Discrete-Time Uncertain LQ Optimal Control with Indefinite Control Weight Costs

Yuefen Chen; Liubao Deng

doi:10.20965/jaciii.2016.p0633

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JACIII Vol.20 No.4 pp. 633-639

doi: 10.20965/jaciii.2016.p0633

(2016)

Paper:

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Discrete-Time Uncertain LQ Optimal Control with Indefinite Control Weight Costs

Yuefen Chen^*,† and Liubao Deng^**

^*College of Mathematics and Information Science, Xinyang Normal University
Xinyang 464000, China

^**School of Finance, Anhui University of Finance and Economics
Bengbu 233030, China

^†Corresponding author

Received:

January 10, 2016

Accepted:

May 6, 2016

Published:

July 19, 2016

Keywords:

uncertain LQ optimal control, indefinite control weight costs, well-posedness

Abstract

This paper deals with a discrete-time uncertain linear quadratic (LQ) optimal control, where the control weight costs are indefinite . Based on Bellman’s principle of optimality, the recurrence equation of the uncertain LQ optimal control is proposed. Then, by using the recurrence equation, a necessary condition of the optimal state feedback control for the LQ problem is obtained. Moreover, a sufficient condition of well-posedness for the LQ problem is presented. Furthermore, an algorithm to compute the optimal control and optimal value is provided. Finally, a numerical example to illustrate that the LQ problem is still well-posedness with indefinite control weight costs.

Cite this article as:

Y. Chen and L. Deng, “Discrete-Time Uncertain LQ Optimal Control with Indefinite Control Weight Costs,” J. Adv. Comput. Intell. Intell. Inform., Vol.20 No.4, pp. 633-639, 2016.

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References

[1] R. E. Kalman, “Contributions to the theory of optimal control,” Boletin de la Sociedad Matematica Mexicana, Vol.5, No.2, pp. 102-119, 1960.
[2] X. Su, L. Wu, P. Shi, et al., “A novel approach to output feedback control of fuzzy stochastic systems,” Automatica, Vol.50, No.12, pp. 3268-3275, 2014.
[3] R. Chaichaowarat and W. Wannasuphoprasit, “Linear quadratic optimal regulator for steady state drifting of rear wheel drive vehicle,” J. of Robotics and Mechatronics, Vol.27, pp. 225-234, 2015.
[4] W. M. Wonham, “On a matrix Riccati equation of stochastic control,” SIAM J. on Control and Optimization, Vol.6, No.4, pp. 681-697, 1968.
[5] Y. Hu and X. Y. Zhou, “Constrained stochastic LQ control with random coefficients, and application to portfolio selection,” SIAM J. on Control and Optimization, Vol.44, No.2, pp. 444-466, 2005.
[6] M. Athens, “Special issues on linear-quadratic-gaussian problem,” IEEE Trans. on Automatic Control, Vol.AC-16, pp. 527-869, 1971.
[7] B. Liu, “Uncertainty Theory,” 2nd Ed., Springer-Verlag, Berlin, 2007.
[8] Y. Zhu, “Uncertain optimal control with application to a portfolio selection model,” Cybernetics and Systems, Vol.41, No.7, pp. 535-547, 2010.
[9] L. Sheng, Y. Zhu, and T. Hamalaonen, “An uncertain optimal control with Hurwicz criterion,” Applied Mathematics and Computation, Vol.224, pp. 412-421, 2013.
[10] Y. Chen and B. Li, “Multi-dimension uncertain linear quadratic optimal control with cross term,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.19, No.5, pp. 670-675, 2015.
[11] H. Yan and Y. Zhu, “Bang-bang control model for uncertain switched systems,” Applied Mathematical Modelling, Vol.39, No.10-11, pp. 2994-3002, 2015.
[12] B. Liu, “Some research problems in uncertainy theory,” J. of Uncertain Systems, Vol.3, No.1, pp. 3-10, 2009.
[13] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.
[14] Y. Zhu, “Functions of uncertain variables and uncertain programming,” J. of Uncertain Systems, Vol.6, No.4, pp. 278-288, 2012.
[15] M. Athans, “The matrix minimum principle,” Information and Control, Vol.11, pp. 592-606, 1968.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] R. E. Kalman, “Contributions to the theory of optimal control,” Boletin de la Sociedad Matematica Mexicana, Vol.5, No.2, pp. 102-119, 1960.

[2] [2] X. Su, L. Wu, P. Shi, et al., “A novel approach to output feedback control of fuzzy stochastic systems,” Automatica, Vol.50, No.12, pp. 3268-3275, 2014.

[3] [3] R. Chaichaowarat and W. Wannasuphoprasit, “Linear quadratic optimal regulator for steady state drifting of rear wheel drive vehicle,” J. of Robotics and Mechatronics, Vol.27, pp. 225-234, 2015.

[4] [4] W. M. Wonham, “On a matrix Riccati equation of stochastic control,” SIAM J. on Control and Optimization, Vol.6, No.4, pp. 681-697, 1968.

[5] [5] Y. Hu and X. Y. Zhou, “Constrained stochastic LQ control with random coefficients, and application to portfolio selection,” SIAM J. on Control and Optimization, Vol.44, No.2, pp. 444-466, 2005.

[6] [6] M. Athens, “Special issues on linear-quadratic-gaussian problem,” IEEE Trans. on Automatic Control, Vol.AC-16, pp. 527-869, 1971.

[7] [7] B. Liu, “Uncertainty Theory,” 2nd Ed., Springer-Verlag, Berlin, 2007.

[8] [8] Y. Zhu, “Uncertain optimal control with application to a portfolio selection model,” Cybernetics and Systems, Vol.41, No.7, pp. 535-547, 2010.

[9] [9] L. Sheng, Y. Zhu, and T. Hamalaonen, “An uncertain optimal control with Hurwicz criterion,” Applied Mathematics and Computation, Vol.224, pp. 412-421, 2013.

[10] [10] Y. Chen and B. Li, “Multi-dimension uncertain linear quadratic optimal control with cross term,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.19, No.5, pp. 670-675, 2015.

[11] [11] H. Yan and Y. Zhu, “Bang-bang control model for uncertain switched systems,” Applied Mathematical Modelling, Vol.39, No.10-11, pp. 2994-3002, 2015.

[12] [12] B. Liu, “Some research problems in uncertainy theory,” J. of Uncertain Systems, Vol.3, No.1, pp. 3-10, 2009.

[13] [13] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.

[14] [14] Y. Zhu, “Functions of uncertain variables and uncertain programming,” J. of Uncertain Systems, Vol.6, No.4, pp. 278-288, 2012.

[15] [15] M. Athans, “The matrix minimum principle,” Information and Control, Vol.11, pp. 592-606, 1968.

Discrete-Time Uncertain LQ Optimal Control with Indefinite Control Weight Costs

Yuefen Chen*,† and Liubao Deng**

Yuefen Chen^*,† and Liubao Deng^**