Synthesis of chemical reactor nonlinear control algorithm using synergetic approach Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

Т 60 (2)

ИЗВЕСТИЯ ВЫСШИХ УЧЕБНЫХ ЗАВЕДЕНИИ. Серия «ХИМИЯ И ХИМИЧЕСКАЯ ТЕХНОЛОГИЯ»

2017

V 60 (2)

IZVESTIYA VYSSHIKH UCHEBNYKH ZAVEDENIY KHIMIYA KHIMICHESKAYA TEKHNOLOGIYA

2017

DOI: 10.6060/tcct.2017602.5479

Для цитирования:

Лабутин А.Н., Невиницын В.Ю. Синтез нелинейного алгоритма управления химическим реактором с использованием синергетического подхода. Изв. вузов. Химия и хим. технология. 2017. Т. 60. Вып. 2. С. 38-44. For citation:

Labutin A.N., Nevinitsyn V.Yu. Synthesis of chemical reactor nonlinear control algorithm using synergetic approach. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 2017. V. 60. N 2. P. 38-44.

Александр Николаевич Лабутин, Владимир Юрьевич Невиницын (EI)

Кафедра технической кибернетики и автоматики, Ивановский государственный химико-технологический университет, пр. Шереметевский, 7, Иваново, Российская Федерация, 153000 E-mail: [email protected], [email protected] (И)

СИНТЕЗ НЕЛИНЕЙНОГО АЛГОРИТМА УПРАВЛЕНИЯ ХИМИЧЕСКИМ РЕАКТОРОМ

С ИСПОЛЬЗОВАНИЕМ СИНЕРГЕТИЧЕСКОГО ПОДХОДА

В работе решена задача аналитического синтеза синергетической системы управления химическим реактором для реализации сложной последовательно-параллельной экзотермической реакции. Синтез законов управления осуществляется с использованием метода аналитического конструирования агрегированных регуляторов. Химический реактор является одним из основных аппаратов в химической промышленности. Несмотря на значительное количество работ, связанных с автоматизацией и управлением химическими реакторами, проблема синтеза систем управления, обеспечивающих поддержание оптимальных режимов их работы, остается до конца не решенной. Это объясняется основной особенностью химических реакторов как объектов управления: многомерностью, нелинейностью и многосвязностью. Выходом из данной ситуации является развитие физической теории управления и в частности синергетической теории управления. Использование идей синергетики в задачах управления предполагает разработку и реализацию способа направленной целевой самоорганизации диссипативных нелинейных систем "объект-регулятор". При этом цель движения системы формулируется в виде желаемого инвариантного многообразия в фазовом пространстве объекта, выполняющего роль целевого аттрактора. В работе рассмотрен химический реактор емкостного типа, снабженный механической мешалкой и теплообменной рубашкой. Аппарат функционирует в политропическом режиме. В реакторе реализуется многостадийная последовательно-параллельная экзотермическая реакция. Целью функционирования химического реактора является получение целевого продукта заданной концентрации. Задача системы управления реактором заключается в стабилизации концентрации целевого компонента и температуры реакционной смеси в аппарате на заданных значениях в условиях действия возмущений на объект. Используя метод аналитического конструирования агрегированных регуляторов на основе параллельно-последовательной совокупности инвариантных многообразий, синтезирован нелинейный алгоритм управления, решающий задачу стабилизации концентрации целевого компонента и температуры смеси. Компьютерное моделирование

УДК: 66.011.001:681.51

А.Н. Лабутин, В.Ю. Невиницын

замкнутой системы "объект-регулятор" подтвердило такие свойства синтезированной системы управления, как инвариантность к возмущениям, ковариантность с задающими воздействиями и асимптотическая устойчивость. Данные обстоятельства делают синер-гетическую теорию управления весьма перспективной применительно к таким сложным, многосвязным и нелинейным объектам химической технологии, как химические реакторы.

Ключевые слова: аналитический синтез, система управления, химический реактор, синергети-ческая теория управления, компьютерное моделирование

UDC: 66.011.001:681.51

A.N. Labutin, V.Yu. Nevinitsyn

Alexander N. Labutin, Vladimir Yu. Nevinitsyn (M)

Department of Technical Engineering Cybernetics and Automation, Ivanovo State University of Chemistry and Technology, Sheremetievskiy ave., 7, Ivanovo, 153000, Russia E-mail: [email protected], [email protected] (M)

SYNTHESIS OF CHEMICAL REACTOR NONLINEAR CONTROL ALGORITHM USING

SYNERGETIC APPROACH

The problem of analytical synthesis of synergetic control system of chemical reactor for realization of a complex series-parallel exothermal reaction has been solved. The synthesis of control principles is performed using the analytical design method of aggregated regulators. A chemical reactor is one of the common apparatuses in chemical industry. Despite a large number of the works related to automation and control of chemical reactors, the problem of synthesizing control systems that provide the maintenance of optimal modes of their operation remains practically unsolved. This is related to the principal feature of chemical reactors as controlled objects, namely, manifold, non-linearity, and multi-coupling. An outcome from this situation is to develop a physical theory of control, in particular synergetic control theory. The use of synergism ideas in the problems of control assumes the development and realization of a method of directed target self-organization of dissipative non-linear systems "object-regulator". Furthermore, the aim of the motion of a system is formulated as the desired invariant manifold in a phase space of the object, which acts as a target attractor. The paper deals with continuous stirred tank reactor equipped with a mechanical stirrer and cooling jacket. The reactor operates in the polytropic mode. The multistep seriesparallel exothermic process is carried out in the reactor. The objective of chemical reactor operation is to obtain the key product of specified concentration. The aim of chemical reactor control system is to maintain both concentration of desired product and temperature of reaction mixture in the device at the given values under the action of disturbances on the object. Using the analytical design method of aggregated regulators on the basis of parallel-series combination of invariant manifolds, a non-linear control algorithm was synthesized, which solves the problem of stabilization of the concentration of the target component and mixture temperature. Computer simulation of the object-regulator isolated system confirmed these properties of synthesized control system as the disturbance invariance, covariance to the given actions, and asymptotic stability. These facts make synergetic control theory very promising for application for such complex, manifold, and nonlinear objects of chemical engineering as chemical reactors.

Key words: analytical synthesis, controlled system, chemical reactor, synergetic control theory, computer simulation

INTRODUCTION

During the step of the design of chemical production, which is related to the conversion of initial substances to final products, the problem of optimal synthesis of a reactor unit and problem of synthesis of the process control algorithms should be solved; and, at the step of industrial operation, the sub-problem of organization of optimal functioning of the object under the effect of parametrical and signal disturbances must be solved too [1-5].

Despite the large number of the works related to the automation and control of chemical reactors [69], the problem of control systems synthesis that provide the maintenance of optimal modes of their work remains practically unsolved. This can be explained by the principal features of chemical reactors as controlled objects, namely, manifold, nonlinearity, and multi-coupling.

A possible outcome from this situation is to develop a physical theory of control, in particular syner-getic control theory, the principal features of which were formulated in [10-12].

The use of synergism ideas in the problems of control assumes the development and realization of the directed target self-organization of "object-regulator" dissipative nonlinear systems. It is supposed that the aim of a system motion is formulated as the desired invariant manifold in the phase space of an object, which acts as a target attractor [12].

In general, the problem of synergetic synthesis of the control system is formulated as follows: the control principle, u = («1,..., Um)T should be determined as the function of state variables of object U1(x1,..., Xn), ., Um(x 1,., Xn), which transforms the representation point (RP) of a system in the phase space from the random initial state to the neighborhood of the given invariant manifolds i^s(x1,..., Xn) = 0, S = 1,., m, and subsequent motion along the intersection of the manifolds to any stationary point or to any dynamic mode. In the given relationships, n is the dimension of the state vector and m is the number of external controls. On the trajectory of the motion, the minimum of the criterion of optimality of system (J) must be reached and its stability should be ensured as follows:

f • 2 Y

'=I

£

s=1

Ts¥s + W2s

dt

(1)

jective functional (1). The condition of asymptotic stability of the entire system generally has the form Ts>0.

The efficiency of the method of analytical design of control algorithms for nonlinear objects by means of synergetic principle (the method of analytical design of aggregated regulators (ADAR)) is shown in [13-16].

In this work, the problem of synthesis the effective control algorithms for the chemical reactor in the realization of the complex series-parallel reaction is stated. The synthesized control system should provide stabilization of the concentration of the target component of a chemical reaction in the apparatus outflow and mixture temperature in the apparatus under the action of disturbances on object.

OBJECT DESCRIPTION AND STATEMENT OF CONTROL PROBLEM

A chemical reactor is a volume-type apparatus equipped with a mechanical stirrer and cooling jacket (Fig. 1). It is functioning in the polytropic mode. The multistep series-parallel exothermic reaction of oxy-ethylation of butyl alcohol is occurs in the reactor as follows:

' ®Pз,

A + B —®P, A + P1 ——®P2, A + P2 where A and B are the initial reagents; P1, P2, and P3 are the products of the reaction; k1, k2, and k3 are the rate constants of the stages. The key component is P2 substance. The initial reagents A and B are fed to the device by separate flows.

The motion of RP in the phase space follows the functional equation

TSYS +¥s = 0, S = 1,..., m, (2)

where Ts is the time constant. This is the equation of the stable extremal, which gives minimum to the ob-

Fig. 1. The schematic diagram of chemical reactor Рис. 1. Принципиальная схема химического реактора

In Fig. 1, the following nomenclature is used: xin, X2in are the concentrations of initial reagents; «1, U2 are the flow rates of initial reagents; xein1, xein2 are the temperatures of initial reagents; xY, X7 are the coolant temperatures at the inlet and outlet of reactor; Uc is the coolant flow rate at the inlet and outlet of reactor; xe is the temperature of reaction mixture in the apparatus; и is the use of reaction mixture on the exit from device;

0

x1, x2, x3, x4, and x5 are the concentrations of components A, B, Pi, P2, and P3 in reactor; Vis the volume of reaction mixture in the operating volume; and V is the volume of coolant in jacket.

The objective of chemical reactor operation is to obtain the key component of specified concentration. Therefore, the aim of chemical reactor control system is to maintain both concentration of desired product and temperature of reaction mixture at the given set points (values) under the action of disturbances, i.e. x4 = X4, x6 = x6 where x4 and x6 are the

given values of concentration and temperature, respectively. The flow rate of the initial reagent U2 and the coolant flow rate vc at the input to the device are chosen as the control parameters for concentration and temperature regulation, respectively.

The mathematical model of chemical reactor has the following form: dxi

-= Ri + MA - b2 xi - b3xiui

d t

dx

—- = R2 - b2 x2 + (MB - b3 x2 )ui d t

dx. d t dx d t dx dt dx

3 = R3 - b2 x3 - b3x3ui

4 = R4 - b2 x4 - Ьз x4ui

5 = R5 - b2x5 - b3x5u1

(3)

— (XiA^ix2 I (X2x^ix3 I ^^зI b2x^ I

d t

+bl - (bl + b2 )*6 + (2 - X6 )b3Ul

dx

-T = P2 (X6 - X7 ) + bi (x'7 - X7 )U2 d t

where R1 = -k1 x1 x2 -k2x1 x3 -k3x1 x4, R2 =-k1 x1 x2,

3 — k 1X1X2 k 2 X1 X3 , R4 — k 2 X1 X3 k3 X1 X4 , r^R5 — k3 X1 X4

is the rate of reaction for components; MA — u1 xi"/ V ; MB = x2"/V; b = 1/Vc ; b2 — u1 /V ; b3 = 1/V ; a = Afl\ /(pC), i —1,...,3 ; b = KrFr /(pCV) ; b2 = KrFr /(pcCcVc) ; A^i, i = 1,...,3 is the heat of reaction for its corresponding stage; p, C are the density and heat capacity of reaction mixture; pc, Cc are the density and heat capacity of coolant; Kt is the heat transfer coefficient; Ft is the heat transfer surface; ki = ki0 • exp(-Ei /R(x6 + 273)), i —1,...,3 is the reaction rate constant; ki0, i = 1,...,3 is the pre-exponen-tial multiplier of rate constant; Ei, i = 1,... ,3 is the activation energy; R is the gas constant; and U1 = «2, U2 = Uc are the control parameters.

The analysis of structure of the set of equations of mathematical model of reactor (3) shows that control parameter ui acts on variable x4 directly. The control parameter u2 acts on the variable x6 indirectly through the variable X7. Thus, the control channels for the concentration and temperature of reaction mixture in the apparatus can be represented as follows: ui^X4, u2^x7^x6.

SYNTHESIS OF CONTROL PRINCIPLES BY MEANS OF ADAR METHOD

Because the mathematical model of the object (3) contains two external control parameters ui = U2 and U2 = Uc, we use the ADAR method on the basis of parallel-series combination of invariant manifolds [12]. The procedure for synthesis the control principle consists of the following steps. At the first step, the invariant manifolds are considered as shown below:

^S(xi,...,x7) = 0, S = 1, 2, which determines the given relationships between phase coordinates of object, which, in turn, reflects the features of the controlled object and requirements of a designer to the system. The control principle u = (ui,U2)T is synthesized so as to perform the transition of the representation point of system in phase space from arbitrary initial position to the intersection of manifolds, ^i,2(Xi,...,X7) = 0.

Let us introduce two aggregated macro-variables, the first of which reflects the technological requirement to the concentration of target component and the second determines the relationship of x7 with controlled variable x6 as follows:

y = X4 -X4, y2 = X7 +Vi(x6>, (4) where vi(x6) is somewhat function, which should be determined at subsequent procedure of synthesis. Macrovariables (4) should follow the solution of principal functional equation of ADAR method (2).

Let us substitute the macro-variables ^i and ^2 of Eq. (4) to the functional equation (2) for the synthesis of control principle u = (ui, U2)T. As the result, we obtain the following equations:

dx. _

Ti—— + x4 -x4 = 0 d t

T

dx, d t

7 + dVi dx

dx.

_6

dt

+ x7 + Vi = 0

Due to the equations of the object (3) these relationships become:

Ti (R4 - b2x4 - b3x4ui) + x4 - X4 = 0

P2 (X6 - X7) + bi (x7" - X7 u + • (f + (5)

dx6

+ biX7 + (x6n2 - X6 )b3ui)] + X7 + Vi = 0

where f6 (X-y x1 x 2 + a2 k 2 x1 x3 + X3 k3 x1 x4

+

+ b2 x6n1 - (b + b2) x6.

The following relationships for the control principle follow from Eq. (5):

b

u _ ( x4 x4 ) + _ R4

Tp^ xA

b x4 b

(x7 + V1 ) b2 (x6 - x7 )

T 2 bi ( x^^r x7 ) bi ( x7 )

9Vl [Í6 +Pi x- +(x6"2 - Хб )b3Ui ] Эх,

(6)

— _ R1 + MA — b2 x1 - b3x1u1

b1 (x^^r X7)

Control parameters u1 and u2 transfer the RP of the system in the phase space to the intersection of manifolds = 0 and = 0, where the relationships x4 = X4, X7 = -V1 are realized and "the compression of phase space" can be observed, i.e., the decrease of dimension of the set of equations (3) occurs. The equations of decomposed system, taking into account the relationship X7 = -V1, takes the following form: dx1 d t dx

—- = R2 - b2 x2 + (MB - b3 x2 )u1 d t

dx3

-= R - bx — bx-u.

dt 3 2 3 3 3 1 (7)

dx. dt dx

-t = f-P1V1 + (x6n2 - x6 )b3U1 dt

The function V1 (x6) in the decomposed system (7) can be considered as the "internal" control, under the action of which the motion of object (7) along the intersection of manifolds ^1,2 = 0 takes place.

At the second step of procedure, the investigation of the expression for V1(x6) is performed. For this purpose, the aim of the motion of system (7) is considered in the form of invariant manifold, which describes the technological requirement to the system as follows:

5 _ R5 — b2 x5 — b3 x5u1

Уз _ Хб — Хб _ 0

(8)

Macro-variable ^3 meets the solution to the functional equation T3\j/3 + y3 = 0. Taking into account the relationship (8) and the model of decomposed system (7), it can be written as follows:

T3 (f -fan + (xf - x6)b3«)+ x6 - x6 = 0. (9) "The internal control" due to Eq. (9) can be written as:

(x6 - x6) , f6 , (x6n2 - x6)b3U1

v _ - 6 _ ^ + ^ + -

T3P1 b

A

Final expression for the control principle u = (Mi,M2)T can be obtained by substitution of the function vi (10) and its partial derivative dvi/dx^ to Eq. (6). The parameters of the control law adjustment, which affect the quality of the dynamics of the processes in object-regulator isolated system, are the time constants Ti, T2 and T3. The conditions of asymptotic stability have the following form: T1>0, T2>0, T3>0.

RESULTS AND DISCUSSION

In order to verify the workability of the synthesized control law for the chemical reactor, the computer simulation of the object-regulator closed system was performed. Properties of the control system, such as the disturbance invariance, covariance to the given disturbance, and the asymptotic stability of closed system were also studied.

T, min

Fig. 2. The change in controlled variables after the inflow U1 decrease (10%)

Рис. 2. Изменение регулируемых переменных при уменьшении расхода U1 на 10%

щ, L/min

4.5

u2, L¡min

3.5

1 1 1 1

г \ -

_

и, /

1 1 1 i

о

100

200

300

400

4

3

2

1

0 500

г, mm

Fig. 3. The change in control actions after the inflow U1 decrease (10%)

Рис. 3. Изменение управляющих воздействий при уменьшении расхода U1 на 10%

The simulation was performed with the following technological and design parameters of the object: V = 500 L, Vc = 290 L, x1in = 19.74 mol/L, x2in = (10) =10.93 mol/L, «1 = 1.5 L/min, « = 3.5 L/min, и = 5.0 L/min, Uc = 3.84 L/min, x6m = 20 °C, x6in2 = 30 °C, x7n = 20 °C,

Kt = 12 kJ/(m2 min K), Ft = 2.9 m2, p = 0.9 kg/L, C = 2 kJ/(kg K), pc = 1 kg/L, Cc = 4.18 kJ/(kg K), AH1 = ДН2 = ДН = 80 kJ/mol, activation energy E1 = 48635 J/mol, pre-exponential multiplier of the rate constant k1 kw = 109860 L/(mol min), ratio of the rate constants for consecutive stages k2/k1 = 2, k3/k1 = 2.5, given concentration of target component X4 = 0.652 mol/L, and

the given temperature of reaction mixture X6 = 140 °C.

The parameters of adjustment of regulators are T = = 50 min, T2 = 15 min, and T3 = 15 min.

In Figs. 2, 3, the examples of transient controlled processes in the object-regulator in the system are shown. The simulation results presented here deal with the effect of step disturbance with change inflow «1. At the initial time the chemical reactor operates in static mode. There is a step increase inflow «1 at the time point т = 50 min. From this time the control parameters U1 = «2 and U2 = «c are computed according to Eq. (6). Then there are concentration and temperature control processes are in progress. In the course of time the controlled variables X4 and X6 take desired values. The steady-state error is zero. As follows from Figs. 2, 3, the synthesized control law of the chemical reactor ensures disturbance invariance with change inflow «1.

CONCLUSIONS

In this work, the problem of the analytical synthesis of the control law of chemical reactor was solved by means of the methods of synergetic theory. Synthesized nonlinear control law solves the problem of stabilization of the concentration of the target component and mixture temperature under the action of disturbances in the object. Computer simulation of the object-regulator closed system confirmed these properties of synthesized control system as the disturbance invariance, covariance to the given actions (set points), and

ЛИТЕРАТУРА

1.

2.

3.

4.

5.

6.

asymptotic stability. These facts make synergetic control theory very promising for application to such complex, manifold, and nonlinear objects of chemical engineering as chemical reactors.

NOMENCLATURE

V - volume of reaction mixture in device, L; Vc - volume of coolant in jacket, L;

u1, v2 - consumption of initial reagents on the enter to device, L/min;

v - consumption of reaction mixture on the exit from reactor, L/min;

vc - coolant flow rate at the inlet and outlet of reactor, L/min;

ki, k2, and k3 - rate constants of steps, L/(mol min);

ki0 - preexponential multiplier of rate constant ki, L/(mol min);

Ri, R2, R3, and R4 - rate constant on components, mol/(L min);

ui, u2 - controlling actions, L/min;

xiin, x2in - concentrations of initial reagents, mol/L;

x6ini, X6m2 - temperatures of initial reagents, °C;

x7in - coolant temperature at the inlet of reactor, °C;

xi, x2, x3, x4, and x5 - concentrations of components A, B, Pi,

P2, and P3 in reactor, mol/L;

KT - heat transfer coefficient, kJ/(m2 min K);

Ft - heat transfer surface, m2;

p, pc - densities of reaction mixture and coolant, kg/L; C, Cc - heat capacities of reaction mixture and coolant, kJ/(kg K);

AHi, AH2 and AH3 - heat of reaction for its corresponding stage, kJ/mol;

Ei, E2, and E3 - activation energies, J/mol; R - the gas constant, J/(mol K);

X4 - given value of concentration of target component, mol/L; X6 - given value of temperature of reaction mixture in the apparatus, °C;

Ti, T2 and T3 - time constants, min; t - time, min.

REFERENCES

Kafarov V.V. Cybernetic methods in chemistry and chemical engineering. M.: Khimiya. i985. 448 p. (in Russian). Ostrovskiy G.M., Volin, Yu.M. Chemical reactor optimization methods. M.: Khimiya. i976. 248 p. (in Russian). Labutin A.N. Optimization of flexible multiproduct in continuous reactor systems. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. i999. V. 42. N i. P. ii7-i22 (in Russian). Labutin A.N., Isaenkov A.E., Volkova G.V. Optimal synthesis of flexible reactor systems. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 20i0. V. 53. N i2. P. i25-i27 (in Russian). Dvoretskiy D.S., Dvoretskii S.I., Ostrovskii G.M. Integrated design of power- and resource-saving chemical processes and process control systems: strategy, methods, and application. Theor. Found. Chem. Eng. 2008. V. 42. N i. P. 2939 (in Russian). DOI: i0.ii34/S00405795080i004i. Computer-Aided manufacturing in the chemical industry. Ed. by Dudnikov E.G. M.: Khimiya. i987. 368 p. (in Russian).

Кафаров В.В. Методы кибернетики в химии и химиче- 1. ской технологии. М.:Химия, 1985. 448с. Островский Г.М., Волин Ю.М. Методы оптимизации 2. химических реакторов. М.: Химия. 1976. 248 с. Лабутин А.Н. Оптимизация гибких многопродуктовых 3. реакторных систем непрерывного типа. Изв. вузов. Химия и хим. технология. 1999. Т. 42. Вып. 1. С. 117-122. Лабутин А.Н., Исаенков А.Е., Волкова Г.В. Оптималь- 4. ный синтез гибкой реакторной системы. Изв. вузов. Химия и хим. технология. 2010. Т. 53. Вып. 12. С. 125-127. Дворецкий Д.С., Дворецкий С.И., Островский Г.М. 5. Интегрированное проектирование энерго- и ресурсосберегающих химико-технологических процессов и ситстем управления: стратегия, методы и применение. Теор. основы хим. технологии. 2008. Т. 42. № 1. С. 29-39. Автоматическое управление в химической промышлен- 6. ности. под. Ред. Дудникова Е.Г. М.: Химия. 1987. 368 с.

7. Беспалов А.В., Харитонов Н.И. Системы управления 7. химико-технологическими процессами. М.: Академкнига. 2007. 696 с.

8. Сухарев А.В., Лабутин А.Н., Головушкин Б.А., Еро- 8. феева Е.В. Вопросы управления химическим реактором. Изв. вузов. Химия и хим. технология. 2011. Т. 54. Вып. 11. С. 119-122.

9. Lopez-Perez P.A., Neria-Gonzalez M.I., Aguilar-Lopez 9. R. Non-linear controller design with application to a continuous bioreactor. Theor. Found. Chem. Eng. 2013. V. 47.

N 5. P. 585-592. DOI: 10.1134/S0040579513050199.

10. Кольцова, Е.М., Гордеев Л.С. Методы синергетики в 10. химии и химической технологии. М.: Химия. 1999. 256 с.

11. Синергетика и проблемы теории управления. Под ред.

А. А. Колесникова . М.: Физматлит, 2004. 502 с 11.

12. Колесников А.А. Синергетическая теория управления.

М.: Энергоатомиздат. 1994. 344 с. 12.

13. Лабутин А.Н., Невиницын В.Ю., Волкова Г.В. Синтез и моделирование многомерной системы управления 13. каскадом химических реакторов. Изв. вузов. Экономика, финансы и управление производством. 2012. № 2.

С. 150-157.

14. Лабутин А.Н., Кукушкин А.В. Векторное управление реактором на основе синергетического подхода. Изв. 14. вузов. Экономика, финансы и управление производством. 2011. № 3. С. 81-86.

15. Лабутин А.Н., Невиницын В.Ю. Синергетический синтез регулятора концентрации целевого компонента в кас- 15. каде химических реакторов. Изв. вузов. Экономика, финансы и управление производством. 2011. № 3. С. 86-92.

16. Лабутин А.Н., Невиницын В.Ю. Синергетический синтез системы управления химическим реактором. 16. Изв. вузов. Химия и хим. технология. 2012. Т. 55. Вып. 11.

С. 104-107.

Bespalov A.V., Kharitonov N.I. Control systems for chemical technology processes. M.: Akademkniga. 2007. 696 p. (in Russian).

Sukharev A.V., Labutin A.N., Golovushkin B.A., Erofeeva, E.V. Chemical reactor control issues. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 20ii. V. 54. N ii. P. ii9-i22 (in Russian).

Lopez-Perez P.A., Neria-Gonzalez M.I., Aguilar-Lopez R.

Non-linear controller design with application to a continuous bioreactor. Theor. Found. Chem. Eng. 20i3. V. 47. N 5. P. 585-592. DOI: i0.ii34/S00405795i3050i99. Kol'tsova E.M., Gordeev L.S. Synergistic methods in chemistry and chemical engineering. M.: Khimiya. i999. 256 p. (in Russian).

Synergetics and Problems of Control Theory. Ed by A.A. Kolesnikov. M.: Fizmatlit. 2004. 502 p. (in Russian). Kolesnikov A.A. Synergistic theory of control. M.: Ener-goatomizdat. i994. 344 p. (in Russian). Labutin A.N., Nevinitsyn V.Yu., Volkova G.V. Synthesis and modeling of a multidimensional control system of a chemical reactors cascade. Izv. Vyssh. Uchebn. Zaved. Ser. Ekon. Finansy Upravl. Proizvodstvom. 20i2. N 2. P. i50-i57 (in Russian).

Labutin A.N., Kukushkin A.V. Synergistic approach to reactor temperature control. Izv. Vyssh. Uchebn. Zaved. Ser. Ekon., Finansy Upravl. Proizvodstvom. 20ii. N 3. P. 8i-86 (in Russian).

Labutin A.N., Nevinitsyn V.Yu. Synergistic synthesis of concentration regulator for the target product in a cascade of chemical reactors. Izv. Vyssh. Uchebn. Zaved. Ser. Ekon. Finansy Upravl. Proizvodstvom. 20ii. N 3. P. 86-92 (in Russian). Labutin A.N., Nevinitsyn V.Yu. Synergistic synthesis of a chemical reactor control system. Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol. 20i2. V. 55. N ii. P. i04-i07 (in Russian).

Поступила в редакцию 23.09.2016 Принята к опубликованию 26.12.2016

Received 23.09.2016 Accepted 26.12.2016