多输入多输出线性定常系统稳定裕度的分析与改进

Analysis and improvement of stability margin for multi-input multi-output linear time-invariant systems

针对多输入多输出(MIMO)控制系统的稳定裕度求解问题,首先分析了现有的回差阵奇异值法这一计算方法,并得到其解决单输入单输出(SISO)系统的稳定裕度结论,在此基础上,提出两种基于系统回差阵的稳定裕度改进方法;一种是在有限条件下利用矩阵的特征值代替奇异值来建立与稳定裕度关系的策略,另一种是利用系统逆回差阵的行列式,通过求其奇异值来计算系统稳定裕度;最后结合工程实例,通过数值仿真验证两种稳定裕度计算方法相比原方法都有不同程度的改进,而且三种方法可以结合起来进行分析,最大化的减小系统稳定裕度结果的保守性.

A method based on the singular value of the return difference matrix is investigated for calculating the stability margin of multi-input multi-output （MIMO） linear time invariant systems. Results are first obtained for single-input single- output linear systems. Then two improved methods are proposed for calculating the stability margin of MIMO linear systems based on the return difference matrix. In the first improved method, the minimum eigenvalue magnitude is used for replacing the minimum singular value of the return difference matrix, and a new relationship between the stability margin and the minimum eigenvalue magnitude is developed. In the other improved method, the stability margin is calculated through the minimum singular value of the inverse return difference matrix. Numerical simulation results clearly show the improvement in the solutions obtained from the improved methods, and the conservativeness of stability margin can be reduced effectively through the combination of three methods.