What are the key stability challenges in high-bandwidth, non-minimum phase systems with time-varying, and non-smooth delays?

The analysis and control of stability in high-bandwidth systems characterized by non-minimum phase delays represent a formidable challenge within the realm of control theory and engineering. This research aims to address the pivotal question of whether it is feasible to enhance the stability of such intricate systems. These systems inherently possess uncertain and swiftly changing delay characteristics, rendering them exceptionally demanding to control effectively. In the course of this investigation, we embark on a comprehensive exploration of the theoretical underpinnings of the stability of high-bandwidth, non-minimum phase delay systems. This encompassing inquiry encompasses a meticulous consideration of both derivative-delay and piecewise continuous delay components. To underpin our analysis, we judiciously incorporate feedback mechanisms, drawing upon mathematical tools such as the Jensen inequality and Lyapunov-based methodologies to rigorously establish stability conditions. Furthermore, our exploration extends to encompass the concept of input-output stability and complements it with the notion of asymptotic stability, thereby ensuring that the systems in question exhibit uniform stability across diverse temporal domains. The outcomes of our investigation furnish compelling evidence that by harnessing the power of discrete-time Lyapunov-Krasovskii functionals, it becomes conceivable to circumscribe the maximum delay within predefined thresholds. This achievement holds the promise of enhancing stability in non-minimum phase delay systems characterized by high bandwidth. These findings have far-reaching implications, profoundly influencing the design and control paradigms across a spectrum of engineering applications. Notably, this impact extends to areas such as communication networks, real-time control systems, and robotics, where the mitigation of instability due to non-minimum phase delays has been an enduring challenge.