Discrete-Time Linear-Quadratic Dynamic Games |
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Authors: | M Pachter K D Pham |
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Institution: | (1) Univ. Toledo, Toledo, OH, USA;(2) Univ. Geneva, Geneva, Switzerland |
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Abstract: | Finite-dimensional, time invariant, linear quadratic dynamic games are perhaps the best understood and researched class of
dynamic games. This is particularly true for continuous-time linear quadratic differential games. In this paper, the application
of the theory of dynamic games to signal processing is considered. We are interested in digital signal processing and therefore
we confine our attention to discrete-time linear-quadratic dynamic games (LQDG). In discrete-time the cost function contains
product terms between the decision variables which complicates the analysis compared to its continuous-time analogue. With
a view to facilitate the application of the theory of dynamic games to digital signal processing, and in particular, disturbance
rejection, the complete solution of the discrete-time LQDG is worked out and explicit results are obtained. Thus,discrete-time
LQDGs have the distinct advantage of being amenable to analysis, closed-form solutions are possible, and one is in tune with
modern digital signal processing techniques. In this paper, minimal necessary and sufficient conditions for the existence
of a solution to the discrete-time LQDG are provided and its explicit, closed-form, solution is worked out. This opens the
way to designing novel digital signal processing algorithms for disturbance rejection. Information plays a critical role in
game theory and in particular in dynamic games. Using our explicit solution of the deterministic LQDG, a hierarchy of three
zero-sum stochastic LQDGs characterized by a sequence of information patterns which increase in complexity is analyzed. |
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