Minimization of energy costs for UAV management in a conflict task

Federation Abstract. This article considers the method of developing an evader control strategy in the non-linear differential pursuit-evasion game problem. It is assumed that the pursuer resorts to the most probable control strategy in order to capture the evader and that at each moment the evader knows its own and the enemy’s physical capabilities. This assumption allows to bring the game problem down to the problem of a unilateral evader control, with the condition of reaching a saddle point not obligatory to be fulfilled. The control is realised in the form of synthesis and additionally ensures that the requirements for bringing the evader to a specified area with terminal optimization of certain state variables are satisfiedt. The solution of this problem will significantly reduce the energy losses for controlling an unmanned vehicle, the possible effect is to save 15-20 % of fuel with a probability of 0.98, to solve the problem of chasing the

1 Introduction [1][2][3][4] presents a method of an optimal evader control strategy in the pursuit-evasion game problem, which does not require attaining the global extremum and complying with the condition of reaching a saddle point. The method was developed based on, first, the assumption that the evader knows its own and the enemy's physical capabilities and, second, on the condition that the enemy resorts to the most probable strategy in order to intercept the evader. The above assumptions made it possible to determine an optimal evader control strategy in the analytical form in the non-linear differential game problem. The final values of some of the evader's state variables were terminally limited, and the total time for solving the game problem   k t t t , 0  was assumed to be preset and fixed.
At the same time, it is known that finite values of phase variables are not always possible to reach within a given time period const 0    t t T k in control problems [5][6][7]. In this case, having performed an effective evasive manoeuvre, the evader may not have enough resources to deliver the useful load to the specified area. Accordingly, the condition of the fixed time interval T set in [1] may make it principally impossible to reach terminal values in certain subproblems, which substantially reduces the practical value of the approach suggested in [1].
Besides, additional requirements of terminal optimization may be set for certain state variables of the evader. E.g., for many practical applications, the evader's trajectory is to satisfy the requirements of passing through a given terminal area with optimization of certain functions of state variables at a finite time period. The method presented in [1-4] also does not provide the possibility of terminal optimization of certain evader's state variables.
To this end, below is presented the method of developing an evader control strategy in the nonlinear differential game problem, which factors in an additional condition requiring the evader to be brought to a given area with the optimization of certain state variables for an arbitrary time . The developed method, as well as that presented in [1], does not require reaching the global extremum, but rather takes an approach which sorts criteria by preference [1, 4,6,8]. It was again assumed that at every moment the evader knows its own physical capabilities and those of the pursuer, and the optimal control laws are admissible and unique, at least for all time values preceding the moment of the encounter.
The problem statement and the assumptions adopted distinguish the problem from the classical pursuit-evasion conflict problems of the differential game theory [6].

Problem Formulation
The current position of the evader is determined by phase vector   t y , and the enemy's positionby vector z(t) ( The dynamics of both objects in the phase space is described by the system of nonlinear differential equations [4]:   [6,9,10,11,12] where i Ф is an dimensional vector function and a requirement of terminal optimization (suppose, maximization) of a certain known scalar function is imposed on the rest of the The pursuer's goal is to minimize the distance between the players, while the evader's goal is to maximize this distance. Therefore, during the entire time of the game, the vector control functions u(t) and v(t) must simultaneously provide optima (maxima and minima) of a certain given non-negative scalar function

The Method of Problem Solving
Consider that it is necessary to find a solution only from the point view of one playerthe evader. Then, using the known approach [1, 4,6], we can bring the game problem (1) - (6) down to the problem of search of a control over the generalized dynamical system is the "best" function of control of the enemy player based on the feedback principle, with the enemy immediately taking advantage of any non-optimal evader's move. The function is constructed in line with the method described in [1]. Then the optimal evader's strategy   t u is realized based on a narrower, compared with (6), condition considering the terminal conditions (3) - (5). In this case the Hamiltonian has the following form: where the optimal phase trajectory   t x and the vector of conjugate variables   t  are determined by the adjoint equations of the canonical two-point boundary value problem where   is the operation of block matrices multiplication introduced in [9].
where , M are 1 n n  , 1 1 n n  dimensional matrices vector correspondingly. Optimal value  taking into account (11) presented in the following form: where I is an 1 1 n n  . identity matrix. The end time of the game k t is found from the additional scalar equation To examine the attainability of the final conditions with the time fixed from the set (18) will be out of reach after the evader y(t) carries out the manoeuvre of "evading" the pursuer z(t), was confirmed on the example of such a simple problem. The condition of fixing the time of solving the problem const  T may lead to its becoming principally impossible to solve.
Further an additional condition of optimizing the evader's velocity in the terminal point was set and the condition of the fixed time interval was eliminated: Computer simulation of a practical example was carried out in a mathematical package Mathcad 15. In this case, the mutual movement of two opposing objects in the lateral plane was simulated.
Model realizations of evasion of the player-ally from the opponent are presented in figures 1, 2 (linear speeds of movement of objects and coordinates respectively). The errors of bringing into the given terminal point of space and the errors of providing the given parameters of the movement of the ally player were 45% -50% less compared to the errors of the traditional method.
The solution was realized in line with the method described above. It was found that the evader carries out the manoeuvre of "fleeing" from the enemy, then executes the operation of coming to any coordinate y1k from the set (4.2) with the maximum possible velocity y2.

Conclusion
The solution to the non-linear differential pursuit-evasion game problem was found based on the assumption of the most probable actions of the enemy in order to intercept the evader and taking into consideration the both objects' limited energy for manoeuvring. The terminal optimization of the evader's n2 state variables is ensured in the finite region of the phase space, which, in turn, is determined by the constraints in the form of equalities imposed on the remaining   n n n п   2 1 1 variables. The results received in the simulation allow to conclude that the developed method is efficient.
The issues of limited availability of energy resources for bringing the evader to the terminal area were not considered. The solution of this problem will significantly reduce the energy losses for controlling an unmanned vehicle, the possible effect is to save 15-20 % of fuel with a probability of 0.98, to solve the problem of chasing the enemy.