From Robust Control to Cyber-Resilience: A Comprehensive Overview of the Polytopic Framework for Safety-Critical Systems

This overview paper highlights the idea that the polytopic approach is more than a modeling technique. It proposes a unified perspective in which polytopic representations constitute a conceptual bridge between complex system dynamics and convex analysis. From modeling to control, the polytope is not only limited to a geometric interpretation but is used as a methodological principle—an intelligent sensor system for structuring uncertainty, representing a nonlinear behaviors, and enabling decision-making, even for critical safety systems. Indeed, the polytopic approach should not be viewed merely as a convexification tool, but as a way of thinking about complexity, i.e., a structured methodology linking representation, uncertainty management, and control synthesis. For this purpose, this work focuses on safety-critical systems, i.e., cyber-physical systems operating under malicious cyber-attacks, where ensuring resilience has become a major challenge. The review highlights recent developments that extend classical polytopic approaches beyond robust control and toward cyber-resilient estimation and control, including false-data injection attack modeling, simultaneous state and attack estimation, resilient observer-based control, and event-triggered control under communication constraints. The paper also discusses how these developments position the polytopic framework with respect to recent advances in cyber-physical security and resilient control. Overall, the proposed perspective illustrates how polytopic representations have evolved into a unifying paradigm for addressing nonlinear dynamics, cyber-attacks, uncertainties, and network-induced constraints while enhancing the resilience, reliability, and operational safety of safety-critical cyber-physical systems.