This paper presents a succinct review of attempts in the literature to use game theory to model decision-making scenarios relevant to defence applications. Game theory has been proven as a very effective tool in modelling the decision-making processes of intelligent agents, entities, and players. It has been used to model scenarios from diverse fields such as economics, evolutionary biology, and computer science. In defence applications, there is often a need to model and predict the actions of hostile actors, and players who try to evade or out-smart each other. Modelling how the actions of competitive players shape the decision making of each other is the forte of game theory. In past decades, there have been several studies that applied different branches of game theory to model a range of defence-related scenarios. This paper provides a structured review of such attempts, and classifies existing literature in terms of the kind of warfare modelled, the types of games used, and the players involved. After careful selection, a total of 29 directly relevant papers are discussed and classified. In terms of the warfares modelled, we recognise that most papers that apply game theory in defence settings are concerned with Command and Control Warfare, and can be further classified into papers dealing with (i) Resource Allocation Warfare (ii) Information Warfare (iii) Weapons Control Warfare, and (iv) Adversary Monitoring Warfare. We also observe that most of the reviewed papers are concerned with sensing, tracking, and large sensor networks, and the studied problems have parallels in sensor network analysis in the civilian domain. In terms of the games used, we classify the reviewed papers into papers that use non-cooperative or cooperative games, simultaneous or sequential games, discrete or continuous games, and non-zero-sum or zero-sum games. Similarly, papers are also classified into two-player, three-player or multi-player game based papers. We also explore the nature of players and the construction of payoff functions in each scenario. Finally, we also identify gaps in literature where game theory could be fruitfully applied in scenarios hitherto unexplored using game theory. The presented analysis provides a concise summary of the state-of-the-art with regards to the use of game theory in defence applications and highlights the benefits and limitations of game theory in the considered scenarios.

In this paper, we derive a novel batch Bayesian weighted instrumental variable estimator for the threedimensional (3D) target motion analysis (TMA) problem using bearing and elevation measurements. Unlike most existing 3D estimators based on instrumental variables, the proposed approach is able to incorporate a priori information into the estimation process and is proven to be approximately asymptotically unbiased. An approximate asymptotic covariance matrix is also presented to evaluate the performance of the estimator. Simulation results show that the proposed estimator outperforms its non-Bayesian counterpart and has an estimation performance on par with the conventional iterative maximum a posteriori (MAP) estimator, while being orders of magnitude faster in a passive sonar TMA problem. Moreover, it was shown that the proposed approach can provide an accurate initialisation for recursive estimators.

Accurate 3D position and velocity estimates of an aircraft are important for air traffic control (ATC) systems. An ATC 2D radar measures the slant range and azimuth of an aircraft. Thus, a single measurement from a 2D radar is not sufficient to calculate the 3D position of an aircraft. Previous researchers have used the multiple model based heightparametrized (HP) extended Kalman filter with Cartesian state vector (HP-CEKF) with one or two 2D radars for an aircraft with nearly constant velocity and altitude. However, the filter initialization algorithms contain errors. In this paper, in addition to the HP-CEKF, we present the HP Cartesian unscented Kalman filter (HP-CUKF) and HP Cartesian cubature Kalman filter (HPCCKF). We also present two new nonlinear filters for the tworadar problem. The first filter uses modified spherical coordinates based HP UKF (HP-MSCUKF) where the range and azimuth are components of the target state. The second filter uses a cubature Kalman filter with filter initialization by the bias-compensated pseudo linear estimator. We consider the climbing motion of an aircraft with nearly constant climbing rate which has not been studied before.

This paper develops Bayesian weighted instrumental variable (WIV) estimators under equality and inequality speed constraints for the three-dimensional (3D) bearings-only target motion analysis (TMA) problem, which show improved estimation accuracies compared to their unconstrained counterpart. Incorporating the speed constraint information into our previously proposed Bayesian WIV estimator [36] imposes a quadratic constraint on this problem. Finding an optimal solution for the resulting non-convex problem is by no means straightforward. The main problem with iterative optimisation methods such as the maximum a posteriori (MAP) is that they require initialisation and may converge to a local minimum if poorly initialised. Using advanced linear and nonlinear algebra techniques, non iterative accurate estimators under equality and inequality constraints are developed for this problem. Furthermore, it is proven that the proposed equality constrained estimator is approximately asymptotically unbiased and that the inequality constrained estimator is the optimal solution. An approximate covariance matrix has also been developed for the constrained WIV under equality constraint. Simulation results indicate that the equality and inequality constrained estimators outperform their unconstrained counterpart in the simulated scenarios.

Bearings-only target motion analysis with fusion of heterogeneous passive sensors with and without propagation delay attempts to estimate a target trajectory by fusing bearings from two sensors: a sensor whose measurements have negligible propagation delay and a sensor with measurements that have non negligible propagation delay. This problem is highly nonlinear and challenging, particularly due to the fact that the propagation delay is unknown and is highly coupled with the unknown target states. This paper proposes a novel closed-form instrumental variable based batch estimator for this problem that is computationally fast. The estimator is derived by writing pseudo-linear type equations for the error vector difference between the true target state and a nominal solution. A theoretical bias and mean square error analysis of the proposed estimator is carried out which provides insight into the performance of this estimator. Results indicate that this is a computationally efficient algorithm that is able to produce comparable performance to the maximum likelihood estimator and achieves the Cramer Rao lower bound.

In this manuscript, an underwater target tracking problem with passive sensors is considered. The measurements used to track the target trajectories are (i) only bearing angles, and (ii) Doppler-shifted frequencies and bearing angles. Measurement noise is assumed to follow a zero mean Gaussian probability density function with unknown noise covariance. A method is developed which can estimate the position and velocity of the target along with the unknown measurement noise covariance at each time step. The proposed estimator linearises the nonlinear measurement using an orthogonal polynomial of first order, and the coefficients of the polynomial are evaluated using numerical integration. The unknown sensor noise covariance is estimated online from residual measurements. Compared to available adaptive sigma point filters, it is free from the Cholesky decomposition error. The developed method is applied to two underwater tracking scenarios which consider a nearly constant velocity target. The filter’s efficacy is evaluated using (i) root mean square error (RMSE), (ii) percentage of track loss, (iii) normalised (state) estimation error squared (NEES), (iv) bias norm, and (v) floating point operations (flops) count. From the simulation results, it is observed that the proposed method tracks the target in both scenarios, even for the unknown and time-varying measurement noise covariance case. Furthermore, the tracking accuracy increases with the incorporation of Doppler frequency measurements. The performance of the proposed method is comparable to the adaptive deterministic support point filters, with the advantage of a considerably reduced flops requirement.