RESEARCH OF THE OF IDENTIFICATION ALGORITHM OF CONTROL OBJECT OF SECOND-ORDER LINKS WITH A DELAY TIME

6. Dashkevich L. V., Nemtseva L. D., Berdnikov S. V. Otsenka ledovitosti Azovskogo morya v XXI veke po sputnikovym snimkam Terra/Aqua MODIS i rezul’tatam matematicheskogo modelirovaniya // Sovremennye problemy distantsionnogo zon­ dirovaniya zemli iz kosmosa. 2016. Vol. 13, Issue 5. P. 91–100. 7. Zamerzanie Azovskogo morya i klimat v nachale XXI veka / Matishov G. G. et. al. // Vestnik Yuzhnogo nauchnogo tsentra RAN. 2010. Vol. 6, Issue 1. P. 33–40. 8. Baran J., G recka A. Seaport efficiency and productivity based on Data Envelopment Analysis and Malmquist Productivity Index // Logistics & Sustainable Transport. 2015. Vol. 6, Is­ sue 1. P. 25–33. doi: http://doi.org/10.1515/jlst­2015­0008 9. Definition of Efficiency and Safety Criteria for Icebreaker in Ice Management Operations / Karulin E. et. al. // Vo­ lume 8: Polar and Arctic Sciences and Technology; Petroleum Technology. 2018. doi: http://doi.org/10.1115/omae2018­77404 10. Dergausov M., Justification of the choice of an icebreaker for winter navigation in the Azov Sea // Shipbuilding and Marine Infrastructure. 2018. Issue 1 (9). P. 108–114. 11. Pravyla lodovoho provedennia suden: Nakaz Ministerstva infra­ struktury Ukrainy No. 14 vid 12.03.2011. Ministerstvo yustytsii Ukrainy No. 447 (19185). 04.04.2011. 15 p. 12. Zinchenko S. G., Yanchetskyі O. V. Analysis of ice conditions of winter navigation in the Azov sea for the substantiation of the icebreaker selection // Collection of Scientific Publications NUS. 2018. Issue 1–2. doi: http://doi.org/10.15589/jnn20180102 13. Pro znyzhennia stavok portovykh zboriv: Nakaz Ministerstva infrastruktury Ukrainy 27.12.2017. No. 474. URL: http://zakon. rada.gov.ua/laws/show/z0046­18


. Introduction
The rising cost of raw materials on world markets causes a rapid increase in the cost of production of Ukrainian industries. Thus, at present, the share of the cost of natu ral gas in chemical products reaches 75 %. So, in order for Ukrainian products to be competitive in the global market, there is an acute need for more efficient use of raw materials, energy, and the like. That is, it is neces sary to carry out optimization of technological processes.

ISSN 2226-3780
At the majority of enterprises, technical modernization was carried out, including control systems. But, as it turns out, this may not be enough, if at the lowest level of the control system the main device in the automatic control system (automatic control system) is the controller, it is incorrectly configured. The controller generates a control signal in order to obtain the required accuracy and quality of the transition process. The proportion of incorrectly tuned controllers used in industry is more than 50 % [1].
Determining the optimal parameters of the controller by conducting an experiment at the facility itself can lead to a loss in the quality of the finished product, damage to raw materials, and catalysts. And even to the occurrence of emergencies, including fires, explosions, emissions of harmful substances into the environment. Therefore, the develop ment of theoretical methods for calculating the optimal controller settings is a very important and urgent task [2].

The object of research and its technological audit
The object of research is the optimal controller infusions and transient quality indicators. The subject of research is singleloop automatic control systems (ACS).
One of the most problematic places is that modern technological processes are complex control objects, when designing automation systems, it becomes important to iden tify the control object and calculate the controller settings and optimize them. Optimal adjustments of the control ler will ensure the highest possible product quality in the conditions of this technology and its minimum cost with a given production volume. Determining the optimal setting parameters of the controller by conducting an experiment at the facility itself can lead to a loss in the quality of the finished product, damage to raw materials, and catalysts.
The existing methods have a number of significant draw backs, which are the reason why at the present time the most effective, from the point of view of optimal control, method of finding controller settings is an experimental search.
The quality of any control system is determined by the magnitude of the error, but the error function for any point in time is difficult to determine because it is described using a high order differential equation and depends on a large number of system parameters. There fore, assess the quality of control systems for some of its properties, which are determined using quality criteria.
Among all known quality criteria, the most universal is the integral quality criterion, which evaluates the generalized properties of the ACS: accuracy, stability margin, speed.

The aim and objectives of research
The aim of research is development of an algorithm for identifying the control object; it has a delay time along the acceleration curve by a link of the secondorder with a delay time.
To achieve the aim it is necessary to solve the fol lowing objectives: 1. To find the optimal controller settings based on the integral quadratic optimization function with a restriction on the overshoot of the transition process.
2. To make a comparison of the quality indicators of transient processes of the investigated automatic control systems with the settings obtained by different methods.

Research of existing solutions of the problem
Among the scientific works devoted to this topic, it is possible to distinguish the work [1]. In this paper, the high speed of technological processes, the presence of a large number of disturbances caused by the interaction of individual parts of the production process and changes in external conditions are considered. As well as the de pendence of the operating modes of the equipment on time, necessitates the creation of highquality automatic control systems.
The efficiency of the production process directly depends on the operation of control systems. In some cases, new units, in principle, cannot function without highquality automatic control systems (ACS) [2].
In [3], a transient oscillating process is also approxi mated by an oscillating link and a delay link in a simi lar way as in [4]. According to the obtained model, the frequency ωπ is found, at which the phase of the system is π, and the amplitude at this frequency, on the basis of which four parameters of the secondorder model of the control object (CO) are calculated with a delay.
There are directions in which methods of identification of controlled objects are explored, which are based on: -method of least squares [5]; -method of instrumental variables [6]; -identification by frequency characteristics [7]; -randomized identification algorithms [8]; -active identification using an additional test sig nal [9,10]. The common problems of applying these methods are guaranteed the convergence of dynamic identification pro cesses, the weak applicability of these methods in the case of largescale problems (a large number of unknown parame ters), and the significant cost of the corresponding software.
Thus, the results of the analysis allow to conclude that the development of theoretical methods for calculating the optimal controller settings is a very important and promising task.

Methods of research
The calculation algorithm was implemented using the «Maple» software package.
Transients of control objects can be aperiodic or oscil latory. It is known that both processes with a sufficient accuracy degree can be described by a secondorder dif ferential equation [1].

Research results
Let's consider the block diagram of a singleloop ACS, which is shown in Fig. 1. When removing an acceleration curve on a real control object, the transient process of an equivalent control object (an openloop system from IC -an intermediate converter to NC -normalizing converter is provided, provided that the transfer function of the secondary device is 1). That is, if identifying an equivalent secondorder control object by the acceleration curve, then the functional scheme of a singleloop ACS can be given as follows (Fig. 2).
Transformed block diagram of a single-loop automatic control system The differential equation of the secondorder control link is: where ′′ T , ′ T -time constants; K p -coefficient. The nature of the transition process of this link depends on the magnitude of the relationship ′ ′′ T T . If ′ ′′ ≥ T T 2, then the transition process will be ape riodic, and when ′ ′′ < T T 2 -oscillatory. Let's find the roots of the differential equation (1): If ′ ′′ > T T 2, then the roots Р 1 and Р 2 will always be real and negative. Then the equation of the transition function will be: where a 1 1 = −P ; a 2 2 = −P ; u 0 -step perturbation. When ′ ′′ < T T 2 the roots will be complex: where a ω In this case, the transition function is described by the equation: Let's consider the identification of control objects on the example of a link of the fifth order, has a transfer function: .
Let's build again the acceleration curve of the fifth order link only with the delay time (Fig. 4). When the tangent was rebuilt (Fig. 4), the delay time for the secondorder link was found.
From Fig. 4 it is shown that the acceleration curve is aperiodic, therefore, in order to find the equation for the acceleration curve, it is possible to use equation (3).
The coefficient K is found by the acceleration curve (K = 1). In this equation there are two more unknown parameters a 1 and a 2 . In order to find them, let's take two points on the acceleration curve, select these points approximately as shown in Fig. 4.
Let's make equations for these two points. As a result, let's exp .
The system of two equations thus formed is solved for a 1 and a 2 . The easiest way to find these variables is using the Maple math package.
Let's find the variables a 1 and a 2 . Let's substitute these values into equation (3)  . e xp . .
Let's build a fifthorder link acceleration curve on a single graph and a curve that corresponds to the ob tained equation (9) Thus, at two points of the acceleration curve of an aperiodic control object with a delay time, its second order aperiodic link with a delay time can be identified quite accurately. . .
Let's build again the acceleration curve of the fifth order link only with the delay time in Fig. 7. When the tangent was rebuild (Fig. 7), the delay time for the secondorder link was found.
From Fig. 7 it is shown that the acceleration curve is oscillatory, therefore, in order to find the equation for the acceleration curve, it is possible to use equations (5). TECHNOLOGY AUDIT AND PRODUCTION RESERVES -№ 1/2(45), 2019

ISSN 2226-3780
The coefficient K is found by the acceleration curve (K = 1). In this equation there are two more unknown parameters a 0 and ω 0 . In order to find them, let's take two points on the acceleration curve (Fig. 7), choose these points approximately, as shown in Fig. 7.
Let's make equations for these two points. As a result, let's obtain the system of equations: .
The system of two equations thus formed is solved with respect to a 0 and ω 0 . The easiest way to find these variables is using the Maple math package.
Let's find the variables a 0 and ω 0 . Let's substitute these values into the equation: Thus, in the study of automatic control systems, con trol objects in which complex technological processes, it is possible to conclude that the equivalent transfer function can be represented in the case of an aperiodic acceleration curve with a delay time by an aperiodic secondorder link with a delay time, and in the case of an oscillatory acceleration curve, a vibrating secondorder link with a delay time. This will greatly facilitate the process of analyzing and optimizing the ACS dynamic characteristics.
Having obtained the transfer function of an equivalent object from an experimental acceleration curve, it is pos sible to synthesize ACS. Let's consider singleloop ACS. Such an ACS, taking into account the transfer function of an equivalent object, can be given in the form of an ACS with a single feedback (Fig. 2).
The existing methods have a number of significant drawbacks, which are the reason why at the present time the most effective from the point of view of the optimal control method for finding the settings of the controller is an experimental search.
The quality of any control system is determined by the magnitude of the error: where u(t) -the reference signal; y(t) -the output sig nal (Fig. 9). But the error function ε(t) for any point in time is difficult to determine, since it is described using a high order differential equation and depends on a large number of system parameters. Therefore, let's assess the quality of control systems for some of its properties, which are determined using quality criteria.

ISSN 2226-3780
Among all known quality criteria, the most universal is the integral quality criterion, which evaluates the generalized properties of the ACS: accuracy, stability margin, speed.
Therefore, the essence of this work lies in the fact that the algorithm based on the integral quadratic opti mization function was developed, with which the optimal controller settings were calculated. The integral criterion proposed in [6,7] gives a generalized estimate of the dam ping rate and the deviation of the controlled variable in the form of a single numerical value. It is according to the formula [5,6]: where T -the regulation time. This integral defines the square of the plane between the problem u(t) and the transition curve у(t). This integral will depend on the controller settings, that is, in the case of the PID controller (proportionalintegraldifferential controller) on the regulation coefficient . The proposed algorithm is based on the solution of the optimization problem: finding such values of K r , T i , T d for which the quadratic integral criterion would be minimal: These values of K r , T i , T d will be the optimal tuning parameters of the controller. For most processes, the in tegral criterion is a unimodal function, which makes it possible to apply the proposed algorithm.
The Pcontroller (proportional controller) has one tun ing parameter -the coefficient of regulation K r , there fore the quadratic integral criterion will be a function of one variable I f K r = ( ). Let's determine the coefficient of regulation K r 0 , at which this integral will be minimal, it is possible to solve the equations dI dK r 2 0 = . Also, the optimal regulation coefficient can be determined by plot ting the dependence I f K r = ( ) and determining the value of K r 0 , at which I = min, directly according to the graph (Fig. 10). This value of the control coefficient will be optimal, and accordingly, the system with this value of the controller gain will have a minimum dynamic error. The PI controller (proportionalintegral controller) has two tuning parameters -the regulation coefficient K r and the integration time T i , therefore the quadratic integral criterion will be a function of two variables I f K T r i = ( , ), and the graph of this function will be some kind of surface. To find the values of K r and Т i , at which the method of the fastest descent is applicable [7,8].
The essence of this method is that one of the variable parameters is fixed, that is, an arbitrary numeric value is assigned to one of the tuning parameters, for exam ple, K r = K r 0 , so I turns into a function of one variable I f T i = ( ). Then find the value Т i0 , at which the quadratic integral criterion will be minimal I = min. This can be done by solving the equation dI dT i 2 0 = or directly on the graph I f T i = ( ) (Fig. 11). At the next step, the second variable parameter is fixed -the integration time T i , assigning to it the value found at the previous step Т і = Т і0 . Then find the value of K r1 , at which the condition would be fulfilled I 2 = min by solving the equation dI dK r 2 0 = or directly on the graph I f K r = ( ) (Fig. 12). After that, the whole cycle repeats. The number of necessary iterations can be determined, for example, from the condition that the change in the quadratic optimization function during the last iteration will not exceed 5 %. As a rule, 3-5 iterations are sufficient to find such values K r and T i for which the quadratic integral criterion will be minimal. These values will be the optimal settings of the PI controller. The PID controller (proportionalintegraldifferential controller) has three tuning parameters -the regulation coefficient K r , the integration time T i and the differentiation time T d , therefore the quadratic integral criterion will be a function of three variables I f K T T r i d = ( , , ). Unlike systems with a P controller and PI controller, the graph of this function is a hypersurface, which can obviously be given. To find the values of K r , T i , T d , for which I = min, let's also apply the steepest descent method. The number of iterations can be determined in the same way as in the previous case. The values of K r , T i , T d found in this way, for which the quadratic integral criterion will be minimal, will be the optimal settings of the PID controller. The quality indicators of ACS transient processes (overshoot, regulation time, static and dynamic errors), in which the optimal settings of the controllers were calculated using this algorithm, as well as by the method of triangles and the ZieglerNichols method are given in compara tive Tables 1, 2.
The research results (Fig. 7, b-d and Tables 1, 2) show an improvement in the dynamic properties of the system when using optimal controller settings calculated by the pro posed method as compared to the most common engineering methods of searching for controller settings for ACS with aperiodic and oscillatory COs. The overshoot has decreased by 10 times, the regulation time has decreased to 30 %, the static and dynamic errors have decreased by 2-3 times.
A characteristic feature of the oscillatory process is overshoot. High overshoot is considered a lack of auto matic control systems, and for some systems is completely unacceptable because it causes system overload, etc. The permissible overshoot value is determined by the specific working conditions and the ACS purpose. Therefore, an important task is the synthesis of systems with given (limi ted) indicators of the quality of the transition process.
In this paper, let's propose an algorithm to search for controller settings with the introduction of restrictions on the transient overshoot. This algorithm consists in the fact that according to the transformed formula (22) a possible overshoot area is build: After that, the area is limited to the desired overshoot value (the line for the P controller, and the plane for the PI and PID controllers) (Fig. 13).
The point of intersection of the two planes (lines) will be the optimal tuning parameters of the controller with the specified value of the overshoot of the transient process.  system when using the controller parameters calculated according to the proposed algorithm: -reduction of overshoot by 10 times; -reduction of regulation time up to 10 times. In the study of systems using the P controller, it is necessary to note an increase in deregulation, but at the same time the static error of the system decreases by 2-3 times compared with other methods.

SWOT analysis of research results
Strengths. In this paper, an algorithm for identifying control objects with different types of transients by second order links with a delay time is proposed and investigated. The identification error does not exceed 3 %, which is quite acceptable for calculations of this type. According to the results of a comparative analysis, it is concluded that the found parameters of the controller according to the proposed algorithm significantly improved the dynamic properties of the system (overshoot, regulation time, static and dynamic errors). Also in this work, an algorithm is proposed for searching the controller settings with the introduction of a restriction on the overshoot of the tran sition process, which also shows a positive result.
Weaknesses. The quality of any control system is deter mined by the magnitude of the error, but the error function for any point in time is difficult to determine because it is described using a high order differential equation and depends on a large number of system parameters. There fore, assess the quality of control systems for some of its properties, which are determined using quality criteria.
Opportunities. The task of further research will be the development and improvement of the search algorithm for the controller settings with the specified (limited) quality indicators of transient processes.
The research results show an improvement in the dy namic properties of the system when using the optimal controller settings calculated by the proposed method as compared to the most common engineering methods for searching the controller settings for ACS with aperiodic and oscillatory COs. The overshoot has decreased by 10 times, the regulation time has decreased to 30 %, the static and dynamic errors have decreased by 2-3 times.
Threats. When implementing this algorithm for iden tifying control objects, significant additional equipment costs are not required. Today, there are many theoretical and experimental methods for finding PID controller set tings. However, there is no universal method that would allow determining the optimal PID controller settings for systems and objects of various types.

Conclusions
1. An algorithm for identifying control objects with different types of transients by secondorder links with a delay time is proposed and investigated. The identifica tion error does not exceed 3 %, which is quite acceptable for calculations of this type. On the basis of the transfer functions of equivalent objects obtained in this way, the settings of the P, PI and PID controllers for the ACS are found by the triangle method, the sustained oscil lation method (ZieglerNichols method) and using the proposed algorithm.
2. A comparative analysis of the quality indicators of transient processes of the investigated ACS with the settings obtained by different methods. The results show an improvement in the dynamic properties of the system when using the optimal controller settings calculated by the proposed method, compared to the most common en gineering methods for searching the controller settings for the automatic control system with aperiodic and oscilla tory COs. The overshoot has decreased by 10 times, the regulation time has decreased to 30 %, the static and dynamic errors have decreased by 2-3 times.