Markov跳跃系统中的Riccati方程的超松弛迭代算法

发布时间：2018-03-07 09:39

本文选题：Markov跳跃系统　切入点：耦合代数Riccati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】：Markov跳跃系统是一类常见的随机控制系统,在多模态的控制系统中各子系统之间的跳跃转移具有马尔科夫性,因此可以用Markov模型描述系统的变化过程。当控制系统的状态发生跳跃转移时,为了使系统的代价函数最小,通常需要求解一个耦合代数矩阵Riccati方程,求解该方程的正定解就可以得到该控制系统在各个状态下的最优控制输入,使系统的代价函数最小。本文提出了两种求解上述Riccati方程的迭代算法,主要内容如下:把耦合代数矩阵Riccati方程转化为非耦合代数矩阵Lyapunov方程,再结合最新估计新息和超松弛迭代算法,得出了超松弛形式的Lyapunov迭代算法。当迭代初值满足一定的条件时,可以通过数学归纳法从理论上给出该算法的收敛性,经过多次迭代可以得到耦合代数矩阵Riccati方程的唯一可镇定的正定解。对于超松弛形式的Lyapunov迭代算法,本文给出了一种选择合适迭代初值的算法。耦合代数矩阵Riccati方程通过解耦也可以写成普通Riccati方程的迭代形式,迭代初值可以选为零矩阵。结合数学归纳法和代数Riccati方程的比较定理可以从理论上证明该迭代算法收敛,如果松弛因子选择合适,收敛精度和速度与Lyapunov迭代算法相差不大。针对以上两种求解耦合Riccati方程的超松弛迭代算法,通过MATLAB仿真可以发现,如果松弛因子选择合适,这两种算法都能加快收敛速度,通过选取不同的松弛因子多次仿真,可以得出最优的松弛因子的取值。
[Abstract]:Markov jump system is a kind of common stochastic control system. In order to minimize the cost function of the system, a coupled algebraic matrix Riccati equation is usually solved. By solving the positive definite solution of the equation, the optimal control input of the control system in each state can be obtained, and the cost function of the system can be minimized. In this paper, two iterative algorithms for solving the above Riccati equation are proposed. The main contents are as follows: the coupled algebraic matrix Riccati equation is transformed into the uncoupled algebraic matrix Lyapunov equation, and then combined with the latest estimated innovation and overrelaxation iterative algorithm, The Lyapunov iterative algorithm in the form of overrelaxation is obtained. When the initial value of the iteration satisfies certain conditions, the convergence of the algorithm can be obtained theoretically by mathematical induction. The unique stable positive definite solution of coupled algebraic matrix Riccati equation can be obtained by multiple iterations. For the Lyapunov iterative algorithm in overrelaxation form, In this paper, an algorithm for selecting suitable initial iterative values is given. The coupled algebraic matrix Riccati equation can be written as an iterative form of ordinary Riccati equation by decoupling. The initial value of iteration can be chosen as zero matrix. The convergence of the iterative algorithm can be theoretically proved by combining the comparison theorem of mathematical induction and algebraic Riccati equation, if the relaxation factor is suitable, The convergence accuracy and speed of the two algorithms are not different from those of the Lyapunov iterative algorithm. In view of the two overrelaxation iterative algorithms for solving coupled Riccati equations, it is found by MATLAB simulation that if the relaxation factor is chosen properly, both algorithms can accelerate the convergence speed. The optimal relaxation factor can be obtained by multiple simulations with different relaxation factors.
【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】：O231

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