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

线性分布式(也称为N-D)系统是在图像处理、地震学、电路理论、流量控制和迭代学习控制等领域中，对既依赖时间又依赖于空间的物理现象的建模而产生的。在本文提出的研究中，我们旨在研究由线性常系数偏微分/微分方程组描述的开放分布系统。这类系统由一组线性的常系数偏微分/差分方程组描述，从动力系统的性质反映在多项式矩阵的代数性质的意义上讲，它与N个不确定项中的多项式矩阵自然地联系在一起。系统动力学和泛函在许多应用中的相互作用(例如稳定性理论、最优控制等)。也可以用多项式矩阵(在2N个不定式中)来有效地描述。利用上面概述的框架，我们的目标是使用多项式代数方法来计算一阶表示，并用于研究物理系统分析中出现的泛函。在这两个领域，首席调查员和东道国调查员的专门知识是相当大的和相辅相成的。该项目分为两个工作包。1)计算2-D系统的一阶表示。由于易于更新，与一维情况下的状态空间方程相对应的一阶表示对于仿真、滤波等是最重要的。然而，它们不能被认为是描述系统的起点，但它们需要从更简单的子系统的互连组成的更高阶模型中构建，例如，从某些组件的标准模型库中获得。由此产生了如何计算由一组高阶偏微分方程组或差分方程组描述的系统的一阶表示的问题。在这个工作包中，我们旨在研究分布式系统的一阶表示的计算。这项研究的关键是由首席调查员在1-D案例中开发的状态图的概念，以及由主持人调查员研究的2-D系统的马尔可夫性的概念。从一组高阶偏微分差分方程组中自动导出一阶表示的可能性对于2-D系统模拟中的应用特别相关。2)泛函在物理系统的描述和分析中动力学和泛函的相互作用在系统和控制中经常被考虑，例如在稳定性分析、物理量建模和最优控制中。在这一研究流中，我们研究了与这一领域相关的一些主题，特别是计算2-D系统守恒量和零均值的多项式代数算法的发展；在2-D系统的分析和控制中出现的重要的多项式矩阵方程(例如Lyapunov方程)的求解算法的发展；以及哈密顿系统的无表示方法。这些量的自动化计算为将计算机代数用于基于能量的建模和控制方法(如工程实践中使用的方法)提供了可能性，并将其集成到计算机辅助模拟和分析工具中。

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.