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Infinite horizon \(H_2/H_\infty \) optimal control for discrete-time Markov jump systems with (\(x,u,v\))-dependent noise

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Abstract

In this paper, an infinite horizon \(H_2/H_\infty \) control problem is addressed for a broad class of discrete-time Markov jump systems with (\(x,u,v\))-dependent noises. First of all, under the condition of exact detectability, the stochastic Popov–Belevich–Hautus (PBH) criterion is utilized to establish an extended Lyapunov theorem for a generalized Lyapunov equation. Further, a necessary and sufficient condition is presented for the existence of state-feedback \(H_2/H_\infty \) optimal controller on the basis of two coupled matrix Riccati equations, which may be solved by a backward iterative algorithm. A numerical example with simulations is supplied to illustrate the proposed theoretical results.

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Acknowledgments

This work was supported by the Mathematical Tianyuan Youth Foundation of China (No. 11126094), the National Natural Science Foundation of China (No. 61174078), the Key Project of Natural Science Foundation of Shandong Province (No. ZR2009GZ001), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20103718110006), the Research Fund for the Taishan Scholar Project of Shandong Province of China, and the SDUST Research Fund (No. 2011KYTD105).

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Authors and Affiliations

  1. College of Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Ting Hou & Hongji Ma

  2. College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao, 266590, China

    Weihai Zhang

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  1. Ting Hou
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  2. Weihai Zhang
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  3. Hongji Ma
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Correspondence to Weihai Zhang.

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Hou, T., Zhang, W. & Ma, H. Infinite horizon \(H_2/H_\infty \) optimal control for discrete-time Markov jump systems with (\(x,u,v\))-dependent noise. J Glob Optim 57, 1245–1262 (2013). https://doi.org/10.1007/s10898-012-0027-9

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  • DOI: https://doi.org/10.1007/s10898-012-0027-9

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