Filtering Discrete-Time Systems with Multiplicative Noise in L2 Spaces with Applications

In this work we adopt a novel formulation of the distributed parameters recursive filter for discrete-time systems evolving in $L_{2}$ spaces to widen the class of systems that can be processed by a state estimation algorithm. Starting from a rigorous definition of Kronecker algebra on $L_{2}$ spaces that involves both elements and bounded operators of $L_{2}$ , we provide a computationally efficient solution in the case of linear systems with multiplicative noises. We illustrate the potential application of the approach by developing a case-study concerning the conceptual design of a distributed thermo-couple in the presence of the Nyquist-Johnson noise.