Abstract
In this chapter we focus on the construction of robust Nash strategies for a class of multimodel games described by a system of ordinary differential equations with parameters from a given finite set. Such strategies entail the “Robust equilibrium” being applied to all scenarios (or models) of the game simultaneously. The multimodel concept allows one to improve the robustness of the designed strategies in the presence of some parametric uncertainty. The game solution corresponds to a Nash-equilibrium point of this game. In LQ dynamic games the equilibrium strategies obtained are shown to be linear functions of the so-called weighting parameters from a given finite-dimensional vector simplex. This technique permits us to transform the initial game problem, formulated in a Banach space (the control functions are to be found) to a static game given in finite-dimensional space (simplex). The corresponding numerical procedure is discussed. The weights obtained appear in an extended coupled Riccati differential equation. The effectiveness of the designed controllers is illustrated by a two-dimensional missile guidance problem.
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Notes
- 1.
Notice that this approach differs from the similar concept developed in van den Broek et al. (2003) and Amato et al. (1998) where the players take into account the game uncertainty represented by a malevolent input which is subject to a cost penalty or a direct bound. Then they used as a base the \(\mathcal{H}^{\infty }\) theory of robust control to design the robust strategies for all players. Here we follow another concept based on the Robust Maximum Principle discussed in the previous chapters.
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Boltyanski, V.G., Poznyak, A.S. (2012). Multimodel Differential Games. In: The Robust Maximum Principle. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8152-4_14
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DOI: https://doi.org/10.1007/978-0-8176-8152-4_14
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