m.The dimension of the deck is 60m×60m.And the section dimensions of vertical,horizontal and diagonal cylindrical steel tubes are Φ1.6×0.04m,Φ0.8×0.02m and Φ0.8×0.02m,respectively.The dimension of the legs cross section is Φ1.46×0.04m.The equivalent distributed weights of each storey from bottom to top are respectively:3 000t,2 500t,2 000t and 12 000t.The sketch of the simplified plane rigid frame model for calculated platform is shown in Fig.5.

4.2 Response of the offshore jacket platform under loads

For the numerical simulation of wave load,the improved P-M wave spectrum[10]is chosen as wave force spectrum with significant wave height 7.5m and wave period 8s.For the numerical simulation of wind load,the stipulated Davenport spectrum in Chinese wind load regulations[11-12]is chosen as power spectra density function of the fluctuating wind velocity.The ground roughness factor is 0.001 29,the ground roughness index is 0.12,the standard wind velocity of 10m elevation is 25m/s and the basic wind pressure is 0.5kN/m2.According to numerical stimulation of the platform,the time histories of the adopted 100s wave and wind loads are shown in Fig.6 and Fig.7 respectively.

The time history of displacement response of the top of the offshore jacket platform under wave and wind loads is shown in Fig.8.The solid line is the time history of displacement response under wave load,and the dotted line represents the time history of displacement response under both wave and wind loads.Under wind load disturbance,it can be seen that the displacement response of the top of the offshore jacket platform changed obviously com-paring to the case where only wave load acted.

4.3 Active control of the offshore jacket platform under wind disturbance

(1)Inverse model of the offshore jacket platform identified by the RNN

The inputs of RNN are the displacement response y（k）,velocity response） and acceleration response） of the top of the platform;the output is the control force u（k） which acts on the top of the offshore jacket platform.Data set with length of 2 500 are extracted by the numerical simulation in this paper.The extracted training data of input variables and output variables of RNN are discretized into seven sections according to the same width.Then taking the input variables as condition attributes and the output variables as decision attributes,the information system decision table can be constructed based on the rough set theory.After analyzing each decision rule,the unnecessary condition attributes,low support degree rules and superfluous rules will be deleted.Finally,62 rules can be obtained and used to construct the simplest decision table.The typology structure of RNN is obtained according to the above data processing method.For this structure of RNN,the input layer has 3 neurons,the classifica tion layer of the input equivalence class has 21 neurons,the rule layer has 62 neurons,the classification layer of the output equivalence class has 7 neurons and the output layer has 1 neuron.The initial values of cijand σijin the network structure are determined in the light of the discretization region,the support degree of each rule is taken as the initial value of weight value,and random numbers are used as the initial value of the weight value.At last data sets with length of 2 500 are taken as the training examples,and after 45 times training,another 1 000 data sets are used to test the generalization capability of RNN,as shown in Fig.9.The solid line is the numerically simulated output of RNN,while the dotted line is the output of the trained RNN.It can be seen that RNN has strong generalization capability and the identified output of the platform inverse system is very close to the real platform inverse system.After connecting the identified inverse system and the controlled platform,the inverse control system can be finally established.

(2)Active control of the offshore jacket platform

The displacement response curve of the top of the platform under both the wave load(shown in Fig.6)and the wind load(shown in Fig.7)is shown in Fig.10 as dotted line.The corresponding displacement response curve after feed-forward control is also shown in Fig.10 as dash-dotted line.The solid line represents the time history of the displacement response of the top of the platform after both feed-forward control and disturbance elimination control.Thus,the control model proposed in this paper can significantly reduce the displacement response of the top of the platform and has excellent anti-disturbance capability.

5 Conclusions

Each time the grey prediction controller samples the data,a new model can be built,which enables the control system to have superior characteristics of adaptability,real-time and realization.Combining the advantages of rough set and neural network,RNN has powerful identification and generalization capability.Moreover,the DSM method has the advantage of fast calculation ability.This paper proposes a new method of inverse control based on grey prediction,DSM and RNN,and applies this method to actively control the harm vibration of offshore jacket platform.The numerical results show that the proposed method has strong stability and robustness,can effectively control the vibration response of the offshore jacket platform,successfully solve the instability and the system surge problem caused by the delay of control signal transmission.

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