In 1975, Kenneth Alan De Jong produced a PhD thesis which analysed the per- formance of a class of genetic adaptive systems [116]. Within the thesis, De Jong proposed five functions which could be used to benchmark the performance of the genetic algorithms. The benchmark functions consisted of the Sphere, Rosenbrock, Step, Quartic and Foxholes functions. The difficulty in optimising each function varied from easy in the case of the Sphere function to very difficult in the case of the Foxholes function. The Sphere function is very easy to optimise, because it is smooth, unimodal and has a definite optima and minima. However, the Foxholes function has many local optima which means that optimisation al- gorithms can become stuck at the first optima that they find.
In 2002, J.G. Digalakis et al. proposed a further 8 algorithms which could be used for the purposes of benchmarking optimisation algorithms each with a spe- cific difficulty. The focus of this paper was once again on the performance of Genetic Algorithms. However, the same functions can be used to test perfor- mance of other types of optimisation algorithm [117].
It would appear from these references and associated works [107, 108, 114] that the choice of optimisation algorithm in any problem should be determined based on the difficulties envisaged from the complexity of the function which is to be optimised. If the function has similar characteristics to the Sphere optimisation problem then a simple algorithm such as Newton’s method will be sufficient to either maximise or minimise the function.
If the function has many local optima such as the Foxholes function, then Meta- heuristic algorithms such as Simulated Annealing and Genetic Algorithms are the most appropriate algorithm to use, as identified by the previous literature review.
For this particular optimisation problem, the complexity of the intercept prob- ability distribution must be considered in order to determine what type of op- timisation algorithm is required i.e. does the distribution have a single global maxima or are their a large number of local maxima as in the Foxholes function?
The trajectory of the weapon may be potentially shaped by the integrated fire control system at several points in time. However it is very difficult to visualise the intercept probability distribution in more than 3 dimensions. The complexity of the intercept probability distribution will therefore be evaluated considering
two possible trajectory shaping approaches. In the first approach, a single off- boresight angle will be applied at various points in time. In the second approach, two off-boresight angles will be transmitted to the weapon at fixed points in time. In both cases, the distribution of possible target trajectories and associated prob- abilities is calculated from an initial target state [x, vx, y, vy]0 of [10000,-26,0,0]’.
The intercept probability distributions for each approach are displayed in Figures 5.3 and 5.4 respectively. 0 5 10 15 20 25 −10 −5 0 5 10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Time (s) Off−boresight angle (degs)
Probability of Intercept
0 0.02 0.04 0.06 0.08 0.1 0.12
Figure 5.3: Intercept Probability Distribution for a Single Off-boresight Angle Applied at Times T=0s....T=25s
In Figure 5.3 an off-boresight command ±10◦ was initially programmed into the weapon at T=0s, the process was repeated increasing the off-boresight angle by 1◦ for each off-boresight angle in this range. The process was then repeated by transmitting the off-boresight command to the weapon at T=1s, T=2s....T=25s.
In Figure 5.4, two off-boresight commands are used to shape the trajectory of the weapon. The first command is programmed into the weapon at T=0s, which is when the weapon is still on the launcher. The second command is transmitted to the weapon at T=10s. The off-boresight commands are an integer value in the range of ±10◦.
−10 −5 0 5 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Off−boresight 2 (degs)
Off−boresight 1 (degs)
Probability of Intercept
0.02 0.04 0.06 0.08 0.1 0.12
Figure 5.4: Intercept Probability Distribution for Two Off-boresight Angles Ap- plied at T=0s and T=10s
In both figures, the intercept probability distribution has a large number of local maxima. As the number of off-boresight commands increases the complexity and the number of local maxima in the intercept probability distribution will most likely also increase. This is because their will be greater variation in the possible shaped weapon trajectories that can be generated.
The greater variation would allow a larger number of trajectories within the predicted target distribution to be overlapped by the weapon scan area. This increases the likelihood of being able to detect the target along more possible tra- jectories, resulting in an increased number of local maxima within the intercept probability distribution.
It is therefore evident that a Metaheuristic optimisation algorithm is required as both Derivative and Derivative-free optimisation algorithms will become stuck in local maxima of the intercept probability distribution. The literature review
identified two potential Metaheuristic algorithms which could be used for this particular optimisation problem, Simulated Annealing (SA) and Genetic Algo- rithms (GA).
Simulated Annealing was selected for the following reasons :
1. If the algorithm is correctly tuned i.e. sensible values are used for the start and end temperature and an appropriate cooling schedule determined the global optimum should be obtained.
2. The time taken for a genetic algorithm to find an optimal solution is gen- erally longer than using a simulated annealing process [118].