Admissible Set for Linear Systems under Linear State Constraints

Abstract

This paper presents a method for computing inner polytopic approximations of admissible sets for continuous-time linear control systems subject to affine state constraints. Building upon barrier theory and the explicit solution of linear systems, a structured sampling procedure is developed to generate boundary samples of the admissible set directly, thereby avoiding the discretization of large regions of the state space required by many viability and reachability methods. The proposed framework computes admissible sets for the individual state constraints separately and combines the resulting inner approximations by intersection, avoiding the explicit treatment of simultaneous active constraints. The boundary samples are converted into a half-space representation, and runtime complexity analyses are derived for both the sampling and polytope reconstruction algorithms. An extension of the framework to Hammerstein systems is also presented. The proposed approach is demonstrated on a triple integrator and a higher-dimensional mass-spring-damper chain, illustrating the geometric construction of the admissible set approximation and demonstrating scalability to higher-dimensional systems.