Application of Wavelet Denoising in the Modeling of Ship Manoeuvring Motion

ZHANG Xin-gung ,ZOU Zo-jin,b

(a.School of Naval Architecture,Ocean and Civil Engineering;b.State Key Laboratory of Ocean Engineering,Shanghai Jiao Tong University,Shanghai 200240,China)

1 Introduction

Computer simulation combined with mathematical models is one of the most popular and effective methods to predict ship manoeuvrability.Two kinds of mathematical models of ship manoeuvring motion are used,i.e.,the hydrodynamic model[1-2]and the response model[3].To use this prediction method,accurately determining the hydrodynamic derivatives and manoeuvring indices in the mathematical models is vital to the prediction accuracy.

Nowadays,system identification by analyzing the free-running model test data or full scale test data is an effective method to obtain the hydrodynamic derivatives and manoeuvring indices.However,the test data always includes some unknown noises which affect the identification accuracy of the hydrodynamic derivatives and manoeuvring indices.How to remove these noises in the test data is one of the decisive factors in enhancing the accuracy of ship manoeuvrability prediction.

Denoising methods can be used to remove the noises in the test data.These methods mainly include empirical mode decomposition(EMD),Kalman filter and wavelet denoising,etc.Sun et al achieved the denoising of laser ultrasonic signal based on EMD[4].Lu et al applied Kalman filter to reduce the influence of the outliers[5].However,EMD has some inherent defects,such as problems of local mean,border effects,stopping criteria for decomposition and stopping criteria for sifting,etc.Kalman filter also has some inherent defects,such as heavily depending on the initial value of variables and parameters and the type of noise,etc.Compared with other denoising methods,wavelet denoising with a detailed mathematical underpinning has the advantages of multi-resolution analysis and the ability of displaying the local information in time domain and frequency domain.It has been widely applied in many fields,such as bioengineering,chemistry,mechanical engineering and automation,etc.Wu and He applied wavelet threshold denoising model to infrared spectral signal processing[6].Sun and Deng used wavelet denoising to improve transfer alignment of strapdown inertial navigation systems[7].However,wavelet denoising has not yet been applied in the field of ship manoeuvring.

In this paper,to study the feasibility of wavelet denoising in the field of ship manoeuvring,wavelet denoising is applied in the modeling of the second-order linear response model of ship steering.The 20°/20° zigzag test is simulated by using the second-order linear response model,in which the manoeuvring indices are obtained from the free-running model test[8].By adding some random noise into the simulated zigzag test data,the polluted test data are obtained.Then wavelet denoising is applied to denoise the polluted test data.By analyzing the denoised test data and the polluted test data,LS-SVR is applied to identify the manoeuvring indices of the second-order linear response model.The prediction results of 20°/20° zigzag test by using the denoised test data and the polluted test data are compared with the simulated test data to verify the validity of wavelet denoising in the denoising of the polluted test data.

2 Method description

The second-order linear response model can be written in the form:

By using the forward difference discretization,the following discretized form of Eq.(1)is obtained:

Let T1T2=h,T1+T2=g;a1,a2,b1and b2are unknown parameters which have a relation with h,g and K,T:

Judging from Eq.(2),we can get the training sample couples:

Input:［r（k）,δ（k）,r （k- 1）,δ（k- 1）］;

Output:r （k+ 1）;k=1,…,l-1;l is the number of training samples.

The algorithm of LS-SVR is as follows:

(1)Training sample set T=｛（x1,y1）,…,（xl,yl）｝∈ （χ×y）l,xi∈χ=Rn,yi∈R,i=1,…,l

(2)Choosing proper penalty factor C and kernel function K（x,x′）

(3)Constructing and solving the under optimalization problem:

αiis Lagrange multiplier;the optimum solution

(4)Constructing decision function:

The noisy model can be written in the form:s=f+σ*w,where f is the actual signal,s is the polluted signal,w is the noise signal,σis the noise level.According to this noisy model,the process of wavelet denoising is shown in Fig.1.

In Fig.1,fΔis the denoised signal.Lψare the decomposition coefficients of the polluted signal by using wavelet transform.Functional threshold value F is to use soft-thresholding or hard-thresholding to eliminate the wavelet coefficients disturbed by the noise signal.Softthresholding can be written in the following form:

Hard-thresholding can be written in the following form:

where W is the vector of the coefficients including the measure coefficients and the wavelet coefficients.Δis the threshold value.Functional Mask is the promotion of functional threshold value F and is to retain some particular coefficients and let the other coefficients be zero.

The steps of wavelet denoising are as follows:

(1)Choosing proper wavelet and optimal decomposition level of wavelet to decompose the polluted signal by using wavelet transform;

(2)Computing wavelet threshold value for each level;

(3)Using soft-thresholding or hardthresholding to eliminate the wavelet coefficients disturbed by the noise signal;

(4)Utilizing the worked wavelet coefficients to reconstruct the polluted signal by using the wavelet inverse transform.

3 Verification of the method

First of all,20°/20° zigzag test is simulated by using the second-order linear response model,where the manoeuvring indices are obtained from the freerunning model test[8].By adding some random noise into the simulated test data,the polluted test data are obtained.The polluted test data are compared with the simulated test data in Fig.2.

Biorthogonality Daubechies 4 wavelet[9-10]is chosen to decompose the polluted test data.The optimal decomposition level of wavelet is 4.Wavelet threshold valueΔis obtained using a wavelet coefficients selection rule based on Birge-Massart strategy.The wavelet threshold values are 12.164,11.384,8.488 5 and 6.599 5 for each level from level 1 to level 4 in the denoising of the polluted heading angle test data.The wavelet threshold values are 2.432 9,2.278 1,1.698 6 and 0.954 52 for each level from level 1 to level 4 in the denoising of the polluted yaw rate test data.The denoised test data are compared with the simulated test data in Fig.3.

By analyzing the denoised 20°/20°zigzag test data and the polluted 20°/20°zigzag test data,the first 100 sample couples are trained by LS-SVR using the linear kernel function K （xi,x）=（xi,x）to identify manoeuvring indices of the second-order linear response model.The penalty factor C is 105.In Tab.1,the identification results of manoeuvring indices by using the denoised test data and the polluted test data are compared with the manoeuvring indices obtained from the free-running model test[8].The prediction results of 20°/20° zigzag test by using the polluted test data are shown in Fig.4 in comparison with the simulated 20°/20° zigzag test data.The prediction results of 20°/20° zigzag test by using the denoised test data are compared with the simulated 20°/20° zigzag test data in Fig.5.

Tab.1 Comparison of manoeuvring indices

It can be seen from the table and the figures above,the identification results of manoeuvring indices and the prediction results of 20°/20° zigzag test by using the denoised test data are in good agreement with the manoeuvring indices obtained from the free running test and the simulated 20°/20° zigzag test data respectively.This shows that the proposed wavelet denoising is an effective method to denoise the polluted test data.

4 Conclusion

By analyzing the denoised 20°/20° zigzag test data and the polluted 20°/20° zigzag test data,LS-SVR is applied to identify the manoeuvring indices of the second-order linear response model.The prediction results of 20°/20° zigzag test by using the denoised zigzag test data and the polluted zigzag test data are compared with the simulated zigzag test data.Satisfactory agreements are obtained,which shows the validity of the wavelet denoising method in denoising the zigzag test data.Further study is to apply the wavelet denoising to process the noisy data of free-running model tests and full-scale trials.

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