A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Volume 6 Issue 1
Jan.  2019

IEEE/CAA Journal of Automatica Sinica

• JCR Impact Factor: 6.171, Top 11% (SCI Q1)
CiteScore: 11.2, Top 5% (Q1)
Google Scholar h5-index: 51， TOP 8
Turn off MathJax
Article Contents
Zhan Shi and Zhanshan Wang, "Optimal Control for a Class of Complex Singular System Based on Adaptive Dynamic Programming," IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 188-197, Jan. 2019. doi: 10.1109/JAS.2019.1911342
 Citation: Zhan Shi and Zhanshan Wang, "Optimal Control for a Class of Complex Singular System Based on Adaptive Dynamic Programming," IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 188-197, Jan. 2019.

# Optimal Control for a Class of Complex Singular System Based on Adaptive Dynamic Programming

##### doi: 10.1109/JAS.2019.1911342
Funds:

the National Natural Science Foundation of China 61473070

the National Natural Science Foundation of China 61433004

the National Natural Science Foundation of China 61627809

SAPI Fundamental Research Funds 2018ZCX22

• This paper presents a new design approach to achieve decentralized optimal control of high-dimension complex singular systems with dynamic uncertainties. Based on robust adaptive dynamic programming (robust ADP) method, controllers for solving the singular systems optimal control problem are designed. The proposed algorithm can work well when the system model is not exactly known but the input and output data can be measured. The policy iteration of each controller only uses their own states and input information for learning, and do not need to know the whole system dynamics. Simulation results on the New England 10-machine 39-bus test system show the effectiveness of the designed controller.

•  [1] S. Singh and R. W. Liu, "Existence of state equation representation of linear large-scale dynamical systems, " IEEE Trans. Circuit Theory, vol. 20, no. 3, pp. 239-246, May 1973. http://www.researchgate.net/publication/3440274_Existence_of_state_equation_representation_of_linear_large-scale_dynamical_systems [2] H. H. Rosenbrock, "Structural properties of linear dynamical systems, " Int. J. Control, vol. 20, no. 2, pp. 191-202, Aug. 1974. [3] D. G. Luenberger and A Arbel, "Singular dynamical Leontief systems, " Econometrica, vol. 45, no. 4, pp. 991-995, May 1977. https://www.researchgate.net/publication/4813825_Singular_Dynamic_Leontief_Systems [4] E. Yip and R. Sincovec, "Solvability, controllability, and observability of continuous descriptor systems, " IEEE Trans. Autom. Control, vol. 26, no. 3, pp. 702-707, Jun. 1981. http://www.researchgate.net/publication/3029121_Solvability_controllability_and_observability_of_continuous_descriptor_systems [5] D. Cobb, "Controllability, observability, and duality in singular systems, " IEEE Trans. on Autom. Control, vol. 29, no. 12, pp. 1076-1082, Dec. 1984. https://www.researchgate.net/publication/3029868_Controllability_observability_and_duality_in_singular_systems [6] L. V. Dai, Singular Control Systems. Berlin, Germany: Spring-Verlag, 1989. [7] D. J. Bender and A. J. Laub, "The linear-quadratic optimal regulator for descriptor systems: Discrete-time case, " Automatica, vol. 23, no. 1, pp. 71-85, Jan. 1987. https://dl.acm.org/citation.cfm?id=25326 [8] D. J. Bender and A. J. Laub, "The linear-quadratic optimal regulator for descriptor systems, " IEEE Trans. Autom. Control, vol. 32, no. 8, pp. 672-688, Aug. 1987. https://www.sciencedirect.com/science/article/pii/0005109887901191 [9] J. Y. Lin and Z. H. Yang, "Optimal control problems for singular systems, " Int. J. Control, vol. 47, no. 6, pp. 1915-1924, Jun. 1988. [10] X. P. Lin and S. Y. Zhang, "Optimal control problem for linear time-varying descriptor systems, " Int. J. Control, vol. 49, no. 5, pp. 1441- 1452, May 1989. http://www.researchgate.net/publication/245321551_Optimal_control_problem_for_linear_time-varying_descriptor_systems [11] H. H. Rosenbrock and A. C. Pugh, "Contributions to a hierarchical theory of systems, " Int. J. Control, vol. 19, no. 5, pp. 845-867, May 1974. https://www.researchgate.net/publication/245318935_Contributions_to_a_hierarchical_theory_of_systems [12] B. Stott, "Power system dynamic response calculations, " Proc. IEEE, vol. 67, no. 2, pp. 219-241, Feb. 1979. https://www.researchgate.net/publication/2995402_Power_System_Dynamic_Response_Calculations [13] J. K. Mills and A. A. Goldenberg, "Force and position control of manipulators during constrained motion tasks, " IEEE Trans. Rob. Autom., vol. 5, no. 1, pp. 30-46, Feb. 1989. https://www.researchgate.net/publication/3298136_Force_and_position_control_of_manipulator_during_constrained_motion_tasks [14] R. Gani and I. T. Cameron, "Modelling for dynamic simulation of chemical processes: the index problem, " Chem. Eng. Sci., vol. 47, no. 5, pp. 1311-1315, Apr. 1992. https://www.sciencedirect.com/science/article/pii/0009250992802528 [15] E. C. Zeeman, "Duffing$'$s equation in brain modelling, " Bull. Inst. Math. Appl., vol. 12, no. 7, pp. 207-214, 1976. [16] W. Xing and Q. L. Zhang, Research on Problems Related to Stability and Optimal Control for Descriptor Systems. Shenyang, China: Northeastern University Press, 2011. [17] Q. L. Zhang, W. Xing, and G. F. Zhang, "Optimal control for linear descriptor systems, " Journal of Northeastern University (Natural Science), vol. 21, no. 6, pp. 672-674, Dec. 2000. http://en.cnki.com.cn/Article_en/CJFDTotal-DBDX200006026.htm [18] S. Q. Zhu and Z. L. Cheng, "A descripor system approach to linear singularly perturbed optimal control problem with perturbed quadratic cost, " Journal of Shandong University, vol. 38, no. 4, pp. 34-38, Oct. 2003. http://en.cnki.com.cn/Article_en/CJFDTotal-SDDX200304006.htm [19] L. Chen, "Singular LQ problem and monotonicity of the optimal cost of descriptor systems, " Journal of Shandong University, vol. 40, no. 5, pp. 61-65, Jul. 2005. % vol. 41, no. 5, pp. 80-83, Oct. 2005. http://en.cnki.com.cn/Article_en/CJFDTotal-SDDX200605018.htm [20] X. H. Zhang and Q. L. Zhang, "Optimal control for discrete-time singular system, " Journal of Northeastern University (Natural Science), vol. 19, no. 4, pp. 435-438, Aug. 1998. [21] L. Y. Sun and S. P. Ma, "Singular LQ problem for time-varying singular system, " Journal of Jinan Communications College, vol. 8, no. 2, pp. 5 -10, Jun. 2000. [22] P. J. Werbos, "Approximate dynamic programming for real-time control and neural modeling, " in Handbook of Intelligent Control: Neural Fuzzy and Adaptive Approaches, D. A. White and D. A. Sofge, Eds. New York, USA: Van Nostrand Reinhold, 1992, pp. 493-525. http://citeseerx.ist.psu.edu/showciting?cid=258656 [23] Z. S. Hou and J. X. Xu, "On data-driven control theory: the state of the art and perspective, " Acta Automatica Sinica, vol. 35, no. 6, pp. 650- 667, Jun. 2009. http://www.cnki.com.cn/Article/CJFDTotal-MOTO200906006.htm [24] Q. L. Wei, D. R. Liu, Y. Liu, and R. Z. Song, "Optimal constrained self-learning battery sequential management in microgrid via adaptive dynamic programming, " IEEE/CAA J. Autom. Sinica, vol. 4, no. 2, pp. 168-176, Apr. 2017. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=zdhxb-ywb201702003 [25] N. Malla, U. Tamrakar, D. Shrestha, Z. Ni, and R. Tonkoski, "Online learning control for harmonics reduction based on current controlled voltage source power inverters, " IEEE/CAA J. Autom. Sinica, vol. 4, no. 3, pp. 447-457, Jul. 2017. http://www.cnki.com.cn/Article/CJFDTotal-ZDHB201703005.htm [26] N. Cao, H. G. Zhang, Y. H. Luo, D. Z. Feng, and Y. Liu, "Suboptimal control of a class of nonlinear singularly perturbed systems, " Control Theory Appl., vol. 28, no. 5, pp. 688-692, May 2011. http://en.cnki.com.cn/Article_en/CJFDTotal-KZLY201105012.htm [27] D. Q. Zhang, M. M. Li, and J. N. Li, "A data-driven methodology for solving the control strategy of descriptor systems, " in Proc. 9th Asian Control Conf., Istanbul, Turkey, 2013, pp. 1-6. http://www.researchgate.net/publication/261350612_A_data-driven_methodology_for_solving_the_control_strategy_of_descriptor_systems [28] Y. Jiang and Z. P. Jiang, "Computational adaptive optimal control for continuous-time linear systems with completely unknown dynamics, " Automatica, vol. 48, no. 10, pp. 2699-2704, Oct. 2012. https://dl.acm.org/citation.cfm?id=2364760 [29] Y. Jiang and Z. P. Jiang, "Robust adaptive dynamic programming, " in Reinforcement Learning and Approximate Dynamic Programming for Feedback Control, F. L. Lewis and D. R. Liu, Eds. New York, United States: John Wiley and Sons, 2013. http://www.researchgate.net/publication/316658357_Robust_Adaptive_Dynamic_Programming [30] W. T. Guo, F. Liu, S. W. Mei, J. Si, D. W. He, and R. Harley, "Approximate dynamic programming based supplementary frequency control of thermal generators in power systems with large-scale renewable generation integration, " in Proc. IEEE PES General Meeting, Conf. and Expo., National Harbor, USA, 2014, pp. 1-5. http://www.researchgate.net/publication/286669105_Approximate_dynamic_programming_based_supplementary_frequency_control_of_thermal_generators_in_power_systems_with_large-scale_renewable_generation_integration [31] D. M. Yang, Q. L. Zhang, and B. Yao, Singular Systems. Beijing, China: Science Press, 2004. [32] D. Vrabie, O. Pastravanu, M. Abu-Khalaf, and F. L. Lewis, "Adaptive optimal control for continuous-time linear systems based on policy iteration, " Automatica, vol. 45, no. 2, pp. 477-484, Feb. 2009. https://www.sciencedirect.com/science/article/pii/S0005109808004469 [33] F. L. Lewis, D. L. Vrabie, and V. L. Syrmos, Optimal Control, 3rd ed. New Jersey, United States: John Wiley and Sons, Inc., 2012. [34] D. Kleinman, "On an iterative technique for Riccati equation computations, " IEEE Trans. Autom. Control, vol. 13, no. 1, pp. 114-115, Feb. 1968. http://amrx.oxfordjournals.org/external-ref?access_num=10.1109/TAC.1968.1098829&link_type=DOI [35] Q. L. Wei and D. R. Liu, "Adaptive dynamic programming for optimal tracking control of unknown nonlinear systems with application to coal gasification, " IEEE Trans. Autom. Sci. Eng., vol. 11, no. 4, pp. 1020- 1036, Oct. 2014. http://www.researchgate.net/publication/266378299_Adaptive_Dynamic_Programming_for_Optimal_Tracking_Control_of_Unknown_Nonlinear_Systems_With_Application_to_Coal_Gasification [36] Q. L. Wei, D. R. Liu, and H. Q. Lin, "Value iteration adaptive dynamic programming for optimal control of discrete-time nonlinear systems, " IEEE Trans. Cybern., vol. 46, no. 3, pp. 840-853, Mar. 2016. https://www.ncbi.nlm.nih.gov/pubmed/26552103 [37] J. L. Zhang, H. G. Zhang, and T. Feng, "Distributed optimal consensus control for nonlinear multiagent system with unknown dynamic, " IEEE Trans. Neural Netw. Learn. Syst., vol. 29, no. 8, pp. 3339-3348, Aug. 2018. https://www.ncbi.nlm.nih.gov/pubmed/28783647 [38] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed. Cambridge, England: Cambridge University Press, 2013. [39] N. R. Sandell, P. Varaiya, M. Athans, and M. Safonov, "Survey of decentralized control methods for large scale systems, " IEEE Trans. Autom. Control, vol. 23, no. 2, pp. 108-128, Apr. 1978. http://www.researchgate.net/publication/3028133_Survey_of_decentralized_control_methods_for_large_scale_systems [40] L. Bakule, "Decentralized control: an overview, " Ann. Rev. Control, vol. 32 no. 1, pp. 87-98, Apr. 2008. https://www.sciencedirect.com/science/article/pii/S1367578808000096 [41] F. Milano, Power Systems Analysis Toolbox. UCD School of Electrical and Electronic Engineering, Dublin, Ireland, 2016.[online]. Available: http://faraday1.ucd.ie/psat.html [42] F. Dörfler and S. Grammatico, "Gather-and-broadcast frequency control in power systems, " Automatica, vol. 79, pp. 296-305, May 2017. https://www.sciencedirect.com/science/article/pii/S0005109817300572 [43] P. Kundur, Power System Stability and Control. New York, United States: McGraw Hill, 1994. http://www.researchgate.net/publication/224324230_A_generalization_of_the_nonlinear_small-gain_theorem_for_large-scale_complex_systems

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

Figures(6)