Constrained profile estimation for distributed parameter system in one dimension using orthogonal collocation

Abstract

Several industrial processes fall under the category of Distributed parameter systems (DPS) which are represented as coupled partial differential equations (PDEs). Obtaining accurate and physically meaningful profile estimates of states despite only a limited number of spatially distributed measurements, is a pre-requisite for adequately controlling such systems. In this work, we propose an online constrained state profile estimation approach for DPS. The approach is based on reduced-dimension model represented by differential and algebraic equations (DAE) which in turn is developed by discretization of the PDEs along the spatial coordinate using the orthogonal collocation (OC) method. A profile prediction error based approach is proposed for systematically fixing the dimension of the OC model to be used. Further, to compensate for the inevitable model plant mismatch (MPM) arising in the process of the dimensionality reduction, additive zero mean Gaussian white noise processes are introduced in both state dynamics as well as measurement model. Statistics of these noises are obtained based on simulation based realizations of approximation errors at collocation points and measurement locations. To incorporate the identified correlation between the resulting state and measurement noises in the estimation method, we propose a constrained DAE-EKF (extended Kalman Filter) extension to deal with such correlations. The estimation approach also involves an appropriately formulated optimization based update step to ensure constraint satisfaction over the entire spatial profile leading to physically meaningful state profile estimates. The constrained profile estimator is also integrated with an online moving window based parameter estimation, which enables tracking of both time varying parameters and state profiles. The efficacy of the proposed approach is demonstrated by conducting stochastic simulation studies using two tubular reactor systems. The results show that the proposed constrained online state estimation approach is able to obtain accurate profile estimates at low computational cost, while ensuring that the profile estimates satisfy the given constraints.