*Article* **Marks: A New Interval Tool for Uncertainty, Vagueness and Indiscernibility**

**Miguel A. Sainz 1, Remei Calm 1,2, Lambert Jorba 3, Ivan Contreras 1 and Josep Vehi 1,2,\***


**Abstract:** The system of marks created by Dr. Ernest Gardenyes and Dr. Lambert Jorba was first published as a doctoral thesis in 2003 and then as a chapter in the book Modal Interval Analysis in 2014. Marks are presented as a tool to deal with uncertainties in physical quantities from measurements or calculations. When working with iterative processes, the slow convergence or the high number of simulation steps means that measurement errors and successive calculation errors can lead to a lack of significance in the results. In the system of marks, the validity of any computation results is explicit in their calculation. Thus, the mark acts as a safeguard, warning of such situations. Despite the obvious contribution that marks can make in the simulation, identification, and control of dynamical systems, some improvements are necessary for their practical application. This paper aims to present these improvements. In particular, a new, more efficient characterization of the difference operator and a new implementation of the marks library is presented. Examples in dynamical systems simulation, fault detection and control are also included to exemplify the practical use of the marks.

**Keywords:** modal interval analysis; interval arithmetic; marks; uncertainty modeling; indiscernibility
