DETERMINING CONFIGURATION PARAMETERS FOR PROPORTIONALLY INTEGRATED DIFFERENTIATING CONTROLLERS BY ARRANGING THE POLES OF THE TRANSFER FUNCTION ON THE COMPLEX PLANE

the arrangement of zeros and poles in the transfer function of the system. The synthesis of the automatic control system involves not only the choice of its structure but also determining its parameters, first of all, the parameters to configure PI and PID controllers. Thus, it is a relevant task to determine the configuration parameters for PID controllers by arranging the poles on a complex plane since it makes it possible to reduce the problem to solving a system of linear algebraic equations.


Introduction
The task to synthesize an automated control system for a particular object is one of the most important in its design. The purpose of synthesis is to construct a system that would meet certain requirements, namely: the system must be robust; demonstrate the predefined control quality indicators such as overshoot and regulation time.
The system synthesis can be carried out using both frequency and time methods.
Frequency methods of synthesis are based on the use of frequency characteristics such as Bode, Nyquist diagrams, or coordinates built in polar systems. They make it possible to synthesize controlling devices in order to achieve certain quality indicators of the control process, such as the maximum value of the frequency performance amplitude, resonance frequency, bandwidth, and margin of stability for the amplitude and phase.
Methods of system synthesis in the time domain make it possible to evaluate the quality indicators of the system according to its transition characteristic, which depends on

DETERMINING CONFIGURATION PARAMETERS FOR PROPORTIONALLY INTEGRATED DIFFERENTIATING CONTROLLERS BY ARRANGING THE POLES OF THE TRANSFER FUNCTION ON THE COMPLEX PLANE M y k h a i l o H o r b i y c h u k
Corresponding author Doctor of Sciences, Professor* Е-mail: mi_profgorb@ukr.net

N a t a l i i a L a z o r i v
Рostgraduate Student*

L i u d m y l a C h y h u r
PhD*

І h o r C h y h u r
PhD* *Department of Automation and Computer Integrated Technologies Ivano-Frankivsk National Technical University of Oil and Gas Karpatska str., 15, Ivano-Frankivsk, Ukraine, 76019

This paper reports a solution to the problem of determining the configuration parameters of PID controllers when arranging the poles of the transfer function of a linear single-circuit automated control system for a predefined set of control objects.
Unlike known methods in which the task to find the optimal settings of a PID controller is formed as a problem of nonlinear programming, in this work a similar problem is reduced to solving a system of linear algebraic equations.

The method devised is based on the generalized Viète theorem, which establishes the relationship between the parameters and roots of the characteristic equation of the automatic control system. It is shown that for control objects with transfer functions of the first and second orders, the problem of determining the configuration parameters of PID controllers has an unambiguous solution. For control objects with transfer functions of the third and higher orders, the generated problem is reduced to solving the redefined system of linear algebraic equations that has an unambiguous solution when the Rouché-Capelli theorem condition is met. Such a condition can be met by arranging one of the roots of the characteristic equation of the system on a complex plane. At the same time, the requirements for the qualitative indicators of the system would not always be met. Therefore, alternative techniques have been proposed for determining the configuration parameters of PID controllers. The first of these defines configuration parameters as a pseudo solution to the redefined system of linear algebraic equations while the second produces a solution for which the value of the maximum residual for the system of equations is minimal. For each case, which was used to determine the settings of PID controllers, such indicators of the control process as overshooting and control time have been determined Keywords: automatic control, PID controller, system of equations, configuration parameters, process quality indicators
function of the automatic control system was partially considered in [12] for the oil separation process.
Thus, the issue of determining the configuration parameters for PID controllers remains relevant, as evidenced by numerous publications. The methods and algorithms offered by their authors are reduced to solving problems of nonlinear programming, which imposes certain restrictions on the class of mathematical models that describe the dynamics of control objects. At the same time, little attention is paid in scientific papers to methods based on determining the parameters of configuration of PID controllers by arranging roots on a complex p-plane.

The aim and objectives of the study
The purpose of this work is to build a method for determining the parameters of PID controllers by arranging poles on a complex plane (p-plane) of the transfer function of a single-circuit automatic control system. This will make it possible to develop effective algorithms for determining the configuration parameters of PID controllers for the predefined set of transfer functions of control objects.
To accomplish the aim, the following tasks have been set: -to form the concept for determining the configuration parameters of PID controllers for control objects with transfer functions up to the third order inclusive; -to devise a procedure for determining the configuration parameters of PID controllers, focused on the predefined class of transfer functions of control objects.

The study materials and methods
We consider a single-circuit automatic control system ( Fig. 1), which consists of a control object with the transfer function W оb (p) and a controlling element whose transfer function is W cl (p).
and the controller has the following transfer function: In (1), (2), the following notations are adopted: a i , = 0, ; i n b j , = 0, j m are the parameters of the transfer function of the object; C 0 , C 1, and C 2 are the controller's configuration options.
In addition, we assume n<m.
The stability of an automatic control system and the quality of the control process largely depend on the configuration parameters of the PID controller.
One of the first studies that addressed solving the problem of determining the configuration parameters for PID controllers was reported in [2]. The essence of the Sigler-Nichols method is that the two parameters, L and R, were determined based on the transitional characteristic of the object. To determine them, the inflection point was determined on the acceleration curve and a straight line was drawn through it, which is tangent to the acceleration curve. Such a straight line cut from the axis of abscissa the value of L while the angle of inclination to the same axis defined the value for R. The Sigler-Nichols method can only be used for a limited class of objects; it is largely subjective in determining the L and R parameters. Further research on solving the task of finding optimal configuration parameters for PID controllers was carried out in two directions -the time [3] and frequency domains [4].
A characteristic feature of works [3] and [4] is that the task of determining the optimal configuration parameters of PID controllers is stated as a problem of nonlinear programming. In addition, the authors confined themselves to objects with transmission functions of the first order with a delay.
A method for determining the configuration parameters of PID controllers in the time domain is based on minimizing the function of the control rms error [5]. The configuration parameters for a PID controller, obtained in this way, typically ensure the stability of the automatic control system but the transition processes in such a system are oscillatory in nature, whose intensity of attenuation is unsatisfactory in most cases. Therefore, it is necessary to introduce an additional component to the criterion of optimality based on the speed of change in the control error with a certain weight coefficient, whose value is selected on the basis of the intuition of the researcher followed by the subsequent repeated machine experiments.
The task of determining the configuration parameters of PID controllers in the temporal domain is a nonlinear optimization problem; solving it requires genetic algorithms [6], neural network technologies [7], and the so-called bacteria reproduction algorithm [8] (Bacterial Foraging Optimization Algorithm, BFOA-algorithm), which is proposed by the authors of work [9].
The use of so many algorithms for finding a minimum of the control rms error indicates the difficulty of solving the task, which is even more complicated when considering the restrictions for the configuration parameters of PID controllers.
The second direction in determining the configuration parameters of PID controllers is based on the use of frequency characteristics of automatic control systems. Among the methods used, the most common is the method of extended amplitude-phase characteristics [10] and the method of determining the configuration parameters of PID controllers using conventional frequency characteristics [11]. The disadvantage of frequency methods is that they are graphic-analytical, which causes certain difficulties in their computer implementation.
A method for determining the parameters of PID controllers by arranging the poles of the transfer function of the closed automatic control system on a complex plane is proposed, which makes it possible to reduce the problem to finding a solution to the system of linear algebraic equations.
The task of calculating the configuration parameters for a PID controller by arranging the poles of the transfer Expression (3) is to be recorded in a slightly different form where β j =b m-j , = 0, ; j m α i =α n-i , = 0, . i n The denominator of the transfer function W yu (p) contains the product of two polynomials. Multiplication of two polynomials is reduced to finding the product of their coefficients [13] according to the following formula: where π 0 =α n ; π 1 =α n-1 ; …; π n-m =α m ; π n-m+1 =s m +α m-1 …; π n =s 1 +α 0 ; π n+1 =s 0 . Summarizing the result, we can write down: For ratio (8), the following condition must be met: If n-m-2<0, then the π i value should be calculated using the following formula: We assume that p 1 , p 2 , …, p n+1 roots of characteristic polynomial (7) contain no multiples.
Then, in accordance with Vieta's theorem, we obtain the following system of relations between the roots of characteristic equation (7) and its coefficients:  In the system of equations (11), its right-hand sides are the functions of the controller configuration parameters, which makes it possible to select the C 0 , C 1, and C 2 values so that the quality of the control process meets certain criteria.
In this case, we shall consider the following. For the automatic control system ( Fig. 1) to be stable, it is necessary and sufficient that all the roots of characteristic equation (6) should belong to the left semi-plane of the p-plane (the complex plane of roots). Therefore, first, the C 0 , C 1, and C 2 values must be such to ensure that the system is robust; second, the system should have the desired properties [14] -the degree of stability η and fluctuations μ.
The degree of stability is determined by the nearest left real root to the imaginary axis In a general case, when the n<m condition is met, the characteristic polynomial (6) of the automatic control system has n+1 simple roots. Therefore, in accordance with ratios (11), we obtain a system of linear algebraic equations whose dimensionality is n+1. Such a system contains three unknown C 0 , C 1, and C 2 , that is, the system of equations (11) belongs to the class of redefined equations [15]. In the case where the characteristic polynomial of the controlled object has a second power, then the system of equations (11) is a second-order system with three unknowns. Such a system is indeterminate [15]. And, only for the characteristic second-order polynomial, we obtain a system of three equations with three unknowns.
If we form matrix A of the coefficients of unknowns in the system of equations (11), then the rank of such a matrix ( ) ( ) where matrix A has a dimensionality of (n+1)×3. Matrices for which condition (12) is met are the matrices of full rank [15]. The system of linear algebraic equations with matrices of full rank for n=1 is indeterminate and always compatible but does not have a single solution. If n=2, then, for the non-special matrix A, we obtain a single solution. The redefined system (n>3) is only compatible if the Rouché-Capelli theorem [16] is met, that is, the following equality should hold: where B is the vector of free terms of the right-hand sides of equation system (11).

2. Procedure for determining the configuration parameters of PID controllers focused on the predefined class of transfer functions of control objects 2. 1. An object's first-order transfer function
Assume the transfer function of a controlled object is as follows: where k is the transmission coefficient; T -an object's time constant. The dynamic properties of the controller are characterized by transfer function (2).
If we take into consideration the corresponding values s 0 , s 1 and s 2, and α 0 і α 1 , we obtain Of course, in this simple case, the characteristic polynomial (16) could be obtained directly from transfer function (12). The built algorithm for determining the characteristic polynomial shows its superiority over the technique of multiplication of polynomials (opening brackets) in the cases where we have high-power polynomials in the numerator and denominator of transfer function (3).
For characteristic polynomial (15), the following system of equations is constructed: Considering values for π 0 , π 1, and π 2 , the following ratios have been derived: We have two equations and three unknowns. One of them can be chosen arbitrarily. Let C 2 =0. Then, from the system of equations (17), we find: Thus, formulas (18) and (19) make it possible to determine the setting parameters for a PI-controller by arranging the roots p 1 and p 2 of the characteristic equation on the complex p-plane. Obviously, to ensure that the system is robust, the roots p 1 and p 2 must be left.
Then, as it follows from formulas (18) and (19), we have: In most cases, the transition process can be considered complete if the component of the transition process, which is determined by the degree of stability η, is damped. That means that α=η and the degree of damping Taking into consideration the found values, formulas (20), (21) take the following form: In a general case, the time of control t p can be approximately estimated [14] by the following formula: After selecting the desired t p value, one can find the degree of stability of the system: A numerical experiment was carried out. The programming environment MATLAB (The MathWorks, USA) was used to develop a program for building the transition characteristics of the system at a single jump-like input action. The parameters of the system's transfer function were as follows: k=2.5; T=12 s; t p =18 s; and the values C 0 and C 1 were calculated from formulas (22), (23).
The results from the numerical experiment are given in Table 1. Table 1 Values of the PI-controller configuration parameters and the control process quality indicators Analysis of Table 1 shows that the synthesized automatic control system allows for a satisfactory quality of controlthe overshoot value does not exceed the permissible norms. The control time is slightly higher than the specified value. This is because (24) is approximate.

2. 2. An object's second-order transfer function
Now a more complicated case is being considered. Let the transfer function of the control object be as follows: The dynamic properties of the controller are characterized by transfer function (2).
The characteristic equation of a closed system is to be written in the form of ratio (7). Since n=2, ratio (7) is a third-power polynomial: ( ) = π + π + π + π Next, we find the polynomial's coefficients (26). The lefthand side of inequality (9) at n=2 and m=1 will be negative. Therefore, the values π i , i=0, 1, 2, 3 are calculated from formula (10). We obtain Considering the values of π і , i=0, 1, 2, 3, we obtain ( )( ) ( ) ( ) Assume that one root of polynomial (26) is left and real, and the other two are the left and complex-coupled ones: p 1 =-α c, 1 , p 2 =-α c, 2 +jζ c, 2 , p 3 =-α c, 2 -jζ c, 2 . Then The results make it possible to write down a system of equations (27) in the following form: We obtained a system of three linear algebraic equations with three unknowns. From the last equation of system (28), we determined The C 0 value is put in the second equation of system (28). As a result, we have a system of two equations with two unknowns After obvious algebraic transformations, the transition to the following system of linear algebraic equations is carried out: where A 11 =-b 0 ; A 12 =(α c, 1 +2α c, 2 )b 0 -b 1 : .

A B A B С A A A A
The known C 2 value is used to find C 0 from formula (29). The next step in calculating the parameters of the controller is to select the value for the roots of the characteristic equation in such a way that, first, the system is robust, and, second, the system should demonstrate the desired quality indicators of the control process.
The first condition is met automatically by arranging the roots of the characteristic equation in the left semi-plane of the p-plane.
To form the conditions for meeting the second requirement, consider the p-plane (Fig. 3), which hosts the roots of characteristic polynomial (24).
Let the following ratio between the α c,1 and α c,2 values hold (Fig. 3).
At k α >1, the real root is closer to the imaginary Imp axis (Fig. 3, a); in the case where k α <1 (Fig. 3, b) a complex-coupled root. As before, the degree of system stability η is determined from formula (24), and the degree of fluctuation is ζ µ = α   For values of the degree of fluctuations μϵ{0,2; 0,4; 0,6; 0,8}, by using the software developed in the MATLAB programming environment, we built the plots of the transition characteristics of the system at a single jump-like input value (Fig. 4). Since, for this case, k α >1, the closest to the imaginary axis is the real root p 1 . Fig. 4 shows that increasing the degree of fluctuation μ entails an increase in overshoot ( Table 2).
In the case where k α =0,8 (k α <1), the transition characteristics of the system for different values of μ (Fig. 5) are constructed. This is the case when the closest to the imaginary axis is the left complex-coupled root.
Analysis of the results given in the Table 2 shows that the overshoot value increases at k α <1. The maximum value of σ increased from 14.7032 % at k α =1.2 to 27.8766, that is the increase in σ was 10.2 %. The control time, in this case, increased in comparison with the case where k α >1. Table 2 Values of the PI-controller configuration parameters and the control process quality indicators at k α =0,9 The degree of fluctuation, μ

2. 3. An object's third-order transfer function
We shall apply the method built for determining the parameters of setting the controller with transfer function (2) when the object (Fig. 1) is characterized by the following transfer function: ( ) Using an earlier constructed algorithm, we find a characteristic polynomial of the system depicted in Fig. 1. The power of such a polynomial is equal to four. So ( ) = π + π + π + π + π Since n=3, m=2, then n-m-2<0. That is why the value for π i , i=0,...,4 are found from formula (10). We obtain Applying the Viète theorem to polynomial (34), a system of equations is built that relates the roots of polynomial (34) to its coefficients π + + + = − π The roots of characteristic polynomial (34), which are the functions of the controller's setting parameters (Fig. 1), can be arranged in different ways in the lines of the p-plane. As an option, we can assume that there are two real and two complex-coupled left roots. Their arrangement is shown in Fig. 6.  The degree of fluctuation of the system is calculated as the ratio of the imaginary part of the complex-coupled root to its real part ζ µ = α . c c Considering the value for α c,3 , ζ c, 3 =μk α, 1 α c, 1 Choosing certain values of k α and k α,1 allows us to obtain the desired arrangement for the roots of polynomial (33) on a complex p-plane.
To arrange the roots as shown in Fig. 6  , Upon obvious transformations, we obtain:  (13) is not met and it is incompatible. One of the possible solutions to the redefined equation (35) is to minimize the residual function [15] ( ) = − 2 , J C AC B (38) relative to vector , C which yields the following: Note that the MATLAB software contains a built-in lsqr function that uses iterative procedures based on the bidiagonalization of Golub and Kohan [17], which is equivalent to the method of conjugate gradients but has better convergence. The method generates a sequence of approximations of the desired such that the residual is 2 , k r decreased monotonously. In fact, the built-in lsqr function generates a pseudo-solution to equation (37) in a form similar to formula (39), the only difference is that such a built-in function is oriented to solving systems of linear equations of great dimensionality.
Another approach to solving the system of equations (37) is to search for a corrective matrix ΔA to the right-hand side of the matrix equation such that the following system of equations is compatible, and the ΔA matrix is to meet the criterion of "smallness" [17,18]. Under such a statement of the problem, it is unclear what values the elements of the ΔA matrix should accept in order to satisfy the criterion of "smallness". In addition, if rang(a)<n, then the correction problem has no solution.
Alternatively, in [19], it is proposed to minimize the sum of squares of residuals where a k0 =B k , a l0 =B l , C b0 =-1, C bj =C j-1 , j=1,2,3, a kl are the elements of matrix A, = , 1, . k l m Introduce designations: The essence of expression (41) is that the following residuals are formed = + + − The sum of the squares of the difference in residuals z k -z l leads to expression (41).
We minimize ( ) 2 J C for the desired values C b1 , C b2 , C b3 by using the condition of the minimum function of many variables As a result, we obtain the following system of linear equations: To compare the efficiency of calculating the parameters for the PID algorithm from formula (39) and by solving the system of linear algebraic equations (43), the MATLAB environment was employed to construct the software for problems (39) and (43).
For values a 0 =6, a 1 =4, a 2 =7, The parameters of the characteristic polynomial of transfer function (34) of the object are selected so that the conditions of stability are met. The degree of stability η=α c,1 is taken from the condition that the value α c,1 is to the left of the pole of transfer function (33), which is closer to the imaginary axis of the p-plane. When one takes into consideration the selected values of the parameters of the characteristic polynomial of transfer function (33), the nearest pole to the imaginary axis would accept the following value: p 1 =-0.1532. Therefore, we chose α c,1 =0. 9. Other values for the algorithm were: k α =1.25, k α,1 =1,4. The values for the degree of fluctuation μ are taken so that the closed system (Fig. 1) is robust -μϵ{0.2;0.4;0.6;0.8}.
First, the parameters for the PID algorithm were calculated from formula (39), which determines the pseudo-solution to equation (40). The calculation results are shown in Fig. 7 and are given in Table 3. Table 3 Values of the PID-control algorithm configuration parameters and the control process quality indicators (pseudo-solution) Now, the parameters for the PID algorithm can be found as a solution to the system of equations (43).
Plots of the transitional characteristics of an automatic control system (Fig. 1) are shown in Fig. 8. The result of our calculations has established parameters for setting up the control PID algorithm, as well as the control quality indicators (Table 4). Table 4 Values of the control PID-algorithm configuration parameters and the control process quality indicators at k α =1.25 and k α,1 =1.4 The degree of fluctuation, μ The comparative analysis of the two methods for calculating the configuration parameters of the PID algorithm reveals that the transformation of the initial problem to the system of equations (43) provides better quality indicators than the pseudo-solution (39) to equation (40). Thus, in the first case, the overshoot and control time do not exceed 18 % and 6.6 s; in the second case, 5.3 % and 5.6 s, which is much better than in the first case. Now we can consider the possibility of achieving the compatibility of the equation system (40) by determining such a value for α c,1 that the Rouché-Capelli theorem conditions are met.
Since the rank of matrix A is equal to three, and the rank of the extended matrix [ ] A Bto four, we shall find such a value for α c,1 that the rank of the extended matrix has a dimensionality of 3. The extended matrix is square, the size of 4×4. Taking into consideration that rang(A)=3, and when adding column B to matrix A, we obtain an extended square matrix, the rank of which is equal to four, the determinant of the matrix [A|B] will be different from zero. The determinant of the extended matrix will be the function of the α c,1 variable. The opening of the determinant of the extended matrix generates a polynomial of the fourth power of the α c,1 variable, equating it to zero produces the following algebraic equation: η α + η α + η α + η α + η = Condition (45) is selected for reasons of ensuring the stability of the automatic control system whose structural diagram is shown in Fig. 1. The result of calculating the configuration parameters for the PID algorithm according to our method is shown in Fig. 9 and given in Table 5. Table 5 Values of the control PID-algorithm configuration parameters and the control process quality indicators at k α =1,25 and k α,1 =1,4 Now, assume b 0 =0. The other parameters of transfer function (33) were: b 1 =3, b 2 =1; a 0 =6, a 1 =4, a 2 =7, a 3 =1. The parameters for transfer function (31) are selected so that the control object is robust, that is, the following ratio holds: a 1 a 2 -a 0 a 3 >0. The parameters at which the values of the roots can be changed on the p-plane were: k α =1.25 and k α,1 =1.4.
The computational experiments have shown that when determining the parameters of the PID algorithm, it is necessary to take into consideration not only the compatibility of equation system (40) but also the requirements for the stability of the automatic control system. Under such an integrated approach to solving the problem of calculating the configuration parameters for the PID algorithm, it is necessary to choose the method that primarily ensures the stability of the automatic control system.
Applying the method for transforming the system of equations (40) to a compatible form has shown that the left root, α c,1 , is close to the imaginary axis, which, first, does not allow for the necessary margin of system robustness, and, second, a value for the setting parameter C 0 is close to zero. This low C 0 value causes an unacceptably large static error of control.  Fig. 1 Therefore, to calculate the configuration parameters for the PID algorithm, a method was chosen that is based on solving the system of equations (43).
The closest to the imaginary axis was the real left root |α c,1 |=0,9. For the set of values of the degree of fluctuation μϵ{0,2;0,4;0,6;0,8}, the values of the parameters of the control algorithm were calculated and the quality indicators of the control process were determined ( Table 6).
The transitional characteristics of a closed automatic control system (Fig. 1) with a single jump-like input value u(t) are shown in Fig. 10.
The analysis of our results shows that the transition process in the automatic control system is oscillatory, which is explained by the presence of a pair of left complex-coupled roots. Table 6 Values of the PID-controller configuration parameters and the control process quality indicators at k α =0.6 and k α,1 =1.4 The degree of fluctuation, μ  Consider the case where b 0 =b 1 =0, and b 2 =k, where k is the transfer factor of the object. For this case, equation system (37) takes the following form: 1 .
The first equation of system (46) does not contain the desired values C 0 , C 1, and C 2 . This means that the values of quantities k α and k α,1 , which determine the arrangement of roots p 2 , p 3 and p 4 on the p-plane (Fig. 6), cannot be selected arbitrarily; one of them, k α or k α,1 , depends on the other. Assume that the selected value is k α,1 . Then, from the first equation of system (46), we determine The k α value determines the position of the root p 2 on the p-plane. To ensure the stability of the automatic control system (Fig. 1), the root p 2 must be left. This means that k α >0, or α > + α For the selected values a 0 =3, a 1 =8, a 2 =2, a 3 =1; k=5, α c,1 =0,4, k α,1 =1,2, using the software developed in the MATLAB environment, at values μϵ{0.2; 0.4; 0.6;0.8}, we built the plots of transition processes (Fig. 11) and determined quality indicators of the control process (Table 7). Table 7 Values of the control PID-algorithm configuration parameters and the control process quality indicators at k α,1 =1.2 The degree of fluctuation, μ In conclusion, we consider the case where the roots of characteristic polynomial (34) are real numbers, located in the left semi-plane of the p-plane, and the transfer function of an object is described by formula (33). As before, we believe that there are the following ratios between the roots of the characteristic equation: p 1 =-α c,1 , p 2 =-k α α c,1 , p 3 =-k α,1 α c,1 , p 4 =-k α,2 α c,1 , where α c,1 >0, k α >0, k α,1 >0, k α,2 >0.
The system of equations (36) takes the following form: ( ) The system of algebraic equations (47) is represented in a matrix-vector form:

АС=В.
The elements of matrix A and vector B are as follows:  So, we have a rectangular matrix of size n×m, where n=4 and m=3. Since the condition n>m holds, the system of equations (47) is redefined [20]. Fig. 10 shows the transition characteristics of a closed system (Fig. 1) for the following data: The parameters of the algorithm k α , k α1 , k α2 , which determine the position of the roots on the p-plane were as follows: 3. The value α c,1 was selected from the set α c,1 ϵ{0,9;0,8;0,7}.
The configuration parameters of the control PID algorithm are calculated by solving the system of algebraic linear equations (47) using the built-in lsqr function, which is part of the MATLAB software [21]. Transitional characteristics of the system are shown in Fig. 12.
Thus, our results confirm the possibility of determining the configuration parameters of controllers by arranging the roots of a characteristic polynomial on a complex p-plane.

Discussion of the study results on calculating the configuration parameters of PID controllers by arranging poles on the p-plane
One of the possible directions of searching for a solution to the problem of determining the configuration parameters of PID controllers is a method based on the generalized Viète theorem, which makes it possible to reduce the formed problem to solving the system of linear algebraic equations, which is a significant advantage over those methods where the solution is based on the methods of nonlinear programming [6][7][8][9].
In the case where the transfer function of the control object is of the first and second orders, the problem of deter-mining the configuration parameters of a PID controller is solved unequivocally as the solution to the systems of linear algebraic equations (17) and (27).
If the transfer function of the automatic control system is of the third and higher orders, then we obtain the redefined system of linear algebraic equations, which, in a general case, does not have an unambiguous solution. Only when the conditions of the Rouché-Capelli theorem are met, one can derive a solution to the algebraic equation system relative to the configuration parameters of the PID controller. For an object's third-order transfer function, the condition of the Rouché-Capelli theorem can be satisfied by selecting the degree of stability of the system. This method of calculating the configuration parameters of the PID controller does not always guarantee the achievement of the necessary quality indicators of the control process.
Other alternatives may be to search for a pseudo-solution to the redefined system of linear algebraic equations, or a method that provides for a minimum of the criterion (42).
Our conclusions are based on the numerical experiments, the results of which are represented in the form of Tables 1-6. The quality of the control process was determined by constructing the plots of transitional characteristics ( Fig. 2-12).
The analysis of our results shows that the overshoot and control time are within the permissible limits (overshoot does not exceed 30 %).
Further research into related topics implies determining the parameters for setting up industrial controllers for multiconnected systems.

1.
A concept of the method for calculating the configuration parameters of PID controllers has been proposed. It is shown that for the control object's transfer functions of the first and second orders, the problem has an unambiguous solution. In the case where the order of the transfer function of the control object is of the third and higher orders, we obtain the redefined system of linear algebraic equations relative to the parameters of the controller configuration. For such a case, three methods for calculating the configuration parameters of PID controllers have been built. An appropriate method is selected by considering the requirements for the stability of the automatic control system and the quality indicators of the control process.
2. A procedure has been devised that makes it possible to select the necessary method of solving the problem in iterative mode, based on the order of the transfer function, requirements for the stability of the system, and indicators of the control process. The effectiveness of the method for calculating the configuration parameters of PID controllers for the predefined set of transfer functions of an object has been evaluated. Based on the transition characteristics of the automatic control system, the indicators of the control process have been determined: overshoot and time of control. According to the results of simulation modeling for the predefined class of transfer functions of objects, it was established that the overshoot is in the range of 0 to 27.9 %, which corresponds to the current norms.