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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。
In this paper, state-space method is used to design a H ~ ∞ low-sensitivity state feedback system. Based on the relationship between the H ~ ∞ norm and the system state space, the H ~ ∞ control problem of state feedback system under the pole fixed condition is transformed into the problem of robustness in the time domain, The objective function of number. The objective function is a function of the feedback matrix F and the eigenvector matrix V of the closed-loop system matrix A + BF. Under the fixed limit condition, P and V can be parameterized as a function of -U∈R ~ (m × a) by the mapping of -R ~ (m × a) → R ~ (m × a). In this way, the objective function is U’s functional, and аJ / аU can be calculated. Therefore, the gradient method can be used to optimize J so that the H ~ ∞ norm decreases. In the gradient method, only 2n n-th order algebraic equations need to be solved for each iteration, which is much simpler than the traditional H ~ ∞ method for solving the Riccati equation. The example shows that the gradient method converges fast and the optimization effect is good.