COMPUTATION OF SOLUTIONS TO POLYNOMIAL MATRIX RICCATI EQUATIONS AND OPTIMAL REGULATOR FOR TIME-DELAY SYSTEMS

时滞系统多项式矩阵Riccati方程解的计算及最优调节器

• 批准号: 14550446
• 负责人: ITO Naoharu
• 金额: $ 0.9万
• 依托单位: NARA UNIVERSITY OF EDUCATION
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14550446/
• 关键词: polynomial Riccati equation; computer algebra; time delay systems; Laurent polynomials; spectral factorizations; optimal regulators; minimal realizations; Jacobson normal forms; ヤコブソン標準形; ローラン多項式環; 多項式行列リャプノフ方程式

This research paper gives a method for computing solutions of polynomial matrix Riccati equations and optimal regulator for linear time delay systems. First, matrix Riccati equations over a ring are studied. In particular, matrix Riccati equations over Laurent polynomial rings are also investigated. Then, we discuss systems over ring and time delay systems. Next, stability of independent of delay and pointwise stability are considered. A problem of stabilization independent of delay for time-delay systems is investigated. Time-delay systems are regarded as systems over the ring of real polynomials, and the corresponding matrix Riccati equations over Laurent polynomial ring are studied. Polynomial matrix Riccati equation approach is considered for the stabilization problem. We derive a procedure for a minimal state space realization of a rational transfer matrix over an arbitrary field. The procedure is based on the Smith-McMillan form and leads to a state transition matrix in Jacobson normal form. Finally, The problem of disturbance rejection by observation feedback for linear time delay systems is investigated. The time delay systems are regarded as systems over the ring of real polynomials, and the problem is formulated within the framework of a geometric approach. Then, a necessary and sufficient condition for the problem to be solvable is obtained.

本文给出了线性时滞系统多项式矩阵Riccati方程和最优调节器的一种求解方法。首先，研究了环上的矩阵Riccati方程。特别地，还研究了Laurent多项式环上的矩阵Riccati方程。然后，我们讨论环上系统和时滞系统。其次，考虑了时滞无关的稳定性和点态稳定性。研究了时滞系统与时滞无关的镇定问题。将时滞系统看作实多项式环上的系统，研究了Laurent多项式环上相应的矩阵Riccati方程。对于镇定问题，考虑了多项式矩阵Riccati方程方法。我们推导了任意域上有理转移矩阵的最小状态空间实现过程。该过程基于Smith-McMillan形式，并导致雅各布森标准形式的状态转移矩阵。最后，研究了线性时滞系统的观测反馈扰动抑制问题。将时滞系统视为实多项式环上的系统，并在几何方法的框架内对问题进行了描述。然后，得到了该问题可解的充要条件。

项目成果

伊藤: "多項式行列Riccati方程式とむだ時間システムの安定化に関する一考察"第32回制御理論シンポジウム資料. 273-276 (2003)

伊藤：“多项式矩阵Riccati方程和时滞系统稳定性的研究”第32届控制理论研讨会材料273-276（2003）。

N.Ito, W.Schmale, H.K.Wimmer: "computation of a minimal state space realization in Jacobson normal form"CONTEMPORARY MATHEMATICS. 323. 221-232 (2003)

N.Ito、W.Schmale、H.K.Wimmer：“雅各布森范式中最小状态空间实现的计算”当代数学。

N.Ito, W.Schmale, H.K.Wimmer: "Minimal state space realizations in Jacobson normal form"International Journal of Control. 75. 1092-1099 (2002)

N.Ito、W.Schmale、H.K.Wimmer：“Jacobson 范式中的最小状态空间实现”国际控制杂志。

N.Ito: "A study on polynomial Riccati equations and stabilization for time delay systems"Proc.32^<nd> SICE Symposium on Control Theory. 273-276 (2003)

N.Ito：“多项式 Riccati 方程和时滞系统稳定性的研究”Proc.32^<nd> SICE 控制理论研讨会。

伊藤: "多項式行列Ricatti方程式とむだ時間システムの安定化に関する一考察"第32回制御理論シンポウジウム資料. 273-276 (2003)

伊藤：“多项式矩阵 Ricatti 方程和死区时间系统稳定性的研究”第 32 届控制理论研讨会材料 273-276 (2003)。