Universal Method of Static Output Feedback Pole Placement for Fourth-Order Linear Time-Invariant Systems with Two Inputs and Two Outputs

A universal analytical method of static output feedback pole placement (for constructing controllers by output) for fourth-order linear time-invariant systems with two control inputs and two observable outputs is proposed, which does not depend on the relations between the controllability and observability indices. The method does not require decomposing or reducing the system and may be applied to any fourth-order systems modally controllable by static output feedback, including systems irreducible to control (observation) with a single input (output). The method is based on linear matrix dependence of the characteristic polynomial of a closed-loop control system on the coefficients and determinant of a matrix of controller by output. The proposed approach gives the necessary and sufficient condition for static output feedback pole placement—it allows defining all possible matrices of controllers by output and the conditions of their existence. A new algebraic criterion of complete modal controllability by output is formulated and proved. Examples of applying the approach to controlling the aviation and space systems based on linearized models both in numerical and symbolic form are given.