Polytope Volume Monitoring Problem: Formulation and Solution via Parametric Linear Program Based Control Barrier Function

2025-03-16

AI Summary

This paper addresses the polyhedral volume monitoring (PVM) problem for nonlinear systems, aiming to design safety-critical control laws that prevent the state-dependent feasible region—represented as a polyhedron—from collapsing to zero volume. To ensure a lower bound on the polyhedral volume under multiple constraints, we propose a nonsmooth control barrier function (CBF) based on parametric linear programming (PLP), establishing—for the first time—a rigorous connection between nonsmooth CBFs and parametric optimization theory. We characterize the nondifferentiability of the PLP-based barrier function via directional derivatives and derive strict sufficient conditions for its validity as a CBF. Furthermore, we construct a quadratic-programming (QP)-based safety filter with guaranteed feasibility. Simulation results demonstrate that the proposed method effectively maintains the polyhedral volume bounded away from zero, overcoming the nonsmoothness issues inherent in conventional Chebyshev-ball-based approaches.

Technology Category

• Natural Language Processing: Safety and Robustness
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

Motivated by the latest research on feasible space monitoring of multiple control barrier functions (CBFs) as well as polytopic collision avoidance, this paper studies the Polytope Volume Monitoring (PVM) problem, whose goal is to design a control law for inputs of nonlinear systems to prevent the volume of some state-dependent polytope from decreasing to zero. Recent studies have explored the idea of applying Chebyshev ball method in optimization theory to solve the case study of PVM; however, the underlying difficulties caused by nonsmoothness have not been addressed. This paper continues the study on this topic, where our main contribution is to establish the relationship between nonsmooth CBF and parametric optimization theory through directional derivatives for the first time, so as to solve PVM problems more conveniently. In detail, inspired by Chebyshev ball approach, a parametric linear program (PLP) based nonsmooth barrier function candidate is established for PVM, and then, sufficient conditions for it to be a nonsmooth CBF are proposed, based on which a quadratic program (QP) based safety filter with guaranteed feasibility is proposed to address PVM problems. Finally, a numerical simulation example is given to show the efficiency of the proposed safety filter.

Problem

• Design control law to prevent polytope volume decrease
• Address nonsmoothness in Polytope Volume Monitoring problems
• Propose safety filter with guaranteed feasibility for PVM

Innovation

• Parametric linear program for nonsmooth CBF
• Quadratic program safety filter design
• Directional derivatives in parametric optimization