A Novel Method to Find the Optimal Range of Robust Stability in Interval-Uncertain Unstable Systems Using the Kharitonov Theorem

This paper explores the application of Kharitonov's theory to determine the range of robust stability in unstable systems characterized by intervals of uncertain parameters. Traditionally, Kharitonov's theory primarily focuses on checking robust stability; however, this study introduces a novel approach to identifying the optimal stability range within the uncertainty range. As uncertainty in system parameters can significantly affect stability, the need for robust stability analysis becomes paramount, particularly in engineering and control systems. A systematic approach to determining the stability boundaries of an unstable system is presented through Kharitonov's polynomial method, with the analytical formulation explicitly developed for second-, third-, and fourth-order systems. Through rigorous mathematical modeling and simulation of four numerical examples, we illustrate the effectiveness of Kharitonov's theory in quantifying stability regions and demonstrate its applicability to real-world systems. The results of the method have also been verified using particle swarm optimization. Our findings provide valuable insights into enhancing system reliability and performance in the presence of uncertainties.