状态反馈系统的H∞低敏感性的设计*

H∞-Control By State Feedback Under the Constraint of pole assignment

中文关键词  H∞控制  H∞范数  状态反馈  极点配置  梯度法

英文关键词  H∞-control  H∞-norm  state feedback  pole assignment  gradient method

中文摘要

      本文用状态空间方法对状态反馈系统进行H∞低敏感设计。利用H∞范数与系统状态空间实现的关系，将极点固定条件下的状态反馈系统的H∞控制问题转化为时域上的鲁棒性问题，并由此提出了反映H∞范数的目标函数。该目标函数为反馈矩阵F与闭环系统矩阵A+BF的特征向量矩阵V的函数，在极点固定的限制条件下，F与V可通过一Rmxn→Rmxn的映射参数化为一U∈Rmxn的函数。这样，目标函数为U的泛函，并且?J/?u可以求出。因此，可用梯度法优化J，从而使H∞范数降低。在梯度法优化中，每项迭代只须求解2n个n阶代数方程，与传统的H∞方法求解Riccati方程相比，要简单许多。实例说明，梯度法收敛速度较快，优化效果良好。

英文摘要

      A simple relation between H∞-norm of a transfer matrix and its state space representation is developed. Through this relation, the H∞-control problem by state feedback under the constraint of pole assignment is reduced to a problem similar to robust design, and a performance index related to the H∞-norm is naturally derived. The set of feedback matrices that assign the poles of a system is determined through an Rmxn→Rmxn function. Using this function, the performance index related to -norm can be minimized by gradient method. Thus reduce the H∞-norm. In each step of the gradient method, only two special Sylvester equations, which are equivalent to n n-ordered linear algebraic equations, are included. Example show that the convergence speed of the gradient algorithm is fast.