A multi-rate sampled-data observer for nonlinear systems under asynchronous sampling is developed in this work. The proposed multi-rate observer is based on a continuous-time design coupled with inter-sample output predictors for the sampled measurements. The sampled-data system with the multi-rate observer forms a hybrid system and it is shown that the error dynamics of the overall system is input-to-output stable with respect to measurement errors, by applying the Karafyllis-Jiang vector small-gain theorem. This sampled-data design also offers robustness with respect to perturbations in the sampling schedule. The performance of the proposed method is evaluated through a chemical batch reactor example.

A multi-rate sampled-data observer for nonlinear systems under asynchronous sampling is developed in this work. The proposed multi-rate observer is based on a continuous-time design coupled with inter-sample predictors for the sampled measurements. The sampled-data system with the multi-rate observer forms a hybrid system and it is shown that the error dynamics of the overall system is input-to-output stable with respect to measurement errors, by applying the KarafyllisJiang vector small-gain theorem. This sampled-data design also offers robustness with respect to perturbations in the sampling schedule. A numerical example is given to show the effectiveness of the proposed method.

Reversible addition–fragmentation chain–transfer (RAFT) polymerization of methyl methacrylate (MMA) is modeled and monitored using a multi-rate multi-delay observer in this work. First, to fit the RAFT reaction rate coefficients and the initiator efficiency in the model, in situ ${}^1$ H nuclear magnetic resonance (NMR) experimental data from small-scale (400 mL) reflux reactions is then used to validate the fitted model. The fitted model accurately predicts the polymer properties of the large-scale reactions with slight discordance at late reaction times. Based on the fitted model, a multi-rate multi-delay observer coupled with an inter-sample predictor and dead time compensator is designed, to account for the asynchronous multi-rate measurements with non-constant delays. The multi-rate multi-delay observer shows perfect convergence after a few sampling times when tested against the fitted model, and is in fair agreement with the real data at late reaction times when implemented based on the experimental measurements.

A nonlinear reduced-order state observer is applied to estimate the degree of polymerization in a series of polycondensation reactors. The finishing stage of polyethylene terephthalate synthesis is considered in this work. This process has a special structure of lower block triangular form, which is properly utilized to facilitate the calculation of the state-dependent gain in the observer design. There are two possible on-line measurements in each reactor. One is continuous, and the other is slow-sampled with dead time. For the slow-sampled titration measurement, inter-sample behavior is estimated from an inter-sample output predictor, which is essential in providing continuous corrections on the observer. Dead time compensation is carried out in the same spirit as the Smith predictor to reduce the effect of delay in the measurement outputs. By integrating the continuous-time reduced-order observer, the inter-sample predictor and the dead time compensator together, the degree of polymerization is accurately estimated in all reactors. The observer performance is demonstrated by numerical simulations. In addition, a pre-filtering technique is used in the presence of sensor noise.