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 9 Issue 8
Aug.  2022

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 7.847, Top 10% (SCI Q1)
    CiteScore: 13.0, Top 5% (Q1)
    Google Scholar h5-index: 64, TOP 7
Turn off MathJax
Article Contents
L. Z. Wang, G. Xie, F. C. Qian, J. Liu, and  K. Zhang,  “A novel PDF shape control approach for nonlinear stochastic systems,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 8, pp. 1490–1498, Aug. 2022. doi: 10.1109/JAS.2022.105755
Citation: L. Z. Wang, G. Xie, F. C. Qian, J. Liu, and  K. Zhang,  “A novel PDF shape control approach for nonlinear stochastic systems,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 8, pp. 1490–1498, Aug. 2022. doi: 10.1109/JAS.2022.105755

A Novel PDF Shape Control Approach for Nonlinear Stochastic Systems

doi: 10.1109/JAS.2022.105755
Funds:  This work was supported in part by the National Natural Science Foundation of China (61903298, 62073259, 61773016)
More Information
  • In this work, a novel shape control approach of the probability density function (PDF) for nonlinear stochastic systems is presented. First, we provide the formula for the PDF shape controller without devising the control law of the controller. Then, based on the exact analytical solution of the Fokker-Planck-Kolmogorov (FPK) equation, the product function of the polynomial and the exponential polynomial is regarded as the stationary PDF of the state response. To validate the performance of the proposed control approach, we compared it with the exponential polynomial method and the multi-Gaussian closure method by implementing comparative simulation experiments. The results show that the novel PDF shape control approach is effective and feasible. Using an equal number of parameters, our method can achieve a similar or better control effect as the exponential polynomial method. By comparison with the multi-Gaussian closure method, our method has clear advantages in PDF shape control performance. For all cases, the integral of squared error and the errors of first four moments of our proposed method were very small, indicating superior performance and promising good overall control effects of our method. The approach presented in this study provides an alternative for PDF shape control in nonlinear stochastic systems.

     

  • loading
  • [1]
    B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods, Englewood Cliffs NJ: Prentice-Hall, 1990.
    [2]
    M. K. Sain, “Control of linear systems according to the minimal variance criterion: A new approach to the disturbance problem,” IEEE Trans. Autom. Contr., vol. 11, no. 1, pp. 118–122, 1966. doi: 10.1109/TAC.1966.1098228
    [3]
    D. Li, F. C. Qian, and P. L. Fu, “Mean-variance control for discrete time LQG problems,” in Proc. Am. Control Conf., Denver, Colorado, USA, 2003, no. 5, pp. 4444–4449.
    [4]
    D. Li and F. C. Qian, “Closed-loop optimal control law for discrete time LQG problems with a mean-variance objective,” in Proc. 43rd IEEE Conf. Decis. Control, Paradise Island, Bahamas. 2004, pp. 2291–2296.
    [5]
    H. Wang, “Robust control of the output probability density functions for multivariable stochastic systems,” in Proc. 37th IEEE Conf. Decis. Control, Tampa, Florida, USA. Philadelphia: Omni Press, 1998, pp. 1305–1310.
    [6]
    H. Wang, “Robust control of the output probability density functions for multivariable stochastic systems with guaranteed stability,” IEEE Trans. Autom. Contr., vol. 44, no. 11, pp. 2103–2107, Nov. 1999. doi: 10.1109/9.802925
    [7]
    H. Wang, “Control for bounded pseudo ARMAX stochastic systems via linear B-spline approximations,” in Proc. 39th IEEE Conf. Decis. Control, Sydney, NSW, Australia, 2000, pp. 3369–3374.
    [8]
    H. Wang and J. H. Zhang, “Bounded stochastic distributions control for pseudo-ARMAX stochastic systems,” IEEE Trans. Autom. Contr., vol. 46, no. 3, pp. 486–490, 2001. doi: 10.1109/9.911429
    [9]
    H. Wang, “Minimum entropy control of non-Gaussian dynamic stochastic systems,” IEEE Trans. Autom. Contr., vol. 47, no. 2, pp. 398–403, 2002. doi: 10.1109/9.983388
    [10]
    H. Wang and X. B. Sun, “Neural network based probability density function shape control for unknown stochastic systems,” in Proc. IEEE Int. Symp. Intell. Control, Taipei, China, 2004, pp. 120–125.
    [11]
    H. Wang, J. F. Zhang, and H. Yue, “Multi-step predictive control of a PDF-shaping problem,” Acta Autom. Sinica, vol. 31, no. 2, pp. 274–279, 2005.
    [12]
    L. Guo and H. Wang, “PID controller design for output PDFs of stochastic systems using linear matrix inequalities,” IEEE Trans. Syst. Man Cybern. B Cybern., vol. 35, no. 1, pp. 65–71, 2005. doi: 10.1109/TSMCB.2004.839906
    [13]
    X. L. Luan and F. Liu, “Robust resilient optimal tracking control for output probability density function,” Control Engineering of China, vol. 15, no. 5, pp. 493–496, 2008.
    [14]
    X. L. Luan and F. Liu, “Finite time stabilization for output probability density function of stochastic system,” Control Decis., vol. 24, no. 8, pp. 1161–1166, 2009.
    [15]
    H. J. Yang and J. K. Liu, “An adaptive RBF neural network control method for a class of nonlinear systems,” IEEE/CAA J. Autom. Sinica, vol. 5, no. 2, pp. 457–462, 2018. doi: 10.1109/JAS.2017.7510820
    [16]
    N. Bu, W. Chen, L. G. Jin, and Y. D. Zhao, “Robust control for uncertain nonlinear feedback systems using operator-based right coprime factorization,” IEEE/CAA J. Autom. Sinica, vol. 6, no. 3, pp. 824–829, 2019. doi: 10.1109/JAS.2017.7510895
    [17]
    A. K. Jain and S. Bhasin, “Tracking control of uncertain nonlinear systems with unknown constant input delay,” IEEE/CAA J. Autom. Sinica, vol. 7, no. 2, pp. 420–425, 2020. doi: 10.1109/JAS.2019.1911807
    [18]
    B. Zhou and X. F. Yang, “Global stabilization of discrete-time multiple integrators with bounded and delayed feedback,” Automatica, vol. 97, pp. 306–315, 2018. doi: 10.1016/j.automatica.2018.08.015
    [19]
    B. Zhou, “Construction of strict Lyapunov-Krasovskii functionals for time-varying time-delay systems,” Automatica, vol. 107, pp. 382–397, 2019. doi: 10.1016/j.automatica.2019.05.058
    [20]
    Z. W. Ping, Y. Y. Li, Y. Z. Huang, J. G. Lu, and H. Wang, “Global robust output regulation of a class of MIMO nonlinear systems by nonlinear internal model control,” Int. J. Robust Nonlinear Control, vol. 31, pp. 4037–4051, 2021. doi: 10.1002/rnc.5456
    [21]
    W. Xu, L. Du, and Y. Xu, “Some recent developments of nonlinear stochastic dynamics,” Chinese J. Engineering Mathematics, vol. 23, no. 6, pp. 951–960, 2006.
    [22]
    L. Socha, “Probability density statistical and equivalent linearization techniques,” Int. J. Syst. Sci., vol. 33, no. 2, pp. 107–127, 2000.
    [23]
    L. Socha and M. Blachuta, “Application of linearization methods with probability density criteria in control problems,” in Proc. Am. Control Conf., Chicago, IL, USA, 2000, pp. 2775–2779.
    [24]
    M. G. Forbes, J. F. Forbes, and M. Guay, “Control design for discretetime stochastic nonlinear processes with a non-quadratic performance objective,” in Proc. 42nd IEEE Conf. Decis. Control, Maui, Hawaii, USA, 2003, pp. 4243–4248.
    [25]
    M. G. Forbes, “Performance characterization and regulatory feedback, control design for time-invariant discrete-time stochastic processes,” Ph.D. dissertation, Univ. of Alberta, Edmonton, Canada, 2003.
    [26]
    M. G. Forbes, M. Guay, and J. F. Forbes, “Control design for firstorder processes: Shaping the probability density of the process state,” J. Process Control, vol. 14, no. 4, pp. 399–410, 2004. doi: 10.1016/j.jprocont.2003.07.002
    [27]
    G. F. Michael, G. Martin, and J. F. Forbes, “Probabilistic control design for continuous time stochastic nonlinear systems: A PDF shaping approach,” in Proc. IEEE Int. Symp. Intell. Control, Taipei, China, 2004, pp. 132–136.
    [28]
    H. Z. Yang, Y. Y. Fu, S. Gao, and F. C. Qian, “PDF control of nonlinear stochastic systems based on the MGC method,” Control Decis., vol. 34, no. 7, pp. 1463–1468, 2019.
    [29]
    L. Z. Wang, F. C. Qian, and J. R. Huang, “The PDF tracking control for nonlinear stochastic system,” in Proc. World Congr. Intelligent Control Autom., Shenyang, China, 2014, pp. 684–689.
    [30]
    L. Z. Wang, F. C. Qian, and J. Liu, “The PDF shape control of the state variable for a class of stochastic systems,” Int. J. Syst. Sci., vol. 46, no. 12, pp. 2231–2239, 2015. doi: 10.1080/00207721.2013.860201
    [31]
    L. Z. Wang and F. C. Qian, “Shaping PDF of the state variable based on piecewise linear control for nonlinear stochastic systems,” Sci. China Inf. Sci., vol. 59, no. 11, pp. 106–116, 2016.
    [32]
    L. Z. Wang, G. Xie, F. C. Qian, and A. Q. Shangguan, “Developing an innovative method to control the probability density function shape of the state response for nonlinear stochastic systems,” Int. J. Robust Nonlinear Control, vol. 31, no. 16, pp. 7904–7919, 2021.

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

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Figures(5)  / Tables(5)

    Article Metrics

    Article views (236) PDF downloads(36) Cited by()

    Highlights

    • The proposed method is suitable for any nonlinear stochastic system
    • The approach is more accurate and dependable than other approximate methods
    • This method can make the PDF of state response match different target PDFs

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return