Studies on Analysis and Synthesis of Control Systems Based on Infinite-Dimensional Linear Matrix Inequalities

在这项研究中，我们建立了一个统一的分析和控制综合的线性系统的不确定参数，不确定但可测量的参数（调度参数），和空间分布的参数的方法。该算法是基于无限维线性矩阵不等式的充分条件，不仅为分析规定的性能，如稳定性和输入输出L2增益，但也为可解的控制综合，实现规定的性能。带调度参数的线性系统是一类特殊的非线性系统，它可以用一些内变量作为调度参数来描述一类非线性系统，但这种描述通常会导致分析和控制综合中的一个困难问题，称为“隐回路问题”。该算法通过引入线性矩阵不等式来刻画状态的可达集，成功地克服了隐环问题。作为具有空间分布参数的线性系统，特别地，我们讨论了线性时滞系统，并基于依赖于时滞参数的特定类型的无穷维线性矩阵不等式，提出了一种算法。给出了一种将无穷维线性矩阵不等式化为有限维矩阵不等式的方法。该约简方法的主要思想是在参数空间中构造一个凸多面体，是本文研究者以前在增益调度问题中所用思想的推广。通过数值仿真和实验验证了该算法的有效性。

In this research we established a unified methodology of analysis and control synthesis for linear systems with uncertain parameters, uncertain but measurable parameters (scheduling parameters), and spatially distributed parameters. The proposed algorithm is based on infinite-dimensional linear matrix inequalities which are sufficient conditions not only for analysis of prescribed performances, e.g. stability and input-output L2 gain but also for solvability of control synthesis which realizes prescribed performances. The Linear system with scheduling parameters, which is a special case of the systems we consider, can describe a class of nonlinear systems by taking some internal variables as scheduling parameters, but this description leads generally to a hard problem in analysis and control synthesis, called the "hidden-loop problem". The proposed algorithm succeeds in overcoming the hidden-loop problem by introducing the linear matrix inequality that characterizes a reachable set of state. As linear systems with spatially distributed parameters, in particular, we discuss linear time-delay systems, and propose an algorithm based on the particular type of infinite-dimensional linear matrix inequalities that depend on the time-delay parameter. We also proposed a reduction method of the infinite-dimensional linear matrix inequality to a finite-dimensional matrix inequality. The main point of the reduction method is to construct a convex polyhedron in parameter space, and is an extension of the idea that the present researcher used previously in the problem of gain scheduling. Efficacy of the proposed algorithm was demonstrated through numerical simulations and experiments.