(523h) Design of Sampled Data Nonlinear Observers for Processes Monitoring

State observers provide a powerful tool for model-based process monitoring. On-line estimation of unmeasured states provides critical information for the early detection of operational problems that could affect safety or product quality.  

Chemical processes typically evolve continuously in time, but process measurements are discrete in time and the implementation of the observer is realized in discrete time using the sampled data. Therefore, time discretization issues naturally arise in specific applications. If the sampling period of all measurements is relatively small, one possible approach is to design the observer in continuous time and discretize the observer equations at the implementation stage. Another possibility is to discretize the process dynamics model at the beginning and perform the observer design in discrete time. Both approaches are reasonable and there no clear superiority of one over the other in the case of linear systems.

For nonlinear processes, time discretization of a continuous-time observer, or a priori discretization of the process dynamics with time step equal to the sampling period, generally involves numerical errors, whose effect on the accuracy of the state estimates needs to be evaluated.

In the presence of infrequent measurements with large sampling periods, special care is needed in the design of the observer, to account for system dynamics in between sampling instants.  In particular, inter-sample behaviour of the system states should be predicted and accounted for in the observer calculations ([1]).

The present paper will explore all three approaches in the design of sampled data observers for nonlinear processes. In order to have a common basis of comparison, the same observer synthesis point of view will be applied: nonlinear Luenberger observer based on exact error linearization with eigenvalue assignment ([2], [3]).

After a discussion on the general theoretical issues involved, the paper will proceed with a detailed evaluation and comparison in a batch reactor case study. In particular, a chemical reaction system following the Lotka-Volterra mechanism will be considered. One of the concentrations will be assumed to be measured and the other will need to be estimated on line. This system is chosen for two reasons:  First, it is an oscillatory system, so that discretization errors have a pronounced effect on observer performance. Second, it is a system where the synthesis of a nonlinear Luenberger observer based on exact error linearization and eigenvalue assignment can be performed analytically.

The results for the Lotka-Volterra reaction system have shown that both the discretized continuous-time observer and the discrete observer design based on a priori discretization, lead to accurate state estimates under fast sampling. However, estimation accuracy decreases quickly with increasing size of the sampling period. Moreover, discrete observer design based on a priori discretization suffers from poor robustness to errors in the sampling period. The ultimate solution for the case of large sampling periods and in the presence of possible errors in the sampling schedule is to use the continuous-time observer in conjunction with an inter-sample output predictor, to predict the evolution of the output during the time period in between two consecutive measurements ([1]).

References:

[1] I. Karafyllis and C. Kravaris, “From continuous-time design to sampled-data design of observers”, IEEE Transactions on Automatic Control, Vol. AC-54, pp.2169-2174, 2009.

[2]  C. Kravaris, J. Hahn and Y. Chu, “Advances and selected recent developments in state and parameter estimation”, Computers & Chemical Engineering, Vol. 51, pp.111-123, 2013.

[3]  N. Kazantzis and C. Kravaris, “Nonlinear observer design using Lyapunov's auxiliary theorem”, Systems & Control Letters, Vol. 34, pp.241-247, 1998.