Identification of Markov Matrices of Milling Models

Detailed modeling of a grinding mill can be achieved through Markov chain models without involving lengthy computations. However, estimation of the key parameters of the model, elements of the Markov transition matrix, using observable quantities is not trivial. This powerful modeling tool can find wide applicability in operation and control of industrial mills if this set of parameters can be estimated using suitably observed quantities. In this study we model a complex multiregion mill using Markov chain and propose a general technique for estimation of the Markov transition matrix for breakage problems. This technique estimates the transition matrix from observed evolution of particle size distributions in various regions of the mill, based on extracting spectral information of the transition matrix from the data. It has been shown in this study that a specific grouping of states can lead to lower triangular block structure of the Markov transition matrix for a breakage problem. Detailed analysis of such a block matrix reveals that simple semilogarithmic plots of a set of observed quantities can be used in order to extract the spectral information of the transition matrix which in turn can be used to reconstruct the Markov matrix. A numerical example has been presented to illustrate various ideas which demonstrate that the proposed technique can be used successfully to estimate the Markov transition matrix very accurately from observations.