Mathematical Approach for Robust Stability and for Robustly Strictly Positive Real on an Uncertain Plant Family of Complex Polynomials

ABSTRACT In this article, we present a robust control problem for a family of uncertain interval plants . We proposed a novel method introducing eight complex‐coefficient Kharitonov polynomials uniquely by minimizing and maximizing the concept of multilinear functions with four uncertain parameters. These polynomials define the vertices of a Kharitonov cuboid, which characterizes both robust stability and the robustly strictly positive real (SPR) property of the plant family. Thus, the introduction of this Kharitonov cuboid stands as a novel innovation in this article. The proposed approach offers two key contributions. Initially, we demonstrate the stability of the eight unique Kharitonov polynomials employing Hurwitz stability criteria, followed by the utilization of Kharitonov's theorem to ensure robust stability of . Subsequently, we establish the robustly SPR property by ensuring that the minimum values of the real and imaginary parts of the plant's frequency response remain positive, and we also analyze the stability of the numerator and denominator of each interval plant (). Numerical examples and simulations illustrate the effectiveness of the method, showing the motion of the Kharitonov cuboid and confirming the robust stability of the system.