Necessary and sufficient condition for existence of two-degree-of-freedom feedback loop factorisation and comparison of zeros in compensator strategies

Abstract

Two cases of the two-degree-of-freedom (2DOF) feedback loop are compared after applying them to the third-order system with uncertain time delay, the fourth-order system with astatism and uncertain time delay and the oscillating system with astatism and uncertain time delay. All systems have periodic changes of some of their parameters. The necessary and sufficient condition for the existence of 2DOF factorisation is formed and proven. The uncertain time delay is treated using multiplicative uncertainty; the periodic changes of parameters are modelled using a general interconnection for the systems with parametric uncertainty in the numerator and denominator. The structured singular value denoted μ is used as a measure of robust performance and stability. For comparison, the D-K iteration and algebraic μ-synthesis are used for simple feedback loop controller derivation. The algebraic μ-synthesis is a new method for robust controller derivation comprising the structured singular value, algebraic control theory and metaheuristic algorithm solving multimodality of the cost function. Minimisation of the μ-function in the algebraic μ-synthesis is treated using the Differential Migration algorithm as a tool for global optimisation with subsequent tune-up using the Nelder–Mead simplex method. The final controllers are verified using the μ-plots and simulations for the worst-case perturbation and periodic changes of parameters with the maximum time delay. © 2026 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.