On one problem of control of voltage during signal transmission in a long line

In the study of controlled processes of electromagnetic oscillations in long lines with distributed parameters, mathematical problems of control of hyperbolic equations arise [1–7]. There are problems of controlling the signal transmission process when the exact value of external influences is not known, in practice. External disturbances that generate travelling wave in the lines distort the transmitted signal. When studying problems of this kind, you can apply the method of optimizing the guaranteed result [8]. This method is based on the theory of differential games [9].

In this paper, we consider the problem when the control is the limited in magnitude voltage of the signal generator at the left end of the long line. The interference consists of external disturbances and limited voltage at the right end of the conductor. The exact value of the magnitude of the external disturbance acting on the conductor is not known. Its limits of change are known. The purpose of the control process is that at a given moment in time the average value of the voltage value is in a given interval. The average is calculated using the specified function The problem is reduced to a control problem of the same type in the presence of interference by changing the variable. For problems that are considered in the theory of differential games, optimal controls of the players are constructed [10].

Formulation of the problem

Consider a homogeneous, long line length is equal to I with a given resistance R, inductance L, electrical capacity C, leakage factor G [11]. The signal generator is at the left end of the line, and the right end of the conductor is grounded. When transmitting a signal in a long-distance transmission line, the current in the wires is not the same in different sections of the line. It causes a voltage drop in the active resistance of the wires and creates an alternating magnetic field, which in turn induces selfinduction EMF along the entire line. Therefore, the voltage between the wires also does not remain constant along the line [12, 13].

Let's associate a coordinate system with a long line, the X axis of which is directed along the wire. The densities of the aggregate of external disturbances on current and voltage are given by continuous functions f t (t,%),i = 1,2, where we assume % - abscissa of a certain cross-section of the conductor when the long line is at rest. We denote J(t, %) change in the current and V(t,%) voltage of the line at the time t. The control is limited and is the voltage at the left end of the long line. The system of differential equations describing voltage and current fluctuations takes the form of telegraph equations [14].

{ dj ⁽ t,x) dt dV⁽t,x") dt

—

—

1 dV(t,x)

L dx 1 dj⁽t,x)

C dx

—

—

where % e [0, Z], t e [0, p].

-LRt, %) + №%), lv(t,%) + f2(t,%),

This system of equations is considered under the given initial conditions

J(0,%) = 51 (%), V(0,%) = 52 (%),(2)

where the functions 5 ; (%), i = 1,2 are continuous on the segment [0, Z]. By condition, the voltage at the ends of the line is limited. Therefore, they can be written as

V(t,Z) = Ai(t) — «i(t)5, I5I < 1,a1(t) > 0,(3)

V(t, 0) = Л2 (t) — a2(t)f, If I < 1, a2 (t) > 0.(4)

The parameter f is a control, and 5 is a interference.

We assume that the densities f t (%,t) of the magnitudes of external disturbances are not exactly known. The following estimates are known

Ш %) < ft(t,%) < /t(t,%), i = 1,2,=% e [0, Z], t e [0,p].(5)

Where Z(t, %) ; : [0, p] x [0, Z] ^ R, and Z(t, %): [0, p] x [0, Z] ^ R are continuous functions.

Let's set the number к e R, £ > 0. The purpose of the choice of control f (4) is to implement the inequality

|joz(V(p, %)O1(%) +/(p,%)^2(%))d% — k| < £ for any realization of external disturbances, the density of which satisfies the condition (5). Here the functions o;: [0,1] ^ R, i = 1,2 are continuous and satisfy the conditions

O;(0) = O;(Z) = 0.(7)

Formalization of the problem

Let us describe the admissible rule for the formation of control f (4). It means that each moment of time 0 < ^ < p and each possible function V(^,%), J (^,%) is assigned a function f: [^,p] ^ [0,1]. This rule will be denoted

f(t) =Xt,V(tf/ )JG9/)),te [tf,p].

We fix the partition of the segment [0, p]

ш: 0 < to < t1 < - < tj < tj+1 < - < tm+1 = p, where a diameter d(w) = maxo

Assume 5;_o(%) = 5;(%), i = 1,2 at 0 < % < Z. The functions Vo(%, t) и /(t, %) at t_o< t < t₁,

0 < % < Z are defined as a solution to the following problem: ' У Lf.x' .^J -■ %)■/_;'^t,% ^d          ■,._.%,.;. %)

ММ Л   M

Jw(tj,%) = 51_j⁽%⁾, Vu(tj, %) = g2_j⁽%), V_w(t,0)=A(t) —a(t)f,V_w(t,Z) = 0,

^i = ^(t,vш(d,• )jto(d/)). (12) Where j = 0, x e [0, l], t e [t0, t1].

Let the functions V_to(x, t), J^(t, x) be defined for t₀< t< tt-^O Suppose ^-(x) = = Jm( tj_₁,x),g₂j(x) = Vo( tj_₁,x). We construct the functions Vo( t,x),J_to(t,x) at tj_₁< t < tj using formulas (9)–(12).

We say that control (8) guarantees the fulfillment of the set goal (6) if for any number w > e there is a number p > 0 such that for any partition ш with diameter d(w) < p and for any continuous functions ft(t, x), satisfying the condition (5), the inequality

| Jq(Vo(P, x^^x) + Jto(p,x')O2(x)')dx - k\ < w. (13)

Transition to a one-dimensional problem

Let the functions ^(t, x), ф(т, x) at 0 < x < l,0 < т < p be a solution to the following problem:

{9ф(т,х) _ 1 9ф(т,х) 9тL dx

-^⁽т, x), -^⁽t, x).

ф(т,0) = 0,ф(т,Г) = 0,0 <т < p,(15)

ф(0,x) = a1(x),^(0,x) = a2(x), 0 < x < l.(16)

It follows from equality (7) that the matching conditions at the ends of the segment are satisfied. From equalities (5) we obtain that

i0)f1(t,x) ^(p - t,x) dx = P1(t) + Y1(t)P1,\P1\ < 1,

^0lf2(t,x)ф(p — t,x)dx = ^2(t) +72(t)P2, \ib \ < 1.

When at i = 1,2

№) = lJo(7i(t,x) + ft(t,x)^^(p - t,x)dx, YiU) =2lj01^fl(t,x)-ft(t,x)^^(p - t,x)dxl > 0.

Suppose

6to(t) = j0(Jto(t,x) ^(p-t,x) + Vto(t,x)ф(p - t,x)) dx. That

6„(t) = .ТД92^ ^ - t.x^ +^^ф(р - t,x))dx-

-^.x) 'V ..   ■ V'e/o'1' 9/'dx

Let us take into account equations (9), (14) and (17), (18). We get

9to(t) = £1 (t) + Y1 (t)P1 + ^2(t) + Y2(t)P2 - JO (Jto(t, x) d^(Pd__t,x) + Vto(t, x) d*(p-tx^ dx +

+ JO ((-ZdZ^dXtX2 - a2J^(t, x)) ^(p - t, x) + (-C19-2!^- b2Vto(t, x)) ф(р - t, x)) dx,   (20)

at tj < t < tj+1.

Further, integrating by parts and taking into account the boundary conditions (11), (15), we obtain

Jo (d■^dXtX2^(P - t,x)) dx = (A1(t) - a1(t)p)^(p -t,l)-

-(^2(^) - «2(^ЖР -t,^)- /J (Vd(t,x)d^(5_,x2)dx,

Jo' (92!ХХФ(Р "t,x))dx = - JjZ Qw(t, x) d^9_X} dx.

From this and (20) it follows that at tj j₊₁,

£(t) = £1(t) + £2(t) + A1(t),Y(t) = Y1(t) + Y2(t) + «1(t) > 0, ri(t)      . y2(t)      . ai(t) I I ч

vi = —TTP1 +--TTP2 +--TTP, l^/l < 1, j Y(t) Y(t) 2 Y(t) 1

9to(t) = - |1a2(t)^(p - t, 0)luj + Y(t)Vj + p(t) +1A2(t)^(p - t,0).

When

^ = sign ^^a2(t)^(p - t,0))u.(21)

Suppose, that sign0 = 1.

Denote zw(t) = fUt) + fp (^ (r) + -LA2(rMp - r, 0)) dr - k.(22)

That for tj < t < tj+1, zto(t) = -a(t)Uj + b(t)Vj, |uj < 1, |vf| < 1.(23)

Here it is denoted

a(t) = \-La2(tMp — t,0)| > 0,b(t) = y(t) > 0.(24)

Further, taking into account condition (11) and formulas (19), (22), we rewrite inequality (13) in the following form:

|zw(p)|

Termination possibility conditions in a one-dimensional problem

Consider the one-dimensional problem (23), (25). Note that functions (24) are continuous. Let's build broken lines zw (7) = zw(tj) - f. a(r)dr Uj + f. b(r)dr Vj, tj < t < tj+1,(26)

where z^(0) = z(0) is the initial condition. The family of these broken lines is uniformly bounded and equicontinuous, defined on the segment [0, p] [10, p. 46]. By Arzela's theorem [15, p. 104] from any sequence of polygonal lines (26), one can select a subsequence that converges uniformly to the segment [0, p].

Let in (26)

Uj = sign z_to(tj),j = ^m,                                                                  (27)

and the function z(t) at 0 < t < p is the uniform limit of the polygonal sequence zWn(t) (26), for which the diameter of the partition d(mn) ^ 0. Then [10, Theorem 8.1] the inequality

|z(p)| < F(z(0)).

Here it is denoted

F⁽z) = max(|z| + ^(b^ - a⁽r))dr; max₀<_T<_p f^P(b(r) - a⁽r))dr ).

Let the number £ > F(z(0)). Then it can be shown that for any number w > £ there is a number 8 > 0 such that inequality (25) holds for any broken line (26) with the partition diameter d(m) < 8 and control (27).

Let in (26)

Vj = sign z_to(tj),j = 07m,                                                                  (28)

and the function z(t) at 0 < t < p is the uniform limit of a sequence of polygonal lines z_Wn(t) (26), for which d(w_n) ^ 0. Then [10, Theorem 8.2], the inequality

|z(p)| > F(z(0)).

From this it can be obtained that if the numbers £ < w < F(z(0)), then there exists a number 8 > 0 such that то |z_w(p)| > w, for any broken line z_w(t) (26) with the partition diameter d(m) < 8 and with Vj (28).

Thus, it is possible to construct control (8), which guarantees the fulfillment of the set goal (13) if and only if F(z(0)) < £.

From formulas (21), (27) we obtain that f = sign (z a2 (t) f ^(p - t, %) d%) u.

When z is defined by formulas (19) and (22) with given in (22) Ц„ (%, t), Jw (t, %) on У (t, %), / (t, %).

Example

Let the function

^^T^vsinCH a2 U when a = VbC’b =

^(rc+gl)2-lrcg

lc

,

then condition (7) is satisfied. Consider the function

-VLC⁽_RC+GL)r     ,         „        ,

^(t,x)= bze     2          2 sin ^7 x) + ^-^1п(т)^ , that satisfies Eq. (14) and conditions (15), (16). Substitute the function ^(x, t) into formula (19) at

V_to(x, t) = V (t, x). Then it follows from (22) that

z(t) = b2 /J (V(t, x)         /чу + ^Л +

J°                                   ^^2-(t)2     °2-1

+ Jtp ^P(r) + ^-|a(r)e-VI^(^£1r£)^ sin(p - r)) dr - k.

Conclusion

The impact of interference on a long line leads to distortion of information, to a decrease in the quality of transmission and subsequent processing of data until the destruction of the communication lines themselves. However, it is possible to construct such a law of voltage variation at the left end, which will lead to the achievement of the goal with any admissible interference, with known estimates of the set of negative effects and the fulfillment of the necessary and sufficient conditions found. The analyzed example clearly demonstrates how the corresponding law of voltage change is constructed.

The research was funded by Russian Foundation for Basic Research and Chelyabinsk Region, project number 20-41-740027.