Formalization of target and assessment of quality of the cyber-physical control systems for emissions reduction technologies

Monitoring and inventory is a necessary but not sufficient condition to reduce emissions in the energy and industry. The ultimate task of controlling air purification technologies in combination with monitoring is effectively achieved while minimizing the human factor. The creation and application of cyber-physical technology management systems provide a solution to a set of related environmental problems. In addition, the development of models based on cyber-physical systems forms new requirements for the development of technologies for their creation. A special feature of such technologies is close interaction between computing methods with design and production facilities. These issues have not been sufficiently reflected in publications. The article presents the target setting for the creation of cyber-physical technology control systems based on local laws of preservation and the decomposition of the target into macroscopic quality assessments. A mathematical model and methods of quality assessment at the life cycle stages of such systems based on recalculation of probability densities of a priori data are presented. Examples of cyber-physical emission reduction systems in the energy and industry developed under the nature-technology concept are given.


Introduction
Problems and tasks in the air quality problem area include monitoring and management of harmful emissions. Relevant emission monitoring and inventory programs and projects have been established both at the global and regional level [1], and for specific energy and industrial facilities. Well-known publications are usually limited by references to the application of cyber-physical monitoring results for following technologies [2,3]. Monitoring is necessary, but the use of cyber-physical systems (CPS) also involves feedback control tasks. Correct and rational penetration into the physical environment using methods and tools of cybernetics must be accompanied by an assessment of the reactions of the physical environment during its control. In the environment, such ultimate objectives are minimization, neutralization, recycling and other problem and objectoriented air quality assurance technologies. In this article, it is proposed that the target of air quality assurance should be presented in local forms on the basis of conservation laws. Air quality assurance technologies are proposed to be implemented at the life cycle stages of the CPS for control of the achievement of the required values of quality indicators based on the two-level model of the specification characteristics and ensuring their parameters. The article proposes to evaluate the quality of systems based on the main characteristics of the requirements specification at the stages of the life cycle. During the design, construction and technological preparation stages of production, there are all kinds of deviations from the limits of established norms and tolerances, having a random character. The article suggests an approach to formalization and decomposition of the CPS objective, a mathematical model and a method for determination (recalculation) of the density distribution of probability characteristics and parameters at any point in time on given initial a priori data. When recalculating parameters at this stage, not only time can be taken into account, but also the functions of dynamic errors, inconsistencies in changes in parameters and characteristics of technological processes.

Materials and methods
The developing class of systems called CPS is characterized by deep penetration and close, including feedback, connection of methods and means of cybernetics and physical environment [6,9]. The physical environment is subject to research and monitoring for the following control of its facilities. Emissions from energy and industrial units can be fully represented in the physical environment of the CPS.
Moreover, the purpose of the CPS as a necessary attribute of any control system can be formulated with sufficient certainty, including on the basis of objective conservation laws [10]. At the same time, the implementation of the life cycle of the CPS in relation to the above objects is related to the presence of uncertainties and random events. The implementation of the development, design and technological preparation stages of production should take into account random factors to ensure the quality of the CPS. The article consistently considers the approach to formalization of the target based on the models of physical processes and their interpretation in terms of control and formal representation of the target in the form of probability functionality of the requirements of the specification (quality indicators). Further decomposition of the target to the level of parameters that form these quality indicators allows to move to probabilistic quality estimates based on given initial a priori data -probability densities.

Formalization of the CPS target
The target of the CPS in the form of reducing emissions in the energy and industry through technology management can be formulated on the basis of physical laws that summarize many experiments and describe the evolution of the sought quantities, both in space and time. Most problems in physics lead to the need to solve differential equations [11]. It was shown in [4] that the equation of turbulent diffusion and convection corresponds to the character of the emission processes. This equation represents the local form of the law of conservation of mass and, after transformations, is reduced to the "input-output" form necessary for the implementation of the control functions for emission reduction technologies. "Output" is the emissions that are mitigated by the technology control system. The system, in turn, is subject to requirements, the fulfillment of which is represented by the generalized functional of the "maximum probability of quality assurance" [12] including the probability P of meeting the requirements of the technical task (TT) where V0and W0 are "initial" characteristics and criteria. At subsequent stages, they are coordinated up to the "n-th" stage of production preparation.

Decomposition of the target into macro and micro levels
The target in the CPS is decomposed into macroscopic characteristics that serve to assess the quality. Behind these basic characteristics of the CPS there are many different parameters, which are responsible for numerous designers and manufacturers of individual subsystems, units and parts of the CPS. The quality of the CPS is determined by compliance with the requirements. The main characteristics, called here "macroscopic", should not go beyond the limits established by the TT. At the stages of the life cycle, nonconformities are assessed in the form of deviations and errors. Macroscopic assessments at the stages of the life cycle, including the processes of design and production of CPS, are carried out on the measured coordinates depend not only on the parameters of the object itself, but also on the parameters of the design and production process. This is especially true for the stages of technological preparation of production and testing and selection of technological and testing equipment. At these stages, the "developer enterprise" receives from the "manufacturer" notifications about changes in design documentation. These notifications are often spontaneous and random.

Assessment of characteristics
Deviations from TT and errors have the character of random functions, the ordinates of which go beyond the established norms and tolerances. In the theory of random processes, this is reduced to the so-called "outlier problem", the solution of which does not lend itself to correlation theory.
The proposed approach is reduced to solving the following problems.
Determination in the design process of the probability of the ordinates of the measured variables going beyond the limit values of the TT and the duration of exceeding these values in these cases, correlation analysis is inapplicable and it is necessary to obtain the density of probability distributions in time and in time itself. This is due to the fact that these probabilistic characteristics at time tk+sare not related to characteristics at time tk. Assessments obtained in the early stages of design should provide an assessment of possible product defects in production.
Assessment of the state of technological equipment in preparation for production and in the production process.
Monitoring the state of the "project" during its movement from the technical assignment to the installation according to the probabilistic characteristics of the output variables.
Reduction of labor costs for testing by correcting design documentation and improving the technological preparation of production.
When solving these problems, it is proposed to proceed from the representation of the CPS as a dynamic system with the components Y(t) = (yi…yn) and For further transformations, you need to go to the new variables and parameters -Pyand P  . When considering at the beginning a particular case, when λ are deterministic, and the initial conditions Y0 are random values, the initial probability density Py,0(y10,…,yn0,t0)

Assessment of quality at the life cycle stages
At the initial stages, it is advisable to develop an enlarged metamodel of the product, according to which a differential equation is constructed for the new variables Py. In the future, this model is transformed into intermediate mathematical models [12]. The initial ( ) 00 , , t PY is set according to a priori data obtained from previous studies, analogs and results of a preliminary analysis of design problems, accumulated in databases [13…15]. In particular, if it is possible to obtain statistical characteristics obtained at the stages of testing elements or according to the data of the manufacturer of component parts, then the "recalculation" of the probability densities of these data in P  will allow setting ( ) Equation (2) in symmetric form takes the form Assuming that Eq.  The average duration of the ejection,  , is estimated by the well-known formulas of the theory of random functions. Estimates using formulas similar to formula (4) are carried out throughout the selected life cycle and represent the probability of achieving the goal formulated as functional (1). The recalculation of the parameters obtained at these stages of the life cycle into the estimated characteristics is carried out on the basis of the known in the theory of the probability of the relationship where Pz-is the probability density of the parameter z, Pxis the known probability density of the parameter x, () zx  = is the given design characteristic.

Results
The results obtained make it possible to obtain probabilistic assessments of the quality of systems according to the main characteristics of the technical specifications at the stages of the life cycle. In this case, various kinds of deviations from the boundaries of the established norms and tolerances, which are random in nature, can be taken into account when performing the stages of design, construction and technological preparation of production. The proposed mathematical model makes it possible to determine the probability density distribution of characteristics and parameters at an arbitrary moment in time for a given initial a priori characteristic. When recalculating the parameters at the stage under consideration, not only time can be taken into account, but also the functions of dynamic errors, inconsistencies in parameter changes, and characteristics of technological processes.

Discussion
The tasks of obtaining a priori information remain relevant. Their solution may be associated with the creation and accumulation of the content of knowledge bases in the relevant problem areas.

Conclusion
The given approach, mathematical models and methodology make it possible to ensure the quality control of CPS, the characteristic feature of which is the uncertainty of behavior and the random nature of the processes. The materials in this article are not intended to be complete and will be developed for receiving a priori data, a detailed description of the life cycle and in practical applications.