【作者】 祝庚；

【导师】 孙振东；

【作者基本信息】 华南理工大学 ， 控制理论与控制工程， 2012， 博士

【摘要】 切换系统作为一类特殊的混合系统，它是由若干个子系统和协调子系统之间进行切换的切换律组成。因为理论和应用的价值，切换系统越来越受到研究者的重视，它在许多的工业领域也有广泛的应用和发展潜力。共同Lyapunov函数、二次型Lyapunov函数、多Lyapunov函数和切换Lyapunov函数均是分析切换系统稳定性、可镇定性及性能优化的重要工具。由于可渐近镇定的切换系统必定存在一个切换Lyapunov函数，寻找合适的切换Lyapunov函数和设计基于该切换Lyapunov函数下的状态反馈切换律是一项有意义的工作。可渐近镇定的离散时间切换线性系统存在一个非凸的最小二次型切换Lyapunov函数，利用代价函数作用下得到的这类切换Lyapunov函数，可以设计出离散时间切换系统的一个最优状态反馈切换律。如果连续时间切换线性系统是可渐近镇定的，那么存在若干个分段可压缩切换路径θ_i及其对应收缩域Ω_i，并且可以利用它们设计出一个分段状态反馈切换律∧_i=1^kθ_i^Ω_i来渐近镇定该切换线性系统。为了优化连续时间切换系统在无穷时间上的代价函数，作者在论文中开展了下面的工作：一、研究有限时间段上的梯度优化条件和基于Hamilton函数的优化方法，提出基于Armijo步长的共轭梯度搜索方法来寻找分段可压缩切换路径对应的分段优化切换路径θ_i^o。为了确保分段优化切换路径也是可压缩的，提出了代价函数中矩阵特征值约束的充分条件。二、根据优化后的分段可压缩切换路径生成一个聚合系统（离散时间切换系统），利用切换Riccati映射生成非凸的最小二次型切换Lyapunov函数V_k（x）=min{x^TPx:P∈Z_k}。由于计算过程中产生的矩阵集Z k的元素数量很多，并且大量是冗余的，为了减少冗余的矩阵，剔除算法将被提出和使用。根据切换Lyapunov函数V_K（x）θo生成θ_i^o对应的优化切换收缩域Ω_i^o，利用它们可以设计出一个优化的分段可压缩切换路径和对应的状态反馈切换律∧_i=1^ko（θ_i^o）^Ω_io，其中k^o为优化后收缩域的个数。三、利用优化的状态反馈切换律∧_i=1^ko（θ_i^o）^Ω_io指导聚合系统的最优代价切换，以及对应连续时间切换系统的次优代价切换。为了使得代价误差更小一些，k^o取的值要大一些，但是仿真程序运行的时间也会更久。为了克服这些不足，提出了另一种同步优化的状态反馈切换律。论文围绕切换Lyapunov函数的分析和计算、分段可压缩切换路径的计算和优化，将切换线性系统在有限时间段的性能优化扩充到其无限时间段上的代价函数优化。突破了以往无限时间段上性能优化的计算瓶颈，利用计算机程序产生切换Lyapunov函数和状态反馈切换律，也在一定程度上回避了复杂的理论分析和研究。文中给出了一些仿真实例，实验数据表明基于分段可压缩切换路径和聚合系统的优化思路及相关结论的正确性。更多还原

【Abstract】 A switched system is an especial class of hybrid systems. It consists of severalsubsystems and a rule that orchestrates the switching between them. For theory andapplications reasons, the switched system has been attacting more and more interests ofinvestigators. It has been used widely in many industry fields and has some potentialdevelopments. Common Lyapunov functions, multiple Lyapunov functions, quadraticLyapunov functions and switched Lyapunov functions are important tools for stability,stabilization and performance optimization of switched systems.If the switched system is asymptotically stabilizable, then there is a switched Lyapunovfunction. It is a significative work to search a proper switched Lyapunov function and designa state-feedback switching law based on the switched Lyapunov function. If a discrete-timeswitched linear system is asymptotically stabilizable, then there is a nonconvex minimumquadratic Lyapunov function. Uing the switched Lyapunov function based on cost function,we can design an optimal state-feedback switching law.When a continuous-time switched linear system is asymptotically stabilizable, severalpiecewise contractive switching pathsθ_i and contractive regionsΩ_ican be found, and apiecewise state-feedback switching law called by∧_i=1^kθ_i^Ω_ican be designed to stabilize thesystem. To optimize the infinite horizon cost function of continuous-time switched systems, inthis work we do the following jobs.Firstly, we investigate a gradient optimization condition on finite interval and methodsbased on Hamilton functions, and present a conjugate gradient algorithm with armijo steps tosearch the optimal piecewise switching pathsθ_i^o. To ensure that the optimal piecewiseswitching paths are also contractive, we present a sufficient condition about restrictedeigenvalues of cost function.Secondly, an aggregated system derived from optimal piecewise switching paths is adiscrete-time switched system. A nonconvex minimum quadratic Lyapunov functionV_k（x）=min{x^TPx:P∈Z_k}can be found by a switched Riccati mapping. We can outline apruning procedure for removing the redundant elements from setsZ_kand obtaining theequivalent subsets with smaller cardinalities. An optimal piecewise state-feedback switchinglaw called by∧_i=1^ko（θ_i^o）^Ω_iocan be designed by these optimal pathsθ_i^oand contractive regionsΩ oiderived from V_k（x）, where k^o is the number of optimal contractive regions.Thirdly, the optimal state-feedback switching law∧_i=1^ko（θ_i^o）^Ω_ioguides an optimal costswitching of the aggregated system and a sub-optimal switching of the correspondingcontinuous-time switched system. To attain a smaller cost error, we use a biggerk^o, but therunning time of the procedure is longer. Another synchronous state-feedback switching law isused to overcome these demerits.The switched Lyapunov functions and piecewise contractive switching paths are analyzedand computed. The performance optimization of switched linear systems is extended formfinite interval to infinite horizon. To break the computational bottleneck, the procedures bringa switched Lyapunov function and a state-feedback switching law, and avoid the complicatedtheoretical analysis. Some simulated examples are shown, and the experiment data validatesthe optimal idea based on piecewise contractive switching path and aggregated system, andcorresponding conclusions.更多还原