Investigation of the unsteady-state hydraulic networks by means of singular systems of integral differential equations Текст научной статьи по специальности «Математика»

MSC 65R20

DOI: 10.14529/mmp 160105

INVESTIGATION OF THE UNSTEADY-STATE HYDRAULIC NETWORKS BY MEANS OF SINGULAR SYSTEMS OF INTEGRAL DIFFERENTIAL EQUATIONS

E. V. Chistyakova, Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Irkutsk, Russian Federation, [email protected],

Nguyen Due Bang, National Research Irkutsk State Technical University, Irkutsk, Russian Federation, [email protected]

Analysis of complex hydraulic networks, electric circuits, electronic schemes, chemical processes etc. often results in a system of interconnected differential and algebraic equations. If the process under study has after-effect, then the system includes integral equations. This paper addresses simulation of hydraulic networks by means of the theory for singular systems of integral differential equations. We present theoretical tools that help investigate qualitative properties of such systems and search for effective methods of solution. A mathematical model for the straight through boiler circuit has been developed and a numerical method for its solution has been constructed. Experimental results showed that the theory for singular systems of integral differential equations performs well when applied to simulation of the hydraulic networks.

Keywords: differential-algebraic equations; integral-algebraic equations; hydraulic network; hydraulic circuit; index; numerical methods.

Introduction

Consider systems of ordinary differential equations

Ax + F(x,t,v) = 0,t e It = (-ж, +ж), (1)

where A is a (q x Q)-matrix with constant components, F(x,t,v) is an n-dimensional vector-function, x = x(t) is a desired vector-function, v e К = (-v0,v0) is a numeric variable. For the sake of simplicity, it is assumed that F(x,t,v) e C^(Re x It x К) and

det A = 0. (2)

Systems (1) satisfying (2) are commonly called differential algebraic equations (DAEs) [1], other frequently used terms include singular systems [2] and algebraic differential systems [3]. Such systems appear in many applications, for example, in electronic schemes, electric and hydraulic networks, mechanical systems etc. [1-5]. In the works [6-8], systems (1),

A

in infinite-dimensional spaces.

By the solution of system (1) for a fixed value of v = v* on T = (а, в) ^ It we mean the vector-function x = x(t) e C1(T) which reduces (1) into identity on T.

Here and in what follows, we use the uniform norm of the m-dimensional vector b e Re and the corresponding norm of the (q x Q)-matrix B = {b ij, i,j = 1, q} which are found by the rules ||b|| = max {\bi\ ,i = 1, 2,..., q} , b = (bi,b2, ...,be)T, \\B|| = max{^J=1 \bj\,i = 1, 2,..., q}, where T stands for conjugation.

t

not lead to misunderstanding.

1. Mathematical Models for Hydraulic Networks

A hydraulic network is a system comprising an interconnected set of discrete components that transport media (such as gas, liquid or a mixture of gas and liquid). The network usually comprises the following components: active components (hydraulic power pack, e.g. pumps), transmission lines (e.g. pipes) and passive components (hydraulic cylinders). Its scheme can be presented in the form of a finite oriented graph with explanatory statements if required. The number of nodes n and the number of lines m are said to be the parameters of the hydraulic network under study.

The hydraulic network graph can be presented by a full (m x n)-matrix A of nodes and lines that identically describes the structure and the orientation of the network: aji = 1, if the line i comes from the node j aji = — 1, if the line i comes into node j aji = 0 if the node j does not belong to the line i. The finite closed set of the oriented lines, where only the start and the end nodes coincide, is called a simple circuit. We say that the line is active if it connects nodes of active components, otherwise we call this line passive [9]. The matrix of nodes and lines should be considered along with the (l x n)-matrix B of

bji = 1

i

bji = — 1 bji = 0 i Introduce the following vector-functions: X(t) = (^x1(t) x2(t)■■■ xn(t)j is the flow

rate in pipelines; P(t) = (p1(t) p2(t) ■ ■ ■ pl(t)^ denotes pressures at nodes; yi(t) = pi(t)

pi+1 (t) is a pressure drop in the i-th pipeline; P*(t) = (p*+1(t) p*+2(t) ■ ■ ■ p*m(t)) denotes

the known pressures; H(t) = (h1(t) h2(t) ■ ■ ■ hn(t)j is the hydraulic heads in pipelines;

Q(t) = (q1(t) q2(t) ■ ■ ■ qm(t)^ is the inflows (outflows) at nodes; hi(t) = yi(t) + hi(t) is a

pressure loss in the i-th line due to friction, yi(t) = hi(t) for a passive line. Here we establish two KirchhofPs laws:

AX(t) = Q(t), BY(t) = 0, (3)

where Y(t) is a vector-function of the pressure drops in pipelines. The first law reflects

m

the fact that for any node j the inflow equals to the outflow: J2xi(t) = qi(t), qi(t) = 0.

j i=1

q

zero: J2yi(t) = 0. In system (3), the number of variables is greater than the number of

q

equations. To amend this, it is common to use so called closing relations which describe the movement of the media along the pipelines:

yi(t) + hi(t) = soiXi(t) + S1ix2(t), soi > 0, S1i > 0, (4)

where s0i, s1i are the pipe frictions corresponding to the stream-line and turbulent flows [9]. After some obvious transformations, we arrive at the closed system of nonlinear equations

( So —AH ) ( X(t) ) + ( S1 \X(t)\X(t) ) = ( H(t) + ¿¡P*(t) ) V A1 0 ) I P(t) ) + \ 0 ) = { Q1(t) ) , (5)

where \X(t)\ X(t) = diag{xi(t)|xi(t)\,x2(t)\x2(t)\, • • • ,xn(t)\xn(t)\}, Q1(t) is a vector-function of inflows at the nodes with unknown pressures, So = diag{so1, so2, • • •, son} and S1 = diag{s11; s12, • • •, s1n}, (AJ AJ) = AT, the matrix A1 is full rank. If we find H(t), P*(t), Q1(t) at a given time t and solve (5) by Newton's method, we obtain distribution for the flows X(t) and the pressures P(t).

Notice that the pipe frictions may depend on X(t), P(t) [10,11]. In the works by A.P. Merenkov, it was proposed to replace x2(t) by \xi(t)\xi(t) in (4). This enables us to carry computations when yi(t) + hi(t) alternates in signs. For example, set soi = 0, [yi(t) + hi(t)] < 0. Then (4) does not have a real solution with respect to x,i, whereupon system yi(t) + hi(t) = su\xi(t)\xi(t) does.

In the monograph [5], the following closing relations were obtained by integrating over the space of the general motion equations: yi(t) + hi(t) = ri(t)cci(t) + s0ixi(t) + s1ix2(t), ri(t) > K = const > 0, t E It, i = In accordance with the technique developed by A.P. Merenkov, we replace this with

yi(t) + hi(t) = ri(t)x i(t) + soixi(t) + su\xi(t)\xi(t), (6)

which corresponds to the DAE

( R0t) 0 ) ( Xt ) + ( Ao - AJ ) ( Xt) ) + ( S1 \X(0) X(t) ) =

= ( H (t) + AjP* (t)

V Q1(t)

where R(t) = diag{r1(t), r2(t), • • • ,rn(t)} are the momentum parameters that depend on the geometric features of the given part of the circuit. Some pipelines may have regulator components and, therefore, taking into account (6), the line equation has the form

t

yi(t) + hi(t) = ri(t)xi(t) + soixi(t) + [su + Ki j(i>i(xi(r)) - 9i)dr]xi(t) \xi(t)\ , (8)

o

where Ki is a proportionality factor, 9i is a value for the regulator component, ^i(•) is the regulator function.

Hence, taking into account (8), system (7) with automatic regulator components can be presented in the form of a singular vector-valued integral differential equation

W(y) := R(t)y + ^(y, Vy,t) = 0, t E T =[a^] C [0, x>), (9)

t

where y = (XJ PT)T, Vy = JK(t,r,y(r))dr is the Volterra operator, R(t) =

o

diag(R(t), 0).

Example 1. Consider the straight-through boiler circuit (see Fig. 1). Along the circuit, the water is pumped at node 5, then it is heated and turns into steam. Afterwards, the steam is heated to 545oC and passes to the turbine through the valve. It is shown on Fig. 1 that the circuit has 6 segments: x1, x2, x3, x4, x5, x6 ■ Water flows through the segments x1 , x3

(7)

Fig 1. Diagram for the straight-through boiler circuit

segment x2 has the water-steam mixture coming through it, whereas steam passes through x4 and x6. The control valve of the turbine is located at x5. The inflow q(t) simulates the media density variation at the boiling segment.

Write down the flow rate equations using the first IvirchhofFs law for the nodes Pi,P2,P3,P4 (Fig. 1):

xi — x2 — x3 = 0; x2 + x3 — x4 = q; x4 — x5 = 0; x5 — x6 = 0. (10)

The pressure loss equations have the form

t

P*5 — Pi = rixi + [sii + Ki j(xi(t) — &i)dr] |xi| xi; Ki = 0, 003, pi — p2 = ^x2 + Si2 |x21 x2;

0

t

Pi — P2 = r:ixx3 + [si3 + K3 j(x3(r) — 03)dr] |x3| x3;

0

K3 = 0, 001, P2 — P3 = T3xc3 + Si4 |x4| x4/(p2 + P3); P3 — P4 = T5x5 + Si5 |x5| x5/(P4 — 0, 09P3); P4 — P*a = r6x6 + Si6 |xa| xe/(P6 + P4). (11) In (7) we have:

0 \

0

0. 0 .

0

—1

Values for the regulators are set: 0i = 75, 03 = 9. The flow rate is measured in kilograms per second, pressure is measured in units of atmosphere, time is measured in seconds.

( —1 0 0 0 \

1 —1 0 0

1 —1 0 0

0 1 —1 0

0 0 1 —1

K 0 0 0 1 )

1 0 0 0

0 0

2. Basic Definitions

Before we start to analyse the system obtained, let us introduce the essential for the future investigation notation.

Definition 1. Let

Ai(x) := Ax + &(x,t) = 0, Ф(х,t) = F(x,t,v*), t e It = (-то, +то), (12) (2 2)

where Ф(х, t) e Ct^ (Re x It), v* e N = (—v0, v0) ¿5 a fixed value of the parameter, and, as in [12], the following conditions hold:

1. All solutions x(t; t0; x0), where x0 e M, M С Re is some mapping, are defined when t0 < t < то;

2. There exists a unique solution n(t), t e It bounded along the entire real axis: sup {||n(t)|| ,t e It} = 1 < то;

3. For each solution x(t; t0; x0), [x(t; t0; x0) — n(t)] = 0.

Then, system (12) possesses the convergence property.

By combining definitions from [2,3,13], introduce the following notion and a statement.

Definition 2. 1. The sum XA + B, where A, B are matrices of the equal dimensions, X is a scalar (generally, complex) parameter, is called a matrix pencil;

2. The pencil of (q x q)-matrices XA + B is regular if there exists a value for the parameter X0 such that det(X0A + B) = 0;

3. The smallest possible integer positive number k, starting with which

rankWk = rankWfc+1, W = [(XoA + B^A] , (13)

is called an index of the matrix pencil XA + B;

4- The regular matrix pencil XA + B is said to satisfy the rank-degree criterion if

deg det [XA + B] = rankA = r.

Lemma 1. If the pencil of square matrices XA + B is regular, then there exist such matrices P and Q with constant components that

P(XA + B)Q = x(e °) + ( 0 El) ,

where N is a nilpotent matrix (i.e. for some j > k, Nj = 0).

DAEs possess a complex inner structure. To measure this complexity to some extent, it is common to use the notation of index. Various definitions of index can be found in literature (see, e.g., [1], [3], [14]), however, below we use the definition from [14].

Definition 3. Let there exists a differential operator A^(z) :=

Zj=0 Lj(t, z,..., z(l~2^)(d/dt)j, where Lj e C(T x Re(—2)) are (q x Q)-matrices with the property Al(z) о A1(y) = A(z,t)Z + $(z,t) ^z = z(t) e Cl+1(T), where det A(u,t) = 0 y(v,t) e Re x T. The smallest possible l is said to be the index of system (12) on T. When l — 2 < 0, matrices Lj do not depend on t and z.

For the time-invariant system (12)

Ax(t) + Bx(t) = f (t), (14)

where the pencil of square matrices XA + B is regular and f (t) is a known vector-function,

l

we can assume, by means of Lemma 1, Al = Qdiag{Ed, (-N)j(d/dt)j+1}P. In other

j=0

l=k

Definition 4. Let there exist an integral differential operator

i

A i (z):=J2 Lj z(l-2),V, Vi, -> Vi-2)(d/dt)j,

j=0

where Lj E C(T x R^(21-2)) are (g x g)-matrices, Vj are the Volterra operators with

the property A i (z) ◦ W (z) = K(z,t, V, Vi,..., V-)z + ^(z,t, V, Vi,..., V-i) Vz = z(t) E

Cl+1(T), where det n(v,t,w,wi,..,wi-2) = 0 V(v,t,w,wu ..,w-2) E Rn(2i—2) x T). The

smallest possible I is called the index of system (9) on T. If l — 2 < 0, the matrices Lj

t

depend only on t. In some cases it is safe to assume Vjz = f [djK(t,r,y(r))/dtj}dr.

0

3. Linear DAEs with the Convergence Property

Lemma 2. If system (12) possesses the property of convergence and §(x,t) = §(x,t + u), then the bounded solution n(t) is also u-periodic with respect to t.

Proof. Indeed, let &(x,t) = &(x,t + u). We have A(d/dt)n(t + u) = Arq(t + u) = &(n(t + u),t + u) = §(n(t + u), t). Therefore, n(t + u) is a bounded solution on It. By definition, n(t) is unique. He nee, n(t + u) = n(t).

□

Theorem 1. Let system (14) satisfy the conditions:

1. The matrix pencil XA + B is regular;

2. f(t) E Ck(It), ||f(j)(t)\ < k, k = const > 0, j = 0,k — 1, f(0)(t) = f(t), where k is the index of XA + B;

3. All roots of the polynomial det(XA + B) have negative real parts. Then, system (14) possesses the convergence property and

k-1

n(t) = Y, Sj f U)(t)+ Gi(t — t)f (t)dr. (15)

j=0

Here Gi(v), Sj are sorne (g x g)-matrices defined below.

Proof. Multiply system (14) on the left by P and introduce the change of variables x = Qy, where P and Q are the matrices from Lemma 1. We obtain

n)K0 ¿d)y=rn = (f$)-fv = prn-

(i6)

Introduce the splitting y = (yj, yJ)T ■ Consider the first subsystem of (16) yi + Jyi = f]_(t). In accordance with [12], under condition 3 of the theorem all real numbers of the matrix J

formula

t

yi(t)= i e—J(t—T)fi(T)dT.

For the second subsystem Ny2 + y2 = f2(t), a simple substitution verifies that

k—1

y2(t) = f2 (t) + J>N )j f2J)(i)-j=i

Then, Gi(v) = Qdiag {e J, 0} P; Sj = Qdiag {0, (-N)j}.

□

Similarly, we can prove a more general statement using (16).

Theorem 2. Let system (14) satisfy conditions 1 and 2 of Lemma 3 and all roots of the polynomial det(AA + B) have non-zero real parts.

Then, there exists some matrix G(t) E C^(0 < \t\ < to) with the properties:

1. G(+0) — G(-0) = M, where M is a (q x g)-matrix defined below;

2. ||G(t)|| < ce—a\t\, (t = 0), where c and a are positive constants;

3. AG(t) + BG(t) = 0 t = 0; 4- The expression

k-i

n(t) ^ Sj fj (t) + G(t — t)f (t)dr (17)

j=0 J

is a unique bounded solution of (14) on It.

Lemma 3. For the bounded solution n(t) of system (14) under the conditions of Theorems 1 and 2, the following estimation holds

k-1

sup

< к sup ^^ Ц/(t) У , к = const > 0.

t t . n 3=0

If the free term f (t) of (14) is an u-periodic function f (t + u) = f (t)(u > 0), then the bounded solution n(t) is also u-periodic.

4. Quasilinear DAEs with the Convergence Property

Definition 5. (see, e.g., [2]) The (g\ x g)-matrix A- is said to be semi-inverse for the (q x Qi)-matrix A if AA-A = A.

The semi-inverse matrix is defined for any (q x Q^-matrix A and techniques for its computation are well developed [2].

Lemma 4. [17] Let:

1. In system (1) F(x,t,v) = F(x);

2. F (x) G C2(U ), U = {x : \\x - a\\ < p}, where a E Re and F (a) = 0;

3. The matrix pencil XA + Fx(a), where Fx = dF(x)/dx, satisfies the rank-degree criterion. Then, the first r roots of the polynomial det [XEe — dF(a)/dx] are equal to the roots of det [XA + Fx(a)]; and the rest q — r roots are equal to -1, where

F (x) = — [A + SFx(x)]-1 F (x),S = Ee — AA-.

Lemma 5. Let in system (12) &(x,t) E C(2'2)(RQ x It). Then, system (12) has index 1 if det[XA + ^x(x,t)} = ar(x,t)Xr + ..., where r = rankA, = /(x,t)dx, and ar(x,t) = 0 y(x,t) E RQ x It.

Proof Consider the equality

P [A + SP-1P$x(Qy, t)] Qy + P$(Qy,t) + P$t(Qy, t) = 0, where = d$/dt, x = Qy, PAQ = diag {Er, 0} , det(PQ) = 0. It is clear that

det( B.2E;,t) B2,0y,t))=det P [A + SP~lP*x(Qy-t)]Q,

where ( B"(y,t) ^^ ) = P&x(Qy,t)Q. Since the matrix S = Ep - AA- is a V B2i(y,t) B22(y,t) ) Q

S2 = S

such P that PSQ = diag {0,EQ-r}. It follows that ar(x,t) = det B22(Q-1x,t)/det(PQ). Therefore, by applying EQ + S(d/dt) to (12) and multiplying by [A + S^x(x,t)]-1, we obtain a system in the normal form

x = F(x, t) = - [A + S$x(x, t)]-1 [&(x, t) + $t(x, t)]. (18)

□

Theorem 3. Let system (18) satisfy the conditions:

1. sup {F(0,t), t E It} = k < to;

2. The biggest eigenvalue Xmax(x,t) of the matrix \Fx(x,t) + Fj(x,t)] /2,Fx(x,t) = dF/(x,t)dx is such that Xmax(x,t) < t, where t is a positive number.

Then, system (18) possesses the convergence property.

Proof. System (18) satisfies that conditions of the theorem from [12]. Therefore, it has a

unique bounded solution n(t) that attracts all solutions x(t; t0; x0) of system (18). Since all solutions of (12) are the solutions to (18), they are also attracted to n(t)- The set M from Definition 1 is defined by rankA = rank(A|F(x0, t0)) and, in virtue of ar(x,t) = det B22(Q-1x,t)/ det(PQ) = 0, is nonempty.

□

Let us give an auxiliary statement from [17] for a nonlinear system with a distinguished linear part.

Theorem 4. Let there be given a system

Ax(t) + Bx(t) = <p(t,x), (19)

and the following conditions be satisfied:

1. The matrix pencil XA + B satisfies the rank-degree criterion;

2. All roots X\, X2,..., Xr of the polynomial det(XA + B) have nonzero real parts;

3. <(t,y) E Cl(It x \\y\\ < œ) u supt \\<(t, 0)| = Y < &;

4- The Lipschitz condition holds: \\<(t,y) — <(t,z)\\ < L \\y — z\iy,z E Re, and L is

sufficiently small.

Then:

1. There exists a solution n = n(t) to (19), defined and bounded on It, and n(t) = n(t + u) if $(x, t) = t + u);

2. System (19) possesses the convergence property if ReAj < 0, (j = 1,... ,r).

Theorems similar to Theorems 1, 2, 4 for the infinite-dimensional case can be found in [18,19].

5. Investigation of the Hydraulic Circuit. Numerical Experiment

Let us establish some qualitative properties of system (7).

Si

P(t)

Proof. Consider the product

y• **">=(An 0)(X) + (B—l0t)So ^r*)(X)+

+ ( R—l(t)Si \X(t)\X(t) ) Y = ( R—l(t) 0 ) + V 0 J ' Y V 0 (d/dt)Em ) '

(20)

where Q(X,P) is an operator of system (7). If S1 in (7) does not depend on P, then, transforming (20) by means of the operator Y2 = ^ (d/dit) A (d/dt)E ) ' We ^

En 0

Y 0 Yl 0 Q(x>p) ^ U(x) A1R-i(t)Aj)\ P ) + ■ ■ ■ '

where U(x) is some block of appropriate dimension. The matrix A\ is full rank if the matrix R-1(t) is diagonal with positive elements. Therefore, AiR-1(t)A^ is nonsingular for all t E T. The product Y2 о Y1 is the second order differential operator.

□

Below, we will need the following statement. Theorem 5. Let the system

S(t)y + Г(у, Wy,t) = 0, t E T =[а,в], (21)

t

where Wy = f K(t,r,y(r))dr is the Volterra operator, satisfy the conditions:

1. S(t) E C2(T), Г(у, t, z) E C2({y : ||a - y\\ < pi} x {z : ||z|| < p2] x T, pi,p2 > 0), a E RK, K(t, т, y) E C1(T x T x {y : Ца - yW < pi});

2. rank S(t) = const, t E [a, a + p1];

3. rank S(a) = rank {S(а)|Г(а, 0,a)};

4- The polynomial XS(a) + D(a, 0, a), where D(y,z,t) = dr(y, z,t)/dy, satisfies the rank-

degree criterion.

Then:

1. the solution y(t) E C1[a,a + e] to system (21) with the initial data y(a) = a is defined

on some segment [a, a + e],e > 0;

2. For the sufficiently small h < h0, there exist solutions to the difference scheme

S (t+ ) Vi+1h Vi +r(t*1,y*1,h £ K(t3, tj ,yj yo

■ji'-jiUj) I ~ u> yo — (222)

where h = e/N, ti+i = a + ih, i = 0, N — 1, Tj = a + jh, and the following estimation holds:

hi — y(U)\\ < Ch, C = const > 0. More over, (22) can be replaced by the non-iterative scheme

S (ti+i) Wl+1h Wi + r(& + V(Ci)(wz+i — Wi) — 0, & — [ii+i,wt,hY^K(ij ,Tj ,wj I , (23)

(^ti+i,Wi,h ^ K(tj ,Tj

with the estimation \wi — y(ti)\ < Clh, C1 = const > 0.

Theorem 5 is a special case of statements proved earlier in [20].

Lemma 7. Let system (7) H(t) = 0, P*(t) = 0, Ql(t) = 0, t E It, have no regulator-components. Then (7) possesses the convergence property.

Proof. Define the set M from Definition 1. It is clear that under the current conditions AlX(t0) = 0. Moreover, system (7) does not satisfy the forth condition of Theorem 5, meanwhile, system Yl o Q(X,P) = 0 does. By direct calculation, it can be shown that det[AS(a) + D(a, 0,a)] = [AiR-l(t)Aj]An + • • • , and the forth condition is satisfied for t

equality holds at the point t0

[AlR-l(t)A]]P(t) = —AlR-l(t)Sl \X(t)\X(t). (24)

The set of solutions to AlX(t0) = 0 is nonempty. Since the matrix AlR-l(t)AJ is nonsingular, for any X(t0) = 0 there exists an allowable initial value P(t0) = 0. Hence, the set M is nonempty, and, by virtue of Theorem 5, there exists a neighbourhood of t0 where the solution to (7) is defined for allowable initial values.

Let y(t) = ^ ^t) ^ be a solution to (7). Consider the inner product

n J n

(y, Q(X,P)) = (1/2) Y, ri(t)dlxKt)] + Y, [so.x2(t) + shix2(t) \Xi(t)\] = 0. (25)

i=l i=l

Here, we took into consideration that R(t) is diagonal and —(X, AjP) + (P, AlX) = 0. Assume that (7) has a nonzero bounded solution n(t) defined on It. Equation (25) implies that for any nonzero solution

n d

Yri(t) Jt X2 (t)] < 0. (26)

i=l

Therefore, when t ^ — œ we have ||rç(t)|| ^ œ. We arrived at a contradiction. The nonzero solution is a unique bounded solution on It. The validity of the lemma follows from (24) and (26).

□

Let us now present results of numerical experiments. It follows from (23) that

= M-1(£)[[S(ti+i) + hV(Cl)]wl - hr(&)] , where M(&) = S(tm) + hD(&).

The forth condition of Theorem 5 implies that matrix Mis invertible for the sufficiently small h. In Example 1, set 61 = 75, 93 = 9 R = diag {1; 1; 1; 1; 1; 1}, Xo = (xio; X20; ...,x6c)T = (65; 45; 20; 65; 65; 65)T, p* = (p5; p6)T = (200; 0,035)T, Po = (pio;P20;P3o;P4o)T = (185; 170; 160; 120)T, q(t) = 0,

S0 = 0,S1 = diag {s11; s12;...; s16} = diag {0,0005; 0, 0014; 0, 0444; 0,1825; 0, 4508; 0, 3336}.

In scheme (23), we set h = 0, 001 t E [0..50],

.......X..........

\

.4..............

X Х2 Xj — Xq — Xg — хб

X?

te [0..50]

Fig 2. Diagram for the flow rates in pipelines with regulators

Fig. 2 shows that the regulators brought the flow rates to the given values. This work has been supported by the Russian Foundation for Basic R.esearch, grant No. 15-01-03228-0,.

References

1. Brenan K.E., Campbell S.L., Petzold L.R. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Classics in Applied Mathematics. Vol. 14. Philadelphia, SI AM, 1996. 256 p. DOI: 10.1137/1.9781611971224

2. Boyarinsev YU.E. Regulyarnye i singuliyarnye sistemy lineynyh obyknovennyh differencial 'nyh uravneniy [Regular and Singular Systems of Linear Ordinary Differential Equations]. Novosibirsk, Nauka, 1980. 222 p.

3. Chistyakov V.F. Algebro-differencial'ye operatory co konechnym yadrom [Algebraic-Differential Operators with Finite Kernel]. Novosibirsk, Nauka, 1996. (in Russian)

4. Ushakov E.I. Staticheckaya uctoichivoch elektricheskih sistem [Static Stability of Power Systems]. Novosibirsk, Nauka, 1988. (in Russian)

o. Balyshev O.A., Tairov E.A. Analis perehodnyh i stacionarnyh processor v truboprovodnyh sistemah [Analysis of Transient and Steady-State Processes in Pipeline Systems]. Novosibirsk, Nauka, 1999. 164 p. (in Russian)

6. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type. Russian Academy of Sciences. Izvestiya Mathematics 1993, vol. 57, no. 3, p. 601-614. DOI: 10.1070/IM1994v042n03ABEH001547

7. Sviridyuk G.A., Zagrebina S.A. Verigin's Problem for Linear Equations of the Sobolev Type with Relatively p-Sectorial Operators. Differential Equations, 2002, vol. 38, no. 12, pp. 1745-1752.

8. Manakova N.A., Bogatyreva E.A. On the Solution of the Dirichlet Problem for the Cauchy - Barenblatt - Gilman. The Bulletin of Irkutsk State University. Series: Mathematics, 2014, vol. 7, pp. 52-60. (in Russian)

9. Merenkov A.P., Hasilev V.Ya. Teoriya gidravlichekih cepey [Theory of Hydraulic Circuits], Moscow, Nauka, 1985. 277 p. (in Russian)

10. Chistyakova E.V., Levin A.A., Chistyakov V.F. [Analysis of the Hydraulic Circuits in the Quasi-Unsteady Mode]. Matematicheskoye modelirovanie truboprovodnyh sistem energetiki: trudy XII vserossiiskogo nauchnogo seminara s mezhdunarodnym uchastiem "Matematicheskie rnodeli i metody analiza i optimalnogo sinteza razvivayushchihsya truboprovodnyh i gidravlicheskih sistem1'. Irkutsk, MESI SB RAS, 2010. pp. 17-27

11. Levin A.A., Tairov E.A., Chistyakov V.F. [Solutions of Powerflow in the Coal Pulverization System]. Matematicheskoye modelirovanie truboprovodnyh sistem energetiki: trudy XII vserossiiskogo nauchnogo seminara s mezhdunarodnym uchastiem "Matematicheskie rnodeli i metody analiza i optimalnogo sinteza razvivayushchihsya truboprovodnyh i gidravlicheskih sistem". Irkutsk, MESI SB RAS, 2010. pp. 27-39

12. Demidovich B.P. Lekcii po teorii matematicheskoy ustoychivosti [Lectures on Mathematical Theory of Stability]. Moscow, Nauka, 1967.

13. Gantmacher F.R. The Theory of Matrices. N.Y., Chelsea Publishing Company, 1959.

14. Chistyakov V.F. [On a Rough Index of Nonlinear Differential-Algebraic Systems]. Trudy XII Baikal'skoy mezdunarodnoy konferencii "Metody optimizacii i ih prilozeniya" II, Irkutsk, 2001, pp. 213-218.

15. Hassard B.D., Kazarinoff N.D., Wan Y.-H. Theory and Applications of Hopf Bifurcation. Cambridge, London, N.Y., Rochelle, Melbouren, Sydney, Cambridge University Press, 1981.

16. Bulatov M.V. Transformation of Algebro-Differential Systems of Equations. Computational Mathematics and Mathematical Physics 1994, vol. 34, no. 3, pp. 301-311.

17. Chistyakov V.F., Chistyakov E.V. On the Existence of Periodic Solutions of Differential-Algebraic Equations. Sib. Zh. Ind. Mat., 2006, vol. 9, no. 3, pp. 148-158.

18. Keller A.V. Issledovanie ogranichennyh risheniy lineynyh uravneniy tipa Soboleva [The study of Bounded Solutions of Linear Equations of Sobolev Type. Dissertation of the Candidate of Physical and Mathematical Sciences]. Chelyabinsk, 1997, 115 p.

19. Sagadeeva M.A., Fedorov V.E. Solutions, Bounded on the Line, of Sobolev-Type Linear Equations with Relatively Sectorial Operators. Russian Mathematics (Izvestiya VUZ. Matematika), 2005, vol. 49, no. 4, pp. 77-80.

20. Chistyakov E.V., Chistyakov V.F. On the Solvability of Degenerate Quasilinear Systems of Integro-Differential Equations of the General Form. Computational Technologies, 2011, vol. 16, no. 5, pp. 100-114. (in Russian)

Received August 4, 2015

УДК 519.711.3 DOI: 10.14529/mmpl60105

ИССЛЕДОВАНИЕ МОДЕЛЕЙ НЕСТАЦИОНАРНЫХ ГИДРАВЛИЧЕСКИХ ЦЕПЕЙ НА ОСНОВЕ ТЕОРИИ ВЫРОЖДЕННЫХ СИСТЕМ ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

Е.В. Чистякова, Нгуен Дык Банг

Анализ сложных гидравлических систем, электрических цепей, электронных схем, химических реакций и т.д. часто приводит к необходимости решать системы взаимосвязанных дифференциальных и алгебраических уравнений. Если процесс обладает последействием, то такие системы могут включать интегральные уравнения. Данная работа посвящена моделированию гидравлических цепей с помощью вырожденных интегро-дифференциальных уравнений. Приводятся теоретические результаты, с помощью которых изучаются качественные свойства рассматриваемых систем и строятся эффективные численные методы. В работе рассмотрена модель гидравлической цепи пароводяного тракта прямоточного котла, предложен численный метод решения. Экспериментальные результаты показали, что теория вырожденных систем интегро-дифференциальных уравнений хорошо работает при моделировании процессов, протекающих в гидравлических системах.

Keywords: дифференциально-алгебрачиеские уравнения; интегро-алгебраические уравнения; гидравлические цепи; индекс, численные методы.

Литература

1. Brenan, К.Е. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations / K.E. Brenan, S.L. Campbell, L.R. Petzold // Classics in Applied Mathematics. -Philadelphia: SIAM. - 1996. No 14. - 256 p.

2. Бояринцев, Ю.Е. Регулярные и сингулярные системы линейных обыкновенных дифференциальных уравнений / Ю.Е. Бояринцев. - Новосибирск: Наука, 1980. - 222 с.

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

4. Ушаков, Е.И. Статическая устойчивость электрических систем / Е.И. Ушаков. - Новосибирск: Наука, 1988.

5. Балышев О.А. Анализ переходных и стационарных процессов в трубопроводных системах (Теоретичесие и экспериментальные аспекты) / О.А. Балышев, Э.А. Таиров. -Новосибирск: Наука, 1999. - 164 с.

6. Свиридюк, Г.А. Квазистационарные траектории полулинейных динамических уравнений типа Соболева / Г.А. Свиридюк // Известия РАН. Серия математическая. - 1993. -Т. 57, № 3. - С. 192-207.

7. Свиридюк, Г.А. Задача Веригина для линейных уравнений соболевского типа с относительно р-секториальными операторами / Г.А. Свиридюк, С.А. Загребина // Дифференциальные уравнения. - 2002. - Т. 38, № 12. - С. 1646-1652.

8. Манакова, Н.А. О решении задачи Дирихле - Коши для уравнения Баренблатта - Гиль-мана / Н.А. Манакова, Е.А. Богатырева // Известия Иркутского государственного уни-верситетата. Серия: Математика. - 2014. - Т. 7. - С. 52—60.

9. Меренков, А.П. Теория гидравлических цепей / А.П. Меренков, В.Я. Хасилев. - М.: Наука, 1985. - 277 с.

10. Чистякова, Е.В. Расчет гидравлических цепей в квазистационарном приближении / Е.В. Чистякова, A.A. Левин, В.Ф Чистяков // Математическое моделирование трубопроводных систем энергетики: труды XII всероссийского научного семинара с международным участием «Математические модели и методы анализа и оптимального синтеза развивающихся трубопроводных и гидравлических систем:». — Иркутск, ИСЭМ СО РАН. - 2010. - С. 17-27.

11. Левин, A.A. Расчет потокораспределения в системе пылеприготовления ТЭС / A.A. Левин, Э.А. Таиров, В.Ф. Чистяков // Математическое моделирование трубопроводных систем энергетики: труды XII всероссийского научного семинара с международным

участием «Математические модели и методы анализа и оптимального синтеза разви-

:

2010. - С. 27-39.

12. Демидович, Б.П. Лекции по теории математической устойчивости / Б. П. Демидович. -М.: Наука, 1967.

13. Гантмахер, Ф.Р. Теория матриц / Ф.Р. Гантмахер. - М.: Наука, 1967.

14. Чистяков, В.Ф. О грубом индексе нелинейных алгебро-дифференциальных систем / В.Ф. Чистяков // Труды XII Байкальской международной конференции «Методы оптимизации и их приложения», II Иркутск, 2001. - С. 213-218.

15. Хэссард, Б. Теория и приложения бифуркации рождения цикла / Б. Хэссард, Н. Каза-ринов, И. Вэн. - М.: Мир, 1985.

16. Булатов М.В. О преобразовании алгебро-дифференциальных систем уравнений / М.В. Булатов // Журнал вычислительной математики и математической физики. -1994. - Т. 34, № 3. - С. 360-372.

17. Чистякова, Е.В. К вопросу о существовании периодических решений дифференциально-алгебраических уравнений /Е.В. Чистякова, В.Ф. Чистяков // Сибирский журнал индустриальной математики. - 2006. - Т. 9. - № 3. - С. 148-158.

18. Келлер, A.B. Исследование ограниченных решений линейных уравнений типа Соболева:

физ.-мат. наук / A.B. Келлер. - Челябинск, 1997. - 115 с.

19. Сагадеева, М. А. Об ограниченных на прямой решениях линейных уравнений соболевского типа с относительно секториальными операторами / М.А. Сагадеева, В.Е. Федоров // Известия вузов. Математика. - 2005. - № 4. - С. 81-84.

20. 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.

шгена

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

Нгуен Дык Банг, аспирант, Национальный исследовательский Иркутский государственный технический университет (г. Иркутск, Российская Федерация), [email protected].

Поступила в редакцию 4 августа 2015 г.