Ellipsoidal Set-Theoretic Design of Robust Safety Filters for Constrained Linear Systems

Abstract: This paper presents an ellipsoidal set-theoretic framework for robust safety filter synthesis in constrained linear systems subject to additive bounded disturbances and input constraints. We formulate the safety filter design as a convex linear matrix inequality (LMI) optimization problem that simultaneously computes a robust controlled invariant (RCI) ellipsoidal set and its associated state-feedback control law. The RCI set is characterized as an ellipsoidal set, enabling computational tractability for high-dimensional systems while providing formal safety guarantees. The safety filter employs a smooth mixing strategy between nominal and backup controllers based on distance to the invariant set boundary, facilitating minimal intervention when the system operates safely. The proposed method extends to nonlinear systems by treating nonlinear terms as bounded disturbances with rigorous approximation bounds. Numerical validation on a six-degree-of-freedom quadrotor system demonstrates the filter's effectiveness in maintaining stability under external disturbances and aggressive maneuvers while preserving nominal performance during safe operation. The approach provides a constructive and computationally efficient solution for safety-critical control applications requiring real-time implementation.