Networked Control Systems as Stochastic Team Decision Problems: A General Introduction
In this chapter, a general probability theoretic framework for stochastic team decision problems is established, by defining and classifying information structures, interaction dynamics, policy spaces, and objective functions. A number of examples are inc
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Networked Control Systems as Stochastic Team Decision Problems: A General Introduction
2.1 Introduction Networked control systems can be viewed as stochastic decision problems with dynamic decentralized information structures or as stochastic dynamic teams, with each subcontroller viewed as an agent in a dynamic team. The goal of this introductory chapter is accordingly to introduce the reader to a general mathematical formulation of stochastic teams, first with static and then with dynamic information structures, and to discuss some salient features of these decision problems and associated solution concepts through some simple but illustrative examples. The chapter discusses both static stochastic teams (i.e., team decision problems where the information signals received by the decision makers are not affected by actions) and dynamic stochastic teams (where the information of at least one decision maker is affected by action). Sections 2.2, 2.3, and 2.6 deal with static teams, whereas Sects. 2.4 and 2.5 discuss dynamic teams. Section 2.2 provides a general formulation for static teams, which is followed by a complete analysis of a finite stochastic team problem under various information patterns, in Sect. 2.3. Section 2.6 provides some general explicit results on existence, uniqueness, and characterization of optimal solutions first for general static teams and then for special classes of teams with Gaussian statistics: those with quadratic and exponentiated quadratic costs. Sections 2.4 and 2.5 can be viewed as the counterparts of Sects. 2.2 and 2.3 for dynamic teams. First a precise mathematical formulation for dynamic team decision problems is given, in Sect. 2.4, along with various dynamic information structures and appropriate solution concepts, and then an illustrative example of a finite dynamic team is provided in Sect. 2.5, within the framework of which some important features of optimal solutions in teams are discussed. The chapter concludes with Sect. 2.7 which provides some bibliographical notes and guidelines for further reading on the topics covered herein.
S. Y¨uksel and T. Bas¸ar, Stochastic Networked Control Systems, Systems & Control: Foundations & Applications, DOI 10.1007/978-1-4614-7085-4 2, © Springer Science+Business Media New York 2013
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2 Networked Control Systems as Stochastic Team Decision Problems. . .
2.2 A Mathematical Framework For Static Decision Problems Multiple person stochastic decision problems could be formulated with varying degrees of generality, abstraction, and rigor, depending on the types of problems to be solved (i.e., the scope of coverage) and the level of mathematical sophistication to be expected from the reader. Common to all possible formulations, however, is the specification of five basic ingredients which are essential for a well-founded mathematical treatment of decision making under uncertainty. These are: 1. The number of decision makers (synonymously, agents or controllers) and the sets of alternative actions (synonymously, decisions or controls
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