Discrete-time mean-field stochastic \(H_2/H_\infty\) control. (English) Zbl 1380.93295
Summary: The finite horizon \(H_2/H_\infty\) control problem of mean-field type for discrete-time systems is considered in this paper. Firstly, the authors derive a mean-field Stochastic Bounded Real Lemma (SBRL). Secondly, a sufficient condition for the solvability of discrete-time mean-field stochastic Linear-Quadratic (LQ) optimal control is presented. Thirdly, based on SBRL and LQ results, this paper establishes a sufficient condition for the existence of discrete-time stochastic \(H_2/H_\infty\) control of mean-field type via the solvability of coupled matrix-valued equations.
MSC:
| 93E20 | Optimal stochastic control |
| 93E03 | Stochastic systems in control theory (general) |
| 93B36 | \(H^\infty\)-control |
| 49N10 | Linear-quadratic optimal control problems |
| 93C05 | Linear systems in control theory |
References:
| [1] | Khargonekar P P and Rotea M A, Mixed H2/H∞ control: A convex optimization approach, IEEE Trans. Automatic Control, 1991, 36(7): 824-837. · Zbl 0748.93031 · doi:10.1109/9.85062 |
| [2] | Limebeer D J N, Anderson B D O, and Hendel B, A nash game approach to mixed H2/H∞ control, IEEE Trans. Automatic Control, 1994, 39(1): 69-82. · Zbl 0796.93027 · doi:10.1109/9.273340 |
| [3] | Wu C S and Chen B S, Adaptive attitude control of spacecraft: Mixed H2/H∞ approach, J. Guid. Control Dyn., 2001, 24(4): 755-766. · doi:10.2514/2.4776 |
| [4] | Doyle J C, Glover K, Khargonekar P P, et al., State-space solutions to standard H2 and H∞ control problems, IEEE Trans. Automatic Control, 1989, 34(8): 831-847. · Zbl 0698.93031 · doi:10.1109/9.29425 |
| [5] | Chen B S, Tseng C S, and Uang H J, Mixed H2/H∞ fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach, IEEE Trans. Fuzzy Syst., 2000, 8(3): 249-265. · doi:10.1109/91.855915 |
| [6] | Chen B S and Zhang W, Stochastic H2/H∞ control with state-dependent noise, IEEE Trans. Automatic Control, 2004, 49(1): 45-57. · Zbl 1365.93539 · doi:10.1109/TAC.2003.821400 |
| [7] | Zhang W and Chen B S, State feedback H∞ control for a class of nonlinear stochastic systems, SIAM J. Control Optim., 2006, 44(6): 1973-1991. · Zbl 1157.93019 · doi:10.1137/S0363012903423727 |
| [8] | Gershon E and Shaked U, Stochastic H2 and H∞ output-feedback of discrete-time LTI systems with state multiplicative noise, Syst. Control Lett., 2006, 55(3): 232-239. · Zbl 1129.93372 · doi:10.1016/j.sysconle.2005.07.010 |
| [9] | EL Bouhtouri A, Hinrichsen D, and Pritchard A J, H∞-type control for discrete-time stochastic systems, Int. J. Robust Nonlinear Control, 1999, 9(13): 923-948. · Zbl 0934.93022 · doi:10.1002/(SICI)1099-1239(199911)9:13<923::AID-RNC444>3.0.CO;2-2 |
| [10] | Muradore R and Picci G, Mixed H2/H∞ control: The discrete-time case, Syst. Control Lett., 2005, 54(1): 1-13. · Zbl 1129.93435 · doi:10.1016/j.sysconle.2004.06.001 |
| [11] | Zhang W, Huang Y, and Zhang H, Stochastic H2/H∞ control for discrete-time systems with state and disturbance dependent noise, Automatica, 2007, 43(3): 513-521. · Zbl 1137.93057 · doi:10.1016/j.automatica.2006.09.015 |
| [12] | Zhang W, Huang Y, and Xie L, Infinite horizon stochastic H2/H∞ control for discrete-time systems with state and disturbance dependent noise, Automatica, 2007, 44(9): 2306-2316. · Zbl 1153.93030 · doi:10.1016/j.automatica.2008.01.028 |
| [13] | Hou T, Zhang W, and Ma H, Finite horizon H2/H∞ control for discrete-time stochastic systems with Markovian jumps and multiplicative noise, IEEE Trans. Automatic Control, 2010, 55(5): 1185-1191. · Zbl 1368.93661 · doi:10.1109/TAC.2010.2041987 |
| [14] | Lasry J M and Lions P L, Mean-field games, Jpn. J. Math., 2007, 2(1): 229-260. · Zbl 1156.91321 · doi:10.1007/s11537-007-0657-8 |
| [15] | Elliott R, Li X, and Ni Y H, Discrete-time mean-field stochastic linear-quadratic optimal control problems, Automatica, 2013, 49(11): 3222-3233. · Zbl 1358.93189 · doi:10.1016/j.automatica.2013.08.017 |
| [16] | Kac M, Fundations of kinetic theory, Proceedings of 3rd Berkeley Symposium on Mathematical Statistics and Probability, 1956, 3: 171-197. · Zbl 0072.42802 |
| [17] | Buckdahn R, Djehiche B, Li J, et al., Mean-field backward stochastic differential equations: A limit approach, Ann. Probab., 2009, 37(4): 1524-1565. · Zbl 1176.60042 · doi:10.1214/08-AOP442 |
| [18] | Buckdahn R, Li J, and Peng S, Mean-field backward stochastic differential equations and related partial differential equations, Stoch. Proc. their Appl., 2009, 119(10): 3133-3154. · Zbl 1183.60022 · doi:10.1016/j.spa.2009.05.002 |
| [19] | Crisan D and Xiong J, Approximate Mckean-Vlasov representations for a class of SPDE, Stoch.: Int. J. Probab. Stoch. Process., 2010, 82(1): 53-68. · Zbl 1211.60022 |
| [20] | Buckdahn R, Djehiche B, and Li J, A general maximum principle for SDEs of mean-field type, Appl. Math. Optim., 2011, 64(2): 197-216. · Zbl 1245.49036 · doi:10.1007/s00245-011-9136-y |
| [21] | Li J, Stochastic maximum principle in the mean-field controls, Automatica, 2012, 48(2): 366-373. · Zbl 1260.93176 · doi:10.1016/j.automatica.2011.11.006 |
| [22] | Shen Y and Siu T K, The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonlinear Analysis, 2013, 86: 58-73. · Zbl 1279.49015 · doi:10.1016/j.na.2013.02.029 |
| [23] | Andersson D and Djehiche B, A maximum principle for SDEs of mean-field type, Appl. Math. Optim., 2011, 63(3): 341-356. · Zbl 1215.49034 · doi:10.1007/s00245-010-9123-8 |
| [24] | Huang M, Caines P E, and Malhamé R P, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized e-Nash equilibria, IEEE Trans. Automatic Control, 2007, 52(9): 1560-1571. · Zbl 1366.91016 · doi:10.1109/TAC.2007.904450 |
| [25] | Huang M, Caines P E, and Malhamé R P, Social optima in mean field LQG control: Centralized and decentralized strategies, IEEE Trans. Automatic Control, 2012, 57(7): 1736-1751. · Zbl 1369.49052 · doi:10.1109/TAC.2012.2183439 |
| [26] | Yong J M, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM J. Control Optim., 2013, 51(4): 2809-2838. · Zbl 1275.49060 · doi:10.1137/120892477 |
| [27] | Du C, Xie L, Teoh J N, and Guo G, An improved mixed H2/H∞ control design for hard disk drives, IEEE Trans. Control Syst. Technol., 2005, 13(5): 832-839. · doi:10.1109/TCST.2005.852116 |
| [28] | Ma H, Zhang W, and Hou T, Infinite horizon H2/H∞ control for discrete-time time-varying Markov jump systems with multiplicative noise, Automatica, 2012, 48(7): 1447-1454. · Zbl 1246.93074 · doi:10.1016/j.automatica.2012.05.006 |
| [29] | Chen X and Zhou K, Multiobjective H2/H∞ control design, SIAM J. Control Optim., 2001, 40(2): 628-660. · Zbl 0997.93033 · doi:10.1137/S0363012998346372 |
| [30] | Lin C and Chen B S, Achieving pareto optimal power tracking control for interference limited wireless systems via multi-objective H2/H∞ optimization, IEEE Trans. Wireless Commun., 2013, 12(12): 6154-6165. · doi:10.1109/TWC.2013.103113.130016 |
| [31] | Hafayed M, Singular mean-field optimal control for forward-backward stochastic systems and applications to finance, Int. J. Dynam. Control, 2014, 2(4): 542-554. · doi:10.1007/s40435-014-0080-y |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
