Opacity is an important information-flow security property in the analysis of cyber-physical systems. It captures the plausible deniability of the system's secret behavior in the presence of an intruder that may access the information flow. Existing works on opacity only consider non-metric systems by assuming that the intruder can always distinguish two different outputs precisely. In this paper, we extend the concept of opacity to systems whose output sets are equipped with metrics. Such systems are widely used in the modeling of many real-world systems whose measurements are physical signals. A new concept called approximate opacity is proposed in order to quantitatively evaluate the security guarantee level with respect to the measurement precision of the intruder. Then we propose a new simulation-type relation, called approximate opacity preserving simulation relation, which characterizes how close two systems are in terms of the satisfaction of approximate opacity. This allows us to verify approximate opacity for large-scale, or even infinite systems, using their abstractions. We also discuss how to construct approximate opacity preserving symbolic models for a class of discrete-time control systems. Our results extend the definitions and analysis techniques for opacity from non-metric systems to metric systems.
The increasing complexity in nowadays engineered systems requires great attention to safety hazards and occurrence of faults, which must be readily detected to possibly restore nominal behavior of the system. The notion of diagnosability plays a key role in this regard, since it corresponds to the possibility of detecting within a finite delay if a fault, or in general a hazardous situation, did occur. In this letter, we introduce and characterize the notion of approximate diagnosability for the general class of metric systems, which are typically used in the research community working on hybrid systems to deal with complex heterogeneous processes in, e.g., cyber-physical systems. This notion captures the possibility of detecting faults on the basis of measurements corrupted by errors, always introduced by non-ideal sensors in a real environment. Relations are established between approximate diagnosability of a given metric system and approximate diagnosability of a system that approximately simulates the given one. Application of the proposed results to the analysis of approximate diagnosability of nonlinear systems is finally discussed. Abstract In this paper we introduce and characterize the notion of approximate predictability for the general class of metric systems, which are a powerful modeling framework to deal with complex heterogeneous systems such as hybrid systems. Approximate predictability corresponds to the possibility of predicting the occurrence of specific states belonging to a particular subset of interest, in advance with respect to their occurrence, on the basis of observations corrupted by measurement errors. We establish a relation between approximate predictability of a given metric system and approximate predictability of a metric system that approximately simulates the given one. This relation allows checking approximate predictability of a system with an infinite number of states, provided that one is able to construct a metric system with a finite number of states and inputs, approximating the original one in the sense of approximate simulation. The analysis of approximate predictability of Piecewise Affine (PWA) systems is carried out as an application of the proposed approach.