Multi-time scale zero-sum differential games with perfect state measurements |
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Authors: | Zigang Pan Tamer Ba?ar |
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Affiliation: | (1) Decision and Control Laboratory, Coordinated Science Laboratory, University of Illinois, 61801 Urbana, IL, USA;(2) Department of Electrical and Computer Engineering, University of Illinois, 61801 Urbana, IL, USA |
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Abstract: | We obtain necessary and sufficient conditions for the existence of approximate saddle-point solutions in linear-quadratic zero-sum differential games when the state dynamics are defined on multiple (three) time scales. These different time scales are characterized in terms of two small positive parameters 1, and 2, and the terminology approximate saddle-point solution is used to refer to saddle-point policies that do not depend on 1 and 2, while providing cost levels withinO(1) of the full-order game. It is shown in the paper that, under perfect state measurements, the original game can be decomposed into three subgames-slow, fast and fastest, the composite saddle-point solution of which make up the approximate saddle-point solution of the original game. Specifically, for the minimizing player, it is necessary to use a composite policy that uses the solutions of all three subgames, whereas for the maximizing player, it is sufficient to use a slow policy. In the finite-horizon case this slow policy could be a feedback policy, whereas in the infinite-horizon case it has to be chosen as an open-loop policy that is generated from the solution and dynamics of the slow subgame. These results have direct applications in theH
-optimal control of three-time scale singularly perturbed linear systems under perfect state measurements.Research supported in part by the National Science Foundation under Grant ECS 91-13153, and in part by the U.S. Department of Energy under Grant DE-FG-02-88-ER-13939. This paper is dedicated to the memory of John Breakwell. |
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