A predictor-corrector approach for multi-rate sampled-data control of spatially distributed systems

This work presents a methodology for the design of model-based output feedback controllers for spatially distributed systems modeled by highly-dissipative partial differential equations (PDEs) with multiple measured outputs that are sampled at different sampling rates. Initially, an approximate finite-dimensional system that captures the dominant dynamics of the infinite-dimensional system is obtained and used to design an observer-based output feedback controller. Due to the lack of continuous measurements, an inter-sample model predictor is included in the controller and used to provide the observer with estimates of the unavailable outputs. The model predictions are then updated and corrected at each time that a measurement becomes available. Owing to the different sampling rates of the available measurement sensors, the model update is performed using different outputs, or combinations of outputs, at each update time. A hybrid system formulation that captures the model update pattern is used to analyze the stability properties of the sampled-data finite-dimensional closed-loop system and derive a necessary and sufficient condition for closed-loop stability. The condition is used to explicitly characterize the interdependence between the different sampling rates, the size of the model uncertainty, the controller and observer design parameters, and the spatial locations of the control actuators and measurement sensors. Finally, the theoretical results are illustrated using a diffusion-reaction process example.