CC BY
2Muhammad 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 FORMATION OF CONTROL EFFECTS IN CONDITIONS OF
UNOBSERVABLE DISTURBANCES
Mamirov Uktam Farkhodovich,
doctor of technical sciences, Associate Professor of the department "Information Processing Systems and Control" at Tashkent state technical university named after Islam Karimov,
uktammamirov@gmail .com
Buronov Bunyod Mamurjon ugli
senior lecturer of the department "Information Processing Systems and Control" at Tashkent state technical university named
after Islam Karimov, buronovbunyod@gmail .com
Abstract: this article develops an algorithm for synthesizing a complex-shaped command-tracking system based on modal considerations by compensating for interference signals when there is an effect of unknown unobservable disturbances on the control object. In this case, the control signal is selected in such a way that the output of the object must accurately and inertibly monitor each command signal. By predicting the future character of the object when the state of the system can be measured directly, we can ensure the observation of command signals using linear feedback to the state.Using Cauchy's formula, the matrix obtained by zeroing the co-head due to the galaions in the equation representing the exact output of the object is an poor-conditioned matrix. A modified Greville's constructive algorithm was used to determine these pseudo-inverse matrices. The structure of the resulting tracking system is built. Its main elements are the identifier of the command signal, the identifier of the interrupts and the state of the object. If the matrices in the identifiers are selected correctly, the system provides high-precision tracking of command signals even in the presence of any interference.
Keywords: non-measurable disturbances, indirect measurement, object condition estimation, combined control systems, and disturbance compensation.
I. Introduction
On the basis of modal considerations, we will consider the synthesis of systems that track commands of complex form in the presence of unknown and unmeasurable disturbances. We consider the unknown disturbance signal and the command signal as the output of some identifiable dynamical system with a known structure and an arbitrary initial state. Then we see a monitoring system for the ideal state, we assume that the ideal state is as follows: we can accurately measure the states of a fictitious (imaginary) dynamic system whose outputs coincide with the interference and command signals at any time, and then we construct the state identity of these dynamic systems, in which the ideal in state formulas, we replace the state with its value, these values are obtained at the output
of the identifier, thereby forming a physically realized construction of the monitoring system.
Let some controlled and identifiable object be represented by the following equations: {x(t) = Ax (t) + Bu(t) + tw(t),
1 y(t ) = Cx(t)' (1)
Where x(t)e Rn - state vector, u(t)e -vector of input effects, y(t) e R - vector of outputs,
w(t) e R - vector of external disturbances, A , B , T, C - invariant matrices. If the outputs of objects are linearly independent, thenrankC = p . The problem of
controlling the object is as follows: the input u(t) should be selected in such a way that the output of the object is equal to the previously given command
123
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
signals of size y(t) is equal to the previously given p
- sized command signals y*(t). In addition, it should be possible to provide monitoring of command signals when the object is affected by unknown external
interference w(t).
Suppose that the command signal (t) is the output of an imaginary dynamic system \r(t )= Rr(t),
1 (t )= Gr(t), (2)
it is possible to measure directly where y*(t).
R , G matrices are known, v - dimensional vector r(t) refers to the state of the command process, which can change at any instant of time when the initial conditions
change. The external disturbance w(t) appears at the output of the imaginary dynamic system jz(t ) = Dz (t),
W ) = Hz(t), (3)
where w(t) is assumed to be impossible to measure directly, D , H matrices are given, the state
of the p - dimensionalz(t) vector disturbance process, the initial conditions in (3) are unknown and may change at arbitrary time instants. Here, the functions
y*(t) and w(t) can be continuous due to changes in the initial conditions. (2) and (3) dynamic processes can be used to model many types of realistic interference and command signals.
Control u(t) should be selected in such a way
that the output y(t) of the object follows each
command signal y*(t) precisely and without inertia.
y*(t), (2) can occur when there is any interference at the output of the system (3) that the system produces,
w(t) cannot be measured. In addition, control u(t) must be physically implemented in the form of
feedback on the output, i.e. u(t) = (y(t)'(t)).
We assume that the state of systems (1), (2), (3) can be directly measured, and we have all the information necessary to predict the future behavior of the object, and thereby define arbitrary good dynamics for tracking command signals using linear feedback on
the state. We can ride. In this case, the control that solves the problem consists of two joiners
u(t) = un (t)+ uK (t) , (4)
Where u°(t) interrupt compensates for the
effect of w(t) on the object, and is a command signal
when there is no interference uK (t) and in the absence
of interference, the command signal y*(t) is observed.
II. Methods of evaluation of non-measurable disturbances
We insert object control (4) into equation (1)
and using the Cauchy formula put w(t) = Hz(t) in place
and exit y(t) the exact expression for can be written as [1 ].
t t y(t; x, t„, z(t)) = CeA{t-t') x + C J eA(,-,)Bu (r )dr + C J eA(t-,) [Bun (r) + THz (r )]dr
. (5)
So that the output
y(t) does not depend on the
interference w(t) the last addend in (5) must be zero. In order to completely eliminate the effect of
the disturbance w(t) on the output
y(t) of object (1),
we find a constant matrix A that satisfies the following condition
CeA(t-T\BK + TH] = 0 all t0 < r, t < « . (6)
In this case, the display of control will be un (t ) = Az (t)
. From (6), it seems to correspond to: C[B, AB, A2B,..., A-1 B] = 0, b = BA + fh , (7)
which in turn gives rise to the following system of equations:
CA [BA + TH] = 0 when 5 = 0,1,2,...,n -1.
If we consider the following matrix as well, its degree is equal to n because the pair of identifying
matrices is
(A, C}
condition (7) can be written as follows ™nk [W , B, TH] = rank \wT, b] .
we use the following formula to calculate the matrix A
a = -(wtb )+ wt th + [i - (wt b)+ wt bJq
124
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
here Qa - arbitrary parametric matrix, 1 -unity matrix,W - n dimensional matrix
W = [ct, ATCT......]
In the case when the matrix F is not a full-rank matrix, then the problem under consideration is ill-posed. To give numerical stability to the procedure of pseudo-inversion of the F matrix, it is advisable to use the concepts of regular methods [2, 3, 5].
Let's consider some of the most constructive algorithms for determining pseudoinverse matrices [6, 7-14, 15].
Let us define a matrix of size n x m,
JJ _ / r r r \
m ^ ^ 2v?JmJ the columns of which are the
f. - = 17
vectors J, 1 m . Using the obvious notation ,
this matrix can be represented as
Fm = (Fm-\ ' fm ), m = 2,3v.
Pseudo-inversions of the matrixD\, are obviously carried out according to the formula
F+ = fT / f\Tf\.
To sequentially find the pseudo-inverse matrix D we will use the Greville method [16, 17]. From this we get
F + [i - f kT "h
1 m\? J m+\ km+\J
F + =
m+1
where
km+1
kT
(I - F F + ) f
\ m m J J m+1
||(I FmFm ) f m
( F + )T F + f
V m / m J m+1
if (I FmFm )fm+1 '
in other cases
\ + F + J
|| m J m+l ||
Accurate compensation of disturbances using
control Un (t) = ^z(t) is suitable even for continuous
function when the state of the interrupting z (t ) process is known. This condition is appropriate only when it is possible to accurately measure the interference signal.
Now we will pay attention to the problem of assessing the state of systems (1), (2), (3). We believe that it is possible to actually measure only two
quantities, the output of the object y(t ) and the
command signal. When developing (constructing) the structure of the initial (zero) state adjuster,
x(t), z(t),
r (t) the state vectors are obtained from the output of the identifiers, the ideal and only the measured
quantities y(t), y (t) found using information about
the optimal values , x(t), z(t), r(t) replacing (in the ideal case) the dynamic system as desired selection allows to build a monitoring system that can be put into practice For simplicity, we use state-estimating n -dimensional identifiers (Kalman filters). We write down the equations of the identifiers from which the
estimate of the state of the object x(t) and the value of
the state of interference is z(t) as follows:
^B
, (8)
Where y(t), u(t) - are the actual output of the object and the actual input to (1), respectively. We
assumeL, L matrices are chosen so that the value
[x(t), z(t)] and grade [x(t)> z(t)] let the difference between asymptotically approach zero: k (t), -z (tff =[x(t), z(t)]T -[x(t), z(t)]T ^ 0
t ^ro
x(t ) " A + LC tH ' 'x(t Ï " L1
_ f (t ). LC D _ f(t )_ _ L2
^(t ) +
it )
if
IILCondition assessment error.
The state estimation error equation has the
14 (t )!
ß. (t )_
A + LC . LC
tH D
ß (t ) ß (t ).
are
(9)
Since the pairs of matrices
(A, C}
and
(D, H}
identifiers, it is possible to choose the
corresponding matrices L\, L in such a way that the desired dynamics can be achieved, ensuring that the error of the identification state tends to zero. The following formulas can be used to construct the
physically constructible state identifier r(t) from the dimensions yK (t ) :
r(t )=[* + NG ]r (t )-Ny K (t )
(10)
125
2 '
<
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
where, R , G (2) is given in, N - is the matrix, which provides the rate of error tending to zero, which is chosen by the designer. The estimation error
sr(t ) = r (t)- r(t) satisfies the following equation: ^r (t) = [R + NG]er (t). (ii)
t ^^ da £r (t) ^ 0 the matrix can always be chosen since the pair is identifiable if the dynamic of the pursuit is given.
Let's copy and write the equation of the observing system that can be physically implemented.
To do this, we replace the un(t ) = Az(t),
uK(t) = x(t) + K2r(t) in the (z(t), x(t), r(t)} formulas
with their values (z(t), x(t), r(t)} and create a physically implemented control with the following appearance:
u (t) = un (t)+uK (t) = Az(t)+ KX(t)+K2r(t)
(12)
It can be shown that this control is actually a control that allows monitoring the signal appearing at the output of (2). In this case, we consider the dynamic
behavior Sy (t) yK (t) y(t) of the actual error of observation in the control (12) in the presence of arbitrary external disturbances produced by the system (3). For this, we put (12) into (1) and from the following expressions: x(t )= x(' )-ex (t)
z(t )= z(t)-Sz(t)
r(t ) = r (t )-er (t )
using we get the following: X (t) = [A + BK ]x(t) + [BA + TH ]z(t) + BK2r (t) +
. (13)
When a real object is affected by perturbations
w(t) and it is controlled in the form of feedback (12), the movement of the object is described by equation (13). Using the above, it is possible to consider the
change of state variable ^(t) given by expression (10).
Using (13), the equation for ^(t) will have the following form:
g{t) = [A + BK i ]g(t) - Fr (t) - Bz(t) + b[Kxsx (t) -. (14)
If the, Kl, K, N matrices in the identifiers are chosen correctly, the errors, x (t), ^ (t), ^ (t)
tend to
zero, and the variable ^(t) (2) jumps of the system between the initial conditions can be determined from the following formula:
ç(t ) = [A + BK i t(t )-Kr (t )-Bz(t ),
this is exactly consistent with equation (13).
IV. Conclusion
Therefore, the error asymptotically approaches zero, which means that the system provides high-precision tracking of command signals even in the presence of any disturbances. The structure of the resulting tracking system is shown in Fig. 1.
Its main elements are the identifier of the command signal, the identifier of the interrupts and the state of the object. The developed algorithms make it possible to synthesize a system that monitors complex commands based on modal considerations by compensating interference signals when there is an BlA^enF- Kfs (u)jobservable noises on the
control object.
References:
1. Glad, T., & Ljung, L. (2000). Control Theory (1st ed.). CRC Press. https://doi.org/10.1201/9781315274737.
2. Strejc V. State Space Theory of Discrete Linear Control. IEEE Transactions on Systems, Man, and Cybernetics. Publisher: IEEE, 1982.
3. Simagina. O.V. Control theory: textbook, KNovÍ6S■^Af^c('SlbAGS publishing house, 2014.-135 p.
4. Afanasyev V.N. "Control of indefinite dynamic objects", Moscow, Fizmat-ref, 2008. - 208 p.
126
Muhammad al-Xorazmiy nomidagi TATU Farg'ona filiali "Al-Farg'oniy avlodlari" elektron ilmiy jurnali ISSN 2181-4252 Tom: 1 I Son: 4 I 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
5. Ogarkov M.A. Methods of statistical estimation of parameters of random processes. Moscow, Energoatomizdat, 1990. 208 p.
6. Filtering and stochastic control in dynamic systems. / Ed. K. T. Leondes Trans. from English, - M.: Mir, 1980. - 407 p.
7. Pelzverger S.B. Algorithmic support of assessment processes in dynamic systems under conditions of uncertainty. -M.: Nauka, 2004. - 116 p.
8. Mamirov U.F. Regular synthesis of adaptive control systems for uncertain dynamic objects. -Tashkent: Publishing house. "Knowledge and intellectual potential", 2021. -215 p.
9. Karabutov N.N. Structural identification of static objects. -Librocom. 2011 -152 p.
10. Mamirov U.F., Azamkhonov B.S. Algorithms for stable estimation of parameters and state of nonlinear control objects // Scientific-technical journal (STJ FerPI, FarPI ITZh, NTZh FerPI, 2020, T.24, special issue No. 1). -P.274-278.
11. Kisenkova N.A., Joint estimation of object parameters and statistical characteristics of non-Gaussian disturbances, Avtomat. and Telemekh., 1991, issue 11, 71-80.
12. Mamirov U.F., Buronov B.M. Systematic analysis of methods of control of dynamic objects in conditions of non-measurable disturbances // Chemical technology control and management. 2023, No. 4 (112) pp.49-63.
13. Igamberdiev Kh.Z., Kholkhodzhaev B.A., Mamirov U.F. Formation of stable algorithms for estimating unknown input signals in dynamic control systems // Journal of Technical Sciences and Innovation. Tashkent. 2019. No. 1. -WITH. 63-67.
14. F.R.Gantmaher, "Matrix theory", Moskva, Nauka, 1988, 552 p.
15. James W., Demmel, "Applied numerical linear algebra", Berkeley, California, University of California, 1997, 184p.
16. V.M.Verzhbitsky, "Computational linear algebra", Moscow, Higher School of Economics, 2009. 351 p.
17. A.I.Zhdanov, "Introduction to methods for solving ill-posed problems", Samara, Aerospace University, 2006, -87 p.
18. Ch.Lawson, R.Henson, "Numerical solution of problems in the method of least squares", 1986, 232 p.
19. Yusupbekov, N.R., Igamberdiev, H.Z., Mamirov, U.F.: Adaptive Control System with a Multilayer Neural Network under Parametric Uncertainty Condition. In: Russian Advances in Fuzzy Systems and Soft Computing: selected contributions to the 8-th International Conference on Fuzzy Systems, Soft Computing and Intelligent Technologies (FSSCIT-2020), Vol. 2782, pp. 228-234. CEUR Workshop Proceedings, Aachen, Germany. doi: http://ceur-ws.org/Vol-2782/paper_32.pdf.
127
