Linearization of differential algebraic Equations with integral terms and their application to the thermal energy Modelling Текст научной статьи по специальности «Математика»

MSC 65L80, 65L05, 65L06, 65L20

DOI: 10.14529/ mmp180407

LINEARIZATION OF DIFFERENTIAL ALGEBRAIC EQUATIONS WITH INTEGRAL TERMS AND THEIR APPLICATION TO THE THERMAL ENERGY MODELLING

E.V. Chistyakova1, V.F. Chistyakov1, A.A. Levin2

1Matrosov Institute for System Dynamics and Control Theory of Siberian Branch

of Russian Academy of Sciences, Irkutsk, Russian Federation 2

of Sciences, Irkutsk, Russian Federation

E-mails: [email protected], [email protected], [email protected]

Modelling of various natural and technical processes often results in systems that comprise ordinary differential equations and algebraic equations This paper studies systems of quasi-linear integral-differential equations with a singular matrix multiplying the higher derivative of the desired vector-function. Such systems can be treated as differential algebraic equations perturbed by the Volterra operators. We obtained solvability conditions for such systems and their initial problems and consider possible ways of linearization for them on the basis of the Newton method. Applications that arise in the area of thermal engineering are discussed and as an example we consider a hydraulic circuit presented as a system comprising an interconnected set of discrete components that transport liquid. Numerical experiments that employed the implicit Euler scheme showed that the mathematical model of the straight-through boiler with a turbine and a regeneration system has a solution and this solution tends to the stationary mode preset by regulators.

Keywords: differential algebraic equations; Fredholm operator; Volterra operator; initial problem; consistency problem; index.

1. Problem Statement and Auxiliary Information

Modelling of natural and technical processes often yields systems that comprise ordinary differential equations (ODEs) of various order and algebraic equations (cf. [1-6]). Their combination can be written in a form of quasi-linear vector ODEs with a singular matrix multiplying the higher derivative of the desired vector-function

Afc (u)u := A(u(k-1), ••• , u(1), u, t, X)u(k) + B(u(k-1), ••• , u(1) ,u,t,X) = 0, m

k = 1, 2, ••• , 1 j

where A(gk-1, ••• ,g1,g0,t,X), B(gk-1, ••• , g1, g0,t, X) are given (v x n)-matrix and an v-dimensional vector-function, correspondingly, gk-1, ••• ,g1 ,g0 E Rra, t E T = [a, ¡3 ] C R1, u = u(t) is n-dimensional vector-function u(%i>(t) = (d/dt)lu(t), i =1, 2, • • • , u(0\t) = u(t), X is a scalar parameter, and matrix A is such that

rank A < min (n, v) (2)

for all values of arguments from the domain. For the case of closed systems (v = n), this condition takes the form det A = 0. Such systems are commonly referred to differential algebraic equations (DAEs). If the process under study has a so-called aftereffect, then

the system may include integral equations. Therefore, in this work we focus our attention on the systems with Volterra and Fredholm operators

Jk (u)u := A(u(k-1), •••,u(1),u,t, Vu, Ku,A)u(k)+B(u(k-1), • • •,u(1),u,t, Vu, Ku,A) = 0, (3)

where A(g-, ••• , gb go,Y1,Yo,t, A), B(g—, ••• , 91, go, Y1, Yo, t, A) are given (v xn)-matrix and an v-dimensional vector-function, correspondingly, j1,j0, G Rra,

t в

V u = J к (t,,M»*b, Ku = !

a a

are the Volterra and Fredholm operator, K, K1 : T x T x Rra ^ Rra, and matrix A is such that

rank A < min(n, v) (4)

for all values of arguments from the domain. Linear DAEs (1), (3) for n = v have the following form

k

Лк x :=Y, Ai(t)x(i)(t) = f (t), t G T, (5)

i=0

where А() are (n x n)-matrices, x(t) and f (t) are the desired and the given vector-functions, correspondingly, x(%i>(t) = (d/dt)%x(t), x(0\t) = x(t),

k t в (Лк + AF + ¡i^)z Ai(t)z(i\t) + X t K(t,s)z(s)ds + ¡it K(t,s)z (s)ds = f (t), (6)

i=o J J

aa

where A, i are scalar parameters (possibly, complex cones), K(t,s), K(t, s) are (n x n)-T x T z(t)

entries are smooth enough and that the following condition is satisfied

det Ak(t) = 0 yt G T. (7)

DAEs with k = 1 have been fairly well studied (see the monographs [6-8] and the bibliography listed therein). Any equation (5) can be reduced a first order DAE using a change of variables. However, if k > 1, DAEs possess a number of interesting properties that disappear after such reduction.

It is assumed that for each system (5), (6), a set of initial data is given

x(j)(a) = a^, z(j) (a) = b3, (8)

where aj, bj are the given vectors from Rra.

By the solution to systems (5), (6) we understand any k—times differentiable on T vector-functions x(t). z(t) that turn the systems under study into identity on T. If these vector-functions are solutions and satisfy (8), then they are solutions to the corresponding initial problems.

At present, there are available only a few works addressing higher order DAEs (cf. [911]). To study (5), we will employ the tools and results that had been previously developed for the first order DAEs.

For the sake of simplicity, the dependence on t sometimes will be omitted, if does not cause misunderstanding. Inclusions V(t) E Ci(T), i > 1, where V(t) is a matrix or a vector-function, mean that all its elements are differentiate on T up to the order i. The continuity will be denoted as V(t) E C(T); symbol CA(T) stands for the space of real analytical matrices. Below, we will also employ denotation r[V(t)] = maxjrank V(t), t E

T }

Here we also use the norms of ^-dimensional vector Z = ((1,(2, ■■■,Zg)T E Rg, and vector-function Z(t) = ((i(t),(2(t), ■ ■■,Zg(t))T, t ET rules

9 f

E = £ j IIZh = . maxAjI , IIZml2iT) = lie(s)\\2E ds,

3 = 1 a

C(T > t&T

where T stands for transposition.

Definition 1. [7] The (n x m)-matrix M +(t) is said to be the pseudoinverse to (m x n)-matrix M (t) if Vt E T

M(t)M+(t)M(t) = M(t), M+(t)M(t)M+(t) = M+(t),

(M+ (t)M(t))T = M+(t)M(t), (M(t)M+(t))T = M(t)M+(t).

t E T (m x n) M(t)

and is unique. If M(t) is square and nonsigular, then M-1(t) = M + (t). Accodring to [7], there exists M +(t) E Cg(T), if M E Cg(T) and rank M(t) = r = const Vt E T. If rank M(t) = const, t E T, then at least one element of M + (t) has a second kind T

Below we will use the following operators

di[M ]

f M \

(d/dt)M

\(d/dt)iM)

Mi[M ]

( C00M 0 C0MC\M

0 0

(9)

\C0M(i) C}M(i-1) ••• CiMJ

where M = M(t) is some matrix from Cl(T), Cj = j!(i-j)!/i! are the binomial coefficients. The operators are related by formula

Mi[M (t)F (t)] — Mi[M (t)]di[F (t)],

(10)

where F(t) is some matrix of the appropriaie size from Ci(T). Formula (10) follows from the Leibniz general rule.

I

2. Properties of Linear Systems

In this section we modify concepts that were introduced earlier in [10]. We single out a class of DAEs, which solution properties are very much similar to those of normal form ODEs.

Definition 2. Equation (5) has a Cauchy type solution if it is solvable for any f (t) E Ckn(T) and its solutions can be represented as a linear combination

x(t,c) = Xd(t)c + t), (11)

where Xd(t) is an (n x d)-matrix from Ck(T), with the property rank dk_i[Xd(t)j = d Wt E T, dk-1[.] is the operator from the formulas (9), c is an arbitrary constant vector, ip(t) is the vector-function with the property Akip(t) = f (t), t E T. Additionally, on any subsegment [a0,[0] C T there is no solution different to x(t,c).

Solution x(t) that passes through point x(ii)(Y) = ai? i = 0,k — 1, y E T, is unique if there exists c such that dk-1[Xd(y)]c = a — dk-1[-0(Y)], where a = (aj aj ... aj_ 1)T. The

c

d = nk, det dk-1[Xd(t)] = 0 Wt E T.

i

Definition 3. If there exists operator Ql = E Lj(t)(d/d)j, where Lj(t) are (n x n)

j=0

k

matrices from C(T), such that Q o Aky = E Mt)v{i)(t) Wy(t) E Cl+k(T), where A()

i=0

are some (n x n)-matrices from C(T), i = 0,q, det Ak (t) = 0 Wt E T, then opera tor Ql is said to be the left regularizing operator (LRO) for the system (5). The smallest possible l is said to be the index of the system.

Definition 4. The combination of (5) and its i derivatives di[Akx — f] = 0, t E T, where d^.] is the operator defined by (9), is called i-extended system (5).

i

k

Di[A(t)]di+k[x] = J] (Oj Mi[Aj(t)] Oj) di+k[x]=di[f (t)], (12)

j=0

where A = Ak Ak-1 ... A0), D4A(t)] is a [(i + 1)n x (i + k + 1)n]-matrix, Oj, Oj are zero blocks of dimension [(i + 1)n x jn] and [(i + 1)n x (k — j)n], j = 0,k, correspondingly. In what follows, we will use splitting

Di[A(t)] = (Bi(t) r[A(t)]) , (13)

where ri[A(t)^ ^^ ^ ^^^^^^^^^^^ular matrix with Ak(t) standing on the diagonal.

The concept of index is quite complex and can be approached in several ways (see, for example, monographs [6-8] and references listed there). Here we employ the definition that was introduced in [13] for index one DAEs and modify it for DAE (5).

Definition 5. Assume that set of solutions X = {x = x(t) : Akx — f = 0, t E T} to DAE (5) is non-empty, and, starting with some natural number l, for any vector-function x£ = x£(t) : ||dl-1[Akx£ — f]||¿2(T) < £ there exists solution x(t) E X : \\x(t) — x£(t)\\L2(T) < ke where k is some constant. Then, we say that DAE (5) is index l

The similar notion but defined in Ci(T) instead of L2(T) is called perturbation index [6,8]. Below we give some results from [12].

Theorem 1. Let

1) DAE Akx = f, t e T, be index l;

x(t) = №= (En 0 ... 0) D+-k[A(t)]di-k[f(t)] = Y, Cj(t)fj(t);

2) Ai(t) e Cm(T), i = 0,k, m = max{(k - l)n + r + 1, 21}, r = r[Ak(t)]. Then,

1) there exists the Cauehy type solution and $(t) from (11) has form

m=i K (t,s)f (s)ds+^ Cj (t)f (j)(t), l > k, m=i K (t,s)f (s)ds, l < k, (14)

j=0

a J a

where K(t, s), Cj (t) are some (n x n) -matrices, t e T, (t,s) e T x T;

2) if fi = 0 and l < k, K(t, s) e Cl(T x T), system (6) is solvable for any X and its general solution has form

z(t,c)= Yd(t)c + g(t), t e T, (15)

where Yd(t) = (En + XV)Xd(t), g(t) = (En + XV)ip(t), V is some Voletrra operator, En is n-dimensional identity matrix;

(d = 0) l-k

E

j=0

i=l

rank ri[A(t)] = const, r+[A(t)]ri[A(t)] = , t e T,

where Z22(t) is some block of the appropriate dimension, and first n rows of matrix r+[A(t)], split into (n x n)-blocks, can be taken as LRO coefficients.

Lemma 1. Let

1) the Cauchy type solution x(t,c) e Cmi (T) to the DAE (5) be defined on T;

2) Ak(t),Ak-1(t),...,A0(t) e Cm2(T), where mx = (k - 1)n + r + 2, m2 = 2((k -1)n + r) + 3. Then, DAE (5) has an LRO on T.

Below we prove the following statement.

Theorem 2. Let

1) system (5) satisfy Theorem 1;

2) X = 0 and K(t, s) e Cl(T x T) in (6).

Then, (6) is solvable for all f, except maybe count a ble set {fi, i = 0,1, 2, • • •}, and its general solution with ff = fi has the form

z(t,c) = Zd(t)c + $(t), t e T, (16)

where Zd(t) = (En + ffW)Xd(t), $(t) = (En + ffW)^(t), W is a Fredholm operator.

k

Proof. Rewrite (6) for A = 0: Akz = —¡$z + f : E Ai(t)z(i)(t) = w(t), t e T, where

i=0

w(t) = —¡j K(t, s)z(s)ds + f (t). Using (14) and (11), write down the expression:

a

z(t,c)= Xd(t)c + i K (t,s)w(s)ds + ^ Cj (t)wj)(t), t e T. (17)

a j=0

Due to the fact that the product of the Volterra and Fredholm operators is a Fredholm

operator, we obtain a system of second kind Fredholm equations

¡

z(t) = ¡i W (t,s)z(s)ds + v (t), (18)

where

в t i W(t,s)z(s)ds = f

в

K(t,s) f K(s, т)z(t)dr

i-k в

ds + Y^ / С(t)[djK(t, s)/dtj]z(s)ds, j=0

a

t k- 1

v(t) = Xd(t)c + K(t,s)f (s)ds + £ Cj(t)f (j)(t).

a j=0

For system (18), except maybe countable set {¡i, i = 0,1, 2, • • • }, we can use the known inversion formula [14]:

¡

z(t) = (En + ¡W)v(t) = v(t) + i W(t, s, i)v(s)ds, (19)

where W(t, s,y) is the resolvent kernel for (18). The validity of the statement follow from (19).

□

Corollary 1. Let

1) Theorem 1 be satisfied;

2) X, л = 0, l < к and K(t,s), K(t,s) e Cl(T x T) in (6).

Then, (6) is solvable for all ¡л, except maybe a count able set {¡i, i = 0,1, 2, • • •}, its general solution has the following form for ¡л = ¡i:

z(t,c) = Zd(t)c + Z(t), t e T, (20)

where Zd(t) = (En + ¡¡W)Xd(t), (f(t) = (En + ¡W)^(t), W is some Fredholm operator.

Lemma 2. Let Theorem 1, Theorem 2 and Corollary 1 be satisfied . Then, initial problems (5), (6), (8), have solution x(t), y(t), z(t) e Ck(T), if and only if systems

dk-i [Xd(a)]c = dk-i^(t)] - a, dk-i[Zd(a)]c = dk-i^(t)] - J), dk-i[Zd(a)]c = dk-i[(f)(t)] - b

are solvable with respect to c, these solutions are unique, if the matrices that multiply c have full rank. Here b = (bj bj ... bJ-l)T.

It is well-known that in practice we usually address not with ideal problem (6), (8), but its perturbed version:

(Ak + AF + fi$)z = f,t e T,

z(j)(a) = bj, j = 0,k - 1, where bj are the given vectors from Rra.

Theorem 3. Let

1) Corollary 1 be satisfied;

2) problems (6), (8), and (21), (22) satisfy Lemma 2. Then, the following estimates hold

||v(t)||C(T) < K2 Mj + K3 IM^WciT) , WY(t)\\l2(T) < Mil + K3

2

L2(T )

where v(t) = z(t) - z(t), c = ([bo - bo]T [bi - bi]1 f (t), Kj, Kj are some positive constants, j = 2, 3.

[bk-i - bk-i]T)T ,v(t) = f (t) -

(21) (22)

(23)

Proof. Using (6), (8), (21), (22), we can write down the following initial problem (Ak + AF + [i$)v = M(t), t e T, v(j)(a) = bj - bj, j = 0,k - 1,

(24)

where v = y(t) = z(t) — z (t). By integrating the system from (24) k times, we obtain the system of integral equations

ki

(Ak + e + XV + ß$)v = h(t) + J2(t - a)jCj, t e T,

j=o

(25)

where

Ak = Ak(t), Vv= Wk-i(t - a)

ki

K(a, s)v(s)ds]da

ev= Q(t,s)v(s)ds, &v= wk-i(t - a)

ki

K(a, s)v(s)ds]da

ki

Q(t, s) = J2 wj-i(t - s)jWj(s), h(t) = wk-i(t - s)k-ip(s)ds, s e T

j=o

J a

Wj(s) are linear combinations of matrices Ak(t), Ak-i(t),..., A0(t) and their k derivaties, Cj are constant vectors in the form the linear combinations of initial data vj(a), j = 0,k - 1, Wj = 1/j!, j > 1. For example, if k = 2, then Wo(s) = Ai(s) - 2A2\s), Wi(s) = Ao(s) - Ai^s) + A<i)(s), Co = A2(a)v(a), ci = A2(a)v(i)(a) + [Ai(a) - A{21)(a)]v(a). Substitute Zj - into (25) instead of v(j)(a). We get

(Ak + e + XV + = h(t) + H(t)c, t e T,

(26)

t

a

ß

t

t

t

where H(t) is (n x kn)-matrix with the polynomial elements depending on t. The

solution to (26) coincides with the solution to initial problem (24). Since the operations

of differentiation and integration are interchangeable, there exists operator QQ = l

E Lj (t)(d/d)j, l < k, where Lj (t) are (n x n)-matrices from C(T), with the property j=0

Q i[(Ak + 6 + AF + i^)v]= (27)

t

= Ak(t)v(t) + j Q(t, s)v(s)ds + AQl o = Ql[h(t) + H(t)c], t e T, 0

where det Ak(t) = 0 Wt e T. In other words, Ql is a version of LRO for operator Ak + 6: ^l o (Ak + 6) = Ak + 6. System (27) is the system of the second kind Fredholm equations with a continuous kernel and a continuous free term. According to [14], there exists the Fredholm operator $R with a continuous kernel, such that v(t) = [En + l[h(t) + H(t)c], t e T. Trivial estimates and computations yield inequalities (23).

□

We should note that if l > k, the solution to (24) includes the derivatives of p(t). Therefore, there is always exists such vector-function f(t), that for the fixed initial data

\\^(t)\\* < £, \\x(t) — x(t)\\* ^ tt, £ ^ 0,

where * stands for one of the spaces: L2(T) or C(T).

3. Linearizartion of Nonlinear System

Models from applications are usually described by nonlinear DAEs and singular systems quasi-linear integral differential equations. Consider some closed (n = v) nonlinear systems of form(3). Let there be given problem

Jk(u)u := A(u{k-1),• • •,u(1),u, t, Viu,A)u(k)+B(u{k-1),• • •,u{1), u, t, V2u,A) = 0, t e T, (28)

u(j)(a) = aj, j = 0,k — 1, (29)

where Vi, i = 1, 2 are some sets of Volterra operators: Vi = {Vi,j, j = 1, 2, • • • qi}, matrix A(.) and vector-function B(.) are defined on sets

Ui = {i : gk-i, ••• ,gi,go e Rn ,yi e Rni ,t e T, A e [j,S] c R1},

= {? : gk-i, • • • ,gi,go,Y2,t,A}, Y2 e Rn2,

Vi : w = (t,s,g0) e T x T x Rn. It is assumed that entries of (28) are sufficiently smooth. There exist several approaches to the study of nonlinear DAEs. They are based on the analysis of the matrix pencils structures, extended systems and application of the techiniques from algebraic geometry (see, for example, [6-8,15,17,18]).

Definition 6. The combination of (28) and its derivatives up to order i: di[Jk(u)u] = 0, t e T, where di {.] is operator from (9), is said to be the i-extended system (28).

Definition 7. Let there exist operator

i

Qi (v) :=Y1 Lj (tU,v(1), ••• ,u(m), V3v,A)(d/d)j, j=0

where Lj£) are some (n x n) smooth matrices from the domain, V3 is a set of Volterra operators, ^ = (t,g0, • • • , gi, gm,j3, A), with property

Qi(v) o Jk(v)v = A(v(k-1), • • • ,v(1),v,t, Viv,A)u(k) +

+B(v(k-1), ••• ,v(1),v,t, V2v,A) Vu = v(t, A) e cmax{l'm}+k'°(T x [7,i]),

where A(££>) is (n x n)-matrix, B(c) is some vector-function, both continuous in the domain, £ = (gk-1, ••• ,g1,go,Y1,t,A), q = (g—, ••• , g1, go,j2,t, A), Vi are some sets of Volterra operators in initial point £ = (ak-1, • • • ,a1,a0, 0, a, A),

detA£ = 0.

Then, operator Ql(v) is said to be the left regularizing operator (LRO) for the system (29). The smallest possible integer number max{l,m} is said to be index of (29).

Example 1. Let there be given two systems

J1(u)u=(-"now 0) ft)+ft+u;;- l)=0.u=ft)

(30)

J1(u> =(0 ft) + ft) =0-' e T. (3D

t t where V1u = f(l 0)u(s)ds, V2u = f(0 l)u(s)ds. It is easy to verify that systems (30),

a a

(31) have only a zero solution on an arbitrary subset T C R1. Trivial computations show that quasi-linear operators

„ ( ) = /0 — cos(V1 u)u2u 1 — sin(V1u)uA (l — sin(V1u)uuA d ih(u)=\0 0 yl + ^0 l )Jt,

n f \ (0 -uA . fl -u2\ d & 1(u) ={0 0 ) + {0 l ) dt

are the LEOs for the systems (30), (31), respectively, and

, . , . (u 1 + u2V2ueV2U\ SMuh o J1(u)u = ,

( U2 J

& 1 (u) 1 o J1 (u)u = (u^ Vu e C2(T).

Therefore, according to Definition 7, systems (30), (31) have index 2.

It is worth noting that it is common to differentiate algebraic relations when solving applied problems. Id we do that to (30), we obtain a system with a matrix

diag{u2 sin(Vi'u), 1} singular at initial point u(a) = 0. However, we can build an LEO for (31). Here

Mdd DC ?)С D-(0->)d+«?)

Define the following neighbourhoods:

z = {£: u — el< pi },

Z2 = {? : - q\\ < pi }, Z3 = {w : \w — w\ < p2 },

where £ = (ak-1, • • • ,a1,a0,0, a, A), q = (ak-1, • • • ,a1,a0, 0, a, A), w = (a, a, a0) are the initial points.

Theorem 4. Let the following conditions be satisfied:

1. A(£) e Cm+1(Z1), B(q) e Cm+1(Z2), the kernels of the operators V1, V2 belong to the class Cm(Z3), m > 1;

2. rank A(q) = max {rank A(£), £ e Z1};

3. rank A(£) = rank (A(£)| — B(<f)): initial data (29) should be chosen so that linearsystem A(£)y = —B(q) would fulfill the Kronecker-Capelli criterion;

4. rank A(£) = deg det[AA(£) + B] [17], where

d

B = CC) = №) + A(£)y], y = —A+(OB(s).

ogk-1

Then, there exists segment T0 = [a,a + e] C T, e > 0, with a unique solution to (28), (29) u = u(t, A) G Cm(T0).

The proof is based on the switching to a first order system by the change of variables and application of corresponding theorem from [19].

Theorem 5. Let problem

A(t)U + B(u, V2U, t) = 0, u(a) = ao, t e T, (32)

satisfy Theorem 4■ Consider iterative process

t

A(t)uj+1(t) + Co(t)uj+1(t)+ Qe(t,s)uj+1(s)ds = G(t,uj(t)), t e To, uj+1(a) = ao, (33)

t

where j = 0,1, 2, ••• , G(t,Uj (t)) = -B(uj (t), V2Uj ,t) + Ce (t)uj (t) + f Qe (t,s)uj (s)ds,

a

д

Ce(t) = C(d(t), V2d,t), C(s) = j— B(s), Qe(t,s) = C(d(t), V2e,t)K2fi(t,s), _д9о_

d d C(q) = j^B(q), K2,e(t,s) = K(t,s,d(t)), K(t,s,go) = j-K2(t,s,go),

dj2 dgo

K2(t, s, g0) is the kernel of operator V2, 9(t) is some smooth vector-function C1(T0), 9(a) = a0, for a sufficiently small value of q = \\9(t) - u*(t)||C(To), u*(t) is a solution to (32). The iterative process (33) fulfill estimate

\\uj(t) - u*(t)\\C(To) < c • Kj, c = const, k = const < l. (34)

Proof. Iterative process (33) was obtained by linearizing (32) at point 9(t). Matrix pencil AA(t)+C<9(t) satisfies the rank-degree criterion on T0 and the corresponding DAE has index 1 [7]. Equation (33) satisfies Theorem 1. Initial vector a0 satisfies the Kronecker-Capelli criterion for any j, whence it follows that (33) is solvable on T0. Trivial computations and estimates with the use of (23) justify relation Zj+1 < к1WG(t,Uj(t)) - G(t,u*(t))\\C(To) < K1Zj, Zj(t) = uj(t) - u*(t), k1 < l, for a sufficient small value of parameter q = \\9(t) -u*(t)\\c(To) (see [7] ).

□

4. Mathematical Models Based on the DAEs Perturbed by Integral Operators

As was mentioned above, DAEs are widely used in mathematical modelling of various dynamic processes [1-8]. We focus our research on the models for hydraulic circuits. A hydraulic circuit is a system comprising an interconnected set of discrete components that transport liquid. There are four types of hydraulic-circuit diagrams: block, cutaway, pictorial and graphical. Block diagrams show the components of a circuit as blocks joined by lines, which indicate connections and/or interactions, and can be interpreted as an oriented graph. The liquid movement is directed by the following rule

AX = Q, AJP = Y, (35)

where A is an (m x n) adjacency matrix, which elements take values 0, l,-l, X = (x1,x2,..., xn) is the flow rate vectors for circuit branches, Q = (q1,q2, ...,Qm) is the vector of inflows in the circuit nodes. The vector of pressures P is split into the subvectors: P = (p1,p2, ...,Pl) and Pm = (Pi+1,Pi+2, ...,V*m) w^h the desired and known pressures,

correspondingly. In matrix A the columns correspond to the splitting of the vector P. Vector Y = (y1,y2, ...,yn) denotes heads (the pressure differences at the opposite of the branch with the corresponding number). Equations (35) are nothing else but the first and second Kirchhoff laws. Due to the first law, we have Yj=o Qj = 0-

Now make matrix A % omitting the last m - r rows from A and do the same with the last m - r elements of vector Q to obtain Q. Write down the new system

AX = Q, ATP + ATPm = Y, (ATAT) = AT. (36)

Matrix A is full rank: rank A = r. In additional to the Kirchoff laws, we assume that the flow rates on branch v e [l, 2, • • • ,n] are directed by the rule, which follows from Darcy's law [16]:

hv + yv = h + (Pi(v) - Pj(v)) = sv (Pi(v),Pj(v) ,xv )xv\xv \, (37)

where i(v),j (v) are node numbers, hv E (hi,v2, ••• ,hn)T = H are hydraulic heads at branches, sv(pi(v),Pj(v),Xv) are positive resistance function (below we assume that they are constant: sv = const). In other words, the fluid moves from the node with a bigger pressure to the node with a smaller pressure and the resistance functions are bounded for any values of arguments.

Note that (37) is solvable with respect to xv for any signs of hv+yv. However, sometimes it is assumed that the pressure drops are described the by formula [16]

Lv

yv = Pi(v) - Pj(v) = j bv (s,Xv (s),Pv (s))ds, (38)

0

where Lv is the length of the corresponding line, bv(s,xv(s),pv(s)) are some functions derived from Darcy's laws and resistance functions. Taking into account the integration of the general motion equations with respect to the special variable, we can move from equations (37), (38) to time-dependent equations

t

hv + yv = Qv Xv (t) + so,v Xv (t) + [si,v + Kv J (9v — Xv (s))ds]Xv (t)\Xv (t)\, (39)

a

Lv

hv + yv = QvXv(t) + so,vXv(t) + J bv(t,s,Xv(s),Pv(s))ds, (40)

0

where Qv, so,v, so,v are the parameters that define the process inertia and resistances (the laminar and turbulent flow components), 9v are the values of the automatic regulators, Kv are the regulators amplification coefficients, bv(t,s,Xv(s),pv(s)) are some functions obtained by integrating bv(s,Xv(s),pv(s)). Taking into account (36), (39), (40), we get DAEs with integral operators: ф(х P)

= (R 0) (X(t)) + (So АЛ (X(t)) + (F(t, X)) = (ATP*m(t) + H(t))

\0 0) \p(t)) + \A 0) [p(t)) + { 0 J = V Q(t) J' (41)

*i(X, P) =

= (R 0) (X(t)) + (So Ал (X(t)) + (B(t, X.P)) = (ATPm(t) + H(t)) = \0 0) \P(t)) + \A 0) \P(t)) + { 0 ) = [ Q(t) )' ^

where t E [а, ж), R = diag{Qi, q2, ••• .Qn}, So = diag{so,i, so,2, • • • -so,n},

t t

F (t,X )={[si,i + Kv J (6i — Xi(s))ds]Xi(t)\Xi(t)\, [si,2 + K2 J (02 — X2(s))ds\X2(t)\X2(t)\,

at a

••• , [si,n + Kn / (0n — Xn(s))ds\Xn(t)\ Xn (t)^ , a

B(t,X, P) =

Li L2 L„ 4 ^

J bl(t'S'Xl(s)'Pl(s))dS' J b2(t'S'X2(s)'P2(s))dS' ••• , j bn(t, s, Xn(s),Pn(s))ds Ko o o

Some properties of (40) were studied in [20]. It can be shown that (41) has index 2, at least in the neighbourhood of the initial point.

The extended model includes components of vector Pm = (p*+1,p*+2, ...,P*m)1 which are found when solving (41) and the nonlinear system of differential equations describing mass balance and enthalpy in heat exchange units

d r ]

Jt [Vi(t)p'(P*(t)) + (Vi - Vi(t))p,,(p*(t))\ = xhbx(t) - xhbiX(t), (43)

d

dt

Vi(t)p'(p*(t))1 (p*(t)) + (Vi - Vi(t))p''(p*(t)) 1'(p*(t)) + MiCiT'(p*(t))\ = (44)

xi,bx(t) 1i,bx(t) xi,bix (t)l' (P*(t)) - xi,pr (t)li,pr (t),

where p*(t),Vi(t) are the desired pressures and water values in heat exchange units, xi,bx(t),xi,bix(t),xi,pr(t) are the components of X = X(t) (water or steam flows entering and leaving the node i), i e {l + l,l + 2, ■ ■ ■ , m}, t'(.), p'(.), p''(.) 1 (.), l" (.) are the known functions connecting parameters of heat exchange units, such as temperature, pressures,

Vi, Mi, Ci

heat exchanger units, iitbx, Li>pr are the given enthalpy of the steam entering heat exchanger unit and the enthapy of the heated water.

If we differentiate the second block equation of (41), we obtain a new system ^(X, P) =

0

experiments that employed the implicit Euler scheme showed that the mathematical model of the straight-through boiler with a turbine and a regeneration system (see the hydraulic circuit graph from [21]) has a solution and this solution tends to the stationary mode preset by regulators.

Acknowledgements. This work has been supported by the Russian Foundation for Basic Research, Grants Nos. 8-01-00643, 18-51-54001 and by the Basic Research Program of the Siberian Branch of the Russian Academy of Sciences No. AAAA-A17-117030310443-5.

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Received August 08, 2018

УДК 519.62 DOI: 10.14529/mmpl80407

ЛИНЕАРИЗАЦИЯ ДИФФЕРЕНЦИАЛЬНО-АЛГЕБРАИЧЕСКИХ УРАВНЕНИЙ С ВОЗМУЩЕНИЯМИ В ВИДЕ ИНТЕГРАЛЬНЫХ ОПЕРАТОРОВ И ПРИЛОЖЕНИЯ К МОДЕЛЯМ ТЕПЛОЭНЕРГЕТИКИ

Е.В. Чистякова1, В. Ф. Чистяков1, А. А. Левин2

1Институт динамики систем и теории управления им. В.М. Матросова СО РАН,

г. Иркутск, Российская Федерация

2

Российская Федерация

Моделирование различных естественных и технических процессов часто приводит к системам, которые включают в себя обыкновенные дифференциальные уравнения и связанные с ними алгебраические соотношения. В данной работе изучаются системы квазилинейных интегро-дифференциальных уравнений с вырожденной матрицей в области определения при производной искомой вектор-функции. Такие системы можно рассматривать как дифференциально-алгебраические уравнения, возмущенные операторами Вольтерра. Получены условия разрешимости возмущенных систем и начальных звдвл ДЛЯ них, обсуждается влияние малых возмущений входных данных на решение начальных задач. Рассмотрены варианты линеаризации таких задач на основе метода Ньютона. Обсуждаются модели из области теплоэнергетики, и как пример рассматривается гидравлическая цепь, представленная в виде набора взаимосвязанных элементов, по которым течет жидкость. Численные эксперименты на основе неявной схемы Эйлера показали, что модель прямоточного котла с турбиной и системой регенерации имеет решение, которое стягивается к стационарному режиму, заданному регуляторами.

Ключевые слова: дифференциально-алгебраические уравнения; оператор Фредголь-ма; оператор Вольтерра; начальная задача; условия согласования; индекс.

Литература

1. Reich, S. Differential-Algebraic Equations and Applications in Circuit Theory / S. Reich. -Potsdam: Universität Potsdam, 1992.

2. Eich-Soellner, E. Numerical Methods in Multibody Systems / E. Eich-Soellner, C. Führer. -Stuttgart: Teubner, 1998.

3. Балышев O.A. Анализ переходных и стационарных процессов в трубопроводных системах / O.A. Балышев, Э.А. Таиров. - Новосибирск: Наука, 1999.

4. Vlach, J. Computer Methods for Circuit Analysis and Design / J. Vlach, K. Singhal. - N.Y.: Van Nostrand Reinhold, 1983.

5. Sviridyuk, G.A. Linear Sobolev Type Equations and Degenerate Semigroups of Operators / G.A. Sviridyuk, V.E. Fedorov. - Utrecht; Boston; Köln; Tokyo: VSP, 2003.

6. Brenan, K.E. Numerical Solution of Initial-Value Problems in Differential Algebraic Equations / K.E. Brenan, S.L. Campbell, L.R. Petzold. - Philadelphia: SIAM Publications, 1996.

7. Бояринцев, Ю.Е. Алгебро-дифференциальные системы. Методы решения и исследования / Ю.Е. Бояринцев, В.Ф. Чистяков. - Новосибирск: Наука, 1998.

8. Lamour, R. Differential-Algebraic Equations: a Projector Based Analysis / R. Lamour, R. Marz, С. Tischendorf. - Berlin: Springer, 2013.

9. Лузин, H.H. К изучению матричной теории дифференциальных уравнений /H.H. Лузин // Автоматика и телемеханика. - 1940. - № 5. - С. 3-66.

10. Чистяков, В. Ф. Алгебро-дифференциальные операторы с конечным ядром / В.Ф. Чистяков. - Новосибирск: Наука, 1996.

11. Mehrmann, V. Transformation of High Order Linear Differential-Algebraic Systems to First Order / V. Mehrmann, С. Shi // Numerical Algorithms. - 2006. - № 42. - P. 281-307.

12. Chistyakov, V.F. Linear Differential-Algebraic Equations Perturbed by Volterra Integral Operators / V.F. Chistyakov, E.V. Chistyakova // Differential Equations. - 2017. - V. 53, № 10. - P. 1-14.

13. Булатов, M.B. Один метод численного решения линейных сингулярных систем ОДУ индекса выше единицы / М.В. Булатов, В.Ф. Чистяков // Численные методы анализа и их приложения. - Иркутск: СЭИ СО АН СССР, 1987. - С. 100-105.

14. Краснов, М.Л. Интегральные уравнения. Введение в теорию / М.Л. Краснов. - М.: Наука, 1975.

15. Пазий Н.Д. Локальная аналитическая классификация уравнений соболевского типа: ДИС « * « « ТуТI,Л, « физ.-мат. наук / Н.Д. Пазий. - Екатеринбург, 1999.

16. Меренков, А.П. Теория гидравлических цепей / А.П. Меренков, В.Я. Хасилев. - М.: Наука, 1985.

17. Чистяков, В.Ф. О линеаризации вырожденных систем квазилинейных обыкновенных дифференциальных уравнений / В.Ф. Чистяков // Приближенные методы решения операторных уравнений и их приложения. - Иркутск: СЭИ СО АН СССР, 1982. -С. 146-157.

18. Чистяков, В.Ф. О связи структуры пучка матриц с существованием решений неявной системы ОДУ / В.Ф. Чистяков // Методы оптимизации и исследования операций. -Иркутск: СЭИ СО АН СССР, 1984. - С. 194-202.

19. Чистякова, Е.В. О разрешимости вырожденных систем квазилинейных интегро-дифференциальных уравнений общего вида / Е.В. Чистякова, В.Ф. Чистяков // Вычислительные технологии. - 2011. - Т. 16, № 5. - С. 100-114.

20. Chistyakova, E.V. Investigation of the Unsteady-State Hydraulic Networks by Means of Singular Systems of Integral Differential Equations / E.V. Chistyakova, Nguyen Due Bang // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2016. -Т. 9, № 1. - С. 59-72.

21. Левин, A.A. Об использовании структуры системы нелинейных уравнений, описывающих гидравлические цепи энергоустановок при численных расчетах / A.A. Левин, В.Ф. Чистяков, Э.А. Таиров // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2016. - Т. 9, № 4. - С. 53-62.

сшена Викторовна Чистякова, К^НДИДсХТ физико-математических наук, научный сотрудник, Институт динамики систем и теории управления им. В.М. Матросова СО РАН (г. Иркутск, Российская Федерация), [email protected].

Виктор Филимонович Чистяков, доктор физико-математических наук, главный научный сотрудник, Институт динамики систем и теории управления им. В.М. Матросова СО РАН (г. Иркутск, Российская Федерация), [email protected].

Анатолий Алексеевич Левин, кандидат технических наук, ведущий научный сотрудник, заведующий лабораторией, Институт систем энергетики им. Л.А. Мелен-ТЬбВсХ СО РАН (г. Иркутск, Российская Федерация), [email protected].

Поступила в редакцию 8 августа 2018 г.