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Zbl 0951.93002
Branicky, Michael S.; Borkar, Vivek S.; Mitter, Sanjoy K.
A unified framework for hybrid control: Model and optimal control theory. (English)
[J] IEEE Trans. Autom. Control 43, No.1, 31-45 (1998). ISSN 0018-9286

Summary: Complex natural and engineered systems typically possess a hierarchical structure, characterized by continuous variable 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.\par 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 ot 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.

MSC 2000:: *93A13 Hierarchical systems
49L20 Dynamic programming method (infinite-dimensional problems)
93A30 Mathematical modelling of systems
49J40 Variational methods including variational inequalities
90C05 Linear programming

Keywords: dynamic programming; hierarchical systems; optimal control; state-space methods; hybrid system; automata; hybrid controller; generalized Bellman equation

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Zentralblatt MATH Berlin [Germany]