Analysis of approaches to solving problems of building models and control systems of technological processes in conditions of uncertainty



Currently, the most relevant is a solution to the problem of modeling and optimal control, which is made taking into account the uncertainty of the initial information. Uncertainty may be due to various reasons, including inaccuracy of mathematical models. When solving problems of optimal control, possible deviations from the nominal values of a number of parameters affecting the process, but not measurable, should also be taken into account.

In the practice of designing systems for automatic and automated control of technological objects, quite often a situation occurs when the real values of individual parameters of control objects are unknown and there are no statistical descriptions of them. Uncertainties in the parameters may appear due to various reasons:

– using simplified models that approximate a real physical process;

– incomplete estimation and identification of unknown variables;

– the presence of non-stationary coefficients in the equations of the mathematical model of the object.

In addition, the accuracy of the description of control objects can be affected, for example, by such technological factors as equipment defects, vibrations, uneven running, changes in operating modes, errors in measuring instruments, inaccuracy of scales. Operational features, such as aging and wear of equipment items, fluctuations in temperature, humidity, pressure and other external influences, can also lead to a deviation of real characteristics from nominal values. In all the cases listed above, the nature of the variations of the unknown parameters of the control systems can be considered undefined, since their changes are subject only to a priori restrictions. To solve problems of managing objects with uncertain parameters, they usually involve minimax methods, methods of stochastic control, fuzzy logic, invariance, and adaptive control. The use of a particular approach depends on the type of uncertainties and requirements for the structure and quality of management systems.

In the practice of designing systems for automatic and automated control of technological objects, quite often a situation occurs when the real values of individual parameters of control objects are unknown and there are no statistical descriptions of them. Uncertainties in the parameters may appear due to various reasons:

– using simplified models that approximate a real physical process;

– incomplete estimation and identification of unknown variables;

– the presence of non-stationary coefficients in the equations of the mathematical model of the object.

In addition, the accuracy of the description of control objects can be affected, for example, by such technological factors as equipment defects, vibrations, uneven running, changes in operating modes, errors in measuring instruments, inaccuracy of scales. Operational features, such as aging and wear of equipment items, fluctuations in temperature, humidity, pressure and other external influences, can also lead to a deviation of real characteristics from nominal values. In all the cases listed above, the nature of the variations of the unknown parameters of the control systems can be considered undefined, since their changes are subject only to a priori restrictions. To solve problems of managing objects with uncertain parameters, they usually involve minimax methods, methods of stochastic control, fuzzy logic, invariance, and adaptive control. The use of a particular approach depends on the type of uncertainties and requirements for the structure and quality of management systems.

The analysis of the scientific and technical literature of recent years concerning research on the development of methods for the parametric identification and synthesis of control systems for technological objects indicates the achievement of significant theoretical and practical results in this field. There are and are developing various ways of building control systems that operate under conditions of a priori parametric uncertainty [1].

At the same time, the possibilities of interval methods in the problems of synthesis of mathematical models and control systems are not sufficiently appreciated in the literature [2]. This is due to the fact that so far no concepts and a constructive methodology have been developed for constructing models and process control systems under the conditions of interval-parametric uncertainty. Methods and algorithms for formalizing and synthesizing mathematical models of control objects, estimating parameters of technological control objects, and analyzing the stability of controlled systems under interval uncertainty have been insufficiently developed. Adaptive-interval methods and algorithms for the synthesis of control systems of technological objects with complete and simplified models of control objects also require their development. These circumstances are the reason for the difference between the theoretical and practically obtained characteristics of the estimates. Sometimes these differences become significant and are of fundamental importance for deciding whether to use the results of the work of interval estimation algorithms.

Thus, at present, the level of detail, the trade-off side between the complexity, accuracy and feasibility of mathematical models of technological processes and control systems, as well as the consideration of uncertainties, are mostly expressed by heuristic considerations of the designer based on experimental data, if any. Under these conditions, it is rather problematic to talk about how effective one or another interval method is used and whether the best options for the synthesis of models and control systems are selected. In connection with the aforementioned, the development of effective interval methods for parametric identification and synthesis of control systems for technological objects in conditions of uncertainty cannot yet be considered complete.

The causes and sources of uncertainties in dynamic systems include the following:

– input uncertainty;

– inaccurate parameter measurements and fuzzy assignments of numerical coefficients of the mathematical model due to instrumental error of measuring devices (errors of data pickup sensors) or methodological experimental error;

– inaccurate (indistinct) knowledge of the numerical value of the range of parameter variation, although quite representative experimental material has been compiled;

– change of parameters in known numerical ranges (intervals) or areas (ellipsoid, parallelepiped, etc.), for example, (parameter tolerances are set);

– unrecorded errors of calculations and methodological errors of calculations carried out when identifying the coefficients of the mathematical model of the system;

– an approximate mathematical description of the physical process under study; use of approximate solution methods;

– intentional simplification of the mathematical model: linearization of the model, neglect of the interaction between the subsystems in a complex system, decomposition of a multiply connected system;

– uncertainty of the structure of the mathematical model;

– difficult realizability, high labor intensity or unreliability of other methods, etc.

Theoretical and practical approaches to the formalization and use of high-quality information under conditions of uncertainty are being developed to select optimal management solutions for complex systems. For dynamic systems in conditions of uncertainty, specialized estimation methods are also developed in the problems of identification and filtering. At the same time, both already developed methods of game theory and new methods related to information processing and interval analysis are also widely used.

Each of the above mathematical methods is focused on certain classes of mathematical models:

– probabilistic models;

– range;

– linguistic;

– models allowing to obtain ellipsoidal estimates of the reachability regions of phase coordinates;

– multimode models of dynamic control systems;

– interval models; mathematical models that combine both interval parameters and probabilistic characteristics.

For a more complete description of the properties of control objects in mathematical models of dynamic systems and quality criteria for the operation of closed circuits, it is also advisable to take into account the uncertainty contained in the model parameters. In addition, the adequacy of the description of the mathematical model of a dynamic system in some cases is achieved only by taking into account uncertain parameters. Often, uncertainty is an integral part of a formalized mathematical model of a controlled dynamic system.

In this paper, we will stick to the point of view that if the exact numerical value of a certain parameter is unknown (for example, measured or specified with an error), then such parameters will be considered as belonging to certain numerical intervals with known boundaries. The numerical value of this parameter will be determined by the interval — a bounded set on the number line. We will speak of a mathematical model containing indefinite parameters as an interval-indefinite model or as a mathematical model that is under conditions of interval uncertainty. The presence of interval-indeterminate parameters (interval numbers) in a mathematical model makes it difficult to apply the known methods of analysis and synthesis of control systems for dynamic objects. Therefore, the further development of interval versions of the methods for analyzing and synthesizing dynamic systems with interval uncertainty seems very relevant.

In practice, the solution of the control problem is sought according to a certain mathematical model, the accuracy of which is conditional. Various models can describe uncertainty in the values of the parameters:

1. Probabilistic (stochastic) model. It is used when an uncertainty factor can be attributed to a probabilistic, random nature. Random factors are described as probabilistic if probability density is given.
2. Statistical model. It is used when an object model is determined from the results of selective experiments under conditions of random noise and errors. In the statistical description of uncertainty factors, instead of the true moments, only estimates are obtained, the accuracy of which is determined by the experimental design, the number of experiments, interference dispersion, estimation method, etc. Thus, it is obvious that all the difficulties associated with operating with estimates of random parameters, stand up here in full.
3. Fuzzy (blurred) model. To describe the uncertainty factor in this situation, use the methods of the theory of fuzzy sets, the main provisions of which and decision-making methods are set forth. Currently, methods for solving such problems are poorly developed.
4. Interval model. Used when the range of possible parameter values is known. Interval analysis is used as a mathematical tool for solving a problem, the use of which for solving control problems became especially noticeable in the last century.

To achieve this goal the following tasks were solved:

– analysis of approaches to solving problems of building models and process control systems in conditions of uncertainty;

– formalization of the tasks of analysis and optimal control of the object under study in the context of parametric uncertainty of the initial information.

– development of the concept and methodology for constructing models and process control systems in the conditions of interval-parametric uncertainty;

– development of methods and algorithms for estimating the parameters and state of technological control objects;

– development and practical implementation of software and algorithmic support systems for solving problems of identification and synthesis of control actions under the conditions of interval uncertainty of the initial information;

– practical testing of the developed methods and computational schemes for the synthesis of process control systems based on the developed algorithms and software.

Solving these problems allows us to develop constructive methods, algorithms and computer models for parametric identification and synthesis of control systems for technological objects, contributing to an increase in the efficiency of functioning of control systems for technological objects of various functional purposes [3].

References:

1. Khlebalin N. Ah. Pyatikh D. S. Modeling of automatic control systems with interval uncertainty of parameters. Conf. ICCM-2004. Novosibirsk: — p. 258–266.
2. Ostrovsky, G. M. Technical systems in conditions of uncertainty: flexibility analysis and optimization // M.: BINOM, 2008. — 319 p.
3. Khalilov A. J. Computer modeling of control systems of technological processes in the conditions of parametric uncertainty. Scientific and technical journal. 2018 № 4. Mining Bulletin of Uzbekistan. P. 86–91.