船舶非线性横摇运动的负反馈控制算法
关克平 张新放
摘要：
为描述船舶横摇运动的非线性现象，找出其横摇运动的规律，建立船舶横摇运动的非线性数学模型.借助Lyapunov指数曲线来判断船舶非线性横摇运动的混沌现象；采用负反馈算法的非线性方法来控制船舶横摇运动中的混沌现象；求出适宜的负反馈系数k的范围，减轻船舶横摇运动的非线性现象.结果表明，该方法取得了较满意的控制效果.
关键词：
船舶横摇运动； 非线性； 混沌； Lyapunov指数曲线； 负反馈控制
0Introduction
The research of nonlinear phenomena of ship rolling motion[1] includes： various nonlinear factors when ship is sailing， such as nonlinear restoring torque and damping torque； coupling effect of rolling and pitching， and of rolling and heaving； establishing nonlinear rolling coupling mathematical models； stability， frequency ratio， saturation， crossing and chaos phenomena of the roots of the coupling equation； the frequency of ship capsizing in the random waves[2].
The nonlinear research of ship rolling motion dates back to the famous initiative research of scholar Euler， and then， scholar Krylov who developed the theory further. With the improvement of the technique， domestic and oversea scholars have made great progresses and they adopted many theories， such as differential dynamic system theory， crossing theory and chaos theory. For example， NAYFEH et al[3] adopted Floquet theory[4] to analyze the crossing and chaos phenomena of ship motions； under the condition of the known system attractor， LEE[5] attempted to adopt the cellmapping synthetic method to analyze the heavingrolling coupling system of attractor. Due to the limitation of calculation， his research range is small. MULK et al[6] used the pathtracking technology. They obtained the bifurcation set of sixDOF nonlinear motion， found various forms of oscillation of solutions and worked out the chaos attractor occurring in the surrounding. BIKDASH et al[7] adopted Melnikov analysis method. Taking the maximum wave steepness as a variable parameter， they researched the security of ship rolling motion. These researches indicate that chaos phenomena could be one of the most crucial factors that lead to shipping capsizing.
In the paper， the mathematical model of ship nonlinear rolling motion is built up， the nonlinear method of the negative feedback control algorithm is employed to solve the model， and different negative feedback coefficients are used to judge whether ship is in the chaotic state or not. Finally， the most appropriate range of the negative feedback coefficient is obtained. Thus， the chaos phenomena are got rid of and the necessary foundation for researching ship capsizing is laid.
1Establishing mathematical model of ship nonlinear rolling motion
The existing research indicates that even though some basic factors are ignored， such as the coupling motion of sixDOF and the accurate determination of the hydrodynamic coefficients[8]， the ship motion is very complex when is sailing. However， the large rolling motion of nonlinear restoring torque and nonlinear damping torque should not be ignored. NAYFEH et al[9] took the nonlinear restoring torque and damping torque into consideration and researched the stability and the complex problem of dynamics of rolling motion under the different slope of wave surfaces and encounter frequencies.
The research in this paper is based on the motion equation established by NAYFEH and SANCHEZ according to Newtons second law. This paper hypothesizes that the disturbance torque of ship rolling motion has some bearing on the motions of ships themselves. In regular waves， the mathematical model of ship nonlinear rolling motion[10] can be represented by the following differential equation：
2Judging chaos phenomena of ship nonlinear rolling motion
The principal characteristic of chaos system is that it is extremely sensitive to initial values. As time going on， traces generated by the little difference between the two initial values will disperse in the index way. Duffing equation is a typical nonlinear equation and is widely used in detecting weak signal of chaos system. It is a simple mathematical model that describes resonance phenomenon， harmonic vibration， periodic vibration， strange attractor， random process and chaos phenomena， so it is of great significance to research Duffing equation in nonlinear vibration theories[11]. Eq. （10） is similar to Duffing equation. Now whether the chaos phenomena appear in the ship model is analyzed.
Letting x=， y=′， and z=ωt， Eq. （10） can be represented as
When αm=1.38 and ω=1， the phasespace diagram and frequency spectrum diagram of the nonlinear system are shown in Fig. 1.
With the value of αm becoming bigger， the dispersion phenomena of system are more obvious， and the rolling angle of ship motion becomes bigger correspondingly.
When αm=1.44 and ω=1， the phasespace diagram and the frequency spectrum diagram of the nonlinear system are shown in Fig. 2.
a） Phasespace diagram
b） Frequency spectrum diagram
Fig.1Phasespace diagram and frequency spectrum diagram （αm=1.38， ω=1）
a） Phasespace diagram
b） Frequency spectrum diagram
Fig.2Phasespace diagram and frequency spectrum diagram （αm=1.44， ω=1）
Considering Figs. 1 and 2 comprehensively， we find that with the increase of αm， the rolling angle of ship motion becomes bigger. These results show that the degree of dispersion becomes bigger. By analyzing Figs. 1 and 2， the conclusion that the system of nonlinear rolling motion of ships is a chaos system can be drawn.
3Negative feedback controlling of ship nonlinear rolling motion
3.1Constructing negative feedback control model
The state of chaos can be controlled by a negative feedback coefficient. A negative feedback control parameter is constructed and a negative feedback coefficient is introduced. The control parameter， which is designed by the conventional backstepping method[12]， will counteract the nonlinear term in the system. If the controlled term is complicated， there will be more unknown parameters， so a simple Lyapunov function is constructed， control processes are simplified， the number of unknown parameters is reduced， and the control parameters which dont counteract nonlinear terms are designed. Based on a simple robust control method， the parameter x is controlled in this paper， and then this algorithm is of certain robustness.
In Eq. （11）， introducing the negative feedback parameter u into x′=y， it can be hypothesized as
where k is a negative feedback coefficient； v is the reference value that is without the loss of generality， so hypothesize v=0. Thus， x′=y will be turned into
Because the primary system can be controlled， an appropriate k has to be found and used to make the primary system into the state of stability. The state of chaos can be judged from the general knowledge that the Lyapunov exponent is greater than 0 directly. If all Lyapunov exponents are negative， the system will be in the state of stability. If positive Lyapunov exponents occur， the system will be in the state of chaos[13]. In this case， appropriate measures could be taken to control the chaos system.
3.2Calculating negative feedback coefficient
Because the state of system can be judged from the Lyapunov exponent， the Lyapunov exponent curve will change with the change of k. If all Lyapunov exponents are negative， the system will be in the state of stability， namely， the range of k is appropriate. This paper uses the method of Jacobian matrix to solve k. Eq. （15） is turned into the form of Jacobian matrix as
3.3Controlling the state of chaos
The purpose of controlling the state of chaos is to eliminate the chaos phenomena and to reduce the amplitude. For different k， MATLAB is used to draw Lyapunov exponent diagrams （see Fig.3）.
When k=-0.35 （see Fig.3a）， there exists an exponent curve that gradually becomes steady at zero， which indicates that the system just breaks away from the chaos phenomena.
When k>-0.35 （see Fig.3b）， there exists an exponent curve that is greater than zero， which indicates that chaos phenomena occur in the system. The chaos phenomena are more obvious with k becoming bigger.
When k<-0.35 （see Fig.3c）， all exponent curves are less than zero， which indicates that the system completely breaks away from the chaos phenomena. With k becoming smaller， the system will be more stable.
These simulation results reveal that using different k， we can get different Lyapunov exponent diagrams， such as Fig. 3， so we can judge whether the system is under the state of chaos or breaks away from it. Within k≤-0.35， when αm=2.0 and ω=1， the phasespace diagram and the frequency spectrum diagram of the nonlinear system can be shown in Fig. 4.
a） Phasespace diagram
b） Frequency spectrum diagram
Fig.4Phasespace diagram and frequency spectrum diagram （k≤-0.35，αm=2.0，ω=1）
As is shown in Fig. 4， the system of ship nonlinear rolling motion breaks away from the chaos phenomena. The result validates that the negative feedback coefficient k has good controlling result.
4Conclusions
Based on the Lyapunov exponent curve， whether the system in the state of chaos or not is judged， the nonlinear phenomena of ship swaying motion （especially， the nonlinear rolling motion） are analyzed， the mathematical model of ship nonlinear rolling motion is established， the chaos theory is used to explain this phenomena， the negative feedback algorithm is adopted to control the state of chaos， the range of the negative feedback coefficient is solved， MATLAB is used to draw different Lyapunov exponent diagrams， and finally the system breaking away from the state of chaos is validated. By analyzing and validating the coefficient， it can be found that the coefficient has good controlling results. This paper provides further analysis and research for eliminating capsizing accidents caused by large rolling motion.
References：
[1]ZHANG Guanghui， LIU Shupeng， MA Ruixian. Nonlinear dynamic characteristics of journal bearingrotor system considering the pitching and rolling motion for marine turbo machinery[J]. Journal of Engineering for the Maritime Environment， 2015， 229（1）： 95107.
[2]FALZARANO J， VISHNUBHOTLA S， CHENG J. Nonlinear dynamic analysis of ship capsizing in random waves[C]//Preceedings of the Fourteenth （2004） International Offshore and Polar Engineering Conference. Toulon， France， 2004： 479484.
[3]NAYFEH A H， KHDEIR A A. Nonlinear rolling of ships in regular beam seas[J]. International Ship Building Progress， 1986， 33： 4049.
[4]CHEN Hongkui， HEGAZY U H. Nonlinear dynamic behavior of a rotor active magnetic bearing[J]. Mathematics， Interdisciplinary Applications， 2010， 20（12）： 39353968. DOI： 10.1142/S0218127410028124.
摘要：
为描述船舶横摇运动的非线性现象，找出其横摇运动的规律，建立船舶横摇运动的非线性数学模型.借助Lyapunov指数曲线来判断船舶非线性横摇运动的混沌现象；采用负反馈算法的非线性方法来控制船舶横摇运动中的混沌现象；求出适宜的负反馈系数k的范围，减轻船舶横摇运动的非线性现象.结果表明，该方法取得了较满意的控制效果.
关键词：
船舶横摇运动； 非线性； 混沌； Lyapunov指数曲线； 负反馈控制
0Introduction
The research of nonlinear phenomena of ship rolling motion[1] includes： various nonlinear factors when ship is sailing， such as nonlinear restoring torque and damping torque； coupling effect of rolling and pitching， and of rolling and heaving； establishing nonlinear rolling coupling mathematical models； stability， frequency ratio， saturation， crossing and chaos phenomena of the roots of the coupling equation； the frequency of ship capsizing in the random waves[2].
The nonlinear research of ship rolling motion dates back to the famous initiative research of scholar Euler， and then， scholar Krylov who developed the theory further. With the improvement of the technique， domestic and oversea scholars have made great progresses and they adopted many theories， such as differential dynamic system theory， crossing theory and chaos theory. For example， NAYFEH et al[3] adopted Floquet theory[4] to analyze the crossing and chaos phenomena of ship motions； under the condition of the known system attractor， LEE[5] attempted to adopt the cellmapping synthetic method to analyze the heavingrolling coupling system of attractor. Due to the limitation of calculation， his research range is small. MULK et al[6] used the pathtracking technology. They obtained the bifurcation set of sixDOF nonlinear motion， found various forms of oscillation of solutions and worked out the chaos attractor occurring in the surrounding. BIKDASH et al[7] adopted Melnikov analysis method. Taking the maximum wave steepness as a variable parameter， they researched the security of ship rolling motion. These researches indicate that chaos phenomena could be one of the most crucial factors that lead to shipping capsizing.
In the paper， the mathematical model of ship nonlinear rolling motion is built up， the nonlinear method of the negative feedback control algorithm is employed to solve the model， and different negative feedback coefficients are used to judge whether ship is in the chaotic state or not. Finally， the most appropriate range of the negative feedback coefficient is obtained. Thus， the chaos phenomena are got rid of and the necessary foundation for researching ship capsizing is laid.
1Establishing mathematical model of ship nonlinear rolling motion
The existing research indicates that even though some basic factors are ignored， such as the coupling motion of sixDOF and the accurate determination of the hydrodynamic coefficients[8]， the ship motion is very complex when is sailing. However， the large rolling motion of nonlinear restoring torque and nonlinear damping torque should not be ignored. NAYFEH et al[9] took the nonlinear restoring torque and damping torque into consideration and researched the stability and the complex problem of dynamics of rolling motion under the different slope of wave surfaces and encounter frequencies.
The research in this paper is based on the motion equation established by NAYFEH and SANCHEZ according to Newtons second law. This paper hypothesizes that the disturbance torque of ship rolling motion has some bearing on the motions of ships themselves. In regular waves， the mathematical model of ship nonlinear rolling motion[10] can be represented by the following differential equation：
2Judging chaos phenomena of ship nonlinear rolling motion
The principal characteristic of chaos system is that it is extremely sensitive to initial values. As time going on， traces generated by the little difference between the two initial values will disperse in the index way. Duffing equation is a typical nonlinear equation and is widely used in detecting weak signal of chaos system. It is a simple mathematical model that describes resonance phenomenon， harmonic vibration， periodic vibration， strange attractor， random process and chaos phenomena， so it is of great significance to research Duffing equation in nonlinear vibration theories[11]. Eq. （10） is similar to Duffing equation. Now whether the chaos phenomena appear in the ship model is analyzed.
Letting x=， y=′， and z=ωt， Eq. （10） can be represented as
When αm=1.38 and ω=1， the phasespace diagram and frequency spectrum diagram of the nonlinear system are shown in Fig. 1.
With the value of αm becoming bigger， the dispersion phenomena of system are more obvious， and the rolling angle of ship motion becomes bigger correspondingly.
When αm=1.44 and ω=1， the phasespace diagram and the frequency spectrum diagram of the nonlinear system are shown in Fig. 2.
a） Phasespace diagram
b） Frequency spectrum diagram
Fig.1Phasespace diagram and frequency spectrum diagram （αm=1.38， ω=1）
a） Phasespace diagram
b） Frequency spectrum diagram
Fig.2Phasespace diagram and frequency spectrum diagram （αm=1.44， ω=1）
Considering Figs. 1 and 2 comprehensively， we find that with the increase of αm， the rolling angle of ship motion becomes bigger. These results show that the degree of dispersion becomes bigger. By analyzing Figs. 1 and 2， the conclusion that the system of nonlinear rolling motion of ships is a chaos system can be drawn.
3Negative feedback controlling of ship nonlinear rolling motion
3.1Constructing negative feedback control model
The state of chaos can be controlled by a negative feedback coefficient. A negative feedback control parameter is constructed and a negative feedback coefficient is introduced. The control parameter， which is designed by the conventional backstepping method[12]， will counteract the nonlinear term in the system. If the controlled term is complicated， there will be more unknown parameters， so a simple Lyapunov function is constructed， control processes are simplified， the number of unknown parameters is reduced， and the control parameters which dont counteract nonlinear terms are designed. Based on a simple robust control method， the parameter x is controlled in this paper， and then this algorithm is of certain robustness.
In Eq. （11）， introducing the negative feedback parameter u into x′=y， it can be hypothesized as
where k is a negative feedback coefficient； v is the reference value that is without the loss of generality， so hypothesize v=0. Thus， x′=y will be turned into
Because the primary system can be controlled， an appropriate k has to be found and used to make the primary system into the state of stability. The state of chaos can be judged from the general knowledge that the Lyapunov exponent is greater than 0 directly. If all Lyapunov exponents are negative， the system will be in the state of stability. If positive Lyapunov exponents occur， the system will be in the state of chaos[13]. In this case， appropriate measures could be taken to control the chaos system.
3.2Calculating negative feedback coefficient
Because the state of system can be judged from the Lyapunov exponent， the Lyapunov exponent curve will change with the change of k. If all Lyapunov exponents are negative， the system will be in the state of stability， namely， the range of k is appropriate. This paper uses the method of Jacobian matrix to solve k. Eq. （15） is turned into the form of Jacobian matrix as
3.3Controlling the state of chaos
The purpose of controlling the state of chaos is to eliminate the chaos phenomena and to reduce the amplitude. For different k， MATLAB is used to draw Lyapunov exponent diagrams （see Fig.3）.
When k=-0.35 （see Fig.3a）， there exists an exponent curve that gradually becomes steady at zero， which indicates that the system just breaks away from the chaos phenomena.
When k>-0.35 （see Fig.3b）， there exists an exponent curve that is greater than zero， which indicates that chaos phenomena occur in the system. The chaos phenomena are more obvious with k becoming bigger.
When k<-0.35 （see Fig.3c）， all exponent curves are less than zero， which indicates that the system completely breaks away from the chaos phenomena. With k becoming smaller， the system will be more stable.
These simulation results reveal that using different k， we can get different Lyapunov exponent diagrams， such as Fig. 3， so we can judge whether the system is under the state of chaos or breaks away from it. Within k≤-0.35， when αm=2.0 and ω=1， the phasespace diagram and the frequency spectrum diagram of the nonlinear system can be shown in Fig. 4.
a） Phasespace diagram
b） Frequency spectrum diagram
Fig.4Phasespace diagram and frequency spectrum diagram （k≤-0.35，αm=2.0，ω=1）
As is shown in Fig. 4， the system of ship nonlinear rolling motion breaks away from the chaos phenomena. The result validates that the negative feedback coefficient k has good controlling result.
4Conclusions
Based on the Lyapunov exponent curve， whether the system in the state of chaos or not is judged， the nonlinear phenomena of ship swaying motion （especially， the nonlinear rolling motion） are analyzed， the mathematical model of ship nonlinear rolling motion is established， the chaos theory is used to explain this phenomena， the negative feedback algorithm is adopted to control the state of chaos， the range of the negative feedback coefficient is solved， MATLAB is used to draw different Lyapunov exponent diagrams， and finally the system breaking away from the state of chaos is validated. By analyzing and validating the coefficient， it can be found that the coefficient has good controlling results. This paper provides further analysis and research for eliminating capsizing accidents caused by large rolling motion.
References：
[1]ZHANG Guanghui， LIU Shupeng， MA Ruixian. Nonlinear dynamic characteristics of journal bearingrotor system considering the pitching and rolling motion for marine turbo machinery[J]. Journal of Engineering for the Maritime Environment， 2015， 229（1）： 95107.
[2]FALZARANO J， VISHNUBHOTLA S， CHENG J. Nonlinear dynamic analysis of ship capsizing in random waves[C]//Preceedings of the Fourteenth （2004） International Offshore and Polar Engineering Conference. Toulon， France， 2004： 479484.
[3]NAYFEH A H， KHDEIR A A. Nonlinear rolling of ships in regular beam seas[J]. International Ship Building Progress， 1986， 33： 4049.
[4]CHEN Hongkui， HEGAZY U H. Nonlinear dynamic behavior of a rotor active magnetic bearing[J]. Mathematics， Interdisciplinary Applications， 2010， 20（12）： 39353968. DOI： 10.1142/S0218127410028124.
