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.

In a new collaborative project involving the University of California, Berkeley, NASA Ames Research Center, and Honeywell Systems Research Center, we have begun the study of hierarchical, hybrid control systems in the framework of air traffic management systems (ATMS). The need for a new ATMS arises from the overcrowding of large urban airports and the need to more efficiently land and take off larger numbers of aircraft, without building new runways. Technological advances that make a more advanced air traffic control system a reality include the availability of relatively inexpensive and fast real time computers both on board the aircraft and in the control tower. The usefulness of these technological advances is currently limited by today's air traffic control system, which involves the use of "freeways" in the Terminal RADAR Approach Control (TRACON) region around urban airports. These freeways are set approach patterns to runways which do not allow for the possibility of so-called...

Abstract. Embedded systems possess a hybrid structure, combining continuous-variable dynamics and logical decision-making. We study such real-world hybrid control systems and introduce a mathematical model of them. Then, we develop tools for analyzing their stability. Finally, we review the theory of optimal hybrid control and present algorithms for synthesizing optimal hybrid control systems. 1

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The problem of feedback linearization of nonlinear systems in the presence of input saturation is considered in this paper. Although different approaches for stabilization of linear and nonlinear systems under bounded controls have been suggested in the literature, the need to perform output tracking for nonlinear systems dictates the choice of feedback linearization in our control design. This method results in linear error dynamics allowing the estimation of the convergence rate. Explicit characterization of the space of feasible trajectories that can be tracked using bounded inputs is provided followed by simulation results. Keywords: Input saturation, trajectory tracking, feedback linearization. 1 Introduction In this paper, the problem of stabilization and trajectory tracking for feedback linearizable systems under input constraints is considered. We are motivated in part by the successful application of feedback linearization techniques to flight control problems. In an aircraft...

In previous work I have concentrated on formalizing the notion of a hybrid system as switching among an indexed collection of dynamical systems. I have also studied in some depth the modeling, analysis, and control of such systems. Here, I give a quick overview of the area of hybrid systems. I also briefly review the formal definition and discuss the main approaches taken in the study of hybrid systems. Finally, I elucidate issues in each of the research areas in light of previous results. 1