由回差阵奇异值求稳定裕度的退化算法
Degraded algorithm for determining stability margin by using singular value of the return difference matrix

本文研究使用回差阵奇异值求多输入多输出(MIMO)线性定常系统稳定裕度的方法. 首先对此方法进行退化分 析, 得到求解单输入单输出(SISO)线性定常系统稳定裕度的算法步骤. 在此基础上, 讨论退化所得算法与传统稳定裕度 的关系; 进一步地, 详细分析此退化算法相比传统稳定裕度的优势, 进而指出当系统的增益和相位同时变化时, 系 统Nyquist曲线g(j!)到临界点(??1; j0)的最短距离min j1 + g(j!)j可作为一种更加合理的稳定裕度指标. 最后, 通过对 实例进行数值仿真, 说明本文所提退化算法可以克服传统稳定裕度局限性，同时与传统稳定裕度结合得到比较完整 的SISO线性系统稳定裕度衡量体系.
We investigate the algorithm for determining the stability margins for multi-input multi-output (MIMO) linear time-invariant systems by using singular values of the return difference matrix. First, we consider the degraded algorithm to develop the procedures for determining stability margins for single-input single-output (SISO) linear time- invariant systems. On this basis, we investigate the relationship between the results obtained from the degraded algorithm and the traditional stability margins. Next, we analyze in detail the advantages of the degraded algorithm over the traditional methods in determining the stability margins and point out that, when the gain and phase are varying simultaneously, the shortest distance min j1 g(j!)j between the Nyquist plot g(j!) and the critical point (??1; j0) can be considered as a more appropriate stability margin index. Finally, the numerical simulation results of practical examples demonstrate that the proposed algorithm is able to overcome the limitations of the traditional methods. Meanwhile, a more perfect stability margin measurement system can be obtained by incorporating the proposed degraded stability margins with the classical stability margin.