CC BY
19INFORMATION TECHNOLOGY & ENGINEERING GEOMETRY
УДК 622.046.
© Usmanov K.I.
SYNERGISTIC INTELLIGENT CONTROL OF NONLINEAR DYNAMIC OBJECTS
Usmanov K.I. - senior lecturer of department "Informatics, automation and control" of the Tashkent chemical-technological institute.
Abstract: The article deals with the synthesis of effective algorithms for controlling a chemical reactor and developed a fuzzy synergistic controller for a class of indefinite nonlinear dynamic systems. The synthesis of control laws is performed by the method of analytical design of aggregated controllers (ADAR). Intelligent systems are used to assess the unknown nonlinear behavior of an object, and a new adaptive fuzzy controller is developed based on synergetic control theory. It consists of a fuzzy system for approximating unknown system dynamics using adaptive synergistic control to archive the desired characteristics. Keywords: synergistic synthesis, intelligent synergetic controller, fuzzy logic system, synergetic control theory, ADAR method.
Аннотация: Ма^олада кимёвий реакторни бош^аришнинг самарали алгоритмларини синтез ^илиш масалалари ва ноани^ синфли дискрет ва^тли ночизи^ли динамик тизимлар учун но^атъий синергетик ростлагич ишлаб чи^илган. (АКАР) усулидан фойдаланиб синтез ^илинди. Интеллектуал тизим объектнинг ноани^ ночизи^ли х,олатини бах,олаш учун ишлатилади, янги адаптив но^атъий контроллер эса синергетик бош^ариш назарияси асосида ишлаб чи^илган. У зарурий характеристикаларни архивлашдаги адаптив синергетик бош^ариш ёрдамида номаълум динамик тизимларни аппроксимациялашда ^улланиладиган но^атъий тизимлардан иборатбулади.
Калит сузлар: синергетик синтез, ночизи^ли синергетик ростлагич, но^атъий манти^ тизими, синергетик бош^ариш назарияси, АКАР усули.
Аннотация. В статье рассмотрены вопросы синтеза эффективных алгоритмов управления химическим реактором и разработан нечеткий синергетический регулятор для класса неопределенных нелинейных динамических систем. Синтез законов управления выполнен методом аналитического конструирования агрегированных регуляторов (АКАР).
Интеллектуальные системы используются для оценки неизвестного нелинейного поведения объекта, а новый адаптивный нечеткий контроллер разработан на основе синергетической теории управления. Он состоит из нечеткой системы для аппроксимации неизвестной динамики системы с помощью адаптивного синергетического управления для архивирования желаемых характеристик. Ключевые слова: синергетический синтез, нечеткий синергетический регулятор, система нечеткой логики, теория синергетического управления, метод AKAP.
One of the main apparatuses of chemical industry is a chemical reactor, the purpose of which is to ensure at its output a predetermined optimal concentration value provided for in the technological regulations of the target product. It is known that a chemical reactor is an energy-consuming object. In
this regard, the economic efficiency of the entire production largely depends on ensuring the normal functioning of the chemical reactor and its performance. The main feature of chemical reactors as control objects is their multidimensional^, as well as the uncertainty of the concentration of the initial mixture. The indicated super systems are nonlinear, multidimensional, and multiply connected, in which complex transient processes occur and critical and chaotic regimes arise. The control problems of such dynamic systems are very relevant, difficult and practically inaccessible to the existing control theory [1-4].
Research Methods and the Received Results. A
chemical reactor is a volume-type apparatus equipped with a mechanical stirrer and cooling jacket (Fig.1). The device operates in isothermal mode. The multistep series-parallel reaction is carried out in the reactor as follows: A + B ^ C A + C ^ C2
A + C2 ^ C3
(1)
The kinetics of the reaction is described by a system of equations
(2)
-— —kj • Xj • X2 — k"2 ■ Xj • X3 — k-3 • Xj • X4
dt
0X2
-— —k • Xi • X-)
dt j j 2
dX3
- - kj • Xj • X2 - k2 • Xj • X3
dt
d*4 , ,
-— k2 • Xj • X3 — кз • Xj • X4
dt
dX5
-— • Xj • X4
dt
where x , x2 - are the concentrations of reagents A
x x x
and B; 3 , x4 , 5 - concentration of reaction
k k k -products; i' 2' 3 stage speed constants [5].
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Fig.1. Flow scheme of chemical reactor.
The apparatus implements a three-stage seriesparallel exothermic reaction:
A+B ——^C, A + C, —^ C 2 A+C2 ——^ C3
(3)
where A and B are initial reagents;
■ Cj C2 C3 -
reaction products; k i , k2 , 3 - stage speed constants. The target component is the substance
C.
. k 1 k 2 , k 3
rget component 2 Starting reagents A and B with concentrations
xin xin _
i ' 2 served in the device in separate streams
xinl x in 2 _
2 and temperatures 6 ' 6
gj,G2 -
G, -
refrigerant consumption at the
with costs
respectively.
xin x _
inlet and outlet of the apparatus; 7 ' 7 refrigerant temperature at the inlet and outlet of the apparatus; G_ mixture consumption at the outlet of
x, X2, X3, x^
component
x _
2 in the reactor; 6
G, • xj G • Xj
-—R. - G*-,
dt V
dx6 G1 • xf G2 • xf G • x6 AH, • k, • x, • x2 + AH2 • k2 • x, • x3 + AH-3 • k3 • x, • x4 KT ■ FT ■ (x6 - x7 )
dt V V V
dx7 _ Gr • x7 Gv • x'f KT ■ FT ■ (x6 - x7)
d" ~ ~V V~+ K-p-c,
pC
KT ' T (x6 x7 )
V^ pC
where R, _ kl xi x2 k2 'xJ 'x- k- 'xJ 'x.
^R— — k, • x, • x2 k2 • x, • x— • x.
control principle, u (Ui,-Um) , should be determined as the function of state variables of object
ui = (ui,-u„X ..., um = (ui,-u„X which transforms the representative point of system in phase space from the random initial state to the environment of
the given invariant manifolds ^s (xi'"'x) — 0 S = 1,..., m and subsequent motion along the intersection of manifolds to somewhat stationary point or to somewhat dynamic mode [10].
Macro variables ^s (xi''''xn) must satisfy the
functional equation r^(i)+^(t) =0 , (10) which at
q>(V)V > 0 and T > 0. Because the mathematical model of object (8) contains two external controlling
effects ui = G2 and u2 = Gr , we use the ADAR method on the basis of parallel-series combination of invariant manifolds [11].
= (x7 + Vi) x4 Rrx4 _ G xi dvi (R5 ■ x4 _ x5 G )
the apparatus;
concentrations A B Ci' C the temperature of the reaction mixture in the
apparatus; V = x _ apparatus volume; Gsv _ shirt refrigerant volume [6].
The mixture from the reactor is taken by the pump. Since an exothermic reaction proceeds in the apparatus, a coolant is fed into the reactor jacket to cool the reaction mass [7].
A mathematical model of the dynamics of a chemical reactor consists of material balance equations for each component in the reactor, heat balance equations of the reaction mixture and the coolant in the shirt [8-9]:
T yir
T 2 • x7
dx„
Fig. 2. Simulation model of the control system.
Let us introduce aggregate macrovariables to consideration, the first of which determines the relationship of x with controlled variable x and the second reflects the technological requirement to the volume of reaction system as follows
¥x _ x4 - x ■
y/2 — x7 + V ( x6)
(11)
where
V( x6)
is somewhat function, which should be
R2 — _k2 ■ xi ■ x2 R = k2 ^ _k3 _ is the rate of reaction
on components. 'z = i, ':3 thermal effect of the
K F _
corresponding reaction stage; kt ' ft heat transfer coefficient through the wall and heat transfer surface
of the apparatus; p'c _ density and heat capacity of the reaction mixture; p,cr _ density and heat capacity of the refrigerant.
In general, the problem of synergetic synthesis of the control system is formulated as follows: the
determined at subsequent procedure of synthesis [12]. Macrovariables (11) should follow the solution of principal functional equation of ADAR method (10).
Let us introduce the macrovariables and of equation (11) to functional equation (10) for the synthersis of
control principle, u = ) . As result, we obtain the following equations [13]:
dx —
T dT + x4 - x4 = 0, and
dx7 + dv,, dv,
dz dx dz
6
+ x7 + Vn — 0.
(12)
2
x
in
x
x
7
dx
G •x'f G^x
2 — R +^^2
G x
3 — R,
T
2
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We obtain the following relationships for the control IF Error is NM AND Change in Error is NB THEN principle from equations (12): Output is NS
(x - x ) — u =—4 + G -u15
(13)
150 140 ^30 120 110 100
2, /
/ r 1 L r
L
0
50
100
150
t, MHH
200
250
300
6 |4
1 V r
Y
0
50
100
150
t, MHH
200
250
300
Fig.3. Transients of the output variable and control at the initial deviation of state variables from statics: 1 - the first embodiment of the control algorithm, 2 - the second option.
A fuzzy system is a collection of IF-THEN rules in the form:
R(l} : IF x1 is F1l and ... and xn is Fln THEN y is
x = (x x )T where v 1V"' n) - is the input of the fuzzy
systems.
The chemical rector model has input linguistic variables:
The linguistic rules for such a PID-like fuzzy controller are given in table.1.
table.1.
Fig. 4 Degree MF inputs and outputs FLC
The block diagram of the mathematical model of the system in the MatLab complex is shown in Fig.5.
i=£>
G'
Q-p
Fig.5. Block diagram of the mathematical model of the tension control system with a fuzzy controller
Step response of various fuzzy binary distillation column controllers, individually tunable GA [15]; GA - Genetic Algorithm; FLC- Fuzzy Logic Control; PSO-; MF;
Outputs Errors
(D TO NB NM NS SS PS PM PB
NB SS NS NM NM NB NB NB
U) E NM PS SS NS NM NM NB NB
-Q (O NS PM PS SS NS NM NM NB
(D ^ C SS PM PM PS SS NS NM NM
CO -C O PS PB PM PM PS SS NS NM
PM PB PB PM PM PS SS NS
PB PB PB PB PM PM PS SS
PB: Positive big NS: negative small
PM: positive mean NM:negative environment
PS: positive small NB: negative big SS: steady state
The rules are applied in the IF-Then form as follows [14]:
f^ IF Error is NB AND Change in Error is NB THEN rH Output is SS
/
/
\
--- Sel-poinl PSO-PD FLC J'SÜ-J'J ]■']_(' PSO-PID FLC GA-PDFLC GA-PI FLC GA-PŒ) FLC 1 —
-
1
0 10 20 30 40 50 60 70 80 90 100
Fig.4. Simulink block diagram of a MIMO PD-like FLC with 12 scaling factors.
CONCLUSION
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In this paper, we have developed a fuzzy synergetic controller for regulating and tracking control of a class of nonlinear systems. The control law has been introduced by using methods of synergetic theory and fuzzy logic control which can handle the nonlinear systems with system uncertainties and external disturbances. The problem of analytical synthesis of nonlinear control laws, which stabilizes the temperature and concentration of the process in the chemical reactor by means of synergistic control methods, is solved.
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