李雅普诺夫函数优化: 正交矩阵构造方案

Lyapunov function optimization: an orthogonal matrix construction scheme

中文关键词  李雅普诺夫函数  黎卡提不等式  线性系统  鲁棒稳定性  H∞范数

英文关键词  Lyapunov function  Riccati inequality  linear system  robust stability  H∞ norm

中文摘要

      本文研究了李雅普诺夫函数的优化问题. 提出了一种正交矩阵构造方案, 用于求解黎卡提不等式中的最优 李雅普诺夫函数. 通过分析系统H∞范数的几何特征, 本文将黎卡提不等式转换为近似等式, 进而给出了最优李雅 普诺夫函数的存在条件. 基于所给最优李雅普诺夫函数存在条件, 所提正交矩阵构造方案利用旋转变换, 将非线性 方程组的求解问题转换为幅值和角度的线性优化问题, 进而实现李雅普诺夫函数参数的优化. 研究结果弥补了目 前的研究无法求解最优李雅普诺夫函数的不足, 对系统性能分析和非保守控制的设计具有建设性. 算例验证了所 提正交矩阵构造方案的有效性.

英文摘要

      This paper focuses on the Lyapunov function optimization problem. An orthogonal matrix construction scheme is proposed to solve the optimal Lyapunov function in the Riccati inequality. By analyzing the geometric characteristics of the system H∞ norm, this paper transforms the Riccati inequality into an approximate equation, and gives the existence condition of the optimal Lyapunov function. Based on the given optimal Lyapunov function existence condition, the proposed orthogonal matrix construction scheme transforms the problem of solving nonlinear equations into a linear optimization problem of the amplitude and phase angle by using the rotation transformation, and realizes the Lyapunov function parameter optimization. The research result makes up for the deficiency in the current researches that the optimal Lyapunov function cannot be solved, and therefore is constructive for the system performance analysis and the non-conservative controller design. The effectiveness of the proposed orthogonal matrix construction scheme is verified by an example.