Analytical Synthesis of Modal Control by Output for Fourth-Order Systems with Two Inputs and Two Outputs at Equal Indices of Controllability and Observability
Alexey V. Lapin, Nikolay E. Zubov, Eugene Yu. Zybin · 2025
Analytical solutions of the problems of modal control by output are obtained for a wide class of fourth-order systems with two inputs and two outputs where the controllability index is equal to the observability index. The conditions for the existence of analytical solutions are found. Formulas for calculating controllers by output are presented explicitly and do not require supplementary solution of polynomial or matrix equations. The conditions for the existence of analytical formulas for modal control by output and the formulas themselves are obtained both in direct version via modal control by state and in dual version via modal observation. The analytical solutions are based on the Cayley-Hamilton theorem, the relationship between matrix polynomials of open-loop and closed-loop control systems, as well as on matrix canonization technology. The efficiency of the approach is confirmed by the example of controlling the longitudinal motion of a civil aircraft via flight linearized model.