Notion of Opacity Considering Security Levels for Piecewise Affine Systems

Cyber-physical systems (CPSs) integrate physical processes and information components through communication networks and are therefore vulnerable to cyber attacks. Opacity is a security property that prevents an adversary from inferring sensitive information from observations, and it has been studied mainly for discrete-event systems. In this paper, we extend this concept to discrete-time piecewise affine (DT-PWA) systems, which constitute an important class of hybrid systems used to model CPSs. In conventional opacity analysis, the result is typically binary, i.e., a system is either opaque or not. For systems with continuous dynamics, however, such a binary characterization may be insufficient, and it is desirable to evaluate the degree of security. To address this issue, we introduce a notion of opacity that incorporates security levels. We first formulate opacity for DT-PWA systems and then derive a necessary and sufficient condition for opacity. Based on this condition, we present a verification method using polytope computations and discuss the interpretation of the proposed notion. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.