TY - JOUR
T1 - Multi-time scale zero-sum differential games with perfect state measurements
AU - Pan, Zigang
AU - Başar, Tamer
PY - 1995/1
Y1 - 1995/1
N2 - 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 within O(∈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 the H∞-optimal control of three-time scale singularly perturbed linear systems under perfect state measurements.
AB - 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 within O(∈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 the H∞-optimal control of three-time scale singularly perturbed linear systems under perfect state measurements.
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U2 - 10.1007/BF01968532
DO - 10.1007/BF01968532
M3 - Article
AN - SCOPUS:0029209529
SN - 0925-4668
VL - 5
SP - 7
EP - 29
JO - Dynamics and Control
JF - Dynamics and Control
IS - 1
ER -