ALGORITHMS FOR SYNTHESIS OF ADAPTIVE CONTROL SYSTEMS WITH IMPLICIT REFERENCE MODELS BASED ON THE SPEED GRADIENT METHOD Текст научной статьи по специальности «Компьютерные и информационные науки»

Muhammad al-Xorazmiy nomidagi TATU Farg'ona filiali "Al-Farg'oniy avlodlari" elektron ilmiy jurnali ISSN 2181-4252 Tom: 1 | Son: 4 | 2023-yil

"Descendants of Al-Farghani" electronic scientific journal of Fergana branch of TATU named after Muhammad al-Khorazmi. ISSN 2181-4252 Vol: 1 | Iss: 4 | 2023 year

Электронный научный журнал "Потомки Аль-Фаргани" Ферганского филиала ТАТУ имени Мухаммада аль-Хоразми ISSN 2181-4252 Том: 1 | Выпуск: 4 | 2023 год

ALGORITHMS FOR SYNTHESIS OF ADAPTIVE CONTROL SYSTEMS WITH IMPLICIT REFERENCE MODELS BASED ON THE SPEED GRADIENT METHOD

Sevinov Jasur Usmonovich

Doctor of technical sciences, prof.

Tashkent State Technical University named after

Islom Karimov E-mail: sevinovj asur@gmail .com,

Boborayimov Okhunjon Khushmurod ogli,

PhD student of Tashkent State Technical University named

after Islom Karimov E-mail:[email protected].

Abstract: Algorithms for the synthesis of adaptive control are considered, in which the reference model acts as a certain reference equation. To obtain precise statements about the properties of a generalized custom object, some auxiliary conditions such as smoothness and regularity are introduced. The use of such systems has made it possible to reduce the requirements for the structure of the main circuit and for the completeness of the measured information, which is advisable to use in the presence of strong unmeasured disturbances, as well as in problems of controlling high-order multidimensional objects, when the implementation of a reference model in the system is impossible or difficult.

I Key words: adaptive control, speed gradient algorithm, control object, reference model, unmeasured disturbance, control action.

Introduction. The current stage of development of control theory and technology is characterized by increasing requirements for control systems, increasing complexity of controlled objects, and high rates of design and commissioning of systems. These circumstances lead, as a rule, to the fact that the available initial information is insufficient to build systems with high quality indicators and it is necessary to replenish the information during the operation of the system. Such systems, in which the information necessary to improve functioning is collected during operation, immediately processed and used for control, are called adaptive. Adaptive systems are currently finding increasing use for managing objects and processes under conditions of uncertainty. An adaptive system uses a control law with variable coefficients. The coefficients are changed by a special algorithm (adaptation algorithm) based on current information about the state of the process obtained during normal operation of the installation. The adaptation algorithm is constructed in such a way as to adapt to a specific situation and ensure the achievement

of the control goal for any possible value of the unknown parameters of the object [1-7].

The gradient principle is widely used to synthesize the adaptation algorithm. In this case, the evaluation function, the gradient of which determines the direction of change in the adjustable parameters, may coincide with the evaluation function that sets the control goal, or may differ from it. If the control goal is given to the designer of the adaptive system from the outside, then the adaptation goal is set by the designer himself when synthesizing the adaptation algorithm. The separation of control and adaptation goals, generally speaking, expands the designer's capabilities. However, this makes it more difficult to justify the system's performance, since achieving the adaptation goal may not directly result from achieving the original control goal. The above considerations explain the origin of speed gradient algorithms. However, our task is to establish the general properties of algorithms that allow us to judge the stability and quality of specific adaptive systems [3,8-14,21].

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Muhammad al-Xorazmiy nomidagi TATU Farg'ona filiali "Al-Farg'oniy avlodlari" elektron ilmiy jurnali ISSN 2181-4252 Tom: 1 | Son: 4 | 2023-yil

"Descendants of Al-Farghani" electronic scientific journal of Fergana branch of TATU named after Muhammad al-Khorazmi. ISSN 2181-4252 Vol: 1 | Iss: 4 | 2023 year

Электронный научный журнал "Потомки Аль-Фаргани" Ферганского филиала ТАТУ имени Мухаммада аль-Хоразми ISSN 2181-4252 Том: 1 | Выпуск: 4 | 2023 год

Formulation of the problem

Let the equation of a generalized custom object be given:

dx / dt = F (x, c, t,h)

x = {x (i) x(n)} where x {x x } - is the state vector of

the generalized customizable object (GCO);

c = (c (1) c (N)}

c {c ,•••,c } - vector of adjustable parameters;

h = f/(i) p(q)\

h \h ,---,h } - vector of unknown object parameters and external influences. Let us assume that the adaptation goal is specified using the evaluation

functional Qt, and approaching the goal corresponds to a decrease in the values. We will consider two cases

Qt [2,3,10]:

a) the estimated functional Qt is a non-negative function of the phase coordinates of GCO:

Qt = Q(x(t),t). We will call such a functional Qt local.

b) Qt is the integral functional:

R( x, c, t )

Qt =| R(x(s), c(s), s)ds

0 , where "v-'-'-v - is some

non-negative function..

In each of these cases, it is possible to calculate

function Qt - the rate of change of functional Qt by virtue of equation (1) for a fixed c. Obviously, in case a)

Qt = dQ(x(t), t)/ dt + F(x(t), c(t), t, h)T VxQ(x(t), t),

and in case b), Qt = R( x(t ^ c(t ^t) i.e. in both cases:

Qt = y(x(t), c(t), t),

where ^t) - is some function that we will assume is continuously differentiable with respect to the components of vector c.

Let's call the following adaptation algorithm the speed gradient algorithm:

dc / dt = - rVciy( x, c, t )

(2)

where r =r > 0 - is a positive definite N x N - is the matrix of gain factors.

It should be noted that the right-hand side in relation (2) may turn out to depend on unknown

parameters h or on phase coordinates of the GCO that are inaccessible to measurement, and then algorithm (2) will be unrealizable. The question of the feasibility of the algorithm and the class E of adaptability of the system (1), (2) must be solved separately in each specific problem.

To obtain precise statements about the properties of the system (1), (2), it is necessary to impose some auxiliary conditions such as smoothness and regularity on the right-hand sides of the system and

the functional Qt, excluding "pathological" cases [1,2,15,16]. We will assume that the right-hand sides of (1), (2) are locally bounded uniformly in t > 0, i.e.,

for any P > 0 the following inequality holds:

||F(z, t,h)|| + IVc¥(z, t,)|| <Kp<»

(3)

INI ^p, t > о

at \r\\~ , where z = {x, c} is the state

vector of the system (1), (2). Note that condition (3) does not exclude the possibility of discontinuities along

t in the right-hand sides of (1), (2). Function Q(^t) in the case of local functional Qt will be considered

x t

uniformly continuous in x, t in any domain of the form

{x'11 'Ixt > 0}. In addition, we will require sufficient smoothness of the functions

Q(x,t), F(x,c,th), R(x,c,t) so that all their derivatives arising in the formulations of the statements exist and are continuous in x, c .

Let us assume that the adaptation goal to be achieved by algorithm (2) is given by the relation: lim Qt = 0

(4)

(in the case of local functional Qt ) or the

relation:

lim R(x(t), c(t), t) = 0

(5)

(in the case of integral functional Qt ). We are interested in the conditions under which goals (4), (5) in systems (1), (2) are achieved for any initial values

c(0). The first of these conditions is the convexity

of the c function X'C't ), i.e., the fulfillment of the inequality [21]:

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"Descendants of Al-Farghani" electronic scientific journal of Fergana branch of TATU named after Muhammad al-Khorazmi. ISSN 2181-4252 Vol: 1 | Iss: 4 | 2023 year

Электронный научный журнал "Потомки Аль-Фаргани" Ферганского филиала ТАТУ имени Мухаммада аль-Хоразми ISSN 2181-4252 Том: 1 | Выпуск: 4 | 2023 год

y/(x, C, t) - y/(x, C', t) > (c - C') Vcy(x, c', t) (6)

for any c, c , x, t. Condition (6) is satisfied, for example, in the frequently encountered case when the right side of the GCO equation (1) linearly depends on the adjustable parameters. The second main condition is the fundamental achievability of the goal, i.e. the

existence of an "ideal" vector c* (depending, perhaps,

on %), such that in system (1) at c = c* goal (4) or (5) is achieved. More precisely, by the condition of attainability we will understand the fulfillment of the inequality [21]:

y/(x, c*, t) <-ßQt + d(t),

œ

ß> 0, ¡(t) > 0, Jd(t)

(7)

< œ

where

Solution of the task

It can be shown that from relation (7) it follows that goals (4), (5) are achieved (in the case of goal (4)

with the additional requirement $ > 0 ).

Let the functional Qt be local and the convexity

condition (6), the reachability condition (7) for $ > 0, and the growth condition are satisfied [2,10]: inf Q(x, t) ^ œ

x

■ œ

at ""'' ' . (8)

Then all trajectories of system (1), (2) are bounded and satisfy (4) [21].

Let the control object be described by the equations:

dx / dt = Ax + bu, y = L x, u = cTy ^

where x e R", u e r1, y e R, - vector of measured outputs; c - vector of adjustable parameters. It is required to find an algorithm for adjusting

vector c(t) so as to ensure achievement of the goal: x(t) ^ 0, c(t) ^ const ^o)

Let us choose the evaluation function

Q = 0,5x'^x, where H = HT > 0, velocity gradient principle.

Qt = xTH(Ax + bcTy), VcQt = xTHby

and use the We have

Since the

value XHb should depend only on the measured

quantities, we arrive at condition ~ , and if it is satisfied, we write the speed gradient algorithm in the form:

dc / dt = - g уГу

(11)

where r = r > 0.

The only condition (8) that needs special verification is the solvability condition (7). It will be

satisfied if there is a vector c* such that x HA*x < 0,

where A* = A + bc*L . Therefore, to check (7) we need

conditions for the existence of a matrix H = H > 0

and a vector c* such that

HA + Al H < 0, Hb = Lg, A = A + bcl LT (12)

For the existence of a matrix H = H > 0 and a vector c* satisfying (12), it is necessary and sufficient

that the polynomial = g Wbe a Hurwitz polynomial of degree n -1 with positive coefficients,

where

W(Ä) = LT (ÄIn - A)-1 b

- is the transfer vector

function of the object, = A) - is the

common denominator of its components.

As is known [17, 18], systems in which the numerator of the transfer function is a Hurwitz polynomial are called minimum-phase. Minimumphase systems, the numerator of the transfer function of which has the maximum possible (equal to n-1) degree and positive coefficients, will be called strictly minimum-phase. The transfer function of a strictly minimum-phase system will also be called strictly minimum-phase. Taking into account the introduced ones, (12) can be formulated as follows: for the

existence of a matrix H = H > 0 and a vector c* satisfying (12), it is necessary and sufficient that the

function g Wbe strictly minimal-phase.

From (12) and (8) it follows that for a given vector, the adaptation goal (10) in the system (9), (11) is achieved for any object (9) such that function

gTW(Ä) •

is strictly minimal-phase. If adaptability class

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"Descendants of Al-Farghani" electronic scientific journal of Fergana branch of TATU named after Muhammad al-Khorazmi. ISSN 2181-4252 Vol: 1 | Iss: 4 | 2023 year

Электронный научный журнал "Потомки Аль-Фаргани" Ферганского филиала ТАТУ имени Мухаммада аль-Хоразми ISSN 2181-4252 Том: 1 | Выпуск: 4 | 2023 год

E is specified, then the synthesis of an adaptive stabilization system is reduced to finding a vector g that provides for any object from class E strict minimum

phase function g W(A).

Let's consider a common special case when the object is described by equation

A(P)l = B(P)u (13)

A(A) = An + an-1An-1 + ••• + a0, B(A) = bmAm +... + b0

and the output variable 1 along with its l -1 derivatives is available for observation, i.e.

y = [f,fj,...,f(l

. The controller will be described by

(-i

C (A) = £ ciAi equation u =C(p)l, where i=0 . The

r = vI

adaptation algorithm (at l ) is written by the

equations:

c. =-vG(p)n.n{i \ i = 0,1,•••, l -1

in which the coefficients of the polynomial

G(p correspond to the components of the vector g from (11).

The transfer vector function of the object has the form:

w (A) = ba&A'";*-1} A(A) ,

where

gTW(A) = G(A)B(A)/A(A), S(A) = A(A) Therefore, the strict minimum phase property of function

gTW (A)

means that the polynomials

G(A), B(A)

are

Hurwitz [1, 19], the signs of their coefficients coincide m +l = n. Consequently, by choosing a Hurwitz polynomial G(A) of degree n - m -1 with coefficients of the same sign as those of B^AA, we can guarantee

the achievement of the adaptation goal ^ 0 for any minimum-phase object of the form (13). To implement an adaptive controller, an n - m -1 derivative of the controlled object coordinate is required [13,20]. The results of modeling such systems for various special cases show that the adaptation process in them proceeds several times faster than the

transition process according to x(t), even in the case of an unstable object (13). In this case, the value

G (after adaptation becomes close to zero, which

allows us to interpret equation G(pll =0 as a reference characterizing the quality of the object's processes after adaptation is completed.

The described approach can be extended to

systems for monitoring the reference influence r (t). In this case, Qt =[x x0(t)] H[x x°] should be taken

evaluation

function,

as

where

an

x°it) - is the equilibrium state of the system with "ideal" regulator coefficients, calculated under the assumption that the setting influence is established at

the level of r (t).

For a special case of object (13), the adaptive controller will have the form:

u = C(p)q- Cl(p)r, dci /dt = -y'1S{t)v(l\ i = 0,...,n -m -\, dcu /dt = -/"S(t)r(i), i = 0,...,k, S(t) = G(p)n~G1(p)r(t),

(14)

where the degree of polynomial

С (A)

is equal

to n m 1; the power of C A is equal to the number

of k available derivatives of r(t); C(AA - Hurwitz

polynomial; Vi,Vi> 0.

Equation G(p)lM = Gi(p)r can be considered

as a reference and polynomials C(AA, C1(AA can be selected based on the desired quality of the system after adaptation. It can be shown, using (8), that

l(t) -Vm (t) ^ 0 if the setting influence r(t) is established more accurately, if

]\r (i\t )

dt <œ, i = 1„..,( +1

Conclusion

Thus, the use of such systems makes it possible to reduce the requirements for the structure of the main circuit and for the completeness of the measured information. It is advisable to use them in the presence of strong unmeasured disturbances, as well as in control problems of high-order multidimensional

175

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Muhammad al-Xorazmiy nomidagi TATU Farg'ona filiali "Al-Farg'oniy avlodlari" elektron ilmiy jurnali ISSN 2181-4252 Tom: 1 I Son: 4 | 2023-yil

"Descendants of Al-Farghani" electronic scientific journal of Fergana branch of TATU named after Muhammad al-Khorazmi. ISSN 2181-4252 Vol: 1 | Iss: 4 | 2023 year

Электронный научный журнал "Потомки Аль-Фаргани" Ферганского филиала ТАТУ имени Мухаммада аль-Хоразми ISSN 2181-4252 Том: 1 | Выпуск: 4 | 2023 год

objects, when implementation of a reference model in the system is impossible or difficult.

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