Polynomial Algebraic Methods for Modeling, Analysis and Control of Distributed Physical Systems

本研究旨在提供基于多变量多项式代数的理论工具和算法，以解决分布式系统的稳定性、建模和控制等具体问题。（也称为“N-D”）系统在考虑其演化不仅取决于时间而且取决于空间的物理现象的建模时自然出现，例如在图像处理中，地震学、电路理论、流控制和迭代学习控制。在这里提出的研究中，我们的目的是研究开放的分布式系统所描述的线性常系数偏差分/微分方程。这种系统由一组线性常系数偏微分/差分方程描述，并且它自然地与N个不定式中的多项式矩阵相关联，在这个意义上，动力系统的属性反映在多项式矩阵的代数属性中。系统动力学和泛函的相互作用在许多应用中是典型的（例如在稳定性理论中，在最优控制中等）。也可以有效地描述多项式矩阵（在2N待定）。使用上述框架，我们的目标是使用多项式代数方法的计算一阶表示，并在物理系统的分析中产生的泛函的研究。在这两个领域，主要研究员和东道研究员的专门知识相当丰富，相互补充。该项目分为两个工作包。1)二维系统一阶表示的简化一阶表示对应于一维情况下的状态空间方程，由于易于更新，对于仿真、滤波等是非常重要的。然而，它们不能被认为是描述系统的起点，而是需要从由较简单的子系统互连组成的高阶模型构建，例如从某些组件的标准模型库中获得。因此，出现了如何计算由一组高阶偏微分或差分方程描述的系统的一阶表示的问题。在这个工作包中，我们的目标是调查分布式系统的一阶表示的计算。在这项调查中至关重要的是状态映射的概念，在1-D的情况下，由主要研究者，和Markovianity的概念，2-D系统，研究的主机研究员。从一组高阶偏微分/差分方程自动导出一阶表示的可能性对于2-D系统的模拟中的应用特别相关。2)在描述和分析的物理过程中的泛函动力学和泛函的相互作用经常在系统和控制中被考虑，例如在稳定性分析中，在物理量的建模中，以及在最优控制中。在这个研究流中，我们调查了一些主题相关的这一领域，特别是多项式代数算法的发展，以计算保守和零平均量的2-D系统;算法的发展，为解决重要的多项式矩阵方程中产生的分析和控制的2-D系统，例如李雅普诺夫方程;和一个表示自由的方法来哈密顿系统。这些量的自动化计算开辟了使用计算机代数的可能性，基于能量的建模和控制方法，如在工程实践中使用的，被集成在计算机辅助仿真和分析工具。

The research described in this proposal aims at providing theoretical tools and algorithms based on multivariable polynomial algebra to deal with some specific problems regarding the stability, modeling, and control of distributed systems.Linear distributed (also called ``N-D'') systems arise naturally when considering the modeling of physical phenomena whose evolution depends not only on time but also on space, for example in image processing, seismology, circuit theory, flow-control, and iterative learning control. In the research proposed here we aim at studying open distributed systems described by systems of linear constant-coefficient partial difference/differential equations. A system of this sort is described by a set of linear, constant-coefficient partial differential/difference equations and it is naturally associated with a polynomial matrix in N indeterminates, in the sense that properties of the dynamical system are reflected in the algebraic properties of the polynomial matrix. The interplay of systems dynamics and functionals typical of many applications (for example in stability theory, in optimal control, etc.) also can be described effectively by polynomial matrices (in 2N indeterminates). Using the framework outlined above, we aim at using polynomial algebraic methods for the computation of first-order representations; and for the study of functionals arising in the analysis of physical systems. In both areas the expertise of the Principal Investigator and of the Host Investigator is considerable and complementary. The project is articulated in two work packages. 1) COMPUTATION OF FIRST-ORDER REPRESENTATIONS OF 2-D SYSTEMS First-order representations, corresponding to state-space equations in the 1-D case, are of primary importance for simulation, filtering, and so forth because of the ease of update. However they cannot be considered a starting point for the description of a system, but they need to be constructed from a higher-order model consisting of the interconnection of simpler subsystems, for example obtained from a library of standard models for certain components. The problem thus arises of how to compute a first-order representation of a system described by a set of higher-order partial differential or difference equations. In this work package we aim at investigating the computation of first-order representations for distributed systems. Crucial in this investigation is the notion of state map, developed in the 1-D case by the principal investigator, and the notion of Markovianity for 2-D systems, studied by the Host Investigator. The possibility of deriving automatically a first order representation from a set of higher-order partial differential/difference equations is particularly relevant for applications in simulation of 2-D systems. 2) FUNCTIONALS IN THE DESCRIPTION AND ANALYSIS OF PHYSICAL SYSTEMSThe interplay of dynamics and functionals is often considered in systems and control, for example in stability analysis, in the modeling of physical quantities, and in optimal control. In this research stream we investigate some themes related to this area, in particular the development of polynomial algebraic algorithms to compute conserved- and zero-mean quantities for 2-D systems; the development of algorithms for the solution of important polynomial matrix equations arising in the analysis and control of 2-D systems, for example the Lyapunov equation; and a representation-free approach to Hamiltonian systems. Automating the computation of these quantities opens up the possibility of using computer algebra for energy-based modeling and control methods such as those used in engineering practice, to be integrated in computer-aided simulation and analysis tools.