离散随机Markov 跳跃系统的广义Lyapunov 方程解的性质

针对离散随机Markov 跳跃系统, 基于?表示方法研究了其对应的广义Lyapunov 方程解的性质. 首先, 证明了广义Lyapunov 方程存在唯一实对称矩阵序列解的充分必要条件是系统的谱不包含零特征值; 然后, 在系统的谱包含零特征值的情况下, 分析了广义Lyapunov 方程解的结构; 最后, 通过数值仿真表明了所得结论的正确性.

Properties of solutions to generalized Lyapunov equations for discrete stochastic Markov jump systems

For discrete stochastic Markov jump systems, the properties of solutions to generalized Lyapunov equations(GLEs) are investigated based on the ?-representation method. Firstly, it is proved that the GLE has a unique solution in the form of a sequence of real symmetric matrices, if and only if the spectrum of the system does not contain zero eigenvalue. Moreover, the structure of solutions to GLEs is also discussed when there exist zero eigenvalue in the spectrum of the system. Finally, a numerical example is given to show the validity of the obtained results.