Markov跳跃系统Lyapunov矩阵方程的显式迭代算法

发布时间：2018-03-10 23:17

本文选题：Markov跳跃系统　切入点：Lyapunov方程　出处：《哈尔滨工业大学》2017年硕士论文　论文类型：学位论文

【摘要】：随着科学技术的迅速发展,构建出的系统愈加复杂,很多系统的内部结构会发生突变。Markov跳跃系统用来对这样的系统进行数学建模,模型中用多个模态表示系统可能的各个状态,Markov链表示系统各个模态之间的转换关系。Lyapunov矩阵方程在Markov跳跃系统的稳定性分析,最优控制和鲁棒控制中都有着重要作用。本文将围绕该系统的耦合Lyapunov矩阵方程的显式迭代算法进行研究,主要研究内容和结果包括以下几个部分。首先是显式迭代算法的推导,通过等价变换将Lyapunov矩阵方程转变为显式,即求解矩阵单独在方程一侧,由此写出显式迭代算法。然后给出了相关定理及其证明。算法推导过程中多出一个参数,通过仿真选取最优值。最后仿真结果发现最优的显式迭代算法与隐式迭代算法相比,前者精度更高,但是收敛明显慢。然后将显式迭代算法进行改进,应用最新信息估计的方法,给出了相关的定理及其证明。同样通过仿真选取最优参数。仿真结果发现,最优的基于最新信息估计的显式迭代算法收敛速度与改进前相比有显著提高,但是仍没有隐式迭代算法收敛快,精度不变。最后将算法加入权重参数,使得显式迭代算法与基于最新信息估计的显式迭代算法成为特例,即对应权重参数的两个取值。然后给出了相关的定理及其证明,通过仿真选取两个参数的最优值。最后仿真结果发现,最优的基于权重的显式迭代算法收敛速度和精度均优于隐式迭代算法。
[Abstract]:With the rapid development of science and technology, the system is becoming more and more complex, many systems will have a sudden change in internal structure. Markov jump system will be used to model the mathematical model of such a system. In the model, several modes are used to represent the possible states of the system. The stability analysis of the Lyapunov matrix equation in the Markov hopping system is presented. Both optimal control and robust control play an important role. In this paper, the explicit iterative algorithm for coupled Lyapunov matrix equations of the system is studied. The main contents and results include the following parts. Firstly, the explicit iterative algorithm is derived. The Lyapunov matrix equation is transformed into an explicit one by equivalent transformation, that is, the solution matrix is separately on one side of the equation, and then an explicit iterative algorithm is written. Then, the correlation theorem and its proof are given, and an additional parameter is given in the derivation of the algorithm. The simulation results show that the optimal explicit iterative algorithm is more accurate than the implicit iterative algorithm, but the convergence is obviously slow. Then the explicit iterative algorithm is improved. By using the method of estimating the latest information, the relevant theorems and their proofs are given. In the same way, the optimal parameters are selected by simulation. The simulation results show that, The convergence speed of the optimal explicit iterative algorithm based on the latest information estimation is significantly improved compared with that before the improvement, but there is no implicit iterative algorithm that converges quickly and the accuracy remains unchanged. Finally, the weight parameter is added to the algorithm. The explicit iterative algorithm and the explicit iterative algorithm based on the latest information estimation become special cases, that is, two values of the corresponding weight parameters. Then, the relevant theorems and their proofs are given. The results of simulation show that the optimal explicit iterative algorithm based on weight has better convergence speed and precision than implicit iterative algorithm.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.62;O241.6

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