RU2690234C1 - Method of automatic group target allocation of fighters based on priority of targets - Google Patents

Abstract

FIELD: aviation.SUBSTANCE: invention relates to aircraft control systems and can be used in a complex of functional programs for control and guidance of aircraft systems for assigning targets to interceptors when aircraft groups are confronted. Disclosed method enables to determine purpose of targets for interceptors in a group confrontation. Method is based on using an interception efficiency functional. Functional is determined for each pair of "interceptor-target" and takes into account both time and energy costs for interception. Then among a certain class of trajectories one is selected, which ensures minimum of specified functional at fulfillment of restrictions on maximum speed and acceleration of interceptor. Intercept matrix is composed of obtained minimum values of functionals. For important purposes, probability of interception of a target by one interceptor is determined and required probability of their interception is set. Required number of interceptors for each of the important targets is determined based on the required probability of their destruction. Then, such distribution of targets by interceptors is sought, which minimizes total quality functional and which provides required probability of interception of important targets. Depending on the number of interceptors, the minimization problem is formulated either as an assignment problem with an interception matrix or with some other matrix obtained from an interception matrix, or in the form of a problem of searching for a minimum value flow in a graph. Solving this task determines assignment of targets to interceptors together with corresponding trajectories of interception.EFFECT: technical result is providing automatic optimum assignment of targets to interceptors based on priority of targets.1 cl, 3 dwg

Description

The invention relates to aircraft control systems (LA) and can be used in a complex of functional control and guidance programs for aircraft systems for assigning targets to interceptors while confronting aircraft groups.

An analysis of the characteristics of the conduct of hostilities within the framework of the strategy of contactless network-centric wars [1] suggests that the main type of aerospace confrontation is group use of both means of attack and defense. In this regard, the assessment of the capabilities of the group of LA to solve the problems of group fighting is very important.

на j-е цели

, изложенный в [2], основан на задании вероятностей поражения каждой цели каждым типом оружия. Для всякого возможного назначения типов оружия на цели определяют вероятность их выживания. Выбирают такое назначение x_ij, которое минимизирует ожидаемую суммарную опасностьThe analogues [2, 3] of the present invention mainly consider the selection of targets that are best for interception, based on solving a very complex nonlinear integer programming problem based on the calculation of the probabilities of hitting individual targets by individual objects. The method of assigning various types of weapons

on the jth goal

, stated in [2], is based on the task of probabilities of hitting each target with each type of weapon. For any possible assignment of types of weapons on targets determine the probability of their survival. Choose an assignment x _ij that minimizes the expected total hazard.

unaffected targets where

V _j is the hazard coefficient of the jth goal,

q _ij - the probability of survival of the j-th target when using the i-th type of weapon,

x _ij is the number of copies of the i-th type of weapon assigned to the j-th target.

From a mathematical point of view, such a task is a complex nonlinear integer programming problem. Finding its exact solution is almost impossible when considering two dozen objects [2].

When targeting to maneuvering targets, it is necessary to know the lifetime of the hypotheses of changing the target speed (usually a few seconds). After this interval, it is necessary to solve the problem of target distribution again and formulate the accompanying control law.

In addition, among the goals, some can be very important, they need to be destroyed with a fairly high degree of probability.

The aim of the invention is to develop a simpler method of target distribution in a group confrontation, the effectiveness of which is determined by taking into account the fulfillment of the real limits on interception during the flight to the target chosen for the defeat. This method should take into account the possibility of having important targets that need to be intercepted with a high degree of probability.

As a prototype, a target distribution method was chosen, which was set forth in [4], in which all goals are assumed to be equivalent.

The specifics of the problem being solved predetermines the need to take into account both the time and energy costs of intercepting.

Proposed in the prototype approach to the formation of the proposed trajectory of interception, taking into account these requirements, based on the use of functional

time and energy costs for interception for each pair of interceptor with the number n and the target with the number m, where

T is the total flight time of the interceptor along the path,

K is a constant coefficient chosen for reasons of balance between the time of interception and the cost of flight with acceleration,

J is the interceptor acceleration vector,

t is the acceleration time.

In (2), the second term takes into account the costs of generating the interceptor control signal.

Based on the minimum of this functional, found among a certain class of trajectories, an interception efficiency matrix is constructed, the procedure for constructing which is given below. On the basis of the obtained matrix with the help of a known algorithm, the optimal distribution is found, ensuring a minimum of the total quality functional.

among all the possible assignments of m (n) targets to interceptors.

The technical result that can be obtained from the use of the present invention is the ability to automatically optimally assign targets to interceptors, taking into account the priority of targets, which reduces the information load on the operators (navigators pointing).

The stated technical result, which can be obtained from the implementation of the proposed technical solution, is achieved by solving the problem of finding the optimal value of the total quality function, based on time and energy costs, taking into account the real limitations on the ability of interceptors.

The possibility of achieving a technical result is due to the following reasons:

- the individual functional efficiency of interception (2) for each pair of interceptor-goal is based on the consideration of time and energy costs, taking into account the real limitations on the capabilities of interceptors;

- the task of finding the minimum value of the interception functional (2) is reduced to the problem of finding the roots of a polynomial, the solution of which is known [5];

- depending on the ratio of the number of targets and interceptors, as well as the requirements in the probabilities of interception of important targets, the task of finding the minimum of the total functional of interception efficiency (3) is reduced to either the assignment task [6], which is effectively solved by the Hungarian algorithm [6 ], or to the more complex problem of finding the minimum flow in the network [7, 8], which can be solved with a moderate number of goals.

The essence of the invention is to develop a fundamentally new way of automatically assigning a group of targets to an interceptor group, in which a pre-selected quality functional (2), taking into account both time and energy costs for interception, is calculated for each pair of interceptor-target, then looking for its minimum among a given class of trajectories, taking into account the given restrictions on the speed and acceleration of interceptors, after which the target distribution is determined by solving the “assignment problem” [6] or solving the problem of finding the minimum flow in the network [7, 8].

перехватчику m-ю

цель, наилучшую по минимуму суммарного функционала эффективности перехвата (3), в котором I_n,m представляет собой функционал, соответствующий траектории перехвата n-м перехватчиком m-й цели. Минимизация функционала (3) производится по всем возможным назначениям m(n) n-го перехватчика на m-ю цель, удовлетворяющим определенным условиям. Эти условия позволяют учесть важность целей следующим образом. Будем рассматривать один из самых простых способов, когда для важной цели с номером m задается требуемая вероятность ее поражения Q_m. Также для важной цели с номером m предполагается известной вероятность P_m ее поражения отдельным перехватчиком, определяемая соотношение ЛТХ самолетов и оружия противоборствующих сторон [9, 10]. Будем считать, что вероятности перехвата цели отдельными перехватчиками независимы друг от друга. Тогда размер k_m группы перехватчиков, реализующих общую вероятность поражения не менее требуемой, определяется из условияIn the problem to be solved, for the group consisting of N randomly located interceptors and M targets, it is necessary to assign the n-th

interceptor mth

the goal is the best in the minimum of the total interception efficiency functional (3), in which I _{n, m} is a functional corresponding to the interception trajectory by the nth interceptor of the m-th target. The minimization of the functional (3) is performed according to all possible assignments m (n) of the n-th interceptor to the m-th target, satisfying certain conditions. These conditions make it possible to take into account the importance of the objectives as follows. We will consider one of the easiest ways, when for an important target with the number m we set the required probability of its defeat Q _m . Also for an important goal with number m, it is assumed that the probability P _{m of} its defeat is known as a separate interceptor, the defined ratio LTH of the aircraft and weapons of the opposing sides [9, 10]. We assume that the interception capability of the target by individual interceptors is independent of each other. Then the size k _{m of the} group of interceptors, realizing the general probability of hitting not less than required, is determined from the condition

In accordance with these conditions, it is required that k _m interceptors be assigned to the important target with the number m in the desired assignment m (n). It is assumed that the total number of interceptors is sufficient to provide the required probability of hitting for all important purposes. For the remaining normal targets, it is required to assign one interceptor for each target, if there is a sufficient number of interceptors for it, if there are not enough interceptors, then a target is assigned to each remaining interceptor, while interceptors will not be assigned for some purposes.

The problem will be solved provided that the following assumptions are fulfilled:

- targets and interceptors are randomly located in space, have different initial speeds and flight directions;

- Interceptors are equivalent;

- some targets may be important and must be destroyed with a probability not less than the required one;

- for important purposes, interception probabilities are set by individual interceptors;

- targets do not maneuver and fly at constant speeds;

- all interceptors have sufficient fuel;

- the trajectory of each interceptor consists of two sections: on the first one, dovorot is performed on the target to the required lead angle with constant acceleration, and on the second - a straight flight to the pre-empt meeting point;

- given the maximum allowable values of speeds and accelerations of interceptors.

The solution of the problem will consist of the following steps.

1. At the first stage, the class of trajectories is selected, with the help of which interceptors should intercept targets. Based on this, an individual quality functional (2) of interception is calculated for each interceptor-target pair, taking into account the specified maximum allowable values of the speeds and accelerations of the interceptors.

2. At the second stage, the solution of the problem of finding the minimum of an individual quality functional (2) is reduced to solving several minimization problems with equality type constraints.

3. At the third stage, depending on the number of targets and interceptors, as well as on the availability of important targets, the type of optimization problem and the method of its solution are determined. Its solution determines the purpose of the interceptors, providing a minimum of the total functionality (3) in accordance with the type of the task.

The first stage is illustrated by figure 1. We select a specific interceptor and a specific target. At the initial moment, the interceptor is at point A and flies at speed V ₀ , and the target is at point B and flies at speed V. The estimated trajectory of interception, consisting of two segments, is constructed as follows: in the first segment the interceptor flies with constant acceleration J, performing a dovorot on the target, until the moment t, when the interceptor is at point C, and the target is at point D, then in the second segment the interceptor flies at a constant speed until the moment T.

(фиг. 1). Тогда перехватчик и цель встретятся в точке Е. Если обозначить в момент t относительное положение цели

и относительную скорость перехватчика V_t, то из условия перехвата следует, что для некоторого τ≥0 выполнено r_t = τV_t.The intercept condition in the case where the interceptor and the target fly at constant speeds is that the relative speed of the interceptor’s flight is directed along the line of sight of the target. This means that at the time t of the end of the acceleration action the relative flight speed should be directed along the vector

(Fig. 1). Then the interceptor and the target will meet at point E. If we designate at the moment t the relative position of the target

and the interceptor's relative speed V _t , it follows from the intercept condition that for some τ≥0, r _t = τV _t .

, после выражения r_t, V_t через начальные величины получим:Here τ is the time interval between the moment of termination of the acceleration and the moment of interception. Indicating the position of the target relative to the interceptor at the initial time

, after the expression r _t , V _t through the initial values we get:

Transforming (5), we get

The sum t + τ represents the total flight time T. Then

According to the accepted assumptions, the interceptor speed V _t = V ₀ + Jt at the time of termination of the acceleration cannot exceed V _max , and its acceleration - J _max . From (6) it follows that the constraint | J | ≤J _max defines the inequality

and the constraint | V _t | ≤V _max - inequality

at

Determining Jt from (6) and substituting in (2), we obtain the function of two variables I _{m, n} (T, t), which must be minimized under constraints (7) - (10).

Suppose that the functional takes the minimum value for some values of T, t, so that all inequalities (7) - (10) are strict. It can be argued that for some values of J * and t ^{*, the} value of | J ^* | t ^* will not increase, the speed limit will be met and the target will be intercepted at time T ^* ≤T. As a result, the value of the functional (2) decreases. Therefore, the minimum value of the functional (2) should be sought provided that one or two inequalities from (7) - (10) become equalities.

At the second stage of the algorithm, the following conditions are sequentially checked.

1. If (8) is an equality, then the condition | V _t | = | V ₀ + Jt | = V _max is satisfied. With its help, functional (2) can be represented as

After substituting Jt from (6) we get

After replacing the variable T with z = 2T-t, the functional takes the form

and equality (8) is

We introduce the notation

Then, after building (12) in the square, we get

and (11) takes the form

After changing the variables x = 1 / z; y = t / z the search for minimum (11) reduces to minimizing the functional

while limiting

If we introduce the Lagrange multiplier λ, then the necessary condition for the minimum is

Let P = 2 a y + b + dx; Q = 2ƒx + e + dy; H = (d-e) x + (2 a -b) y.

After eliminating λ and calculating the derivatives, we obtain the equation

Getting rid of the root using squaring, we get equality

which, after simplification using (16), takes the form

Thus, the minimization problem (15) is reduced to solving the system of equations (16), (17). Since the coefficient ƒ = | r ₀ | ^{2 is} positive; by replacing x = (2ξ-dy-е) / 2ƒ, we can reduce (16) to the form ξ ² = a ₁ y ² + b ₁ y + с ₁ . At the same time (17) can be written in the form:

where h _k (y) are some polynomials of degree k. After squaring, only the even powers of ξ will remain, which are expressed in terms of y. The result is an equation of the twelfth degree with respect to y. Numerically, we find all its real roots using well-known algorithms for finding the roots of polynomials [5]. Substitute the found roots in (16) and from the obtained quadratic equation we find the real values of x, if they exist. As a result, we obtain several pairs of x, y values. We turn back to the variables T, t, and discard those values that do not satisfy (7), (9) and (10). The remaining pairs are listed in the general list of candidates for a minimum of functional (2).

2. If equality is (7), i.e. | J | = J _max , then from it we can express t:

where the notation (13) is used. The sign before the second term is selected on the basis of the condition t≤T. In the functional (2) we put | J | = J _max and substitute in it the expression found for t. As a result, the quality functional becomes a function of T:

Calculate the derivative dI _{m, n} (T) / dT and equate it to zero. If to designate

then after a series of transformations, the condition for the zero derivative is determined by

which, after opening the parentheses, leads to the eighth degree equation in the variable T. We solve it numerically. For each T obtained, we find t from (18). Of all the obtained pairs of real values of (T, t), we leave only those that satisfy (8), (9) and (10). Put them in the general list of candidates for a minimum of functional (2).

3. If equality is (9) (T = t), i.e. from the beginning of the interception of the target to its end, the uniformly accelerated flight of the interceptor is realized, then we express Jt from (6) and substitute it into (2):

If we go to z = 1 / T, then in the notation (13) we get

The equation dI _{m, n} (z) / dz = 0 reduces to a sixth degree equation

We solve it numerically and those roots for which (7), (8) and (10) are fulfilled, add to the general list of candidates for a minimum of the functional (2).

при

в обозначениях (13). Если

и величина промаха r_min находится в допустимых пределах, то добавим пару значений

; t=0 в общий список кандидатов на минимум с соответствующим значением функционала

.4. If equality is (10) (t = 0), i.e. a flight at a constant speed takes place, then the distance from the interceptor to the target at time T * will be | r ₀ + (VV ₀ ) T ^* |. It takes the minimum value.

at

in the notation (13). If a

and the miss value r _min is within acceptable limits, then we add a couple of values

; t = 0 to the general list of candidates for a minimum with the corresponding value of the functional

.

5. In the case when of the four inequalities (7) - (10), two are equalities, the following situations are possible.

5.1. Let the equalities be expressions (7) and (8), i.e. | V _t | = V _max , | J | = J _max . Make the replacement z = 2Т-t. Then, after squaring in the notation (13), equality (8) takes the form (14), and equality (7) is written as

Add (19) and (14), we get

Can be reduced by z because z = T + (T-t) ≥T> 0:

We express z from here and substitute it into (14). Get the sixth degree equation for t:

where indicated

Numerically, we find all the real roots t of equation (21), then we find the corresponding values of z from (20) and the values T = (z + t) / 2.

Satisfying inequalities (9) and (10), the values are entered into the general list of candidates for a minimum of the functional (2).

5.2. Let the equalities be (7) and (9). Then, after substitution of t = T in (7) and squaring, we get the equation

His solutions, satisfying inequalities (8) and (10), will be added to the general list of candidates for a minimum of functional (2).

5.3. Suppose now that the equalities are (8) and (9). Substitute t = T in (8) and squared. In the notation (13) we obtain the equation

His solutions, satisfying inequalities (7) and (10), are added to the general list of candidates for a minimum of functional (2).

5.4. The degenerate case, when one of the equalities is (10), has already been considered earlier in Section 4.

Now we find the global minimum of the functional (2). For all pairs of values (T, t) from the general list of candidates for the minimum of the functional (2) constructed at the previous stages, we calculate Jt using (6) and substitute the obtained value into (2). Choose those values that give the minimum value. The obtained value of T together with the corresponding value of t, the value of the functional I _{m, n} and the vector J determine the best trajectory of interception of the target and the cost of its implementation.

Having solved the problem of finding the minimum (2) for a single interceptor and a target, we proceed to the third stage of the algorithm. For each interceptor with the number n and the target with the number m, determine the best intercept trajectory and the corresponding value I _{n, m of the} functional (2). For important purposes, the required number of interceptors k _{m is} determined by the formula (4). The task of finding target distribution falls into one of four classes.

1. There are no important goals, all goals are equivalent. The task of searching for target distribution in this case is formulated as the “assignment problem” [6], with the cost matrix I _{n, m} . The way to solve this problem is known as the “Hungarian algorithm” [6]. This algorithm consists of consecutive iterations, at each of which the assignment m (n) is formed so that the total cost of the choice at each subsequent iteration decreases. The result is a variant of the appointment of interceptors on the target, which gives the minimum value of the total cost of its implementation. If M> N, then each interceptor will be assigned its own goal, but for some purposes interceptors will not be assigned. If N> M, then an interceptor will be assigned to each target and some interceptors will remain without an assignment.

2. Some of the targets are important, but the number of interceptors is enough to assign the required number of interceptors to important targets and one of the interceptors to normal ones. For the solution, we construct the same matrix I _{n, m} as in the previous paragraph.

Then, instead of the column with the number m, corresponding to the important goal, we insert k _m copies of this column. The application of the "Hungarian algorithm" to the expanded matrix constructed in this way will give the desired target distribution. Each important target will be assigned k _m interceptors, the other targets will be assigned one interceptor, the extra interceptors, if any, will be free.

3. Some of the targets are important, and the number of interceptors is not enough to assign the required number of interceptors to important targets. Then the task has no solution.

4. Some of the targets are important, the number of interceptors is enough to allocate k _m interceptors for important targets, but the remaining interceptors are not enough to assign one interceptor for ordinary targets. This task can no longer be formulated as an “assignment task”. However, it can be formulated as a more general problem of finding the flow of minimum value in a graph. A detailed description of this problem and ways to solve it can be found in [7, 8]. As a rule, all input data are assumed to be integer, so you should first convert the matrix I _{n, m} , for which you need to find the maximum element I _{max of the} matrix I _{n, m} , and then multiply all the elements of I _{n, m} by 10000 / I _max and round up to integers

The task of searching for target distribution in the form of the task of finding a stream of minimum value in a graph is formulated as follows. The target distribution graph (Fig. 2) is a directed graph containing one vertex for each interceptor and target, as well as two special vertices: the source (I) and the drain (C). From each interceptor n to each target, m leads an edge of cost I _{n, m} (integer) and with a unit bandwidth. From the source to each interceptor leads edge zero cost and unit bandwidth. From each goal to the drain goes edge. For normal purposes, its cost is given by a value that significantly exceeds the maximum of the I _{n, m} values, and its carrying capacity is equal to one. For important purposes, the cost of the edge is zero, and the bandwidth is equal to k _m . You want to skip the stream size N from the source to the drain. The solution to the problem is the magnitude of the flow for each edge. The assignment of targets to interceptors is determined by edges leading from interceptors to targets for which the value of the flow found is equal to one.

The assignment of targets to interceptors via communication lines, formed as a result of the solution of the problem, is transmitted to the command control system, in which they generate interceptor control signals that ensure their guidance to selected targets.

It should be noted that to use the algorithm, the following estimates are needed: the velocity vectors of all interceptors and targets; relative position vectors for each pair of interceptor - goal; maximum limits on the speed and acceleration of interceptors. These data can be presented in any form: in Cartesian or polar coordinates, in absolute or relative values. It is only necessary to specify the method of calculating the coefficients (13).

перехватчиками

.The efficiency of the developed algorithm (2) - (22) was studied in the process of simulation. As an example, consider the procedure for intercepting goals

interceptors

.

The layout of N = 6 interceptors and M = 3 targets in simulating target distribution with respect to priority is shown in FIG. 3. Target number 4 is considered important. The probability of its defeat by a separate interceptor is taken equal to 0.5, and the required probability of interception is set equal to 0.9. In accordance with formula (4), four interceptors are required to intercept it. Therefore, the total number of interceptors is not enough to intercept all targets and the task in question corresponds to the fourth paragraph of the third step of the algorithm. As a result of solving the minimum flow problem in the graph, the first, second, third, and fourth interceptors were assigned to target number 4, the fifth interceptor was assigned to the second target, and the sixth to the third target. Interceptors were not assigned to the first target. Estimated intercept trajectories corresponding to this assignment are shown in FIG. 3

The obtained algorithm of group targeting confirmed its effectiveness in a wide field of application conditions. Its advantage is that it allows not only the assignment of interceptors to the targets, taking into account the importance of the targets, but also the construction of the proposed intercept trajectories, taking into account the real restrictions on the maximum allowable speeds and accelerations.

The proposed algorithm can be used to implement various methods of targeting.

Industrial applicability of the proposed technical solution is also confirmed by the possibility of realizing its purpose using standard onboard computational tools.

It should be noted that the proposed algorithm follows the general scheme used in the domestic aviation complexes of the radar watch and guidance.

List of used sources

1. E.A. Fedosov. The implementation of the network-centric technology of warfare will require the creation of a new generation of radar. // Phasotron. 2007. № 1, 2.

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3. J. Zhang, C. Hu, X. Wang, D. Yuan. ACGA Algorithm // Open journal of applied sciences, 2012.

4. V.I. Merkulov, A.C. Little dancer Group target allocation in aerial confrontation. // Information and measuring and control systems. 2016. №7. Pp. 59-63.

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Claims (67)

The method of automatic group targeting of fighters, taking into account the priority of targets, is that for each n-th interceptor

and mth goal

form functional

interception efficiency where T - time interception K - coefficient of the fine, J is the interceptor acceleration vector, t is the duration of the interceptor acceleration, further in the interceptor filters form the optimal parameter estimates

Where V is the velocity vector of the target, V ₀ - interceptor velocity vector, r ₀ is the position vector of the target relative to the interceptor, V _{m a x} - the maximum allowed speed of the interceptor, J _{m a x} - the maximum allowable acceleration rate of the interceptor, further form the conditions for the implementation of restrictions in the form

after which they form a list of pairs (T, t), successively solving the following seven tasks with the help of well-known algorithms for finding the roots of polynomials numerically: a) solve the system of equations

Where P = 2 a y + b + dx; Q = 2ƒx + e + dy; H = (d-e) x + (2 a -b) y, which replacing x = (2ξ-dy-e) / 2ƒ takes the form

where h _k (y) are polynomials of degree k, then squared (10) and all even powers of ξ are expressed in terms of y using (9), then the numerically obtained equation is solved for y, the found roots are substituted in (7) and from the resulting equations find the corresponding values of x, then for all the resulting pairs x, y find the pairs T, t by the formulas T = (y + 1) / 2x; t = y / x and those pairs that satisfy (3), (5) and (6) are listed in the general list; b) solve the equation numerically ((16-h) T ² ƒ ₁ (T) + ƒ ₂ (T ² ) ² -ƒ ₁ (T) (8Tƒ ₂ (T) -hƒ ₁ (T)) ² = 0, Where h = 16 (1 + 1 / KJ _{m a x} ) ₂ , ƒ ₁ (T) = (4aT ² + 2dT + f) / g, ₂ (T) = (8 a T + 2d) / g, and, calculating T in this way, find the corresponding values of t using the formula

then, of all the resulting pairs (T, t), those that satisfy (4), (5) and (6) are listed in the general list; c) solve the equation numerically K ² z (ƒz + d) ² = ƒz ² + 2dz + 4 a , find the corresponding values of T = 1 / z and those pairs of values of T; t = T for which (3), (4) and (6) are fulfilled, add to the general list; d) find

and

if a

and the miss value r _mjn is within acceptable limits, then add a couple of values

t = 0 to the general list; e) solve the equation numerically cp ₁ (t) ² -p ₁ (t) p ₃ (t) p ₂ (t) + p ₄ (t) p ₂ (t) ² = 0, Where p ₁ (t) = (b-2 a ) t + (ed); p ₂ (t) = gt ² + (c- a ); p ₃ (t) = bt + e; p ₄ (t) = a t ² + dt + ƒ, find the corresponding values T = (t - ((ca) z + (b-2a) t + (ed)) / gt ² ) / 2 and put in the general list the pairs (T, t) satisfying inequalities (5) and (6); e) solve the equation numerically gT -4 ⁴ and T ² -2dT-ƒ = 0 and the pairs T; t = T, satisfying inequalities (4) and (6), are added to the general list; g) solve the equation numerically ( a + b + c) T ² + (e + d) T + ƒ = 0 and pairs T; t = T, satisfying inequalities (3) and (6), are added to the general list, then for all pairs of values (T, t) from the constructed list, the interceptor acceleration vector is calculated J = 2 (r ₀ + r (VV ₀ )) / t (2T-t) and the value of the functional (1), then choose the pair (T, t) for which the functional (1) takes the minimum value, thereby obtaining the minimum value I _{m, n of the} functional (1) and the interception trajectory defined by the values (T, t , J), on which the specified minimum is realized, then from the values of I _{m, n} constitute an N × M matrix, characterized in that for each priority target with number m, according to a given probability P _{m of} its interception by one interceptor and the required interception probability Q _m calculating a required number k _m interceptor based on conditions

then analyze the possible cases: 1) if there are no priority targets, then the “assignment problem” with the matrix I _{m, n is} solved using the “Hungarian algorithm”, thereby obtaining the optimal assignment of m (n) targets to interceptors together with the corresponding interception trajectories from the minimum total cost point of view, 2) if there are priority targets and the number of interceptors is not less than the sum k _m for all priority targets plus the number of normal targets, then in the matrix I _{m, n} for each number m setting the priority target, instead of one column with number m, insert k _m copies of this column, then solve the “assignment problem” with the obtained extended matrix using the “Hungarian algorithm”, thereby obtaining the optimal (from the point of view of the minimum total cost, the assignment of m (n) targets to interceptors together with the corresponding interception trajectories, 3) if there are priority targets and the number of interceptors is less than the sum of k _m for all priority targets, then the consumer is informed about the absence of a solution to the target allocation task with the required interception probabilities, 4) if there are priority targets and the number of interceptors is not less than the sum of k _m for all priority targets, but less than the sum of k _m for all priority targets plus the number of normal targets, then find the maximum element I _{max of the} matrix I _{n, m} , after which all the elements matrices I _{n, m} are multiplied by the value of 10,000 / I _max and rounded to whole numbers, then a target distribution graph is constructed, containing one vertex for each interceptor and target, as well as two special peaks: the source (I) and the drain (C), from each interceptor n to each target m build an edge worth I _{n, m} (c A single number with a unit capacity, from the source to each interceptor, an edge of zero cost and unit capacity is built, an edge is built from each target to the drain, for ordinary purposes its cost is set to a value significantly exceeding the maximum of the values In _{, m} , and bandwidth is believed to be unity for priority purposes ribs value is set equal to zero, and equal bandwidth believed k _m, then known algorithms solve the problem of finding the minimum flow size N of the source in the Art k, its solution is the quantity of flow for each edge, the purpose of the goals of the hooks define a set of edges leading from the interceptor to the purposes for which the value found by the flow is unity, then, the generated assignment of targets to interceptors is transmitted via command radio links to the command control system, in which they generate interceptor control signals, which ensure their guidance to selected targets.

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