CC BY
28
At two-fold drying of raw cotton (pic. 5-8) with output On the basis of the received results it is possible to make
P=3,5 t/h and moisture of raw cotton below Wx/c=15,8%, with practical recommendations on choosing the drying conditions
output P=5 t/h and moisture of raw cotton below Wx/c=14%, with depending on initial moisture of raw cotton, output of the dryable
output P =7 t/h and moisture of raw cotton below Wx/c=12,1% it is material, which guarantee maximal keeping of natural properties
not recommended to put raw cotton to two-fold drying. of fiber and seeds.
References:
1. Boltabaev S. D., Parpiev A. P. Drying of raw cotton. - Tashkent: «Ukituvchi», 1980.
2. Khadzhinova M. A. Research of properties and structure of cotton fiber in the course of drying. - Tashkent: Fan, 1966.
3. Alfei T. Mechanical characteristics of polymers. - M.: Inostrannaya literatura (Foreign literature), 1952. P. 305.
4. Bu m. G. Dobb and m. z.Safain. The effect of thermal treatment on the cruscerized cotton. J. of the textile Institute. - V. N 7/8. 1976, P. 229-234.
5. Edith Honold, Frederic R., Ondrers and James N. Crand Heating, Cleaning and mechanical procossing effect and cotton, Partil; Fiber chages as measured by achali Centuge Test. Text Reas. j. 1963, jannary, - N1, P. 51-60.
6. Kucherova L. I. Assessment of influence of drying on structure and properties of the cotton fiber and produced yarn and fabric: Ph. D. thesis in Engineering Science - M., 1981.
7. The "Cotton gin and oil mill Press" 22.11. 86. P. 8-9.
Mirzabaev Akram Makhkamovich, International Solar Energy Institute, Senior research assistant, the Faculty of PVSolar Plants E-mail: [email protected] Makhkamov Temur Akramovich, Tecon Group, Leading engineer, Research and Development Department, E-mail: [email protected]
The investigation of invariance of the output of complex electric power system with application system's embedding approach
Abstract: In article is considered the problem of providing the invariance of the output of dynamic system to external disturbances. As the dynamic system is considered the model of electric power system (EPS), provided for the small oscillation conditions. If the necessary and sufficient conditions of invariance required on the base of system's embedding approach are provided, then invariance of the exploring system's output to external disturbance is also provided. Keywords: Matrix approach, system's embedding approach, steady-state stability, invariance.
Invariance is one of the most important properties of the dynamic system. The problem of invariance, according to [1, 12], is the problem of identification structures and parameters of controlling system where the impact of spontaneous changes of the external disturbances and the system's own parameters to dynamic performance of the controlling process could be in part or in whole compensated.
This problem was formulated for the first time by G. V. S.C Hi-panov [2, 49-66] and the extensive discussions about its application have been going on up to now [3, 43-49; 4,21-29; 5, 34-41; 6, 61-67 etc.].
It must be noted that the different type of invariance systems are existed. They differ both in terms of functional capability and design concept [7, 23].
Below is considered the application of system's embedding approach to study the invariance of the output of controlled complex EPS at small disturbances as stationary determined multidimensional dynamic system. This is due to the fact that the input-output range of complex EPS are subjected to non-unique changings due to the existence of zero devisors and noncommutative operators [8,25], put in other words due to the algebraic singularity of the exploring system which is typical only for the multidimensional systems [9, 177].
In the case of representation of the exploring system in the state space [9, 185; 10, 23]:
x = Ax + Bu + Sw, (1)
u = -Kx, (2)
y = Cx, (3)
where x, u, y, and w are vectors of state, control, output and disturbance of the system, respectively; A, B, C, and S are matrices with constant digital elements of the respective size; K is regulator matrix, with constant digital elements. For invariance of the output of controlled system being studied, the transfer matrix from disturbance w(p) to the system output y(p) with the model in state space shall identically equal zero:
Fy (p) = C(pI„ - Ay )-1S = 0, (4)
where Ay = A + BK is matrix of dynamics of the system with the controller. The main problem is to find the controller (synthesis), ensuring the fulfillment of the condition (4). However, as indicated in [10, 25], solving this problem poses certain difficulties, as (4) has the operation of matrix inversion, and, as a rule, it is polynomial.
Necessary and sufficient conditions, under which the equality (4) is just, is ensured, when fulfilled the conditions of the theorem [10, 28], where established that the system (1)-(3) for specified matrices A, B, C and S is invariance to disturbances in the sense of
the fulfillment of theorem (4), if and only if the following condition is fulfilled:
= 0, (5)
where n — matrix of maximum column rank, complying to condition:
_' (6)
(7)
C^A CRn= 0,
wherein the following identity is satisfied:
CRn B CRn ACRn = 0, and the system is closed by any controller from the set:
{K}rz=-(CnLB) CrnLACRx(cRxy + C"ni 11 B x + yCn1 ,(8) where % and j are matrices of the set size with arbitrary digital elements; CR is right zero divisor of the matrix C; CRn is left zero divisor of the matrix C n; matrices with the upper mark (~) are summed-up canonizers of the respective matrices; double and triple bars above matrices designate the repeated definition of the respective zero divisor of the maximum rank out of the combination of matrices standing under that bar.
Below is given algorithm of generation of maximum rank matrix n, which satisfies the condition (6), in a finite number of steps [10, 27]:
A =
Step 1. Testing of the condition:
CBLCACr = 0. (9)
If the condition is fulfilled, then assumes n = n0 = I(n _rankC (. Step 2. If the condition (9) is not fulfilled, then matrix n is determines from the formula:
CBLCACr = 0. (10)
If = 0, then the system is not invariance and algorithm should be stopped. In the contrary case the condition (6) should be tested on the assumption n =nl.
Step 3. Matrix n. at i>1 is defined as:
CRni_l B CRn~l1ACR , (11)
and then the fulfillment of condition (7) must be checked.
Step 4. Algorithm will stop at the k-th step on the first fulfillment of the condition (7). Matrix n of maximum rank has value nk.
Let's apply presented method of determination of invariance of the output of dynamic system by the example of the three-generator system without due regard to damper coefficient of the generator.
The problem of invariance is solved on the basis of the matrix canonization method.
Matrix of own dynamics of the model of EPS being studied is written as:
(12)
0 0 0 1 0 0" " 0 0 0 1 0 0
0 0 0 0 1 0 0 0 0 0 1 0
0 0 0 0 0 1 0 0 0 0 0 1
-©11 ©12 ©13 0 0 0 -57,2 27,53 33,42 0 0 0
©21 -©22 ©23 0 0 0 39,33 -92,08 50,53 0 0 0
©31 ©32 -©33 0 0 0 23,61 50,53 -95,15 0 0 0
C = [ 0 0 0 0 0] = [1 0 0 0 0 0], S =
0 0 0 1 0 0 10
00 01 00
(13)
0 0 0
dp m
dE, T ,
ql -1
—^ 0
dpl m dE T
0
dPl m0
dEl T0
0 0 0
-2,0837 0 0
0 0 0 0
-2,0687 0
0 0 0
0 0
0,9706
(14)
R
B
Computation shows that at the adopted operating parameters (on the base case), the system is unstable, what can be seen from the matrix spectrum (12) of own dynamics of EPS being studied: 0,0000 +11,9757«, -0,0000 + 9,7538«, 0,0000 + 2,4239«.
To verify the required conditions of invariance of the output of EPS and as the result to find the controller's parameter (8) we consistently will find the conformable matrices.
Condition (9) requires determination of the right devisor of matrix C and the left devisor of matrix CB, which could be found by canonization of these matrices:
CK =
0 0 0 0 0 1 0 0 0 0 0 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 0 1
CB = [0 0 0], CB =[0 0 0] = 1, CACR =[0 0 1 0 0],
CBLCACR = [0 0 1 0 0] * 0.
Condition (9) is not fulfilled, and then the matrix n1 will be determined by (10):
^0 10 0" 10 0 0 0 0 0 0 0 0 10 0 0 0 1
Verification of fulfillment of condition (7) at n = n : "0 0 0 0"
n = CB CACR =
cRn =
CRn LB =
0 10 0 10 0 0 0 0 0 0 0 0 10 0 0 0 1 0
cRn =
1 0 0 0 0 0 0 0 0 1 0 0
00 2,0837 0 0
, CRn1 B =[1 0],
CRn ACRn =
0 0 0 0 33,42 27,53 0 0
—-1
CRn1 B CRn1 ACRn1 =[0 0 0 0] In this way, the condition (7) is fulfilled. Now can verify fulfillment of condition (5):
-L
с n s =
"0 0 0"
1 0 0
1 0 0 0 0 0" 1 0 1 "0 0 0"
0 0 0 1 0 0 0 0 0 0 0 0
0 1 0
0 0 1
Hence, the condition (5) is fulfilled, it is possible to form controller's coefficient matrix (8), for which purpose it is necessary to determine matrices in this formula: "0 0,4799"
(C\ B)~ =
(C4)~ =
0 0 00 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
A = A + BK =
C n. B =
By substituting the obtained numerical value of matrices into formula (8):
(K}rz =-(cr^lB) cRnLACRn(cRn)~ + CV^B x+ УC
п =
Yii
-13,2116 -16,0383
Y 21
0 0
(15)
(^21 + Xl) X2 X Y22 + X4 ^ X5 X6
_ Y 0 0 y32 0 0
where X and Y are forming matrices with the random numerical values specified as:
/11 /12
X=[ X X3 X4 X X ] / = /21 /22
Y,1 /32 _
Controller for the three-generator system is given by:
K =
С 0
0 k tSl
0 kt'' 0
1
0 0 k t'1
0 0 • kf- 0 0 • kf
(16)
-78,0370 27,5300 39,3300 -108,6224 23,6100 50,5300 with the spectrum: -1.0790 ± 4.2951i, —0.6007 ± 12.2361i, —1.4379 ± 10.3156i. With the selected parameters of controller (17) EPS becomes steady with the one electromechanical frequency 0,6839 Hz and two electromagnetic frequency: 1,6426 and 1,9484 respectively.
It is evident that the presence of Ay makes possible the comprehensive investigation of dynamical properties of controlled three-generator system by changing the parameters of controller
Synthesized controller (16) with the matrices X and Y with random numerical values of elements must be structured in such a way that necessary technical requirements such as stability, damping the low frequency oscillations etc., should be provided in dynamic system.
It should be noted that formula (15) should be fit with formula
(16), that is, it is conceivable that k"1 = Yu, K^ = Y12, kfj = X2, kf^ = X5, and the rest of elements are equal to zero. The final matrix of controller's coefficients (15) will be as follows:
>„ 0 0 y21 0 0" (K}M =0 x2 0 0 X5 0 . (17)
0 0 0 0 0 0
It is characteristic that at the selected matrix the output of the exploring system C and matrix of disturbances S, the third controller is not involved in the mode controlling of complex EPS.
Previously it was shown that the matrix of own dynamics of the model of EPS A is equal to (12).
Now we check the impact of the regulator (17) to the spectrum of the matrix of own dynamics of controlled EPS, which has following matrix:
Ay = A + BK. (18)
At = yu = -10, = ya = -2, =X2 = 8, k£ = X = 1 the matrix (18) is equal to:
0 1,0000 0 0 " 0 0 1,0000 0 0 0 0 1,0000 33,4200 -4,1674 0 0
50,5300 0 -2,0678 0 -95,1500 0 0 0
(17) and also to determine the invariance condition of the output to disturbances that take place in the system being studied.
In Fig.1, listed are characteristics of the change of deviation of the first generator's angle AS1 = f (t), the stable, controlled
(18) EPS (Fig.1, А) at the synthesized parameters of the controller (17) and the stable, uncontrolled EPS (12) (Fig. 1, B). The process attenuates relatively quickly and bears virtually aperiodic character.
к
L
R
Fig.1. Change characteristics of the angle of the first generator A81 =f (t) of the three-generator electric system at:
A: = Yu = -10 ; k£ = y21 = -2 ; k£ = Xl = 8; k£ = = 1
B: k? = kt" = № = kt" = 0 and Pd, = Pd, = Pd, = 0.
Since the provided technology of EPS controller synthesis is based on the modern theory of matrices, which is in its turn adapted for computer processing and therefore has high computational performances, it could be recommended for the analysis of controlled complex EPS.
As the result, it may be noted that on the basis of matrix canonization approach, which provide a basis for the system's embedding approach, the invariance conditions of EPS output to the spontaneous external disturbances were specified. In order to solve this problem the typical controller was synthesized.
The obtained results of determination the impact of controller's and EPS's operating parameters to dynamics of EPS are absolutely same with the all known classic results what also proves the adequate of the mathematical models of controlled complex EPS.
Since the system's embedding approach is based on the matrix analysis then many software applications are available for work with it, for which reason, the principal difference of this method is the decreasing the computational. The analytical descriptions of controllers' type that provide the required dynamics of exploring systems are also important.
References:
1.
The advanced method of automatic control system's design. Analysis and synthesis/Edited by: B. N. Petrova, V. V. Solodovnikova, YU. I. Topcheev. - Moscow, Mashinostroenie [Machine engineering], - 1967.
The theory and methods of automatic control system's design./Avtomatika i telemekhanika [Automatics and telecontrol], 1939, - № 1. Lusin L. L., Kuznetsov P. I. To absolute invariance and invariance through e in differential equation theory//Doklady Akademii Nauk SSSR [Proceedings of the Academy of Science of USSR], - 1946. T.51. - № 4, 5.
Petrov B. N. About the realizability of invariance conditions/Conference "The invariance theory and its application for automatic control", - Kiev, 1958.
Ivakhnenko A. G. Combine the invariance theory with the theory of deferential equation. - Moscow: Avtomatika [Automatics], 1961, -№ 1.
Aliev R. A. The invariance principle and its application. - Moscow: Energoizdat, 1985.
Novikov M. A. Mathematical modeling and reexpression in the problems of stability of steady-state motion of mechanical and controlled systems./Authors abstract, St. Petersburg, - 2012.
Fazilov Kh. F., Nasirov T. Kh. Steady-state modes in electric power systems and their optimization. - Tashkent, "Molniya", - 1999. Bukov V. N. System's embedding. The analytical approach to analysis and synthesis of matrix sysmtems//Edited by N. F. Bochkareva, -Kaluga, - 2006.
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