On Robust Controlled Invariants for Continuous-time Monotone Systems

TL;DR

This work addresses robust controlled invariants for monotone continuous-time systems under lower-closed specifications by introducing SM and CSM classes, establishing structural and feasibility-based characterizations, and presenting a practical algorithm to compute the maximal invariant set $K$ within constraints $(X,U,D)$. It leverages monotonicity to reduce verification to feasibility of trajectories from extremal points and utilizes a two-stage algorithm that refines feasible and unsafe regions via Hausdorff distance, with improvements for CSM and Lipschitz dynamics. The coupled-tanks numerical example demonstrates fast, open-loop invariant computation and feasibility certificates under disturbances $d \in D$, underscoring the method’s practicality for safety verification in open-loop settings. The results pave a path toward closed-loop invariance analysis using tangent cones and to broader specifications such as signal temporal logic.

Abstract

This paper delves into the problem of computing robust controlled invariants for monotone continuous-time systems, with a specific focus on lower-closed specifications. We consider the classes of state monotone (SM) and control-state monotone (CSM) systems, we provide the structural properties of robust controlled invariants for these classes of systems and show how these classes significantly impact the computation of invariants. Additionally, we introduce a notion of feasible points, demonstrating that their existence is sufficient to characterize robust controlled invariants for the considered class of systems. The study further investigates the necessity of reducing the feasibility condition for CSM and Lipschitz systems, unveiling conditions that guide this reduction. Leveraging these insights, we construct an algorithm for the computation of robust controlled invariants. To demonstrate the practicality of our approach, we applied the developed algorithm to the coupled tank problem.

Figures (3)

• Figure 1: Coupled Tanks setup
• Figure 2: The Robust controlled invariant is in brown. The Unsafe set is in teal. The non-explored region of the state constraint set is in yellow. Three feasible trajectories initiated at: $[30,18]$ in green, $[29,19]$ in brown and $[20,20]$ in light blue, are shown. Feasible trajectories are determined for minimal control inputs.
• Figure 3: The Robust controlled invariant is in brown. The non-explored region of the state constraint set is in yellow. Four feasible trajectories from the following initial conditions: $[30,18.5]$ in red, $[29.5, 19]$ in green, $[26.5, 19.5]$ in brown, $[20,20]$ in blue . All feasible trajectories are determined for minimal control inputs.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Remark 2
• Proposition 1
• Definition 5
• Definition 6
• Lemma 1