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Indefinite LQ optimal control for discrete-time uncertain systems

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Abstract

This paper is concerned with a linear quadratic (LQ) optimal control for discrete-time uncertain systems, with indefinite state and control weighting matrices in the cost function. Firstly, a recurrence equation of general optimal control problem for discrete-time uncertain systems is obtained by applying Bellman’s principle of optimality. Then, the optimal state feedback control is obtained based on the recurrence equation. Moreover, a sufficient condition of well-posedness for the LQ problem is proposed and a general expression for the optimal control set is given. Furthermore, a numerical example is presented by using the obtained results. Finally, as an application of the indefinite LQ optimal control, an optimal production inventory problem of uncertain environment is solved.

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References

  • Athans M (1968) The matrix minimum principle. Inf Control 11:592–606

    Article  MathSciNet  Google Scholar 

  • Athans M (1971) Special issues on linear-quadratic-Gaussian problem. IEEE Trans Autom Control 16(6):527–869

    Google Scholar 

  • Chen Y, Zhu Y (2018) Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. J Ind Manag Optim 14(3):913–930

    MathSciNet  MATH  Google Scholar 

  • Chen S, Li X, Zhou X (1998) Stochastic linear quadratic regulators with indefinite control weight costs. SIAM J Control Optim 36(5):1685–1702

    Article  MathSciNet  Google Scholar 

  • Davis M (1977) Linear estimation and stochastic control. Chapman and Hall, London

    MATH  Google Scholar 

  • Deng L (2012) Multidimensional uncertain optimal control of linear quadratic models with jump. J Comput Inf Syst 8(18):7441–7448

    Google Scholar 

  • Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New York

    Book  Google Scholar 

  • Hinrichsen D, Prichard AJ (1998) Stocastic \(H^{\infty }\). SIAM J Control Optim 36:1504–1538

    Article  MathSciNet  Google Scholar 

  • Kahneman D, Tversky A (1979) Prospect theory: an analysis of decision under risk. Econometrica 47(2):263–292

    Article  MathSciNet  Google Scholar 

  • Kalman RE (1960) Contributions to the theory of optimal control. Bol Soc Mat Mex 5(2):102–119

    MathSciNet  Google Scholar 

  • Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  • Liu B (2009) Some research problems in uncertainy theory. J Uncertain Syst 3(1):3–10

    Google Scholar 

  • Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, Berlin

    Book  Google Scholar 

  • Penrose R (1955) A generalized inverse of matrices. Proc Camb Philos Soc 51:406–413

    Article  Google Scholar 

  • Perez T, Haimovich H, Goodwin GC (2004) On optimal control of constrained linear systems with imperfect state information and stochastic disturbances. Int J Robust Nonlinear Control 14(4):379–393

    Article  MathSciNet  Google Scholar 

  • Rami MA, Zhou X (2000) Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls. IEEE Trans Autom Control 45(6):1131–1143

    Article  MathSciNet  Google Scholar 

  • Rami MA, Chen X, Moore JB, Zhou X (2001a) Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls. IEEE Trans Autom Control 46(3):428–440

    Article  MathSciNet  Google Scholar 

  • Rami MA, Moore JB, Zhou X (2001b) Indefinite stochastic linear quadratic control and generalized differential Riccati equation. SIAM J Control Optim 40(4):1296–1311

    Article  MathSciNet  Google Scholar 

  • Rami MA, Chen X, Zhou X (2002) Discrete-time indefinite LQ control with state and control dependent noises. J Glob Optim 23(3):245–265

    Article  MathSciNet  Google Scholar 

  • Sheng L, Zhu Y (2013) Optimistic value model of uncertain optimal control. Int J Uncertain Fuzziness Knowl Based Syst 21(supp01):75–87

    Article  MathSciNet  Google Scholar 

  • Shu Y, Zhu Y (2018) Stability analysis of uncertain singular systems. Soft Comput 22(17):5671–5681

    Article  Google Scholar 

  • Wonham WM (1968) On a matrix Riccati equation of stochastic control. SIAM J Control Optim 6(4):681–697

    Article  MathSciNet  Google Scholar 

  • Yan H, Zhu Y (2015) Bang–bang control model for uncertain switched systems. Appl Math Model 39(10–11):2994–3002

    Article  MathSciNet  Google Scholar 

  • Yang X, Gao J (2016) Linear-quadratic uncertain differential game with application to resource extraction problem. IEEE Trans Fuzzy Syst 24(4):819–826

    Article  Google Scholar 

  • Zhou XY, Li D (2000) Continuous-time mean-variance portfolio selection: a stocastic LQ framework. Appl Math Optim 42(1):19–33

    Article  MathSciNet  Google Scholar 

  • Zhu Y (2010) Uncertain optimal control with application to a portfolio selection model. Cybern Syst 41(7):535–547

    Article  Google Scholar 

  • Zhu Y (2012) Functions of uncertain variables and uncertain programming. J Uncertain Syst 6(4):278–288

    Google Scholar 

Download references

Acknowledgements

This study was funded by the National Natural Science Foundation of China (No. 61673011), the Training Plan of Young Key Teachers in Universities of Henan Province (No. 2018GGJS096) and the Nanhu Scholars Program for Young Scholars of XYNU.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China

    Yuefen Chen

  2. School of Science, Nanjing University of Science and Technology, Nanjing, 210094, China

    Yuanguo Zhu

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  1. Yuefen Chen
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  2. Yuanguo Zhu
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Correspondence to Yuefen Chen.

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Communicated by A. Di Nola.

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Chen, Y., Zhu, Y. Indefinite LQ optimal control for discrete-time uncertain systems. Soft Comput 24, 267–279 (2020). https://doi.org/10.1007/s00500-019-04350-3

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  • DOI: https://doi.org/10.1007/s00500-019-04350-3

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