基于混沌分析的船舶参数激励横摇运动及其鲁棒控制
|
摘要
船舶作为一种重要的交通工具,在实施设计中虽然考虑了国际公约和国内法规有关稳性标准的规定,但波浪造成船舶失稳从而引发船舶倾覆的现象时有发生。目前船舶静态稳性纳入了稳性标准而动态稳性却未予以考虑,也就是说对于船舶在波浪中的稳性损失、运动瞬时状态及船舶与波浪遭遇状况缺少相关的考虑和研究。因此研究船舶在波浪激励下的复杂运动特性及鲁棒控制工程意义极为显著。
船舶参数激励横摇可能导致船舶的大幅度横摇运动,威胁船舶、货物和海上人命安全。为了减小参数激励带来的不利因素,通过李亚普诺夫指数和功率谱对船舶在规则纵浪中运动的稳定性进行分析。从而分析船舶参数激励横摇运动的产生机理,不仅分析了船舶参数激励横摇产生混沌现象的条件,而且确定出船舶参数激励横摇运动的安全与危险区域。然后基于Backstepping算法、闭环增益成形算法及Lyapunov稳定性设计出减摇鳍控制器,并且在考虑一定的干扰后进行了仿真试验。仿真结果表明该控制策略对于消除船舶参数激励横摇系统的混沌现象是十分有效的,并且鲁棒性能令人满意。
船舶参—强激励横摇比纯参数横摇可能导致船舶更大幅度的横摇运动,在主参数共振区内,较小的参数激励和强迫激励仍可能引起船舶大幅横摇,甚至倾覆,威胁船舶、货物和海上人命安全。为了减小参—强激励带来的不利因素,通过李亚普诺夫指数和功率谱对船舶在规则纵浪中运动的稳定性进行分析,找到参—强激励横摇运动的稳定和危险区域,以及对于船舶初始运动参数的变化关系。然后基于Backstepping算法、闭环增益成形算法及Lyapunov稳定性设计出非线性鲁棒控制器,并且在考虑一定的干扰后进行了仿真试验。仿真结果表明该控制策略对于消除船舶参—强横摇系统的混沌现象是十分有效的,并且鲁棒性能令人满意。
在船舶初稳性高的时变性的影响下,随机纵浪也会对船舶的横摇稳定性造成较大的影响,特别在特征波长接近船长,且参数激励频率与船舶横摇固有频率之比为2时,船舶会发生大幅横摇不稳定运动,即为随机波条件下的参数激励横摇或参—强激励横摇。通过求解船舶在随机纵浪中主参数共振时的Lyapunov (?)旨数,分析船舶在随机纵浪条件下的主参数共振稳定性,并由此计算了船舶运动的稳定域与不稳定域,讨论了船舶横摇阻尼、随机波浪的中心频率,带宽等对船舶参数激励横摇运动稳定性的影响。同时,针对随机波条件下参数激励与参—强激励非线性运动设计出鲁棒控制器,从而减小船舶横摇危险,提高航行稳定性。
As an important means of transport, the ship can satisfy the stability criteria required by the provisions of international conventions and domestic regulations in the design process, but often may capsize in the wave due to instability. At present, the stability criterion only considers the static stability without considering the dynamic stability and the research for stability loss, the instantaneous movement state and the wave encounter situation during navigating in the sea is lacked. Therefore the research for complex motion characteristics and robust control under the wave excitations has important engineering significance.
Parametric excitation rolling can lead to dangerously large rolling motions, endangering the ship, cargo and crew. To decrease the influence caused by parametric resonance, stability of ships in longitudinal waves was then analyzed using Lyapunov characteristic exponents and power spectrum.The mechanism of parametric excitation rolling was explained by chaos theory. Not only the condition of chaos in parametric rolling was discussed, but also the safe and unsafe regions of target ships were then identified. Then the fin stabilizer controller was designed by using backstepping algorithm, closed-loop gain shaping algorithm,Lyapunov stability and simulation tests were carried out under certain disturbances. The simulation results show that the control effects are very effective for the elimination of chaos in the ship parametric rolling system and have satisfactory robust stability.
Parametric-forced excitation rolling is more likely to cause dangerously large rolling motions, comparing with parametric excitation rolling, posing a danger to the ship, cargo and crew. In the principal parametric resonance zone, small parametric excitation and forced excitation can still generate large-amplitude roll, or even cause overturning. To decrease the influence caused by parametric and forced resonance, stability of ships in longitudinal waves was then analyzed using Lyapunov characteristic exponents and power spectrum. At the same time, stability and dangerous areas and the relationship between initial motion parameters and rolling movement can be discovered. Then the nonlinear robust controller was designed by using backstepping algorithm,closed-loop gain shaping algorithm,Lyapunov stability and simulation tests were carried out under certain disturbances. The simulation results show that the control effects are very effective for the elimination of chaos in the ship parametric and forced rolling system and have satisfactory robust stability.
Under the effects of time-varying metacentric height of the ship, the random longitudinal wave can also a great impact on the ship's rolling stability. Especially under the condition that the characteristic wavelength is close to the length of ship and the parameters excitation frequency double the rolling natural frequency, the ship will occur large and unstable rolling motion. That is parametric rolling and parametric-forced rolling under the random waves. By obtaining Lyapunov index during principal parametric resonance and analyzing principal parametric resonance stability, the stability domain and unstable domain were found.The influence of ship rolling damping, center frequency of random wave and bandwidth to parametric and parametric-forced excitation was discussed. Then the nonlinear robust controller was designed to reduce ship rolling risk and improve the stability of navigation.
船舶参数激励横摇可能导致船舶的大幅度横摇运动,威胁船舶、货物和海上人命安全。为了减小参数激励带来的不利因素,通过李亚普诺夫指数和功率谱对船舶在规则纵浪中运动的稳定性进行分析。从而分析船舶参数激励横摇运动的产生机理,不仅分析了船舶参数激励横摇产生混沌现象的条件,而且确定出船舶参数激励横摇运动的安全与危险区域。然后基于Backstepping算法、闭环增益成形算法及Lyapunov稳定性设计出减摇鳍控制器,并且在考虑一定的干扰后进行了仿真试验。仿真结果表明该控制策略对于消除船舶参数激励横摇系统的混沌现象是十分有效的,并且鲁棒性能令人满意。
船舶参—强激励横摇比纯参数横摇可能导致船舶更大幅度的横摇运动,在主参数共振区内,较小的参数激励和强迫激励仍可能引起船舶大幅横摇,甚至倾覆,威胁船舶、货物和海上人命安全。为了减小参—强激励带来的不利因素,通过李亚普诺夫指数和功率谱对船舶在规则纵浪中运动的稳定性进行分析,找到参—强激励横摇运动的稳定和危险区域,以及对于船舶初始运动参数的变化关系。然后基于Backstepping算法、闭环增益成形算法及Lyapunov稳定性设计出非线性鲁棒控制器,并且在考虑一定的干扰后进行了仿真试验。仿真结果表明该控制策略对于消除船舶参—强横摇系统的混沌现象是十分有效的,并且鲁棒性能令人满意。
在船舶初稳性高的时变性的影响下,随机纵浪也会对船舶的横摇稳定性造成较大的影响,特别在特征波长接近船长,且参数激励频率与船舶横摇固有频率之比为2时,船舶会发生大幅横摇不稳定运动,即为随机波条件下的参数激励横摇或参—强激励横摇。通过求解船舶在随机纵浪中主参数共振时的Lyapunov (?)旨数,分析船舶在随机纵浪条件下的主参数共振稳定性,并由此计算了船舶运动的稳定域与不稳定域,讨论了船舶横摇阻尼、随机波浪的中心频率,带宽等对船舶参数激励横摇运动稳定性的影响。同时,针对随机波条件下参数激励与参—强激励非线性运动设计出鲁棒控制器,从而减小船舶横摇危险,提高航行稳定性。
As an important means of transport, the ship can satisfy the stability criteria required by the provisions of international conventions and domestic regulations in the design process, but often may capsize in the wave due to instability. At present, the stability criterion only considers the static stability without considering the dynamic stability and the research for stability loss, the instantaneous movement state and the wave encounter situation during navigating in the sea is lacked. Therefore the research for complex motion characteristics and robust control under the wave excitations has important engineering significance.
Parametric excitation rolling can lead to dangerously large rolling motions, endangering the ship, cargo and crew. To decrease the influence caused by parametric resonance, stability of ships in longitudinal waves was then analyzed using Lyapunov characteristic exponents and power spectrum.The mechanism of parametric excitation rolling was explained by chaos theory. Not only the condition of chaos in parametric rolling was discussed, but also the safe and unsafe regions of target ships were then identified. Then the fin stabilizer controller was designed by using backstepping algorithm, closed-loop gain shaping algorithm,Lyapunov stability and simulation tests were carried out under certain disturbances. The simulation results show that the control effects are very effective for the elimination of chaos in the ship parametric rolling system and have satisfactory robust stability.
Parametric-forced excitation rolling is more likely to cause dangerously large rolling motions, comparing with parametric excitation rolling, posing a danger to the ship, cargo and crew. In the principal parametric resonance zone, small parametric excitation and forced excitation can still generate large-amplitude roll, or even cause overturning. To decrease the influence caused by parametric and forced resonance, stability of ships in longitudinal waves was then analyzed using Lyapunov characteristic exponents and power spectrum. At the same time, stability and dangerous areas and the relationship between initial motion parameters and rolling movement can be discovered. Then the nonlinear robust controller was designed by using backstepping algorithm,closed-loop gain shaping algorithm,Lyapunov stability and simulation tests were carried out under certain disturbances. The simulation results show that the control effects are very effective for the elimination of chaos in the ship parametric and forced rolling system and have satisfactory robust stability.
Under the effects of time-varying metacentric height of the ship, the random longitudinal wave can also a great impact on the ship's rolling stability. Especially under the condition that the characteristic wavelength is close to the length of ship and the parameters excitation frequency double the rolling natural frequency, the ship will occur large and unstable rolling motion. That is parametric rolling and parametric-forced rolling under the random waves. By obtaining Lyapunov index during principal parametric resonance and analyzing principal parametric resonance stability, the stability domain and unstable domain were found.The influence of ship rolling damping, center frequency of random wave and bandwidth to parametric and parametric-forced excitation was discussed. Then the nonlinear robust controller was designed to reduce ship rolling risk and improve the stability of navigation.
引文
[1]苏作靖,张显库.“育鲲”轮参数横摇的数值模拟及分析.哈尔滨工程大学学报,2012,23(5)：590-594.
[2]SCANCHEZ N E, NAYFEH A H.Nonlinear rolling motion of ships in longitudinal waves. ISP, 1990,37:247-272.
[3]张延峰,董艳秋,唐友刚.纵浪上基本参数共振对船舶稳性的影响.船舶力学,1998,2(3)：6-12.
[4]王立军,张显库.减摇航向自动舵的简捷鲁棒优化控制算法.大连海事大学学报,2013,39(1)：1-5.
[5]任海鹏.混沌控制方法及其应用研究.西安理工大学博士论文,2003.
[6]Li T Y,Yorke J A.Period three implies chaos.A mer. Math. Monthly.1975,82:985-992.
[7]Hao B.L.Chaos.Singapore:World Scientific,1984.
[8]Hao B.L.Chaos Ⅱ.Singapore:World Scientific,1990.
[9]陆君安,陈士华.混沌动力学初步.武汉,武汉水利水电人学出版社,1998.
[10]王杰智,陈增强,袁著祉.一个新的混沌系统及其性质研究.物理学报,2006,55(8)：3956-3964.
[11]李鹤.工程中的混沌的识别、利用和控制的研究.东北大学博士论文,2005.
[12]王瑞平.某些常微和偏微系统的混沌控制与反控制.中山大学硕士论文,2003.
[13]Lorenz E.N.Deterministic non-periodic flow.J.Atmos.Sci.1963,20:130-141.
[14]Sivakumar B.Chaos theory in geophysics:past, present and future. Chaos Solitons&Fractals, 2004,19 (2):441-462.
[15]Freeman W.H.,Mandebort B.B. Fractal,form,chance and dimension. San Francisco,1977.
[16]Rossler O.E. An equation for continuous chaos. Physical Leters A,1976,57:397-398.
[17]Chua L.O.,Komuro M., Matsumoto T. The double scroll family. Part Ⅰ:rigorous proof of chaos. IEEE Trans. On Circuits & Systems-Ⅰ,1986,33:1072-1096.
[18]Chen G.R., Ueta T. Yet another chaotic attractor. Int. J. Bifurcation & Chaos,1999,9: 1465-1466.
[19]Lu J.H., Chen G.R., Cheng D.Z. Bridge the gap between the Lorenz system and the Chen system. Int. J. Bifurcation & Chaos,2002,12:2917-2926.
[20]Liu C.X., Liu L., Liu T. A new chaotic attractor. Chaos, Solitons & Fractals,2004,22: 1031-1038.
[21]Liu C.X., Liu T., Liu L. A new butterfly-shaped attractor of Lorenz-like system. Chaos, Solitons & Fractals,2006,28:1196-1203.
[22]Wang F.Q., Liu C.X. Hyperchaos evolved from the Liu chaotic system. Chinese Physics, 2006,15(5):963-968.
[23]KERWIN J E. Notes on rolling in longitudinal waves.Int Shipbuild Progr,1955,2(16): 597-614.
[24]HADDARA M R. Effect of parametric excitation on ship motion in random waves. Ship Res, 1973,7(5):30-37.
[25]SILVA S, SOARES C. Time domain simulation of parametrically excited roll in head seas.7th International Conference on Stability of Ships and Ocean Vehicles. Tas-mania, Australia,2000: 652-664.
[26]NEVES M, VALERIO L. Parametric resonance in waves of arbitrary heading. Int Shipbuild Progr,1990,35(320):230-241.
[27]MATUSIAK J. Parametric resonance of the conical buoy in regular waves. Int Shipbuild Progr, 2000,37(350):100-121.
[28]BOUKHANOVSKY A, DEGTYAREV A. Peculiarities of motion of ship with low buoyancy on asymmetrical random waves. Int Shipbuild Progr,2001,38(367):350-364.
[29]SILVA S R, SANTOS T A, SOARES C G. Parametrically excited roll in regular and irregular head seas. Int Ship-build Progr,2005,52(1):29-56.
[30]闵美松,丁勇,赵晓东,等.随浪中船舶横甩试验研究.哈尔滨船舶工程学院学报,1993,14(3)：12-18.
[31]张延峰,董艳秋,唐友刚.纵浪上基本参数共振对船舶稳性的影响.船舶力学,1998,2(3)：6-12.
[32]唐友刚,田凯强,张泽盛.船舶参数激励非线性横摇运动方程.船舶工程,1998,6：16-18.
[33]唐友刚,林维学,董艳秋,等.船舶参数激励和强迫激励作用下的非线性运动响应.中国造船,2001,42(2)：34-39.
[34]Sogawa Y, Umeda N, Hashimoto H. Parametric roll of a post Panamax containership in regular waves:Experiment, ana lytical method and simulation. Proc 5th APHydro. Osaka, Japan,2010: 99-102.
[35]张显库.非脆弱鲁棒PID控制器设计.重庆工学院学报(自然科学版).2009,23(5)：28-30.
[36]Zhang X K,Jin Y C.Control of A Multivariable High Purity Distillation Column Based on Closed-loop Gain Shaping Algorithm.International Journal of Information Technology.2005,11 (5):116-123.
[37]Zhang X K,Jia X L.Simplification of H mixed sensitivity algorithm and its application. Automatic Control and Computer Sciences.2002,36 (3):28-33.
[38]贾欣乐,张显库.H∞控制器应用于船舶自动舵.控制与决策.1995,10(3)：250-254.
[39]张显库,贾欣乐.船舶自动舵鲁棒控制权函数综合选择方法.大连海事大学学报.1999,25 (3)：21-25.
[40]张显库,贾欣乐,杨承恩,等.船舶鲁棒自动舵产品化研究.系统工程与电子技术.1999,21(2)：75-81.
[41]张显库,贾欣乐,王兴成.一种变形的回来成形控制器及其应用.控制与决策.2000,15(1)：116-118.
[42]张显库,贾欣乐.求PID参数新方法.系统工程与电子技术.2000,22(8)：4-5.
[43]张显库,贾欣乐.用镜像映射方法求非稳定过程的鲁棒控制器.系统工程与电子技术.2000,22(4)：10-12.
[44]张显库.用镜像映射方法求纯不稳定过程的鲁棒控制器.系统工程与电子技术,2004,26(10)：1466-1467.
[45]Zhang X K.New method on design of robust controller for unstable process.Proceeding of International Conference on Machine Learning and Cybernetics,643-648,2005.
[46]张显库,金一丞.基于信息对称的简捷控制.控制与决策.2007,22(11)：1255-1258.
[47]张显库.不对称信息理论在船舶运动控制中的应用.中国造船.2006,47(1)：55-59.
[48]张显库,杨盐生.不对称信息理论与非线性鲁棒控制算法.控制与决策.2005,20(11)：1241-1244.
[49]张显库,郭晨,杜佳璐.船舶航向不对称信息理论与非线性逆推鲁棒控制算法.交通运输工程学报.2006,6(2)：47-50.
[50]M.J. Grimble,S.A. Carr,M.R. Katebi.Integrated ship control using H∞ robust design techniques.Proceedings of the IEEE Conference on Control Applications,1994,Vol.2:1087-1091.
[51]D.S. Desanj,M.J. Grimble,M.R. Katebi.State-space adaptive H∞ controller with application to a ship control system.Transactions of the Institute of Measurement and Control,1997,19 (3):139-153.
[52]V.A. Abramovsky,V.M. Abramovsky,E.B. Kat.Robust stabilization of SES heaving and pitching.Ship Control Systems Symposium,Proceedings,1997,Vol.1:551-565.
[53]Hwang Cheng-Neng, Yang Joe-Ming, Chiang Chung-Yen.The design of fuzzy collision-avoidance expert system implemented by H-autopilot.Journal of Marine Science and Technology.2001,9 (1):25-37.
[54]Wang XingCheng,Wang Ran.Two-Degree-of-Freedom H∞ control of ship steering. Proceedings of the World Congress on Intelligent Control and Automation (WCICA),2002,Vol.1: 763-767.
[55]Wang XingCheng,Ren Ying.H∞ control of ship steering 2004 IEEE Conference on Robotics.Automation and Mechatronics,2004,Vol.2:1198-1202.
[56]Yang Pao-Hwa,Hu Shr-Shiung,Juang J.Y.Design of a nonlinear H∞ controller applied to a ship control system.IEEE Conference on Control Applications Proceedings,2000,Vol.1:349-354.
[57]Hu Shr-Shiung,Yang Pao-Hwa,J.Y.Juang,et al.Robust nonlinear ship course-keeping control by H∞ I/O linearization and μ-synthesis.International Journal of Robust and Nonlinear Control.2003,13 (1):55-70.
[58]Yang Pao-Hwa,Hu Shr-Shiung,Juang J.Y.Design of a nonlinear H∞ controller applied to a ship control system.IEEE Conference on Control Applications Proceedings,2000,Vol.1:349-354.
[59]Hu Shr-Shiung,Yang Pao-Hwa,J.Y.Juang,et al.Robust nonlinear ship course-keeping control by H∞ I/O linearization and μ-synthesis.International Journal of Robust and Nonlinear Control.2003,13 (1):55-70.
[60]Tan Feng-yang.Yamato Hiroyuki,Koyama Takeo.H∞control design to include nonlinearities. Second report:Nonlinearities in equations of motion. Journal of Marine Science and Technology. 2002,6 (3):148-157.
[61]N.A. Fairbairn,M.J. Grimble.H∞ robust controller for self-tuning applications part 3 self-tuning controller implementation.International Journal of Control.1990,52 (1):15-36.
[62]D.S. Desanj,M.J. Grimble,M.R. Katebi.State-space adaptive H∞ controller with application to a ship control system.Transactions of the Institute of Measurement and Control.1997,19 (3):139-153.
[63]D.C. Donha,D.S. Desanj,M.R. Katebi,M.J. Grimble.H∞ adaptive controllers for auto-pilot applications.International Journal of Adaptive Control and Signal Processing.1998,12 (8):623-648.
[64]袁士春,郭晨,史成军.基于线性变参数的船舶运动H∞控制及仿真.大连海事大学学报.2007,33(2)：88-91.
[65]N.A. Hickey,M.J. Grimble,M.A. Johnson,et al.Robust fin roll stabilization of surface ships.Proceedings of the IEEE Conference on Decision and Control,1997,Vol.5:4225-4230.
[66]Liu Sheng.Sun Jingchuan,Chen Shengzhong.Ship's fin stabilizer H∞ control under sea wave disturbance.Canadian Conference on Electrical and Computer Engineering,1999,Vol.2:891-895.
[67]刘彦文,王广雄,李栋良.鳍/舵减摇双重控制的H_∞设计.电机与控制学报,2013,18(1)：102-107.
[68]K.D. Do,J. Pan.Nonlinear robust fin roll stabilization of surface ships using neural networks.Proceedings of the IEEE Conference on Decision and Control,2001,Vol.3:2726-2731.
[69]Yang Yansheng,Zhou Changjiu.Design of fuzzy adaptive robust control algorithm via small gain approach.IEEE International Conference on Plasma Science,2002,Vol.1:650-655.
[70]姜礼平,王玉珍,胡伟文.风浪中船舶横摇仿真与预报研究.控制工程,2013,26(2)：344-347.
[71]许可建,刘维亭,朱志宇,等.船舶减摇控制方法综述.船舶.2004,Vo1.5：14-17.
[72]P. Tristan. Ship motion control-course keeping and roll stabilization using rudder and fin.Springer,2005.
[73]金鸿章,李国斌.船舶特种装置控制系统.北京：国防工业出版社,1995.
[74]A.E. Baitis.The development and evaluation of a rudder roll stabilization system for the WHEC Hamilton Class.Report DTNSRDC/SPD-0930-02,Bethesda,Md,USA,1980.
[75]E. Baitis,D.A. Woolaver,T.A. Beck.Rudder roll stabilization for coast guards cutters and frigates.Naval Engineering Journal.1983,Vol.91:267-282.
[76]杨承恩,田园,毕英君.船舶舵阻摇技术的回顾与展望.世界海运.2002,25(4)：4-7.
[77]葛德宏,高启孝,陈永冰,李安.舰船舵阻摇技术的研究现状及展望.舰船科学技术.2007,29(4)：22-26.
[78]J.B. Carley.Feasibility Study of Steering and Stabilizing by Rudder.Proceeding of SCSS75,2,The Hague,The Netherlands,1975:172-194.
[79]J.B. Carley,A. Duberley.Design considerations for optimum ship motion.3rd Ship Control System Symposium,Bath,UK,1972.
[80]P.H. Whyte.On the application of modern control theory on the ship roll stabilisation.Canada Drb Drea Report,79/2,1979.
[81]Van Amerongen. Adaptive Steering of Ship,a Model-Reference Approach to Improved Manoervring and Economical Course Keeping,Ph.D.thesis, Delft University,1982.
[82]R.Femat,J.Capistran-Tobias,G.Solis-Perales.Laplace domain controllers for chaos control, Physics Letters A252(1999):27-36.
[83]马磊,张显库.规则纵浪中船舶参数激励横摇运动的鲁棒控制.中国航海,2012,35(4)：54-57.
[84]Spanos D, Papanikolaou A. Benchmark study on numerical simulation methods for the prediction of parametric roll of ships in waves.10th Inter. Conf. on Stability of Ships and Ocean Vehicles. St. Petersburg, Russia,2009.
[85]唐友刚,田凯强,张泽盛.船舶参数激励非线性横摇运动方程.船舶工程,2004.
[86]Hu Kaiye,Ding Yong,Wan Hongwei.Chaotic roll motions of ships in regular longitudinal waves.Journal of Marine Science and Application,2010,9(2):32-34.
[87]McCue LS. Applications of finite-time Lyapunov exponents to the study of capsize in beam seas.Proceeding of 8th International Ship Stability Workshop,Istanbul,Turkey,2007,1-9.
[88]张延峰,董艳秋,等.纵浪上基本参数共振对船舶稳性的影响.船舶力学,1998,2(3)：16-18.
[89]张显库,贾欣乐.船舶运动控制.北京：国防工业出版社,2006.
[90]王新屏,张显库.基于反馈线性化与闭环增益成形的减摇鳍控制.中国航海.2007,30(4)：5-8.
[91]王新屏,张显库.具有航向保持非线性的舵鳍非线性鲁棒控制.系统工程与电子技术.2008,30(8)：1549-1552.
[92]马磊,张显库.基于混沌分析的船舶参数激励横摇运动及其减摇鳍控制.船舶力学.2013,32(7)：48-52.
[93]马磊,张显库.基于Lyapunov稳定性的船舶参数激励横摇非线性简捷鲁棒控制.大连海事大学学报.2013,39(2)：43-47.
[94]唐友刚,林维学,董艳秋,等.船舶参数激励和强迫激励作用下的非线性运动响应.中国造船,2001,42(2)：34-39.
[95]胡开业,丁勇.船舶在随机纵浪中参数激励横摇稳定性研究.船舶力学,2011,15(2)：11-16.
[96]唐友刚,李红霞.随机斜浪中船舶参数-强迫激励横摇运动计算.中国舰船研究,2008,3(1)：5-8.
[97]鲁江,顾民,马坤,黄武刚.随机波中船舶参数横摇研究.船舶力学,2012,16(8)：893-900.
[98]Bulian G, Francescutto A, Umeda N, Hashimoto H. Qualitative and quantitative characteristics of parametric ship rolling in random waves in the light of physical model experiments. Ocean Engineering,2008,35(17-78):1661-1675.
[99]张显库,金一丞.控制系统建模与数字仿真(第2版).大连：大连海事大学出版社,2013.
[100]张显库.基于李雅普诺夫稳定性的船舶航向保持非线性简化控制程.西南交通大学学报,2010(1)：34-39.
[101]张显库.船舶航向保持的非线性逆推鲁棒控制算法.大连海事大学学报,2007,33(2)：80-83.
[2]SCANCHEZ N E, NAYFEH A H.Nonlinear rolling motion of ships in longitudinal waves. ISP, 1990,37:247-272.
[3]张延峰,董艳秋,唐友刚.纵浪上基本参数共振对船舶稳性的影响.船舶力学,1998,2(3)：6-12.
[4]王立军,张显库.减摇航向自动舵的简捷鲁棒优化控制算法.大连海事大学学报,2013,39(1)：1-5.
[5]任海鹏.混沌控制方法及其应用研究.西安理工大学博士论文,2003.
[6]Li T Y,Yorke J A.Period three implies chaos.A mer. Math. Monthly.1975,82:985-992.
[7]Hao B.L.Chaos.Singapore:World Scientific,1984.
[8]Hao B.L.Chaos Ⅱ.Singapore:World Scientific,1990.
[9]陆君安,陈士华.混沌动力学初步.武汉,武汉水利水电人学出版社,1998.
[10]王杰智,陈增强,袁著祉.一个新的混沌系统及其性质研究.物理学报,2006,55(8)：3956-3964.
[11]李鹤.工程中的混沌的识别、利用和控制的研究.东北大学博士论文,2005.
[12]王瑞平.某些常微和偏微系统的混沌控制与反控制.中山大学硕士论文,2003.
[13]Lorenz E.N.Deterministic non-periodic flow.J.Atmos.Sci.1963,20:130-141.
[14]Sivakumar B.Chaos theory in geophysics:past, present and future. Chaos Solitons&Fractals, 2004,19 (2):441-462.
[15]Freeman W.H.,Mandebort B.B. Fractal,form,chance and dimension. San Francisco,1977.
[16]Rossler O.E. An equation for continuous chaos. Physical Leters A,1976,57:397-398.
[17]Chua L.O.,Komuro M., Matsumoto T. The double scroll family. Part Ⅰ:rigorous proof of chaos. IEEE Trans. On Circuits & Systems-Ⅰ,1986,33:1072-1096.
[18]Chen G.R., Ueta T. Yet another chaotic attractor. Int. J. Bifurcation & Chaos,1999,9: 1465-1466.
[19]Lu J.H., Chen G.R., Cheng D.Z. Bridge the gap between the Lorenz system and the Chen system. Int. J. Bifurcation & Chaos,2002,12:2917-2926.
[20]Liu C.X., Liu L., Liu T. A new chaotic attractor. Chaos, Solitons & Fractals,2004,22: 1031-1038.
[21]Liu C.X., Liu T., Liu L. A new butterfly-shaped attractor of Lorenz-like system. Chaos, Solitons & Fractals,2006,28:1196-1203.
[22]Wang F.Q., Liu C.X. Hyperchaos evolved from the Liu chaotic system. Chinese Physics, 2006,15(5):963-968.
[23]KERWIN J E. Notes on rolling in longitudinal waves.Int Shipbuild Progr,1955,2(16): 597-614.
[24]HADDARA M R. Effect of parametric excitation on ship motion in random waves. Ship Res, 1973,7(5):30-37.
[25]SILVA S, SOARES C. Time domain simulation of parametrically excited roll in head seas.7th International Conference on Stability of Ships and Ocean Vehicles. Tas-mania, Australia,2000: 652-664.
[26]NEVES M, VALERIO L. Parametric resonance in waves of arbitrary heading. Int Shipbuild Progr,1990,35(320):230-241.
[27]MATUSIAK J. Parametric resonance of the conical buoy in regular waves. Int Shipbuild Progr, 2000,37(350):100-121.
[28]BOUKHANOVSKY A, DEGTYAREV A. Peculiarities of motion of ship with low buoyancy on asymmetrical random waves. Int Shipbuild Progr,2001,38(367):350-364.
[29]SILVA S R, SANTOS T A, SOARES C G. Parametrically excited roll in regular and irregular head seas. Int Ship-build Progr,2005,52(1):29-56.
[30]闵美松,丁勇,赵晓东,等.随浪中船舶横甩试验研究.哈尔滨船舶工程学院学报,1993,14(3)：12-18.
[31]张延峰,董艳秋,唐友刚.纵浪上基本参数共振对船舶稳性的影响.船舶力学,1998,2(3)：6-12.
[32]唐友刚,田凯强,张泽盛.船舶参数激励非线性横摇运动方程.船舶工程,1998,6：16-18.
[33]唐友刚,林维学,董艳秋,等.船舶参数激励和强迫激励作用下的非线性运动响应.中国造船,2001,42(2)：34-39.
[34]Sogawa Y, Umeda N, Hashimoto H. Parametric roll of a post Panamax containership in regular waves:Experiment, ana lytical method and simulation. Proc 5th APHydro. Osaka, Japan,2010: 99-102.
[35]张显库.非脆弱鲁棒PID控制器设计.重庆工学院学报(自然科学版).2009,23(5)：28-30.
[36]Zhang X K,Jin Y C.Control of A Multivariable High Purity Distillation Column Based on Closed-loop Gain Shaping Algorithm.International Journal of Information Technology.2005,11 (5):116-123.
[37]Zhang X K,Jia X L.Simplification of H mixed sensitivity algorithm and its application. Automatic Control and Computer Sciences.2002,36 (3):28-33.
[38]贾欣乐,张显库.H∞控制器应用于船舶自动舵.控制与决策.1995,10(3)：250-254.
[39]张显库,贾欣乐.船舶自动舵鲁棒控制权函数综合选择方法.大连海事大学学报.1999,25 (3)：21-25.
[40]张显库,贾欣乐,杨承恩,等.船舶鲁棒自动舵产品化研究.系统工程与电子技术.1999,21(2)：75-81.
[41]张显库,贾欣乐,王兴成.一种变形的回来成形控制器及其应用.控制与决策.2000,15(1)：116-118.
[42]张显库,贾欣乐.求PID参数新方法.系统工程与电子技术.2000,22(8)：4-5.
[43]张显库,贾欣乐.用镜像映射方法求非稳定过程的鲁棒控制器.系统工程与电子技术.2000,22(4)：10-12.
[44]张显库.用镜像映射方法求纯不稳定过程的鲁棒控制器.系统工程与电子技术,2004,26(10)：1466-1467.
[45]Zhang X K.New method on design of robust controller for unstable process.Proceeding of International Conference on Machine Learning and Cybernetics,643-648,2005.
[46]张显库,金一丞.基于信息对称的简捷控制.控制与决策.2007,22(11)：1255-1258.
[47]张显库.不对称信息理论在船舶运动控制中的应用.中国造船.2006,47(1)：55-59.
[48]张显库,杨盐生.不对称信息理论与非线性鲁棒控制算法.控制与决策.2005,20(11)：1241-1244.
[49]张显库,郭晨,杜佳璐.船舶航向不对称信息理论与非线性逆推鲁棒控制算法.交通运输工程学报.2006,6(2)：47-50.
[50]M.J. Grimble,S.A. Carr,M.R. Katebi.Integrated ship control using H∞ robust design techniques.Proceedings of the IEEE Conference on Control Applications,1994,Vol.2:1087-1091.
[51]D.S. Desanj,M.J. Grimble,M.R. Katebi.State-space adaptive H∞ controller with application to a ship control system.Transactions of the Institute of Measurement and Control,1997,19 (3):139-153.
[52]V.A. Abramovsky,V.M. Abramovsky,E.B. Kat.Robust stabilization of SES heaving and pitching.Ship Control Systems Symposium,Proceedings,1997,Vol.1:551-565.
[53]Hwang Cheng-Neng, Yang Joe-Ming, Chiang Chung-Yen.The design of fuzzy collision-avoidance expert system implemented by H-autopilot.Journal of Marine Science and Technology.2001,9 (1):25-37.
[54]Wang XingCheng,Wang Ran.Two-Degree-of-Freedom H∞ control of ship steering. Proceedings of the World Congress on Intelligent Control and Automation (WCICA),2002,Vol.1: 763-767.
[55]Wang XingCheng,Ren Ying.H∞ control of ship steering 2004 IEEE Conference on Robotics.Automation and Mechatronics,2004,Vol.2:1198-1202.
[56]Yang Pao-Hwa,Hu Shr-Shiung,Juang J.Y.Design of a nonlinear H∞ controller applied to a ship control system.IEEE Conference on Control Applications Proceedings,2000,Vol.1:349-354.
[57]Hu Shr-Shiung,Yang Pao-Hwa,J.Y.Juang,et al.Robust nonlinear ship course-keeping control by H∞ I/O linearization and μ-synthesis.International Journal of Robust and Nonlinear Control.2003,13 (1):55-70.
[58]Yang Pao-Hwa,Hu Shr-Shiung,Juang J.Y.Design of a nonlinear H∞ controller applied to a ship control system.IEEE Conference on Control Applications Proceedings,2000,Vol.1:349-354.
[59]Hu Shr-Shiung,Yang Pao-Hwa,J.Y.Juang,et al.Robust nonlinear ship course-keeping control by H∞ I/O linearization and μ-synthesis.International Journal of Robust and Nonlinear Control.2003,13 (1):55-70.
[60]Tan Feng-yang.Yamato Hiroyuki,Koyama Takeo.H∞control design to include nonlinearities. Second report:Nonlinearities in equations of motion. Journal of Marine Science and Technology. 2002,6 (3):148-157.
[61]N.A. Fairbairn,M.J. Grimble.H∞ robust controller for self-tuning applications part 3 self-tuning controller implementation.International Journal of Control.1990,52 (1):15-36.
[62]D.S. Desanj,M.J. Grimble,M.R. Katebi.State-space adaptive H∞ controller with application to a ship control system.Transactions of the Institute of Measurement and Control.1997,19 (3):139-153.
[63]D.C. Donha,D.S. Desanj,M.R. Katebi,M.J. Grimble.H∞ adaptive controllers for auto-pilot applications.International Journal of Adaptive Control and Signal Processing.1998,12 (8):623-648.
[64]袁士春,郭晨,史成军.基于线性变参数的船舶运动H∞控制及仿真.大连海事大学学报.2007,33(2)：88-91.
[65]N.A. Hickey,M.J. Grimble,M.A. Johnson,et al.Robust fin roll stabilization of surface ships.Proceedings of the IEEE Conference on Decision and Control,1997,Vol.5:4225-4230.
[66]Liu Sheng.Sun Jingchuan,Chen Shengzhong.Ship's fin stabilizer H∞ control under sea wave disturbance.Canadian Conference on Electrical and Computer Engineering,1999,Vol.2:891-895.
[67]刘彦文,王广雄,李栋良.鳍/舵减摇双重控制的H_∞设计.电机与控制学报,2013,18(1)：102-107.
[68]K.D. Do,J. Pan.Nonlinear robust fin roll stabilization of surface ships using neural networks.Proceedings of the IEEE Conference on Decision and Control,2001,Vol.3:2726-2731.
[69]Yang Yansheng,Zhou Changjiu.Design of fuzzy adaptive robust control algorithm via small gain approach.IEEE International Conference on Plasma Science,2002,Vol.1:650-655.
[70]姜礼平,王玉珍,胡伟文.风浪中船舶横摇仿真与预报研究.控制工程,2013,26(2)：344-347.
[71]许可建,刘维亭,朱志宇,等.船舶减摇控制方法综述.船舶.2004,Vo1.5：14-17.
[72]P. Tristan. Ship motion control-course keeping and roll stabilization using rudder and fin.Springer,2005.
[73]金鸿章,李国斌.船舶特种装置控制系统.北京：国防工业出版社,1995.
[74]A.E. Baitis.The development and evaluation of a rudder roll stabilization system for the WHEC Hamilton Class.Report DTNSRDC/SPD-0930-02,Bethesda,Md,USA,1980.
[75]E. Baitis,D.A. Woolaver,T.A. Beck.Rudder roll stabilization for coast guards cutters and frigates.Naval Engineering Journal.1983,Vol.91:267-282.
[76]杨承恩,田园,毕英君.船舶舵阻摇技术的回顾与展望.世界海运.2002,25(4)：4-7.
[77]葛德宏,高启孝,陈永冰,李安.舰船舵阻摇技术的研究现状及展望.舰船科学技术.2007,29(4)：22-26.
[78]J.B. Carley.Feasibility Study of Steering and Stabilizing by Rudder.Proceeding of SCSS75,2,The Hague,The Netherlands,1975:172-194.
[79]J.B. Carley,A. Duberley.Design considerations for optimum ship motion.3rd Ship Control System Symposium,Bath,UK,1972.
[80]P.H. Whyte.On the application of modern control theory on the ship roll stabilisation.Canada Drb Drea Report,79/2,1979.
[81]Van Amerongen. Adaptive Steering of Ship,a Model-Reference Approach to Improved Manoervring and Economical Course Keeping,Ph.D.thesis, Delft University,1982.
[82]R.Femat,J.Capistran-Tobias,G.Solis-Perales.Laplace domain controllers for chaos control, Physics Letters A252(1999):27-36.
[83]马磊,张显库.规则纵浪中船舶参数激励横摇运动的鲁棒控制.中国航海,2012,35(4)：54-57.
[84]Spanos D, Papanikolaou A. Benchmark study on numerical simulation methods for the prediction of parametric roll of ships in waves.10th Inter. Conf. on Stability of Ships and Ocean Vehicles. St. Petersburg, Russia,2009.
[85]唐友刚,田凯强,张泽盛.船舶参数激励非线性横摇运动方程.船舶工程,2004.
[86]Hu Kaiye,Ding Yong,Wan Hongwei.Chaotic roll motions of ships in regular longitudinal waves.Journal of Marine Science and Application,2010,9(2):32-34.
[87]McCue LS. Applications of finite-time Lyapunov exponents to the study of capsize in beam seas.Proceeding of 8th International Ship Stability Workshop,Istanbul,Turkey,2007,1-9.
[88]张延峰,董艳秋,等.纵浪上基本参数共振对船舶稳性的影响.船舶力学,1998,2(3)：16-18.
[89]张显库,贾欣乐.船舶运动控制.北京：国防工业出版社,2006.
[90]王新屏,张显库.基于反馈线性化与闭环增益成形的减摇鳍控制.中国航海.2007,30(4)：5-8.
[91]王新屏,张显库.具有航向保持非线性的舵鳍非线性鲁棒控制.系统工程与电子技术.2008,30(8)：1549-1552.
[92]马磊,张显库.基于混沌分析的船舶参数激励横摇运动及其减摇鳍控制.船舶力学.2013,32(7)：48-52.
[93]马磊,张显库.基于Lyapunov稳定性的船舶参数激励横摇非线性简捷鲁棒控制.大连海事大学学报.2013,39(2)：43-47.
[94]唐友刚,林维学,董艳秋,等.船舶参数激励和强迫激励作用下的非线性运动响应.中国造船,2001,42(2)：34-39.
[95]胡开业,丁勇.船舶在随机纵浪中参数激励横摇稳定性研究.船舶力学,2011,15(2)：11-16.
[96]唐友刚,李红霞.随机斜浪中船舶参数-强迫激励横摇运动计算.中国舰船研究,2008,3(1)：5-8.
[97]鲁江,顾民,马坤,黄武刚.随机波中船舶参数横摇研究.船舶力学,2012,16(8)：893-900.
[98]Bulian G, Francescutto A, Umeda N, Hashimoto H. Qualitative and quantitative characteristics of parametric ship rolling in random waves in the light of physical model experiments. Ocean Engineering,2008,35(17-78):1661-1675.
[99]张显库,金一丞.控制系统建模与数字仿真(第2版).大连：大连海事大学出版社,2013.
[100]张显库.基于李雅普诺夫稳定性的船舶航向保持非线性简化控制程.西南交通大学学报,2010(1)：34-39.
[101]张显库.船舶航向保持的非线性逆推鲁棒控制算法.大连海事大学学报,2007,33(2)：80-83.