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

当面对一个具有未知参数的非线性动态系统的控制问题时，很容易计算一族线性化近似，推导出每个工作点附近的LQ、H-无穷或Mu控制器，然后将所有局部稳定的补偿器“缝合”在一个全局工作方案中。一种这样的方法最近被称为线性参数变(LPV)控制，其中参数是未知的动态，但被约束在有界集合中，其中它们的变化应该足够慢，以确保自适应方案的稳定性。受传递轨道跟踪等问题的启发，该方案发展了所谓的线性动态变化(LDV)方法，该方法对参数进行动态建模，将参数变化的动力学融入到线性LQ或H无穷大设计中，从而得到不需要局部稳定但保证全局稳定的设计。数学上，以离散时间为单位(代表连续时间)，方法的特征是泛函(代表。与传统LPV理论中的“状态依赖”的Riccati方程形成鲜明对比的是偏微分)Riccati方程和线性矩阵不等式。主要关注的是参数在紧集上运行的情况，这很自然地赋予了LDV系统遍历性质。遍历理论被用来开发一种求解泛函Riccati方程的计算方案，该方案主要依赖于Poincare递推方案。文中还提出了其他不动点方法、连续选择方法、可微选择方法和Leray-Schauder次数方法。展示设计遍历性的一个例子是泛函Riccati方程的“自相似”解。接下来，虽然LPV方法关注的是在欧氏空间的子集上运行的参数，其中参考轴是理所当然的，但另一方面，LDV方法关注运行在非平凡(不可压缩)流形上的参数，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化的，即在与流形相切的空间中存在光滑的正交化参照系，并由此写出线性化的状态空间方程。在状态流形不可并行化的情况下，找到LDV系统在其上运行的可并行覆盖流形的指导思想。最后，提出了一种利用高余维叶化的God十亿-Vey特征类对这类控制问题进行分类的尝试。最后，提出了用Gelfand-Feigin-Fuks特征类变异理论来衡量系统的鲁棒性和控制权威性。***

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. ***