【作者】 于家凤；

【导师】 严质彬；

【作者基本信息】 哈尔滨工业大学 ， 控制科学与工程， 2014， 博士

【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。更多还原

【Abstract】 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.更多还原