On the Completeness and Ordering of Path-Complete Barrier Functions

TL;DR

It is proved that under some conditions, such as when there exists a simulation relation between two path-complete graphs, it is possible to conclude that one graph is always able to provide less conservative safety conditions than another, independent of the algebraic properties of the switched system and the template of the barrier function under consideration.

Abstract

This paper is concerned with path-complete barrier functions which offer a graph-based methodology for verifying safety properties in switched systems. The path-complete framework leverages algebraic (barrier functions) as well as combinatorial (graph) components to characterize a set of safety conditions for switched systems, thus offering high flexibility (two degrees of freedom) in searching for suitable safety certificates. In this paper, we do not propose any new safety criteria. Instead, we further investigate the role that the combinatorial component plays in the safety verification problem. First, we prove that path-completeness, which is a property on a graph that describes the switching sequences, is necessary to obtain a set of valid safety conditions. As a result, the path-complete framework is able to provide a complete characterization of safety conditions for switched systems. Furthermore, we provide a systematic methodology for comparing two path-complete graphs and the conservatism associated with the resulting safety conditions. Specifically, we prove that under some conditions, such as when there exists a simulation relation between two path-complete graphs, it is possible to conclude that one graph is always able to provide less conservative safety conditions than another, independent of the algebraic properties of the switched system and the template of the barrier function under consideration. Such a result paves the way for a systematic use of the path-complete framework with barrier functions, as one can then consistently choose the appropriate graph that provides less conservative safety conditions.

Figures (8)

• Figure 1: Example for a path-complete graph for a dt-SwS $S$ with $\Sigma = \{1,2\}.$
• Figure 2: Example for a non path-complete graph for a dt-SwS $S$ with $\Sigma = \{1,2\}$. The sequence $\bm{\sigma}_3 = (121)$ cannot be accepted.
• Figure 3: $\hat{\mathcal{G}}_{np}$ corresponding to graph $\mathcal{G}_{np}$ for Example \ref{['ex:nec']}
• Figure 4: Path-complete graph $\bar{\mathcal{G}}$ that is simulated by the graph $\mathcal{G}$ in Figure \ref{['fig:pcgraph']}.
• Figure 5: PC graph $\mathcal{G}$ utilized for the examples in Section \ref{['subsec:cs1']} and Section \ref{['subsec:cs_comp']}.

Theorems & Definitions (20)

• Definition 2.1: System Safety
• Definition 3.1: Path-Complete Graphs
• Remark 3.2
• Definition 3.3: Graph-Based Barrier Function
• Theorem 3.4
• proof
• Theorem 4.1: Necessity of Path-Completeness
• Definition 4.2: dt-SwS Construction
• Lemma 4.3
• proof