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УДК 517.92
doi:10.18720/SPBPU/2/id21-159
Лэ Ван Хань1, аспирант;
Фирсов Андрей Николаевич1,
д-р техн. наук, профессор
ОБРАТНАЯ ЗАДАЧА УСТОЙЧИВОСТИ ДИНАМИЧЕСКОЙ СИСТЕМЫ ПРИ МАЛЫХ ВОЗМУЩЕНИЯХ ЕЁ ПАРАМЕТРОВ: АНАЛИТИЧЕСКИЕ И ВЫЧИСЛИТЕЛЬНЫЕ АСПЕКТЫ
1 2
' Россия, Санкт-Петербург, Санкт-Петербургский политехнический
университет Петра Великого,
1 2 levankhanhth@gmail.com; anfirs@yandex.ru
Аннотация. В работе представлены аналитические и численные методы решения некоторых обратных задач управления динамическими системами при малых возмущениях параметров этих систем. В результате был построен математический алгоритм вычисления диапазонов малых возмущений параметров динамической системы, удобный для его реализации с помощью пакета MatLab; построенный алгоритм реализован в среде пакета MatLab; приведен пример расчета диапазона устойчивости для модельной схемы каскада химических реакторов с заданными параметрами.
Ключевые слова: динамические системы, устойчивость, многопараметрическое возмущение, знакоопределённость, собственные значения оператора, обратная задача устойчивости.
Le Van Khanh\
Graduate student;
Andrei N. Firsov , Dr. of Tech. Sciences, Professor
THE INVERSE PROBLEM FOR SMALL PERTURBATIONS OF DYNAMICAL SYSTEMS: ANALYTICAL AND COMPUTATIONAL ASPECTS
Peter the Great St.Petersburg Polytechnic University,
St. Petersburg, Russia, 1 2 levankhanhth@gmail.com; anfirs@yandex.ru
Abstract. The work presents analytical and numerical methods for solving some inverse problems of control of dynamic systems under small perturbations of the parameters of these systems. As a result, a mathematical algorithm for calculating the ranges of small perturbations of the parameters of the dynamical system was built, which is convenient for its implementation using the MatLab package; the constructed algorithm is implemented in the environment of the MatLab package; an example of calculating the stability range for a model scheme of a cascade of chemical reactors with given parameters is given.
Keywords: dynamical system, stability, multiparameter perturbation, determine signs, eigenvalues operators, inverse problem of stability.
Introduction
One of the important problems in a technical system is analyzing and assessing the stability of dynamical systems with parameters admitting small random perturbations. Over time, parameter changes can be linear or nonlinear over time. Practice shows that it is impossible to show the necessary and sufficient permissible ranges of variation of the corresponding parameters. A serious interest in such problems appeared in the middle of the last century, however, the construction of the corresponding theory ran into difficulties since the methods of the classical theory of dynamical systems turned out to be ineffective in solving the corresponding inverse problems.
This paper proposes a mathematical algorithm for solving the inverse problem of stability of dynamical systems under small perturbations of its pa-
rameters. And in this paper, we study a mathematical model that describes dynamical processes in a cascade of chemical reactors; the problems arising in the course of the study are of a fairly general nature for dynamical systems. Thus, the method of solving them can be successfully used in the analysis of other technical systems.
Consider a linear dynamical system defined by the following system of differential-operator equations:
^ = Ax(t) + b , (1)
at
where the operator A0 is represented by the matrix (atj ) ^_1,
X (t ) = ( x1 (t), x2 (t),..., xn (t ))T, and B = (b. )n=1. Let's pretend that
Aj, j = 1,2,...,n - the eigenvalues of the matrix (a j)n,j=1,different and
Re (As ) < 0, j = 1,2,..., n,i.e. the system (1) is stable.
Along with system (1), let us consider the "perturbed" system. Let be A(s) = A0 + E, E - perturbation operator s = (s^... ,em ) e Cm, where sj -
perturbations of the parameters of the original matrix of the operator. We will assume that the value of sy3 can be neglected in comparison with sj: sy3 ^ sj.
1. Formulation of the problem
The problem of analyzing the stability of a dynamical system under small perturbations of its parameters is urgent. Consider differential-operator equations of the form:
= a(e )X(t) + B (2)
dt
In what follows, we will assume that
A (s1,.,Sm ) = A + S1A1 + S2 A2 +--.SmAm
A V A (3)
= A0 + LsA, j=1
where Aj - given matrices. We also assume that X] (s), j = 1,2,..., n - operator eigenvalues A(s) - are sufficiently smooth functions of the parameters Sj, j = 1,2,...,m in the neighborhood of zero. The operator A0 will be called "unperturbed", or basic, and s1A +s2A2 +...SmAm - "perturbed". We will call
such systems with weakly undefined parameters. By the norm of an operator and vectors, we mean the Euclidean norm.
The problem is to estimate the stability region of solutions to system (1) when replacing the "unperturbed" operator A in it by an operator A (s).
Estimation of the stability region presupposes finding a set \sj j for which the
real parts of all eigenvalues Xj, j = 1,2,...,n of the "perturbed" operator A(s)
do not exceed zero.
2. Algorithm for finding the admissible boundaries of perturbations
of the matrix of the operator of a dynamical system Let's start with the lemma that will allow us to propose the method for solving the problem of determining the stability range for a dynamical system under small perturbations of its parameters. Below, it is assumed that the following conditions are satisfied with respect to the parameters \s] j, j = 1,2,..., n,
associated with the system of inequalities f.(sps2,...,sn) < 0, i = 1,2,..., m,
moreover, each function fi is assumed to be continuous in the region Qi c Rn,0 eQi Vi = 1,2,...,m.
Lemma.[6] Let be (-si,s),s > 0, i=1,2,...,m - family of intervals, all points s of which are solutions of the inequalities f(s,s,...,s)<-s/2, i = 1,2,...,m for some 0 < 8 ^ 1 for the common to all i. Then m- measuring cube
m
Q = {(s1,s2,..,sm ): sj e (-S,S) = n (-S ,SS ); j = 1,2,..., m}
i=1
gives one of the solutions to the system of inequalities f (s1,s2,...,sn) <-8/ 2 i = 1,2,...,m .
Using the theorems of J.G Sun to derive formulas for the coefficients of the perturbed eigenvalues indicated in [8], we write down an algorithm for finding the admissible boundaries of perturbations of the operator of the parameters of a dynamical system.
Step 1: Calculate the eigenvalues As of the matrix of the unperturbed operator A0, and the corresponding right and left eigenvectors xs, ys.
(A, -V) xt = 0
, where yTsXs = 1
y (A0 -V ) = 0
Step 2: Construct matrices X, Y and A . According to the hypothesis (see [8]) we build the matrix X = (,XX) and Y = (ys, Y). Step 2.1.
Case 1: If the right eigenvectors have the form xs = (0.1.0)T i = 1,n. Thus,
i-u
can immediately form the matrix
Y = ( ,Y)
yi 1 .. 0 0 .. 0
ys1 0 .. 1 0 ^ 0
1 0 .. 0 0 ^ 0
y:+1 0 .. 0 1 .. 0
0
0 0
i-u (i+1)-u
J nxn
i.e Y Tx = 0
(n-1)x1 •
Case 2: Right eigenvector coordinates x's ^ 1, i = 1, n. Let the matrix
Z ' =
x
1 0 0 1
0 0 0 0
0 0
1 0
' x
x
J nxn V xs
n-1
x
1 0 0 1
0 0 0 0
0 0
1 0
Applying the Gram - Schmidt orthogonalization to the matrix Z', we obtain the matrix Z = (xs, Y)e Cnxn is orthogonally, i. e YTxs = 0(n-1)x1 and matrix
Y = ( ,Y).
Step 2.2: form the matrix X = (, XX).
We have det (7) = ±1, whence it follows that there is an inverse matrix for Y. Calculate the matrix
H = K j ..= Y-1>
whence we get the
i, j = 1
X (
<(n-1))
'fa j
i=2,j=1
(h )T (h, )T ... (h„)
where h, i = 2...n- matrix rows H. i.e ysX = 01x(n-1).
Step 3: We construct linear systems of inequalities for Xs(s), s = 1,n.
Xs(s) = AS + yTsEXs + yTsEXt(\I-4)-1YTE: <0 We solve this system to get the permissible range for s . 3. Calculation example
We consider the nonlinear dynamical system that describes the process of monomerization of a substance in a cascade of chemical reactors with a stirrer
Fig. 1. Three-stage cascade of reactors with countercurrent cooling
T
The mathematical description of the process is presented as follows:
vdcL = F(0 -C1)-Vk(T)C1,
pcpVdT = FPcp (T0 - T1) + Vk(T )QAH - KtA( - Tc3) p^V dTc3 = FcPcCpc (Tc2 - Tc3) - KtA(T - Tc3)
VdCC1 = F (1 - C2)-Vk (T2 )C2,
pcpVOT = Fpcp (T - T2) + Vk(T2 )C2AH - KtA(T2 - Tc2) (4)
pcpZ OTj2 = FcPfpc (Td - Tc2) - KtA(T2 - Tc2)
V^CT = F(2 -C3)-Vk(T3)C3,
pc/^at= Fpcp (T2 - T3) + Vk(T3 )C3AH - KtA(T3 - Tc1)
PccpcVc = Fcpccpc (Tc0 - Tc1) - KtA (T3 - Tc1)
Linearization of the system (4) of differential equations of the dynamics of chemical reactors. We'll get
— = AX + B, (5)
dt W
where: X = (,...,x9)T, B = (b1,.,b9)T, matrix A = |aj^ ^ 1. In practice, it turns out that the coefficients
S1, S2, S3, S4 can have undefined values. Therefore, we put:
A (s ) = A +]T SjAj. (6)
j=1
or A(s1,.,s4 ) = A) + (s1 A1 + s2 A2 + s3 A3 + s4 A4), where: a° - elements of the matrix A0, and a\, a^, aj, ai4 - elements of the matrices A1, A2, A3, A4 respectively.
Substituting in the numeric values from the data (see [5]), we get
the following matrices:
A =
-1993.9G3 G G -6.G889 G G G G
G.GG1125 -1994.5215 G G -5.4721 G G G
G G.GG1125 -1994.558 G G -5.4354 G G
779.3G28 G G 2.3786 G G G G
G 7GG.3583 G G.GG1G1 1.92G3 G G G.GGGG2G1
G G 695.66G2 G G.GG11 1.8946 G.GGGG2 G
G G G G G G.GGG4 G.G2G27 G
G G G G G.GGG4 G G.G2 -G.G2G27
G G G G.GGG36 G G G G.G2
' 6.G889 G G - 6.G7G3 G G G G G
G 5.4721 G G - 5.4571 G G G G
G G 5.4354 G G - -5.42G6 G G G
-2.3798 G G 2.3725 G G G G G
A = G -1.9215 G G 1.9162 G G G G
G G -1.8957 G G 1.89G6 G G G
G G G G G G G G G
G G G G G G G G G
, G G G G G G G G G ;
f-1993.9G2 G G -6.G889 G G G G G1
G -1994.52 G G -5.4721 G G G G
G G -1994.557 G G -5.4354 G G G
779.3G28 G G 2.3798 G G G G G
Ai = G -695.66G2 G G 1.9215 G G G G
G G 695.66G2 G G 1.8957 G G G
G G G G G G G G G
G G G G G G G G G
G G G G G G G G G,
G G G
G.GGGG224 G G G G
-G.G2G27
A3 =
f G G G G G G G G G
G G G G G G G G G
G G G G G G G G G
779.3G28 G G 2.3798 G G G G G
G 7GG.3583 G G 1.9215 G G G G
G G 695.66G2 G G 1.8957 G G G
G G G G G G G G G
G G G G G G G G G
V G G G G G G G G G
A4 =
f G G G G G G G G G
G G G G G G G G G
G G G G G G G G G
G G G 2.3798 G G G G G.GGGG2
G G G G 1.9215 G G G.GGGG2 G
G G G G G 1.8957 G.GGGG2 G G
G G G G G G.GGG4 -G.G2G3 G G
G G G G G.GGG4 G G -G.G2G3 G
,G G G G.GGG36 G G G G -G.G2G3
To analyze the stability of the solution of the dynamical system (6), we use the algorithm for finding the range of possible changes in the perturbing parameter of the dynamical systems (in section 3), and the rooted stability criterion.
First, we find the eigenvalues of the unperturbed matrix A
V =-1991.5231
V =-1992.6
V =-1992.6626 V4 =-0.00107
V =-0.0012 + 0.000069i V6 =-0.0012 + 0.000069i
V =-0.02035
V =-0.0202 + 0.000069i
V =-0.0202 + 0.000069i
According to the algorithm, we get the system of inequalities:
2.3769s2 - 1980.6835s -1991.523 < 0 1.9196s2 - 1983.2909s -1992.600 < 0 1.8939s2 - 1983.4415s -1992.662 < 0 295.7039s2 + 2.0540s- 0.001067 < 0 < -8518.8378S2+ 0.9921S-0.001198 < 0 -8518.8378s2+ 0.9921S-0.001198 < 0 -1.0247s2 - 0.01088s - 0.020354 < 0 -134292.858s2 -0.01454s -0.0202238 < 0 -134292.858s2 -0.01454s -0.0202238 < 0
We solve the system. The solution to this system is the following interval:
-0.0074 <s<0.00048 Consequently, under disturbances from these interval, our dynamical system is guaranteed to remain stable.
We check the fidelity of the interval, putting perturbations on these parameters lying inside it. Now, we sequentially take several values of the perturbation of the allowable range, and impose them on the perturbed design pa-
rameters of the system, we recalculate the eigenvalues for the perturbed matrices.
When s = -0.0004 we have:
V =-1991.5233
V =-1992.6003
V =-1992.6628 V4 =-0.0016816
V = -0.001707+0.00001i V6 = -0.001707+0.00001i
V =-0.020659 + 0.00001i
V =-0.020659 + 0.00001i
V = -0.020684
Re(V) < 0, j = 1.9, therefore, the system retained stability.
When s = 0.0004 we have:
V = -1991.5228
V =-1992.5228
V = -1992.6622
V =-0.0000119
V = -0006949 + 0.000296i V6 = -0006949 + 0.000296i
V = -0.0202535
V = -0.019569+0.000297i
V =-0..019569+0.000297i
Re(V) < 0, j = 1.9, therefore, the system retained stability.
When s = 0.0005 we have:
V =-1992.5134
V2 =-1993.5916
V = -1993.6542 14 = +0.0000454
V =-0.0002054 + 0.00003i V6 = -0.0002054 + 0.00003i
V =-0.02036
V = -0.020236+0.00006i
V = -0.020236+0.00006i
Because V4 > 0 then the system is not stable.
That is, under disturbances from this interval, our dynamical system is guaranteed to remain stable.
Conclusions
In the presented work, the problem of determining the boundaries of the ranges of small perturbations of the elements of the matrix of the system of differential equations, under which the stability property of the system is preserved
The solution to this problem is based on the theory of perturbations of linear Rellich-Kato-Lancaster operators and the results on the dependence of the eigenvalues on the perturbing parameters were obtained by J.G. Sun's.
*Listing from MatLab:_
function Algorithm(A,E) clc
% Initial variables: % Matrix A and perturbed matrix E % Determine the size of the original matrix N=size(A); n=N(:,1);
% determination of the eigenvalues and vectors % Let l: vector of eigenvalues % V: matrix of right eigenvectors
% W: matrix of left eigenvectors
[V,D]=eig(A); W=((VA-1)'); V=(real(V)); W=(real(W)); % (W'*V)
l=vpa(real(diag(D))) for k=1:n
% Declaring Variables
X=zeros(n);Y=zeros(n);X2=zeros(n,n-1);Y2=zeros(n,n-1);Z=zeros(n);index=0;
% Checking the Elements of Right Eigenvectors for i=1:n
if V(i,k)==1 index=i;
end
end
if index == 0 % Case 2
% Construct the orthogonal matrix Z
Z=MatrixZ(V(:,k),n) ;
% Construct the matrix Y
[Y,Y2]=MatrixY(W(:, k),Z,n);
else % Case 1
% Construct the matrix Y
[Y,Y2]=MatrixY 1(W( :,k),n,index);
end
% Construct the matrix X
[X,X2]=MatrixX(Y,V(:,k) ,n);
% Construct the matrix Y'A(0)X:
% the matrix A2
A2=MatrixA2(A,X,Y,n);
% Construct the inequality
L = l(k) +
W(:,k)'*E*V(:,k)+W(:,k)'*E* X2*((l(k)*eye(n-1)-A2)A-
1)*Y2'*E*V(:,k);
solve(L)
end
end
References
1. Sun Ji-guang. Eigenvalues and eigenvectors of a matrix dependent on several parameters. // J. Comput. Math. - 1985. - Vol. 3. - Pp. 351-364.
2. Kato T. A Short Introduction to Perturbation Theory for Linear Operators. - N.-Y.: Springer-Verlag, 1982.
3. Beklemishev D.V. Kurs analiticheskoj geometrii i linejnoj algebry. [A course in Analytical Geometry and Linear Algebra.] - Мoscow: FIZMATLIT, 2005 (in Russian).
4. Lankaster P. Teoriya matric. [Matrix theory.] - Мoscow: Nauka Publ., 1978. -280 p. (in Russian).
5. Firsov A.N, Chulin S.L. Construction of an analytical solution for a system of nonlinear differential equations describing dynamical processes in chemical reactors. // System Analysis in Engineering and Control. Proceedings of the XVI international scientific and practical conference. Part 1. - SPb.: SPb. Polytechnic University publishing house, 2012. -Pp. 145-151 (in Russian).
6. Bulkina E, Firsov A. Numerical-analytical method for solving the inverse problem of stability for technical systems with multiple uncertain parameters. // Machines. Technologies. Materials. International journal for science, technics and innovations for the industry. -2016. - Issue 11. - Pp. 12-14.
7. Firsov A.N., Le Van Khanh. Substantion of J.-G. Sun's hypothesis, which lies in the basic of the theory of analytical dependence of eigenvalue of matrix from "Disturbing" parameters under multi-parametric perturbation of the matrix elements // Mathematical modeling. Int. Sci. J. - 2018. - Issue 3. - Pp. 88-90.
8. Le Van Khanh, Firsov A.N. Numerical algorithm for solving the inverse problem of stability of the dynamical system under a multi-parametric perturbation of its elements. // System Analysis in Engineering and Control. Proceedings of the XXIII International Scientific and Practical Conference. Part 2. - St. Petersburg: SPbPU Publishing House, 2019. -Pp. 43-45.
