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

TL;DR

This work addresses safety guarantees for constrained linear systems subject to bounded disturbances by constructing ellipsoidal robust controlled invariant (RCI) sets and an associated state-feedback controller. It formulates safety-filter synthesis as a convex semidefinite program that simultaneously maximizes the ellipsoid volume (via $\text{trace}(\mathbf{Q})$) and enforces input/output constraints through LMIs derived with the S-procedure, yielding a computationally tractable real-time solution. A smooth mixing strategy between nominal and backup controllers ensures minimal intervention while maintaining forward invariance of the ellipsoidal safe region. The approach is extended to nonlinear systems by bounding linearization errors and incorporating them into the augmented disturbance, with numerical validation on a quadrotor demonstrating maintained stability under disturbances and aggressive maneuvers while preserving nominal performance when safe.

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.

Figures (4)

• Figure 1: Block diagram of the hierarchical control structure for the quadrotor. The outer loop controls position $p$ by generating desired thrust $F_d$, roll, and pitch commands, while the inner loop tracks these commands using thrust $F$ and torques $\tau$. External disturbances $d(t)$ affect the quadrotor dynamics as external torques. The robust safety filter modifies the desired torque commands $\tau_d$ to ensure safety.
• Figure 2: The safety filter does not significantly alter performance in this safe scenario. (solid line: with safety filter, dotted line: without safety filter)
• Figure 3: The safety filter prevents instability caused by large initial position errors. (solid line: with safety filter, dotted line: without safety filter)
• Figure 4: The safety filter maintains stability during high-frequency trajectory tracking. (solid line: with safety filter, dotted line: without safety filter)

Theorems & Definitions (33)

• Definition 1: Safe Set
• Definition 2: Viable Set lygeros2004reachability
• Definition 3: Safe Robust Controlled Invariant Set
• Remark 1
• Definition 4: Bouligand's Tangent Cone blanchini_set-theoretic_2015
• Remark 2
• Proposition 1
• proof
• Theorem 1
• proof