Abstract
Optimal control problems governed by two different types of uncertain discrete-time singular systems are investigated under expected value criterion. The objective function including uncertain variables is optimized with the help of expected value method provided that the singular systems are both regular and impulse-free. At first, based on the principle of dynamic programming, a recurrence equation is derived to simplify an optimal control model for a class of uncertain discrete-time singular systems. After that, according to uncertainty theory and the recurrence equation, two kinds of optimal control problems subject to an uncertain linear singular system and an uncertain singular system with quadratic input variables are considered in order, and the optimal solutions are both presented by accurate expressions. A numerical example and a dynamic input-output model are settled to illustrate the effectiveness of the results obtained.
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Chen, Y., & Zhu, Y. (2016). Indefinite LQ optimal control with equality constraint for discrete-time uncertain systems. Japan Journal of Industrial and Applied Mathematics, 33(2), 361–378.
Choi, C. (1990). A survey of numerical methods for solving matrix Riccati differential equations. IEEE Proceedings on Southeaston, 2, 696–700.
Cobb, J. (1984). Controllability, observability, and duality in singular systems. IEEE Transactions on Automatic Control, 29, 1076–1082.
Dai, L. (1989). Singular Control Systems. Berlin, Germany: Springer-Verlag.
Federico, S. (2011). A stochatic control problem with delay arising in a pension fund model. Finance and Stochastics, 15, 421–459.
Ge, X., & Zhu, Y. (2013). A necessary condition of optimality for uncertain optimal control problem. Fuzzy Optimization and Decision Making, 12(1), 41–51.
Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. New York: Wiley.
Ishihara, J., & Terra, M. (2002). On the Lyapunov theorem for singular systems. IEEE Transactions on Automatic Control, 47(11), 1926–1930.
Kang, Y., & Zhu, Y. (2012). Bang-bang optimal control for multi-stage uncertain systems. International Journal on Information, 15(8), 3229–3237.
Kirk, D. E. (2012). Optimal control theory: an introduction. Prentice Hall.
Konstantin, A., Oussama, H., & Alexey, P. (2015). Infinite horizon optimal impulsive control with applications to Internet congestion control. International Journal of Control, 88(4), 703–716.
Kumaresan, N., & Balasubramaniam, P. (2009). Optimal control for stochastic linear quadratic singular system using neural networks. Journal of Process Control, 19, 482–488.
Lewis, F. L. (1989). A survey of singular systems. Circuits Systems and Signal Processing, 8, 3–36.
Li, B., & Zhu, Y. (2017). Parametric optimal control uncertain linear quadratic models. Applied Soft Computing, 56, 543–550.
Liu, B. (2007). Uncertainty Theory (2nd ed.). Berlin, Germany: Springer-Verlag.
Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3–10.
Liu, B. (2010). Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Berlin, Germany: Springer-Verlag.
Merton, R. C. (1971). Optimal consumption and portfolio rules in a continuous time model. Journal of Economic Theory, 3(4), 373–413.
Peng, H., Fei, L., Liu, J., & Ju, Z. (2019). A symplectic instantaneous optimal control for robot trajectory tracking with differential-algebraic equation models. IEEE Transactions on Industrial Electronics, 67(5), 3819–3829.
Shu, Y., & Zhu, Y. (2017). Stability and optimal control for uncertain continuous-time singular systems. European Journal of Control, 34, 16–23.
Xu, S., & Liang, Y. (2013). Sliding mode control for a class of nonlinear systems with a quadratic input form. Journal of Computational and Theorectical Nanoscience, 10(4), 1048–1054.
Yao, K., & Qin, Z. (2010). An uncertain control model with application to production-inventory system. In: Proceeding of the Twelfth Asia Pacific Industrial Engineering and Management Systems Conference (pp. 972–977). China: Beijing.
Zhang, W., Hu, J., & Lian, J. (2010). Quadratic optimal control of switched linear stochastic switched systems. Systems and Control Letters, 59, 736–744.
Zhong, J., Cheng, D., & Hu, X. (2008). Constructive stabilization of quadratic-input nonlinear systems with bounded controls. Journal of Dynamic and Control Systems, 14(4), 571–593.
Zhu, Y. (2010). Uncertain optimal control with application to a portfolio selection model. Cybernetics and Systems, 41(7), 535–547.
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This work is supported by the Startup Foundation for Introducing Talent of NUIST (No. 2018r097) and the National Natural Science Foundation of China (No. 61673011).
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Shu, Y., Li, B. & Zhu, Y. Optimal control for uncertain discrete-time singular systems under expected value criterion. Fuzzy Optim Decis Making 20, 331–364 (2021). https://doi.org/10.1007/s10700-020-09346-5
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DOI: https://doi.org/10.1007/s10700-020-09346-5