In recent years, algorithms based on Computer Algebra (1-3) have been introduced into a range of cuntrol design problems because of the capacity to handle unknown parameters as indeterminates. This feature of algorithms in Computer Algebra reduces the costs of computer simulation and the trial and error process involved, enabling us to design and analyze systems more theoretically with the behavior of given parameters. In this paper, we apply Computer Algebra algorithms to H_∞ control theory, representing one of the most successful achievements in post-modern control theory. More specifically, we consider the H_∞ norm minimization problem using a state feedback controller. This problem can be formulated as follows: Suppose that we are given a plant described by the linear differential equation dx/dt = Ax + B_1ω + B_2u, z = Cx + Du, where A, B_1, B_2, C, D are matrices whose entries are polynomial in an unknown parameter k. We apply a state feedback controller u = -Fx to the plant, where F is a design parameter, and obtain the system dx/dt = (A - B_2F)x + B_1ω, z = (C - DF)x. Our task is to compute the minimum H_∞ norm of the transfer function G(s) (= (C - DF)(sI - A+ B_2F)~(-1)B_1) from ω to z achieved using a static feedback controller u = -Fx, where F is a constant matrix. In the H_∞ control theory, it is only possible to check if there is a controller such that ‖G(s)‖_∞ ＜ γ is satisfied for a given number γ, where ‖G(s)‖_∞ denotes the H_∞ norm of the transfer function G(s). Thus, a typical procedure to solve the H_∞ optimal problem would involve a bisection method, which cannot be applied to plants with parameters. In this paper, we present a new method of solving the H_∞ norm minimization problem that can be applied to plants with parameters. This method utilizes QE (Quantifier Elimination) and a variable elimination technique in Computer Algebra, and expresses the minimum of the H_∞ norm as a root of a bivariate polynomial. We also present a numerical example to illustrate each step of the algorithm.
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机译：近年来，基于计算机代数（[1]-[3]）的算法已被引入到一系列cuntrol设计问题中，因为其能够将未知参数处理为不确定参数。计算机代数算法的这一特性降低了计算机模拟的成本和所涉及的试错过程，使我们能够使用给定参数的行为更理论地设计和分析系统。本文将计算机代数算法应用于H_∞控制理论，代表了后现代控制理论最成功的成果之一。更具体地说，我们使用状态反馈控制器来考虑H_∞范最小化问题。这个问题可以表述如下：假设我们得到一个由线性微分方程 dx/dt = Ax + B_1 ω + B_2u， z = Cx + Du 描述的工厂，其中 A、B_1、B_2、C、D 是矩阵，其条目在未知参数 k 中为多项式。我们将状态反馈控制器 u = -Fx 应用于被控对象，其中 F 是设计参数，并得到系统 dx/dt = （A - B_2F）x + B_1 ω， z = （C - DF）x。我们的任务是计算使用静态反馈控制器 u = -Fx 实现的传递函数 G（s） （= （C - DF）（sI - A+ B_2F）~（-1）B_1） 从 ω 到 z 的最小H_∞范数，其中 F 是一个常数矩阵。在H_∞控制理论中，只能检查是否存在一个控制器，使得给定数γ满足 ‖G（s）‖_∞ < γ，其中 ‖G（s）‖_∞ 表示传递函数 G（s） 的H_∞范数。因此，求解H_∞最优问题的典型过程将涉及二分法，该方法不能应用于具有参数的植物。在本文中，我们提出了一种求解H_∞范数最小化问题的新方法，该方法可以应用于具有参数的植物。该方法利用计算机代数中的QE（量词消除）和变量消除技术，并将H_∞范数的最小值表示为二元多项式的根。我们还提供了一个数值示例来说明算法的每个步骤。
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