УПРАВЛІННЯ У ТЕХНІЧНИХ СИСТЕМАХ УПРАВЛЕНИЕ В ТЕХНИЧЕСКИХ СИСТЕМАХ СО N Т ROL IN TECHNICAL SYSTEMS OPTIMAL CONTROLLING PATH DETERMINATION WITH THE HELP OF HYBRID OPTIONAL FUNCTIONS DISTRIBUTIONS

Context. The problem of the determination of the optimal value of the augmentation coefficient of a proportional governor included into an inertness-less linear object control system on the basis of a synthesized model is solved. The object of the presented study is the optimal control process. Objective. The goal of the work is a creation of a method for a problematic situation of the optimum definition, evaluation, and determination solving at the control system. Method. A rough model of the phenomenon, and simplified dependence of optimal controlling trajectory upon the cost, of control in an inertness-less linear controlling system equipped with a proportional governor are proposed. The accuracy of the behavior of the investigated linear object of control has been chosen in the given consideration as an initial target value which needs to be minimized. The method of the model building with regards to an expenditures principle is offered. It provides taking into account the cost of controlling process. It allows finding the optimal controlling value on the multi-optional basis. There applied a certain analogue to the subjective entropy maximum principle of the subjective analysis in order to obtain a specific optimal distributions for the objective value in the view of the composed functional. The method of the uncertainty degree of the options extremization is improved by a continuous optional value introduction that allows forming the value distribution density. The optional synthesized model of the control process is built. Results. The developed theoretical models allow obtaining, and have been implemented in, finding the hybrid optional density as an optimal solution of a variational problem with two independent variables, which maximal value is the sought optimal controlling path delivering minimum to the integrated expenses pertaining with the process. Conclusions. The numerical experiments on the proposed methods studying in the problem of optimization are conducted. The discovered dependencies are substantiated as a result of these experiments. Their use in practice makes it possible, and is recommended, to carryout optimal control in the described systems. The prospects for further research may include creations of models for the optimal control trajectories findings on conditions involving rates of the considered values varying and in probabilistic, stochastic, undetermined problem settings.


NOMENCLATURE
y is an outlet value of the control system; t is a functions independent argument (time); 0 k is a coefficient characterizing the control influence effectiveness; u is a control function; f k is a coefficient characterizing the disturbance influence effectiveness; f is a disturbance function; р k is a coefficient of a governor augmentation; a function of time; a functions independent argument; ε is an error function; x is a given action function (an inlet value of the control system); ε L is a rate of the losses stipulated by the error; ε C is a coefficient of the error; n is a power index of the governor augmentation coefficient; J is an objective functional of the total expenses related to the process of control optimization; УПРАВЛІННЯ У ТЕХНІЧНИХ СИСТЕМАХ 0 t is an initial time of the process; 1 t is a terminal time of the process; F is an under-integral function of a functional; р k′ is a first complete derivative of an unknown (free) function of the governor augmentation coefficient with respect to the independent variable (time); * p k is an optimal function (extremal) delivering an optimal (minimal/maximal) value to an integral functional; h is a hybrid optional function distribution density; β is an internal structural parameter of the system optimal behavior; γ is an internal structural parameter for a normalizing condition; р k tΔ Δ is a degree of accuracy at the hybrid optional function distribution density entropy determination; h H is an entropy of the hybrid optional function distribution density; * h is an extremal control surface; t h′ is a first partial derivative of the hybrid optional function distribution density with respect to time (the first independent variable); р k h′ is a first partial derivative of the hybrid optional function distribution density with respect to the coefficient of the governor augmentation (the second independent variable).

INTRODUCTION
According to the contemporary progress in the development of the diagnosing and recognizing models synthesis it is still an actual scientific problem (task) in general to implement the modern achievements in the field of information technologies [1]. The importance of the issue lies in the plane of connections of the up to date technologies between themselves (a compatibility aspect) and putting them into practice.
Therefore scientific basis for the development of the presented theme will deal with modeling the optimal controlling process in regards to hybrid optional functions distributions densities taking into account the expenses related with the process.
Thus, this justifies the study of optional hybrid approaches combining different theoretical concepts and urgency of finding elements of generalization on the basis of the critical analysis and comparison with already known solutions for the synthesized models.
Therefore, the object of the presented study is the optimal control process which generates a problematic situation of the optimum definition, evaluation, and determination. And the subject of the study contained within the object is the rough model of the phenomenon, and simplified dependence of optimal controlling trajectory upon the cost of control in an inertness-less linear controlling system equipped with a proportional governor.
The aim of the work is to build up an adequate model of the mentioned above process development. And the tasks needed to be solved to achieve this aim are formulated as follows: to develop methods for quantitative and qualitative evaluations of relationships between: 1) the system accuracy behavior, 2) optimal values obtained on the basis of an expenditures principle, and 3) optimal controlling trajectory determination on the basis of optional hybrid functions densities entropy paradigm; with further justification of the methods through computer simulation and calculation experimenting, as well as the obtained results interpretation and discussion.
1 PROBLEM STATEMENT Suppose a linear inertness-less object of control with the time dependent outcome value: ( ) t y is given. As for a linear control system the structure of ( ) t y also depends upon the influences of control ( ) t u and disturbance ( ) t f and it is regularly represented with their sum: , where 0 k , f k -coefficients which characterize the effectiveness of the corresponding influence exerted upon the object. If the system is equipped with a proportional governor, which means ( ) ( ) then we come to the problem of the error minimization with the help of the р k coefficient increase since max р → k However, the value of the coefficient р k is restricted. Therefore, through the prism of the system accuracy cost analysis the synthesized control trajectory will have some optimal value: giving the minimal cost to the system accuracy.
In turn, the problem of the optimal function of ( ) as well as minimum to the cost to the system accuracy.

REVIEW OF THE LITERATURE
Theoretical core of the study is a synthesis of a hybrid approach on the basis of an optional uncertainty pattern. Mentioned in the introductory section of this paper problems of diagnostics and recognition models [1] has a variety in use.
It touches in actual fact the widest application of that kind of modeling, optimization, control, and all types of artificial intelligence involvements, starting with that for aeroengines [2] and their maintenance and diagnosing [3,4] respectively, as well as robust design works likewise [5] up to other aviation problems such as aviation noise and radio equipment qualities [6,7] and finally ending with the social and economical ones [8][9][10].
Informativeness of diagnostic attributes let us say for their optional use like formulated in patent [11] nevertheless requires every time redefining the proposed criteria. There must be a certain general optional based parameter with a character of optimality in conditions of uncertainty of the considered options.
Uncertainty of some intrinsic value, preferences functions, in the view of their entropy is proposed in Subjective Analysis [12][13][14][15] and its applications [16][17][18][19][20][21][22]; and those analogues will be widely used in this work, however in the view of optional hybrid functions distributions densities entropy paradigm since the humanbeing influence is excluded from consideration rather the objectively existing matter is taking into account only.
The mathematical background for the presented paper is the cornerstones of fundamental theories such as [23][24][25][26][27] but not restricted just to those.

METHODS
Let us consider a linear inertness-less object of control. Supposedly, its behavior is described with the equation of [23, P. 162, (1)]: where y -outlet value ( ) If there is a proportional governor in the control system, then its behavior equation has the view of [23, P. 163, (2)]: where р k is a coefficient of the governor augmentation; ε is an error value.
Traditionally, the accuracy of a control system is assessed with the value of, [23, pp. 160-165]: where ( ) t x is a given action which is predetermined by the task being solved.
The considered approach developed of the Eq. (1)-(3) yields the theoretical result represented in the following sequence of formulas.
From the relation (3) we can express the given predetermined value of ( ) t x on the basis of the relation (4): The error ( ) t ε , of the dependences (3), (4), in its turn from the interrelation (2), for the Eq. (4) will be expressed with the help of the equation (5): where the control ( ) t u , from the formula (1), is written as the expression (6): Hence, on condition of the formula (6) the error ( ) t ε from Eq. (5) becomes a functional dependence defined by the formula (7): Thus, substituting the expression (7) value into the Eq. (4), one can obtain the interrelationship between the input - influencing the object in the view of the next formula (8): Adducing the right hand part of the Eq. (8) to common denominator it gets it the other notation (9): Developing the obvious transformations for Eq. (9) we obtain dependence (10): Taking into account that from formula (3), (4) and substituting the value of the Eq. (11) for its value into the dependence (10) it yields the needed expression for the relation (11), between the error, inlet, and disturbance: After several simplest transformations of the Eq.
And finally, from the Eq. (13)- (15), the workable view relationship with the explicitly expressed objective values will be the dependence (16): As far as we can judge from the relation (16) that increasing the coefficient ( ) t k р it is possible to decrease the value of the error ( ) t ε . In actual fact, the increase of the governor augmentation coefficient ( ) t k р cannot be unlimited or endless.
Thus, we come to the problem of the optimal value of the coefficient ( ) t k р of the control system. Application of an analogue to the Subjective Entropy Maximum Principle [12][13][14][15][16][17][18][19][20][21][22] helps us solving such a problem with respect to distributions that might be considered an elements approach in the given case study.
Let us apply an Expenditures Principle which assumes that the rate of the losses stipulated by the error of the control system is proportional to the absolute value of the error ( ) where ε C is a coefficient.
The cost of the coefficient where р k C is a coefficient; n is a power index.
Then, the total expenses related to the process of the control optimization will be found as the integral functional (19) of the sum of these two mentioned above components (17) and (18): Then the functional (19) with respect to Eq. (20) becomes The functional (21) optimal value with taking into consideration the governor augmentation coefficient will be found on the basis of the Euler-Lagrange equation (22): where ( ) Since by assumption the under-integral function of the functional (19), (21) does not depend upon the first complete derivative of ( ) and for the entire Euler-Lagrange Eq. (22) we have one very important partial case of ( ) ( ) The optimal function (extremal) of ( ) t k p * delivering the optimal (minimal) value to the integral functional (19), (21) is obtained on the necessary condition of the functional extremum existence in the view of Eq. (27).
In order to shorten the notation starting from now we denote ( ) From the expression (28) we get the relation (29): (29) and finally the formula (30): Or the relation (31): For recursion or recurring or iteration it is interpreted with For the relationships (29) Since it is difficult to find the explicit analytical function of the optimal controlling path determination, any of the numerical solutions in the view of the expressions (29) 0 t t … . The control system optimal dynamic characteristics here will also be the extremal of ( ) t k p * (as that one yielded in the procedures of Eq. (1)-(33)) but for the considered case with taking into account the optional nature of р k as well as a degree of uncertainty of specifically introduced Hybrid Functions. A presented study is a development of Subjective Analysis, Active System, Multi-alternativeness Concepts, Subjective Preferences Approach, and Subjective Entropy Paradigm, Subjective Entropy Maximum Principle considered in the sequence of publications [12][13][14][15][16][17][18][19][20][21][22].
Here we consider a hybrid functions apparatus equivalent to the preferences functions one; however the hybrid functions are an intrinsic property of the system, being pertained to the system itself on the contrary with the preferences functions pertaining to the subject (active element of the system, person responsible for making controlling decisions, individual). Thus, we remove from taking into account a human-being, considering technically objectively existing optima instead although.
Here in the functional (34) the first underintegral member is the entropy of the hybrid optional function distribution conventionally without the logarithm of the degree of accuracy at the entropy determination р k tΔ Δ .
Also, here, in functional (34), there applied the sign "minus" before the internal structural parameter of the system optimal behavior (likewise endogenous parameter of the active element's psych [12][13][14][15][16][17][18][19][20][21][22]) at the value of 0 > β since we guess it is a better option having the minimal value of the system effectiveness function -the sum of the rates (17) and (18), and which has to be found in case of the two independent variables although.
For obtaining an extremal surface of ( ) р * ,k t h in such a case we will need the Euler-Lagrange equation in the view of the applicable formula for the functional of (34) in the case of the two independent variables, [24]: On the basis of Eq. (38) from the functional (34) we get the relation (39): From Eq. (39) we obtain the expression (40): On the basis of the normalizing condition Substituting the result of (41) into the expression (40) we find the canonical distribution of the hybrid optional function density as the extremal surface of ( ) р * ,k t h , that delivers maximal value to the functional (34) and is in that sense the optimal controlling surface -the dependence (42): (42)

EXPERIMENTS
Calculation experiments illustrate the theoretical speculations (1)-(42) of the above sections and subsections of the presented researches.
The numerical simulation has been performed for both one-dimensional and two-dimensional modeling cases. The accepted conditions were as follows: The obtained results of the mathematical modeling are shown in Fig. 1.
The hybrid optional function distribution density as the extremal surface of ( ) р * ,k t h , obtained by the formula (42), depicted as h" " in Fig. 1, is shown in conjunction with the corresponding surface of the sum of the rates (17) and (18), in the view of the function of the two independent variables of t and p k although, represented with the designation of Sum" " in the view of contour plots for the conveniences. Also in Fig. 1 the surface indicated as Z" " is shown. It illustrates the contour lines obtained from Eq. (33) in yellow color. The contour lines shown are marked for the paces of: " " 1000 − , " " 500 − , " "0 , " "500 , " "1000 . The fragment portrayed in Fig. 1 is represented for the time zone at the abscissa axis 210 0 ≤ ≤ t and governor augmentation coefficient range at the ordinate axis 270 0 ≤ ≤ p k .

DISCUSSION
The concepts described with the formulae of (1)-(42) yield the optimal value for the governor augmentation coefficient (see Fig. 1): which at the same time is with the formula (42), itself in its turn is the optimal argument that maximizes the synthesized objective functional (34), taking into account the uncertainty (that is represented with the entropy member, the formula (35) in the functional) of the normalized optional hybrid value ( ) р ,k t h .
All this allows treating the optional hybrid density as the optimal controlling surface with the relativity of its magnitude.
Critical comparison of the achieved results with the analogues [23] shows advantages of the proposed method as it takes into consideration the cost of the controlling system accuracy and centers a local optimum whereas without such assessments there is none.
Moreover, control in conditions of uncertainty in the given problem setting allows making allowance for the uncertainty of the hybrid optional functions distribution densities with respect to the objective effectiveness functions, which significantly differs from results discussed in monograph [25].
We should also note that the presented method, developed on the basis of variational principles [24] in application to the subjective analysis theory [11][12][13][14] actually dealing with the given sets of both discrete and continuous alternatives as well as uncertainty entropy measures for the system of the two independent variables [27], in fact does not have anything in common with the active system rather than objectively existing intrinsic properties of the controlled system.
It definitely has to have development in terms of mass service systems theory [26] in the direction of the entropy paradigm research.

CONCLUSIONS
The urgent problem of a mathematical model synthesis for the augmentation coefficient optimal value of a proportional governor included into an inertness-less linear object control system determination is solved.
The method of hybrid optional function distribution density entropy is firstly proposed. The discovered value of the hybrid optional function distribution density has a property of, and allows determining, an optimal value with respect to the synthesized objective functional concerning the uncertainty and normalization of such an option.
The practical significance of the obtained results is that the hybrid optional density delivering the maximal value to the synthesized functional has its own maximum that provides minimum in regards with the integrated cost of the controlling process. That must be taken as the optimal controlling path.