logo

SCIENCE CHINA Information Sciences, Volume 59, Issue 11: 112207(2016) https://doi.org/10.1007/s11432-015-0401-9

Shaping the PDF of the state variable based on piecewise linear control for non-linear stochastic systems

More info
  • ReceivedNov 10, 2015
  • AcceptedMar 26, 2016
  • PublishedOct 14, 2016

Abstract

In this paper, we propose a shape-control scheme for non-linear stochastic dynamical systems to enable us to shape the probability density function (PDF) of the state variable. First, we derive the PDF analytical expression using the Fokker-Planck-Kolmogorov (FPK) equation obtained from stochastic systems. Then, we control the PDF shape by devising a piecewise linear control law whose parameters are calculated using the conjugated gradient method. Finally, we perform contrast simulation experiments to validate the effectiveness and superiority of the proposed algorithm.


Funded by

Key Program of National Natural Science Foundation of China(61533014)

Key Laboratory for Fault Diagnosis and Maintenance of Spacecraft in Orbit(SDML\\_OF2015004)

National Natural Science Foundation of China(61273127)

National Natural Science Foundation of China(U1534208)

Scientific Research Plan Projects of Shannxi Education Department(16JK1690)


Acknowledgment

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant Nos. 61273127, U1534208), Key Laboratory for Fault Diagnosis and Maintenance of Spacecraft in Orbit (Grant No. SDML\_OF2015004), Key Program of National Natural Science Foundation of China (Grant No. 61533014), and Scientific Research Plan Projects of Shannxi Education Department (Grant Nos. 16JK1690).


References

[1] Li D, Qian F C, Fu P L. Variance minimization approach for a class of dual control problems. IEEE Trans Automat Control, 2002, 47: 2010-2020 CrossRef Google Scholar

[2] Yue H, Wang H. Minimum entropy control of closed-loop tracking errors for dynamic stochastic systems. IEEE Trans Automat Control, 2003, 48: 118-122 CrossRef Google Scholar

[3] Li D, Qian F C, Fu P L. Optimal nominal dual control for discrete-time LQG problem with unknown parameters. Automatica, 2008, 44: 119-127 CrossRef Google Scholar

[4] Li D, Qian F C, Gao J J. Performance-first control for discrete-time LQG problems. IEEE Trans Automat Control, 2009, 54: 2225-2230 CrossRef Google Scholar

[5] Qian F C, Xie G, Liu D. Nonlinear optimal trade-off control for LQG problem. In: Proceedings of American Control Conference, Baltimore, 2010. 5: 1931--1936. Google Scholar

[6] Qian F C, Xie G, Liu D. Optimal control of LQG problem with an explicit trade-off between mean and variance. Int J Syst Sci, 2011, 42: 1957-1964 CrossRef Google Scholar

[7] Qian F C, Gao J J, Li D. Complete statistical characterization of discrete-time LQG and cumulant control. IEEE Trans Automat Control, 2012, 57: 2110-2115 CrossRef Google Scholar

[8] Sain M K. Control of linear systems according to the minimal variance criterion: a new approach to the disturbance problem. IEEE Trans Automat Control, 1966, 11: 118-122 CrossRef Google Scholar

[9] Sain M K, Liberty S R. Performance measure densities for a class of LQG control systems. IEEE Trans Automat Control, 1971, 16: 431-439 CrossRef Google Scholar

[10] Li D, Qian F C. Closed-loop optimal control law for discrete time LQG problems with a mean-variance objective. In: Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, 2004. 3: 2291--2296. Google Scholar

[11] Li D, Qian F C, Fu P. Mean-variance control for discrete time LQG problems. In: Proceedings of the American Control Conference, Denver, 2003. 5: 4444--4449. Google Scholar

[12] Jacobson D H. Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games. IEEE Trans Automat Control, 1973, 18: 124-131 CrossRef Google Scholar

[13] Liberty S R, Hartwig R C. On the essential quadratic nature of LQG control-performance measure cumulants. Inf Control, 1976, 32: 276-305 CrossRef Google Scholar

[14] Whittle P. Risk-sensitive Optimal Control. New York: Wiley, 1990. Google Scholar

[15] Wang H, Zhang J H. Bounded stochastic distributions control for pseudo-ARMAX stochastic systems. IEEE Trans Automat Control, 2001, 46: 486-490 CrossRef Google Scholar

[16] Wang H. Minimum entropy control of non-Gaussian dynamic stochastic systems. IEEE Trans Automat Control, 2002, 47: 398-403 CrossRef Google Scholar

[17] Guo L, Wang H. Fault detection and diagnosis for general stochastic systems using B-spline expansions and nonlinear observers. In: Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, 2004. 5: 4782--4787. Google Scholar

[18] Guo L, Wang H. PID controller design for output PDFs of stochastic systems using linear matrix inequalities. IEEE Trans Syst Man Cybern Part B-Cybern, 2004, 35: 65-71 Google Scholar

[19] Forbes M G, Guay M, Forbes J F. Control design for first-order processes: shaping the probability density of the process state. J Process Control, 2004, 14: 399-410 CrossRef Google Scholar

[20] Guo L, Wang H, Wang A P. Optimal probability density function control for NARMAX stochastic systems. Automatica, 2008, 44: 1904-1911 CrossRef Google Scholar

[21] Zhu C X, Zhu W Q. Feedback control of nonlinear stochastic systems for targeting a specified stationary probability density. Automatica, 2011, 47: 539-544 CrossRef Google Scholar

[22] Pigeon B, Perrier M, Srinivasan B. Shaping probability density functions using a switching linear controller. J Process Control, 2011, 21: 901-908 CrossRef Google Scholar

[23] Zhang J F, Yue H, Zhou J L. Predictive PDF control in shaping of molecular weight distribution based on a new modeling algorithm. J Process Control, 2015, 30: 80-89 CrossRef Google Scholar

[24] Wang L Z, Qian F C, Liu J. Shape control on probability density function in stochastic systems. J Syst Eng Electron, 2014, 25: 144-149 CrossRef Google Scholar

[25] Wang L Z, Qian F C, Liu J. The PDF shape control of the state variable for a class of stochastic systems. Int J Syst Sci, 2013, 46: 1-9 Google Scholar

[26] Xu W, Du L, Xu Y. Some recent developments of nonlinear stochastic dynamics (in Chinese). Chin J Eng Math, 2006, 26: 951-960 Google Scholar

[27] Crespo L G, Sun J Q. Nonlinear control via stationary probability density functions. In: Proceedings of the American Control Conference, Anchorage, 2002. 2: 2029--2034. Google Scholar

[28] Rong H W, Meng G, Wang X D, et al. Approximation solution of FPK equations (in Chinese). Chin J Appl Mech, 2003, 20: 95-98 Google Scholar

Copyright 2019 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备18024590号-1