基于平方和方法的多项式非线性系统控制器设计
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摘要
许多实际的控制系统都有其本质的、不可忽略的非线性。在系统和控制理论中，非线性系统的分析和设计一直被列为最具挑战性的问题之一。本文研究多项式非线性系统，主要采用平方和方法和密度函数研究多项式非线性系统的控制综合问题。
近年来，平方和方法已得到广泛的应用。它为研究多项式非线性系统提供了一个有效的方法。相比二次Lyapunov函数，平方和方法具有能构造并搜索高次Lyapunov函数的优势。已有许多工作将平方和方法用于非线性系统稳定性分析。然而，用平方和方法探讨系统的控制综合问题还不多。因此，本文探究如何将平方和方法用于卫星姿态、船舶航向以及横摇等实际系统的控制器设计，并给出控制器设计的数值解。
在2001年，Rantzer提出了非线性系统的一个新的渐近性准则，此准则可视为Lyapunov第二定理的对偶。Rantzer准则采用密度函数而不是Lyapunov函数来判断系统轨线的渐近性。利用Rantzer准则可将控制器搜索问题转为凸规划问题，而基于原Lyapunov第二定理的控制Lyapunov函数搜索问题是非凸的。本文的主要工作如下：
首先，提出了非线性系统局域渐近稳定和几乎全局稳定的一个充分条件，即如果非线性系统的密度函数及其对时间的导数大于0，则系统是局域渐近稳定和几乎全局稳定。还将此结果用到多项式非线性系统的吸引域估计。
第二、带有三个输入的卫星姿态控制器设计是一个典型的多项式非线性控制系统设计问题。将前述渐近稳定的结果与平方和方法相结合用于卫星姿态控制器的设计，采用平方和方法成功搜索到了闭环系统的四次Lyapunov函数。仿真结果表明了控制器设计方法的有效性。
第三、对确定参数的多项式非线性系统，给出了将平方和技术与Rantzer准则相结合用于控制器设计及其数值解的方法，并应用到带有确定参数的船舶航向的静态反馈控制器设计。基于Rantzer准则设计的控制器只能保证渐近性，其稳定性需要后续验证。采用平方和方法搜索到了闭环系统的Lyapunov函数，从而验证系统全局渐近稳定。
第四、对不确定参数的多项式非线性系统，将平方和技术与Rantzer准则相结合用于鲁棒控制器设计及其数值求解。具体研究了带有不确定参数的船舶航向的鲁棒镇定控制、带有执行器的不确定参数的船舶航向动态反馈控制以及带有不确定参数的船舶横摇鲁棒镇定控制。仿真结果表明设计的控制器针对不确定参数具有强的鲁棒性。
Many practical systems have inherent nonlinearity that cannot be ignored. The anal-ysis and design of nonlinear systems are among the most challenging problems in systemsand control theory. In this thesis, we focus on nonlinear control synthesis for polynomialnonlinear systems using Sum of Squares (SOS) programming.
SOS is a powerful technique which has been widely used in recent years. It providesan efcient way for researchers to explore polynomial nonlinear systems. Compared toquadratic Lyapunov function, it has the advantage of being able to find high-order Lya-punov function through sum of squares programming for polynomial nonlinear systems.SOS has been used for nonlinear systems analysis extensively. However, its potential forcontrol synthesis has not been fully explored. Therefore, this dissertation explores how touse SOS programming for controller design and its numerical solution for the followingthree systems: satellite attitude control, ship course control and ship roll control.
A novel convergence criterion for the nonlinear systems has been recently derived byAnders Rantzer. The criterion uses a density function, instead of a Lyapunov function, toguarantee the systems trajectories asymptotically tending to the equilibrium. The criterionis used for transforming the problem of searching controller into a convex programming.However, it is not a convex programming problem to search control Lyapunov functionoriginally based on Lyapunov’s second theorem. The main work is as follows:
Firstly, we prove that the existence of a density function with its time derivativegreater than zero ensures almost global stability and local asymptotical stability. Theresult has been used for estimating the domain of attraction for polynomial nonlinearsystems.
Secondly, for satellite attitude control with three inputs, the controller design is ofa typical nature of polynomial nonlinear control systems. Nonlinear state feedback con-trollers are designed based on the aforementioned asymptotically stable result combinedwith the sum of squares technique. Lyapunov function of degree4is searched success-fully using sum of squares programming. Simulations demonstrate the validity of thesuggested controller design method.
Thirdly, for systems with certain parameters, we present a nonlinear controller de-sign method and its numerical solution using the sum of squares technique and Rantzer’s convergence criterion. For ship course control, nonlinear static feedback controllers aredesigned based on the suggested method. Lyapunov function of the resulting closed-loopsystem is found using sum of squares programming, which verifies globally asymptoticalstability of the closed-loop system.
Lastly, for systems with uncertain parameters, a robust nonlinear controller designmethod and its numerical solution is proposed based on the sum of squares technique andRantzer’s convergence criterion. For three cases of ship course control with uncertainparameters, uncertain ship course control with dynamic actuator, and ship roll controlwith uncertain parameters, the corresponding robust controllers are designed based on thesuggested design method. The simulation results show that the suggested design methodhas strong robustness for the systems with uncertain parameters.
近年来，平方和方法已得到广泛的应用。它为研究多项式非线性系统提供了一个有效的方法。相比二次Lyapunov函数，平方和方法具有能构造并搜索高次Lyapunov函数的优势。已有许多工作将平方和方法用于非线性系统稳定性分析。然而，用平方和方法探讨系统的控制综合问题还不多。因此，本文探究如何将平方和方法用于卫星姿态、船舶航向以及横摇等实际系统的控制器设计，并给出控制器设计的数值解。
在2001年，Rantzer提出了非线性系统的一个新的渐近性准则，此准则可视为Lyapunov第二定理的对偶。Rantzer准则采用密度函数而不是Lyapunov函数来判断系统轨线的渐近性。利用Rantzer准则可将控制器搜索问题转为凸规划问题，而基于原Lyapunov第二定理的控制Lyapunov函数搜索问题是非凸的。本文的主要工作如下：
首先，提出了非线性系统局域渐近稳定和几乎全局稳定的一个充分条件，即如果非线性系统的密度函数及其对时间的导数大于0，则系统是局域渐近稳定和几乎全局稳定。还将此结果用到多项式非线性系统的吸引域估计。
第二、带有三个输入的卫星姿态控制器设计是一个典型的多项式非线性控制系统设计问题。将前述渐近稳定的结果与平方和方法相结合用于卫星姿态控制器的设计，采用平方和方法成功搜索到了闭环系统的四次Lyapunov函数。仿真结果表明了控制器设计方法的有效性。
第三、对确定参数的多项式非线性系统，给出了将平方和技术与Rantzer准则相结合用于控制器设计及其数值解的方法，并应用到带有确定参数的船舶航向的静态反馈控制器设计。基于Rantzer准则设计的控制器只能保证渐近性，其稳定性需要后续验证。采用平方和方法搜索到了闭环系统的Lyapunov函数，从而验证系统全局渐近稳定。
第四、对不确定参数的多项式非线性系统，将平方和技术与Rantzer准则相结合用于鲁棒控制器设计及其数值求解。具体研究了带有不确定参数的船舶航向的鲁棒镇定控制、带有执行器的不确定参数的船舶航向动态反馈控制以及带有不确定参数的船舶横摇鲁棒镇定控制。仿真结果表明设计的控制器针对不确定参数具有强的鲁棒性。
Many practical systems have inherent nonlinearity that cannot be ignored. The anal-ysis and design of nonlinear systems are among the most challenging problems in systemsand control theory. In this thesis, we focus on nonlinear control synthesis for polynomialnonlinear systems using Sum of Squares (SOS) programming.
SOS is a powerful technique which has been widely used in recent years. It providesan efcient way for researchers to explore polynomial nonlinear systems. Compared toquadratic Lyapunov function, it has the advantage of being able to find high-order Lya-punov function through sum of squares programming for polynomial nonlinear systems.SOS has been used for nonlinear systems analysis extensively. However, its potential forcontrol synthesis has not been fully explored. Therefore, this dissertation explores how touse SOS programming for controller design and its numerical solution for the followingthree systems: satellite attitude control, ship course control and ship roll control.
A novel convergence criterion for the nonlinear systems has been recently derived byAnders Rantzer. The criterion uses a density function, instead of a Lyapunov function, toguarantee the systems trajectories asymptotically tending to the equilibrium. The criterionis used for transforming the problem of searching controller into a convex programming.However, it is not a convex programming problem to search control Lyapunov functionoriginally based on Lyapunov’s second theorem. The main work is as follows:
Firstly, we prove that the existence of a density function with its time derivativegreater than zero ensures almost global stability and local asymptotical stability. Theresult has been used for estimating the domain of attraction for polynomial nonlinearsystems.
Secondly, for satellite attitude control with three inputs, the controller design is ofa typical nature of polynomial nonlinear control systems. Nonlinear state feedback con-trollers are designed based on the aforementioned asymptotically stable result combinedwith the sum of squares technique. Lyapunov function of degree4is searched success-fully using sum of squares programming. Simulations demonstrate the validity of thesuggested controller design method.
Thirdly, for systems with certain parameters, we present a nonlinear controller de-sign method and its numerical solution using the sum of squares technique and Rantzer’s convergence criterion. For ship course control, nonlinear static feedback controllers aredesigned based on the suggested method. Lyapunov function of the resulting closed-loopsystem is found using sum of squares programming, which verifies globally asymptoticalstability of the closed-loop system.
Lastly, for systems with uncertain parameters, a robust nonlinear controller designmethod and its numerical solution is proposed based on the sum of squares technique andRantzer’s convergence criterion. For three cases of ship course control with uncertainparameters, uncertain ship course control with dynamic actuator, and ship roll controlwith uncertain parameters, the corresponding robust controllers are designed based on thesuggested design method. The simulation results show that the suggested design methodhas strong robustness for the systems with uncertain parameters.
引文
[1] Parrilo P A. Structured Semidefinite Programs and Semialgebraic Geometry Meth-ods in Robustness and Optimization[D]. Pasadena: California Institute of Technol-ogy,2000.
[2] Jarvis-Wloszek Z, Feeley R, Tan W. Some Controls Applications of Sum ofSquares Programming[C]//Proceedings of the42nd IEEE Conference on Decisionand Control. Maui: Hawaii,2003:4676–4681.
[3] Prajna S, Parrilo P A, Rantzer A. Nonlinear Control Synthesis by Convex Opti-mization[J]. IEEE Transactions on Automatic Control,2004,49(2):310–314.
[4] Rantzer A. A Dual to Lyapunov’s Stability Theorem[J]. System＆Control Letters,2001,42(3):161–168.
[5] Ataei A, Wang Q. Nonlinear Control of an Uncertain Hypersonic Aircraft Modelusing Robust Sum-of-squares Method[J]. IET Control Theory and Applications,2011,6(2):203–215.
[6]童长飞.基于半定规划的多项式非线性系统镇定控制研究[D].杭州:浙江大学,2008.
[7] Jarvis-Wloszek Z, Feeley R, Tan W, et al. Control Applications of Sum of SquaresProgrmming[M]. Berlin Heideberg: Springer-Verlag,2005:3–22.
[8]王佳,曾鸣,苏宝库.基于平方和的卫星大角度姿态机动非线性H∞控制[J].系统工程与电子技术,2010,32(5):1024–1028.
[9] Chesi G. On the Gap between Positive Polynomials and SOS of Polynomials[J].IEEE Transactions on Automatic Control,2007,52(6):1066–1072.
[10] Prajna S. Optimization-Based Methods for Nonlinear and Hybrid Systems Verifi-cation[D]. Pasadena: California Institute of Technology,2005.
[11] Papachristodoulou A. Scalable Analysis of Nonlinear Systems Using Convex Op-timization[D]. Pasadena: California Institute of Technology,2005.
[12] Jarvis-Wloszek Z. Lyapunov based Analysis and Controller Synthesis for Poly-nomial Systems using Sum-of-squares Optimization[D]. Berkeley: University ofCalifornia,2003.
[13] Tan W. Nonlinear Control Analysis and Synthesis using Sum-of-Squares Program-ming[D]. Berkeley: University of California,2006.
[14] Zheng Q. Sum of Squares Based Nonlinear Control Design Techniques[D].Raleigh: North Carolina State University,2009.
[15] Prajna S, Papachristodoulou A, Parrilo P A. Introducing SOSTOOLS: A GeneralPurpose Sum of Squares Programming solver[C]//Proceedings of the41st Confer-ence on Decision and Control. Las Vegas: Nevada,2002:741–746.
[16] Prajna S, Papachristodoulou A, Seiler P, et al. SOSTOOLS: Sum of Squares Opti-mization Toolbox for MATLAB[M]. Pasadena: California Institute of Technology,2004:1–42.
[17] Toh K C, Todd M J, Tutuncu R H. SDPT3-A Matlab Software Package for Semidef-inite Programming[J]. Optimization Methods and Software,1999,11(1):545–581.
[18] Sturm J F. Using SeduMi1.02, a Matlab Toolbox for Optimization over SymmetricCones[J]. Optimization Methods and Software,1999,11(1):625–653.
[19] Tan W, Packard A. Seaching for Control Lyapunov Functions using Sums ofSquares Programming[J].2004:1–10. http://www.ie.nthu.edu.tw/info/ie.newie.htm(Big5).
[20] Papachristodoulou A, Prajna S. On the Construction of Lyapunov Functions usingthe Sum of Squares Decomposition[C]//Proceedings of the41st IEEE Conferenceon Decision and Control. Las Vegas: Nevada,2002:3482–3487.
[21] Prajna S, Papachristodoulou A, Wu F. Nonlinear Control Synthesis by Sumof Squares Optimization: A Lyapunov-base Approach[C]//Proceedings of the5thAsian Control Conference. Melbourne: Australia,2004:157–165.
[22] Prajna S, Papachristodoulou A, Seiler P. New Developments in Sum of SquaresOptimization and SOSTOOLS[C]//Proceedings of the American Control Confer-ence. Boston: Massachusetts,2004:5606–5611.
[23] Prajna S. Barrier certificates for Nonlinear Model Validation[J]. Automatica,2006,42(1):117–126.
[24] Prajna S, Papachristodoulou A. Analysis of Switched and Hybrid Systems-BeyondPiecewise Quadratic Methods[C]//Proceedings of the American Control Confer-ence. Pasadena: California,2003:2779–2784.
[25] Papachristodoulou A, Parrilo P A. A Tutorial on Sum of Squares Techniques forSystems Analysis[C]//Proceedings of the American Control Conference. Portland:California,2005:2686–2700.
[26] Papachristodoulou A. Analysis of Nonlinear Time-delay Systems using the Sumof Squares Decomposition[C]//Proceedings of the American Control Conference.Boston: Massachusetts,2004:4153–4158.
[27] Papachristodoulou A. Robust Stabilization of Nonlinear Time Delay Systems Us-ing Convex Optimization[C]//Proceedings of the44th IEEE Conference on Deci-sion and Control and2005European Control Conference, CDC-ECC’05. Seville:Spain,2005:5788–5793.
[28] Papachristodoulou A, Doyle J C, Low S H. Analysis of Nonlinear Time-delaySystems using the Sum of Squares Decomposition[C]//Proceedings of the43rdIEEE Conference on Decision and Control. Nassau: Germany,2004:4684–4689.
[29] Papachristodoulou A, Papageorgiou C. Robust Stability and PerformanceAnalysis of a Longitudinal Aircraft Model using Sum of Squares Tech-niques[C]//Proceedings of the IEEE International Symposium on Intelligent Con-trol. Limassol: Cyprus,2005:1281–1286.
[30] Papachristodoulou A, Peet M, Lall S. Constructing Lyapunov-Krasovskii Func-tionals for Linear Time Delay Systems[C]//Proceedings of the American ControlConference. Portland: Oregon,2005:2845–2850.
[31] Tan W, Packard A. Stability Region Analysis using Polynomial and CompositePolynomial Lyapunov Functions and Sum of Squares Programming[J]. IEEE Tran-s. Autom.Control,2008,53(2):565–571.
[32] Topcu U, Packard A, Seiler P. Local Stability Analysis using Simulations andSum-of-squares Programming[J]. Automatica,2008,44(10):2669–2675.
[33] EI-Samad H, Prajna S, Papachristodoulou A, et al. Model Validation and Ro-bust Stability Analysis of the Bacterial Heat Shock Response using SOSTOOL-S[C]//Proceedings of the42nd IEEE Conference on Decision and Control. Maui:Hawaii,2003:3766–3771.
[34] EI-Samad H, Prajna S, Papachristodoulou A, et al. Advanced Methods and Al-gorithms for Biological Networks Analsis[J]. Proceedings of the IEEE,2006,94(4):832–853.
[35] Ebenbauer C, Raf T, Allgower F. A Duality-based LPV Approach to Polynomi-al State Feedback Design[C]//Proceedings of the American Control Conference.Portland: Oregon,2005:703–708.
[36] Ebenbauer C, Ren Z, Allgower F. Polynomial Feedback and Observer Design us-ing Nonquadratic Lyapunov Functions[C]//Proceedings of the44th IEEE Confer-ence on Decision and Control, and European Control Conference. Seville: Spain,2005:7587–7592.
[37] Ebenbauer C, Allgower F. Analysis and Design of Polynomial Control Systemsusing Dissipation Inequalities and Sum of Squares[J]. Computers and ChemicalEngineering,2006,30(10-12):1590–1602.
[38] Ebenbauer C, Allgower F. Stability Analysis of Constrained Control Systems: AnAlternative Approach[J]. Systems＆Control Letters,2007,56(2):93–98.
[39] Krishnaswamy K, Papageorgiou G, Glavaski S, et al. Analysis of AircraftPitch Axis Stability Augmentation System using Sum of Squares Optimiza-tion[C]//Proceedings of the American Control Conference. Portland: Oregon,2005:1497–1502.
[40] Ataei A, Wang Q. Nonlinear Control Design of a Hypersonic Aircraft using Sum-of-squares Method[C]//Proceedings of the American Control Conference. NewYork: USA,2007:5287–5283.
[41] Nauta K M, Weiland S, Backx A C P, et al. Approximation of Fast Dynamicsin Kinetic Networks using Non-negative Polynomials[C]//16th IEEE InternationalConference on Control Applications part of IEEE Multi-Conference on Systemsand Control. Singapore: Singapore,2007:1144–1149.
[42] Sontag E D, Ebenbauer R, Findeisen F, et al. Nonlinear Model Predictive Controland Sum of Squares Techniques[M]. New York: Springer-Verlag,2006:325–344.
[43] Franze G, Casavola A, Famularo D, et al. An of-line Formulation of RecedingHorizon Control for Nonlinear Systems using Sum of Squares[M]. New York:Springer-Verlag,2009:491–499.
[44] Pozo F, Rodellar J. Robust Stabilization of Polynomial Systems with UncertainParameters[J]. International Journal of System Science,2010,41(5):575–584.
[45] Fazel M, Chiang M. Network Utility Maximization with Nonconcave Utilities us-ing Sum-of-Squares Metnod[C]//Proceedings of the44th IEEE Conference on De-cision and Control, and European Control Conference. Seville: Spain,2005:1867–1874.
[46] Yu J, Yan Z, Wang J, et al. Robust Stabilization of Ship Course Via Convex Opti-mization[J]. Asia Journal of Control,2013,5(9):1666–1675.
[47] Henrion D, Lofberg J. Solving Nonconvex Optimization Problems[J]. IEEE Con-trol Systems,2004,24(3):72–83.
[48] Biswas P, Grieder P, Lofberg J, et al. A Survey on Stability Analysis of Discrete-time Piecewise Afne Systems[C]//Proceedings of the16th IFAC World Congress.Prague: Czechoslovakia,2005:331–331.
[49] Johansson M, Rantzer A. Computation of Piecewise Quadratic Lyapunov Func-tions for Hybrid Systems[J]. IEEE Transactions on Automatic Control,1998,43(4):555–559.
[50] Prajna S. Barrier Certificates for Nonlinear Model Validation[C]//Proceedings ofthe42nd IEEE Conference on Decision and Control. Maui: Hawaii,2003:2884–2889.
[51] Prajna S, Jadbabaie A. Safety Verification of Hybrid Systems using Barrier Cer-tificates[M]. New York: Springer-Verlag,2004:477–492.
[52] Prempain E. An Application of the Sum of Squares Decomposition to the L2Gain Computation for a Class of Nonlinear Systems[C]//Proceedings of the44thIEEE Conference on Decision and Control,and the European Control Conference.Seville: Oregon,2005:6865–6868.
[53] Ichihara H, Nobuyama E. L2Gain Analysis and Control Design of Linear Systemswith Input Saturation Using Matrix Sum of Squares Relaxation[C]//Proceedings ofthe American Control Conference. Minneapolis: Minnesota,2006:3837–3842.
[54] Wu F, Yang X H, Pzckard A, et al. Induced L2Norm Control for LPV Systemswith Bounded Parameter Variation Rates[J]. International Journal of Robust andNonlinear Control,1996,6(9):983–998.
[55] Khalil H K. Nonlinear Systems[M]. London: Prentice Hall,1996:261–284.
[56] Fujisaki Y, Miyoshi E. Controller Redesign via Sum of Squares Program-ming[C]//Proceedings of SICE-ICASE International Joint Conference. Busan: Ko-rea,2006:4012–4015.
[57] Monzon P. On Necessary Conditions for Almost Global Stability[J]. IEEE Trans-actions On Automatic Control,2003,48(4):631–634.
[58] Angeli D. Some Remarks on Density Functions for Dual Lyapunov Method-s[C]//Proceedings of the42nd IEEE Conference on Decision and Control. Maui:Hawaii,2003:5080–5082.
[59] Prajna S, Rantzer A. On Homogeneous Density Functions[M]. Berlin: SpringerVerlag,2003:261–274.
[60] Monzon P. Almost Global Attraction in Planar Systems[J]. Systems＆ControlLetters,2005,54(1):753–758.
[61] Rantzer A. A Converse Theorem for Density Functions[C]//Proceedings of the41stIEEE Conference on Decision and Control. Las Vegas: Nevada,2002:1890–1891.
[62] Monzon P, Potrie R. Local and Global Aspects of Almost Global Stabili-ty[C]//Proceedings of the45th IEEE Conference on Decision and Control. SanDiego: California,2006:5120–5125.
[63] Masubuchi I. Analysis of Positive Invariance and Almost Regional Attractionvia Density Functions with Converse Results[C]//Proceedings of the45th IEEEConference on Decision and Control. San Diego: California,2006:5126–5131.
[64] Potrie R, Monzon P. Local Implications of Almost Global Stability[J]. DynamicalSystems,2009,24(1):109–115.
[65] Boyd S, Vandenberghe L. SP: Software for Sedidefinite Programming[D]. Cali-fornia: Stanford University,1994.
[66] Boyd S, Wu S. A Parser/Solver for Sedidefinite Programs with Matrix Struc-ture[D]. California: Stanford University,1996.
[67] Toh K C, Todd M J. A Matlab Software Package for Semidefinite Programming[J].Optimization Methods and Software,1999,11(1):545–581.
[68] Young J C, Rasmussen B P. Stable Controller Interpolation for LPV Sys-tem[C]//Proceedings of the American Control Conference. Seattle: Washington,2008:3082–3087.
[69] Zhou K, Khargonekar P P, Stoustrup J, et al. Robust Performace of Systems withStructured Uncertainties in State-space[J]. Automatica,1995,31(2):249–255.
[70] Brandon H, Andrew A. Robust Gain-Scheduled Control[C]//Proceedings of theAmerican Control Conference. Baltimore: Maryland,2010:3075–3081.
[71] Hencey B, Alleyne A G. A Robust Controller intepolation Design Technique[J].IEEE Transaction on Control Systems Technology,2010,18(1):1–10.
[72] Murty K G, Kahadi S N. Some NP-complete Problems in Quadratic and NonlinearProgramming[J]. Mathematical Programming,1987,39(2):117–129.
[73] Powers V, Wormann T. An algorithm for Sums of squares of real polynomials[J].Journal of Pure and Applied Linear Algebra,1998,127(1):99–104.
[74] Shor N Z. Class of Global Minimum Bounds of Polynomial Functions[J]. Cyber-netics,1987,26(3):731–734.
[75]黄琳.稳定性理论[M].北京:北京大学出版社,2001:1–10.
[76] Krstic M, Kanellakopoulos I, Kokotovic P V. Nonlinear and Adaptive ControlDesign[M]. New York: John Wiley and Sons,1995:1–50.
[77] Isidori A. Nonlinear Control Systems[M]. New York: Springer-Verlag,1995:80–90.
[78] Prieur C, Praly L. Uniting Local and Global Controllers[C]//Proceedings of the38th IEEE Conference on Decision and Control. Phoenix: Arizona,1999:1214–1219.
[79] Stengle G. A Nullstelllensatz and a Positivstellensatz in Semialgebraic Geome-try[J]. Mathematische Annalen,1974,207(2):87–97.
[80] Rantzer A, Parrilo P. On Convexity in Stabilization of Nonlinear System-s[C]//Proceedings of the39th IEEE Conference on Decision and Control. Sidney:Australia,2000:2942–2945.
[81] Rantzer A, Ceragioli F. Smooth Blending of Nonlinear Controllers using DensityFunctions[C]//Proceedings of the European Control Conference. Porto: Portugal,2001:2851–2853.
[82] Khalil H K. Nonlinear Systems[M]. New Jersey: Prentice Hall,2001:49–89.
[83] Kocvara M, Stingl M. PENBMI User’s Guide[M]. online: http://www.penopt.com,2005:Version2.0.
[84] Shuster M D. A Survey of Attitude Representations[J]. The Journal of Astronauti-cal Sciences,1993,41(4):439–517.
[85]周丽明.饱和控制系统理论及应用研究[D].哈尔滨:哈尔滨工程大学,2009:79–80.
[86] Ka¨llstro¨m C G,A stro¨m K J, Thorell N E, et al. Adaptive Autopilots for Tankers[J].Automatica,1979,15:241–254.
[87] Amerongen J. Adaptive Steering of Ships-A Model Reference Approach[J]. Au-tomatica,1984,20(1):3–14.
[88] Kahveci N E, Ioannou P A. Adaptive Steering Control for Uncertain Ship Dynam-ics and Stability Analysis[J]. Automatica,2013,49(1):685–697.
[89] Witkowska A, Tomera M, Smierschalski R. A backstipping Approach to ShipCourse Control[J]. International Journal of Applied Mathematics and ComputerScience,2007,17(1):73–85.
[90]贾欣乐,杨盐生.船舶运动数学模型-机理建模[M].大连:大连海事大学出版社,1999:35–55.
[91]隋江华. AND—OR模糊神经网络研究及在船舶控制中的应用[D].大连:大连海事大学博士论文,2006:78–79.
[92] Nomoto K, Taguchi T, Honda K, et al. On the Steering Quality of Ships[J]. Inter-national Shipbuilding Progress,1957,35(5):354–370.
[93]张显库,贾欣乐,刘川.响应型船舶运动数学模型的构造[J].大连海事大学学报,2004,30(1):18–21.
[94] Amerongen J. Adaptive steering of ship-a model reference approach to improvedmanoeuvering and economical course keeping[D]. Holand: Delft University ofTechnology,1982.
[95] Yang Y S, Zhou C J. Adaptive Fuzzy Control of Ship Autopilots with UncertainNonlinear Systems[C]//Proceedings of the2004IEEE Conference on Cyberneticsand Intelligent Systems. Singapore: Singapore,2004:1323–1328.
[96]杨盐生,贾欣乐.不确定系统的鲁棒控制及其应用[M].大连:大连海事大学出版社,2003:149–179.
[97] Han C S, Zhang G J, Wu L G, et al. Sliding Mode Control of T-S Fuzzy DescriptorSystems with Time-delay[J]. Journal of the Franklin Institute,2012,349(4):1430–1444.
[98] Ting H C, Chang J L, Chen Y P. Output Feedback Integral Sliding Mode Controllerof Time-delay Systems with Mismatch Disturbances[J]. Asian Journal of Control,2012,14(1):85–94.
[99] Zhao D Y, Li S Y, Zhu Q M. Output Feedback Terminal Sliding Mode Control fora Class of Second Order Nonlinear Systems[J]. Asian Journal of Control,2013,15(1):237–247.
[100] AL-Hadithi B M, Jimnez A, Matla F. Variable Structure Control with Chatter-ing Reduction of a Generalized T-S Model[J]. Asian Journal of Control,2013,15(1):155–168.
[101]杨盐生.不确定系统的鲁棒控制及其在船舶运动控制中的应用[D].大连:大连海事大学博士论文,1999:175–178.
[102]金鸿章.减摇鳍变参数最优控制器及其设计[J].中国造船,1992,119(4):12–17.
[103]杨盐生.船舶减摇鳍非线性系统的变结构自适应鲁棒控制[J].大连海事大学学报,2000,26(4):1–5.
[104]杨盐生,贾欣乐.不确定非线性系统变结构鲁棒自适应控制及其应用[J].大连海事大学学报,1998,24(4):1–8.
[105] Zhang X K, Wang K F. Choas of Ships Rolling Motions and Its Nonlinear Simpleand Direct Control[J]. Shipbuilding of China,2010,37:21–27.
[106] Nayfeh A H, Sanchez N E. Stability and Complicated Rolling Responses of Shipin Regular Beam Sea[J]. Shipbuilding Progress,1990,37(410):177–198.
[2] Jarvis-Wloszek Z, Feeley R, Tan W. Some Controls Applications of Sum ofSquares Programming[C]//Proceedings of the42nd IEEE Conference on Decisionand Control. Maui: Hawaii,2003:4676–4681.
[3] Prajna S, Parrilo P A, Rantzer A. Nonlinear Control Synthesis by Convex Opti-mization[J]. IEEE Transactions on Automatic Control,2004,49(2):310–314.
[4] Rantzer A. A Dual to Lyapunov’s Stability Theorem[J]. System＆Control Letters,2001,42(3):161–168.
[5] Ataei A, Wang Q. Nonlinear Control of an Uncertain Hypersonic Aircraft Modelusing Robust Sum-of-squares Method[J]. IET Control Theory and Applications,2011,6(2):203–215.
[6]童长飞.基于半定规划的多项式非线性系统镇定控制研究[D].杭州:浙江大学,2008.
[7] Jarvis-Wloszek Z, Feeley R, Tan W, et al. Control Applications of Sum of SquaresProgrmming[M]. Berlin Heideberg: Springer-Verlag,2005:3–22.
[8]王佳,曾鸣,苏宝库.基于平方和的卫星大角度姿态机动非线性H∞控制[J].系统工程与电子技术,2010,32(5):1024–1028.
[9] Chesi G. On the Gap between Positive Polynomials and SOS of Polynomials[J].IEEE Transactions on Automatic Control,2007,52(6):1066–1072.
[10] Prajna S. Optimization-Based Methods for Nonlinear and Hybrid Systems Verifi-cation[D]. Pasadena: California Institute of Technology,2005.
[11] Papachristodoulou A. Scalable Analysis of Nonlinear Systems Using Convex Op-timization[D]. Pasadena: California Institute of Technology,2005.
[12] Jarvis-Wloszek Z. Lyapunov based Analysis and Controller Synthesis for Poly-nomial Systems using Sum-of-squares Optimization[D]. Berkeley: University ofCalifornia,2003.
[13] Tan W. Nonlinear Control Analysis and Synthesis using Sum-of-Squares Program-ming[D]. Berkeley: University of California,2006.
[14] Zheng Q. Sum of Squares Based Nonlinear Control Design Techniques[D].Raleigh: North Carolina State University,2009.
[15] Prajna S, Papachristodoulou A, Parrilo P A. Introducing SOSTOOLS: A GeneralPurpose Sum of Squares Programming solver[C]//Proceedings of the41st Confer-ence on Decision and Control. Las Vegas: Nevada,2002:741–746.
[16] Prajna S, Papachristodoulou A, Seiler P, et al. SOSTOOLS: Sum of Squares Opti-mization Toolbox for MATLAB[M]. Pasadena: California Institute of Technology,2004:1–42.
[17] Toh K C, Todd M J, Tutuncu R H. SDPT3-A Matlab Software Package for Semidef-inite Programming[J]. Optimization Methods and Software,1999,11(1):545–581.
[18] Sturm J F. Using SeduMi1.02, a Matlab Toolbox for Optimization over SymmetricCones[J]. Optimization Methods and Software,1999,11(1):625–653.
[19] Tan W, Packard A. Seaching for Control Lyapunov Functions using Sums ofSquares Programming[J].2004:1–10. http://www.ie.nthu.edu.tw/info/ie.newie.htm(Big5).
[20] Papachristodoulou A, Prajna S. On the Construction of Lyapunov Functions usingthe Sum of Squares Decomposition[C]//Proceedings of the41st IEEE Conferenceon Decision and Control. Las Vegas: Nevada,2002:3482–3487.
[21] Prajna S, Papachristodoulou A, Wu F. Nonlinear Control Synthesis by Sumof Squares Optimization: A Lyapunov-base Approach[C]//Proceedings of the5thAsian Control Conference. Melbourne: Australia,2004:157–165.
[22] Prajna S, Papachristodoulou A, Seiler P. New Developments in Sum of SquaresOptimization and SOSTOOLS[C]//Proceedings of the American Control Confer-ence. Boston: Massachusetts,2004:5606–5611.
[23] Prajna S. Barrier certificates for Nonlinear Model Validation[J]. Automatica,2006,42(1):117–126.
[24] Prajna S, Papachristodoulou A. Analysis of Switched and Hybrid Systems-BeyondPiecewise Quadratic Methods[C]//Proceedings of the American Control Confer-ence. Pasadena: California,2003:2779–2784.
[25] Papachristodoulou A, Parrilo P A. A Tutorial on Sum of Squares Techniques forSystems Analysis[C]//Proceedings of the American Control Conference. Portland:California,2005:2686–2700.
[26] Papachristodoulou A. Analysis of Nonlinear Time-delay Systems using the Sumof Squares Decomposition[C]//Proceedings of the American Control Conference.Boston: Massachusetts,2004:4153–4158.
[27] Papachristodoulou A. Robust Stabilization of Nonlinear Time Delay Systems Us-ing Convex Optimization[C]//Proceedings of the44th IEEE Conference on Deci-sion and Control and2005European Control Conference, CDC-ECC’05. Seville:Spain,2005:5788–5793.
[28] Papachristodoulou A, Doyle J C, Low S H. Analysis of Nonlinear Time-delaySystems using the Sum of Squares Decomposition[C]//Proceedings of the43rdIEEE Conference on Decision and Control. Nassau: Germany,2004:4684–4689.
[29] Papachristodoulou A, Papageorgiou C. Robust Stability and PerformanceAnalysis of a Longitudinal Aircraft Model using Sum of Squares Tech-niques[C]//Proceedings of the IEEE International Symposium on Intelligent Con-trol. Limassol: Cyprus,2005:1281–1286.
[30] Papachristodoulou A, Peet M, Lall S. Constructing Lyapunov-Krasovskii Func-tionals for Linear Time Delay Systems[C]//Proceedings of the American ControlConference. Portland: Oregon,2005:2845–2850.
[31] Tan W, Packard A. Stability Region Analysis using Polynomial and CompositePolynomial Lyapunov Functions and Sum of Squares Programming[J]. IEEE Tran-s. Autom.Control,2008,53(2):565–571.
[32] Topcu U, Packard A, Seiler P. Local Stability Analysis using Simulations andSum-of-squares Programming[J]. Automatica,2008,44(10):2669–2675.
[33] EI-Samad H, Prajna S, Papachristodoulou A, et al. Model Validation and Ro-bust Stability Analysis of the Bacterial Heat Shock Response using SOSTOOL-S[C]//Proceedings of the42nd IEEE Conference on Decision and Control. Maui:Hawaii,2003:3766–3771.
[34] EI-Samad H, Prajna S, Papachristodoulou A, et al. Advanced Methods and Al-gorithms for Biological Networks Analsis[J]. Proceedings of the IEEE,2006,94(4):832–853.
[35] Ebenbauer C, Raf T, Allgower F. A Duality-based LPV Approach to Polynomi-al State Feedback Design[C]//Proceedings of the American Control Conference.Portland: Oregon,2005:703–708.
[36] Ebenbauer C, Ren Z, Allgower F. Polynomial Feedback and Observer Design us-ing Nonquadratic Lyapunov Functions[C]//Proceedings of the44th IEEE Confer-ence on Decision and Control, and European Control Conference. Seville: Spain,2005:7587–7592.
[37] Ebenbauer C, Allgower F. Analysis and Design of Polynomial Control Systemsusing Dissipation Inequalities and Sum of Squares[J]. Computers and ChemicalEngineering,2006,30(10-12):1590–1602.
[38] Ebenbauer C, Allgower F. Stability Analysis of Constrained Control Systems: AnAlternative Approach[J]. Systems＆Control Letters,2007,56(2):93–98.
[39] Krishnaswamy K, Papageorgiou G, Glavaski S, et al. Analysis of AircraftPitch Axis Stability Augmentation System using Sum of Squares Optimiza-tion[C]//Proceedings of the American Control Conference. Portland: Oregon,2005:1497–1502.
[40] Ataei A, Wang Q. Nonlinear Control Design of a Hypersonic Aircraft using Sum-of-squares Method[C]//Proceedings of the American Control Conference. NewYork: USA,2007:5287–5283.
[41] Nauta K M, Weiland S, Backx A C P, et al. Approximation of Fast Dynamicsin Kinetic Networks using Non-negative Polynomials[C]//16th IEEE InternationalConference on Control Applications part of IEEE Multi-Conference on Systemsand Control. Singapore: Singapore,2007:1144–1149.
[42] Sontag E D, Ebenbauer R, Findeisen F, et al. Nonlinear Model Predictive Controland Sum of Squares Techniques[M]. New York: Springer-Verlag,2006:325–344.
[43] Franze G, Casavola A, Famularo D, et al. An of-line Formulation of RecedingHorizon Control for Nonlinear Systems using Sum of Squares[M]. New York:Springer-Verlag,2009:491–499.
[44] Pozo F, Rodellar J. Robust Stabilization of Polynomial Systems with UncertainParameters[J]. International Journal of System Science,2010,41(5):575–584.
[45] Fazel M, Chiang M. Network Utility Maximization with Nonconcave Utilities us-ing Sum-of-Squares Metnod[C]//Proceedings of the44th IEEE Conference on De-cision and Control, and European Control Conference. Seville: Spain,2005:1867–1874.
[46] Yu J, Yan Z, Wang J, et al. Robust Stabilization of Ship Course Via Convex Opti-mization[J]. Asia Journal of Control,2013,5(9):1666–1675.
[47] Henrion D, Lofberg J. Solving Nonconvex Optimization Problems[J]. IEEE Con-trol Systems,2004,24(3):72–83.
[48] Biswas P, Grieder P, Lofberg J, et al. A Survey on Stability Analysis of Discrete-time Piecewise Afne Systems[C]//Proceedings of the16th IFAC World Congress.Prague: Czechoslovakia,2005:331–331.
[49] Johansson M, Rantzer A. Computation of Piecewise Quadratic Lyapunov Func-tions for Hybrid Systems[J]. IEEE Transactions on Automatic Control,1998,43(4):555–559.
[50] Prajna S. Barrier Certificates for Nonlinear Model Validation[C]//Proceedings ofthe42nd IEEE Conference on Decision and Control. Maui: Hawaii,2003:2884–2889.
[51] Prajna S, Jadbabaie A. Safety Verification of Hybrid Systems using Barrier Cer-tificates[M]. New York: Springer-Verlag,2004:477–492.
[52] Prempain E. An Application of the Sum of Squares Decomposition to the L2Gain Computation for a Class of Nonlinear Systems[C]//Proceedings of the44thIEEE Conference on Decision and Control,and the European Control Conference.Seville: Oregon,2005:6865–6868.
[53] Ichihara H, Nobuyama E. L2Gain Analysis and Control Design of Linear Systemswith Input Saturation Using Matrix Sum of Squares Relaxation[C]//Proceedings ofthe American Control Conference. Minneapolis: Minnesota,2006:3837–3842.
[54] Wu F, Yang X H, Pzckard A, et al. Induced L2Norm Control for LPV Systemswith Bounded Parameter Variation Rates[J]. International Journal of Robust andNonlinear Control,1996,6(9):983–998.
[55] Khalil H K. Nonlinear Systems[M]. London: Prentice Hall,1996:261–284.
[56] Fujisaki Y, Miyoshi E. Controller Redesign via Sum of Squares Program-ming[C]//Proceedings of SICE-ICASE International Joint Conference. Busan: Ko-rea,2006:4012–4015.
[57] Monzon P. On Necessary Conditions for Almost Global Stability[J]. IEEE Trans-actions On Automatic Control,2003,48(4):631–634.
[58] Angeli D. Some Remarks on Density Functions for Dual Lyapunov Method-s[C]//Proceedings of the42nd IEEE Conference on Decision and Control. Maui:Hawaii,2003:5080–5082.
[59] Prajna S, Rantzer A. On Homogeneous Density Functions[M]. Berlin: SpringerVerlag,2003:261–274.
[60] Monzon P. Almost Global Attraction in Planar Systems[J]. Systems＆ControlLetters,2005,54(1):753–758.
[61] Rantzer A. A Converse Theorem for Density Functions[C]//Proceedings of the41stIEEE Conference on Decision and Control. Las Vegas: Nevada,2002:1890–1891.
[62] Monzon P, Potrie R. Local and Global Aspects of Almost Global Stabili-ty[C]//Proceedings of the45th IEEE Conference on Decision and Control. SanDiego: California,2006:5120–5125.
[63] Masubuchi I. Analysis of Positive Invariance and Almost Regional Attractionvia Density Functions with Converse Results[C]//Proceedings of the45th IEEEConference on Decision and Control. San Diego: California,2006:5126–5131.
[64] Potrie R, Monzon P. Local Implications of Almost Global Stability[J]. DynamicalSystems,2009,24(1):109–115.
[65] Boyd S, Vandenberghe L. SP: Software for Sedidefinite Programming[D]. Cali-fornia: Stanford University,1994.
[66] Boyd S, Wu S. A Parser/Solver for Sedidefinite Programs with Matrix Struc-ture[D]. California: Stanford University,1996.
[67] Toh K C, Todd M J. A Matlab Software Package for Semidefinite Programming[J].Optimization Methods and Software,1999,11(1):545–581.
[68] Young J C, Rasmussen B P. Stable Controller Interpolation for LPV Sys-tem[C]//Proceedings of the American Control Conference. Seattle: Washington,2008:3082–3087.
[69] Zhou K, Khargonekar P P, Stoustrup J, et al. Robust Performace of Systems withStructured Uncertainties in State-space[J]. Automatica,1995,31(2):249–255.
[70] Brandon H, Andrew A. Robust Gain-Scheduled Control[C]//Proceedings of theAmerican Control Conference. Baltimore: Maryland,2010:3075–3081.
[71] Hencey B, Alleyne A G. A Robust Controller intepolation Design Technique[J].IEEE Transaction on Control Systems Technology,2010,18(1):1–10.
[72] Murty K G, Kahadi S N. Some NP-complete Problems in Quadratic and NonlinearProgramming[J]. Mathematical Programming,1987,39(2):117–129.
[73] Powers V, Wormann T. An algorithm for Sums of squares of real polynomials[J].Journal of Pure and Applied Linear Algebra,1998,127(1):99–104.
[74] Shor N Z. Class of Global Minimum Bounds of Polynomial Functions[J]. Cyber-netics,1987,26(3):731–734.
[75]黄琳.稳定性理论[M].北京:北京大学出版社,2001:1–10.
[76] Krstic M, Kanellakopoulos I, Kokotovic P V. Nonlinear and Adaptive ControlDesign[M]. New York: John Wiley and Sons,1995:1–50.
[77] Isidori A. Nonlinear Control Systems[M]. New York: Springer-Verlag,1995:80–90.
[78] Prieur C, Praly L. Uniting Local and Global Controllers[C]//Proceedings of the38th IEEE Conference on Decision and Control. Phoenix: Arizona,1999:1214–1219.
[79] Stengle G. A Nullstelllensatz and a Positivstellensatz in Semialgebraic Geome-try[J]. Mathematische Annalen,1974,207(2):87–97.
[80] Rantzer A, Parrilo P. On Convexity in Stabilization of Nonlinear System-s[C]//Proceedings of the39th IEEE Conference on Decision and Control. Sidney:Australia,2000:2942–2945.
[81] Rantzer A, Ceragioli F. Smooth Blending of Nonlinear Controllers using DensityFunctions[C]//Proceedings of the European Control Conference. Porto: Portugal,2001:2851–2853.
[82] Khalil H K. Nonlinear Systems[M]. New Jersey: Prentice Hall,2001:49–89.
[83] Kocvara M, Stingl M. PENBMI User’s Guide[M]. online: http://www.penopt.com,2005:Version2.0.
[84] Shuster M D. A Survey of Attitude Representations[J]. The Journal of Astronauti-cal Sciences,1993,41(4):439–517.
[85]周丽明.饱和控制系统理论及应用研究[D].哈尔滨:哈尔滨工程大学,2009:79–80.
[86] Ka¨llstro¨m C G,A stro¨m K J, Thorell N E, et al. Adaptive Autopilots for Tankers[J].Automatica,1979,15:241–254.
[87] Amerongen J. Adaptive Steering of Ships-A Model Reference Approach[J]. Au-tomatica,1984,20(1):3–14.
[88] Kahveci N E, Ioannou P A. Adaptive Steering Control for Uncertain Ship Dynam-ics and Stability Analysis[J]. Automatica,2013,49(1):685–697.
[89] Witkowska A, Tomera M, Smierschalski R. A backstipping Approach to ShipCourse Control[J]. International Journal of Applied Mathematics and ComputerScience,2007,17(1):73–85.
[90]贾欣乐,杨盐生.船舶运动数学模型-机理建模[M].大连:大连海事大学出版社,1999:35–55.
[91]隋江华. AND—OR模糊神经网络研究及在船舶控制中的应用[D].大连:大连海事大学博士论文,2006:78–79.
[92] Nomoto K, Taguchi T, Honda K, et al. On the Steering Quality of Ships[J]. Inter-national Shipbuilding Progress,1957,35(5):354–370.
[93]张显库,贾欣乐,刘川.响应型船舶运动数学模型的构造[J].大连海事大学学报,2004,30(1):18–21.
[94] Amerongen J. Adaptive steering of ship-a model reference approach to improvedmanoeuvering and economical course keeping[D]. Holand: Delft University ofTechnology,1982.
[95] Yang Y S, Zhou C J. Adaptive Fuzzy Control of Ship Autopilots with UncertainNonlinear Systems[C]//Proceedings of the2004IEEE Conference on Cyberneticsand Intelligent Systems. Singapore: Singapore,2004:1323–1328.
[96]杨盐生,贾欣乐.不确定系统的鲁棒控制及其应用[M].大连:大连海事大学出版社,2003:149–179.
[97] Han C S, Zhang G J, Wu L G, et al. Sliding Mode Control of T-S Fuzzy DescriptorSystems with Time-delay[J]. Journal of the Franklin Institute,2012,349(4):1430–1444.
[98] Ting H C, Chang J L, Chen Y P. Output Feedback Integral Sliding Mode Controllerof Time-delay Systems with Mismatch Disturbances[J]. Asian Journal of Control,2012,14(1):85–94.
[99] Zhao D Y, Li S Y, Zhu Q M. Output Feedback Terminal Sliding Mode Control fora Class of Second Order Nonlinear Systems[J]. Asian Journal of Control,2013,15(1):237–247.
[100] AL-Hadithi B M, Jimnez A, Matla F. Variable Structure Control with Chatter-ing Reduction of a Generalized T-S Model[J]. Asian Journal of Control,2013,15(1):155–168.
[101]杨盐生.不确定系统的鲁棒控制及其在船舶运动控制中的应用[D].大连:大连海事大学博士论文,1999:175–178.
[102]金鸿章.减摇鳍变参数最优控制器及其设计[J].中国造船,1992,119(4):12–17.
[103]杨盐生.船舶减摇鳍非线性系统的变结构自适应鲁棒控制[J].大连海事大学学报,2000,26(4):1–5.
[104]杨盐生,贾欣乐.不确定非线性系统变结构鲁棒自适应控制及其应用[J].大连海事大学学报,1998,24(4):1–8.
[105] Zhang X K, Wang K F. Choas of Ships Rolling Motions and Its Nonlinear Simpleand Direct Control[J]. Shipbuilding of China,2010,37:21–27.
[106] Nayfeh A H, Sanchez N E. Stability and Complicated Rolling Responses of Shipin Regular Beam Sea[J]. Shipbuilding Progress,1990,37(410):177–198.
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