Based on a minimum least squares criterion and without any assumption on process model error and measuring device noise, a sequential estimation algorithm has been developed for nonlinear differential and algebraic systems with the use of variational calculus. Estimators for both continuous and discrete time data are derived, and can be used without explicitly considering the indices of differential and algebraic systems. The estimation algorithm has been applied to a large pilot scale fixed-bed reactor with decaying catalysts for simultaneous estimation of catalyst activity, temperature and concentration profiles. Although a simplified lumped reaction kinetics network and an approximate catalyst deactivation model were utilised by the estimator, the catalyst activity profile was accurately inferred and the dynamic behaviour of the reactor was excellently tracked using only two or three real-time temperature measurements. The estimated variation of the catalyst activity enabled us to fully understand the change of the hot spot magnitude and the shift of the temperature profile. Computational experience indicated that the estimator can be implemented on-line for reactor control and optimisation.