FORMATION OF A REFERENCE MODEL FOR THE METHOD OF INVERSE DYNAMICS IN THE TASKS OF CONTROL OF UNDERWATER COMPLEXES

Main requirements to the systems of automatic control of underwater complexes are specified. Main points of the synthesis of control systems by the method of inverse dynamics are given. The drawbacks of the method of inverse dynamics are defined, which consist in the limiting of the choice of the structure of a reference model and the necessity of bringing a mathematical model of the object of control to the view of a reference model. A mathematical model of an underwater complex is developed for the synthesis and study of the systems of its control that simulates a one-dimensional motion of an underwater vehicle under conditions of perturbing influence of a tether cable. Impossibility of the use of the method of inverse dynamics in the classic form for the synthesis of automatic control systems of underwater complexes is determined because their mathematical models contain elements that are set in arbitrary form. The method of inverse dynamics was improved by applying the principle of decomposition of a reference model, which boils down to its formation as a set of reference submodels based on the actual structure of an object. Decomposition of a reference model makes it possible to synthesize high–precision control systems of underwater complexes by the method of inverse dynamics and to set reference submodels of dynamics of phase coordinates of an object in the form of arbitrary functions of time. Automatic control system of a one-dimensional motion of an underwater complex was synthesized by the method of inverse dynamics using the principle of decomposition of a reference model. The author proposed three variants of a reference model of an automatic control system. The first variant makes it possible not to execute conversion of a mathematical model of the object to the view of a reference model. The second and the third variants are formed on the basis of the structure of a mathematical model of an object, presented in the form of Cauchy. This allows setting the dynamics of transition processes for each phase coordinate of the object in the form, accordingly, of differential equations of the first order and arbitrary functions of time. Computer simulation of an automatic control system revealed that the transitional process of eliminating an error of the controlled magnitude corresponds to the reference one. After eliminating the error, the controlled magnitude strictly follows the set one without delay.


Introduction
A wide range of underwater work is performed with the use of underwater complexes (UC), which usually consist of underwater vehicles (UV) and surface support vessels [1].For rapid information exchange and power supply to underwater component of the complex, flexible connections are used that connect UV with support vessels [2].The main type of flexible connection between the elements of UC is a tether cable (TC).
Solving modern underwater tasks is impossible without efficient systems of automatic control (SAC) of UC.Such SAC must provide coordinated control of several non-linear multidimensional objects, which include surface ships, UV and TC.The tasks of managing UC are formed as a set of tasks of control of its constituents and require synthesis of SAC: -high dynamic accuracy for the processing of tasks, in the form of functions of time; -by one dimensional and multidimensional systems; -with robust properties; -capable to work and maintain stability under conditions of parametric uncertainty.
The presence of mechanical connection via TC between the components of UC significantly complicates the problem of synthesis of SAC.In addition, mathematical models of the elements of UC contain nonlinearities that are set in arbitrary form.
One of the powerful methods, which enables to synthesize high-precision SAC by complex non-linear objects, is the method of inverse dynamics (ID) [3].Inverse problem of dynamics is known from theoretical mechanics as the problem of determining forces acting on a body if the trajectory of its motion is known.In the theory of automatic control, the method of ID implements the concept of inverse control.Its essence consists in the following.Based on the requirements to control quality, a reference model of SAC is synthesized.It is a differential equation with desired dynamic characteristics of control system.The order of the differential equation of a reference model cannot be less than the order of the object of control.Next, a senior derivative is expressed from the reference model.The resulting expression is substituted into the equation of the object instead of a senior derivative and a law of control is expressed from it.
In the theory of automatic control, the method of ID was launched as a method of structural synthesis of nonlinear SAC [4].The papers [5,6] contributed to the wide spread of the method of ID, in which high-precision SAC are synthesized by nonlinear objects with low sensitivity to parametric and coordinate perturbations.These particular problems occur at the synthesis of UC SAC because substantial nonlinearities and parametric uncertainties of the control object are characteristic for them, predetermined by the operation under real sea conditions.Therefore, development of the method of inverse dynamics for the automation of UC control is an actual scientific task.

FORMATION OF A REFERENCE MODEL FOR THE METHOD OF INVERSE DYNAMICS IN THE TASKS OF CONTROL OF UNDERWATER COMPLEXES О . B l i n t s o v
PhD, Associate Professor Department of Information Security Lviv Polytechnic National University Bandera str., 12, Lviv, Ukraine, 79000 E-mail: energybox@mail.ru

Analysis of scientific literature and the problem statement
In the implementation of underwater technologies, high dynamic accuracy of control is crucial.However, the method of ID contains a drawback that makes its application for the control of UC impossible without considerable simplifications of their mathematical models (MM).
The essence of the drawback is in the following.A reference model for a one-dimensional object is chosen in the form of differential equation of the n-th order, which corresponds to the order of the object.However, the structure of the UC MM as the object of control contains nonlinear elements that are set in arbitrary form [7].For example, a MM of propellers and UV hulls contain non-linear coefficients of hydrodynamic nature, which are set in tabular or graphic form [8]. Bringing MM of such objects to the structure of a reference model is impossible.And simplification of UC MM greatly reduces accuracy of control.In this regard, it is impossible to obtain a high-precision UC SAC by the classic method of ID.
The choice of the structure of a reference model is also limited in the method.It must be set in the form of a differential equation.The possibility to set a reference model in the form of arbitrary function of time, for example, in the form of analytical solution of the selected differential equation, is not provided.This limits the type of transitional processes of SAC.
The paper [9] proposed to modify the method of ID for the realization of control system of a manoeuvring aircraft.The system settings that cannot be measured by sensors are calculated by way of solving nonlinear differential equation.But the synthesis of SAC is performed according to the classical procedure.The peculiarities of application of the method of ID to control an object with lower number of controlling influences than the number of degrees of freedom are outlined in [10].However, the synthesis of a regulator is also performed by the classical procedure.The method of ID for synthesis of the system of stabilization of UV pitch [11] proved efficient.However, a mathematical model (MM) of the object was significantly simplified for its application, which reduced the accuracy of control.
To manage the objects under conditions of uncertainty, the method of ID is used in conjunction with the tools of fuzzy logic.The paper [12] proposed a structure of control system of a manipulator, the main component of which is derived by the classical method of ID.The effect of uncertainty is eliminated by fuzzy elements in additional compensation circuits with direct or reverse connection.
The ideas of inverse dynamics are developed in the approach of "direct -inverse control", in particular, for the synthesis of regulators under conditions of parametric uncertainty.One of the elements of these SAC is an inverse model of the object in the form of artificial neural network, which recreates its inverse dynamics [13].However, training neural network is necessary to perform under condition of the absence of external perturbations.The availability of flexible connections in the composition of UC makes the separation of perturbing effect impossible and reduces the accuracy of these regulators.
Based on neural networks, model-less control system of technical objects with forecasting inverse control are synthesized [14].The disadvantage of such systems is significant dependence of the quality of control on the type of a refer-ence transition process, as well as sensitivity to perturbing influences.
The concept of inverse dynamics is also used in the method of adaptive inverse control of nonlinear objects based on artificial neural networks [15].Unlike the classical method of ID, neural network approximates inverse dynamics of the object and functions under conditions of structural uncertainty of the object.The use of this approach for the synthesis of UC SAC does not provide high dynamic accuracy of control.
A system of dynamic stabilization of a sea platform based on the control with forecasting models was proposed in [16].It is based on linearization of the object's MM, which reduces the accuracy of control.The paper [17] designed a system of control of the motion of an autonomous underwater vehicle.But its application for the automation of UC motion will not provide high accuracy of control due to the presence of a flexible connection.
A system of automatic control of the course of UV of working class as part of UC based on a PD-regulator was developed in [18].It provides smooth control of the course at low speed, characteristic for UV of working class, which restricts the application of resulting regulator.In addition, such a SAC is sensitive to parameter errors of the object's MM that does not provide high accuracy of control.
Therefore, the analysis of scientific and technical literature reveals that the method of ID is used for control of non-linear technical objects of different nature mostly in the classic form.The development of other approaches of the concept of inverse control is carried out in the direction of control under conditions of structural and parametric uncertainties of the object.The questions of high precision control of UC, in particular, by the concept of inverse control, are still at initial stage.

The purpose and objectives of the study
The aim of the work is improvement of the method of inverse dynamics by applying the principle of decomposition to form a reference model that enables to synthesize high precision automatic control system of underwater complexes, the non-linear elements of which are set in arbitrary form.
To achieve the set goal, the following problems are to be solved: -development of a MM of one-dimensional motion of UV as part of UC; -design of the principle of decomposition of a reference model for the method of ID; -synthesis of SAC of one-dimensional motion of UV as part of UC; -verifying a synthesized SAC by the method of computer simulation.

1. Mathematical model of one-dimensional motion of an underwater vehicle as an element of an underwater complex
Underwater complexes represent complex multidimensional objects.The main performer of an underwater task is UV.Spatial motion of UV as a solid body in the composition of UC can be presented by a set of elementary one-dimensional motions [19].
Let us consider a mathematical model (MM) of one-dimensional motion of a propelled UV as a component of UC, i. e. under conditions of perturbing effect of TC (Fig. 1).

Fig. 1. Structure of MM of one-dimensional motion of UV as part of UC
Controlled motion of UV is provided by a thruster, which is an electric motor (EM) of direct current with a reducer, connected by a shafting with a screw propeller (SP).The inner space of electric motor is filled with liquid dielectric.A mathematical model of the thruster consists of EM MM and SP MM.Kinematical parameters of the UV motion are calculated in the MM of UV hull dynamics, which takes into account hydrodynamic force of resistance.The main influence of perturbation is the tension force of TC, which is calculated in the MM of the TC dynamics.
Such an object of control is classified as one-dimensional because it contains one control action u and one controlled magnitude y.In this case it consists of two one-dimensional objects, connected in series.Control input of MM of the thruster is the magnitude u, the output is the magnitude F c , control input of MM of the UV hull is the magnitude F c , the output is the magnitude y.
Electrical transient processes that occur in the electric motor are considerably faster than the mechanical ones, which allows accepting an assumption about instant change of current at the change of control action u, and presenting a MM of the thruster by a differential equation of the first order.One-dimensional motion of UV as part of UC, taking into account the adopted assumption, is described by a system of nonlinear differential equations: where u is the control signal; ω is the speed of rotation of a screw propeller; y is the coordinate of UV (controlled magnitude); J is the inertia moment of the elements of the thruster brought to the screw propeller; k u , k ω are the coefficients that characterize electromechanical parameters of the thruster; M d is the hydrodynamic brake moment of SP; m is the mass of UV with added masses of water; F c is the SP thrust (control force); F s is the resistance force of UV hull; F d is the perturbing force of TC.At any point in time the state of UV is characterized by three phase coordinates: y, ẏ and ω.Detailed description of equations of MM of the motion of tethered UV is given in scientific and technical literature, such as in [19].

2. Problems of the synthesis of automatic control systems of underwater complexes by the method of inverse dynamics
According to the classic method of ID, phase coordinates of one-dimensional object must be presented by the values of the object's output y and its (n-1) derivatives.A mathematical model of the object in this case should have the view of a differential equation of the n-th order: where y is the control value, u is the control influence.Phase coordinates of an object uniquely determine its state at any given time, but do not make it possible to forecast the motion of an object, i.e., its status in the next moment of time in the infinitely small time interval dt.To define the motion of an object, it is necessary to know phase coordinates and velocities of their changes.In this case only the speed of change of the phase coordinate y (n-1) , which equals y (n) , depends clearly on the control influence u, i. e. it can be changed in the required direction.The magnitude y (n) represents the highest derivative of the controlled magnitude of the object at any point in time.
Thus, for the motion of the object along required trajectory, the condition must be true in any moment of time: where ( ) y is the desired value of higher derivative of the controlled magnitude.This value is determined from the reference model that connects the controlled magnitude y and its set value y g (t) and presents a differential equation [4]: The set value of the controlled magnitude y g in a general case may be known (program SAC) or unknown (trailing SAC) function of time y g (t).
Next, from a MM of the object we receive its inverse model: Desired value of the controlled influence is found by substituting ( )   n d y instead of ( ) n y in the inverse model of the object: The simplest implementation of UC is a control object of the 3rd order, the dynamics of which is described by a system with one equation of the first order and one equation of the second order (1).According to the classic method of ID, a reference model should represent a differential equation of the third order.As a reference model, let us take a linear differential equation with a single coefficient of amplification and the time constants T 0 , T 1 , and T 2 : where E y is the error of control.
Let us express higher derivative from the reference model: where d y  is the desired value of higher derivative of the controlled magnitude.
The expression for d y  must be substituted in a MM of the object of control.To do this, we apply the operation of differentiation to the second equation of a MM of the object: From the resulting expression we will find necessary value of the first derivative of the control force cg F ,  the action of which to the UV hull will provide a desired value of higher derivative of the controlled magnitude d y : A law of control is obtained from the system of equations: ( ) where d ω  is the desired value of the first derivative of the magnitude ω.In this case d ω  has to be determined based on Ḟ cg .
Under the condition of the problem of synthesis, an inverse dependency is considered known, which connects by the means of approximation the magnitudes ω and F c , but not their derivatives.Therefore, the expression for the control influence cannot be obtained in analytical form, since to do this, it is necessary to have the relationship between the magnitudes d ω  and Ḟ cg in analytical form.The magnitude of control influence can be obtained through consistent calculation of Ḟ cg and d , ω  which eliminates the need to receive the expression for u in analytical form.But it is necessary for it to complete differentiation of the inverse model of a screw propeller, presented by the means of approximation, which is a separate scientific task and is not always possible.
In addition, if one overcomes mentioned difficulties, the implementation of a regulator will contain additional variables: derivatives from the perturbing forces of TC and resistance of UV hull, as well as the magnitude y.
 The presence of additional derivatives increases the volume of computation and significantly reduces the accuracy of control.

3. The principle of decomposition of a reference model
If the object of control is consistently connected one-dimensional objects, then its MM has the view of a system of differential equations, each of which has the order k j : ( ) In the first equation, the controlled magnitude is y 1 , the control influence is y 2 , in the second equation, the controlled magnitude is y 2 , the control influence is y 3 , etc.A controlled magnitude of an object in general is the output of its first component -the magnitude y 1 , a control influence of an object in general is the input of its last component -the magnitude u.
A general order of an object is defined as the sum of the orders of each equation in the system: For the motion of an object along desired trajectory, the system of conditions must be fulfilled in any moment of time:  dynamics for every internal process that runs in the system.In accordance with the method of ID, their values must be determined from the reference model.A reference model for the control of such an object is proposed to synthesize by the principle of decomposition.According to this principle, it is formed as a set of reference submodels based on the actual structure of the object: where y jg is the necessary value of the output of the j-th component of the object.
Using a reference model, formed by the principle of decomposition, is possible with the presence of all values y jg (as well as R j of their derivatives).For the otput of the object of control y 1 (j=1), the magnitude y 1g (t) is known as a controlling task.
The remaining values y jg are defined by inverse models of the components of the object: ( ) A law of control can be written down in the form of the system: The first equation of the system is the set value of the controlled magnitude.The second equation is reference submodels of internal transient processes of the system.The third equation is inverse models of the component of the object.The last equation specifies a desired control action of the object.
Decomposition of a reference model makes it possible to synthesize SAC without the need to bring a MM of the object to the view of differential equation of the n-th order.Instead of combining the equations of MM of the object, the formation of a reference model in the form of a system of reference submodels for each element of the object is fulfilled.Each submodel specifies desired dynamics of internal transient processes of the object of control.

Synthesis of high-precision automatic control system of one-dimensional motion of an underwater vehicle as an element of an underwater complex
Let us form a reference model according to the principle of decomposition.With this purpose we will select the laws of elimination of errors for each equation of a MM of the object in the form of linear differential equations of the 1st and 2nd order with the time constants T ω , T y0 and T y1 : g 0 1 y 0 y y y g T E E 0; E ; T T E T E E 0; E y y, where ω g is the necessary value of the magnitude ω, E ω is the error of control of the magnitude ω, E y is the error of control of the magnitude y.Let us substitute expressions for errors in the equations and define higher derivatives d ω  and d y ,  which represent the desired values relative to the first derivative ω and the second derivative of the controlled magnitude y: A controlling influence is determined from a MM of the object, in which the desired values substitute higher derivatives: ( ) ( ) Thus, the system of automatic control of one-dimensional motion of UV as a part of UC contains a regulator that implements the resulting law of control, and reverse connections by the phase coordinates of the object y, ẏ and ω.Information about current magnitudes of the forces F d , F s and the moment M d (Fig. 2) must be supplied to the input of the regulator.

Fig. 2. SAC structure
In real UC, it is not always possible to measure the magnitudes F d , F s or M d by instruments, but they can be calcu- ,1, 2, , k 1; r 0,1, 2, , R ; y t y , y ; i 0,1, 2, , k 1; r 0,1, 2, , R , lated based on the corresponding MM of the UC elements.If their accuracy is insufficient, then corrections are made with the help of compensation circuits [23].Thus a high dynamic accuracy of UC SAC is maintained.The application of the principle of decomposition made it possible not to fulfill the transformations of MM of the object and receive a law of control that does not require computation of additional variables: derivatives from perturbing forces of TC and UV hull resistance, as well as the magnitude y. 

5. Control of phase coordinates
The principle of decomposition allows setting desired dynamics for each phase coordinate of the object.With this purpose, a reference model should contain submodels, each of which is responsible for managing a specific phase coordinate.Let us explain the essence of the proposal by the example.With this purpose we will present a MM of the object of control in the normal form of Cauchy: Let us form reference submodels for each phase coordinate of the object in the form of the first-order differential equations with the time constants T ω , T y0 and T y1 : T E E 0; E ; T E E 0; E y y ; T E E 0; E y y.
Let us express higher derivatives: Let us find control action by the substitution of higher derivatives in a MM of the object with the desired values: Thus, a reference submodel is selected separately for the task of desired dynamics of each phase coordinate of the object.

5. Assignment of an arbitrary law of eliminating error
The principle of decomposition in conjunction with the method of ID opens up powerful capacities of formation of a reference model of SAC.Due to an independent choice of equations (submodels), which are part of the reference model, one can create a reference model as a system of arbitrary functions of time.Thus the restriction is removed that is in the choosing of the structure of a reference model or its submodels in the form of differential equations.
A law of eliminating error is proposed to form as a certain function of time for each phase coordinate of the object: where x g is the set value of phase coordinate in the current moment of time t c , x is the actual value of phase coordinate in the current moment of time t c , t≥t c .
Then we find a desired trajectory of motion of phase coordinate: After defining x d , we find its derivatives of necessary order.Then the control action is determined by substituting of the desired values of the corresponding derivatives of phase coordinates in the inverse models of the object.
Thus, for example, in program SAC, where the dependency y g (t) is known in advance, instead of the reference submodels in the form of differential equations of the first order, one can apply their analytical solutions.Based on them, the desired values of higher derivatives of a MM of the object are numerically calculated: where t h ={0…t end } is the sequence of values of time, for which the errors are calculated; t end is the horizon of error calculation.
A law of control in this case is defined, as before, from the inverse model of the object.Transient processes of SAC when choosing such a reference model correspond to transient processes of SAC with a reference model, set by a differential equation of the first order for each phase coordinate of the object.But their view can be changed by choosing another law of eliminating error in the function of time.

SAC simulation
To examine the laws of control received by an enhanced method of ID, in this work we performed a computer simulation of SAC of one-dimensional motion of UV as part of UC.
Simulation results of SAC at T ω =T y0 =T y1 =0,5 s with harmonic character y g (t) and zero initial values of phase coordinates of the object are presented in Fig. 3.
Initial values y g and y differ, that is why in the first few seconds SAC eliminates the error of control, then the controlled magnitude coincides with its preset value.The chart of elimination of error E y coincides with the chart of elimination of error of its reference submodel E ey0 .This occurs because the initial values of phase coordinates of the object coincide with their desired magnitudes, that is why a transition process of elimination of error of control matches the reference submodel for the controlled magnitude by a linear differential equation of the first order with the time constant T y0 =0,5 s.
The resulting SAC provides high precision control of UC.In this case, transition process of elimination of error almost completely coincides with the reference one, and after removing the error, the controlled magnitude strictly follows the set one without delay.

Discussion of the results of application of the principle of decomposition of a reference model
The method of ID is known in the theory of control as a method of synthesis of control systems of non-linear objects, including those of the highest order.Its main feature is to provide a high dynamic accuracy of control, which is the key requirement to the majority of modern underwater technologies.
In conjunction with the developed principle of decomposition, the method of ID provides powerful capacities for the synthesis of UC SAC.Formation a reference model by the principle of decomposition provides: -a possibility of synthesis of high-precision SAC by the method of ID for UC, the nonlinearities of which are set in any form (graphically, tabulated, etc); -a possibility to set a reference submodel to control each phase coordinate of UC; -setting submodels in the form of arbitrary functions of time, including analytical, by the means of approximation or tabularly (with interpolation mechanism); -combination of different laws of eliminating errors within the limits of single SAC, that is, in a general case, the choice of submodels is independent and is limited by the set requirements to stability and other criteria of SAC quality.
Defined properties extend capabilities of the method of ID and make it possible to select a reference model depending on the requirements to the set problem and the system of UC control.
Decomposition of the reference model does not guarantee strict following of the phase coordinates of the object of the transitional processes, set by their reference submodels.But the closer the value of the phase coordinates of higher orders to their desired values, the more precisely a controlled magnitude follows the transitional process, set by its reference submodel.
Application of the principle of decomposition to control UV as part of UC make it possible to synthesize SAC without loss of accuracy due to the simplification of a MM of the object, unlike the UV SAC obtained in [11].In addition, the SAC obtained in this paper provides the following of the controlled magnitude by the set one without delay, unlike a PD regulator, designed to control UV as part of UC in [18].
It should be noted that real UC have limitations, in particular, by the control action.In this case, the work of SAC in the mode of saturation leads to the occurrence of over-regulation and oscillation, which manifest themselves on the objects of the second order.In this case it is necessary to adjust the parameters of the reference model or to apply optimization techniques by speed performance.

Conclusions
1. Based on the known mathematical models of the components of an underwater complex, a mathematical model was developed of one-dimensional motion of a propelled underwater vehicle under conditions of perturbing influence of a tether cable for the synthesis and study of automatic control system of its motion.It is a system of nonlinear differential equations with one equation of the first order and one equation of the second order and it includes the curves of action of a screw propeller, which are set in graphic form.Perturbing effect of a tether cable on an underwater vehicle hull is modeled by using a known mathematical model of dynamics of flexible connection.
2. A method of inverse dynamics was improved for the synthesis of high-precision automatic control systems of underwater complexes with low sensitivity to parametric and coordinate perturbations.The improvement is based on the principle of decomposition of a reference model of the control system by which it is formed as a set of reference submodels according to the structure of the object.Decomposition of a reference model provides a possibility to apply the method of inverse dynamics in the problems of synthesis of high-preci-sion automatic control systems of underwater complexes, the nonlinear elements of which are set in arbitrary form.
3. We designed a system of automatic control of one-dimensional motion of underwater vehicle as a part of an underwater complex with the use of the method of inverse dynamics based on the principle of decomposition of a reference model.In the work we presented three variants of a reference model of an automatic control system: based on the accordance with the structure of a mathematical model of the object, based on the accordance with the structure of an object presented in the form of Cauchy, as well as in the form of arbitrary functions of time.The first variant makes it possible not to execute transformations of a mathematical model of the object.The second and the third variants allow setting desired dynamics to each phase coordinate of the object in the form, accordingly, of differential equations of the first order and arbitrary functions of time.
4. Computer simulation of the designed automatic control system of an underwater complex displayed that the transitional process to eliminate error of the coordinate of the vehicle corresponds to the reference one.Insignificant deviations of the real and the reference processes are caused by approximation of the inverse model of a screw propeller and the assumption about the first order of a mathematical model of the thruster.After eliminating the error, the controlled magnitude strictly follows the set one without delay.

Introduction
The category "system" is one of the central concepts of cybernetics as a science about general control principles.The analysis of steady patterns indicates that system objects are differentiated depending on solvable task: digestion system, heating system, crushing system, movement system etc.
It is logical to assume that all these system objects, providing transformation of physical products by interaction with other systems shall have the one internal architecture.
At the same time the studied system has not just to execute the own function, but execute it in the best way, that is optimally.It means that parameters optimization of input system products is its internal function.
Considering that every business connected with transformation of physical products includes a set of the inter-acting systems of the converting class, the synthesis of the internal structure of such systems is an important scientific task.

Literature review and the problem statement
Proceeding from general conceptions about the system as an indivisible essence, it is possible to present its structure in the form of subsystems.Each subsystem has to carry out one or several uniform functions.Nevertheless, these conceptual ideas are very often broken.
So, in the work [1] it is proposed to consider a subsystem as the part of the system, selected by some sign according to the purposes and research problems in which it can independently be accepted as a system.

Представлены результаты исследований, которые были проведены с целью выявления структуры исполнительной системы преобразовательного класса. Установлено, что исполнительная система преобразовательного класса состоит из технологической подсистемы, подсистемы контроля, подсистемы учета, подсистемы координации и подсистемы оптимизации. Установлено, что особенности конкретной исполнительной системы преобразовательного класса проявляются исключительно в рамках особенности функционирования базового технологического механизма
Ключевые слова: система преобразования, подсистема координации, подсистема контроля, подсистема учета UDC 007.52 DOI: 10.15587/1729-4061.2016.74873 Perturbing effect F d is modeled by using a MM of TC dynamics [ω  represent a MM of a screw propeller and are essentially non-linear.They have in their composition ω 2 and coefficients, which nonlinearly depend on ω and ẏ and are defined from the curves of the action of propeller[21].For the synthesis of SAC by the method of ID it is necessary to have an inverse model of SP in the form of dependency ( ) inv c f F , y .ω =  To build it analytically is impossible because the curves of SP action are derived experimentally and are presented in graphical (or tabular) form.However, an inverse model of SP can be set in a tabular form and, in combination with the means of approximation, may be used in the composition of SAC [22].