DEVELOPMENT OF A MATHEMATICAL APPARATUS FOR DETERMINING OPERATOR IMAGES OF THE DESIRED QUANTIZED TRANSITION FUNCTIONS OF FINITE DURATION

An important part of modern methods for automated control systems synthesis is selection of the desired characteristic polynomial. The desired polynomial is usually selected from a known standard set. There are standard Butterworth polynomials [1], polynomials with binomial coefficients [2], Bessel polynomials [3], Graham and Lathrop polynomials [4], and so on. In most cases, the roots locations of characteristic equations for closed-loop systems by standard polynomials have semi-empirical nature: they cannot be considered optimal according to some optimization criterion. For example, when using the binomial characteristic polynomial, all roots of a characteristic equation are selected as identical, negative and real with a value of the module 0 ω , which determines the processing rate of the synthesized system. The Butterworth polynomial puts the roots on the half-circle with a radius 0 ω at equal angular distances (Fig. 1). The coefficients of Graham and Lathrop polynomials are defined by mathematical modelling. In this way, standard polynomials allow us to specify the necessary dynamic properties of the system only roughly. In doing so, almost identical transient processes with various ways of the pole locations can be obtained. Fig. 1. Roots location by the Butterworth polynomial for a third-order system


Introduction
An important part of modern methods for automated control systems synthesis is selection of the desired characteristic polynomial.The desired polynomial is usually selected from a known standard set.There are standard Butterworth polynomials [1], polynomials with binomial coefficients [2], Bessel polynomials [3], Graham and Lathrop polynomials [4], and so on.
In most cases, the roots locations of characteristic equations for closed-loop systems by standard polynomials have semi-empirical nature: they cannot be considered optimal according to some optimization criterion.
For example, when using the binomial characteristic polynomial, all roots of a characteristic equation are selected as identical, negative and real with a value of the module 0 ω , which determines the processing rate of the synthesized system.The Butterworth polynomial puts the roots on the half-circle with a radius 0 ω at equal angular distances (Fig. 1).The coefficients of Graham and Lathrop polynomials are defined by mathematical modelling.
In this way, standard polynomials allow us to specify the necessary dynamic properties of the system only roughly.In doing so, almost identical transient processes with various ways of the pole locations can be obtained.The existing problematic issues in using standard characteristic polynomials for the regulators' synthesis require finding innovative approaches to solving the problem of assigning desired dynamics to the automatic control systems.A mathematical apparatus should be suggested and be devoid of the disadvantages of standard characteristic polynomials.This determines the direction and relevance of the research that is presented in the study.

Analysis of published data and statement of the problem
So far there have been no general guidelines developed for selecting a trajectory at which the poles should be placed for any automated control system.
The study [5] suggests carrying out the synthesis of characteristic polynomials according to certain quality principles (for example, by an overshoot given with a uniform poles distribution on the circle).However, desirable polynomials are considered to be known in advance.
In general, during the automated control system synthesis by a standard characteristic polynomial, dynamic properties of the system are determined by its coefficients that remain unchanged in operating activities.Standard polynomials cannot reflect dynamic features of real automated control systems because their behaviour is predetermined by poles distribution -that is, polynomial coefficients.
In some cases, the problem of control system synthesis is solved locally, only for specific equipment.Thus, the roots of the characteristic equation are located, taking into account the features and indeterminacies of a separate technical object [6,7].
An expansion opportunity that is offered by the standard theory of characteristic polynomials is a generalized characteristic polynomial method [8,9].This method allows an integrated approach to the location of zeros and poles of the transfer functions for automated electromechanical systems and provides an implementation of transition functions in a standard form.
In particular, synthesis of subordinated regulation systems by using a generalized characteristic polynomial can extend their dynamic properties because regulators in loops can be configured to perform not only a modular or symmetric optimum but other standard forms of transition functions that are defined by standard characteristic polynomials.
However, the generalized characteristic polynomial method is still used as a reference to standard characteristic polynomials and, in this sense, does not bring any innovation.Consequently, no matter how progressive a synthesis method is when it is based on standard polynomials, the possibility of the automated control system will be limited by a selected standard polynomial.
It is not necessary that the trajectories of roots movement should be continuous functions.They can be piecewise monotonic functions or discrete argument functions (lattice functions).For example, the study [10] demonstrates a system with a toggle property between two possible positions of poles, one of which is required in the acceleration mode, and the other -in operation.Discontinuous control in systems with a variable structure is connected with a similar approach.These systems have multiple regulators implementations that can change during the control process [11,12].The structure, moments and durations of regulators switching are selected so as to provide the desired control quality [13].The resulting variation in output coordinates will be some time-expanded superposition of individual components influencing the regulator with a variable structure.
A convenient and clear alternative for imparting the desirable properties to systems in static and dynamic modes (without using standard characteristic polynomials) is not the desired trajectory of roots movement definition but the desired transition function determination.This function is not selected by the designer from some list of standard forms during regulators synthesis but is given solely on the basis of technological requirements and possibilities of technical implementation for a defined type of equipment.Such a transition function can be changed in the operation time of the machine, mechanism or process complex, providing quantitatively and qualitatively new properties for an automated electromechanical system [14].
To improve opportunities for software implementation of the regulator, which allows to control the electromechanical system in that way, it is advisable for the desired transition function to be represented in a numerical form, i. e. as a set of operated coordinate values that change with a period T, which is relatively small with regard to the duration of the transition process (Fig. 2).

Fig. 2. An example of the desired transition function presentation
Thus, avoiding the use of standard characteristic polynomials in the synthesis of automated control systems requires their replacement by another mathematical apparatus, which would be the basis for creating regulators with desired quality ratings in static and dynamic modes.The desired transition function in a numerical (discrete) form can be considered as such basis.This approach requires development of an appropriate method for performing synthesis of automated electromechanical systems.

The purpose and objectives of the study
The purpose of the study is to prove the possibility of analytical synthesis of regulators for a quantized form of desired transition functions and develop a mathematical apparatus for composing quantized transition functions of finite duration as entities that can replace standard characteristic polynomials.
The objectives of the study are as follows: -to analyse the possibility of using quantized transition functions of finite duration for discrete regulators synthesis; -to suggest an analytical method for regulators synthesis in open-loop automated control systems by representing the desired quantized transition functions as the amount of time for shifted Heaviside functions; -to develop a mathematical apparatus for analytical determination of operator images for desired quantized transition functions of finite duration, relying only on the values of the signal levels at quantization points and a quantization period value.

The features of discrete control systems
Control systems with signals discretization in some form are called discrete.The realization of the idea of splitting the desired transition function into parts should be solved directly by its application not to analogue systems with continuous signals but to discrete systems that have at one or more points a sequence of impulses or a digital code.
It should be noted that in addition to the term "discrete system" the other terms used are "digital system" and "impulse system".Digital systems are mostly known as systems in which the coded control signals are generated by computer equipment.Impulse systems provide using impulse-amplitude modulation and signal quantization by time.The term "discrete systems" is the most integrant; it describes systems that can have impulse signals as well as digital codes.
In research literature, the term "discrete signal" means a signal sampled in time (Fig. 3, a), whereas the term "digital signal" refers to a signal that is sampled both in time and level (Fig. 3, b Let us consider the concept of "discrete" and "digital" signal capabilities within the context of modern microprocessor technology and software.It can be noted that the quantization signal level almost loses its meaning because of significant bit width for the data types that are implemented programmatically.For example, the data type double, which is used in the programming language C++, allows recording the minimal positive value of 1.7•10 -308 and the maximum positive value of 1.7•10 -308 with 15 significant digits after the decimal point [15].Such capabilities are enough to solve most technical problems without taking into account quantization by its level. Specialized computer engineering and general-purpose computer equipment become more used to control technical objects.Such equipment is used at low levels (for local proj-ects) as well as at higher levels (to control the technological lines or workshops) [16].Microprocessor engineering is often used in control systems as discrete regulators or devices that perform direct digital control of power converters.
One of the most significant advantages of digital regulators is their much greater flexibility in comparison with analogue regulators.The program of a digital regulator that is implemented by using a microcontroller can be changed according to new technological requirements or parameters of a control object at any time, without changing the hardware of the control system.
It can be concluded that discrete regulators provide significantly more opportunities regarding synthesis of control systems for technical objects.Thus, standard polynomials can be abandoned, and it requires searching for new approaches in sampling the desired transition functions and performing synthesis problem solving in a digital form.

Analytical regulators synthesis by a quantized form of the desired transition function
Output transition functions of technical systems, at changing the input actions or external disturbances, should satisfy certain requirements that are presented as factors of quality.These factors are determined by the shape of the transition function.In most methods of synthesis, regulators are selected so as to provide the desired level of two or three quality factors.The whole transition function is not covered.
The regulator transfer function can be obtained for the desired transition function by doing the reverse dynamic conversion; however, if the system is nonlinear, or if the transition function has no expression in elementary functions, direct inverse mathematical transformation is impossible.In this case, one can perform quantization, i. e. separation of the desired transition function on a set of Heaviside functions [17] with a rising time shift.
The study [14] is devoted to proving a possibility of analytical regulators synthesis by the desired transition function represented as a set of time-shifted Heaviside functions.
Let us consider the output coordinate ( ) y t of the automatic control system as a set of discrete values each of which exists during some time 0 T .The quantized transition function ( ) * y t can be represented as a step function, which is the sum of Heaviside functions ( ) that are delayed relative to the zero point by the whole number of periods 0 T . ( ) The inverse Laplace transform of the expression (1) gives the image ( ) * Y p for the signal ( ) h e e h e .p We represent an open-loop system as a constant part  The transfer function of this system: It is possible to attain the necessary dynamics in the open-loop system (Fig. 4).For this purpose, the transfer function Based on the formula (3), the regulator can build as follows: With the unit step action and the input of the automatic control system that have an image ( ) ( Exponential functions from the formula (5) can be represented as expansion in a Maclaurin series: where k is an integer number that reflects the order of the Maclaurin series component.
The sampling period 0 T is a very small value, so to simplify the calculations by the formula (6), the components, including the second and higher powers of 0 T , i. e.In this way, the regulator synthesis is performed by the formula (7).The levels i h and n h are defined by the desired transition function, but ( ) cp W p is defined by the constant part of the system.The parameter 0 T is selected from the technical characteristics of the control object, and it affects the locations of regulator poles in the complex plane.
Thus, for the considered open-loop system, the desired quantized transition function can be used in the regulators synthesis instead of the standard characteristic polynomial.A similar synthesis for a closed-loop system can be performed by setting only levels i h and quantization time 0 T .However, the scope of the proposed method is limited by Heaviside transformation capabilities, i.e. transfer functions should be presented in a balanced ratio of two polynomials that have simple (aliquant) roots [18].
The theory of transient processes of finite duration can provide one of the possible approaches to forming quantized transition functions that has no faults of the Heaviside transformation.

Quantized transition functions with finite duration
In analogue systems, transition functions are theoretically completed when time t → ∞, so in practice some trust zones are added.The transition functions are considered as finished when entering into such zones.
Transition functions of finite duration can be achieved in closed-loop discrete systems, unlike in analogue systems.Such processes are completed during the final number of quantization periods.
If the transfer function of a discrete system is in Z-form -

( )
W z and the input signal is a unit step action, the image of the output signal is represented by the following formula: where z is the operator of Z-transform associated with the Laplace operator p by the equation ( 9): where 0 T is the quantization period in a discrete system.The transitional function for a closed-loop system in a general case will be a ratio of polynomials: where k a , k 1 a − ,…, 1 a , 0 a are numerator coefficients of the transfer function; It should be noted that a system can be realized physically when the order of the transfer function numerator does not exceed the order of the denominator.For most orders, the difference between the denominator and the numerator is 1 or 2. Therefore, some of the coefficients at the high powers of z in the formula (10) may be equal to zero.For example, when the orders' difference between the denominator and the numerator is 2, then k a 0 = and k 1 a 0 − = .The coefficient at z in the highest power in the transfer function (10) denominator should be reduced to one so that the component k z is part of the polynomial.The image of output signal (8), taking into account the transfer function (10), is represented as follows: ( An infinite power series can be obtained by dividing the numerator of (11) by its denominator: ( ) where 0 c , 1 c , 2 c , … , i c are constant coefficients, whereas i is the number of the cycle.
With the determination of Z-transform, it follows that the coefficients of the series (12) are the values of the transitional lattice function ( ) 0 y iT in the moments of quantization, i. e.

( )
c y iT = .In order to provide a finite duration of the transition function, a series (12) at i k = should reach the value that remains unchanged for the next cycles.This condition can be accomplished when all of the characteristic polynomial coefficients, i. e. the denominator of the transfer function (10), reach point zero.
When fulfilling the condition (13), the transfer function (8) takes the following form: The characteristic polynomial k z 0 = of the transfer function ( 14) has k roots 1 2 k z z ... z 0 = = = = , which are located in the middle of the stability circle with a unit radius (Fig. 5).
The image of the output signal in the system with a transfer function (14) in response to the unit step input action is represented as follows: Fig. 5.The stability circle with roots 1 2 k z z ... z 0 The power series is formed by dividing the numerator of ( 15) by its denominator.In this series, the coefficients at i z − change only to the cycle number i k = ; after that, they remain unchanged for the next cycles: ( ) The series ( 16) gives the values of the lattice transition function at quantization points.Let us define the connection between coefficients k a , k 1 a − , … , , … , 1 c , 0 c by dividing polynomials according to the formula (15) -Fig.6.It is possible to set the following formal relations based on Fig. 6: The equation ( 17) can be written in a generalized form to establish a mathematical interpretation of the transition function of finite duration: a when i k, c a when i k.
The advantage of the formulas (18) consists in a possibility of an immediate determination of the values of transition functions of finite duration by transfer functions in the form (14).Each of the values i c is one level of the transition function of finite duration (Fig. 7).
The value 0 c is not equal to zero only when the order of the desired transfer function (14) numerator coincides with the order of the denominator, that is, when k a 0 ≠ .If the order of the numerator is smaller than the denominator order by value 2, the coefficients c 1 and c 0 are equal to zero ( 1 0 c c 0 = = ).The duration of the transition functions shown in Fig. 7 is defined as 0 kT , where k is the order of the desired transfer function denominator (14).The levels of discretized transition functions are changing at sampling moments and run up to the time quantization value of 0,05z 0,1z 0,3z 0,5z 1 W z z In the case under consideration, the greatest degree of the denominator is k 4 = because the transition function is completed in 4 cycles, i. e. the time length is equal to 0 4T .Numerator coefficients have the following values: Let us calculate the transition function of the system in response to the input unit step action by the formulas (18).The results are listed in Table 1.
The transition function of finite duration (Fig. 8) is based on data from Table 1.Table 1 The values of the transition function of finite duration Thus, the theory of transition functions of finite duration gives a quite simple and obvious mathematical apparatus to set the desired quantized transition functions that are based only on the step sizes at quantization moments and a quantization period.

Discussion of the research results of quantized transition functions of finite duration
Standard polynomials allow only rough setting of the required dynamic properties of automatic control systems.It is necessary to perform a mathematical modelling for determining the coefficients of some polynomials.The advantages of the desired dynamics assignment for automatic control systems by using quantized transition functions are the following: visibility of the analytical synthesis process and ease of subsequent technical implementation of discrete regulators programmatically.The use of the desired transition functions of finite duration provides the following benefits: -the time range of levels variation for desired transition functions is known in advance, so there is no need for any trust zones or calculations of the estimated time of the transition function duration with accuracy; -the quantization moments divide the time range into equal segments each of which can be set the required value of the desired transition function; -the relationship between the coefficients of the polynomial transfer function of a system that is configured for transient processes of finite duration and the level values that correspond to the quantized transition function are determined simply and clearly by equations (18); -all direct and inverse transformations that are associated with the passage from the transfer function of the system to the desired transition function are rather easy to represent in the program code; they will be used in further research to perform synthesis of a discrete time equalizer [19].

Conclusion
1. Analysis has enabled us to find a possible use of desired quantized transition functions for regulators synthesis: realization of the idea of splitting the desired transition function into parts should be solved naturally in its application not to analogue systems with continuous signals but to discrete systems that have, at one or more points, a sequence of impulses or a digital code.
2. An analytical method was suggested for regulators synthesis in open-loop automated control systems.This method allows representing the desired quantized transition functions as the number of time-shifted Heaviside functions, but its use is limited by features of the Heaviside transformation.
3. The theory of transient processes of finite duration can provide one of the possible approaches to forming quantized transition functions that has no faults of the Heaviside transformation.The developed mathematical apparatus can perform an analytical determination of operator images for desired quantized transition functions of finite duration, relying only on the values of signal levels at quantization points and a quantization period value.

Fig. 1 .
Fig. 1.Roots location by the Butterworth polynomial for a third-order system

Fig. 3 .
Comparison of discrete and digital signals: a -discrete signal; b -digital signal should provide changing of the output coordinates by the desired quantized transition function ( )

Fig. 4 .
Fig. 4. The open-loop system consisting of a constant part and a regulator considered as constant in the dynamic mode, and the image ( ) U p corresponds to one of the typical input actions, which are used in studying automatic control systems (including the step function and the ramp function).
the transfer function for the regulator is converted to the form . With these expressions, the formula (5) takes the following form: coefficients of the transfer function; k is the order of the characteristic polynomial.
us consider an example.Suppose the desired transfer function of the system ( ) W z is configured for transient processes of finite duration:

Fig. 6 .
Fig.6.The division of polynomials according to the formula(15)

Fig. 7 .
Fig. 7.The transition function of finite duration (built on values c i )