(747c) Multi-Rate Sampled-Data Control of Spatially Distributed Process Systems

With the wide-spread use of digital sensors and controllers in the control of continuous-time process systems over the past few decades, sampled-data control systems have been the focus of considerable research interest. The discrete availability and transfer of information (e.g., sensor measurements, control signals) in these systems typically stems from the intrinsic limitations on the data collection, transmission and processing capabilities of the measurement and control devices, but can also arise due to resource constraints on the communication medium interfacing the process with the control system. It is now well understood that these limitations may have a detrimental influence on the stability and performance properties of the closed-loop system, and that sampling effects need to be taken into consideration in the design of the control system.

These considerations have motivated considerable research work on the analysis and design of sampled-data control systems, with the overwhelming majority of existing studies focused on lumped parameter systems modeled by systems of ordinary differential equations. Many important engineering processes and systems, however, are characterized by significant spatial variations due to the presence of strong diffusive, convective and phase dispersion phenomena and are naturally described by partial differential equations. Compared with the extensive body and long history of research work on control of distributed parameter systems, the design of sampled-data control systems for spatially distributed processes has received only limited attention. Recently, we introduced in [1],[2] a framework for sampled-data control of distributed processes modeled by parabolic partial differential equations (PDEs) with measured outputs that are sampled over a resource-limited communication medium at discrete time instances. The main idea was to include an approximate finite-dimensional model of the infinite-dimensional system to provide the controller with estimates of the dominant slow states to compensate for the discrete availability of the sensor measurements, and to execute model updates using the actual measurements at each sampling time. In these studies, the controller design and analysis were carried out under the assumption that all the measurement sensors have the same fixed sampling rate and are forced to transmit their data to the controller concurrently. In many practical situations, however, limitations on the measurement capabilities of different sensors may result in a significant gap between the sampling rates, and in such cases a synchronized sampling mechanism may not be the best choice. For example, in chemical reactors, the composition and density measurements typically need several minutes of analysis, while the temperature can be measured at a relatively fast rate. Moreover, the importance of the measurement collected is another factor that can trigger the use of multi-rate sampling. It is reasonable to apply a fast sampling rate to the sensors placed at certain critical locations in the process (e.g., where frequent monitoring and tight control are required), while reducing the sampling rates of the other sensors in order to reduce cost and optimize energy resource consumption (e.g., battery life).

In this contribution, we present 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 dictated by the range of sensor sampling rates 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. Singular perturbation techniques that exploit the two time-scale behavior of the spectrum of the differential operator are used to derive conditions for closed-loop stability of the sampled-data closed-loop infinite-dimensional system. Finally, the theoretical results are applied to the control of a low-density polyethylene tubular reactor.

References:

[1] Yao, Z. and N. H. El-Farra, ``On Model-Based Networked Control of Nonlinear Spatially Distributed Process Systems," Proceedings of the 18th Mediterranean Conference on Control and Automation, pp. 898-903, Marrakech, Morocco, 2010.

[2] Yao, Z. and N. H. El-Farra, ``Robust Fault Detection and Reconfigurable Control of Uncertain Sampled-Data Distributed Processes," Proceedings of 50th IEEE Conference on Decision and Control, pp. 4925-4930, Orlando, FL, 2011.