Optimisation of the spatial configuration of microphones for robust virtual sensing in a diffuse sound field

Virtual sensing methods are utilised in active noise control systems where the error sensors cannot be placed at the locations where control is physically required. Their performance critically depends on the spatial configuration of the physical monitoring microphones used to estimate the pressures at the virtual error sensor locations. This paper investigates the use of a genetic algorithm to calculate optimal microphone placements for estimation within a stationary diffuse sound field. A multi-objective optimisation framework is formulated, simultaneously minimising the estimation error and the condition number of the monitoring microphone power spectral density matrix, thereby addressing both estimation accuracy and robustness to uncertainties. Optimisations are carried out for a single frequency and for three representative frequencies spanning three octaves. The resulting Pareto fronts reveal the inherent trade-off between performance and numerical stability. The Technique for Order of Preference by Similarity to Ideal Solution is applied to select a single optimal solution from each Pareto set. These solutions achieve a balanced compromise, offering a small reduction in estimation performance while reducing the condition number by up to an order of magnitude compared with configurations that solely minimise the error. The minimum error and optimal solutions are evaluated over a broad frequency range, where the optimal designs are shown to significantly reduce the conditioning for a modest increase in estimation errors. The study highlights characteristic spatial patterns that promote optimal performance, and demonstrates the effectiveness of a genetic algorithm-based multi-objective optimisation for designing robust microphone configurations for virtual sensing applications.