The pole-placement issue for linear multi-input multi-output (MIMO) dynamic systems with uncertain parameters has been addressed in this article. A static feedback matrix has been designed for minimizing variances of closed-loop poles (CLPs) and for assigning poles to the nominal system at the desired places. It is assumed that the joint probability density function (PDF) of uncertain parameters is known and the system has more than one input. A new unknown vector is used like an eigenvector for a stochastic closed-loop system matrix to state the problem. The variances of poles are considered as cost functions, and the means of poles are termed constraints. This form of the problem statement has helped us to simply find a solution. In the first step, the optimization problem with constraints was handled by solving the equality constraint, and then, the problem was converted to a classic extended eigenvalue optimization problem. Later, the eigenvalue optimization problem was solved by the Rayleigh quotient and the feedback matrix was accomplished. Finally, this approach was simulated and validated using the MATLAB simulations, and the results were compared with a robust pole-placement method, which MATLAB control toolbox uses. The Monte Carlo simulations showed lower covariance for CLPs around the mean poles as compared to the robust pole-placement method.

中文翻译：

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不确定线性控制系统中的最小方差极点布置

本文解决了参数不确定的线性多输入多输出（MIMO）动态系统的极点放置问题。设计了静态反馈矩阵，以最大程度地减小闭环极点（CLP）的方差，并在所需位置将极点分配给标称系统。假设不确定参数的联合概率密度函数（PDF）是已知的，并且系统具有多个输入。对于随机闭环系统矩阵，使用一个新的未知向量（如特征向量）来说明问题。极点的方差被视为成本函数，极点的均值称为约束。问题陈述的这种形式帮助我们简单地找到了解决方案。第一步，通过求解等式约束来处理具有约束的优化问题，然后，该问题被转换为经典的扩展​​特征值优化问题。后来，利用瑞利商解决了特征值优化问题，并完成了反馈矩阵。最后，使用MATLAB仿真对这种方法进行了仿真和验证，并将结果与​​MATLAB控制工具箱使用的稳健的极点放置方法进行了比较。与健壮的极点放置方法相比，蒙特卡洛模拟显示平均极点周围CLP的协方差较低。并将结果与​​MATLAB控制工具箱使用的稳健的极点放置方法进行了比较。与健壮的极点放置方法相比，蒙特卡洛模拟显示平均极点周围CLP的协方差较低。并将结果与​​MATLAB控制工具箱使用的稳健的极点放置方法进行了比较。与健壮的极点放置方法相比，蒙特卡洛模拟显示平均极点周围CLP的协方差较低。