An Optimization Perspective on Discrete-Time Sliding Modes: When Proximal-Point Algorithms Meet Set-Valued Systems

This article investigates how classical discontinuous sliding-mode signals can be more effectively understood and implemented using principles from optimization and convex analysis. For continuous-time systems, the study identifies energy dissipation as a fundamental aspect of the method. In discrete-time implementations, the research shows that employing the well-established backward-Euler discretization allows us to preserve robustness against matched disturbances as well as finite-time convergence whilst drastically reducing high-frequency oscillations (known as the chattering phenomenon). A significant contribution of this work is establishing a clear connection between control methodologies and optimization algorithms, using Proximal-Point Algorithms (PPA) and proximal operators. By creating this link, powerful optimization techniques can be utilized to simplify analysis and enhance the design of control systems.