Active systems with uncertain parameters: an information-theoretic perspective

Active Systems with Uncertain Parameters: an Information-Theoretic Perspective by Gireeja Vishnu Ranade Doctor of Philosophy in Electrical Engineering and Computer Sciences University of California, Berkeley Professor Anant Sahai, Chair The sister fields of control and communication share many common goals. While communication aims to reduce uncertainty by communicating information about the state of the world, control aims to reduce uncertainty by moving the state of the world to a known point. Furthermore, transmitters must communicate over the unreliability in the communication channel, while controllers must overcome unreliability in the sensing and actuation channels of the system to stabilize it. Extensive work in information theory has provided a framework with which we can understand the fundamental limits on the communication capacity of a communication channel. This dissertation builds on the information-theoretic perspective to understand the limits on the ability of controllers to actively dissipate uncertainty in systems. High-performance control systems include two types of uncertainty: noise that might be introduced by nature, and inaccuracy that might be introduced by modeling, sampling errors or clock jitter. The first of these is often modeled as an additive uncertainty and is the object of most prior work at the intersection of communication and control. This dissertation focuses on the multiplicative uncertainty that comes from modeling and sampling inaccuracies. Multiplicative uncertainty could be introduced from the observation mechanism that senses the state (the sensing channel) or from the actuation mechanism that implements actions (the actuation channel). This dissertation examines the control capacity of systems where the parameters of these channels are changing so fast that they cannot be perfectly tracked at the timescale at which control actions must be performed. This parallels the fast-fading models used for wireless communication channels. This dissertation defines a notion of the “control capacity” of an unreliable actuation channel as the fundamental limit on the ability of a controller to stabilize a system over that channel. Our definition builds from the understanding of communication capacity as defined by Shannon. The strictest sense of control capacity, zero-error control capacity, emulates the worst-case sense of performance that the robust control paradigm captures. The weakest sense of control capacity, which we call “Shannon” control capacity, focuses on the typical behavior of the zeroth-moment or the log of the state. In between these two there exists a