# American Institute of Mathematical Sciences

July  2022, 15(7): 1859-1870. doi: 10.3934/dcdss.2022019

## Composite control with observers for a class of stochastic systems with multiple disturbances

 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

* Corresponding author: Yunliang Wei

Received  November 2021 Revised  December 2021 Published  July 2022 Early access  February 2022

Fund Project: This work was supported in part by the Youth Creative Team Sci-Tech Program of Shandong Universities (grant no. 2019KJI007) and the Natural Science Foundation of Shandong Province (grant no. ZR2019MF041)

In this paper, the composite anti-disturbances control problem is considered for a class of stochastic systems with multiple disturbances. The states of the system are assumed to be unavailable. A state observer and a disturbance observer are constructed to estimate the states and the matched disturbance respectively. Based on the estimated values of state observer and disturbance observer, a non-fragile composite controller is designed to achieve disturbance attenuation and rejection. By means of the technique of the disturbance compensation control and stochastic control theory, some sufficient conditions are obtained to guarantee that the closed-loop system is asymptotically bounded in mean square or asymptotically stable in probability. Finally, a numerical example is given to verify the validity of the obtained results.

Citation: Yuanyuan Li, Yunliang Wei. Composite control with observers for a class of stochastic systems with multiple disturbances. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1859-1870. doi: 10.3934/dcdss.2022019
##### References:
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Øksendal, Stochastic Differential Equations-An Introduction with Applications, 6$^{nd}$ edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [12] H. Sun and L. Guo, Neural network-based DOBC for a class of nonlinear systems with unmatched disturbances, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 482-489.  doi: 10.1109/TNNLS.2015.2511450. [13] H. Sun, Y. Li and G. Zong, Disturbance attenuation and rejection for stochastic Markovian jump system with partially known transition probabilities, Automatica J. IFAC, 89 (2018), 349-357.  doi: 10.1016/j.automatica.2017.12.046. [14] W. Sun, Z. Zhao and H. Gao, Saturated adaptive robust control for active suspension systems, IEEE Transactions on Industrial Electronics, 60 (2013), 3889-3896.  doi: 10.1109/TIE.2012.2206340. [15] Y. Wang, Robust control of a class of uncertain nonlinear systems, Systems Control Lett., 19 (1992), 139-149.  doi: 10.1016/0167-6911(92)90097-C. [16] Z. Wang, Y. Liu and X. Liu, Exponential stabilization of a class of stochastic system with markovian jump parameters and mode-dependent mixed time-delays, IEEE Trans. Automat. Control, 55 (2010), 1656-1662.  doi: 10.1109/TAC.2010.2046114. [17] X. Wei and N. Chen, Composite hierarchical anti-disturbance control for nonlinear systems with DOBC and fuzzy control, Internat. J. Robust Nonlinear Control, 24 (2014), 362-373.  doi: 10.1002/rnc.2891. [18] X. Wei, L. Dong and H. Zhang, Adaptive disturbance observer-based control for stochastic systems with multiple heterogeneous disturbances, Internat. J. Robust Nonlinear Control, 29 (2019), 5533-5549.  doi: 10.1002/rnc.4683. [19] X. Wei and L. Guo, Composite disturbance-observer-based control and terminal sliding mode control for non-linear systems with disturbances, Internat. J. Control, 82 (2009), 1082-1098.  doi: 10.1080/00207170802455339. [20] X. Wei and L. Guo, Composite disturbance observer based control and H-infinity control for complex continuous models, Internat. J. Robust Nonlinear Control, 20 (2010), 106-118.  doi: 10.1002/rnc.1425. [21] X. Wei, Z. Wu and H. R. Karimi, Disturbance observer-based disturbance attenuation control for a class of stochastic systems, Automatica J. IFAC, 63 (2016), 21-25.  doi: 10.1016/j.automatica.2015.10.019. [22] Z. Wu and M. Cui, Stability of stochastic nonlinear systems with state-dependent switching, IEEE Trans. Automat. Control, 58 (2013), 1904-1918.  doi: 10.1109/TAC.2013.2246094. [23] J. Xie, D. Yang and J. Zhao, Composite anti-disturbance model reference adaptive control for switched systems, Inform. Sci., 485 (2019), 71-86.  doi: 10.1016/j.ins.2019.02.016. [24] S. Xu, J. Lam and Y. Zou, New results on delay-dependent robust H-infinity control for systems with time-varying delays, Automatica J. IFAC, 42 (2006), 343-348.  doi: 10.1016/j.automatica.2005.09.013. [25] W. Zhang, Y. Huang and H. Zhang, Stochastic $H_2/H_\infty$-infinity control for discrete-time systems with state and disturbance dependent noise, Automatica J. IFAC, 43 (2007), 513-521.  doi: 10.1016/j.automatica.2006.09.015. [26] G. Zong, Y. Li, et al., Composite anti-disturbance resilient control for Markovian jump nonlinear systems with general uncertain transition rate, Science China Information Sciences, 62 (2019), Article number: 22205. doi: 10.1007/s11432-017-9448-8. [27] G. Zong and Q. Wang, Robust resilient control for impulsive switched systems under asynchronous switching, Int. J. Comput. Math., 92 (2015), 1143-1159.  doi: 10.1080/00207160.2014.946413. [28] G. Zong, D. Yang and L. Hou, Robust finite-time $H_\infty$-infinity control for Markovian jump systems with partially known transition probabilities, J. Franklin Inst., 350 (2013), 1562-1578.  doi: 10.1016/j.jfranklin.2013.04.003.

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##### References:
 [1] M. Chen and W. Chen, Disturbance-observer-based robust control for time delay uncertain systems, International Journal of Control Automation and Systems, 8 (2010), 445-453. [2] W. Chen, J. Yang and L. Guo, Disturbance-observer-based control and related methods-an overview, IEEE Transactions on Industrial Electronics, 63 (2016), 1083-1095.  doi: 10.1109/TIE.2015.2478397. [3] F. Deng, W. Mao and A. Wan, A novel result on stability analysis for uncertain neutral stochastic time-varying delay systems, Appl. Math. Comput., 221 (2013), 132-143.  doi: 10.1016/j.amc.2013.05.071. [4] S. Ding, J. Wang and W. Zheng, Second-order sliding mode control for nonlinear uncertain systems bounded by positive functions, IEEE Transactions on Industrial Electronics, 62 (2015), 5899-5909.  doi: 10.1109/TIE.2015.2448064. [5] L. Fridman, J. Moreno and R. Iriarte, Sliding Modes after the First Decade of the 21st Century, 2$^{nd}$ edition, Springer Berlin Heidelberg, 2011. [6] L. Guo and W. Chen, Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach, Internat. J. Robust Nonlinear Control, 15 (2005), 109-125.  doi: 10.1002/rnc.978. [7] X. Li and P. Li, Stability of time-delay systems with impulsive control involving stabilizing delays, Automatica J. IFAC, 124 (2021), Paper No. 109336, 6 pp. doi: 10.1016/j.automatica.2020.109336. [8] X. Li, D. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Transactions on Automatic Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558. [9] X. Li and X. Yang, Lyapunov stability analysis for nonlinear systems with state-dependent state delay, Automatica J. IFAC, 112 (2020), 108674, 6 pp. doi: 10.1016/j.automatica.2019.108674. [10] X. Mao and C. Yuan, Stochastic Differential Equations With Markovian Switching, 2$^{nd}$ edition, Imperial College Press, London, 2006. doi: 10.1142/p473. [11] B. Øksendal, Stochastic Differential Equations-An Introduction with Applications, 6$^{nd}$ edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [12] H. Sun and L. Guo, Neural network-based DOBC for a class of nonlinear systems with unmatched disturbances, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 482-489.  doi: 10.1109/TNNLS.2015.2511450. [13] H. Sun, Y. Li and G. Zong, Disturbance attenuation and rejection for stochastic Markovian jump system with partially known transition probabilities, Automatica J. IFAC, 89 (2018), 349-357.  doi: 10.1016/j.automatica.2017.12.046. [14] W. Sun, Z. Zhao and H. Gao, Saturated adaptive robust control for active suspension systems, IEEE Transactions on Industrial Electronics, 60 (2013), 3889-3896.  doi: 10.1109/TIE.2012.2206340. [15] Y. Wang, Robust control of a class of uncertain nonlinear systems, Systems Control Lett., 19 (1992), 139-149.  doi: 10.1016/0167-6911(92)90097-C. [16] Z. Wang, Y. Liu and X. Liu, Exponential stabilization of a class of stochastic system with markovian jump parameters and mode-dependent mixed time-delays, IEEE Trans. Automat. Control, 55 (2010), 1656-1662.  doi: 10.1109/TAC.2010.2046114. [17] X. Wei and N. Chen, Composite hierarchical anti-disturbance control for nonlinear systems with DOBC and fuzzy control, Internat. J. Robust Nonlinear Control, 24 (2014), 362-373.  doi: 10.1002/rnc.2891. [18] X. Wei, L. Dong and H. Zhang, Adaptive disturbance observer-based control for stochastic systems with multiple heterogeneous disturbances, Internat. J. Robust Nonlinear Control, 29 (2019), 5533-5549.  doi: 10.1002/rnc.4683. [19] X. Wei and L. Guo, Composite disturbance-observer-based control and terminal sliding mode control for non-linear systems with disturbances, Internat. J. Control, 82 (2009), 1082-1098.  doi: 10.1080/00207170802455339. [20] X. Wei and L. Guo, Composite disturbance observer based control and H-infinity control for complex continuous models, Internat. J. Robust Nonlinear Control, 20 (2010), 106-118.  doi: 10.1002/rnc.1425. [21] X. Wei, Z. Wu and H. R. Karimi, Disturbance observer-based disturbance attenuation control for a class of stochastic systems, Automatica J. IFAC, 63 (2016), 21-25.  doi: 10.1016/j.automatica.2015.10.019. [22] Z. Wu and M. Cui, Stability of stochastic nonlinear systems with state-dependent switching, IEEE Trans. Automat. Control, 58 (2013), 1904-1918.  doi: 10.1109/TAC.2013.2246094. [23] J. Xie, D. Yang and J. Zhao, Composite anti-disturbance model reference adaptive control for switched systems, Inform. Sci., 485 (2019), 71-86.  doi: 10.1016/j.ins.2019.02.016. [24] S. Xu, J. Lam and Y. Zou, New results on delay-dependent robust H-infinity control for systems with time-varying delays, Automatica J. IFAC, 42 (2006), 343-348.  doi: 10.1016/j.automatica.2005.09.013. [25] W. Zhang, Y. Huang and H. Zhang, Stochastic $H_2/H_\infty$-infinity control for discrete-time systems with state and disturbance dependent noise, Automatica J. IFAC, 43 (2007), 513-521.  doi: 10.1016/j.automatica.2006.09.015. [26] G. Zong, Y. Li, et al., Composite anti-disturbance resilient control for Markovian jump nonlinear systems with general uncertain transition rate, Science China Information Sciences, 62 (2019), Article number: 22205. doi: 10.1007/s11432-017-9448-8. [27] G. Zong and Q. Wang, Robust resilient control for impulsive switched systems under asynchronous switching, Int. J. Comput. Math., 92 (2015), 1143-1159.  doi: 10.1080/00207160.2014.946413. [28] G. Zong, D. Yang and L. Hou, Robust finite-time $H_\infty$-infinity control for Markovian jump systems with partially known transition probabilities, J. Franklin Inst., 350 (2013), 1562-1578.  doi: 10.1016/j.jfranklin.2013.04.003.
The responses of composite system states
Errors of the estimation of states of system
Errors of the estimation of disturbance
The responses of composite system states
Errors of the estimation of states of system
Errors of the estimation of disturbance
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