摘要
执行器饱和特性广泛地存在于现实控制系统中,但传统的控制理论又常漠视之,直至发生多起灾难事故后才引起学者关注。饱和是硬非线性,具有不光滑特性,它的加入增加了系统分析与设计的复杂性。目前对于执行器幅度饱和的系统研究已取得了许多成果,但对于更难于分析的执行器幅度与速率饱和的系统研究工作却很少。在实际工程中,除了系统存在的执行器饱和外,还存在由于系统建模参数及运行环境的变化等一系列因素造成的系统模型不确定,以及当被控对象进行大范围运动时,系统模型存在的光滑非线性,这些实际问题在进行控制系统设计时都应予以相应的考虑。由于船舶航向控制的操舵系统存在幅度与速率饱和,进行控制器设计时必须给予考虑和很好的处理。然而目前已有的船舶航向控制器设计结果中,一直没有给出很好解决此问题的办法。
对于存在执行器饱和的系统,全局稳定是很难得到保证的,大部分工作集中在系统局部分析,因此本文针对含执行器幅度饱和的线性系统,改进了局部鲁棒稳定性和性能的分析方法,并针对具有特殊形式的含执行器幅度与速率饱和的系统,完善了现有的抗饱和控制理论,从理论上保证了含抗饱和补偿器的闭环系统的代数环良定性,即系统动力学方程解的存在性及唯一性。采用的主要处理手段有：饱和特性的凸包处理方法,结果具有更低的保守性；借助椭球体和参考集估计系统吸引域的大小；采用线性微分包含处理系统模型中的光滑不确定非线性项；通过引入幅度与速率饱和(MRS)模型和状态变量增广法处理了执行器的幅度与速率饱和；且将所有控制问题转化为线性矩阵不等式(LMI)或双线性矩阵不等式(BMI)约束的(凸)优化问题,便于用MATLAB优化求解。
主要研究内容：
对四类饱和控制系统进行了理论研究并给出相应的结论,且将其中一类系统的理论研究结果应用于船舶航向控制的实际中,很好的解决了航向控制中的舵角与舵角速率饱和问题。
(1).对于幅度饱和的反馈矩阵有摄动的线性系统进行稳定性分析,采用凸包方法处理饱和,基于二次Lyapunov函数,给出了判定闭环系统收缩不变椭球的充分性条件,通过选取多面体参考集,进行最大吸引域估计,且将此问题转化为LMI约束的凸优化问题。
(2).对于幅度饱和的参数不确定线性系统进行保成本控制,用凸包方法处理饱和,基于二次Lyapunov和非二次Lyapunov函数,分别给出保证系统鲁棒稳定与鲁棒性能的充分性条件,且将控制器设计问题分别转化为LMI和／或BMI约束的优化问题。
(3).对于一类幅度与速率饱和且执行器满足一定动力学特性的线性系统进行静态抗饱和控制。通过采用状态变量增广法,将幅度与速率饱和问题转化为幅度饱和问题,基于二次Lyapunov函数,采用多面体微分包含和范数有界微分包含,分别给出系统同时满足稳定性和最佳性能的充分性条件,且将抗饱和补偿器设计问题转化为LMI约束的凸优化问题。
(4).对于一类幅度与速率饱和的不确定非线性系统进行静态抗饱和控制。首先采用线性微分包含处理系统模型中的不确定非线性项；之后通过引入MRS模型,将幅度与速率饱和问题转化为幅度饱和问题。给出了闭环系统代数环良定性的充要条件,并将其化作LMI约束,从而去除了代数环良定性假设。基于二次Lyapunov函数,采用范数有界微分包含,给出了闭环系统同时满足鲁棒稳定与鲁棒性能的充分性条件,且将抗饱和补偿器设计问题转化为LMI约束的凸优化问题。
(5).进行船舶航向保持控制器设计,基于线性船舶运动学模型,采用静态抗饱和控制方法,很好地解决了舵角速率饱和问题。
(6).进行船舶转向控制器设计,基于不确定非线性船舶运动学模型,给出了转向静态抗饱和控制器的设计,通过与转向饱和有限时间控制器仿真比较,更好地说明抗饱和控制器很好地解决了舵角与舵角速率饱和问题,提高了系统的控制性能。
最后是全文总结及展望。
Although saturation characteristics exist extensively in practical control systems, it had been rarely studied in conventional control theory until several disasters happened. As a hard nonlinearity, saturation's non-smooth characteristic makes system analysis and design more complex. As for the systems subject to actuator amplitude saturation, many research results have been recently achieved, however, the system analysis of actuator amplitude and rate saturation still remains harder with less results. In addition to practically encountered actuator saturation, system model uncertainties and smooth non-linearities really exist and should be appropriately taken into consideration, due to the factors such as the variation of the system modeling parameters, the variation of the operating environment and so on. Since there exists amplitude and rate saturation in rudder manipulation systems of steering, the appropriate consideration and management must be paid. However, the present results of ship steering controller design can't provide a perfect solution to the rudder saturation.
It's definitely difficult to achieve the global stability of systems subject to actuator saturation, therefore most work concentrate on system local analysis. In this paper, local stability and performance analysis methods are improved for linear systems subject to actuator amplitude saturation. As for the systems with both amplitude and rate saturation, however, the present anti-windup control theory is inapplicable because of the assumption of the algebraic loop well-posedness of the closed-loop system. The theory is improved in this paper that the closed-loop well-posedness is guaranteed for the special formed systems with anti-windup compensator, therefore, also guaranteed is the existence and uniqueness of the solution to the system dynamics. The tricks of this paper include the convex hull manipulation of saturation characteristics with less conservative results, estimating the system domain of attraction via ellipsoids and the reference sets, manipulating the uncertain smooth nonlinear terms of the system model via linear differential inclusions, introducing the magnitude and rate saturation (MRS) model and state variable augmentation to deal with the actuator amplitude and rate saturation, and finally transforming all the control problems into (convex) optimizations subject to linear matrix inequalities (LMI) and/or bi-linear matrix inequalities (BMI) constraints. All the problems can be sufficiently optimized using MATLAB.
The main research contents are as follows:
Four kinds of saturation control systems are concerned in the theory and the corresponding results are presented. Furthermore, the theory research results of one kind of them are applied to well solve the problem of the rudder amplitude and rate saturation in the ship course control.
(1). For linear systems subject to amplitude saturation and feedback matrix perturbation, the stability is analyzed via convex hull method. The sufficient condition of determining the closed-loop system's contractively invariant ellipsoid is obtained based on the quadratic Lyapunov function. Therefore the maximum domain of attraction is estimated by choosing the reference sets as polytopes and the problem can be transformed into the LMI constrained convex optimization.
(2). For linear systems subject to amplitude saturation and parametric uncertainties, the guaranteed cost control problem is solved via the convex hull method. The sufficient conditions are respectively proposed based on the quadratic and non-quadratic Lyapunov functions, with the closed-loop robust stability and the guaranteed robust performance. Furthermore, the controller synthesis problems are transformed into the LMI and/or BMI constrained convex optimization problems.
(3). For linear systems subject to both amplitude and rate saturation, the static anti-windup control problem is discussed with the actuator dynamics of certain form. The amplitude and rate saturation problem is simplified to the amplitude saturation problem by augmenting the state variables. Under the assumption that the closed-loop system's algebraic loop is well-posed, the sufficient conditions are respectively proposed based on the quadratic Lyapunov function and the polytopic/norm-bounded differential inclusions, with closed-loop stability and better performance. Furthermore, the anti-windup compensator design problems are transformed into the LMI constrained convex optimization problems.
(4). For a class of nonlinear uncertain systems subject to amplitude and rate saturation, the static anti-windup problem is addressed. The first step involves the nonlinear uncertain terms' management using linear differential inclusions and the second, via introducing the MRS model, the transformation of the amplitude and rate saturation problem into the amplitude saturation problem. The key point is that the necessary and sufficient condition of the closed-loop algebraic loop well-posedness is obtained and transformed into the LMI constraint, thus the above-mentioned assumption is removed. By employing the norm-bounded differential inclusions, the sufficient conditions are obtained based on the quadratic Lyapunov function, with the closed-loop robust stability and robust performance. Furthermore, the anti-windup compensator design problems are transformed into the LMI constrained convex optimization problems.
(5). Based on the linear ship motion model and the static anti-windup control theory, the course keeping controller is designed to perfectly solve the problem of the rudder rate saturation.
(6). Based on the nonlinear uncertain ship motion model, the ship course changing static anti-windup controller is designed. By comparison with the saturated finite interval course changing controller, simulation results show that the anti-windup controller achieves better ship course changing performance since it can perfectly solve the problem of the rudder amplitude and rate saturation.
The conclusions and perspectives are given in the end of the paper.
引文
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