This paper addresses the least-squares quadratic filtering problem in discrete-time stochastic systems with random parameter matrices in both the state and measurement equations. Defining a suitable augmented system, this problem is reduced to the least-squares linear filtering problem of the augmented state based on the augmented observations. Under the assumption that the moments, up to the fourth-order one, of the original state and measurement vectors are known, a recursive algorithm for the optimal linear filter of the augmented state is designed, from which the optimal quadratic filter of the original state is obtained. As a particular case, the proposed results are applied to multi-sensor systems with state-dependent multiplicative noise and fading measurements and, finally, a numerical simulation example illustrates the performance of the proposed quadratic filter in comparison with the linear one and also with other filters in the existing literature.

Ministerio de Economía y Competitividad (Grant No. MTM2014-52291-P and FPU programme)