Time Allocation of a Set of Radars in a Multitarget Environment

In many applications sensors are nowadays a part of a multisensor system, each sensor bringing its complementarity and its redundancy to the overall system. Year after year the complexity and the performances of many sensors have increased leading to more and more complex multisensor systems which supply the decision centers with an increasing amount of data. This increasing complexity also led to other uses for each sensor and therefore for the multisensor systems. It is no more considered as a passive system the role of which is just limited to simple measurement actions; the many parameters of each sensor and the interactions between all the sensors allow to choose how the measurement action must be done: the sensors need to be managed. The complexity of this problem is such that it is often impossible to a man to find an optimal solution (with respect to the goal of the mission of the multisensor system) and multisensor management strategies must be derived. That is the reason why sensors management has become during the past years an active field of research. From a theorical point of view this problem can be written in the frame of optimal control and the sensor management viewed as a Markov decision problem. Optimal solutions could therefore be found. Unfortunately, the complexity is such that it is impossible in practice to derive these solutions. Suboptimal solutions as well as alternative approaches have then been proposed. In [1] or [2] the authors use reinforcement learning, Q-learning and approximation functions to derive sub-optimal solutions. In many works the choice of the next action is based on information theory and information divergence like the Rényi information divergence and the Kullback Leibler divergence [3], [1], [2]. In [4] Mahler proposes to solve the problem in the frame of random sets. All these works bring a possible solution to the sensor management problem but as far as the authors know, it is often difficult to derive bound of performance which can be a drawback in an operational context. Moreover, these approaches rarely take into account the characteristics of the sensors. The work described in this paper proposes, in the frame of an aerial patrol in charge of the detection of potential targets, to derive radar optimal time allocations (a part of the sensor management problem) which allow to determine such bounds and which are based on the modelling of the detection performances of a radar. It is assumed here that each aircraft is equipped with an ESA (Electronically Steered Antenna) radar. We focus on the detection step for which a fixed duration T has been allocated. Methods exist to optimize the detection of a single target by a single sensor and the frame Search Theory is devoted to such a problem [5], [6]. In this paper we consider a multitarget environment and the optimization process is led by considering the overall targets and not the targets one by one. The problem then becomes: if radars have to observe P targets during T , how do they organize themselves to detect them in the best possible way, i.e. how do they distribute the duration T over the space directions ? The aim of this article is to derive an optimal temporal allocation based on the modelling of the radar detection probability and on an a priori knowledge coming from an ESM type (Electrical Support Measurement) or AEW type (Airbone Early Warning) system of supervision. Along the study, two contexts are considered. The first one is the ideal case: the position of the targets are known and we must detect them. Of course this situation is not realistic but it allows to derive some interesting results for the second context : the position of the targets are known by the mean of probability densities. After having defined the assumptions of our study in the second section, we present in the third section a modelling of the radar detection functions. However the context of this study is multisensor multitarget, we start by a study of the optimization of the detection process in a monosensor monotarget environment. Comparing to existing methods, our aim in this preliminary work is to derive analytically an optimal strategy and the corresponding probability of detection. This last probability will be used along the overall paper. The third section presents analytic results and a performance evaluation. The multitarget environment is tackled in the fourth section but we are still in a monosensor case. Under the assumption of an a priori knowledge, we propose an optimal temporal allocation. The allocation derived in this section uses the results derived in the previous sections. Finally, the last section shows how all the previous results can be used to propose an allocation strategy in the multisensor multitarget case. It is important to understand the needs at the origin of the study, proposed by Thales Optronics, the results of which are written out in this paper. The aim was to found bounds of performanc

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