Abstract-There is an increasing interest in studying control systems employing multiple sensors and actuators that are geographically distributed. Communication is an important component of these distributed and networked control systems. Hence, there is a need to understand the interactions between the control components and the communication components of the distributed system. In this paper, we formulate a control problem with a communication channel connecting the sensor to the controller. Our task involves designing the channel encoder and channel decoder along with the controller to achieve different control objectives. We provide upper and lower bounds on the channel rate required to achieve these different control objectives. In many cases, these bounds are tight. In doing so, we characterize the "information complexity" of different control objectives.

In this paper, we show that the coarsest, or least dense, quantizer that quadratically stabilizes a single input linear discrete time invariant system is logarithmic, and can be computed by solving a special linear quadratic regulator (LQR) problem. We provide a closed form for the optimal logarithmic base exclusively in terms of the unstable eigenvalues of the system. We show how to design quantized state-feedback controllers, and quantized state estimators. This leads to the design of hybrid output feedback controllers. The theory is then extended to sampling and quantization of continuous time linear systems sampled at constant time intervals. We generalize the definition of density of quantization to the density of sampling and quantization in a natural way, and search for the coarsest sampling and quantization scheme that ensures stability. We show that the resulting optimal sampling time is only function of the sum of the unstable eigenvalues of the continuous time system, and that the associated optimal quantizer is logarithmic with the logarithmic base being a universal constant independent of the system. The coarsest sampling and quantization scheme so obtained is related to the concept of minimal attention control recently introduced by Brockett. Finally, by relaxing the definition of quadratic stability, we show how to construct logarithmic quantizers with only finite number of quantization levels and still achieve practical stability of the closed-loop system. This final result provides a way to practically implement the theory developed in this paper.

Complex natural and engineered systems typically possess a hierarchical structure, characterized by continuousvariable dynamics at the lowest level and logical decision-making at the highest. Virtually all control systems today-from flight control to the factory floor-perform computer-coded checks and issue logical as well as continuous-variable control commands. The interaction of these different types of dynamics and information leads to a challenging set of "hybrid" control problems. We propose a very general framework that systematizes the notion of a hybrid system, combining differential equations and automata, governed by a hybrid controller that issues continuous-variable commands and makes logical decisions. We first identify the phenomena that arise in real-world hybrid systems. Then, we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems, evolving on continuous-variable state spaces and subject to continuous controls and discrete transitions. The model captures the identified phenomena, subsumes previous models, yet retains enough structure on which to pose and solve meaningful control problems. We develop a theory for synthesizing hybrid controllers for hybrid plants in an optimal control framework. In particular, we demonstrate the existence of optimal (relaxed) and near-optimal (precise) controls and derive "generalized quasi-variational inequalities" that the associated value function satisfies. We summarize algorithms for solving these inequalities based on a generalized Bellman equation, impulse control, and linear programming.

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