Testing Homogeneity of Coefficients in Distributed Systems With Application to Quality Monitoring

The problem of testing homogeneity of coefficients in distributed parameter systems (DPS) is stated and motivated by an example from the quality monitoring of the copper hot rolling process. When a system described by partial differential equation(s) (PDE) is observed using sensors or digital cameras and its coefficients are suspected to be inhomogeneous in space, then it is rather difficult to estimate them from noisy observations. Such observations can however be sufficient to test whether these coefficients are constant in space. The test for homogeneity is proposed, which is computationally simple and applicable online. The main idea for its construction is based on the expansion of a PDE solution into a series of eigenfunctions. Then, coefficients of this series are estimated from the observations of the process state: directly and indirectly, through estimates of PDE parameters, provided that they are constant. Finally, the squared differences between these two sets of coefficients are used to form the test statistic. The results of verification of the test on simulated and experimental data are also provided.