(183f) A Dead Time Compensation Approach for State Estimation of Sampled-Data Systems in the Presence of Large Measurement Delays

State estimation of continuous-time dynamical systems in the presence of sampled and delayed measurements (e.g., chemical processes, networked control systems) has caught researchers’ attention over the last decade [1-3]. In addition to fast-sampled measurements, it is crucial to incorporate slow-sampled measurements in the observer design framework to make the entire system observable as well as improve the estimation accuracy, despite their low sampling rate and large output delay. Moreover, different sampling rates and different sizes of delays should be handled in the observer design.

In this paper, a multi-rate sampled-data observer design is adopted as a starting point before considering the presence of possible measurement delays. The multi-rate observer design (see [4] for linear systems and [5] for nonlinear systems) is based on an available continuous-time observer design coupled with multiple, asynchronous inter-sample predictors for the sampled measurements. Each predictor generates an estimate of a sampled output between consecutive measurements and gets reinitialized once the associated, most-recent measurement becomes available. Sufficient conditions were derived to guarantee stability property of the error dynamics by using Lyapunov’s second method and the vector small-gain theorem [6], respectively.

To handle measurement delays, we propose a model-based dead time compensation algorithm and show that the stability property of the multi-rate observer will be preserved under arbitrarily large measurement delays. The algorithm is described as follows: 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. By integrating the observer and compensator equations, corrected estimates can be obtained at current time which will subsequently be used in the inter-sample prediction during the time when there is no sampled measurement available. This algorithm inherits all the nice properties (i.e., stability and robustness) of a multi-rate sampled-data observer. Moreover, the convergence property is not affected by perturbations in the size of each measurement delay. An industrial gas-phase polyethylene reactor example will be revisited to illustrate the applicability of the proposed method in both linearized system and nonlinear system.

References:

[1] Ahmed-Ali, T., Karafyllis, I., & Lamnabhi-Lagarrigue, F. (2013). Global exponential sampled-data observers for nonlinear systems with delayed measurements. Systems & Control Letters, 62(7), 539-549.

[2] Khosravian, A., Trumpf, J., Mahony, R., & Hamel, T. (2015). Recursive attitude estimation in the presence of multi-rate and multi-delay vector measurements. Proceedings of the American Control Conference, 3199-3205.

[3] Shen, Y., Zhang, D., & Xia, X. (2017). Continuous observer design for a class of multi-output nonlinear systems with multi-rate sampled and delayed output measurements. Automatica, 75, 127-132.

[4] Ling, C., & Kravaris, C. (2017). Multi‐rate observer design for process monitoring using asynchronous inter‐sample output predictions. AIChE Journal, 63(8), 3384-3394.

[5] Ling, C. and Kravaris, C. (2017). Multi-rate sampled-data observers based on a continuous-time design. Proceedings of the 56^thConference on Decision and Control, 3664-3669.

[6] 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.