On One Approach to the Analytic Synthesis of Modal Control by Output for Fourth-Order Dynamic Systems with Two Inputs and Two Outputs

The problem of modal control by output for a fourth-order dynamic system with two control inputs and two measured outputs is considered, provided that the controllability index is not equal to the observability index. It is shown that this problem is solvable, despite the fact that the total dimension of the input and output vectors does not exceed the dimension of the state vector. Compact analytic solutions to this problem using matrix zero-divisors and solvability conditions of one-sided linear matrix equations are proposed. Theorems are proved that implement direct (control) and dual (observation) approaches to solve the problem for cases when the controllability index is correspondingly greater and less than the observability index. Examples are given confirming the efficiency of each approach.