Mathematical Sciences: Nonlinear Control: Feedback Stabilization and Cardiac Arrhythmia Control

数学科学：非线性控制：反馈稳定和心律失常控制

• 批准号: 9530973
• 负责人: Henry Hermes
• 金额: $ 2.14万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 1998-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9530973&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Control; Feedback

ABSTRACT Proposal: DMS-9530973 PI: Hermes Hermes proposes to continue his research program on the problem of existence and construction of asymptotically stabilizing feedback control (ASFC) for affine control systems, in particular via the use of high order, homogeneous, approximations. The dilations and local coordinates relative to which the approximating systems are constructed can be naturally obtained from the Lie algebra structure of the vector fields defining the original system. Also, an ASFC of appropriate homogeneous order for the approximating system is a local ASFC for the original system. The method will be to extend the classical linear, quadratic, regulator technique to a nonlinear, homogeneous regulator. A first goal is to classify homogeneous, affine, systems which admit a smooth ASFC. Linear, controllable, approximating systems are divided into equivalence classes via the action of the linear feedback group, and canonical representatives of each equivalence class are well known. Hermes proposes tho study the equivalence classes of small time, or large time, locally controllable homogeneous approximations having known smooth ASFC, under the action of a homogeneity preserving group of transformations, and to find canonical representatives of the equivalence classes. The odd power integrators provide an example of such. The above topics have implications to questions of regularity conditions for viscosity solutions of Hamilton-Jacobi-Bellman equations, another topic to be studied. Many mathematical systems, e.g., an inverted compound pendulum, a robotic arm, a high performance aircraft, a communications satellite, either are, or are designed to be, unstable in their uncontrolled mode of operation. Stability of the system is achieved via the introduction of external controls. In past decades, construction of stabilizing controllers was achieved via the use of linear approximations, which had to be "controllable" for the method to work. Many space age prob lems are basically nonlinear, have uncontrollable linear approximations, and their analysis requires high order approximations which retain relevant nonlinear behavior but are still amenable to analysis. The high order, homogeneous approximations are such. The goal of Hermes research is to use these for the construction of stabilizing controls.

摘要提案：DMS-9530973 PI：爱马仕 爱马仕建议继续他的研究计划的问题的存在和建设的渐近稳定反馈控制（ASFC）的仿射控制系统，特别是通过使用高阶，齐次，近似。 构造近似系统时所依据的伸缩和局部坐标可以从定义原系统的向量场的李代数结构中自然地得到。 此外，近似系统的适当齐次阶的ASFC是原系统的局部ASFC。 该方法将是扩展经典的 线性，二次，调节器技术的非线性，齐次调节器。 第一个目标是分类同质，仿射，系统承认一个光滑的ASFC。 线性可控近似系统 通过线性反馈群的作用被分成等价类，并且每个等价类的典型代表是众所周知的。 爱马仕提出研究具有已知光滑ASFC的小时间或大时间局部可控齐次逼近在保齐性变换群作用下的等价类，并找到等价类的规范表示. 奇数功率积分器提供了这样的示例。 上述问题对Hamilton-Jacobi-Bellman方程粘性解的正则性条件问题也有一定的意义，这是另一个有待研究的问题。 许多数学系统，例如，倒立复摆、机械臂、高性能飞行器、通信卫星，在它们的不受控模式下是不稳定的，或者被设计成不稳定的 的操作。 系统的稳定性通过引入外部控制来实现。 在过去的几十年里，稳定控制器的构造是通过使用线性近似来实现的， 是“可控”的方法来工作。 许多空间时代的问题基本上是非线性的，有不可控的线性近似，他们的分析需要高阶近似，保留相关的非线性行为，但仍然适合分析。 高阶齐次近似就是这样。 爱马仕研究的目标是利用这些来构造稳定控制。