(625h) Multi-Rate Sampled-Data Observer Design for Nonlinear Systems with Multiple Measurement Delays

The problem of nonlinear observer design has been intensively studied for systems under fast sampling [1-3], which can be potentially applied to estimation, process control, and fault detection and identification. Motivated by practical implementation needs, however, one of the biggest challenges is to design an observer for general multi-rate sampled-data systems with multiple measurement delays (e.g., chemical processes, biological systems, networked control systems). In the monitoring of polymerization processes, for instance, the slow-sampled measurements with significant output delay often provide important quality information on the products, and hence must be incorporated in an intelligent manner together with the fast-sampled measurements, to make the entire system observable as well as improve the estimation accuracy. Different sampling rates and different sizes of measurement delays, caused by sample preparation, analysis and calculation, need to be accommodated in the observer design framework. Furthermore, the presence of possible perturbations in the sampling schedule as well as in the size of measurement delay makes the real-time monitoring even more challenging.

The observer design for linear multi-rate systems was recently investigated in [4]. The proposed multi-rate observer adopts the idea in [5] of using a state predictor to approximate the inter-sample behavior, and moreover, multiple, asynchronous inter-sample predictors are used for the multi-rate system as a generalization. The proposed multi-rate observer is based on an available continuous-time Luenberger observer design coupled with asynchronous inter-sample predictors. Each predictor generates an estimate of a sampled output in between consecutive measurements, and will get reinitialized once the associated, most-recent measurement becomes available. Sufficient and explicit conditions in terms of maximum sampling period were established to guarantee exponential stability of the error dynamics. Although the presence of possible measurement delays was not considered, it would be straightforward to incorporate multiple output delays by using the algorithm of dead time compensation as will be mentioned later.

In this paper, a nonlinear multi-rate sampled-data observer design is developed as a starting point before taking multiple output delays into account. In the same spirit as in [4], the proposed multi-rate sampled-data observer is based on an available continuous-time design coupled with asynchronous inter-sample predictors for the sampled measurements [6]. 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 vector small-gain theorem [6-7]. This sampled-data design also offers robustness with respect to perturbations in the sampling schedule. To handle the possible measurement delays, a dead time compensation algorithm is developed, which preserves the stability property of the multi-rate sampled-data observer, no matter how large the size of each delay is. Once a delayed measurement is obtained, the state estimates at its sampling instant can be re-calculated by using the multi-rate sampled-data observer. Then the process model can be utilized to predict the state at current time, given the delayed estimates, in the same spirit as in Smith-predictor methods. During the compensation, any available measurement can be treated as a delay-free output and the compensator will get reinitialized at every sampling instant where the measurement is available. This algorithm inherits all the nice properties (i.e., stability and robustness) of a multi-rate sampled-data observer. Moreover, it provides robustness with respect to the perturbations in the size of each measurement delay.

In the present work, we revisit the example of an industrial gas-phase polyethylene reactor in [4], which aims at monitoring the amount of active catalyst sites. The only fast-sampled output without delay is the temperature of the reactor. The gas concentrations in the reactor are measured on line by using gas chromatography, where the sampling normally occurs every 20 min with 8 min measurement delay. Cumulative melt index and density of polyethylene are measured off line in a quality control lab, where the sampling normally occurs every 40 min with 60 min measurement delay. However, perturbations in the sampling schedule as well as in the size of delay have been considered. A continuous-time nonlinear observer is first designed to cope with the nonlinearity of the process [3,8]. With the proposed framework of multi-rate multi-delay observer, we can obtain a continuous estimate of the unmeasured state with a satisfactory convergence rate. In addition, the output behavior between two consecutive samples can be estimated as well by using inter-sample predictors.

References:

[1] Krener, A. J., & Isidori, A. (1983). Linearization by output injection and nonlinear observers. Systems & Control Letters, 3(1), 47-52.

[2] Gauthier, J. P., Hammouri, H., & Othman, S. (1992). A simple observer for nonlinear systems applications to bioreactors. IEEE Transactions on Automatic Control, 37(6), 875-880.

[3] Kazantzis, N., & Kravaris, C. (1998). Nonlinear observer design using Lyapunov’s auxiliary theorem. Systems & Control Letters, 34(5), 241-247.

[4] Ling, C., & Kravaris, C. (2017). Multi-rate observer design for process monitoring using asynchronous inter-sample output predictions. AIChE Journal. DOI: 10.1002/aic.15707.

[5] Karafyllis, I., & Kravaris, C. (2009). From continuous-time design to sampled-data design of observers. IEEE Transactions on Automatic Control, 54(9), 2169-2174.

[6] Ling, C., & Kravaris, C. Multi-rate sampled-data observers based on a continuous-time design. Submitted to the 56th Conference on Decision and Control.

[7] Karafyllis, I., & Jiang, Z. P. (2011). A vector small-gain theorem for general non-linear control systems. IMA Journal of Mathematical Control and Information, 28(3), 309-344.

[8] Kazantzis, N., Kravaris, C., & Wright, R. A. (2000). Nonlinear observer design for process monitoring. Industrial & Engineering Chemistry Research, 39(2), 408-419.