An optimal choice of characteristic polynomial roots for pole placement control design

The problem of finding the arrangement of closed-loop control system poles that minimizes an objective function is considered. The system optimality criterion is the value of the $H_\infty$ norm of the frequency transfer function relative to the disturbance with constraints imposed on the system pole placement and the values of the $H_\infty$ norm of the sensitivity function and the transfer function from measurement noise to control. An optimization problem is formulated as follows: the vector of variables consists of the characteristic polynomial roots of the closed loop system with the admissible values restricted to a given pole placement region; in addition to the optimality criterion, the objective function includes penalty elements for other constraints. It is proposed to use a logarithmic scale for the moduli of the characteristic polynomial roots as elements of the vector of variables. The multi-extremality problem of the objective function is solved using the multiple start procedure. A coordinate descent modification with a pair of coordinates varied simultaneously is used for search.