Switched Control Systems with Limited Information: An Entropy Approach to Stabilization and Disturbance Attenuation

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

This project will extend a deep and intuitive notion of the entropy of a dynamic system to the control of switched systems subject to external disturbances. Similarly to related concepts of entropy from thermodynamics and information theory, the entropy of a dynamic system is a measure of uncertainty about its state -- more specifically, it is a measure of the rate of increase in this uncertainty. The importance of this concept of entropy for control engineering arises in part from its relation to the fundamental control tasks of stabilization and disturbance rejection. Stabilization means that the system will return to its desired operating condition after being perturbed. Disturbance rejection means that a persistent external disruption to the system will either not appear in the system output, or will at least be highly attenuated. For either of these desirable objectives to be achieved, it is required that the uncertainty growth -- due, for example, to open-loop unstable dynamics, external disturbances, or finite control resolution -- must be counteracted using information from measurements of the system. The use of entropy allows important practical limitations on sampling rates, sensor and actuator resolution, and communication bandwidth, to be rigorously incorporated into systematic analysis and design procedures. These critically important elements of real engineering systems are too often treated as an afterthought in current control engineering practice. The results of this project are expected to have significant impact on applications such as autonomous cars and aerial vehicles, chemical process control, and the safety and reliability of the electric power grid. The project also includes components for integrating basic research with personnel training and educational activities. Entropy characterizes the rate at which uncertainty about the state of a dynamical system increases with time. From the point of view of control theory, entropy corresponds to the minimal rate at which information about the system has to be communicated to the controller in order to stabilize the system. The control-theoretic entropy also informs the information rate needed to perform other control tasks, such as disturbance attenuation. In this project, these concepts and connections will be extended in the context of switched systems, for which the analysis of system behavior, as well as the task of controlling it, are considerably more complicated than the unswitched case. The project has the following three main theoretical goals: 1) to identify meaningful classes of switched systems for which the entropy (or reasonably tight bounds on it) can be calculated; 2) to design encoding and control strategies that achieve stabilization and disturbance attenuation for switched systems based on limited data rate feedback; and 3) to systematically use entropy as a tool to measure efficiency and performance of these control schemes. Switched systems describe many processes of industrial relevance, and the extension of the definition of control-theoretic entropy to these systems allows rigorous treatment of realistic limitations on sensing, actuation, and communication. Entropy-based analysis promises a unified analysis and design framework for control of realistic dynamics and control architectures.