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

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,P与V可通过—R^(m×a)→R^(m×a)的映射参数化为—U∈R^(m×a)的函数。这样,目标函数为U的泛函,并且аJ/аU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。

A simple relation between the 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 Rnxr→R(?) 'function. Using this function, the performance index related to H∞-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 arc equivalent to n n-ordcrcd linear algebraic equations, are included. Example show that the convcgcncc speed of the gradient algorithm is fast.