Russia, Petrozavodsk, Petrozavodsk State University (PetrSU)
INFORMATION-MATHEMATICAL ASPECTS OF THE RENEWAL SYSTEMS CONTROL FOR FINITE TIME INTERVAL
The paper presents some approaches to solving the problems that arise in investigation of the problem of optimal control of a renewal system for finite time interval from the stage of statement and analysis of the problem to the stage of software programming.
Россия, Петрозаводск, Петрозаводский государственный университет (ПетрГУ)
ИНФОРМАЦИОННО-МАТЕМАТИЧЕСКИЕ АСПЕКТЫ ПРОБЛЕМЫ УПРАВЛЕНИЯ ВОССТАНАВЛИВАЕМЫМИ СИСТЕМАМИ НА КОНЕЧНОМ ИНТЕРВАЛЕ ВРЕМЕНИ
В докладе рассказывается о некоторых подходах к решению задач, возникающих при полном цикле исследования (от постановки задачи и ее анализа до создания соответствующего программного обеспечения) по организации оптимального управления восстанавливаемыми системами, которые функционируют на конечном интервале времени.
The problem of renewal system control for finite time interval [1, 5, 6 and 8] is attracting increased interest due to its practical applications. Usually the renewal processes, alternating and semi-Markovian processes are used as a model of renewal systems operation. Some models are based on embedded processes [12, 14]. After a model has been developed, its adequacy has to be checked. In order to determine control parameter values a corresponding optimization problem has to be stated and solved. The solution found is analyzed and if this analysis proves that this solution is practically acceptable then the corresponding computer software should be developed. This software must be convenient enough for the managerial personnel who do not possess specialized mathematical and computer skills.
2. Renewal system simulation
Some approaches to simulation of renewal system operation on the example of paper-machines (PM) with the use of embedded alternating processes as a model are presented in [1, 12 and 14]. In those papers five stages have been distinguished for describing the operation efficiency of a paper-machine. These are working, accident by mechanical cause, accident by technological cause, stopping by administrative and economical cause and planned preventive maintenance. The model is based on three embedded alternating processes.
Unfortunately, the data received from this model is censored. It satisfies the following testing plans [n, B, T], [n, T1, …, Tn], [n, A, (F1, T1, …, Tn)], [n, A, (F1, F2, T1, …, Tn)] etc. This fact complicates the checking of model adequacy and requires development of special statistical methods.
In author’s opinion the use of embedded processes as models is more prospective than the use of semi-Markovian processes.
3. Statistical analysis of mathematical model parameters
We now present (perhaps incomplete) list of statistical problems, which arise when renewal processes are used as a model of a renewal system :
Statistical checking of the model described in the previous section and estimation of reliability parameters of the renewal system have required the development of special statistical methods. The results obtained by the author are presented in [4-6, 9-15].
4. Control system simulation
The following problems arise during control system simulation:
The real practice leads to a series of optimization problems for renewal systems functioning. Therefore, synthesis of optimization mathematical model for functioning of a renewal system is a very complicated problem. To get an idea of how to approach this problem, the author suggests taking a look at the results of development and investigation of optimization models for planning preventive measures (PPM) for a complex system of paper-machines on finite time interval [0, T]. These results can be found in [1-3, 6 and 8]. Optimization problems described there are nonlinear problems of separable continuous-discrete programming with constraints of “or-or” type. Besides that, these are multi-criterion problems. As a rule, the number of variables in these problems is too large to be able to solve them by straight item-by-item examination. The dynamic programming method often does not apply, too. Application of some heuristic methods with local optimization gives acceptable results.
5. Requirements to the information system of control support
The following problems arise when the information system of control support is being developed:
Generalizing the results of development and applying the information and mathematical support for solving different optimization problems of control for some enterprises of North-Western region of Russia (Karelia, Arkhangelsk area), we come to the conclusion, that now there is a need in developing special software environment. It should be oriented on solution of practical problems in the following areas: information systems and networks, service of operative repair of technological equipment, robot-technical complex, transport systems, problems of calendar network planning etc .
Let formulate the main requirements of this environment. It must consist of the following units:
Information unit is intended for statistical data entry, lexicographic data entry and database entry. This information about control support system will be used to solve optimization problems.
Mathematical unit will
Mathematical unit has to have modular structure. Every user can use only modules he/she needs.
Service unit is intended to specify characteristics and parameters of the control support system. This unit is used with specialists having mathematical knowledge and experience of mathematical model development. Such specialists also apply user interface, as users are not supposed to have special mathematical and mathematical programming knowledge. If application is correct then specialist-mathematician does not take part in the environment exploitation. Thus the service includes choice of mathematical model of control support system (possibly, several models) and choice of user interface.
Printing unit is intended to prepare and print different reports. Unit has to have graphic means.
Existing software, which can be used for solving the optimization problems, may be divided into two groups. The first, such as MathLab, MathCad, Mathematica, are used by specialist-mathematician in laboratory conditions. The second type – Excel, Quattro Pro – is intended for economical calculation. It has limited mathematical means and can not be used for solving optimization problems. Besides, practically all software does not have convenient tools for customizing the environment for non-specialist users.