Ergodic and Topological Aspects of Linear Dynamically Varying (LDV) Control

线性动态变化 (LDV) 控制的遍历和拓扑方面

• 批准号: 9802594
• 负责人: Edmond Jonckheere
• 金额: $ 20万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-09-01 至 2002-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802594&HistoricalAwards=false
• 关键词: Ergodic; Topological; Aspects; Linear; Dynamically

9802594JonckheereWhen confronted with the problem of controlling a nonlinear dynamical system with unknown parameters, it is tempting to compute a family of linearized approximations, derive the LQ, H-infinity or Mu controller around each operating point, and then "stitch together" all of the locally stabilizing compensators in a globally working scheme. One such approach has recently been referred to as Linear Parametrically Varying (LPV) control where the parameters are of unknown dynamics, but constrained to lie in a bounded set, in which their variation should be slow enough to ensure stability of the adaptive scheme. Motivated by problems as tracking of transitive orbits, this proposal develops the so-called Linear Dynamically Varying (LDV approach, in which the parameters are dynamically modeled, the dynamics of the variation of the parameters is incorporated in the linear LQ or H-infinity design, resulting in a design that need not be locally stable, but that is guaranteed to be globally stable. Mathematically, in discrete-time (rep. continuous time), approach is characterized by functional (rep. partial differential) Riccati equations and linear matrix inequalities in sharp contrast with the "state dependent" Riccati equation of the traditional LPV theory. The major focus of attention is on the case of parameters running in a compact set and this quite naturally endows the LDV svstem with ergodic properties. Ergodic theory is "put to work" to develop a computational scheme for solving functional Riccati equations that relies crucially on the Poincare recurrence scheme. Other fixed point, continuous and differentiable selections, and Leray-Schauder degree methods are proposed as well. An example of the manifestation of the ergodic properties of the design is a "self-similar" solution to the functional Riccati equation. Next, while the LPV approach has focused on parameters running in a subset of the Euclidean space where the reference axes are taken for granted, the LDV approach on the other hand focuses on parameters running on a nontrivial (incontractible) manifold, and existence of the reference axes cannot be taken for granted. The relevant global topological property is parallelizability, that is, existence of a smooth orthonormal reference frame in the tangent space to the manifold, relative to which the linearlized state space equation are written. In case the state manifold is not parallelizable, the guiding idea to find parallelizable covering manifold on which the LDV system runs. Finally, an attempt to classifly these kind of control problems using the Godbillion-Vey characteristic classes of higher-codimenional foliation is proposed. Finally, the Gelfand-Feigin-Fuks theory of variation of characteristic classes is proposed to measure robustness and control authority. ***

9802594Jonckheere当面临控制具有未知参数的非线性动态系统的问题时，很容易计算一系列线性化近似，导出每个操作点周围的LQ、H ∞或Mu控制器，然后将所有局部稳定补偿器"缝合"在一起，形成全局工作方案。 一种这样的方法最近被称为线性参数变化（LPV）控制，其中参数是未知动态的，但被约束为位于有界集合中，其中它们的变化应该足够慢以确保自适应方案的稳定性。 基于传递轨道的跟踪问题，提出了一种线性动态变化（LDV）方法，该方法对参数进行动态建模，将参数变化的动态特性引入线性LQ或H ∞设计，使得设计不需要局部稳定，但保证全局稳定。 在数学上，在离散时间（代表连续时间），方法的特点是功能（代表偏微分）Riccati方程和线性矩阵不等式与传统的LPV理论的“状态相关”Riccati方程形成鲜明对比。 人们关注的主要焦点是参数在紧凑集中运行的情况，这很自然地赋予LDV系统遍历属性。 遍历理论是“投入工作”，以制定一个计算方案，解决功能Riccati方程，关键依赖于庞加莱递归计划。 其他不动点，连续和可微的选择，以及Leray-Schauder度的方法也被提出。 一个例子的表现形式的遍历性的设计是一个“自相似”的解决方案的功能黎卡提方程。 接下来，虽然LPV方法专注于在欧几里得空间的子集中运行的参数，其中参考轴被认为是理所当然的，但另一方面，LDV方法专注于在非平凡（不可收缩）流形上运行的参数，并且参考轴的存在不能被认为是理所当然的。 相关的全局拓扑性质是可并行性，即在流形的切空间中存在一个光滑的标准正交参考系，线性化的状态空间方程相对于该参考系被写。 在状态流形不可并行的情况下，给出了寻找LDV系统运行于其上的可并行覆盖流形的指导思想。 最后，尝试分类这类控制问题使用Godbillion-Vey特征类的高余维叶理。 最后，Gelfand-Feigin-Fuks特征类变分理论被提出来衡量鲁棒性和控制权威性。 ***