SYNERGETIC SYNTHESIS OF CONTROL SYSTEMS FOR NONLINEAR DYNAMIC OBJECTS Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

SYNERGETIC SYNTHESIS OF CONTROL SYSTEMS FOR NONLINEAR DYNAMIC OBJECTS

PhD Khurshida BAKHRIEVA ALFRAGANUS UNIVERSITY ORCID: 0000-0003-2709-1232

Abstract. The paper presents the possibilities of using the methods of the synergetic approach for the synthesis of the control law for nonlinear dynamic objects. To impart robust properties to the control law, it is proposed to apply the principle of integral adaptation, which makes it possible to compensate for the influence of external and parametric disturbances. The implementation of the synergetic control law is carried out when constructing the analytical design of aggregated controllers, which provide the asymptotic stability of the control system for nonlinear dynamic objects. To ensure technological invariance, the principle of synergetic control is proposed. At the same time, an extended dynamic model of the system was formulated, including external and internal perturbations. To determine the trajectory of movement of variable coordinates, a functional relationship of the AKAR method was compiled. The application of the method of analytical design of aggregated controllers makes it possible to eliminate the need for the synthesis of the observer of states, due to the presence in it of the possibility of prompt evaluation of external and internal disturbances. The main idea of the new approach is to use a single high-order perturbation model, consisting of series-connected integrators, instead of separate models for each perturbation separately. The proposed technique is confirmed by examples of digital modeling and the effectiveness of the proposed approach to the problems of synthesis of a nonlinear control system for dynamic objects is shown, which ensures the stability of the control system and compensates for unmeasured and external disturbances. Another approach to the study of nonlinear control systems for dynamic systems is the use of the synergetic method of control theory [4, 5], which is widely used in various branches of modern production. The peculiarities of the synergetic control theory in the problems of synthesis of nonlinear control laws for complex dynamic objects lies in the fact that when forming a new mechanism of the control law.Applications the synergetic control theory makes it possible to ensure the robustness of a nonlinear system to external and parametric disturbances and makes it adaptable to systems through the use of nonlinear integrators that compensate for disturbances.

Keywords: nonlinear control, robust control, synergetic control theory, invariance, integral adaptation, sliding control.

Annotatsiya. Maqolada chiziqli bo'lmagan dinamik ob'ektlar uchun nazorat qonunini sintez qilish uchun sinergeti'k yondashuv usullaridan foydalanish imkoniyatlari kelti'rilgan. Nazorat qonuniga mustahkam xususiyatlarni berish uchun tashqi va parametrik buzilishlar ta'sirini qoplashga imkon beradigan integral moslashuv printsipini qo'llash taklif etiladi. Sinergeti'k boshqaruv qonunini amalga oshirish chiziqli bo'lmagan dinamik ob'ektlarni boshqarish tizimining asimptoti'k barqarorligini ta'minlaydigan yig'ma kontrollerlarning analiti'k dizaynini qurishda amalga oshiriladi. Texnologik invariantlikni ta'minlash uchun sinergeti'k nazorat printsipi taklif etiladi. Shu bilan birga, ti'zimning kengaytirilgan dinamik modeli, shu jumladan tashqi va ichki buzilishlar shakllantirildi. O'zgaruvchan koordinatalar harakati'ning traektoriyasini aniqlash uchun AKAR usulining funktsional aloqasi tuzildi. Yig'ilgan kontrollerlarni analiti'k loyihalash usulini qo'llash tashqi va ichki buzilishlarni tezkor baholash imkoniyati' mavjudligi sababli davlatlar kuzatuvchisini sintez qilish zaruratini bartaraf eti'shga imkon beradi. Yangi yondashuvning asosiy g'oyasi har bir bezovtalanish uchun alohida modellar o'rniga ketma-ket ulangan integratorlardan iborat bitta yuqori tarti'bli bezovtalanish modelidan foydalanishdir. Taklif etilayotgan texnika raqamli modellashtirish misollari bilan tasdiqlangan va dinamik ob'ektlar uchun chiziqli bo'lmagan boshqaruv ti'zimini sintez qilish muammolariga taklif qilingan yondashuvning samaradorligi ko'rsati'lgan, bu boshqaruv tizimining barqarorligini ta'minlaydi va o'lchovsiz va tashqi buzilishlarni qoplaydi. Dinamik ti'zimlar uchun chiziqli bo'lmagan boshqaruv ti'zimlarini o'rganishga yana bir yondashuv zamonaviy ishlab chiqarishning turli sohalarida keng qo'llaniladigan boshqaruv nazariyasining sinergeti'k usulidan [4, 5] foydalanishdir. Murakkab dinamik ob'ektlar uchun chiziqli bo'lmagan boshqaruv qonunlarini sintez qilish muammolarida sinergeti'k boshqaruv nazariyasining

o'ziga xos xususiyatlari shundan iboratki, boshqaruv qonunining yangi mexanizmini shakllanti'rishda.Novalar sinergetik boshqaruv nazariyasi chiziqli bo'lmagan tizimning tashqi va parametrik buzilishlarga chidamliligini ta'minlashga imkon beradi va buzilishlarni qoplaydigan chiziqli bo'lmagan integratorlardan foydalanish orqali uni tizimlarga moslashtiradi.

Kalit so'zlar: chiziqli bo'lmagan boshqaruv, mustahkam boshqaruv, sinergetik boshqaruv nazariyasi, invariantlik, integral moslashuv, toymasin boshqaruv.

Introduction

The basis of nonlinear control systems for dynamic objects is structural and parametric uncertainty. At present, methods for overcoming various types of uncertainties are widely used, which include adaptive and robust control methods that provide adaptability through the hybrid application of classical adaptive and nonlinear control methods using methods of intelligent control theory.

Another approach to the study of nonlinear control systems for dynamic systems is the use of the synergetic method of control theory [4, 5], which is widely used in various branches of modern production. The peculiarities of the synergetic control theory in the problems of synthesis of nonlinear control laws for complex dynamic objects lies in the fact that when forming a new mechanism of the control law.

Applications the synergetic control theory makes it possible to ensure the robustness of a nonlinear system to external and parametric disturbances and makes it adaptable to systems through the use of nonlinear integrators that compensate for disturbances.

It should be noted that the synergistic control system based on the principle of adaptation does not require the use of state and disturbance observers to evaluate these disturbances.

Formulation of the problem

To compensate for various external and internal disturbances, which are the worst disturbances affecting the dynamic properties of the controlled system, it is most convenient to apply the adaptation principle with an integrator, which consists in the construction of guaranteeing controllers.

In this case, as the worst perturbations, we take perturbations of the form

Msup (t) = M. _ sign u(t) = const

' 1 u , having a random character by changing the value iU on

a given interval with a function sign ^^)- Compensation for the measured disturbances is provided by a "guaranteed controller" with an integrator that implements the control law, with astatism ensuring the stability of the control system of complex dynamic objects. The application of the principle of adaptation with the integrator synergistic control theory allows you to compensate for the perturbations of the impacts, by introducing the control loop of the integrators.

It should be noted that the analytical design of aggregated controllers (ACAR) does not completely compensate for the harmonic perturbation, but significantly weakens the influence of this perturbation with three integrators.

Let the dynamic perturbation model be represented as:

zi(t )=z2; z2(t )=z

(t )= zn+r Zn+1(t )=°i(x )'

Z. 0.( X )

where are the dynamic variables of the disturbance model, iV ' -a function of the state

variables of a dynamic source object that reflects the desired invariance.

In the work, as a proposed approach, it is proposed to build not a separate model for each of the

disturbances, but to build one model for the disturbance with the maximum degree. The essence of

the procedure for synthesizing nonlinear control laws lies in the following actuator (electric drive). Let

the dynamics of a heat and power plant be described in the following form:

x1(t):

•Xi \t ' — ;

k x - M - J (t )x

X (t)— m 3_c_2

vtj — j (t)

u - k x - Rx

XX (t)—-^-

3 r (1)

x x x

where 1 -electric drive shaft rotation angle, 2 - rotation speed of the electric drive shaft, 3 -

current in the armature circuit of the electric drive, u - control, J (t) - reduced moment of inertia of

M k k r R

the electric drive, c - moment of resistance, m , e ,r,R - electric drive parameters.

Model (1) characterizes the rotation speed of the superheater actuator and is related to the rotation

angle of the electric drive shaft. Parametric uncertainty of the electric drive (1) arising due to changes

in the reduced moment of inertia J (t) in time and its derivative

j (t)

. The influence of external

M

disturbances is determined by the moment of resistance c . The main technological task of the object (1) is to maintain the desired rotation speed of the superheater electric drive, i.e.

x — x0 — const

2 2 in the presence of the specified disturbances. To solve the problem, the following

options are proposed, which are based on changes J (t) . In the first option, the moment of inertia

J — J + at

changes linearly described by an equation of the form 1 0 , and in the second in the form

J — b sin a t, M — const.

of a harmonic function 2 0 and the moment of resistance c

J(t) M

Depending on the selected torque change option V ' and c based on the AKAR method, synergetic regulators of various types were synthesized, providing compensation for the specified

M * 0

external and parametric disturbances c [ ].

The use of this principle allows each to represent disturbances in its own separate dynamic model.

J — J + at + b sin a t M — const * 0

To compensate for disturbances, 1 0 0 and c the approach of

using the principle of integrated adaptation is considered - integrated high-order adaptation with a unified disturbance model.

Accordingly, using the AKAR method, an extended model for synthesizing the control law will be

formed. In model (1) the moment of inertia J (t) determines only the transient mode of the electric drive and does not affect the stationary state, then in the extended model we represent it with a

J (t)— J0 J (t)— 0

nominal 0 constant value and, accordingly, V/ 'we then have

M ;-(t) 0

Disturbance c and the actual impact of the disturbance w * let's imagine a dynamic model in the form of continuously connected three integrators:

z1(t )=z2; z2(t )=z3;

z3(t Mk- )

2 2' (2)

Zj = M -

where c output state variable of the system (2) which is a dynamic estimate of disturbances

for the system (1), @ - constant coefficient. Combining (1) and (2), we obtain an extended system:

k X - Z

XX

X2 (t )=-J-L

o

u - k x - Rx k (t)= e 2 3-

3W L

Z1(t )= z2; z2 (t)=z3;

k3(t j=4x2 - x2o )

(3)

u = u(x, z ),,

Therefore, model (3) is used to synthesize the synergetic control law V ' /'which ensures

x = X0 = const

the implementation of the technological invariant for the control system 2 2

J = J + at + b sin < t M = const ^ U

U U u c

on the object (1). According to the ACAR method, you can introduce a macro variable of the following type

compensating for unmeasured disturbances 1 U U u c acting

^ = x - x o + y z + y z + y z ,

2 2 1 1 2 2 3 3 (4)

y.

where 1 - constant coefficients.

Substituting v (4) into the main functional equation of the AKAR method

^(t)+^xF(t)+A v(t) = U

1 2 (5)

and solving it together with (3) and (4), we find the control law:

LJ

u(x, z) = k x + Rx +

e 2 3 k

m " 0

J (l - J k - (yi + V2 K - ^2 + ^3 lx2 - x2 ^

—— (y B + A)\k x - z )-Â2 (x - xo + y z + y z +y + z J 1>\ m 3 1' 2 ^ 2 2 '1 1 ' 2 2 '3 3'

o

(6)

A, >U, j = 1,2. A,

The condition for asymptotic stability (6) is the condition J Selecting values J

u V (t )= u

provided by the parameter of the transition process to manifolds ' and V /

^ = 0 ^FiV ) = 0

At the intersection of varieties ' and V / the dynamics of system (3) is described by

the following linear decomposed system:

z(t )=z2;

Z2 (t ) = Z3;

z3(t )=^(-^1z1-^2 z2 —y3 z3 )

(7)

To find the unknown coefficients that ensure the stability of the system (7), we use the modal control method [15]. We write down the state matrix of system (7) and find its characteristic equation: A(p)= det( »E —A) = »3 + Py p2 + Py p + Py = 0.

3 2 1

The desired characteristic equation with a given location of roots can be represented in the following form:

A (p) = \p — p 3 = »3 — 3p »2 + 3»2p — P3 = 0, y- jt 0/ ^o 0

p < 0 — p here 0 desired root. Equating the coefficients of these equations at the same powers we

find

y =— p3 / P,y = 3 p 2 / P,y =—3 p / p.

1 0 2 0 3 0 (8)

1> 0, j = 1,2 y. Thus, the choice of coefficients j Thus, the choice of coefficients 1 according to

(8) will ensure the asymptotic stability of system (1) with the synthesized control law (6).

The example of Figures 1, 2 shows the results of modeling an object (1) with a synthesized control

M = M = const

law (6) under unmeasured disturbances c c 0 and

{J + at + b sin <y t, t > 4;

0 J , 0 t ^ 4; <9>

0

(a + b& cosCT t, t > 4;

0, 0 0 t £ 4. <10>

k - 1- k - I" J - 2

L — 01" R —10" n_

Object parameters: m e ? ■> •> disturbance parameters: 0

M -1; n 0 , ai ^ —10; x0 -3; a inr.

c0 a - 0,2; b - 0,1; 0 control law parameters: 2 100;

p --200; A -100; A - 2500.

0 1 2

As can be seen from the modeling results, the control law (6) ensures the fulfillment of the

x2 - x0 - COnst s(t)< 10 - 4.

technological invariant 2 2 with some error 1

The technological invariant is determined by the assumptions made when forming its disturbance

model (2).

0 4 S 12 16 Fig.1. Graph of error changes

0 4 8 12 16 20 Fig. 2. Management change schedule

s(t ) = x2 (t )- x°2

In theory, this error can be further reduced by increasing the number of integrators used when forming the model (2).

It is possible to reduce this error to zero by using disturbance observers [8-10], but in this case the structure of the control system will become significantly more complicated due to the complication of the structure of the estimation subsystem, since each disturbance is represented by its own distinctive dynamic model. For practical purposes, an error in stabilizing the rotation speed of the superheater

;(t )< 10 - 4

electric drive actuator

is acceptable and, accordingly, the results obtained indicate the

effectiveness of the proposed approach.

In [6] it is noted that if the frequency of the harmonic influence is limited and the relation is

can be represented by a linear

J (t )= J + at + b< t,

v ' n U

satisfied approximation

2x62 >><0, J(t)

0 then the moment of inertia V >

(11)

and its derivative

J (t ) = a + ba0 = const.

0 (12) Figures 3, 4 show the results of modeling an object (1) with a synthesized control law (6) under

M = M = const

unmeasured disturbances c cU , (9) and (10), modified according to (11), (12). In this

x = xU = const

case, the control law (6) ensures error-free execution of the technological invariant 2 2

m

0 4 8 12 16

Fig.3. Error change graph

40

20

0

.mJt

---- 1 _ _ L „ l ..1___ 1

---- 1 _____L. 1 1 _ - L _ I __L___ 1

I _____]_. 1 I _ _L _ 1 _ _ L___ I

1 ____J__ 1 i 1 1 _ _ L___ ! -1-L .

0 4 8 12 16

Fig.4. Management change schedule

s(t) = x (t) — X0

Let the dynamics of an object be described by a system of equations:

X1(t) - X2; X (t) - x ;

2 3

X (t)- -ex - bx - ax + mx2,

3

1

2

3

1

X (t ) = |x1 (t ), x2 (t ), x3 (t )JT —

(13)

here "' 1 ' 2W' 3 WJ state vector, a'b'm positive constants, while ab ^ c From (13) we obtain a perturbed system with control:

X1(t) - X2; X (t) - X ;

2 3

x (t)=—cx — bx — ax + mx2 + Af + d(t)+ U,

3

1

2

3

1

Af

), d (t )— u

(14)

nmeasurable external disturbance,

U —

here object parametric uncertainty (13),

control.

In the general case, disturbances of system (14) are limited:

|A/| |d (t)|

$ positive constants. The task of system control (14) is to ensure a zero value of the error

here vector

lim ||E(t)= lim X(t)—X (t

t ^ œ

t ^ œ

= 0,

desired state vector. Let's consider the construction of a

X d (t )=[xd (t )' xd (t )' xd (t )

h e re d d d d

synergetic control law U U ( X 'Z ^ for object (14), which provides compensation for parametric

x (t) = 0. and external disturbances at d

According to the AKAR method, we will form an extended model for synthesizing the control law. Let us introduce a dynamic disturbance model

zi(t )=z2; z2(t )=z3;

z, (t )= x + y x + y x ,

3 w 3 12 2 1

(15)

yy -

here 1 2 constant coefficients. Thus, combining (14) and (15), we obtain the extended system:

xi(t ):

x^ \ t J - x2 ;

x (t )- x ;

2 3

x (t)--cx - bx - ax + mx2,

3 1 2 3 1

z (t ) - z„

z1(t )-z2; z2(t )- z3;

z~ (t )- x + 'y x + y x .

3 3 1 2 2 1

(16)

x + yx + yx + B z +Bz +Bz ,

3 1 2 2 1 1 1 2 2 3 3

P.

here 1 - constant coefficients.

Using ^ (17) into the main functional equation of the AKAR method

T(t )+¥- 0 T (18) and solving it together with (17) and (16), we find the control law:

(17)

U(X, z) = -y x + y x + cx + bx + ax - mx2 - z - B z - B z -

2213 1 2 3 11 12 23

- B (x + y x + y x )+ — (x + y x + y x + B z + B z + B z )

3V3 12 2 1' TX 3 1 2 2 1 1 1 2 2 3 3'

(19)

Solution(18) ^ = 0 is asymptotically stable for T ^ 0-.

On diversity ^ = 0 the dynamics of system (16) is described by the following linear decomposed system:

xi(t) = x2;

x2(t)= -yx2 - y2 - x - z1 - B2z2 - B3z3;

x (t

3 1 2 3 1

-cx - - bx -

1 2

z1(t >- = z2;

z2 (t )■■ = z3;

M- "B2 z2

~3V / \ "22 "3 3 '

(20)

To find the unknown coefficients that ensure the stability of the system (20), we use the modal control method. We write down the state matrix of the system (20) and find its characteristic equation:

A(p)= det(pE — A)= p + (p +y )p + (p + yp y )p + (p + yp + y p )^2 +

31 2132 11223

+ (yP +y P )p + y p = 0.

12 2 3 2 1

Let us represent the desired characteristic equation with a given location of roots in the form

A (p)- p - p 5 - p5 + 5p p4 +10p2p3 +10p3p2 + 5p4p + p5 - 0,

0 ^ 0; 0 0 0 0 0

—

p0 < 0 here 0 desired root.

F

The coefficients of these equations at the same powers r , are equated, and one can find

_ „2 r _ R _ a „2

y = 2p ,y = p2,P = p3,p = 3p2 ,P = 3p 1 0 2 1 0 2 3 0

0 0

(21)

r., P. -

So the choice T ^ 0 and coefficients ' j according to (21) will ensure the asymptotic stability of system (16) with the synthesized control law (19).

Figures 6-9 show the results of modeling an object (14) with a synthesized control law (19) under unmeasured disturbances according to:

A/- 0,5sin(^x )sin(2^x )sin(3^x );d(t) - 0,2cos(t).

(22)

1

2'

3'

Object parameters:

p =—4.

a = 1,2; b = 2,92; c = 6; m = 1;

0

When modeling up to the moment

t = 5

1 tj(l) ] 1 1 r

1 1 1

1/ 1 _J£

0 10 20 30

Fig.6. Change schedule

control law parameters:

U = 0

T = 1;

c the object is uncontrollable, i.e. %

l.c

20

30

x (t)

Fig.7. Change schedule 2

20

0

-20

t,c

20

Wi)

Fig.8. Change schedule 3V '

21)

Ulli

30 -20

1

1 1 ~ 1 r^ ■ -

/

u -;-j-1-1-

20

30

Fig.9. Control change schedule To solve a similar problem of object control (14) with the same disturbances, adaptive control with a sliding mode is proposed. Omitting the synthesis procedure, we present the final expression for the control law:

U = -c e - c e + x (t)-

2 3 12 dW (23)

e = x (t)-x (t),e = x (t)-X (t),e = x (t)-X (t)-

here 12 d 22 d 33 d

0 = C

X + l X + a X +m X 2

1 2 3 1

+ cc + ß, s = e + ce + ce -

3 2 2 1 1

error vector components; sliding surface;

0 = (1 - exp(-As))(1 + exp(-As))-1; c , c , p

1 2 positive constants. The control law (23) is supplemented with a subsystem for dynamic parameter estimation:

A(t) = -y0 exp(As), As(0) = A ;

C(t) = |s| l(t) = |s| a(t) = |s

m(t) = Isl

, C(0) = Co;

x

2

X

3

, l(0) = b ;

o

, a(0) = a ;

o

x

1

m(o) = m

C(t) = ß(t) = |s|, c(0) = c ß(0) =ß

(24)

A , c , b , a , m ,a , f5 -Here 0 0 0 0 0 0 0 positive and limited initial values of the parameters, - a positive constant. The parameters of the control law are given in formulas (23), (24):

c = 10; c = 6;u = 1,1;r = 1; = 0,5; С = 9; b = 7; a = 5; w = 3;a = 1;/? = 11.

1 2 0000000

The results of modeling an object (14) with an adaptive control law (23), (24) under the same unmeasured disturbances (22) and the same initial conditions for the system state variables (14) have a similar qualitative nature and dynamic characteristics. However, the following advantages of the synergetic control law (19) should be noted: - the control law (19) is structurally simpler, because contains only one nonlinear component; - control law (19) contains three dynamic components -system (15), and control law (23) contains seven dynamic components - system (24); - control law (23) with large deviations from the equilibrium state leads to a high-frequency change in the control amplitude with polarity switching, and control law (19) does not lead to such a negative effect.

Conclusion

The paper presents the development of the principle of integral adaptation of synergetic control theory for the synthesis of synergetic laws of robust control of nonlinear dynamic systems. The main idea of the new approach is to use one high-order disturbance model, consisting of sequentially connected integrators (at least three), instead of separate models for each disturbance separately. Examples of synthesis are presented that demonstrate the effectiveness of the proposed approach -the fulfillment of technological tasks and the asymptotic stability of a closed-loop system are ensured, and unmeasured parametric and external disturbances are compensated with an acceptable error.

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Siddikov Isamiddin Xakimovich, Umurzakova Dilnoza Maxamadjonovna and Bakhrieva Hurshida Askarxodjaevna Adaptive system offuzzy-logical regulation by temperature mode of a drum boiler // IIUM Engineering Journal, Vol.21, №.l, 2020.-P. 182-192. (№6; Scopus; IF:0.281).

Sidikov, Kh. Bakhrieva. Synthesis of the synergeti'c law of control of nonlinear dynamic objects // Technical science and innovation Journal, №1/2023, P.111-121. 86000609