The Mathematical Models of Lattice Functions in Modelling of Control System

The paper proposes an approach to constructing a mathematical model of lattice functions, which are mainly used in the study of discrete control systems in the time and domain of the Laplace transform. The proposed approach is based on the assumption of the physical absence of an impulse element. An alternative to the classical approach to the description of discrete data acquisition - the process of quantization in time, is considered. As a result, models of the lattice function in the time domain and the domain of the discrete Laplace transform are obtained. Based on the obtained mathematical models of lattice functions, a mathematical model of the time quantization element of the system is obtained. This will allow in the future to proceed to the construction of mathematical models of various discrete control systems, incl. expanding the proposed approaches to the construction of mathematical models of multi-cycle continuous-discrete automatic control systems

System of Laser Monitoring of Water Pollution with Application of Relative Description of Signal Shape

As a rule, the wastewater treatment system is not designed to filter substances formed, as a result of beyond design basis accident. The nature of the beyond design basis accident is associated with the shortterm appearance of a clot of these substances in wastewater, determined by the volume of the substance storage tank. Therefore, a rational approach is to divert this portion of the formed substances into a separate branch of the sewage system or sedimentation tanks. The aim of the work is to implement this approach by creating a laser monitoring system for water pollution.The article proposes a system for automatic detection of a clot of emergency discharge of pollutants into the wastewater of an industrial enterprise. The structural diagram of the system and the purpose of its main elements are given. The system should provide clot detection in real time. To ensure this function, a preliminary study is made of the spectral characteristics of all substances that may appear in wastewater in the event of an emergency.Based on these data, the wavelengths of laser radiation in the system are selected. The obtained measurement data from several probes are presented in the form of a lattice function, which is translated into a relative description representing the order relationship matrix on the set of lattice function components. The relative description is invariant to linear changes in the lattice function. The decision to detect any substance from emergency discharges is made based on a comparison of the relative description of the measurements with the standards prepared at the stage of system setup.The article provides an example of the formation of standards for emergency clots from glycerin and allyl alcohol. The graphs of the lattice functions obtained from the IR spectra of emergency discharges of these substances are given; algorithms for constructing a lattice function and comparison of lattice functions. Thus, using the developed mathematical description of the shape of digital signals based on the relative description, the signal of the monitoring curve can be described in the form of a curve of the optical density change of an aqueous medium.

On weak positive predicates over a finite set

Predicates that are preserved by a semi-lattice function are considered. These predicates are called weak positive. A representation of these predicates are proposed in the form of generalized conjunctive normal forms (GCNFs). Properties of GCNFs of these predicates are obtained. Based on the properties obtained, more efficient polynomial-time algorithms are proposed for solving the generalized satisfiability problem in the case when all initial predicates are preserved by a certain semi-lattice function.