Minimum Variance Pole Placement in Uncertain Linear Control Systems

Abstract

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.